IEEE Transactions on Antennas and Propagation [volume 58 number 3]

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982-986
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994-998
999-1002
1003-1008
1009-1010
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1015-1016
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Citation preview

MARCH 2010

VOLUME 58

NUMBER 3

IETPAK

(ISSN 0018-926X)

PAPERS

Antennas RF MEMS Integrated Frequency Reconfigurable Annular Slot Antenna . . . . B. A. Cetiner, G. Roqueta Crusats, L. Jofre, and N. Bıyıklı A Shallow Varactor-Tuned Cavity-Backed Slot Antenna With a 1.9:1 Tuning Range . . . . . . . . . . . . . . . . C. R. White and G. M. Rebeiz Dielectric Loaded Substrate Integrated Waveguide (SIW) -Plane Horn Antennas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . H. Wang, D.-G. Fang, B. Zhang, and W.-Q. Che Planar Annular Ring Antennas With Multilayer Self-Biased NiCo-Ferrite Films Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G.-M. Yang, X. Xing, A. Daigle, O. Obi, M. Liu, J. Lou, S. Stoute, K. Naishadham, and N. X. Sun Compact Loaded PIFA for Multifrequency Applications . . . . . . . . . . . . . . . . . . . . . . . . O. Quevedo-Teruel, E. Pucci, and E. Rajo-Iglesias A Comparison of a Wide-Slot and a Stacked Patch Antenna for the Purpose of Breast Cancer Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. Gibbins, M. Klemm, I. J. Craddock, J. A. Leendertz, A. Preece, and R. Benjamin A Holographic Antenna Approach for Surface Wave Control in Microstrip Antenna Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Sutinjo, M. Okoniewski, and R. H. Johnston Receiving Polarization Agile Active Antenna Based on Injection Locked Harmonic Self Oscillating Mixers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Vázquez, S. Ver Hoeye, M. Fernández, G. León, L. F. Herrán, and F. Las Heras Singly and Dual Polarized Convoluted Frequency Selective Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Sanz-Izquierdo, E. A. Parker, J.-B. Robertson, and J. C. Batchelor Arrays A Linear Rectangular Dielectric Resonator Antenna Array Fed by Dielectric Image Guide With Low Cross Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. S. Al-Zoubi, A. A. Kishk, and A. W. Glisson Beamforming Lens Antenna on a High Resistivity Silicon Wafer for 60 GHz WPAN . . . . . . . . W. Lee, J. Kim, C. S. Cho, and Y. J. Yoon High Permittivity Dielectric Rod Waveguide as an Antenna Array Element for Millimeter Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J. P. Pousi, D. V. Lioubtchenko, S. N. Dudorov, and A. V. Räisänen Linear Sparse Array Synthesis With Minimum Number of Sensors . . . . . . . . . . . . . . . L. Cen, W. Ser, Z. L. Yu, S. Rahardja, and W. Cen Alternating Adaptive Projections in Antenna Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J. L. Araque Quijano and G. Vecchi An Optimum Adaptive Single-Port Microwave Beamformer Based on Array Signal Vector Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S. Farzaneh and A.-R. Sebak Decoupled 2D Direction of Arrival Estimation Using Compact Uniform Circular Arrays in the Presence of Elevation-Dependent Mutual Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. H. Wang, H. T. Hui, and M. S. Leong Ultrawideband Aperiodic Antenna Arrays Based on Optimized Raised Power Series Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. D. Gregory and D. H. Werner Original and Modified Kernels in Method-of-Moments Analyses of Resonant Circular Arrays of Cylindrical Dipoles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G. Fikioris, D. Tsamitros, S. Chalkidis, and P. J. Papakanellos Electromagnetics A New Fast Physical Optics for Smooth Surfaces by Means of a Numerical Theory of Diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F. Vico-Bondia, M. Ferrando-Bataller, and A. Valero-Nogueira Planar Electromagnetic Bandgap Structures Based on Polar Curves and Mapping Functions . . . . . . . . C. B. Mulenga and J. A. Flint Active Phase Conjugating Lens With Sub-Wavelength Resolution Capability . . . . . . . . V. F. Fusco, N. B. Buchanan, and O. Malyuskin A Geometrical Optics Model of Three Dimensional Scattering From a Rough Layer With Two Rough Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . N. Pinel, J. T. Johnson, and C. Bourlier

H

626 633 640 648 656 665 675 683 690

697 706 714 720 727 738 747 756 765

773 790 798 809

(Contents Continued on p. 625)

(Contents Continued from Front Cover) Numerical Methods Dual-Grid Finite-Difference Frequency-Domain Method for Modeling Chiral Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E. Alkan, V. Demir, A. Z. Elsherbeni, and E. Arvas Optimized Analytic Field Propagator (O-AFP) for Plane Wave Injection in FDTD Simulations . . . . . . . . . . . . . . . T. Tan and M. Potter Development of the CPML for Three-Dimensional Unconditionally Stable LOD-FDTD Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I. Ahmed, E. H. Khoo, and E. Li An Auxiliary Differential Equation Formulation for the Complex-Frequency Shifted PML . . . . . . . . . . . . . S. D. Gedney and B. Zhao Integral Equation Analysis of Scattering From Multilayered Periodic Array Using Equivalence Principle and Connection Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F.-G. Hu and J. Song EFIE Analysis of Low-Frequency Problems With Loop-Star Decomposition and Calderón Multiplicative Preconditioner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S. Yan, J.-M. Jin, and Z. Nie A Newly Developed Formulation Suitable for Matrix Manipulation of Layered Medium Green’s Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J. L. Xiong and W. C. Chew Analysis of Frequency Selective Surfaces on Periodic Substrates Using Entire Domain Basis Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Yahaghi, A. Fallahi, H. Abiri, M. Shahabadi, C. Hafner, and R. Vahldieck Further Development of Vector Generalized Finite Element Method and Its Hybridization With Boundary Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . O. Tuncer, C. Lu, N. V. Nair, B. Shanker, and L. C. Kempel Scattering Multiple Loaded Scatterer Method for E-Field Mapping Applications . . . . . . . . . . . . . . . . . . . . . . . . . . M. A. Abou-Khousa and R. Zoughi Deterministic Approach for Spatial Diversity Analysis of Radar Systems Using Near-Field Radar Cross Section of a Metallic Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R. Deban, H. Boutayeb, K. Wu, and J. Conan Wireless Effect of Human Presence on UWB Radiowave Propagation Within the Passenger Cabin of a Midsize Airliner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S. Chiu and D. G. Michelson BAN-BAN Interference Rejection With Multiple Antennas at the Receiver . . . . . . . I. Khan, Y. I. Nechayev, K. Ghanem, and P. S. Hall Characterization of UWB Channel Impulse Responses Within the Passenger Cabin of a Boeing 737-200 Aircraft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S. Chiu, J. Chuang, and D. G. Michelson Landmobile Radiowave Multipaths’ DOA-Distribution: Assessing Geometric Models by the Open Literature’s Empirical Datasets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K. T. Wong, Y. I. Wu, and M. Abdulla

817 824 832 838 848 857 868 876 887 900 908

917 927 935 946

COMMUNICATIONS

The Beam Pattern of Reflector Antennas With Buckled Panels . . . . . . . . A. Greve, D. Morris, J. Peñalver, C. Thum, and M. Bremer The Periodic Half-Width Microstrip Leaky-Wave Antenna With a Backward to Forward Scanning Capability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Y. Li, Q. Xue, E. K.-N. Yung, and Y. Long Differentially-Fed Millimeter-Wave Yagi-Uda Antennas With Folded Dipole Feed . . . . . . . . . . . . . . . R. A. Alhalabi and G. M. Rebeiz A Finite Edge GTD Analysis of the H-Plane Horn Radiation Pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. Ali and S. Sanyal Broadband CPW-Fed Circularly Polarized Square Slot Antenna With Lightening-Shaped Feedline and Inverted-L Grounded Strips . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J.-Y. Sze, C.-I. G. Hsu, Z.-C. Chen, and C.-C. Chang Optimization and Modeling of Sparse Conformal Retrodirective Array . . . . . . . . . . . . . . . . . . . . . . . . . . . J. S. Sun, D. S. Goshi, and T. Itoh Retrodirective Array Performance in the Presence of Near Field Obstructions . . . . . . . . . . . . . . . . . . . . . . . . . V. F. Fusco and N. Buchanan Experimental Two-Element Time-Modulated Direction Finding Array . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Tennant A New Procedure for Assessing the Sensitivity of Antennas Using the Unscented Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . L. R. A. X. deMenezes, A. J. M. Soares, F. C. Silva, M. A. B. Terada, and D. Correia 60-GHz Wideband Substrate-Integrated-Waveguide Slot Array Using Closely Spaced Elements for Planar Multisector Antenna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. Ohira, A. Miura, and M. Ueba Efficient Modeling of Radiation and Scattering for a Large Array of Loops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P. J. Papakanellos, N. L. Tsitsas, and H. T. Anastassiu Asymptotic Extraction Approach for Antennas in a Multilayered Spherical Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S. K. Khamas Miniature Internal Penta-Band Monopole Antenna for Mobile Phones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.-L. Liu, Y.-F. Lin, C.-M. Liang, S.-C. Pan, and H.-M. Chen Printed Single-Strip Monopole Using a Chip Inductor for Penta-Band WWAN Operation in the Mobile Phone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K.-L. Wong and S.-C. Chen

959 963 966 969 973 977 982 986 988 993 999 1003 1008 1011

COMMENTS AND REPLIES

Comments on “Fast Direct Solution of Method of Moments Linear System” . . . . . . . . . . . A. Heldring, J. M. Rius, and J. M. Tamayo Comments on “Decoupling Efficiency of a Wideband Vivaldi Focal Plane Array Feeding a Reflector Antenna” . . . . . S. P. Skobelev Reply to “Comments on ‘Decoupling Efficiency of a Wideband Vivaldi Focal Plane Array Feeding a Reflector Antenna’” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. V. Ivashina, M. Ng Mou Kehn, P.-S. Kildal, and R. Maaskant

1015 1016 1016

CORRECTIONS

Corrections to “Bandwidth Limitations on Linearly Polarized Microstrip Antennas” . . . . . . . . . . . . . . . . . . . . . . . . . . R. A. Abd-Alhameed

1018

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Digital Object Identifier 10.1109/TAP.2010.2044543

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RF MEMS Integrated Frequency Reconfigurable Annular Slot Antenna Bedri A. Cetiner, Member, IEEE, Gemma Roqueta Crusats, Student Member, IEEE, Lluís Jofre, Senior Member, IEEE, and Necmi Bıyıklı, Member, IEEE

Abstract—A new kind of double- and single-arm cantilever type DC-contact RF MEMS actuators has been monolithically integrated with an antenna architecture to develop a frequency reconfigurable antenna. The design, microfabrication, and characterization of this “reconfigurable antenna (RA) annular slot” which was built on a microwave laminate TMM10i ( = 9.8, tan = 0.002), are presented in this paper. By activating/deactivating the RF MEMS actuators, which are strategically located within the antenna geometry and microstrip feed line, the operating frequency band is changed. The RA annular slot has two reconfigurable frequencies of operation with center frequencies low =2.4 GHz and high = 5.2 GHz, compatible with IEEE 802.11 WLAN standards. The radiation and impedance characteristics of the antenna along with the RF performance of individual actuators are presented and discussed. Index Terms—Full-wave analyses, microfabrication, reconfigurable antenna, RF MEMS actuators.

I. INTRODUCTION

T

HE reconfigurable antenna (RA) concept [1], [2] has gained significant interest as a result of two main factors. First, a single RA can perform multiple functions by dynamically changing its properties (operating frequency, polarization, and radiation pattern). This can result in a significant reduction in the overall size of multi-mode multi-band wireless communication systems and replace multiple single-function legacy antennas. Second, the reconfigurable antenna properties of a RA can be used as important additional degrees of freedom in an adaptive system (first proposed in [3] and later in [4]–[7]). In particular it was shown that a RA equipped adaptive multiple-input multiple-output (MIMO) wireless communication system can provide gains up to 30 dB as compared to conventional fixed antenna MIMO systems [7]. These additional gains result from the joint optimization of dynamically reconfigurable antenna properties with adaptive space-time modulation techniques [8] in response to the changes in the propagation environment. Manuscript received January 29, 2009; revised April 28, 2009. First published December 28, 2009; current version published March 03, 2010. This work was supported in part by Army Research Office under Grant W911 NF-07-1-0208 and in part by the National Institute of Justice, Office of Justice Programs, US Department of Justice under Grant 2007-IJ-CX-K025. B. A. Cetiner and N. Bıyıklı are with the Electrical and Computer Engineering Department, Utah State University, Logan, UT 84322 USA (e-mail: [email protected]). L. Jofre and G. Roqueta Crusats are with the Department of Signal Theory and Communications, Technical University of Catalonia, 08034 Barcelona, Spain. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2009.2039300

In order to dynamically change the properties of a RA, the current distribution over the volume of the antenna needs to be changed, where each distribution corresponds to a different mode of operation. To this end, one can change the geometry and feed line of the antenna by switching on and off various geometrical segments that make up the RA and the feed circuitry. For switching, either MEMS or solid-state switching devices can be employed. In this work, we prefer using RF MEMS actuators due to monolithic integration capability with antenna segments along with their low loss and power characteristics. Also, the potential of MEMS in avoiding nonlinearity and intermodulation effects is an important advantage over solid-state switching devices [9]. To cover multiple frequency bands by a single antenna element, various design approaches such as a multi-frequency antenna covering each individual band, a broadband antenna covering the whole frequency bandwidth, or a RA with a narrow instantaneous operating bandwidth that can be tuned over the whole bandwidth can be used. The design of slot-ring antennas for mobile communications was previously studied in [10], [11]. Multi-frequency operation was also presented in several research papers. In [12], [13], the multi-band operation is achieved by using multiple concentric annular-ring slots, which are fed by either a microstrip or coplanar waveguide. Microstrip-fed slot-ring antennas, which consist of a single slot-ring of various geometries, that achieve multi-frequency operation were also investigated [14]. Although these antennas support multiple frequencies, the radiation patterns and gain values corresponding to each frequency can be significantly different. The same problem applies to a broadband antenna, where the degradation in gain can be significant over a broad frequency band. This might be a problem for multi-mode multi-band wireless communications applications where the radiation characteristics and gain performance are required to be similar over multiple frequencies. Recent efforts have been focusing on the tunable slot-ring antennas [15], [16]. The RA annular slot presented in this paper maintains the same radiation characteristics for both modes of operation ( and ) due to a similar tunable highly efficient operating bandwidth over which the radiation behavior remains almost constant. This is accomplished by MEMS reconfiguration which provides a circular one-wavelength-perimeter for both the external longer slot of the lower frequency and the internal shorter one of the higher frequency band. Additionally the use of an annular geometry compared to the conventional factor linear slot reduces the dimension of the antenna by a (perimeter of for the annular slot instead of a length of for the linear slot).

0018-926X/$26.00 © 2010 IEEE

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Fig. 2. Photograph of the RA annular slot. (a) Microstrip feed line integrated with a single-arm MEMS actuator. (b) Annular slot integrated with two double-arm MEMS actuators.

Fig. 1. Architecture of the microstrip-fed RA annular slot. (a) Top view and magnified A-A’ cross section view. (b) Bottom view and magnified B-B’ cross section view. (MEMS actuator width is 350  and length is 650  ).

m

m

II. RECONFIGURABLE ANNULAR SLOT: ARCHITECTURE, DESIGN, CHARACTERIZATION A. Architecture and Working Mechanism The architecture and a photograph of the microstrip-fed RA annular slot are shown in Figs. 1 and 2, respectively. The antenna is built on two separate layers of TMM10i ( , ) microwave laminate each with 0.635 mm thickness. The microstrip feed line is placed on one layer and the annular slot is placed on the other layer, which are bonded together having a total thickness of 1.27 mm. This RA has two concentric circular slots, each of which can be excited individually in order to achieve frequency reconfigurability. The microstrip feed line is broken into two segments, which are spanned by a single-arm cantilever type MEMS actuator, (see Fig. 1(b) and the inset of Fig. 2(a) for a magnified view), which is similar to the MEMS switch presented in [17]. The actuator enables either the outer or the inner slot to be excited by changing the length of the microstrip. When is not activated (actuator up-state), the microstrip segments are disconnected,

thereby the outer slot is fed; the operation frequency is (Mode 1). When is activated (actuator down-state), the microstrip segments become connected, thereby the inner (Mode slot is fed; the operation frequency is 2). Two double-arm cantilever type MEMS actuators, and (see Fig. 1(a) and inset of Fig. 2(b) for a magnified view), are located on the opposite sides of the outer slot as shown in Figs. 1(a) and 2(b). These actuators enable the metallic annular ring, which stays between the outer and inner slots, to be shorted to RF ground so that when the inner slot is excited, it has a continuous ground plane. Therefore the operation of the inner slot is not adversely impacted by the presence of the outer slot. We also , observed that when the outer slot is excited for the influence of the presence of the inner slot combined with the central metallic circular island on the outer slot produces only a minor resonant effect, which is out of the operation range of the higher frequency, thereby not requiring the inner slot to be shorted. The actuation of the MEMS actuators is due to the electrostatic force created by the applied DC voltage between the cantilever’s metallic post (the fixed point of the MEMS actuator) and the pull-down bias electrode which is placed underneath the cantilever. When the applied electrostatic force becomes larger than the mechanical force due to the stress of the metallic cantilever beam, the cantilever moves down and makes a DC contact, thereby connecting the broken microstrip segments and also creating a short to the ground plane. DC bias voltages are applied through high DC resistance Silicon Chrome (SiCr) lines and an with a sheet resistance of approximately 50

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absolute resistance of 500 . Please note that the bias line used and is placed into the outer slot for the comto activate pactness of the design. Due to high resistivity of the bias line, this placement has no detrimental effect to the operation of RA annular slot as demonstrated by the measurements and simulations given in the following sections. An annular slot etched in a large conducting plane can be viewed as an annular distribution of a magnetic current given by the actual mode excited into the circumferential slot. To obtain maximum bandwidth with minimum dimensions, the mode, normally called fundamental mode, tends to be the basic choice [18]. Considering the annular slot as a transmission line, the fundamental resonance appears around the frequency at which the circumference of the annular slot becomes one guided wavelength of the slot ( ). The slot guided wavelength for the frequency and dielectric permittivity ranges of interest can be expressed [19] as Fig. 3. Microfabrication process steps for the double-arm MEMS actuator interconnecting two segments of the RA annular slot.

(1)

where is the slot-line width, the free-space wavelength, and and are the permittivity and thickness of the substrate, respectively. While (1) provides a good first order approximation for the resonance frequency, the actual frequency for the microstrip-fed MRA annular slot will be in close vicinity of that of given by (1) as the penetration distance of the projection of the microstrip onto the slot plays a second order role on the resonance frequency. As seen in Fig. 1(b), this penetration distance . The bandwidth may be adjusted for the inner slot is by selecting the appropriate internal and external radii of the slots as (2)

where , , are the central, upper and lower frequencies for the or and , , are the central, external and internal radius for the (outer slot) or (inner slot) frequency bands. Please note that these definitions indicate that , is the guided wavelength corresponding to where (outer slot) or (inner slot) frequency bands. B. Microfabrication The microfabrication process of the RA annular slot was performed based on the microwave laminate compatible RF MEMS technology [1], [20], [22] that enables the monolithic integration of MEMS actuators with antenna segments. The layout of MEMS actuators and antenna segments including microstrip feed lines and high resistance DC control lines are part of the same lithographic process. Thus, the process flow for a single double-arm MEMS actuator interconnecting two

different antenna metallic segments is given in Fig. 3, which also serves as the description for the complete microfabrication of the RA annular slot. Because this RA has MEMS actuators on both sides of the substrate, the two sides are processed separately and subsequently bonded together. We developed a seven-step microfabrication process using microwave laminate TMM10i substrates. The microfabrication process was started by wet-etching and chemical mechanical polishing -thick Cu layer on top of the microwave (CMP) the 15 . Afterwards, the laminate TMM10i substrate down to annular slot and microstrip line segments were patterned and -thick SiCr bias lines wet-etched selectively. Next, were formed by DC-sputtering which was followed by the thick Ti/Cu formation of Ti/Cu bias electrode pads. A island metal is deposited as the central-base metallic pad for the MEMS cantilever beam. In the fifth step, bias lines and bias dielectric electrodes were passivated with a 250 nm thick ) layer. This was followed by the deposition of a thick ( amorphous Si (a:Si) sacrificial layer which was planarized thick TiW using CMP. In the final seventh step, a layer was sputtered. After dry-release process, the cantilevers and ) curled upwards ( due to the internal stress-gradient within the deposited TiW. Fig. 3 summarizes the microfabrication process steps used for double-arm MEMS actuators interconnecting two metal segments of RA annular slot structure. It is important to note that a MEMS integrated antenna needs to be packaged since it is exposed to harsh environmental conditions when in use. To seal the entire MEMS integrated antenna structure, a wafer scale low-loss packaging technology, which was recently developed by MIT Lincoln Laboratory [23], can be used. 1) RF Characterization of RF MEMS Single- and Double-Arm Actuators: RF performances of the individual single- and double-arm actuators have been characterized by measuring and simulating small-signal -parameters. To this

CETINER et al.: RF MEMS INTEGRATED FREQUENCY RECONFIGURABLE ANNULAR SLOT ANTENNA

Fig. 4. Measured and simulated S

of the double-arm MEMS actuator.

629

= 5:2 GHz = 2 4 GHz (Mode 1

Fig. 5. Measured and simulated reflection coefficients for f (Mode 2 operation, inner slot is excited), and for f : operation, outer slot is excited).

end, an RF-probe station interfaced to a vector network analyzer is used. The results for single- and double-arm actuators are observed to be very similar, therefore only double-arm actuator results are given in Fig. 4. As shown in Fig. 4, the , in the down-state, including the inherent average value of loss of the microstrip ( ) is over the frequency range of 500 MHz to 25 GHz. The corresponding , which can be found by using switch contact resistance is , where is the switch resistance and ( ) is the characteristic impedance of the microstrip. in the up-state ranges from to for the same frequency range. The actuation voltage was at around 20 Volts. C. Reflection Coefficient To measure the antenna reflection coefficients, i.e., the parameter, corresponding to Mode 1 and Mode 2 operations, the RA annular slot was connected to the fully-calibrated single ) port of a VNA. For Mode 2 operation ( MEMS actuators were activated (actuators down-state) by applying DC bias voltages. In the case of Mode 1 operation ), the actuators were in the up-state not ( requiring DC bias. The measured and simulated reflection coefficients, with a good agreement between them, corresponding to two reconfigurable modes of operation, and , are shown in Fig. 5. For a 1:2 VSWR reflection coefficient), the corresponding frequency ( bandwidths are 100 MHz and 650 MHz for 2.4 GHz and 5.2 GHz, respectively, covering the ISM allocated frequency range. For each mode of operation a small amount of residual resonance of the non-activated slot appears, which is due to the different feeding-boundary conditions created by the presence of the activated MEMS. The small residual resonance, however, is out of the active band with negligible effect on the antenna performance. D. Radiation Patterns and Current Distribution Antenna radiation pattern measurements have also been performed for Modes 1 and 2. The measurements took place in an (Mode 1) and anechoic chamber at (Mode 2) frequencies in the principal cuts, - and

Fig. 6. Co- and cross-polar radiation patterns at f -plane(y -z plane,  ), (b) in -plane (x-z plane, 

E

= 90

H

= 2:4 GHz (a) in = 0).

-planes, which correspond to - ( ) and - ( ) planes, respectively. Both co- and cross-polar components were measured and compared to the simulated results, which were obtained using [24]. At each frequency, the radiation pattern in and planes are normalized with respect to the maxthe imum in their crossing points. The results for and are shown in Figs. 6 and 7, respectively. The measured and simulated results agree well. Well-behaved linear polarization and broadside radiation patterns are observed, and the two frequencies have the same polarization planes. Cross-polarization levels at both planes remain quite , due to the geometrical symmetry low, on the order of especially with respect to the -axis. The electric field distributions on the inner and outer slots have also been studied. Fig. 8(a) and (b) show the simulated and , current distribution at respectively. The electric field distributions are similar to each other resulting in similar radiation patterns at both frequencies as shown in Figs. 6 and 7.

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TABLE I DIRECTIVITY, EFFICIENCY AND GAIN VALUES OF THE RA ANNULAR SLOT WITH AND WITHOUT BIAS LINE

Fig. 7. Co- and cross-polar radiation patterns at f -plane( - plane,  ), (b) in -plane (x-z plane, 

E

yz

= 90

H

= 5:2 GHz (a) in = 0).

Fig. 8. Electric field distributions on the RA annular slot for (a) Mode 2, : : . and (b) Mode 1, f f

= 5 2 GHz

= 2 4 GHz

The gain values corresponding to both modes of operation have been determined from the product of ohmic efficiency , ( ), directivity ( ) and the mismatch factor . The directivities have been obtained by integration of the three-dimensional measured radiation patterns (resulting from with 3 angular step with an accuracy of the combination of the measurement and integration relative , obtained errors: from a set of measurements for different distances and angular steps). In order to measure the efficiency of the antenna avoiding the influence of the feeding cables that the limited ground plane (in the order of 0.3 wavelengths at the lower frequency) could produce, the antenna has been vertically mounted into a large ground plane, enclosed by a cap. The Wheeler cap method has then been used to determine the ohmic efficiency with a precision of (3) is the input impedance of the antenna while it is where is the input impedance placed inside the Wheeler cap, and without the cap. The results are summarized in Table I. As expected gain values close to 2.7 dB for the inner and 2 dB for the outer slots are obtained with combined precision resulting from the combination of directivity and of efficiency estimated accuracies. In order to check these results, especially at the lower frequency band where the effects of the cables and finite ground plane may be more significant, a direct

Fig. 9. Measured reflection coefficients with and without the bias lines at Mode 1 and Mode 2 of operation.

band-limited measurement of the gain has been performed with a far-field anechoic chamber. In summary, the RA annular slot has similar gain and well behaved radiation patterns at both frequency bands, which is a required antenna property for multimode multi-band mobile wireless communication applications. E. The Influence of the Bias Circuitry on the MRA Performance A RA with MEMS actuators monolithically integrated within radiating antenna segments requires an effective DC biasing approach to achieve both actuator actuation and decoupling between radiating antenna and DC control signals. In this work, a high DC resistance line approach is implemented as opposed to high RF impedance conductive metal lines, since the latter can cause deleterious electromagnetic interactions degrading the antenna performances. As described in the microfabrication section, high DC resistance lines of Silicon Chrome (SiCr) with a and an absosheet resistance of approximately 50 are used. lute resistance of 500 In order to investigate the influence of DC bias lines on the antenna performances, radiation patterns and reflection coefficients were measured for both modes of operation (Mode 2: and Mode 1: ) for the two cases: with and without bias lines. The comparative results shown in Figs. 9–11 indicate that the influence of the bias line

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REFERENCES

Fig. 10. Band-limited Smith chart representation with and without the bias lines at (a) Mode 1 (2 – 3 GHz) and (b) Mode 2 (4.5 – 5.5 GHz).

Fig. 11. Measured co-polar radiation patterns in E- plane (y -z plane,  = 90) and in H-plane (x-z plane,  = 0) with and without bias lines at (a) 5.2 GHz (Mode 2) and (b) 2.4 GHz (Mode 1).

on both the impedance (magnitude and phase) and the radiation behavior is negligible. III. CONCLUSION A frequency reconfigurable ( and ) annular slot antenna has been designed, microfabricated, and characterized. The reconfigurability is achieved through single- and double-arm DC-contact MEMS actuators, which are strategically located within the antenna geometry and the microstrip feed line. Simulations and mea) surements showed that the antenna has similar gain ( and well behaved radiation patterns at both frequency bands, which is a required antenna property for multi-mode multi-band mobile wireless communication applications. The average value ( in the switch down-state) indicates that the of . Furthermore, the effect of switch contact resistance is SiCr bias circuitry on the radiation and impedance characteristics has been found to be negligible. ACKNOWLEDGMENT This work was performed in part at the Cornell NanoScale Facility (CNF), a member of the National Nanotechnology Infrastructure Network, which is supported by the National Science Foundation.

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Bedri A. Cetiner (M’00) received the Ph.D. degree in electronics and communications engineering from the Yildiz Technical University, Istanbul, in 1999. From November 1999 to June 2000, he was with the University of California, Los Angeles, as a NATO Science Fellow. He then joined the Department of Electrical Engineering and Computer Science, University of California, Irvine, where he worked as a Research Specialist from June 2000 to June 2004. From July 2004 until July of 2007, he worked as an Assistant Professor in the Department of Space Science and Engineering, Morehead State University. In August 2007, he joined Utah State University, Logan, where he is an Assistant Professor of electrical engineering. His research focuses on the applications of micro-nano technologies to a new class of micro-/millimeter-wave circuits and systems, and intelligent wireless communications systems with an emphasis on the multifunctional reconfigurable antennas (MRA) for use in cognitive multi-input multi-output (MIMO) systems. He is the Principal Inventor of three technologies including microwave laminate compatible RF MEMS technology and MRA equipped MIMO systems. Prof. Cetiner is a member of the IEEE Antennas and Propagation, Microwave Theory and Techniques, and Communication societies.

Gemma Roqueta Crusats (S’08) was born in Girona, Spain, in 1983. She received the Telecommunication Engineer degree from the Technical University of Catalonia (UPC), Barcelona, Spain, in 2007, where she is working toward the Ph.D. degree. In 2006, she was involved in body area networks research as foreign student in the Catholic University of Louvain, Louvain, Belgium. Since September 2007, she has been with the UPC where she is involved in developing wideband microwave imaging applications for non-destructive quality testing methods for civil structures with fiber reinforcement. Her research interests include indoor and outdoor propagation, wideband microwave imaging, spiralometric discrimination and ultra wide band antennas.

Lluís Jofre (S’79–M’83–SM’07) was born in Mataró, Spain, in 1956. He received the M.Sc. (Ing) and Ph.D. (Doctor Ing.) degrees in electrical engineering (telecommunications eng.), from the Technical University of Catalonia (UPC), Barcelona, Spain, in 1978 and 1982, respectively. From 1979 to 1980, he was a Research Assistant in the Electrophysics Group,UPC, where he worked on the analysis and near field measurement of antenna and scatterers. From 1981 to 1982, he joined the Ecole Superieure d’Electricite, Paris, France, where he was involved in microwave antenna design and imaging techniques for medical and industrial applications. In 1982, he was appointed Associate Professor at the Communications Department, Telecommunication Engineering School, UPC, where he became Full Professor in 1989. From 1986 to 1987, he was a Visiting Fulbright Scholar at the Georgia Institute of Technology, Atlanta, working on antennas, and electromagnetic imaging and visualization. From 1989 to 1994, he served as Director of the Telecommunication Engineering School (UPC), and from 1994–2000, as UPC Vice-Rector for Academic Planning. From 2000 to 2001, he was a Visiting Professor at the Electrical and Computer Engineering Department, Henry Samueli School of Engineering, University of California. From 2002 to 2004, he served as Director of the Catalan Research Foundation and since 2003 as Director of the UPC-Telefonica Chair. His research interests include antennas, electromagnetic scattering and imaging, and system miniaturization for wireless and sensing industrial and bio applications. He has published more than 100 scientific and technical papers, reports and three books and chapters in specialized volumes.

Necmi Bıyıklı (M’04) was born in Utrecht, The Netherlands, in 1974. He received the B.S., M.S., and Ph.D. degrees in electrical and electronics engineering from Bilkent University, Ankara, Turkey, in 1996, 1998, and 2004, respectively. His M.S. and Ph.D. thesis work concentrated on the design, fabrication and characterization of high-performance photodetectors. During his Postdoctoral research at the Virginia Commonwealth University, he worked on the growth, fabrication, and characterization of AlGaN/GaN heterostructures for various applications including high-performance GaN/AlGaN transistors. He also worked as a Research Scientist at Utah State University. His research at Utah State University focused on RF-MEMS/NEMS integrated multifunctional reconfigurable antennas. Currently he is with the UNAM – Materials Science and Nanotechnology Institute, Bilkent University. His research interests include RF-MEMS/NEMS integrated reconfigurable antennas, III-Nitride materials and devices for optoelectronic and photovoltaic applications, nanofabrication for novel sensor technologies. He is the author of over 70 citation-index journal papers and refereed conference proceedings. Dr. Bıyıklı serves as a reviewer for several scientific journals including Applied Physics Letters, IEEE Photonics Technology Letters, and IEEE Journal of Quantum Electronics.

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A Shallow Varactor-Tuned Cavity-Backed Slot Antenna With a 1.9:1 Tuning Range Carson R. White, Member, IEEE, and Gabriel M. Rebeiz, Fellow, IEEE

Abstract—A shallow (0.025 wavelengths) microstrip-fed cavity-backed-slot (CBS) antenna has been demonstrated that tunes from 1 to 1.9 GHz with better than 20 dB reflection coefficient using a single varactor diode (0.45–2.5 pF). This is possible because the slot and the cavity combine to form a single resonance, and therefore, do not need to be tuned independently. The cavity is 72 72 3.18 mm3 , and is dielectrically loaded ( = 2 1). The impedance match over the tuning range has been achieved by searching for combinations of feed and varactor locations for which the impedance is matched to 50 by a single series inductor, and the input impedance is nearly identical whether the antenna is in free-space or conformally mounted on a 1.2 1.2 m2 ground plane. The cross-polarization is better than 25 dB at 1.0, 1.5, and 1.9 GHz. To the authors’ knowledge, this is the first demonstration of a varactor-tuned CBS antenna. Index Terms—Cavity backed antennas, reconfigurable antennas, slot antennas.

I. INTRODUCTION ANY of today’s advanced communication systems require either frequency-agile or wideband antennas. Software Defined Radios, for example, can be reconfigured to communicate using many different protocols at different frequencies. The instantaneous bandwidth of efficient passive antennas is limited as they become small with respect to the wavelength [1], and tunable narrowband antennas can be advantageous if small efficient antennas are required to cover a large frequency range. In addition, tunable narrowband antennas provide frequency selectivity, relaxing the requirements of the receive filters. Resonant slot antennas are good candidates for frequency tuning because their resonance frequency can be changed easily with varactors. In a previous work, the authors demonstrated single-and dual-polarized tuning of the slot-ring antenna from 0.95 to 1.8 GHz and 0.93 to 1.6 GHz, respectively [2]. Behdad et al. demonstrated a dualband slot antenna where both of the bands can be tuned independently, covering the 1.1–1.34 and 1.74–2.94 GHz bands [3]. Slot antennas have two-sided radiation, however, and are unsuitable for conformal mounting on a ground-plane, printed circuit board, or other scattering object. The cavity-backed-slot (CBS) antenna achieves single-sided radiation by placing a conducting enclosure behind the slot. Although CBS antennas are not usually thought of as being planar,

M

Manuscript received March 13, 2009; revised July 14, 2009. First published December 28, 2009; current version published March 03, 2010. This work was supported by the UCSD Center for Wireless Communications. C. R. White is with HRL Laboratories, LLC., Malibu, CA 90265 USA (e-mail: [email protected]). G. M. Rebeiz is with the Department of Electrical and Computer Engineering, University of California, San Diego, La Jolla, CA 92093 USA. Digital Object Identifier 10.1109/TAP.2009.2039311

Fig. 1. Geometry of the shallow varactor-tuned cavity-backed slot antenna. A microstrip feed at d is enabled by placing a superstrate above the slot (Fig. 6).

two shallow ( , where is the free-space wavelength) cavity-backed slot antennas have recently been demonstrated that are compatible with printed circuit-board fabrication techniques. The first is a circularly-polarized crossed-slot that is fed by a single coaxial probe [4], and the second is a one-wavelength microstrip-fed linear slot [5]. A ferrite-tuned cavity backed slot antenna has been tuned over a small range [6], but no wideband-tunable CBS was found in the literature. In this paper a printed planar CBS is proposed that is tuned by a single varactor diode. The measured performance shows dB reflection coefa tuning range of 1.0–1.9 GHz with dB cross-polarization. ficient and II. THE TUNABLE CAVITY-BACKED SLOT ANTENNA The proposed antenna consists of a shallow rectangular cavity , and height , (Fig. 1) whose with length, , width, top conducting sheet extends an arbitrary distance beyond the cavity walls in the - plane. The cavity is filled with dielectric material with relative permittivity . If is chosen to be a few millimeters or less, the antenna can be fabricated by standard printed-circuit-board fabrication techniques; the vertical cavity walls can be realized by closely spaced via holes. , A slot—with length, , in the direction and width, in the direction—is cut in the center of the top wall, and is from the center. As in [5] and [7], the excited a distance feed is achieved by crossing the slot with a microstrip line, whose ground plane is the cavity wall in which the slot is cut. This requires that a thin superstrate be laminated on top of the cavity wall that contains the slot, and enables the placement of a matching network and or active circuitry directly at the antenna feed. The resonance frequency of the antenna is tuned by a varactor diode across the slot at a distance from the center. The admittances of the slot and cavity are in parallel [8]–[10], and there-

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TABLE I DIMENSIONS OF THE SHALLOW CBS ANTENNA IN BOTH MILLEMETERS AND WAVELENGTHS AT 2 GHZ. REFERRED TO  ; REFERRED TO 

fore, combine to form a single resonance. As a consequence, the cavity and the slot do not need to be tuned independently. A. Design The starting point of the antenna design is to choose resonant dimensions for both the slot and the cavity. The length, , of , where and are the slot is the wavelength and effective relative permittivity of the slot-line mode, respectively [11]. The cavity is square, with , and , where is the is the relative permittivity free-space wavelength, and inside of the cavity. The dimensions are summarized in Table I for an unloaded resonance frequency of 2 GHz. Although it is trivial to tune the resonance frequency of any slot antenna with a variable shunt capacitance, a wideband impedance match is critical for wideband tuning. This impedance match over a large tuning range has been achieved for the slot-ring antenna by operating below resonance for every tuning state at the point where the input resistance is equal to the characteristic impedance of the system [2]. At this point, there is a significant series reactance, which is cancelled at every tuning state by the same microstrip stub. An impedance match over a wide tuning range for a microstrip-fed slot antenna (with a non-tunable stub) was also reported in [3], [12]. These results suggest that there may be combinations of feed point, , and loading point, , that allow the shallow CBS antenna to have a good impedance match over a wide tuning range. A computer program was written to search for combinations and that provide a reflection coefficient less than over the entire capacitance range of the varactor diode. Ten and equally-spaced locations were chosen between , and every combination where was simulated using the IE3D MoM solver [13] (for the purpose of the and are impedance match, the cases and ). The substrate identical if configuration consists of two parallel infinite conducting planes separated by 3.18 mm (the top and bottom of the cavity) with between them (Fig. 2). The cavity dielectric material walls are modelled by vertical conducting plates between the and top and bottom at . The slot is modelled as magnetic a current in the top sheet, and is excited by gap (port 1) and (port 2). Finally, a 0.79-mm-thick ports at dielectric layer is placed above the top sheet in anticipation of feeding the antenna with a microstrip line. These simulations are quite efficient for shallow cavities because only the slot and the sidewalls are meshed; the total time to simulate all 45 cases was less than 40 minutes on a 32-bit 3.2-GHz Intel Pentium IV processor.

Fig. 2. (a) Side view and (b) top view of the CBS structure that was simulated in IE3D. The horizontal layers extend to infinity.

Fig. 3. Circuit model for Z and port 2 with a varactor diode.

Z

when the antenna (Fig. 2) is loaded at

The 2-port S-parameters from each simulation were processed by the following method to find the combinations of that produce a sufficient impedance match over tuning. is chosen ( in this work) and it is decided First, whether the antenna will be matched by a series capacitor or , is then inductor. The antenna impedance, calculated by loading port 2 with a varactor diode as shown in Fig. 3. The varactor diode used in this work is the M/A-Com pF for bias voltage MA46H071 ( V; series resistance ). A series inductance, nH [14], is added to the varactor model because it is connected to the slot through two 0.25-mm via holes in the realization of the antenna. is plotted in Fig. 4 for and . The input resistance at resonance varies from 400 when pF, to 200 when pF. This resistance should not be confused with , where is the radiation conductance, because the feed is offset from the center of the is actually decreasing as the antenna is tuned to lower slot. is scaled in a frefrequencies, but the input conductance at quency dependent way. Furthermore, contributes signifias increases. Although a 200–400- impedance cantly to variation does not produce large reflections (one could, for example, operate at 283 with a maximum reflection coefficient of dB), transforming the impedance to 50 over an octave bandwidth is a problem. Instead, the antenna is matched using , which a series reactance at frequencies where are away from its natural resonance. For each combination of , the frequency points above and below resonance,

WHITE AND REBEIZ: A SHALLOW VARACTOR-TUNED CAVITY-BACKED SLOT ANTENNA

Fig. 4. Simulated R (f ) for different C for d

635

=` = 0:35 and d =` = 0:05.

Fig. 5. Simulated X (f ) for various combinations of (d

=`; d =`).

and , respectively, are found where . is positive, and is negative. Therefore, is is chosen for inchosen for capacitive matching and ductive matching. Inductive matching is used for this antenna, but the same methodology can be used for capacitive matching (which was not successful for this antenna). The ability to match an antenna over the tuning range with either a single inductor or capacitor depends on both the antenna geometry and the of the tuning element(s), and is beyond the scope of this paper. is known, the inductance, , required to Once is calculated for each tuning state. If is cancel constant, the antenna can be matched perfectly over the whole varies with , and therefore, tuning range. In general, the average value over the tuning range is chosen. Finally, the reflection coefficient is evaluated over the tuning range for each , and the combination that meets the requirement with the most convenient is chosen. Fig. 5 shows for four combinations of ( , ). Two solutions satisfy dB, and they are given in Table II. B. Realization A prototype based on Solution 1 has been realized (Table II), and the geometry is shown in Fig. 6. A cavity with dimensions is machined into a block of aluminum, and a card is cut to fit inside the cavity. There are PTFE two minor differences between this cavity and the one in the previous section: first, there are small air-filled cutouts due to the machining process, and second the dielectric constant is 2.1 instead of 2.2. These are believed to have only a minor effect on the impedance characteristics.

Fig. 6. (a) Top and (b) cross sectional view of the shallow cavity backed slot prototype. All dimensions are in millimeters. TABLE II COMBINATIONS OF (d ; d ) THAT RESULT IN A REFLECTION COEFFICIENT OF LESS THAN 20 DECIBELS OVER THE TUNING RANGE OF THE VARACTOR

0

The slot is etched on the bottom of a 0.79-mm-thick Rogers substrate, which is fastened to the cavity by 5880 eight screws. A short-circuited microstrip feed is placed at , and inductive matching is achieved by placing a 7.5-nH surat face-mount inductor (Coilcraft 0603CS-7N5XJL, 1.7 GHz) [15] in series with the microstrip line at the feed point. The microstrip substrate continues slightly beyond the cavity on one side so that an SMA connector can be attached from below, minimizing the interaction with the feed cable and enabling conformal mounting of the antenna. Normally, the varactor diode is connected directly across the slot, but the slot is on the bottom of the superstrate and the cavity is filled with dielectric. Instead, the varactor is mounted on the top, and is connected to the slot edges using via holes. Varactor biasing is achieved by cutting a 150- m-wide slot in the top of the cavity to create a small area that is connected to both the RF line and the cathode of the varactor diode, but is isolated from the rest of the cavity. These gaps are then RF-short-circuited along the slot edge by 22-pF capacitors (AVX ACCU-P, 0603) [16]. The bias voltage is applied to the RF feed-line using a bias tee. There is no appreciable DC resistance between the

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Fig. 7. Measured reflection coefficient for both the free-standing and conformally mounted (1.2 1.2 m ground plane) cases.

2

RF port and the cathode of the varactor diode, and therefore, a 1-k resistor was placed in series with the DC source during measurements to limit any transient current. Although not important for this demonstration, this resistance value is a critical parameter in the tuning speed of the antenna. Two environments in which this antenna may operate are freespace and conformally mounted on a large conducting surface. The second case was tested by mounting the antenna at the center of a 1.2 1.2 m conducting plane made of aluminum mm) electrically foil. A piece of copper foil ( connected the antenna to the ground plane in the plane of the slot, and all seams were short-circuited with copper tape.

Fig. 8. Comparison of the measured and simulated (finite ground-planes and finite dielectrics) reflection coefficient of the free-standing antenna (92 102 mm ground plane).

2

Fig. 9. Measured 10-dB bandwidth of the free-standing and conformally-mounted CBS antennas.

III. RESULTS AND DISCUSSION A. Impedance The impedance of the antenna was measured with an Agilent E5071B network analyzer for both the free-standing and conformally mounted cases for bias voltages between 0 and 20 V. The free-standing condition was achieved by placing the antenna on a foam block about 1 m from any scatterers. The measured reflection coefficient of the two cases is compared in Fig. 7. There is almost no difference between the two cases, suggesting that this antenna is not sensitive to the substrate size. This is consistent with the results in [17], and can be explained by the fact that the susceptance of the cavity is large compared to the susceptance of the radiating side. This also suggests that the antenna is not sensitive to objects placed near it. Before it was mounted on the antenna, the impedance of the varactor diode was measured versus frequency using and Agilent E4491A impedance analyzer for bias voltages . varies from 0.45 pF at 2 GHz and 20 V to 2.7 pF at 1 GHz and 0 V. The reflection coefficient was re-simulated using the simplified infinite-substrate model of Fig. 2 (with for the cavity) and the measured impedance of both the varactor and the matching inductor. The resonance frequencies (not shown) are in agreement for V, but the agreement degrades as the bias voltage V, the simulation predicts GHz, decreases; at whereas the measured is 1.0 GHz. Better agreement has been obtained by including the biasing slot and RF-shorting capacitors in the infinite-substrate model (not shown), and even better agreement has been obtained by simulating the structure in Fig. 6 with finite ground planes and finite dielectrics (Fig. 8).

The microstrip lines were not included in this simulation; the feed and all lumped elements were included using gap-ports. The agreement is good for V, and is satisfactory for V. The most likely causes of the disagreement for V are port parasitics in the simulation, the effect of the microstrip feed, and the fabrication tolerance of the cavity. Although the measured and simulated resonance frequencies differ for some bias voltages, this simulation method led to an excellent wideband design. varies from 1 to 2 The 10-dB impedance bandwidth percent as the antenna is tuned from 1 to 1.9 GHz (Fig. 9). The reduction in bandwidth as the antenna is capacitively loaded is consistent with slot antennas in the literature. For a well matched resonant circuit, the 3-dB bandwidth is related to the quality . The variation of the radiation of a factor as lossless tunable CBS with frequency is more severe than [18], but the measured total of the tunable antenna varies more gently due to loss. B. Radiation Patterns and Efficiency The radiation pattern of the prototype was measured in a Satimo Stargate32 spherical near-field pattern measurement dB over 1–2 GHz). The system [19] ( measured radiation patterns of the free-standing antenna are shown in Fig. 10 for 1.0, 1.5, and 1.9 GHz (0, 4, and 20 V bias, respectively), and agree quite well with the simulated patterns (not shown). The patterns are similar and nearly symmetric in the E- and H-planes, and are linearly polarized with cross-polarization levels of dB. The front-to-back ratio is as low as 5 dB at 1.0 GHz, and as high as 13 dB at

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TABLE III SIMULATED LOSS BUDGET OF THE VARACTOR-TUNED CBS ANTENNA. EACH LOSS TERMS REPRESENTS THE PERCENTAGE OF THE INCIDENT POWER DISSIPATED. THE VARACTOR IS THE LARGEST CONTRIBUTOR TO THE LOSS

Fig. 11. Measured and simulated antenna gain and total efficiency.

Fig. 10. Measured antenna-gain patterns (in dBi) over the tuning range of the tunable CBS antenna. There is no ground-plane extension beyond the 92 102 mm antenna (Fig. 6). (a) 1.0 GHz, V V, (b) 1.5 GHz, V V, (c) 1.9 GHz, V V.

2

= 20

=0

=4

1.5 GHz. The similarity of the E- and H-plane patterns and the low front-to-back ratio occur because the antenna is less than

one wavelength in dimension at the highest frequency. It has been shown that the input impedance is relatively insensitive to the ground plane size, and therefore, increased front-to-back ratio can be achieved by simply increasing the size without any degradation in the impedance match. to The measured gain and total efficiency vary from dB and 18 to 76%, respectively, over the 1.0–1.9 GHz range (Fig. 11). The gain and total efficiency of the free-standing antenna (Fig. 6) were simulated in IE3D using aluminum for the cavity, copper for the top sheet, and the measured impedances of the varactor diode and matching inductor; the results agree quite well with the measured values. The simulated total efficiency was then combined with a linear circuit simulation to determine the loss budget (Table III). The proportions of the incident power dissipated in the varactor diode and the matching inductor were calculated by replacing their respective series resistances with ports and calculating the scattering matrix. The metal/dielectric loss was then calculated by conservation of energy. The largest contribution to the loss is the series resistance of , and the second largest is the varactor diode the metal/dielectric loss. Loss mechanisms that were not simulated and may be significant above 1.5 GHz (where the varactor loss is less dominant) are the resistance in the interface between the superstrate and the cavity—to which force is applied by 8 screws—the finite of the RF-short-circuiting capacitors across the bias gap, and losses in the SMA connector ( dB based on a measured test structure). C. Power Handling The varactor diode is a non-linear device, and therefore, its capacitance depends not only on the bias voltage, but also on the RF voltage across it. As the power incident upon the antenna increases, two things occur that degrade the performance

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ACKNOWLEDGMENT The authors would like to thank Dr. E. Ozaki at Qualcomm for the use of the Satimo antenna measurement system. REFERENCES Fig. 12. Measurement setup for P

.

Fig. 13. Measured input 1-dB compression point.

of the antenna: first, the effective capacitance increases, which de-tunes the antenna, and second, some of the power is converted to harmonics of the signal. The input 1-dB-compres, was measured using the setup in Fig. 12, sion point, dBm at 1 GHz to 10 dBm at 1.9 GHz and varies from is lower at 1–1.5 GHz (lower bias volt(Fig. 13). The ages) because the capacitance varies most rapidly with voltage at 0–4 V, and also due to the higher antenna , which means larger RF voltage swings across the diode. The varactor-diodetuned CBS is therefore most suitable for receive applications; however, higher power handling is achievable with varactors based on RF-MEMS [20].

IV. CONCLUSION A shallow microstrip-fed cavity-backed-slot (CBS) antenna has been demonstrated that tunes from 1 to 1.9 GHz with dB reflection coefficient and dB cross-polarization using a single varactor diode, and the input impedance is nearly identical whether the antenna is in free-space or conformally mounted on a 1.2 1.2 m ground plane. The impedance match over tuning is achieved by operating at frequencies slightly above the slot/cavity resonance where the antenna resistance is 50 , and cancelling the reactance with a . The amount of inductance required for this series inductor impedance match depends on both the frequency and the feed and loading locations; however, two combinations of feed and is constant loading locations have been found for which with frequency. Solution 1 (Table II) has been implemented using a machined aluminum cavity, and future designs may utilize printed-circuit-board fabrication with via-hole technology. This antenna is most suited to receive applications due to the non-linearity of the varactor diode, and higher power levels can be handled using MEMS varactors.

[1] R. C. Hansen, “Fundamental limitations in antennas,” Proc. IEEE, vol. 69, no. 2, pp. 170–182, Feb. 1981. [2] C. R. White and G. M. Rebeiz, “Single- and dual-polarized tunable slot-ring antennas,” IEEE Trans. Antennas Propag., vol. 57, no. 1, pp. 19–26, Jan. 2009. [3] N. Behdad and K. Sarabandi, “Dual-band reconfigurable antenna with a very wide tunability range,” IEEE Trans. Antennas Propag., vol. 54, no. 2, pp. 409–416, Feb. 2006. [4] D. Sievenpiper, H.-P. Hsu, and R. M. Riley, “Low-profile cavity-backed crossed-slot antenna with a single-probe feed designed for 2.34-ghz satellite radio applications,” IEEE Trans. Antennas Propag., vol. 52, no. 3, pp. 873–879, Mar. 2004. [5] A. Vallecchi and G. B. Gentili, “Microstrip-fed slot antennas backed by a very thin cavity,” Microw. Opt. Technol. Lett., vol. 49, no. 1, pp. 247–250, Jan. 2007. [6] A. C. Polycarpou et al., “Radiation and scattering from ferrite-tuned cavity-backed slot antennas: Theory and experiment,” IEEE Trans. Antennas Propag., vol. 49, no. 9, pp. 1297–1306, Sep. 1998. [7] W. Hong, N. Behdad, and K. Sarabandi, “Size reduction of cavitybacked slot antennas,” IEEE Trans. Antennas Propag., vol. 54, no. 5, pp. 1461–1466, May 2006. [8] C. R. Cockrell, “The input admittance of the rectangular cavity-backed slot antenna,” IEEE Trans. Antennas Propag., vol. 24, no. 3, pp. 288–294, May 1976. [9] J. Galejs, “Admittance of a rectangular slot which is backed by a rectangular cavity,” IEEE Trans. Antennas Propag., vol. 11, no. 2, pp. 119–126, Mar. 1963. [10] C.-H. Liang and D. K. Cheng, “Electromagnetic fields coupled into a cavity with a slot-aperture under resonant conditions,” IEEE Trans. Antennas Propag., vol. 30, no. 4, pp. 664–672, July 1982. [11] S. Raman and G. M. Rebeiz, “Single- and dual-polarized millimeterwave slot-ring antennas,” IEEE Trans. Antennas Propag., vol. 44, no. 11, pp. 1438–1444, Nov. 1996. [12] N. Behdad and K. Sarabandi, “A varactor-tuned dual-band slot antenna,” IEEE Trans. Antennas Propag., vol. 54, no. 2, pp. 401–408, Feb. 2006. [13] IE3D 11 Zeland Software Inc. Fremont, CA, USA. [14] M. E. Goldfarb and R. A. Pucel, “Modeling via hole grounds in microstrip,” IEEE Microw. Guided Wave Lett., vol. 1, no. 6, pp. 135–137, Jun. 1991. [15] Coilcraft, Inc. Cary, IL, USA. [16] AVX Corporation. Myrtle Beach, SC, USA. [17] B. Zheng and Z. Shen, “Effect of a finite ground plane on microstrip-fed cavity-backed slot antennas,” IEEE Trans. Antennas Propag., vol. 53, no. 2, pp. 862–865, Feb. 2005. [18] M. H. Cohen, “On the band width of cavity antennas,” J. Appl. Phys., vol. 25, no. 5, pp. 582–587, May 1954. [19] Satimo. Courtaboeuf, France. [20] G. M. Rebeiz, RF MEMS: Theory, Design, and Technology. New York: Wiley, 2003.

Carson R. White (M’07) received the B.S. degree in electrical engineering from the University of Washington, Seattle, in 2002, and the M.S. and Ph.D. degrees in electrical engineering from the University of Michigan, Ann Arbor, in 2004 and 2008, respectively. He is currently a Research Staff Member with HRL Laboratories, LLC., Malibu, CA, where he works in the field of applied electromagnetics and antennas.

WHITE AND REBEIZ: A SHALLOW VARACTOR-TUNED CAVITY-BACKED SLOT ANTENNA

Gabriel M. Rebeiz (F’97) received the Ph.D. degree from the California Institute of Technology, Pasadena. He is currently a Professor of electrical and computer engineering at the University of California, San Diego. Prior to this appointment, he was at the University of Michigan from 1988 to 2004. He has contributed to planar mm-wave and THz antennas and imaging arrays from 1988–1996, and his group has optimized the dielectric-lens antennas, which is the most widely used antenna at mm-wave and THz frequencies. His group recently developed 6–18 GHz and 30–50 GHz 8- and 16-element phased arrays on a single chip, making them one of the most complex RFICs at this frequency range. His group also demonstrated very high-Q RF MEMS tunable filters from 4–6 GHz (Q 300) and the new angular-based RF MEMS capacitive and metal-contact switches. As a consultant, he developed the 24 GHz single-chip automotive radar with USM/ViaSat, X, Ku-Band and

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W-band phased arrays for defense applications, the RFMD RF MEMS switch and the Agilent RF MEMS switch. He leads a group of 18 Ph.D. students and three Postdoctoral Fellows in the area of mm-wave RFIC, microwaves circuits, RF MEMS, planar mm-wave antennas and terahertz systems, and is the Director of the UCSD/DARPA Center on RF MEMS Reliability and Design Fundamentals. He is the author of the book, RF MEMS: Theory, Design and Technology (Wiley, 2003). Prof. Rebeiz is an IEEE Fellow, an NSF Presidential Young Investigator, an URSI Koga Gold Medal Recipient, an IEEE MTT Distinguished Young Engineer (2003), and is the recipient of the IEEE MTT 2000 Microwave Prize. He also received the 1998 Eta Kappa Nu Professor of the Year Award and the 1998 Amoco Teaching Award given to the best undergraduate teacher at the University of Michigan, and the 2008 Teacher of the Year Award at the Jacobs School of Engineering, UCSD. His students have won a total of 17 best paper awards at IEEE MTT, RFIC and AP-S conferences. He has been an Associate Editor of the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, and a Distinguished Lecturer for IEEE MTT and IEEE AP.

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Dielectric Loaded Substrate Integrated Waveguide (SIW) -Plane Horn Antennas

H

Hao Wang, Member, IEEE, Da-Gang Fang, Fellow, IEEE, Bing Zhang, and Wen-Quan Che, Member, IEEE

Abstract—A dielectric loaded substrate integrated waveguide (SIW) -plane sectoral horn antenna has been proposed in this paper. The horn and the loaded dielectric are integrated by using the same single substrate resulting in easy fabrication and low cost. Two antennas with rectangular and elliptical shaped loaded dielectrics were designed and fabricated. These antennas have high gain and narrow beamwidths both in the -plane and in the -plane. The results from the simulation and those from the measurement are in good agreement. To demonstrate applications of the array, the small aperture elliptical dielectric loaded antenna has been used to form an array to obtain higher gain and to form a one-dimensional monopulse antenna array. Index Terms—Array, dielectric loaded, substrate integrated waveguide.

-plane sectoral horn,

I. INTRODUCTION

T

HE rectangular waveguide horn is one of the simplest and probably the most widely used microwave antenna. Recently, the substrate-integrated waveguide (SIW) technique has been investigated and developed to construct the planar rectangular waveguide [1]–[9]. The application of SIW for the design of an integrated -plane sectoral horn antenna was proposed in [10]. This structure is easy to be integrated with the feed network and is a good candidate to feed the surface-wave antennas or the leaky-wave antennas [11]. On the other hand, the patch fed planar dielectric slab extended hemi-elliptical lens antenna was reported in [12]. The SIW millimeter-wave monopulse antenna was introduced in [13]. In this paper, we present a dielectric loaded -plane sectoral SIW horn antenna. This antenna is integrated by using a single substrate. It is easy to fabricate and the structure is compact. To eliminate the higher order modes in the waveguide, the thickness of the substrate is restricted. The loaded dielectric slab in front of the horn aperture can be considered as a dielectric guiding structure excited by the horn aperture resulting in a narrower beamwidth in the -plane. For a horn of maximum gain, the aperture phase distribution along the -plane is nearly uniform

W = 1 mm L = 1:8 mm a = 5 mm D = 14 mm L = 7 mm L = 15 mm R = 0:25 mm R = 0:5 mm

Fig. 1. Geometry of SIW horn antenna without dielectric loading. , , , , , , , .

without the dielectric loading. If the length of the slab is not properly chosen the beamwidth in the -plane will be broadened. To reduce the size, the length of the horn can be shortened, but the quadratic phase error will increase simultaneously. In this case, the dielectric loading may serve as the phase corrector in the -plane. Through proper choosing of the length of the dielectric slab, both of the beamwidths in the -plane and in the -plane will be narrowed and consequently the high gain is obtained. The good agreement between the simulated results and those from experiments confirms the correctness of the proposed idea and the advantage of this horn antenna. In addition, in this paper the single horn is also used to form a high gain array and a one-dimensional monopulse array. of 4.8, In this paper, a substrate with dielectric constant thickness of 2.5 mm and a working frequency of 27 GHz are used in all the simulated and measured results. All the simulated results are gotten from Ansoft HFSS. II. SIW

Manuscript received January 24, 2009; revised July 11, 2009. First published December 28, 2009; current version published March 03, 2010. This work was supported in part by the Nature Science Foundation of China under Grant 60671038 and in part by the Key Laboratory of Target Detection at NJUST. H. Wang, D.-G. Fang, and W.-Q. Che are with the Department of Communication Engineering, Nanjing University of Science and Technology, Nanjing 210094, China (e-mail: [email protected]). B. Zhang was with the Department of Communication Engineering, Nanjing University of Science and Technology, Nanjing 210094, China. He is now with ZTE Company, Shenzhen 51805, China. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2009.2039298

-PLANE SECTORAL HORN ANTENNA WITHOUT DIELECTRIC LOADING

The geometry of the SIW -plane sectoral horn antenna is given in Fig. 1. The rectangular waveguide and sectoral horn antenna are integrated by using the same single substrate based on the SIW technology. As a result they are easy to fabricate and the structure is compact. In the waveguide of the SIW horn, . To ensure the single mode excithe dominant mode is tation of the horn, the width and the thickness should be chosen based on the inequality: . The given dielectric constant and the thickness of the substrate

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Fig. 2. Radiation patterns of SIW horn antenna shown in Fig. 1.

Fig. 4. Dielectric loaded SIW horn antennas. (a) Elliptical dielectric loading. (b) Rectangular dielectric loading.

Fig. 3. Radiation patterns of shortened length SIW horn antenna without dielectric loading.

satisfy the inequality. The equivalent width been derived in [14] as

of the SIW has

(1)

is the spacing between two adjacent vias, and is the where radius of the via. Based on the design rule of SIW, and are chosen to be 0.8 mm and 1 mm [14]. From formula (1), the width of SIW is determined as 5 mm. The equivalent thickness of the SIW is equal to the the thickness of the waveguide . For a given aperture , it was found that the affects not only the quadratic phase error length of the horn on aperture along the -plane but also the higher order mode excitation in the horn. A reasonable length of 15 mm was determined to obtain the acceptable quadratic phase error and single mode on the aperture. The simulated gain and side lobe level ; the simulated beamwidths of the are 6.73 dBi and -plane and the -plane are 180 and 48 respectively. The simulated radiation patterns are shown in Fig. 2.

III. DIELECTRIC LOADED SIW

-PLANE HORN ANTENNAS

The beamwidth in the -plane can be controlled through the aperture size in the -plane. The beamwidth in the -plane is determined by the aperture size in the -plane that is limited. In some applications, a narrow beamwidth in the -plane is also desired. For this purpose, a dielectric slab is placed in front of the aperture of the horn. This slab serves as the dielectric guiding structure in the -plane. In the -plane, for a horn with maximum gain, the aperture phase distribution along the -plane is nearly uniform without the dielectric loading. If the length of the slab is not properly chosen the beamwidth in the -plane will even be broadened. Take the example of the horn in Fig. 1. If it is loaded by a dielectric elliptical slab with a length of 1.4 mm, the beamwidth in the -plane is broadened by 18 . The dielectric loading was used in the -plane sectoral shortened length horn antenna to reduce the size long before [11]. In this case, the dielectric loading serves as the phase corrector in the -plane to compensate the quadratic phase error caused from the shortened length. Through proper choosing of the length of the dielectric slab, both of the beamwidths in the -plane and in the -plane will be narrowed and consequently the high gain is obtained. in Fig. 1 from 15 mm to Now we change the length 10 mm. The simulated radiation patterns of this -plane

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TABLE I COMPARISON OF THE COMPUTING TIME USED USING THE DIFFERENT METHODS.

Fig. 5. Gains versus the length of loaded dielectric.

horn antenna without dielectric sector are shown in Fig. 3. The gain is 5.75 dBi, the back sidelobe level is 11.78 dB and the beamwidths of the - and -plane are 178 and 63 , respectively. An elliptical dielectric slab and a rectangular dielectric slab are now placed in front of the aperture of the horn, respectively. The corresponding structures are shown in Fig. 4. The gains versus the lengths of dielectric slabs are shown in Fig. 5. The variations of beamwidths in the -plane and in the -plane are given in Fig. 6. From these figures, it is seen that on the whole, the beamwidths monotonically decrease with the increase of the dielectric length. For example, when the length of elliptical slab is 9.8 mm, the gain is 11.68 dBi and the beamwidths in the -plane and in the -plane are 60 and 42 . When comparing with the SIW antenna without dielectric loading, the gain has increased by 5.93 dB and the beamwidths in the -plane and in the -plane have reduced by 118 and 21 , respectively. The side lobe levels versus the length of dielectric loading are shown in Fig. 7. Furthermore, different shapes of dielectric loading were investigated. They are ellipse, rectangle, hyperbola and parabola. Table I shows the results. From these results, it is seen that the rectangular slab gives the narrowest beamwidths in both - and -planes and the highest gain with slightly higher side lobe level. The rectangular and elliptical dielectric loaded SIW horn antennas were fabricated and measured. The length of dielectric slab is 9.8 mm. An SMA probe was used to feed the antenna. The photos of antennas are given in Fig. 8. The simulated and are given in Fig. 9. The - and - plane radiameasured tion patterns of rectangular dielectric loaded SIW horn antenna

Fig. 6. Beamwidths versus the length of loaded dielectric. (a) E plane. (b) H plane.

Fig. 7. Side lobe levels versus the length of loaded dielectric.

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Fig. 8. Two kinds of SIW sectoral horn antennas. (a) Rectangular loading. (b) Elliptical loading.

Fig. 10. Radiation patterns of rectangular dielectric loaded SIW horn antenna. plane. (b) plane. (a)

H

E

IV. SIW ANTENNA ARRAY

Fig. 9. Simulated and measured jS11j parameters.

are shown in Fig. 10. The measured gain is 9.7 dBi. The corresponding simulated and measured radiation patterns of elliptical dielectric loaded SIW horn antenna are given in the Fig. 11. The measured gain is 9.3 dBi. The results from the simulation and those from measurement are in good agreement.

To construct the antenna array in -plane, the aperture of SIW -plane horn antenna should be within one wavelength in -plane. The dielectric loaded small aperture SIW -plane horn antenna is introduced for such purpose. The size of the aperture is 10 mm which is less than one wavelength. The simulated results are given when the length of dielectric slab is 7 mm. The gain is 8.83 dBi and the beamwidths of the - and -plane are 80 and 60 . This dielectric loaded small aperture SIW antenna was also fabricated. The photo is given in Fig. 12. The simulated and measured reflection coefficients are shown in Fig. 13, while the simulated and measured radiation patterns are illustrated in Fig. 14. The measured gain is 9.1 dBi. A SIW antenna array formed by four small aperture SIW elements is shown in Fig. 15. As we can see, this dielectric loaded SIW horn element can be integrated in the array easily. The detailed dimensions of the 5-way power divider are also shown in the figure. The photo of this array is given in Fig. 16. The simof SIW antenna array are shown in ulated and measured

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Fig. 13. Simulated and measured jS11j of elliptical dielectric loaded small aperture horn.

Fig. 11. Radiation patterns of elliptical dielectric loaded SIW horn antenna. plane. (b) plane. (a)

H

E

Fig. 12. Elliptical dielectric loaded small aperture SIW horn antenna.

Fig. 14. Radiation patterns of elliptical dielectric loaded small aperture SIW plane. (b) plane. horn antenna. (a)

Fig. 17. The bandwidth ( ) is 1.5% at the central frequency 27 GHz from the simulated results. The measured

curve is shifted to the low frequency due to the fabrication error. The corresponding radiation patterns of the array are

H

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Fig. 18. Radiation patterns of 1

2 = = 4 5 mm = 5 mm = = 10 mm = 12 mm = 1 5 mm

2 4 SIW antenna array.

=w = = 1:1 mm,

Fig. 15. Structure of 1 4 dielectric loaded SIW antenna array. w : ,w w : w w ,w ,w : ,l ,l . w

Fig. 16. Photo of 1

= 1 5 mm

2 4 dielectric loaded SIW antenna array. Fig. 19. Structure of monopulse dielectric loaded SIW antenna array.

1:5 mm, w = 1:6 mm.

l =

= 0:4 mm, s = 0:8 mm, = 0 175 mm, L = 6:5 mm.

Fig. 20. Structure of Riblet short-slot coupler. d w : ,w w : ,w :

= 4 5 mm

Fig. 17. Simulated and measured

jS11j of 1 2 4 SIW antenna array.

shown in Fig. 18. The simulated gain of the array is 14.9 dBi. The beamwidths are 76 in the plane and 14 in the plane, respectively. The measured gain is 13.75 dBi. Furthermore, a

=

= 3 9 mm

one-dimensional SIW monopulse antenna array was developed. The structure of this monopulse array given in Fig. 19 is very compact and simple. This monopulse antenna array is formed by two 1 4 sub-arrays. A Riblet short-slot coupler [15] and 90 phase delay line are used to form a comparator. The detailed dimensions of the Riblet short-slot coupler are shown in

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Fig. 21. Magnitude of simulated S parameters of Riblet short-slot coupler.

Fig. 24. Measured S parameters of monopulse SIW horn antenna.

Fig. 22. Phase of simulated S parameters of Riblet short-slot coupler.

Fig. 23. Photo of 1

2 8 dielectric loaded SIW monopulse antenna array.

Fig. 20. The simulated parameters of coupler are shown in Figs. 21–22. From the simulated results, it is shown that at the frequency of 27 GHz, the phase difference between coupled and . Isolation and return loss for this direct signals is . The photo of this monopulse array coupler are around is given in Fig. 23. The measured parameters of the array are

Fig. 25. Sum and difference radiation patterns of monopulse SIW horn antenna. (a) Sum pattern. (b) Difference pattern.

shown in Fig. 24. The isolation between two ports is at the resonant frequency 27 GHz. The simulated and measured sum and difference patterns of monopulse SIW array are shown in Fig. 25. The simulated gain and the side lobe level of sum pattern are 17.9 dBi and 9.63 dB, respectively. The measured gain

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is 15.65 dBi. The big difference between simulated result and the measured one is believed to be the dielectric loss especially in the case of using the long feed line to form a large array. In the above simulation, the dielectric loss tangent is assumed to be 0.001 according to the company’s data sheet measured at X band. In Ka band the loss tangent will be much larger. As an evidence, if we choose the loss tangent to be 0.003, the gain of the monopulse array reduces to 15.3 dBi that is very closer to the measured gain. The simulated and measured null depths of and , respectively. difference pattern are

Hao Wang (M’08) received the B.E. and Ph.D. degrees from the Nanjing University of Science and Technology (NJUST), Nanjing, China, in 2002 and 2009. He is currently a Lecturer at the NUST. His research is focused on microstrip antennas and related electromagnetic simulation.

V. CONCLUSION This paper presents the optimized design of dielectric loaded substrate integrated waveguide -plane antennas. Through proper choosing of the structure and the parameters, the maximum gain for the given size or the minimum size for a given gain may be achieved. The reasonable agreement between the simulated results and the experimental ones verifies the correctness of the designs. In addition, the examples of two arrays show the potentiality of the proposed antenna for many applications.

Da-Gang Fang (SM’90–F’03) was born in Shanghai, China. He graduated from the graduate school of Beijing Institute of Posts and Telecommunications, Beijing, China, in 1966. From 1980 to 1982, he was a Visiting Scholar at Laval University (Quebec, Canada), and the University of Waterloo (Ontario, Canada). Since 1986, he has been a Professor at the Nanjing University of Science and Technology (NJUST), Nanjing, China. Since 1987, he had been a Visiting Professor with six universities in Canada and in Hong Kong. He has authored and coauthored three books, two book chapters and more than 380 papers. He is also the owner of three patents. His research interests include computational electromagnetics, microwave integrated circuits and antennas and EM scattering. Prof. Fang is a Fellow of the Chinese Institute of Electronics (CIE). He was the recipient of the National Outstanding Teacher Award and People’s Teacher Medal, and the Provincial Outstanding Teacher Award. He is an Associate Editor of two Chinese journals and is on the Editorial or Reviewer Board of several international and Chinese journals. He was TPC Chair of ICMC 1992, Vice General Chair of PIERS 2004, the member of International Advisory Committee of six international conferences and TPC Co-Chair of APMC 2005 and is the General Co-Chair of ICMMT 2008. His name was listed in Marquis Who is Who in the World (1995) and in International Biographical Association Directory (1995)

ACKNOWLEDGMENT The authors would like to thank the reviewers for their constructive comments and their scrutiny in correcting the errors in this paper. REFERENCES [1] D. Deslandes and K. Wu, “Integrated transition of coplanar to rectangular waveguides,” in IEEE MTT-S Int. Microwave Symp. Dig., Feb. 2001, pp. 619–622. [2] Y. Cassivi et al., “Dispersion characteristics of substrate integrated rectangular waveguide,” IEEE Microw. Wireless Compon. Lett., vol. 12, no. 9, pp. 333–335, 2002. [3] C.-H. Tseng and T.-H. Chu, “Measurement of frequency-dependent equivalent width of substrate integrated waveguide,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 4, pp. 1431–1437, 2006. [4] F. Xu and K. Wu, “Guided-wave and leakage characteristics of substrate integrated waveguide,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 1, pp. 66–73, 2005. [5] D. Deslande and K. Wu, “Single-substrate integration technique ofplanar circuits and waveguide filters,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 2, pp. 593–596, 2003. [6] A. Zeid and H. Baudrand, “Electromagnetic scattering by metallic holes and its applications in microwave circuit design,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 4, pp. 1198–1206, 2002. [7] L. Yan et al., “Simulation and experiment on SIW slot array antenna,” IEEE Microw. Wireless Compon. Lett., vol. 14, no. 9, pp. 446–448, 2004. [8] D. Deslandes and K. Wu, “Integrated microstrip and rectangular waveguide in planar form,” IEEE Microw. Wireless Compon. Lett., vol. 11, pp. 68–70, 2001. [9] F. Xu, K. Wu, and W. Hong, “Domain-decomposition FDTD algorithm combined with numerical TL calibration technique and its application in parameter extraction of substrate integrated,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 1, pp. 329–338, 2006. [10] Z. L. Li and K. Wu, “An new approach to integrated horn antenna,” in Proc. Int. Symp. on Antenna Technology and Applied Electromagnetics, Jul. 2004, pp. 535–538. [11] R. E. Collin and F. J. Zucker, Antenna Theory, Part 2. New York: McGraw-Hill, 1969. [12] L. Xue and V. Fusco, “Patch fed planar dielectric slab extended hemielliptical lens antenna,” IEEE Trans. Antennas Propag., vol. 56, no. 3, pp. 661–666, 2008. [13] Y. J. Cheng, W. Hong, and K. Wu, “Design of a monopulse antenna using a dual V-type linearly tapered slot antenna (DVLTSA),” IEEE Trans. Antennas Propag., vol. 56, no. 9, pp. 2903–2906, 2008. [14] W. Che, K. Deng, D. Wang, and Y. L. Chow, “Analytical equivalence between substrate-integrated waveguide and rectangular waveguide,” IET Microw. Antennas Propag., vol. 2, no. 1, pp. 35–41, 2008. [15] D. M. Pozar, Microwave Engineering, 2nd ed. New York: Wiley, 1998.

Bing Zhang received the B.S. and M.S. degrees in electrical engineering from Nanjing University of Science and Technology, Nanjing, China, in 2007 and 2009, respectively. Now, he is working at the ZTE Company, Shenzhen, China.

Wen-Quan Che (M’01) received the B.Sc. degree from the East China Institute of Science and Technology, China, in 1990, the M.Sc. degree from the Nanjing University of Science and Technology (NJUST), China, in 1995; and the Ph.D. degree from City University of Hong Kong (CITYU), in 2003. In 1999, she was a Research Assistant at CITYU. From March 2002 to September 2002, she was a Visiting Scholar at the Polytechnique de Montreal, Canada. She is currently a Professor in NJUST. In 2007–2008, she conducted the academic research with the Institute of HFT, Technische University Munchen. Her interests include electromagnetic computation, planar/co-planar circuits and subsystems in RF/microwave frequency. She has authored or coauthored over 70 articles in refereed journals. Dr. Che was the recipient of the 2007 Humboldt Research Fellowship presented by the Alexander von Humboldt Foundation of Germany, and she was also the recipient of the 5th China Young Female Scientists Award in 2008.

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Planar Annular Ring Antennas With Multilayer Self-Biased NiCo-Ferrite Films Loading Guo-Min Yang, Student Member, IEEE, Xing Xing, Andrew Daigle, Ogheneyunume Obi, Student Member, IEEE, Ming Liu, Jing Lou, Student Member, IEEE, S. Stoute, Krishna Naishadham, Senior Member, IEEE, and Nian X. Sun, Member, IEEE

Abstract—With their high relative permeability, magneto-dielectric materials show great potential in antenna miniaturization. This paper presents an annular ring antenna with self-biased magnetic films loading in the gigahertz frequency range. The annular ring antenna was realized by cascading a microstrip ring and a tuning stub. Self-biased NiCo-ferrite films were adopted to load an annular ring antenna on a commercially available substrate that operates at 1.7 GHz. Novel antenna designs with self-biased NiCo-ferrite films on one side and both sides of the substrate were investigated. Antennas with self-biased magnetic films loading working at 1.7 GHz showed a down shift of 2–71 MHz of the central resonant frequency. An antenna gain enhancement of up to 0.8 dB was observed over the non-magnetic antenna. Index Terms—Annular ring antennas, antenna miniaturization, magnetic films, self-biased ferrite films.

I. INTRODUCTION

T

HE need for antennas with small size, light weight, and low profile have been continuously growing in modern wireless communication systems [1], [2]. Annular ring antennas have been of great interest to many researchers and engineers in recent years [3]–[7], because annular ring antennas have much smaller circumferences compared to circular patch antennas, and they can radiate a linearly polarized wave or a circularly polarized wave by disturbing the symmetry of the ring. The substrates of planar antennas play a very important role in achieving desirable electrical and physical characteristics. For most cases, antennas can be greatly miniaturized by using a substrate with high relative permittivity [8]. However, antennas with high-permittivity substrates will result in decreased bandwidth and the excitation of surface waves leading to lower radiation efficiency. Bulk ferrite materials [9]–[13], composites of ferrite particles in a polymer matrix [14], metamaterials [15]–[17], etc. have been . In [14] ferused in antenna substrates for achieving rite and polymer composite, antenna substrates were produced for the purpose of antenna miniaturization by using their high permeability. It is possible for a ferrite patch antenna to excite Manuscript received January 12, 2009; revised August 08, 2009. First published December 28, 2009; current version published March 03, 2010. This work was supported by the ONR and NSF. G.-M. Yang, X. Xing, A. Daigle, O. Obi, M. Liu, J. Lou, S. Stoute and N. X. Sun are with the Department of Electrical and Computer Engineering, Northeastern University, Boston, MA 02115 USA (e-mail: [email protected]; [email protected]). K. Naishadham is with the Sensors and Electromagnetic Application Laboratory, Georgia Institute of Technology, Atlanta, GA 30332. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2009.2039295

radiation patterns to realize a circularly polarized antenna [10]. A tunable resonant frequency in a patch antenna was obtained by applying an external bias magnetic field to an yttrium iron garnet (YIG) film [12]. It has also been reported that by applying a biasing magnetic field to the ferrite substrate of a patch antenna, two different radiation modes could be observed [13]. However, these materials or composites are too lossy to be used at frequenMHz under a self-biased condition and large biasing cies magnetic fields are needed for these ferrites to operate at higher frequencies. Metamaterials with embedded metallic circuits are good candidates for providing a high permeability in antenna miniaturization [15]–[17]. However these metamaterials with embedded metallic circuits need periodic metallic rings and slabs to produce the relatively high permeability, and are not practical in real applications in modern mobile communication systems. In order to be practically feasible in miniature antenna applications, such as handheld wireless communication devices, it is important for antenna substrates to comprise of magnetic materials without an external bias magnetic field. Magnetic thin films provide a unique opportunity for achieving self-biased magat frequencies larger netic patch antenna substrates with than 1 GHz. The strong demagnetization field for magnetic thin , allows for a self-biased magnetizafilms, tion with high ferromagnetic resonance (FMR) frequencies up to several GHz, making these magnetic thin films great candidates for achieving self-biased magnetodielectric antenna substrates working in the same frequency range. Our most recent work on magnetic films loaded patch antennas showed significantly enhanced antenna gain and bandwidth [18]–[20]. In this paper, we present a planar annular ring antenna with self-biased NiCo-ferrite thin films on the top of the antenna and both on top and beneath the ground plane, thus essentially creating a magneto-dielectric substrate/superstrate for practical applications. A large frequency shift of the central resonant frequency of 2 MHz–71 MHz and an enhanced bandwidth were obtained for the designed magnetic ring antennas over the nonmagnetic counterparts at 1.7 GHz, which shows great potential for applications in mobile wireless communication systems. In addition, these magnetic antennas can be made conformably at a low cost near room temperature. II. ANTENNA CONFIGURATION AND CHARACTERISTICS OF NICO-FERRITE FILM A. Antenna Configuration Fig. 1(a) and (b) show the schematic view and photograph, respectively, of the annular ring antenna. This antenna consisted of

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Fig. 1. Geometry of the proposed annular ring antenna. (a) Top view and side mm, R : mm, R : mm, L : mm and view. R H : mm. (b) Photograph of the fabricated annular ring antenna without films.

= 1 27

= 14

= 12 4

= 11 4

= 6 22

Fig. 2. (a) Hysteresis loop of the NiCo-ferrite film. (b) X-ray diffraction for the NiCo-ferrite film.

a microstrip ring and a tuning stub. Both the microstrip ring and the tuning stub were realized by patterned copper cladding on the top surface of the underlying dielectric substrate. The feed point was located at the junction of the tuning stub and the microstrip ring with a distance of 0.5 mm to the outer edge of the ring. The radius of the outer ring is 12.4 mm, the radius of the inner ring is 11.4 mm, the length of the tuning stub is 6.22 mm, and the width of the tuning stub is 1 mm. We adopted Rogers RT/duroid 6010, with a relative permittivity of 10.2 and a thickness of 1.27 mm, as the substrate in both simulations and fabrication. The proposed annular ring antenna was designed and simulated with the help of High Frequency Structure Simulator (HFSS 10.0). B. The Characteristics of NiCo-Ferrite Films Microwave ferrite ceramics show relatively high permeand high permittivity , as well ability of as low loss at RF/microwave frequencies. These characteristics are highly desirable for the miniaturization of many different RF/microwave devices, including antennas. The operating frequencies of bulk microwave magnetic materials are limited to less than 600 MHz due to the excessive magnetic loss tangents associated with various loss mechanisms, with FMR being one of the major loss mechanisms. The FMR frequency is therefore the upper frequency limit for antenna substrates for achieving . Our recent work on microwave magnetic thin films, including metallic magnetic films and ferrite films, indicates that

these magnetic thin films can readily operate in the gigahertz frequency range under a self-bias condition [21], [22], and have been widely used in RF/microwave devices, including antennas [18]–[20]. In this work, we used self-biased spinel NiCo-ferrite films fabricated by a low-cost spin-spray deposition process [23], a wet chemical synthesis process at a low-temperature of 90 C. NiCo-ferrite films with the composition of Ni Co Fe O were deposited onto a thin transparency. The thickness of ferrite film was about 2 m. The in-plane and out-of-plane magnetic hysteresis loops of the NiCo-ferrite films were measured with a vibrating sample magnetometer (VSM) with the external magnetic field applied in the film plane, out of the film plane, respectively. The hysteresis loops indicate the on the applied magnetic field dependence of magnetization . As shown in Fig. 2(a), the in-plane hysteresis loop shows an in-plane coercivity of 165 Oe as well as the self-biased magnetization of the film under zero applied magnetic fields. There is a huge difference between the in-plane hysteresis and the out-plane hysteresis, indicating that the magnetization stays in the film plane under zero bias magnetic field. The NiCo-ferrite film showed an in-plane homogeneous magnetization with an . The in-plane resistivity in-plane relative permeability of of the NiCo-ferrite film was measured to be cm. An X-ray diffraction (XRD) technique was used to reveal the microstructure of NiCo-ferrite films with a copper Ka X-ray source. The X-ray diffraction pattern is shown in Fig. 2(b). It is

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clear from the XRD data that the NiCo-ferrite film has a single phase spinel structure without preferential orientation. The governing equation for magnetodynamics is the LandauLifshitz-Gilbert (L-L-G) equation [24], [25]. The in-plane susceptibility spectra of the magnetic films can be expressed by the equation [26] (1) where

is the Gilbert damping constant, and ( is the applied field, is the in-plane anisotropy field, is the saturation magnetization and is the gyromagnetic ratio). The relative permittivity of the NiCo-ferrite film is about 13 and the relative permeability is about 10 at the frequency of 2 GHz, similar to what was reported in [23].

Fig. 3. Equivalent circuit of the annular ring antenna.

cuit [1], [29]. As shown in Fig. 3, the value of capacitor will increase if we load the antenna with dielectric material, which in turn will shift the resonance frequency. One has to adjust the inductor to get the resonant frequency in this resonant circuit. If we load the antenna with inductive material (i.e., NiCo-ferrite films), the equivalent value of will increase. We can then decrease the equivalent capacitor for a certain frequency. The benefit of the inductive loading can be explained with the variation of stored energy, as shown in (4)

C. Loading Effects of Magnetic Materials Applying a superstrate is an effective gain enhancement method for microstrip antennas [27]. Instead of the conventional dielectric material, magnetic materials are adopted as a practical means to coat above the annular ring antenna in this paper. For a rectangular patch antenna with the dominant mode , the resonant frequency of the antenna is a function of of its dimension and given by [1] (2) where is the relative permeability and is the relative permittivity of the substrate, is the speed of light in free space, is the effective length of the rectangular patch [1]. Magnetodielectric materials loaded as superstrates can not only improve the antenna gain, but also miniaturize the antenna by the same factor using moderate values of permittivity and perme[28]–[30]. The magnetic films also ability of lead to better wave impedance match between the antenna sub), thus mitigating the negative strate and air (since impact on antenna efficiency from wave impedance mismatch between the substrate and air. The wave impedance mismatch at the substrate/free space interface is reduced by the addition of NiCo-ferrite layer with a relative permittivity of and a permeability of . If the wave impedances are matched on both sides of the interface, maximum transmission occurs, whereas little energy will be transmitted when the wave impedances are significantly different at the interface. The bandfor planar microwave antennas with a magwidth neto-dielectric substrate can be described as [28]

(4) In the above equation, is the current through the inductor of , and is the voltage of the capacitor . For the annular ring antenna, the source voltage is a constant, thus an increase in leads to increased stored energy. If magnetic ferrite films were loaded over the ring antenna, the equivalent value of will be increased, thus we can decrease the equivalent capacitor for a certain frequency. Furthermore, as indicated in (4), the stored energy will be decreased and more energy will be radiated into the free space. D. The Lossy Effects of NiCo-Ferrite Films on the Antenna Bandwidth The self-biased spinel NiCo-ferrite films possess a high relative permeability and can be fabricated by a low-cost spin-spray deposition process. The effects of the magnetic loss tangent of the ferrite films could not be neglected when several layers of NiCo-ferrite film were stacked together as a superstrate loading. The relationship between the antenna bandwidth and the lossy effects of substrate and NiCo-ferrite films could be expressed by the maximum achievable fractional bandwidth (FBW), which is given by [31], [32] (5) where is the total quality factor of the designed antenna and is given by

(3)

(6)

is the wavewhere is the thickness of the substrate, and length at the resonant frequency. Clearly the incorporation of a self-biased microwave ferrite film in antenna substrates leads to an increased permeability of antenna substrates, and thus, a reduced radiation frequency and an enhanced bandwidth. The impedance behavior of the annular ring antenna can be approximated with the behavior of an equivalent resonant cir-

where is the antenna radiation quality factor. is the quality factor due to the surface wave. For the ferrite films loading, the antenna acts as a source of surface waves in the material and the amount of the energy radiate to the excitation of surface wave depends on the thickness of the superstate. is the quality factor due to the dielectric losses, and

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Fig. 5. Measured resonant frequency and antenna gain with single-sided ferrite films loading.

Fig. 4. (a) Annular ring antenna with ferrite films above the ring. (b) Measured = 2; 4; 6; 8; 10 m. reflection coefficient with T

, here is the effective dielectric loss tangent of the composite structure of the Rogers material and the ferrite films. is the quality factor due to the magnetic losses, and , here is the effective magnetic loss tangent of the composite structure of the Rogers material and the ferrite films. The thickness of the ferrite film is only several microns, which is less than one percent of the total thickness of the substrate. The effective magnetic loss tangent would therefore not be a dominant factor to the antenna fractional bandwidth. is the quality factor due to the copper losses. With (5) and (6), we can see that the antenna fractional bandwidth is tightly linked to the material losses (both dielectric and magnetic losses). High material losses lead to large antenna fractional bandwidth but at the cost of the radiation efficiency at the same time. The self-biased spinel NiCo-ferrite film is a low-loss thin film at gigahertz frequencies with a very small thickness fraction, which leads to enhanced antenna bandwidth and improved antenna gain. E. Experimental Results for Single-Sided Ferrite Films Loading To investigate the loading effects of magnetodielectric materials as antenna’s superstrate, five annular ring antennas with ferrite films were designed and fabricated as follows. First, one layer of ferrite thin film with thickness of 2 m was introduced above the microstrip ring, as indicated in Fig. 4(a), in which m. In addition, magnetic antennas with several layers of the NiCo-ferrite film loading on one side were designed with the thickness of ferrite films varied with m, and shown in Fig. 4(a). In order to compare the

results with the non-magnetic antenna, the measured reflection coefficient of five magnetic antennas along with that of nonmagnetic ring antenna were plotted and analyzed. The reflection coefficient curves in Fig. 4(b) were measured with all the geometrical dimensions of the antenna unchanged, except the ferrite film thickness. From Fig. 4(b) we can see that the central resonant frequency dB of the non-magnetic antenna is about 1.72 GHz, and the bandwidth is 5 MHz. When a ferrite film with thickness of 2 m is added above the ring; the resonant frequency shifts down to 1.70 GHz. This indicates a frequency down shift of 20 MHz relative to the non-magnetic antenna. When the thickness of the ferrite film is 4 m, the resonance shifts down to 1.675 GHz, a frequency shift of 45 MHz, or is equivalent to five times of the antenna bandwidth of non-magnetic ring antenna. The resonant frequencies are 1.667 GHz, 1.664 GHz, and 1.649 GHz, when the thicknesses of the ferrite film are 6, 8, 10 m, respectively. The antenna gains are 0.57 dB, 1.23 dB, 1.4 dB, 0.87 dB and 0.57 dB, respectively, for the antenna loaded with 2, 4, 6, 8 and 10 m thick ferrite film. Clearly, ferrite films loading can lead to miniaturized antennas by downward shift of the resonance frequency. A summary of the variation of resonant frequency and antenna gain with ferrite films loading is shown in Fig. 5. The antenna gain and the radiation patterns of E-plane and H-plane were measured in the anechoic chamber and plotted in Fig. 6(a) and (b), respectively. In order to compare the experimental results with the numerical results, the simulated E-plane and H-plane of the annular ring antenna are also given in the same figure. The gain comparison technique was used to determine the gain of the antenna. As shown in Fig. 6(a), when the thickness of ferrite films is 4 m or 6 m, the upper half pattern is almost the same, while the bottom half pattern shrank. This can also be observed in Fig. 6(b). This shows that as an inductive loading acts as a superstrate, more energy will be radiated into forward free space. A maximum antenna gain of 1.4 dBi was obtained when the thickness of ferrite films was 6 m and coated above the ring. All the measured gains were plotted in Fig. 5. However when the thickness is more than 6 m, the antenna gain begins to decrease. From the radiation patterns of E-plane, we can see that more energy may be radiated between the elevation angles of 60 –120 and 240 –300 . The main reason for this may be that with an increase in the thickness of ferrite

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Fig. 7. (a) Annular ring antenna with several layers of ferrite film above the ring and one layer of ferrite film under the ground plane. (b) Measured reflection = 2; 4; 6; 8 m; T = 2 m. coefficient with T

Fig. 6. Normalized radiation patterns of annular ring antennas loaded with ferrite films. Simulated results with T = 0 m only and measured results with T = 0; 2; 4; 6; 8; 10 m, respectively. (a) E-plane. (b) H-plane.

films, the surface wave may radiate more energy, which resulted in decreased antenna gain. The designed annular ring antenna is a linear polarization antenna. From the measured radiation patterns [Fig. 6(a) and (b)] we can clearly see that the radiation patterns are approximately the same shape, which indicates that self-biased NiCo-ferrite films can not excite two near degenerate orthogonal modes with equal amplitude and 90 degree phase difference. However, by applying a biasing magnetic field to the ferrite substrate of an antenna, one may get a circular polarization antenna. F. Experimental Results for Double-Sided Ferrite Films Loading As the radiation pattern of E-plane of the ring antenna is very similar to the shape of the digit “8”, which means the ring antenna will radiate into both forward and backward directions at the same time, it is necessary to investigate the magnetic antenna with ferrite films on both sides. For this purpose, we designed magnetic antennas with ferrite films loaded on both sides of the antenna, with one layer under the ground plane m) and one to four layers of ferrite films above ( the ring ( and 8 m, respectively), as shown in

Fig. 7(a). In order to compare the results with the non-magnetic antenna, the measured reflection coefficient of five antennas were plotted and analyzed. The reflection coefficient curves in Fig. 7(b) were measured under the condition that all the geometrical dimensions of the antenna were kept unchanged, and only the ferrite films were added at different positions. When one layer of ferrite film was added above the microstrip ring and one under the ground plane at the same time, the resonant frequency moved down to 1.70 GHz with a reflection dB. When two layers of ferrite coefficient magnitude of film were added above the ring and one layer under the ground plane, the resonance shifted down to 1.695 GHz with the mindB, a frequency shift imum reflection coefficient of about of 25 MHz compared to the non-magnetic substrate. The bandwidth was 7 MHz with the addition of the ferrite film, an increase of 40% relative to non-magnetic antenna’s bandwidth. As shown in Fig. 7(b), the ferrite film loading led to an enhanced bandwidth and an improved matching. It is notable that as we have analyzed in the Section II, the bandwidth improvement could be partially attributed to the lossy effects of the ferrite films (i.e., the loss tangent of the magnetic material. Also the antenna gain of this case is 1.23 dB. The addition of the three layers of ferrite film above the ring and one layer underneath the ground plane tunes the resonance down to 1.671 GHz. The resonant frequency shift is 49 MHz compared with the non-magnetic antenna. The antenna gain of magnetic antenna with top 3 layers and bottom 1 layer of ferrite film is 1.33 dB. Adding four layers of ferrite film above the patch shifts the resonance down dB. We to 1.664 GHz, with the reflection coefficient of observe a central frequency shift of about 56 MHz. Both sides (top and bottom) of ferrite films loading effects are summarized in Fig. 8. From this figure we can see that with a 2 m thick layer of ferrite film beneath the antenna, the resonant

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of without films, which means more energy will be radiated into free space. The measured antenna gains are also summarized in Fig. 8. The measured antenna gain increases first and with the increase of the thickness of ferrite films. Then the gain decreases when the thickness is more than 6 m. The main reason for this phenomenon may be the problem of surface waves and the lossy effects of the ferrite film, which may have resulted in decreased antenna gain when the thickness is more than 6 m. III. CONCLUSION

Fig. 8. Measured resonant frequency and antenna gain with double-sided ferrite films loading.

NiCo-ferrite films were successfully introduced into antenna substrates, leading to miniaturized magnetic annular ring antennas with enhanced performances at 1.7 GHz. These spinspray deposited NiCo-ferrite films are not as lossy as bulk ferrites materials, due to the strong demagnetization field of the magnetic thin films as well as the large in-plane anisotropy field of the magnetic films in the range of hundreds of Oe. These magnetic antennas show a great promise for achieving miniaturized microstrip antennas on magneto-dielectric superstrate/substrate with enhanced bandwidths, improved gains, and high efficiencies. In addition, these magnetic antennas can be made conformably at a low cost with low-temperature physical vapor deposition method, making these ring antennas with ferrite films very promising for real applications. Measurements on magnetic antennas demonstrated that the central resonant frequency could be varied downward over a range of 50% to 1420% of the , which indicates the self-biantenna bandwidth ased magnetic films do lead to minimized antennas by shifting down the resonance frequency. A maximum enhancement of antenna gain up to 0.8 dBi was obtained with ferrite films loading over the non-magnetic antenna. Antennas with self-biased ferrite films loading show significant improvement in antenna efficiency, gain and omnidirectional performance in gigahertz frequency range. ACKNOWLEDGMENT The authors would like to thank Prof. A. Farhat for allowing the use of the measurement facility. The authors would also like to thank several anonymous reviewers for their constructive and detailed comments which led to important improvements to this paper. REFERENCES

Fig. 9. Normalized radiation patterns of annular ring antennas loaded with ferrite films. (a) E-plane. (b) H-plane.

frequency decreased drastically with the increase of ferrite films loaded on top. Furthermore, an interesting phenomenon is that an enhanced bandwidth are observed with both sides loading, and the maximum antenna bandwidth was obtained in the case dB of top 2 and bottom 1, which showed a 40% enhanced bandwidth over the non-magnetic counterparts. The normalized radiation patterns of E-plane and H-plane are plotted in Fig. 9(a) and (b), respectively. As shown in the figure is 2 m and are 4 m and of the E-plane, when 6 m, the bottom half patterns are a little bit smaller than that

[1] C. A. Balanis, Antenna Theory: Analysis and Design. Hoboken, NJ: Wiley, 2005. [2] K. L. Wong, Planar Antennas for Wireless Communications. Hoboken, NJ: Wiley, 2003. [3] H. Nakano, H. Yoshida, H. Mimaki, J. Yamauchi, and K. Hirose, “Printed loop array antenna radiating a circularly polarized wave,” in Proc. 9th Int. Antennas Propag. Conf., Apr. 1995, vol. 1, pp. 504–507. [4] R. L. Li, N. A. Bushyager, J. Laskar, and M. M. Tentzeris, “Determination of reactance loading for circularly polarized circular loop antennas with a uniform traveling-wave current distribution,” IEEE Trans. Antennas Propag., vol. 53, no. 12, pp. 3920–3929, Dec. 2005. [5] F. Qureshi, M. A. Antoniades, and G. V. Eleftheriades, “A compact and low-profile metamaterial ring antenna with vertical polarization,” IEEE Antennas Wireless Propag. Lett., vol. 4, pp. 333–336, 2005. [6] E. Ojefors, H. Kratz, K. Grenier, R. Plana, and A. Rydberg, “Micromachined loop antennas on low resistivity silicon substrates,” IEEE Trans. Antennas Propag., vol. 54, no. 12, pp. 3593–3600, Dec. 2006. [7] A. L. Borja, P. S. Hall, Q. Liu, and H. Iizuka, “Omnidirectional loop antenna with left-handed loading,” IEEE Antennas Wireless Propag. Lett., vol. 6, pp. 495–498, 2007.

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[8] A. Hoorfar and A. Perrotta, “An experimental study of microstrip antennas on very high permittivity ceramic substrates and very small ground planes,” IEEE Trans. Antennas Propag., vol. 49, pp. 838–840, May 2001. [9] D. M. Pozar and V. Sanchez, “Magnetic tuning of a microstrip antenna on a ferrite substrate,” Electron. Lett., vol. 24, pp. 729–731, Jun. 1988. [10] H. How, T. Fang, and C. Vittoria, “Intrinsic modes of radiation in ferrite patch antennas,” IEEE Trans. Microw. Theory Tech., vol. 42, pp. 988–994, Jun. 1994. [11] R. K. Mishra, S. S. Pattnaik, and N. Das, “Tuning of microstrip antenna on ferrite substrate,” IEEE Trans. Antennas Propag., vol. 41, pp. 230–233, Feb. 1993. [12] P. J. Rainville and F. J. Harackiewicz, “Magnetic tuning of a microstrip patch antenna fabricated on a ferrite film,” IEEE Microw. Guided Wave Lett., vol. 2, pp. 483–485, Dec. 1992. [13] H. How, P. Rainville, F. Harackiewicz, and C. Vittoria, “Radiation frequencies of ferrite patch antennas,” Electron. Lett., vol. 28, pp. 1405–1406, Jul. 1992. [14] T. B. Do and J. W. Halloran, “Fabrication of polymer magnetics,” in Proc. IEEE Int. Symp. Antennas Propag., Jun. 2006, pp. 1709–1712. [15] K. Buell, H. Mosallaei, and K. Sarabandi, “A substrate for small patch antennas providing tunable miniaturization factors,” IEEE Trans. Microw. Theory Tech., vol. 54, pp. 135–146, Jan. 2006. [16] H. Mosallaei and K. Sarabandi, “Design and modeling of patch antenna printed on magneto-dielectric embedded-circuit metasubstrate,” IEEE Trans. Antennas Propag., vol. 55, pp. 45–52, Jan. 2007. [17] K. Sarabandi, A. M. Buerkle, and H. Mosallaei, “Compact wideband UHF patch antenna on reactive impedance substrate,” IEEE Antennas Wireless Propag. Lett., vol. 5, pp. 503–506, 2006. [18] N. X. Sun, J. W. Wang, A. Daigle, C. Pettiford, H. Mosallaei, and C. Vittoria, “Electronically tunable magnetic patch antennas with metal magnetic films,” Electron. Lett., vol. 43, no. 8, pp. 434–435, Apr. 2007. [19] G. M. Yang, A. Daigle, M. Liu, O. Obi, S. Stoute, K. Naishadham, and N. X. Sun, “Planar circular loop antennas with self-biased magnetic film loading,” Electron. Lett., vol. 44, pp. 332–333, Feb. 2008. [20] G. M. Yang, X. Xing, A. Daigle, M. Liu, O. Obi, J. W. Wang, K. Naishadham, and N. Sun, “Electronically tunable miniaturized antennas on magnetoelectric substrates with enhanced performance,” IEEE Trans. Magn., vol. 44, no. 11, pp. 3091–3094, Nov. 2008. [21] S. X. Wang, N. X. Sun, M. Yamaguchi, and S. Yabukami, “Sandwich films: Properties of a new soft magnetic material,” Nature, vol. 407, pp. 150–151, Sep. 2000. [22] J. Lou, R. E. Insignares, Z. Cai, K. S. Ziemer, M. Liu, and N. X. Sun, “Soft magnetism, magnetostriction and microwave properties of FeGaB thin films,” Appl. Phys. Lett., vol. 91, p. 18254, Oct. 2007. [23] K. Kondo, S. Yoshida, H. Ono, and M. Abe, “Spin sprayed Ni(-Zn)-Co ferrite films with natural resonance frequency exceeding 3 GHz,” J. Appl. Phys., vol. 101, p. 09M502, 2007. [24] L. Landau and E. Lifshitz, “On the theory of the dispersion of magnetic permeability in ferromagnetic bodies,” Physik Zeits. Sowjetunion, vol. 8, pp. 153–169, 1935. [25] T. L. Gilbert, “A Lagrangian formulation of the gyromagnetic equation of the magnetization field,” Phys. Rev. B, vol. 100, pp. 1243–1255, 1955. [26] N. X. Sun, S. X. Wang, T. J. Silva, and A. B. Kos, “Soft magnetism and high frequency behavior of Fe-Co-N thin films,” IEEE Trans. Magn., vol. 38, pp. 146–150, Jan. 2002. [27] D. R. Jackson and N. G. Alexopoulos, “Gain enhancement methods for printed circuit antennas,” IEEE Trans. Antennas Propag., vol. 33, pp. 976–987, Sep. 1985. [28] R. C. Hansen and M. Burke, “Antenna with magneto-dielectric,” Microw. Opt. Technol. Lett., vol. 26, pp. 75–78, Feb. 2000. [29] P. M. Ikonen, K. N. Rozanov, A. V. Osipov, P. Alitalo, and S. A. Tretyakov, “Magnetodielectric substrate in antenna miniaturization: Potential and limitations,” IEEE Trans. Antennas Propag., vol. 54, pp. 3391–3399, Nov. 2006. [30] P. M. Ikonen, S. I. Maslovski, C. R. Simovski, and S. A. Tretyakov, “On artificial magnetodielectric loading for improving the impedance bandwidth properties of microstrip antennas,” IEEE Trans. Antennas Propag., vol. 54, pp. 1654–1662, Jun. 2006. [31] K. R. Carver and J. W. Mink, “Microstrip antenna technology,” IEEE Trans. Antennas Propag., vol. 29, pp. 2–24, Jan. 1981. [32] E. Belohoubek and E. Denlinger, “Loss considerations for microstrip resonators,” IEEE Trans. Microw. Theory Tech., vol. 23, pp. 522–526, Jun. 1975.

Guo-Min Yang (S’07) was born in Zhejiang Province, China, in 1979. He received the B.S. degree (with honors) in communication engineering from Xi’an University of Technology, Xi’an, China, in 2002 and the M.S. degree in electronic engineering from Shanghai Jiao Tong University, Shanghai, China, in 2006. He is currently working toward the Ph.D. degree in electrical and computer engineering at Northeastern University, Boston, MA. His current research interests include antenna miniaturization, magneto-dielectric materials, metamaterials, frequency selective surfaces, UWB filters, UWB antennas, computational electromagnetics and inverse scattering problems in EM. He has authored 12 journal publications and 10 conference papers. Mr. Yang is a member of the Editorial Board of the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES. He was the recipient of National Graduate Student Scholarship in 2006.

Xing Xing received the B.S. degree in physics from Nanjing University, China, in 2006 and the M.S. degree in electrical engineering from Northeastern University, Boston, MA, in 2009. She is currently working toward the Ph.D. degree in electrical and computer engineering at Northeastern University, Boston, MA. Her research interests include magnetic materials and devices.

Andrew Daigle received the B. S. and M. S. degree in electrical engineering from Northeastern University, Boston, MA, in 2007 and 2009, respectively, where he is currently working toward the Ph.D. degree.

Ogheneyunume Obi (S’09) was born in Benin City, Edo State, Nigeria, in 1979. She received the B.Eng. degree in electrical and electronics engineering from the University of Benin, Benin City, in 2004 and the M.S. in electrical engineering from Northeastern University, Boston, MA, in 2009, where she is currently working toward the Ph.D. degree. Her principle research interest is fabrication of magnetic thin films for microwave devices. Ogheneyunume is a member of Eta Kappa Nu (HKN).

Ming Liu received the B.S. degree in mathematical physics from Inner Mongolia University, Baotou, Inner Mongolia, China, in 1999 and the M. S. degree in chemical physics in Dalian Institute of Chemical Physics, Chinese Academy Sciences, Dalin, Liaoning, China, in 2004. Currently, he working toward the Ph.D. degree at Northeastern University, Boston, MA. His current research interests are on Processing and characterization of magnetic, ferroelectric, and magnetoelectric materials and the demonstration of composite microstructures and functional coupling.

YANG et al.: PLANAR ANNULAR RING ANTENNAS WITH MULTILAYER SELF-BIASED NICO-FERRITE FILMS LOADING

Jing Lou (S’08) was born on January 20, 1981, in China. He received the B.S. degree in physics from Nanjing University, Nanjing, China, in 2003 and the M.S. degree in physics from Northeastern University, Boston, in 2005, where he is currently working toward the Ph.D. degree. His main research interests include synthesis, microstructure, and properties of magnetic and magnetoelectric materials for applications in RF and microwave devices. Novel devices based on magnetoelectric concept are also his focus.

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electromagnetic modeling of large bodies containing small features, and to RCS characterization using spectral estimation methods. In 2008, he joined Georgia Institute of Technology as a Research Faculty and a Principal Research Scientist at the Sensors and Electromagnetics Laboratory. His current research focuses on multifunctional antenna arrays, co-site interference in arrays, antenna miniaturization, and mitigation of parasitic effects in wearable antennas. He published four book chapters and several papers in professional journals and conference proceedings on topics related to antennas, computational EM and wave-oriented signal processing. Dr. Naishadham is a Member of URSI Commission B. He served the IEEE as Member of the Technical Program Committee for Antennas and Propagation Symposium (1992–1995), and International Microwave Symposium (1994–2005). He was Chair of the IEEE Dayton Joint Chapter of AP and MTT Societies from 1994 to 1999.

Stephen Stoute is currently working toward the B.S. degree at Northeastern University, Boston, MA.

Krishna Naishadham (M’87–SM’94) received the M.S. degree from Syracuse University, and the Ph.D. from the University of Mississippi, both in electrical engineering, in 1982 and 1987, respectively. From 1987 to 1999, he served on the faculty of Electrical Engineering, as an Assistant Professor at the University of Kentucky, Lexington, and as an Associate Professor (tenured) at Wright State University, Dayton. He taught courses in electromagnetics (EM) and antennas, and performed research on microwave circuits, computational EM and microwave characterization of novel electronic materials. From 1999–2002, he was at Philips, working on advanced RF system architectures for broadband mixed signal distribution. From 2002–2007, he was a Research Scientist at MIT Lincoln Laboratory, and contributed to hybrid asymptotic techniques for

Nian X. Sun (S’99–M’02) received the Ph.D. degree from Stanford University in 2002. Previously, he was a Scientist at IBM and Hitachi Global Storage Technologies between 2001 to 2004. Currently, he is an Associate Professor in the Electrical and Computer Engineering Department, Northeastern University, Boston, MA. His research interests include novel magnetic, ferroelectric and magnetoelectric materials and devices, such as antennas, phase shifters, filters, circulators, etc., magnetic sensors; material and device properties at RF/microwave frequencies; energy harvesting materials and devices; micro/nanotechnologies for biomedical magnetic sensing, etc. He has published over 70 technical papers, more than 20 US patents and patent disclosures. Dr. Sun was the recipient of the NSF CAREER Award in 2008, the ONR Young Investigator Award in 2007, and the 1st Prize IDEMA Fellowship in 2000.

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Compact Loaded PIFA for Multifrequency Applications Óscar Quevedo-Teruel, Student Member, IEEE, Elena Pucci, and Eva Rajo-Iglesias, Senior Member, IEEE

Abstract—A new multifrequency microstrip patch antenna is presented. The antenna can be considered a PIFA since it has a metallic wall on one of its sides. The different bands of operation are independent of each other, and different radiation patterns for each band can be achieved if desired. In addition, a circuital model is introduced to explain the operation of the antenna. This model presents some similarities with composite right left handed models presented in the literature. Some prototypes have been manufactured and measurements of return losses, efficiencies and radiation patterns, have been performed for a thorough characterization of the antenna as well as to validate the simulation results. Index Terms—Composite right left handed (CRLH), dual band, multiple band, microstrip patch antenna, PIFA.

I. INTRODUCTION ODERN telecommunication devices are required to be small and able to integrate several functionalities. The antennas used for these wireless systems must hence possess multiband capabilities but yet remain compact. One of the most common types of antenna used for conventional devices are patch antennas, due to their low cost of manufacture and lightweight nature. In addition, they have a high radiation efficiency [1], [2]. Patch antennas are resonant antennas that typically operate at the frequency whereby the length of a dimension is half a wavelength. This requirement may render the antenna too large for low-frequency applications. Consequently, some recent studies have focused on achieving compact patch antennas that can still function at lower frequencies [3]–[5]. A classical and simple method for reducing the operational electrical-size of patch antennas is to implement a modified structure known as PIFA (Planar Inverted F Antenna), which has a metallic wall on one of the sides. In this way, the antenna fulfils the boundary conditions with a quarter wavelength distance between the open and the short-circuit, instead of the half wavelength between two open boundaries in conventional patch antennas [2], [6]. Previous works on compact PIFA antennas with slots for mobile communications have succeeded in providing different

M

Manuscript received February 23, 2009; revised July 29, 2009. First published December 28, 2009; current version published March 03, 2010. This work was supported by the Spanish Government TEC2006-13248-C04-04. O. Quevedo-Teruel is with the Department of Signal Theory and Communications, University Carlos III of Madrid, Madrid 28911, Spain (e-mail: [email protected]). E. Pucci is with the Department of Signal and Systems, Chalmers University of Technology, Göteborg 41296, Sweden. E. Rajo-Iglesias are with the Department of Signal Theory and Communications, University Carlos III of Madrid, Madrid 28911, Spain and also with the Department of Signal and Systems, Chalmers University of Technology, Göteborg 41296, Sweden. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2009.2039305

bands of operation with compact designs [7]–[9]. In addition, other new techniques have also been applied to the design of compact patch antennas. Some of them are based on CRLH (composite right left handed) transmission lines [10]–[14]. These antennas have multiple bands of operation as they allow the excitation of both right handed (RH) modes and left handed (LH) modes. According to this theory, the traditional transversal magnetic (TM) modes are produced by RH modes, and new radiation bands can be obtained from the frequencies at which the new LH modes propagate (being, TM modes too). The main advantage of this design is the possibility of achieving a compact multiband antenna with new resonant frequencies which are lower than the fundamental mode frequency in traditional patches. Different radiation patterns associated with the bands may also be generated, thus offering prospects of catering to various service requirements. This flexibility in radiation pattern for frequencies below the fundamental mode is not common for patch antennas. However, there are two drawbacks of these compact designs. Firstly, the efficiency or gain of the antennas is low if the antenna size is too small; and secondly, the new modes related to the LH modes have narrow bands. The purpose of this paper is to present a novel compact patch antenna with a potential multifrequency response, including the possibility of obtaining a different radiation pattern at each band if desired. The theory of operation is based on [10], [11]. The paper is organized as follows. The description of the antenna modus operandi will be briefly explained in Section II. Examples of designs achieving dual and triple bands will be presented in Section III, including measurements of both return losses and radiation patterns. Section IV contains a study of the efficiency of the antenna, since this is one of the important parameters in compact designs. Finally, in Section V, conclusions will be derived from the results. II. BASIS OF THE ANTENNA The proposed antenna is described in Fig. 1 (with top and side . It views). The basic antenna is a printed semicircle of radius is a PIFA antenna since there is an electric wall that connects this semicircle to the ground plane. In addition, there is an external printed semi-ring which is also connected to the ground plane . This semi-ring is separated and whose external radius is , and it from the inner semicircle by an arbitrary distance has a width. A new resonance is created since there is an intrinsic capacitance between the inner semicircle and the outer semi-ring; and an intrinsic inductance given by the semi-ring and its connections to the electric wall. Consequently, this resonance can be used to match new radiation modes. The equivalent circuit for this antenna is the one represented in Fig. 2

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Fig. 1. Geometrical description of the antenna under study. Top and side views.

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Fig. 3. Simulated band operations of TM and TM modes along with their three replicas, for an antenna with three external semi-rings, with the following dimensions: radius of the inner semicircle: 23 mm, separation between rings: 1 mm and ring widths: 1 mm. The thickness of the substrate is 8 mm, and its dielectric constant 2.7.

Fig. 2. Equivalent circuit for the antenna under study.

(not considering the feeding). The shunt capacitance and the series inductance correspond to the circuital model of ordinary patch modes, whilst the series capacitance and the shunt inductance model the new modes, thereby having the bi-frequency behavior. There is no restriction a priori on the number of new radiation modes. The number of introduced semi-rings determines the number of added resonances, and consequently, the number of new possible radiation frequencies. In order to show this phefor an example with three external nomenon, the simulated semi-rings is included in Fig. 3. The antenna in the example has the following dimensions: a 23 mm inner semicircle radius, a 1 mm separation between rings and a 1 mm width of the rings. The thickness of the substrate is 8 mm and its dielectric constant 2.7. As predicted, three new modes appear below the frequency of the traditional TM mode (which is the fundamental one in these structures), and another three new modes are observed between the frequency of the fundamental mode and that of the second one (TM mode). We have verified that these new modes have similar radiation characteristics as those of the corresponding “conventional” ones which occur above them in frequency. For this reason, they will be denominated as “replicas” of the respective ordinary modes. However, they are in principle not necessarily all simultaneously well-matched if a conventional single-port feeding technique is used. This can be a constraint for using a high number of these new modes in practical designs when a simple feeding technique is used. With a single port, the challenge is then to find out the feeding point

Fig. 4. Electric field distribution for TM replicas.

and TM

ordinary modes and their

where all the modes to be used are matched. This is a complicated task for a high number of replicated modes as the field distribution is not identical for all modes. Fig. 4 shows the simulated absolute value of the vertical elecfor the ordinary TM and TM modes, along tric field with their replicas when only one semi-ring is introduced. The field distribution of a mode resembles its replica, i.e., they have the same number of maxima and minima in the same positions

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Fig. 6. Equivalent evolutionary approximation of the proposed antenna from CRLH lines as in the one proposed in [10].

Fig. 5. Simulated S for prototypes with different feeding positions (d , which represents the distance to the metallic wall), with the following dimenmm, R mm and R : mm in a substrate with sions: S  : and 8 mm thickness.

= 27

=1

= 23

= 24 5

(following the criterion of definition of traditional modes in patch antennas [1], [2], [6]). Therefore, their radiation properties will also be similar. Nevertheless, since each mode has a particular distribution and amplitude of the fields, the optimal matching point of each mode can also be placed at a different position. As a consequence, a preliminary study of these positions is required if a multifrequency response is desired. The optimal feeding position for each mode will depend on the dimensions of the elements, as well as on the thickness and dielectric constant of the substrate (as in conventional microstrip patch antennas [1], [2], [6]). In order to show the effect of the feeding point on the matching of the modes, a simulation study with different feeding positions for an example (with only one external semi-ring) was mm, made. The antenna has the following dimensions: mm and mm in a substrate of (PVC) with an 8 mm thickness. The simulated is plotted in represents the distance from the feeding point Fig. 5, where to the metallic wall. We can conclude that for the fundamental mode (for this particular case), when this distance increases, the matching becomes worse. However, for the replica of this mode, the matching does not have a clear trend as the matching level oscillates up and down with increasing . Thereby, the best matching for this mode is not achieved at the same feeding point. Furthermore, the challenge posed to the antenna designer (as it would be in designing traditional patch antennas) is to find the optimal feeding point to excite the modes required by the given application of the antenna. Obviously, this is more challenging if we are limited to a single port design. Other authors [10]–[14] have denominated the modes of similar structures as CRLH, since their designs are coming from transmission lines where the excited modes are LH. In addition, it is well known that patch antennas can be analyzed from transmission line theory, since the origin of these antennas is found in microstrip lines [6]. The antenna proposed in this paper can

Fig. 7. Operation frequency for the replica of TM , as a function of different parameters. The reference antenna has the following dimensions: S mm, : mm in a substrate of  : and 8 mm R mm and R thickness.

= 23

= 24 5

= 27

=1

also be seen as an extension of those CRLH transmission lines as was the case in [10], where the grounded pins of each cell have been first moved to the edge of the cells, and later on replaced by a metallic wall. This approximation is illustrated in Fig. 6. The main role of the inner semi-circle of the antenna is to define the operation frequency of the ordinary modes excited in a microstrip patch antenna [1], whereas the external semi-rings and their distances to the inner semi-circle dictate the operation frequencies of the new modes denominated replicas. The latter effects are attributed to the influences which those two parameters have on the series capacitance and shunt inductance mentioned earlier. Nevertheless, the inner semi-circle will contribute also to the matching of the new modes, since the feeding point is located on it. Fig. 7 shows the variation of the operation frequency of the first replica mode as a function of the four main parameters which define its operation: permittivity , substrate and . As initial design we have considered thickness, was prethe one analyzed previously in this Section, whose sented in Fig. 5. Finally, in order to show the effect of the capacitance established by the distance between the inner semicircle and the outer semi-ring, lumped capacitors that connect these two parts of the antenna were introduced, and the measured results are plotted in Fig. 8. These results are for a structure with only one mm, mm semi-ring whose dimensions are: mm, in a substrate of (PVC) with an 8 and mm thickness. Although the antennas are not properly matched

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Fig. 9. Simulated and measured S for a prototype with S = 1 mm, R = = 23 mm and R = 24:5 mm, in a substrate of  = 2:7 (PVC) with a 10 mm

Fig. 8. Reduced operation frequencies with increased capacitances. Measured S for different prototypes with S mm, R mm and R : mm and different lumped capacitors, in a substrate of  : and 8 mm thickness.

=1

24 5

= 23

=27

659

(since the authors only expect to show the trend), we can conclude from these results that with higher capacitances the frequency of the new resonance is shifted down, and the traditional mode is not excessively affected. This agrees with the circuital transmission line model presented above. All the simulations presented along this Section have been carried out with CST Microwave Studio. III. DESIGN EXAMPLES In this section, some particular designs will be presented. Firstly, we will focus on the simplest structures, that is, antennas with only two bands of operation. To this aim, only one semiring is necessary. Later on, we will deal with triple band designs. As explained above, the external semi-rings will contribute to the excitation of new modes. The operation frequency of these modes will be determined by the dimensions of each semi-ring. keeping the slot width conThe larger the radius stant, the lower the frequency, and the same happens when the is separation between the inner semicircle and semi-ring increased, since in both cases the shunt inductance and the series capacitance shown in Fig. 2 are modified. Moreover, the operation frequency of the ordinary modes is almost unaffected by the addition of the semi-rings and their dimensions. A. Dual Band Operation First of all, two different examples of the simplest configurations of the antenna are presented. These configurations are dual band antennas, making use of the ordinary mode of operation, and the replica of this mode or the replica of the first high-order . mode 1) TM and Its Replica: The most compact multifrequency antenna that we can achieve with this configuration is the one in and its replica which we excite the fundamental mode that works at lower frequencies. As previously commented, they will have similar radiation patterns since they have similar field

thickness.

distributions. Therefore, if we consider the case of only one external semi-ring, a dual-frequency characteristic is achieved, and these two frequencies are independent of each other. The dimensions chosen for this particular example are the following: mm, mm and mm in a substrate of (PVC) with a 10 mm thickness. The feeding point position is 22 mm from the metallic wall. The total size of at the frequency the antenna is approximately at the freof the replica of the TM , and quency of the TM . The simulated and measured for this case are included in Fig. 9. The simulation predicts properly the working frequencies of the antennas although the bandwidths of the measurements are wider, due to the absence of losses in the simulation. The radiation patterns of both modes were measured. The measurements in E-plane and H-plane are shown in Fig. 10 for both frequencies: 1.38 GHz and 2.4 GHz which correspond to the operation frequencies of the fundamental mode and its replica. Although the gain of the antenna is different for each mode (1.8 dB for the replica and 3.7 dB for TM ), they both exhibit a broadside radiation pattern slightly tilted due to the lateral metallic wall. 2) TM and the Replica of TM : A second example is presented in this subsection, where the fundamental mode of and the replica of the second mode the patch are excited. Thereby, each band of operation will have its own particular radiation pattern. This fact provides flexibility to the design depending on the application ([15], [16]), since various kinds of services requiring different radiation patterns can be provided by each band. A prototype was manufactured and meamm, mm sured, and has dimensions: mm, in a substrate of (PVC) with a and 12 mm thickness. The feeding point was placed at 29 mm from the metallic wall. The total size of the antenna is approximately at the frequency of the replica of the TM , at the frequency of the TM . The graph of and parameter versus frequency obtained from simulations the

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=9

Fig. 11. Simulated and measured S for a prototype with S mm, R mm and R mm, in a substrate of  : (PVC) with a 12 mm thickness.

= 42

= 30

=27

Here again, some measurements of the radiation patterns were also carried out. Fig. 12 shows the radiation patterns at both operation frequencies: 1.84 GHz for the TM mode and 2.34 GHz for the TM replica. They present distinct radiation characteristics since the field distributions are different. At 2.34 GHz, the pattern takes on the typical shape of a TM mode, with a minimum co-polar level in the broadside direction for the H-plane. The E-plane is not represented for the case of the TM mode, since in that plane the radiation pattern has a minimum. Moreover, the H-plane presents a high cross polarization level due to the large thickness of the substrate. Finally, the directivities are 5 dBi for TM and 5.5 dBi for TM . B. Triple Band Operation

Fig. 10. Radiation patterns at 1.38 GHz and 2.4 GHz, where respectively the replica of TM mode and the ordinary TM are excited. (a) E plane (1.38 GHz), (b) H plane (1.38 GHz), (c) E plane (2.4 GHz), (d) H plane (2.4 GHz).

and measurements is shown in Fig. 11. Good agreement of the operation frequencies is achieved.

In addition to the foregoing dual-band examples, we now deal also with triple-band antennas. Two particular examples will be studied. The first one employs a single external semi-ring, for which the three operation bands are associated with the traditional TM mode, its replica and the replica of the TM mode. Following this, two external semi-rings that excite the ordinary TM mode and two of its replicas shall be dealt with. This is in virtue of a replicated mode produced by each semi-ring. 1) TM , its Replica and the Replica of TM : For this first example of triple-band design, we propose the use of only one semi-ring. Thereby, there will be only one replica of each mode, and the most compact design requires the use of the TM mode, . The its replica, and the replica of the second mode mm, dimensions of the designed example are: mm and mm, in a substrate of (PVC) with an 8 mm thickness. The feeding point is placed at 20 mm from the metallic wall. The simulated and measured reflection coefficients of this antenna are shown in Fig. 13. We see how dB). the three bands are properly matched (below 2) TM and Its Two Replicas: As a final example, we propose here a design with two external semi-rings, for obtaining a triple-band operation. With the two semi-rings we obtain two replicas of each mode, but in order to have the most compact

QUEVEDO-TERUEL et al.: COMPACT LOADED PIFA FOR MULTIFREQUENCY APPLICATIONS

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=9

Fig. 13. Simulated and measured S for a prototype with S mm, R : (PVC) with a mm and R mm, in a substrate of  12 mm thickness.

= 42

= 30

=27

Fig. 14. Simulated and measured S for a prototype with R mm, S mm, R mm and R substrate of  : (PVC) with a 12 mm thickness.

= 28

Fig. 12. Radiation patterns at 1.84 GHz and 2.34 GHz, where respectively the ordinary TM mode and the replica of TM are excited. (a) E plane (1.84 GHz), (b) H plane (1.84 GHz), H plane (2.34 GHz).

=27

=2

= 38

= 1 mm, = 21 mm, in a

S

mm, mm, mm and mm, (PVC) with an 8 mm thickness. The in a substrate of feeding point is placed at 20 mm from the metallic wall. The of this antenna are presented in simulated and measured Fig. 14. As in the previous example, the three bands are matched dB) with a simple coaxial probe. The size of this de(below at the frequency of the sign is approximately at the frequency of the first replica of TM the TM and at the frequency of the second replica of the TM . IV. STUDY OF THE RADIATION EFFICIENCY

design, the fundamental mode TM and its two replicas must be used. Since this design has two semi-rings, we need to define and two . We will name them with the subindex 1 two for the internal semi-ring, and subindex 2 for the external one. The dimensions for this example are: mm,

In this Section, the study will be focused on the radiation efficiency of the lowest mode of the antenna, i.e., the replica of the first mode, since this mode will be the most critical in terms of efficiency. For simplicity, an antenna with only one semi-ring was considered. The initial dimensions for this study were:

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Fig. 16. Photo of some of the manufactured prototypes.

Fig. 15. Variation of the operation frequency (black solid line) and radiation efficiency (blue dashed for simulations, red cross for measurements) with different parameters: (a) semi-ring widths R , (b) distances between semi-circular patch and semi-ring S , (c) substrate thicknesses, and (d) substrate permittivities.

mm, strate of

mm and mm, in a subwith an 8 mm thickness. In order to properly

study the structure, different parameters of the antenna were changed and the radiation efficiency was obtained for each design. These parameters were: the thickness of the substrate, the dielectric constant, the distance between the inner semicircle and the width of the semi-ring and the outer semi-ring . With the latter two parameters, the total external radius is varied. This study was developed using CST Microwave Studio and it was validated by some measurements of the radiation efficiency in a reverberation chamber ([17], [18]). Fig. 15 shows the operation frequencies of the mode and the radiation efficiencies for the studied cases, in which the red crosses represent the measured efficiencies for some examples. The measurements present a lower radiation efficiency when compared to the simulations due to uncertainties in the characteristics of the substrates. Nonetheless, the trends are similar to the simulations. The radiation efficiency is also affected by the electrical size of the structure through frequency variation. The lower the frequency, the lower the efficiency. Moreover, the efficiency varies strongly with the thickness, although the operation frequency is not changed. The larger the thickness, the higher the efficiency of this mode. This coincides with what the cavity model of a patch antenna predicts [2]. However, the increase in the substrate thickness can have some well-known drawbacks such as the strong excitation of surface waves, or problems with the matching of ordinary TM modes. Finally, the last parameter that has been studied was the permittivity of the substrate. Like in traditional TM modes, when the dielectric constant increases, the efficiency decreases. The operation frequency of the replicated mode also suffers the same effect. It is also important to point out how the frequency of the ordinary mode (2.1 GHz) is almost unaffected by any of the proposed changes in the antenna geometry (excluding substrate characteristics) and for that reason is not represented in the previous figures. Furthermore, the radiation efficiency of the fundamental mode was also measured, achieving a radiation efficiency above 0.95 for all the examples. Finally, Fig. 16 shows a photo of some of the prototypes which have been manufactured and measured.

QUEVEDO-TERUEL et al.: COMPACT LOADED PIFA FOR MULTIFREQUENCY APPLICATIONS

V. CONCLUSION A compact loaded PIFA has been presented in this paper. The antenna is composed of an inner semi-circle and an arbitrary number of external semi-rings. Each semi-ring produces the excitation of new modes that can be used in radiation. These modes are replicas of the ordinary TM modes of patch antennas, having similar radiation properties. Therefore, the proposed antenna provides multiple frequencies of operation with different types of radiation patterns, giving versatility to the design depending on the application. Four particular examples, using only one or two semi-rings, have been manufactured. The first one uses TM mode in conjunction with its replica. This first design provides two quasibroadside radiation patterns. The second one uses the same ordinary TM mode and the replica of the ordinary TM , obtaining two bands with two different radiation patterns. The third one provides a triple band antenna, with the traditional TM , its replica and the replica of the TM . Finally, the fourth example shows an antenna with triple band response making use of two semi-rings and two replicas of the TM . These four examples were studied by simulations and measurements (in terms of return losses and radiation patterns). The most challenging aspect is the matching of all these modes at the same time, since the optimal point for the matching of each mode is at a different position of the patch and in this work we have assumed a single port feeding technique. Finally, a study of the radiation efficiency was carried out for the replica of the TM with only one semi-ring, since this mode is the one which works at the lowest frequency and therefore its efficiency is the most critical. The efficiency depends on the geometry of the antenna since various parameters of the geometry modify the operation frequency of the new mode. In addition, the permittivity and thickness of the dielectric are important factors, as they strongly affect the radiation efficiency. This study of the radiation efficiency was verified with some measurements. ACKNOWLEDGMENT The authors would like to thank to the Antenna Group of Chalmers University of Technology for supporting the measurements of the antennas in the anechoic and reverberation chambers. The authors want to thank Dr. M. Ng Mou Kehn for his help with the English and technical review. The authors would like to thank also the anonymous reviewers for pointing out the critical aspects of this work. Finally, the authors want to thank to C. J. Sánchez-Fernández for manufacturing the prototypes.

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[5] S. A. Bokhari, J. F. Zürcher, J. R. Mosig, and F. E. Gardiol, “A small microstrip patch antenna with a convenient tuning option,” IEEE Trans. Antennas Propag., vol. 44, pp. 1521–1528, Nov. 1996. [6] J. R. James and P. S. Hall, Handbook of Microstrip and Printed Antennas. New York: Wiley, 1997. [7] Z. D. Liu, P. S. Hall, and D. Wake, “Dual-frequency planar inverted-F antenna,” IEEE Trans. Antennas Propag., vol. 45, pp. 1451–1458, Oct. 1997. [8] R. Hossa, A. Byndas, and M. E. Bialkowski, “Improvement of compact terminal antenna performance by incorporating open-end slots in ground plane,” IEEE Microw. Wireless Compon. Let., vol. 14, pp. 283–285, Jun. 2004. [9] C. D. Nallo and A. Faraone, “Multiband internal antenna for mobile phones,” Electro. Lett., vol. 41, no. 9, pp. 514–515, 2005. [10] A. Lai, K. Leong, and T. Itoh, “Dual-mode compact microstrip antenna based on fundamental backward wave,” presented at the Asia Pacific Microwave Conf., Dec. 2005. [11] J.-Y. Park, C. Caloz, Y. Qian, and T. Itoh, “Composite right/lefthanded transmission line metamaterials,” IEEE Microw. Mag., vol. 5, Sep. 2004. [12] C.-J. Lee, K. M. K. H. Leong, and T. Itoh, “Composite right/lefthanded transmission line based compact resonant antennas for RF module integration,” IEEE Trans. Antennas Propag., vol. 54, pp. 2283–2291, Aug. 2006. [13] A. Lai, K. M. K. H. Leong, and T. Itoh, “Infinite wavelength resonant antennas with monopolar radiation pattern based on periodic structures,” IEEE Trans. Antennas Propag., vol. 55, pp. 868–876, Mar. 2007. [14] F. J. Herraiz-Martínez, V. González-Posadas, L. García-Muñoz, and D. Segovia-Vargas, “Multifrequency and dual-mode antennas partially filled with left-handed structures,” IEEE Trans. Antennas Propag., vol. 56, pp. 2527–2539, Aug. 2008. [15] V. González-Posadas, D. Segovia-Vargas, E. Rajo-Iglesias, J. Vázquez-Roy, and C. Martín-Pascual, “Approximate analysis of short circuited ring patch antenna working at TM mode,” IEEE Trans. Antennas Propag., vol. 54, pp. 1875–1879, Jun. 2006. [16] E. Rajo-Iglesias, O. Quevedo-Teruel, and M. P. Sánchez-Fernández, “Compact multimode patch antennas for MIMO applications,” IEEE Antennas Propag. Mag., vol. 50, pp. 197–205, Apr. 2008. [17] K. Rosengren, P.-S. Kildal, C. Carlsson, and J. Carlsson, “Characterization of antennas for mobile and wireless terminals in reverberation chambers: Improved accuracy by platform stirring,” Microw. Opt. Technol. Lett., vol. 30, pp. 391–397, Sep. 2001. [18] P.-S. Kildal and C. Carlsson, “Detection of a polarization imbalance in reverberation chambers and how to remove it by polarization stirring when measuring antenna efficiencies,” Microw. Opt. Technol. Lett., vol. 34, pp. 145–149, Jul. 2002.

Oscar Quevedo-Teruel (S’05) was born in Madrid, Spain, in 1981. He received the M.Sc. degree in telecommunication engineering from University Carlos III of Madrid, Madrid, Spain, in 2005. Since 2005, he has been working in the Department of Signal Theory and Communications, University Carlos III of Madrid. His research activity has been focused in optimization techniques applied to electromagnetism, analysis and design of compact microstrip patch antennas and metamaterials applied to microwave designs. He has (co)authored more than 15 contributions in international journals and more than 20 in international conferences.

REFERENCES [1] C. Martin-Pascual, E. Rajo-Iglesias, and V. González-Posadas, “Invited tutorial: ‘Patches: The most versatile radiator?’,” presented at the IASTED Int. Conf. Advanced in Communications, Jul. 2001. [2] C. Balanis, Antenna Theory: Analysis and Design.. New York: Wiley Interscience, 2005. [3] J.-Y. Park, C. Caloz, Y. Qian, and T. Itoh, “A compact circularly polarized subdivided microstrip patch antenna,” IEEE Microw. Wireless Compon. Lett., vol. 12, pp. 1531–1309, Jan. 2002. [4] K. Z. Rajab, R. Mittra, and M. T. Lanagan, “Size reduction of microstrip patch antennas with left-handed transmission line loading.,” IET Microw. Antennas Propag., vol. 1, pp. 39–49, Feb. 2007.

Elena Pucci received the degree in telecommunications engineering from University of Siena, Siena, Italy, in September 2008. She is currently working toward the Ph.D. degree at Chalmers University of Technology, Gothenburg, Sweden, since January 2009. Her main research activity includes analysis and design of new gap-waveguide technology, with particular interest to losses characterization and filters design.

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Eva Rajo-Iglesias (SM’08) was born in Monforte de Lemos, Spain, in 1972. She received the Telecommunication Engineering degree from University of Vigo, Vigo, Spain, in 1996 and the Ph.D. degree in telecommunication from University Carlos III of Madrid, Madrid, Spain, in 2002. From 1997 to 2001 she was a Teacher Assistant at the University Carlos III of Madrid. In 2001 she joined the University Polytechnic of Cartagena as Teacher Assistant for a year. She came back to University Carlos III as a Visiting Lecturer in 2002 and since 2004, she is an Associate Professor with the Department of Signal Theory and Communications, University Carlos III of Madrid. After

visiting Chalmers University of Technology (Sweden) as a Guest Researcher, during autumn 2004, 2005, 2006, 2007 and 2008, she is now an Affiliated Professor in the Antenna Group of the Signals and Systems Department of that University. Her main research interests include microstrip patch antennas and arrays, metamaterials and periodic structures and optimization methods applied to Electromagnetism. She has (co)authored more than 30 contributions in international journals and more than 60 in international conferences. Dr. Rajo-Iglesias received the Loughborough Antennas and Propagation Conference (LAPC) 2007 Best Paper Award and “Best Poster Award in the field of Metamaterial Applications in Antennas” sponsored by the IET Antennas and Propagation Network, at Metamaterials 2009: 3rd International Congress on Advanced Electromagnetic Materials in Microwaves and Optics.

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A Comparison of a Wide-Slot and a Stacked Patch Antenna for the Purpose of Breast Cancer Detection David Gibbins, Maciej Klemm, Ian J. Craddock, Jack A. Leendertz, Alan Preece, and Ralph Benjamin

Abstract—A wide-slot UWB antenna is presented for intended use in the detection scheme being developed at the University of Bristol, based on the principle of synthetically focused UWB radar using a fully populated static array. The antenna’s measured and simulated, input and radiation characteristics are presented and compared to an existing, stacked patch antenna that has been designed for the same purpose. The results of this study show that the wide-slot antenna has excellent performance across the required frequency range. Compared to the stacked-patch antenna used in our previous array, the wide-slot antenna can be 3 times smaller (in terms of front surface). The compact nature of the slot antenna means that the detection array can be densely populated. Additionally, this new antenna offers better radiation coverage of the breast. For angles up to 60 away from bore-sight radiated pulses are almost identical (fidelity 95%), whereas for the patch antenna fidelity falls to 58% at the angular extremes. This uniform radiation into the breast should result in focused images with low levels of clutter. Index Terms—Breast cancer, microwave radar, ultrawideband (UWB) antennas.

I. INTRODUCTION REAST cancer is the most common form of cancer in women (excluding skin cancers) [1], [2] however with early detection there is a high chance of successful treatment and long-term survival. The most common method in use for the detection of breast cancer is X-ray mammography and, while it has been an effective tool for detecting breast cancer it is recognized that the technique has a number of limitations including producing a significant number of false-negative and false-positive results [3], [4]. Microwave imaging has gained interest recently due to advances in both hardware and imaging software. The method is a potential alternative imaging technique that would be inexpensive, provide more sensitive 3D imaging data, avoid using ionising radiation and would yield a system that is both quick and comfortable for the patient [5]. Microwave imaging technology relies on there being a detectable difference in the dielectric properties of a tumor and the surrounding breast tissue

B

Manuscript received January 13, 2009; revised June 11, 2009. First published December 28, 2009; current version published March 03, 2010. This work was supported by the EPSRC and Micrima Ltd. The authors are with the Centre for Communications Research, Department of Electrical and Electronic Engineering, University of Bristol, Bristol BS8 1UB, U.K. (e-mail: [email protected]; [email protected]; ian. [email protected]; [email protected]; [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2009.2039296

at microwave frequencies, such that when the breast is illuminated with microwave radiation there is a significant reflection from the tumor [6]. Early work in this area was based on the premise that the structure of the breast is relatively electrically homogeneous and that a contrast of approximately 5:1 exists between malignant and normal tissue [7], [8]. More recent studies [9] have shown that, while the contrast between malignant and normal adipose-dominated tissue could be as large as 1:10, the contrast in denser glandular tissue is much less at around 10%. This presents a significantly more challenging problem than was initially thought and serves to underline the need for an antenna design with the best possible performance in terms of bandwidth, size and pattern characteristics. There are a number of techniques under investigation utilizing microwave signals as a means of detection. One approach considers it as an inverse scattering problem (microwave tomography [10]), in which the breast is illuminated with microwave radiation and the scattered energy is received at a number of remote locations. From this information the permittivity distribution inside the breast may be estimated. However the data processing required when implementing this method is complex and it is not easy to see how information from many different frequencies may be included [11]. An alternative approach is to tackle the problem using a similar architecture to that of ground penetrating radar (GPR) in an approach first introduced independently by Hagness et al. [12] and Benjamin [13], [14]. Ultrawideband (UWB) radar imaging as a means of detecting breast cancer, is a technique that is currently being developed by a number of research teams [12], [15], [16]. In these systems a short pulse, or a synthesized pulse constructed from a frequency sweep, is directed into the breast and the reflected signals are then detected by one or more receive antennas. The sweep is transmitted in turn from a number of different locations and the resulting set of received signals are then time- or phase-shifted and added in order to enhance returns from high contrast objects and to reduce clutter. Such radar-based systems are capable of producing high resolution images without the need for complicated reconstruction algorithms, due to the wideband nature of the UWB signals. With this approach there is a trade-off between simplicity of analysis and information, since—unlike the inverse scattering approach—material properties are not directly recovered [16]. A critical part of any detection scheme is the antenna design. In order to obtain high resolution, accurate images the antennas must be able to radiate signals over a wide band of frequencies while maintaining the fidelity of the waveform over a large angular range [17]. If the antenna is to be used in a fully populated array aperture (as at Bristol) then there is an additional, quite

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Fig. 1. Schematic of the stacked patch antenna (not to scale).

critical, geometrical size constraint placed on the antenna; the antenna’s geometrical dimensions must be as small as possible in order that the maximum number of antennas may accommodated in the array. This will allow as much information to be gathered as is possible which, in turn, will reduce clutter in the results. Existing antenna designs either use resistive loading to improve their UWB performance, e.g., resistively loaded monopoles [18], dipoles [16], [19], [20], bowtie [21], [22], and horn [23] antennas, which result in low efficiencies [6], or antennas are unsatisfactorily large, e.g., stacked patch [24] and planar monopole [25], all of which have dimensions in the order of 25 mm or greater while operating in the same frequency band. This paper presents a new slot antenna designed for use in Bristol’s breast cancer detection system. The dimensions of the slot, feed and ground have been chosen so that the antenna operates optimally while cavity-backed, with the slot in contact with a matching medium that has electrical properties similar to that of normal breast tissue. In terms of it’s electrical size, this antenna has frontal surface area of aproximately half that of the planar monopole in [25] and a third of that of the stacked patch. The structure of this paper is as follows; firstly the new wide-slot antenna will be presented along with a slot-fed, stacked patch antenna that was previously designed for the same application [24]. The performance of the patch will be compared with that of the wide-slot antenna. The method by which measurements were obtained will then be discussed and the paper will then go on to present results obtained by measurement and FDTD computer simulation. These will include return-loss, transfer functions, fidelity and time domain characteristics. II. THE ANTENNAS AND EXPERIMENTAL SETUP The antennas being examined in this paper are: (i) A cavity backed version of a stacked patch antenna (Fig. 1) that has been previously designed at the University of Bristol. This antenna was designed for use in the breast cancer detection system being developed at that institution and was originally presented in [24]. (ii) A fork-fed wide-slot antenna (Fig. 2) that is intended to replace the stacked patch antenna in this application. Both these antennas are in the main constructed of dielectric substrates with a high relative permittivity of 10.2. They have been optimized to operate with the antenna face immersed in a matching medium with dielectric properties similar to that

Fig. 2. Schematic of the wide-slot antenna (not to scale).

of human breast tissue, in order to reduce reflections by eliminating the air/skin interface [6]. The matching medium used is described in [26] and is mainly comprised of oil of paraffin and distilled water. This mixture has a relative permittivity of around between 9 and 10 at frequencies between 2 and 10 GHz and an attenuation of 2 dB/cm at a frequency of 8 GHz As this material has been designed to have similar electrical properties to human breast fat it is also used as a simple breast phantom for the antenna measurements presented in this paper. A. The Stacked Patch The stacked patch antenna consists of a microstrip line feeding a slot, which in turn excites an arrangement of stacked patches. The slot feed was used in order to eliminate the inductance associated with a probe feed. The patches sandwich a lower permittivity substrate and their size was chosen so that a lowest order resonance was achieved at either end of the desired frequency band. The dimensions were then manually optimized by using an FDTD computer simulation [24]. The efficiency of this antenna was found from FDTD to be 88% at 4 GHz, and 97% at 6 and 8 GHz (these values were found by simulation since experimentally obtaining full 3D radiation patterns of the antenna is difficult when submerged in the medium). The stacked patch antenna can be seen in (Fig. 1); the dimensions (in [mm]) being: , , , , , , , , , , , . It should be noted that this figure shows the antenna without the cavity used to back the antenna while taking measurements. B. The Wide Slot The wide-slot antenna consists of an approximately square slot set in a ground plane on one side of a substrate with a relative permittivity of 10.2. On the other side of the substrate is a forked microstrip feed that splits, just below the slot, from a 50 feed into two 100 sections which excite the slot. The fork feed was chosen as a means of increasing the operational bandwidth [27]. As with the patch the efficiency of this antenna has been found by simulation. At 3 GHz the efficiency was calculated as 60%,

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Fig. 4. (a) Experimental arrangement. (b) Close-up of positioning apparatus.

Fig. 3. The wide-slot antenna with a prototype cavity, fed directly from a coaxial cable (left). The SMA fed stacked patch antenna mounted in the cavity used for the measurements (right).

at 6 GHz at 80% and 90% at 9 GHz. The new antenna can be seen in (Fig. 2), its dimensions are: , , , , , , , , (all dimensions in [mm]). This antenna is based on that described in [27] but has been heavily modified and optimized to work efficiently when the slot is in contact with a matching medium with relative permittivity of approximately 10. This optimization was a manual iterative procedure performed using FDTD simulations and included the resizing and rescaling of the slot, fork feed and ground plane and the introduction of the high permittivity substrate on which the current antenna is built. The simulations used in the optimization process were run using FDTD software developed at the University of Bristol. In the simulation the slot side of the antenna radiates into a , ) with an air 50 mm-thick block of dielectric ( gap behind the antenna. The meshes used in the optimization process have cells of sizes varying from 1 to 0.1 mm, while to number of cells in the mesh varied from cells depending on the stage of the optimization. The workspace was terminated by Mur 1st order absorbing boundary conditions. A planar current source between the ground and microstrip was used to excite the feed line with a raised cosine pulse of width 112 ps. The transmission line was terminated inside the workspace using a vertical 50 resistive load. Both antennas were designed to work with a cavity attached to the rear side where the microstrip feed is located. This was done to prevent any radiation from the back surface of each antenna coupling into other antennas in the array. The cavities used consist of an absorber lined, brass enclosure that extends down to the face of the antenna. The two antennas are shown together in Fig. 3. This figure clearly demonstrates the difference in size between them. It should be noted that, while a detailed description is not in the scope of this article, the feed for the wide-slot has been modified to eliminate the SMA connector and, hence, further reduce

the overall dimensions. This was done by connecting the feed coaxial cable directly to the antenna itself. A hole was made in the side wall of the cavity level with the top surface of the antenna substrate and the microstrip line. The outer conductor of the coaxial cable was soldered to the edges of the hole which was just large enough for the inner conductor and dielectric insulator to pass through. Inside the cavity the dielectric is terminated at the edge of the antenna substrate and the inner conductor is soldered to the top surface of the microstrip line. C. Experimental Setup In order to fully characterize the antennas the input and radiation characteristics must be determined. As these results are required with the antennas radiating into a breast phantom, obtaining measurements, especially the radiation data, is quite a challenge. Since it is the transmission performance between two of the antennas that is the critical factor in the imaging application and due to a lack of suitable reference antenna, the radiation characteristics of the antenna under test were found by measuring the transmission between two identical antennas of that type. The experimental setup used to take measurements can be seen in Fig. 4. In this arrangement both receiving and transmitting antennas are immersed in a large tank of the matching medium/phantom, described at the start of Section II (1). The transmitting antenna (2) is fixed close to the tank wall in a stationary position facing out into the medium. The receiving antenna (3) is mounted on a rig (4) that describes an arc of radius 100 mm around a central point at which is the center of the face of the first antenna. The manner in which the second rig follows the arc results in the second antenna always directly facing the first. Measurements were taken in the E and H planes using a VNA (5), the chosen plane selected by attaching the antennas to the measurement rig in the correct orientation before immersion in the phantom medium. III. RESULTS It is important that the input and radiation characteristics of the antenna be known across the whole of the antennas intended operational bandwidth and angular range. As the antenna is intended to radiate a simulated short pulse its transient response

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Fig. 5. Measured and simulated (FDTD) S antenna.

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characteristics of the wide-slot

Fig. 6. Comparison of measured S antennas.

characteristics for the wide slot and patch

Fig. 7. Simulated and measured S the wide slot and patch antennas.

characteristics (bore-sight direction) for

is also important, including the fidelity of the signals that the antenna transmits. A. Input Response The simulated reflection coefficient for the cavity backed wide-slot antenna can be seen in Fig. 5, along with the equivalent measured results. It can be seen that there is a pleasing level of agreement between the two data sets. Both show that the 10 dB bandwidth extends from around 4 GHz (4.5 GHz in the simulated case) to above 10 GHz which is sufficient for this application. In the measured data there are some small rises above 10 dB around 6 discrepancies where the GHz. This frequency region will be examined later to assess its impact on the transmission response. Both simulated and measured results show that there are two major nulls in the response and are in good agreement in their location in the frequency spectrum. The differences between the two plots are probably due to fabrication tolerances particularly the feed, which has dimensions of the order of 0.2 mm. This level of agreement gives confidence that other results obtained using the FDTD model will correctly reflect the true properties of the antenna. The FDTD model used to produce these results had the same setup as that used in the optimization process. and was descretized The workspace was cells with sizes using a mesh made up from varying from 1-0.1 mm. When the measured return loss of the new wide-slot antenna is compared with that of the stacked patch (Fig. 6) it can be seen that the two antennas show comparable performance characteristics, with similar 10 dB bandwidths; the patch demonstrating a 5.5 GHz–10 dB bandwidth between 4.25 and 9.75 GHz. The patch antenna data also shows a drop in performance across the frequency range of interest between 7.3 and 8.3 GHz where the rises to 8 dB at it’s maximum. This discrepancy will also be examined in the transmission response.

the transfer function of the antenna be as flat as possible across the required frequency range [28]. Examining the measured in Fig. 7 (bore-sight being normal to the plane bore-sight of the antennas—see Fig. 1, Fig. 2) shows that both antennas have approximately the same performance; both having a maximum magnitude around 4 GHz which then drops steadily with frequency. The major difference is that the stacked patch has a transfer function magnitude that is consistently 7 dB higher from 4–9 GHz. This is because the main beam of the patch antenna is narrower than that of the wide-slot, concentrating the radiated energy at bore-sight, in other words, the patch has a higher gain due to its larger aperture. This can be illustrated by employing an approximation based on beamwidths to find the directivity at of both antennas at a particular frequency. Asbore-sight suming that each antenna only has one main lobe the directivity at 4.5 GHz, in dBs is given by [29];

B. Transmission Response In order to achieve minimal distortion to UWB signals transmitted into the breast, it is desirable that the magnitude of

(1)

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Fig. 8. Measured S data: (a) E-plane, wide-slot antenna; (b) E-plane, stacked patch antenna; (c) H-plane, wide-slot antenna; (d) H-plane, stacked patch antenna. Angle 0 corresponds to the bore-sight radiation direction.

Where and are the 3 dB beamwidths in degrees for two orthogonal planes (in this case the E and H-planes). Substituting the beamwidth values for the patch and slot gives (2) (3) Hence, the gain would be approximately 2.3 dB less for the wide-slot antenna than for the patch. The transfer functions in Fig. 7 are measured using two antennas, therefore for one antenna the wide-slot’s gain is 3.5 dB lower than that of the patch. Comparing this value to the 2.3 dB difference due to the change in aperture it can be seen that the lower directivity of the wide-slot accounts for the majority of this difference; the remaining 1.2 dB is most likely due to slightly higher back-radiation from the slot or to measurement errors. The wider antenna beam (hence, lower gain) seen in the results for the slot is in fact desirable, since it gives the most uniform illumination of the breast by the antenna array elements, the reduction in bore-sight gain is simply an unavoidable consequence of this. The main reason for the shape of the slope of the measured responses is the attenuation in the matching medium, which increases with frequency. This is borne out in the fact that the for the wide-slot antenna was practically flat besimulated fore losses in the phantom were taken into account. With their inclusion (post simulation using a frequency dependent model) the simulated results again show a good agreement with those

Fig. 9. Examples of pulses synthesized from measured transfer functions normalized to the maximum field strength at 0 . Each pulse is displayed in a 2 ns time window.

measured. The simulated was also found, using an FDTD model with the same general setup and antenna mesh as that used to measure the s11 but with two antennas facing each other through a block of the high dielectric medium, 100 mm thick. data the wideReferring back to the issues raised with the slot antenna seems to have been affected very little by the slight mismatch in the 6 GHz region. The stacked patch does show a slight dip in the transfer function at around 7–8 GHz indicating that in this case the mismatch may be having an effect. Examining only the bore-sight gain gives a limited view of the antenna’s performance. For our application it is required that

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Fig. 10. Comparison of measured pulses’ duration (99% of pulse energy): (a) E-plane, (b) H-plane.

the antenna has as uniform a response as possible from to away from bore-sight, in both the E and H-planes (the E-plane being in the y-direction—see Figs. 1 and 2). In order to best understand what is happening at a variety of angles and frequencies Fig. 8 illustrates how the transfer function data from to , while the contour lines 2–11 GHz varies from indicate signal magnitude in 2 dB steps. Examining the E-plane plots shows that the wide-slot antenna [Fig. 8(a)] maintains its signal strength across the angular range. The antenna has maximum field strength at 0 but never drops below 10 dB of this value. The variation in the magnitude with frequency is similar at all angles. There does seem to be a slight squint in the positive direction at higher frequencies, which is probably a result of the lack of symmetry in this plane due to the location of the feed on the slot’s lower edge. This squint is minimal, manifesting itself as a 3–5 dB difference between values at the angular extremes. Fig. 8(b) shows the equivalent results for the patch. This shows that the main beam of the patch antenna is narrow in comparison and above 7 GHz splits into a number of lobes which vary by up to 25 dB between the nulls and peaks in only 30 . The drop off of signal magnitude at higher frequencies noted in the bore-sight transfer functions (Fig. 7) can again be seen between 9 and 10 GHz, with the addition that at more extreme angles this occurs at lower frequencies. The asymmeof the patch (and as will be seen later, tries in the E-plane pulse duration data) are related and likely due to the feed. The microstrip/slot feed arrangement means that the fields exciting the patches are not symmetrical in the E-plane (a similar effect can also be seen in the form of the squint in the wide-slot radiation characteristics in this plane). These asymmetries in the field feeding the patches lead to asymmetries in the radiated fields. Examining the equivalent H-plane results [Fig. 8(c) and (d)] shows that, as would be expected, the plots are symmetrical due to the fact that the antennas are also symmetrical in this plane. Both antennas show a broad, relatively consistent beam-width across the frequency range. It should be noted that, as in the E-plane, the wide-slot antenna shows a flatter response across frequency and angular ranges, the beam width of the patch becoming narrower at higher frequencies.

C. Pulse Duration For radar-based breast cancer detection it is important that pulses produced by the antenna (or in this case synthesised from a frequency sweep) are as short as possible with minimal latetime ringing [12]. Therefore pulse duration is a good indication of the ability of the antenna to effectively transmit UWB signals into the medium. To test this the measured frequency domain data for the transmission between two antenna elements was obtained from the VNA. Pulses were then synthesized by applying a pulse template to this transfer function. Some examples of received pulses can be seen in Fig. 9, while a detailed description of the time and frequency domain responses of the template pulse can be seen in [30]. The time taken for 99% of the energy of these pulses to be received was then calculated. This process was carried out in the E and H-planes at an angular resolution of 15 from to and results are presented in Fig. 10. Pulse durations for the E-plane [Fig. 10(a)] show that, as in the plots, there is an asymmetry present for both antennas. The performance of the wide-slot shows in general that the pulses produced by this antenna are shorter and there is far less variation with angle (0.8–0.9 ns) than those produced by the patch (0.75–1.45 ns). This can be seen in the example pulses (Fig. 9) where there is little difference at bore-sight but at there is significantly more late-time ringing in the pulse produced by the patch. Examining the equivalent H-plane data [Fig. 10(b)] shows that the wide-slot antenna maintains a pulse width of just above 0.8 ns in the range of , this rises to a value of 0.97 ns at the angular extremes. The patch antenna shows a marginally better performance than the wide-slot between the angles , with a pulse duration of just below 0.8 ns. Beyond these angles the length of the pulses increases to 0.98 ns at and 1.07 ns at . This difference is illustrated in Fig. 9 where, although not as marked as in the E-plane there is still evidence of late-time ringing. The difference in the performance of the two antennas is the result of the poorer transmission properties of the patch antenna at high angular values

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Fig. 11. Comparison of measured fidelities: (a) E-plane, (b) H-plane.

[Fig. 8(d)] when compared to the equivalent transmission characteristics for the wide-slot [Fig. 8(c)] which are much more similar to those at bore-sight. D. Fidelity In order to study the level of distortion in the radiated pulses the fidelity of the signals was computed at the same points for which pulse duration was found. The fidelity is the maximum magnitude of the cross correlation between the normalized observed response and an ideal response [28] which in this case is the signal radiated at bore-sight. The fidelity, F, is given by (4) The fidelity parameter, F, is the maximum of the cross-correlation function and compares only shapes of both waveforms, not amplitudes. The calculated fidelity for the antennas in the E-plane can be seen in Fig. 11(a). The asymmetry noted previously can again be seen in both sets of results. The fidelity of the signals transmitted by the wide-slot antenna is excellent and fidelity remains above 95% for the entire angular range. Near bore-sight the performance of the patch is good, however beyond angles of the fidelity drops dramatically. This distortion is due to the irregularities seen in the transfer function data and borne out in the pulse duration measurements. The distortion at larger anare compared. gles can be seen in Fig. 9 when the pulses at The wide-slot antenna reproduces the signal at bore-sight with reasonable accuracy at this angle whereas the pulse produced by the patch shows significant distortion. For the H-plane [Fig. 11(b)] both antennas perform well, with the fidelity remaining above 90% for almost the entire angular range; the fidelity of signals radiated from the wide-slot never dropping below 98%. Once again the stacked patch performance degrades at high angles. This drop in performance can also be related back to the transfer function data and is most likely due to the rapid drop off seen in the signal content at higher frequencies, subsequent ringing (seen in the lengthening pulse duration) and signal distortion that occurs at these higher angles.

IV. RADAR DETECTION EXPERIMENT A numerical experiment has been conducted to demonstrate the suitability of the wide-slot antenna for the imaging application. The experiment consists of transmitting a UWB pulse into a numerical breast phantom to locate a tumor-like inclusion. The FDTD experimental setup uses the same excitation arrangement, boundary conditions and is based on the same mesh structure as the simulations used to obtain the antenna’s S-parameters. Two antennas are spaced 65 mm apart. Antenna 1 radiates a pulse and reflected signals are received by Antenna 2. , The dielectric properties of the numerical phantom are ( ). A 5 mm diameter spherical inclusion is positioned in the phantom. The relative permittivity of the inclusion is set at 50 giving a contrast of 5:1 with the background medium, similar to the contrast that might be seen between a tumor and adipose breast tissue [9]. The position of the inclusion is varied between , at 10 intervals, along an arc located mid-way between the two antennas as shown in Fig. 12. The simulated coupling between the antenna elements is around 60 to 80 dB. In the detection system the coupling between antennas is obviously dependent on their location in the array. However this issue is not critical as any signals directly coupling between antennas are removed prior to the signal analysis by calibration. For details of the calibration process and more information on the coupling between elements in a previous prototype of the array see [30]. As the signal received at Antenna 2 contains reflections and coupling between the antennas, as well as the response from the inclusion, a calibration is performed to remove these unwanted signals. The calibration signal is obtained by running simulation without an inclusion present. This background signal is subtracted from the signal that is received with the inclusion present, leaving only the response from the target, allowing the effect of the antenna to be clearly seen. As the path length is the same for each inclusion, waveforms from the different inclusion positions, normalized to their maximum value, can be added coherently to produce an aggregate pulse that summarizes the antenna performance over the entire

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Fig. 12. Schematic of the (a) y-z plane and (b) the y-x plane of the FDTD model used in the radar detection experiments (all dimensions in mm).

results it can be concluded that the wide-slot would be a good candidate for use in a radar-based breast imaging system. V. CONCLUSION

Fig. 13. Integrated spatial energy on a radial path outwards from the centre of the inclusion arc.

angular range. Squaring and then integrating this pulse over a sliding window corresponding to the transmit pulse width yields an energy curve that can be calibrated to show the radial distance of the inclusion from the centre of the arc Fig. 13 The curve should peak at the true position of 5 cm. The results of this analysis show that the energy distribution for the wide-slot response has a single peak exactly centred at the radial location of the inclusion. The peak has a width-athalf-height of just over 15.6 mm, allowing the position of the inclusion to be easily identified. In comparison the energy distribution produced by the patch response is much more diffuse with a lower, less well-defined peak and a width-at-half-height of 26.3 mm, nearly twice that of the slot antenna. This is a result of the increased distortion and dispersion of pulses produced by the stacked patch at angles away from boresight. From these

A wide-slot antenna intended for use in a UWB antenna array for breast cancer detection system has been presented and compared to a stacked patch antenna that was previously designed for the same application. Return loss measurements showed that both antennas had suitable bandwidths for use in a UWB detection system and good agreement was found between simulated and measured results for the wide-slot antenna. On examination of the transmission properties it was found that, while the stacked patch performed well at angles close to bore-sight, at wider angles of illumination (which are very important in this application) the transfer function showed significant degradation which manifested itself as significant late-time ringing and distortion of transmitted signals. This was especially notable in the E-plane. The wide-slot antenna performed well over the entire angular/ frequency range and faithfully radiated pulses at angles up to 60 away from bore-sight. This was confirmed by results from a simple radar detection experiment that showed that the wide slot performed well in breast imaging scenario. This, along with the fact that the antenna is approximately half the size of existing antenna designs, suggests that it is an excellent candidate for use in a UWB radar breast cancer detection system ACKNOWLEDGMENT The authors would like to acknowledge K. Stevens for help with the manufacture of the measurement setup and Prof. J. P. McGeehan for the provision of facilities at the Centre for Communications Research, University of Bristol. REFERENCES [1] B. C. Cancer Agency [Online]. Available: http://www.buccancer.bc.ca [2] J. Ferlay, F. Bray, P. Pisani, and D. Parkin, Globocan 2000: Cancer Incidence, Mortality and Prevalence Worldwide. : IARC CancerBase, 2001, Version 1.0(5).

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[3] J. G. Elmore and M. B. Barton et al., “Ten year risk of false positive screening mammograms and clinical examinations,” New England J. Med., vol. 338, no. 16, pp. 1089–1096. [4] P. T. Huynh and A. M. Jarolimek, “The false-negative mammogram,” Radiograph, vol. 18, no. 5, pp. 1137–1154, 1998. [5] E. Fear, S. C. Hagness, P. Meaney, M. Okoniewski, and M. Stuchly, “Enhancing breast tumor detection with near-field imaging,” IEEE Microw. Mag., vol. 3, no. 1, pp. 48–56, Mar. 2002. [6] , B. Allen and M. Dohler, Eds. et al., Ultra-Wideband Antennas and Propagation for Communications, Radar and Imaging. Chichester, U.K.: Wiley, 2007. [7] A. J. Surowiec, S. S. Stuchly, J. R. Barr, and A. Swarup, “Dielectric properties of breast carcinoma and the surrounding tissues,” IEEE Trans. Biomed. Eng., vol. 35, no. 4, pp. 257–263, April 1988. [8] W. T. Joines, Y. Zhang, C. Li, and R. L. Jirtle, “The measured electrical properties of normal and malignant human tissues from 50 to 900 MHz,” Med. Phys., vol. 21, p. 547, 1994. [9] M. Lazebnik et al., “A large-scale study of the ultrawideband microwave dielectric properties of normal, benign and malignant breast tissues obtained from cancer surgeries,” Phys. Med. Biol., vol. 52, p. 6093, 2007. [10] P. M. Meaney, M. W. Fanning, D. Li, S. P. Poplack, and K. D. Paulsen, “A clinical prototype for active microwave imaging of the breast,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 11, pt. 1, pp. 1841–1853, Nov. 2000. [11] Q. Fang, P. M. Meaney, and K. D. Paulsen, “Microwave image reconstruction of tissue property dispersion characteristics utilizing multiple-frequency information,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 8, pt. 2, pp. 1866–1875, Aug. 2004. [12] S. C. Hagness, A. Taflove, and J. E. Bridges, “Two-dimensional FDTD analysis of a pulsed microwave confocal system for breast cancer detection: Fixed-focus and antenna-array sensors,” IEEE Trans. Biomed. Eng., vol. 45, no. 12, pp. 1470–1479, Dec. 1998. [13] R. Benjamin, “Detecting Reflective Object in Reflective Medium,” U.K. patent GB2313969, Dec. 10, 1997. [14] R. Benjamin, “Synthetic, post-reception focusing in near-field radar,” in Proc. EUREL Int. Conf. (Conf. Publ. No. 431) The Detection of Abandoned Land Mines: A Humanitarian Imperative Seeking a Technical Solution, Oct. 7–9, 1996, pp. 133–137. [15] I. J. Craddock, R. Nilavalan, J. Leendertz, A. Preece, and R. Benjamin, “Experimental investigation of real aperture synthetically organised radar for breast cancer detection,” in Proc. IEEE Antennas and Propagation Society Int. Symp., 2005, vol. 1B, pp. 179–182, vol. 1B. [16] E. C. Fear and M. A. Stuchly, “Microwave detection of breast cancer,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 11, pt. 1, pp. 1854–1863, Nov. 2000. [17] C. J. Shannon, E. C. Fear, and M. Okoniewski, “Dielectric-filled slotline bowtie antenna for breast cancer detection,” Electron. Lett., vol. 41, no. 7, pp. 388–390, Mar. 31, 2005. [18] J. M. Sill and E. C. Fear, “Tissue sensing adaptive radar for breast cancer detection-experimental investigation of simple tumor models,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 11, pp. 3312–3319, Nov. 2005. [19] M. Fernández Pantoja, S. González García, M. A. Hernández-López, A. Rubio Bretones, and R. Gómez Martín, “Design of an ultra-broadband V antenna for microwave detection of breast tumors,” Microw. Opt. Technol. Lett., vol. 34, no. 3, pp. 164–166, Aug. 5, 2002. [20] H. Kanj et al., “A novel ultra-compact broadband antenna for microwave breast tumor detection,” Prog. Electromagn. Res., vol. 86, pp. 169–198, 2008. [21] S. C. Hagness, A. Taflove, and J. E. Bridges, “Three-dimensional FDTD analysis of a pulsed microwave confocal system for breast cancer detection: Design of an antenna-array element,” IEEE Trans. Antennas Propag., vol. 47, no. 5, pp. 783–791, May 1999. [22] H. Kanj and M. Popovic, “T- and X-arrangement of “dark eyes” antennas for microwave sensing array,” in Proc. IEEE Proc. Antennas and Propagation Society Int. Symp., Jul. 9–14, 2006, pp. 1111–1114. [23] X. Li, S. C. Hagness, M. K. Choi, and D. W. van der Weide, “Numerical and experimental investigation of an ultrawideband ridged pyramidal horn antenna with curved launching plane for pulse radiation,” IEEE Antennas Wireless Propag. Lett., vol. 2, no. 1, pp. 259–262, 2003. [24] R. Nilavalan and I. J. Craddock et al., “Wideband microstrip patch antenna design for breast cancer detection,” IET Microw. Propag., vol. 1, no. 2, pp. 277–281, 2007. [25] H. M. Jafari et al., “A study of ultrawideband antennas for near-field imaging,” IEEE Trans. Antennas Propag., vol. 55, no. 4, pp. 1184–1188, Apr. 2007.

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[26] J. Leendertz, A. Preece, R. Nilavalan, I. J. Craddock, and R. Benjamin, “A liquid phantom medium for microwave breast imaging,” presented at the 6th Int. Congress of the Eur. Bioelectromagnetics Assoc., Budapest, Hungary, Nov. 2003. [27] J.-Y. Sze and K.-L. Wong, “Bandwidth enhancement of a microstripline-fed printed wide-slot antenna,” IEEE Trans. Antennas Propag., vol. 49, p. 1020-102, Jul. 2001. [28] D. Lamensdorf and L. Susman, “Baseband-pulse-antenna techniques,” IEEE Antennas Propag. Mag., vol. 36, no. 1, pp. 20–30, Feb. 1994. [29] C. A. Balanis, Antenna Theory, 3rd ed. Hoboken, NJ: Wiley, 2005. [30] M. Klemm, I. J. Craddock, A. Preece, J. Leendertz, and R. Benjamin, “Evaluation of a hemi-spherical wideband antenna array for breast cancer imaging,” Radio Sci., vol. 43, 2008.

David Gibbins received the M.Eng. degree (first class honors) in aerospace engineering from Liverpool University, Liverpool, U.K., in 2004. He joined the University of Bristol, Bristol, U.K., in 2005 where he is working toward the Ph.D. degree and is a Research Assistant in the Centre for Communications Research where he is a member of Bristol’s Breast Cancer Imaging Project. His involvement in this project includes UWB antenna design, FDTD electromagnetic simulation and inverse scattering. His other research interests include conformal FDTD meshing techniques and applications of UWB radar.

Maciej Klemm was born in 1978. He received the M.Sc. degree in microwave engineering from Gdansk University of Technology, Poland, in 2002 and the Ph.D. degree from the Swiss Federal Institute of Technology (ETH) Zurich, Switzerland, in 2006. In February 2003, he joined the Electronics Laboratory, Swiss Federal Institute of Technology (ETH) Zurich, Switzerland. At ETH his research interests included small UWB antennas and UWB communications, antenna interactions with a human body, electromagnetic simulations, microwave MCM technologies and millimeter-wave integrated passives (European IST LIPS project). In spring 2004, he was a Visiting Researcher at the Antennas and Propagation Laboratory, University of Aalborg, Denmark, where he was working on the new antennas for UWB radios. In February 2006, he joined the University of Bristol (UoB), Bristol, U.K., where he currently holds a position of a Research Associate. At UoB, he is working on the microwave breast cancer detection and UWB textile antennas. His involvement in the breast cancer project includes antenna design, electromagnetic modeling, experimental testing as well as participating in clinical trial. Dr. Klemm received the Young Scientists Award at the IEEE MIKON 2004 Conference for his paper, “Antennas for UWB Wearable Radios.” For his paper, “Novel Directional UWB Antennas” he won the CST University Publication Award competition in 2006. In 2007, he won the “Set for Britain” competition for the top early-career research engineer and received a Gold Medal at the House of Commons.

Ian J. Craddock is a Reader in the CCR, University of Bristol, Bristol, U.K. His research interests include antenna design, electromagnetics, biomedical imaging and radar, funded by organizations such as EPSRC, QinetiQ, DSTL and Nortel. He leads Bristol’s Breast Cancer Imaging Project, this project winning the IET’s Innovation in Electronics prize in 2006. He has published over 100 papers in refereed journals and proceedings. He has led a workpackage on ground-penetrating radar in an EU Network of Excellence and has a related active research interest in antennas and propagation for instrumentation within the human body. He has delivered numerous invited papers to conferences in Europe, the US and Asia and chaired sessions at leading international conferences.

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Jack A. Leendertz received the B.Sc. degree in physics with mathematics from Bristol University, Bristol, U.K. He is with the Centre for Communications Research, Department of Electrical and Electronic Engineering, University of Bristol. His interests include microwave engineering, coherent optics in engineering, instrumentation for medical research and microwave imaging.

Alan Preece is a Clinical Scientist and Emeritus Professor of Medical Physics at Bristol University, Bristol, U.K., who previously researched biological effects of ionizing and non-ionizing radiation on humans. Current work is applied to practical equipment design and the clinical application of microwave imaging in human subjects for the purpose of identifying and evaluating the imaging possibilities of such microwaves in detection of breast cancer.

Ralph Benjamin received the B.Sc. degree (1st class honors) in electronic engineering from Imperial College, U.K., and invented the single-sideband mixer during his undergraduate course. He joined RN Scientific Service in 1944 and developed first countermeasure resistant 3D radar, and first force-wide integrated CCIS (Command, Control, Communications and Intelligence System) from 1947 to 1957. In 1947, he patented the interlaced cursor, controlled by joy-stick or mouse, to link displays to stored digital information. He also patented the world’s digital compression of video data, and first digital data

link, 1947, still in use NATO-wide as “Link 11”. Following repeated “special merit” promotions, Head of Research and Deputy Director, Admiralty Surface Weapons Establishment, 1961–64. (Evening/night work to lay the theoretical foundations for this field resulted in a Ph.D., then published as the textbook on Signal Processing) 1961 Acting International Chairman NATO “Von Karman” studies on “Man and Machine” and “Command and Control.” In the 1950s and 1960s, leading member of DTI Advanced Computer Techniques Project. Chief Scientist Admiralty Underwater Weapons Establishment (AUWE), 1964 to 1971, combined with Director, AUWE, and MoD Director Underwater Weapons R&D (and member of Navy Weapons Department Board), 1965 to 1971. (Published personal contributions led to the London D.Sc.) Chief Scientist, Chief Engineer and Superintending Director, GCHQ 1971 to 1982. This entailed responsibility for fast-track research, development, procurement, and deployment and use of equipment and techniques for the collection interpretation evaluation and assessment of Electronic or Signals Intelligence information. (Most projects had to create urgent solutions to problems which the opposition’s leading experts thought they had made impossible.) From 1972 to 1982, his functions were combined with those of Chief Scientific Advisor to both the Security Service and SIS, and with acting as Cabinet Office Co-ordinator, Intelligence R&D. Also, Visiting Professor, University of Surrey for two 3-year terms, 1972 to 1978, during which time he helped to start the Surrey University mini-satellite program. Following his first “retirement”, Head of Communications Techniques and Networks and semi-official global research coordinator NATO (SHAPE Tech Centre) 1982 to 1987. Graduate NATO Staff College, 1983. Currently, he is a Visiting Research Professor at University College, London, U.K. and Bristol University, Bristol, U.K. Until recently, he was also a Visiting Professor at Imperial College, London, External Ph.D. Supervisor at Open University, and external Post-Graduate Course Examiner, Military College of Science, Member of Court, Brunel University. During all these appointments, he combined the administration of large scientific/engineering organizations and of their R&D programmes, with creative, innovative up-front leadership, as illustrated by numerous classified and learned-society publications and patents.

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A Holographic Antenna Approach for Surface Wave Control in Microstrip Antenna Applications Adrian Sutinjo, Member, IEEE, Michal Okoniewski, Fellow, IEEE, and Ronald H. Johnston, Life Senior Member, IEEE

Abstract—A holographic antenna inspired structure is used to control the surface wave (SW) excited by a microstrip patch antenna. The hologram is designed to support a periodic leaky-wave which radiates at broadside and enhances the radiation of the patch while suppressing the horizontal lobe. In this design, the holographic approach is adapted for patch antenna applications where the SW wavelengths are comparable to the freespace wavelength. This is achieved by introducing dual phase-shifting metallic dipoles with periodic spacings. This paper discusses a simple and intuitive design method for the holographic surface, as well as its integration with the microstrip patch. The initial design concept was developed by assuming small perturbation to the SW, which was subsequently verified through full-wave simulations and prototype measurements. The results verified the improvements in the broadside gain and SW efficiency of the microstrip patch at the cost of increased area. Index Terms—Leaky wave antennas, microstrip antennas, microwave antennas, periodic structures, surface waves.

I. INTRODUCTION

T

HE problem of SW excitation in microstrip antennas is quite well documented [1]. Of primary concern is the SW mode of the grounded dielectric slab (GS) which has zero cutoff frequency. As a result, this mode is generally excited in patch antennas even for relatively thin substrates [2]. In an infinite GS, the excited SW contributes to antenna inefficiency. However, for practical GS sizes, the SW arriving at the GS edges is scattered resulting in ripples in the radiation pattern and increased back radiation [3]. In microstrip array applications, spurious edge radiation contributes to the degradation in sidelobe level (SLL) performance [1]. Therefore, SW excitation in patch antenna designs is considered undesirable. A number of techniques to control SW effects, ranging from simply placing foam absorbers [1] to designing intricate electromagnetic band gap (EBG) structures, have since been introduced [3], [4]. Recent advances in EBG technology include planar circularly symmetric (PCS) structures [5] which have been successfully applied to improve the performance of slot antennas [6] and 1-D scanning arrays [7]. A notable advantage of Manuscript received March 31, 2009; revised July 15, 2009. First published January 12, 2010; current version published March 03, 2010. This work was supported in part by the Canadian Natural Sciences and Engineering Research Council (NSERC) and in part by the Alberta Informatics Circle of Research Excellence (iCORE). The authors are with the Department of Electrical and Computer Engineering, University of Calgary, Calgary, AB, Canada (e-mail: [email protected]; [email protected]; [email protected]). Digital Object Identifier 10.1109/TAP.2009.2039316

the PCS structure is precisely its circular symmetry which is inherently compatible with the cylindrical SW excited by a patch antenna. For this reason, a similar PCS approach is adopted in this work. In contrast to the PCS-EBG operation (slow wave), we examine the application of a leaky-wave (fast wave) PCS structure for SW control in patch antennas. The main difference between the EBG and the leaky-wave operations is that the attenuation provided by the PCS structure is primarily reactive in the EBG while it is mainly radiative in the leaky-wave configuration. As a result, the antenna gain is expected to asymptotically increase (albeit with decreasing aperture efficiency) to the gain of the structure with infinite periodic rings with the subsequent addition of leaky-wave PCS periodic rings. The design of the surface-wave based leaky-wave antennas (SW-LWA) itself has been a topic of recent studies [10]. In many existing designs, however, the SW-LWA operate at SW wavelengths which are considerably shorter than the freespace wave–0.6 in [9], [10]), which is not comlength (e.g., patible with microstrip patch applications (where ). Hence, in this paper, we propose a technique to adapt the SW-LWA/holographic antenna to be compatible with the operation of a microstrip patch. A similar idea was presented in [11] where SW excited non-periodic scattering strips were designed using a genetic algorithm. The difference here is that we strive to design the PCS scatterers systematically using the holographic concept. The outline of this paper is as follows. The design principle is discussed in Section II. A design example and validation based on full-wave simulations are presented in Section III. Verification of the design using measured data from two prototypes is presented in Section IV. Finally, conclusions are given in Section V. II. DESIGN PRINCIPLE The holographic approach allows us to start the design by assuming that the presence of the metallic scatterers presents small perturbations to the SW on the grounded dielectric slab [8]. This assumption will be removed in the next Section where full-wave analysis of the antenna and computation of the leaky wavenumber will be presented. A. Surface Wave on a Grounded Dielectric Slab Assuming a GS on the -plane where the dielectric surface with a perfect electric conductor (PEC) at , the is at

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radial electric field for the region, is given by [12]

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SW mode in the

(1) is the SW phase constant, is the where attenuation constant in direction, is the -directed phase constant in the dielectric, and is a constant that depends on the excitation. Note that an directed infinitesimal electric dipole at , excites the SW mode with and . Since the primarily concern is the and variations in (2), the rest of the factors are insignificant and will not be from mentioned in the following discussions. To prevent the exciting a -directed (circumferential) current in the metallic strips in the PCS structure, the strips are realized as concentric rings of -directed (radial) dipoles [5], [9].

Fig. 1. Bi-phase double dipole scatterer. Note radial coordinate.

on dipole 1 and 2 linear equation

may be approximated by solving the

(2) where the incident voltages have been assumed to be proporcomponent of the SW at the center of tional to the and are the mutual coupling terms. Nethe dipoles and glecting the mutual coupling for now and solving (2) for the ratio of the currents excited on dipoles 1 and 2 leads to [12]

B. The Dual Phase-Shifting Dipole Scatterers In the specific case of the SW based holographic antenna, the pattern of the holographic surface is determined by the interference pattern of the SW on the grounded slab and the incoming plane wave from the direction of the intended maximum radiation [8]. Since we are concerned with maximum radiation in the broadside direction and the SW is assumed to be cylindrical, the interference pattern on the grounded slab takes the form of conwith a periodic distance centric rings centered at , which is consistent with the factor in (2). Accordingly, the simplest realization of the SW based holographic antennas (or, SW-LWAs) involves a single continuous (or, pseudo-continuous) metallic ring per period [8]–[10]. However, in the range of operation of a microstrip patch, , and consequently, the required periodic spacing , which is close to the grating lobe condition for the is rings. As a result, such single ring per period implementation is expected to exhibit a high sidelobe at or near the horizon when the main beam is at broadside. To suppress the grating lobe, the spacings between successive radiators in to . the radial direction need to be decreased from are nearly out of However, at such spacings, the incident phase at successive radiators. This apparent problem can be successfully resolved by introducing appropriate phase shifts to the ring elements. A dipole scatterer in free-space exhibits phase-shifting propin the vicinity of its half-wave erty as a function of length (fundamental) resonance. Assuming sinusoidal current distribution on the dipole, the ratio of the incident E-field to the excited current is approximately proportional to the self impedance of the dipole. The same principle applies to printed dipoles, as long as the dipoles operate near the fundamental resonance. Following this principle, consider two dipoles of lengths separated from center to center by a distance as indicated in Fig. 1. Assuming the SW incident wave originating from ), the currents excited a source some distance away (i.e.,

(3) (4) To achieve broadside radiation, we need condition leads to the approximate design equation

. That (5)

which implies that the difference in phase of the self impedances needs to compensate for the phase delay of the incident wave. Note that operation with would require , which is not practical since it would yield and , im, suggesting vanishingly plying small radiation. Consequently, in a practical design must be because of this limitation. However, one should still strive for value as close to as possible without making each resonance too narrowband. This will minimize the distance from a dipole to the in the next period while maintaining reasonably good current amplitude and dipoles. These remarks will balance between the be elucidated with a design example in the next Section. The target value of for a given substrate parameters can be inferred from the range of attainable self impedance of a printed dipole which is resonant at the design frequency. Also, notice that for circularly polarized (CP) infinitesimal electric dipole sources, and for left-hand/right-hand (LH/RH) CP in (2). Given the same set of assumptions, (5) remains unchanged, such that the design principle is applicable to linear as well as CP sources. III. DESIGN EXAMPLE A. Design of the Holographic Surface To illustrate the advantages of the holographic technique introduced in this work, the GS and design frequency were selected such that the patch excites ample SW to illuminate the

SUTINJO et al.: A HOLOGRAPHIC ANTENNA APPROACH FOR SURFACE WAVE CONTROL

hologram. However, to still be compatible with the range of operation of microstrip transmission line, the permittivity and thickness of the GS must be low enough such that the higher mode, to a first approximaorder microstrip line modes ( tion), which are leaky in themselves [13], remain insignificant. Furthermore, to avoid design complexity, the GS also needs to SW implying that the frequency of operation only support must be below the SW cutoff [9]. of 8 GHz and In this design, the center frequency (75 mils), the RT/D 6010LM substrate with , and were selected. For this GS, the and (for a microstrip ). Note that the width of 1.9 mm, for which higher order microstrip line mode is the upper frequency limiting factor. At 8 GHz on this GS, the radiation efficiency , which implies that in a lossless substrate, the SW accounts for 60% of the injected power to the antenna. Also, at which corresponds to 8 GHz the . The self impedance of a resonant printed dipole on 1.905 mm RT/D6010LM GS at 8 GHz was simulated using FEKO. It was for a resfound that the range of attainable differences in onant printed dipole at was approximately . The were determined by finding the resonant dipole lengths at 7.5 and 8.5 GHz, , 5.85 mm, respectively as illustrated in Fig. 2(a) for where the actual resonant frequencies were 7.6 and 8.4 GHz. The with is plotted in Fig. 2(a) where the maximum value of 138.9 occurs at 7.9 GHz. Also note that at 7.95 GHz, which implies Fig. 2(b) suggests that that the magnitude of the induced currents on the dipoles are approximately equal at this frequency according to (3). Note that if current amplitudes will each resonance was too narrow,the rapidly become imbalanced as the operation frequency moves away from the center frequency. Using this design, may be determined using (5). At 8 GHz, and resulting in . With this result, the approximate design of the holographic surface is considered complete. The dipoles will be arranged as concentric rings of double dipoles as seen in Fig. 3 where only one quadrant is shown. The angular spacing between the dipoles was designed such that the arc distance be, this was also done tween the neighboring elements is to avoid grating lobes at the horizon. For the excitation, standard patch designs such as rectangular or circular patches, or the circularly polarized versions thereof may be used. The distance , from the center of the patch to the center of the first ring of dipoles is critical and will be addressed next. B. Integration of the Holographic Surface With the Patch Antenna We verified, using full-wave simulation (FEKO), that the incident field is well approximated using (2) [12]. A 1 Am -directed Hertzian dipole was placed in FEKO slightly above the dielectric surface (to avoid a singularity when the source is on , (2) well approxithe surface). It was found that for and from FEKO simulation. mates both

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W = 0:5 mm and L = 6:63, ( = 10:2): (a) phase and (b) amplitude.

Fig. 2. Printed dipoles self impedances with 5.85 mm on 1.905 mm RT/D6010LM

L =

Fig. 3. One quadrant of the bi-phase double dipole hologram excited by a circular patch antenna. Two periodic rings are shown. Dimensions (mm): , , , , , , and for the , 2 periods, and for , 4 periods, such that where is the distance to the center of a dipole element.

6:63 L2 = 5:85 p = 13:25 s = 18:735 a = 2:9 d = 34:55 1 = 15 M =1 1 = 7:5 M = 3 1 r < 0:5 r

Next, the distance indicated in Fig. 3 was determined. This distance was chosen such that the current excited on the first dipole is approximately in-phase with the excitation, which

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for this purpose, was modeled as an -directed Hertzian dipole . At ( at 8 GHz), using FEKO, it was found that the phase of the current excited dipole lags the Hertzian dipole by 36 . While on the the exact in-phase condition was not achieved at this distance, a simple 1-D array factor analysis revealed that relatively high dieither leading or lagging rectivity was still achieved for by approximately 45 . In fact, the peak directivity was found to shift slightly from the exact in-phase condition depending on . For the sign of the leaky-wave phase constant and , the maximum broadside directivity occurs with slightly leading (by ) [12]. In the was selected for the radiation end, pattern performance, and also, since the slight lag in the phase near maximum. of with respect to results in To excite the holographic surface, a circular patch and fed at from the with radius center was selected. For a linearly polarized (LP) patch, the feed . A CP patch was fed in phase position was at quadrature at where the phase of the leads the by 90 for a left-handed (LH) CP. In all cases, the patches were fed using wire probe(s) and the infinite GS , , , parameters are unless otherwise indicated. and broadside gain for periFEKO simulated odic rings surrounding the circular patch are reported in Figs. 4(a) and 4(b), respectively for the cases involving LP cir, 2, and cular patch only and the LP patch surrounded by 4, as well as a quadrature-fed LHCP circular patch surrounded rings of double dipoles. Significant by the identical improvements in both gain (from 3.1 dBi to 17.2 dBi at 8 GHz, ) and (from 43% to 90% at 8 GHz, patch only vs. ) are clearly evident. In fact, notable patch only vs. improvements are observed with just a single periodic ring. Note that the broadside gain increases with additional periodic rings as the hologram radiates more of the SW energy which is in contrast to the PCS-EBG where the gain is maximized for 2 periodic rings [5]. Also, notice that the broadside gain structure are virtually and radiation efficiency of the identical regardless whether it is driven by a LP or CP sources. was also investigated for the The significance of the case by plotting the lossless case in Figs. 4(a) and 4(b). Note that in terms of broadside gain at 8 GHz, the improves the broadside gain by only 0.3 dB, while the improves by 5% compared to the case. These results indicate that dielectric losses of this substrate do not significantly affect the antenna performance. are shown in The E and H-plane gain patterns for components are shown for the E-plane Fig. 5. Only the 50 dB below since the cross-polarized levels are very low ( at 8 GHz). In the H-plane, the cross-polarized levels are 15 dB below the co-polarized patterns, except at where the cross-polarized levels are negligibly small. The H-plane co-polarized patterns also exhibit noticeably wider main beamwidths compared to that of the E-plane, which is due to the circumferential illumination of the holographic surface by the LP patch. As a result, this design is compatible only with applications which can tolerate this mismatch in E and H-plane

M

Fig. 4. FEKO simulation results for a circular patch surrounded with periodic rings of double dipoles: (a) radiation efficiency and (b) broadside gain.





Fig. 5. FEKO simulated radiation pattern for a LP circular patch surrounded with 4 periodic rings at 8 GHz for the E-plane ( = 0 ) and H-plane ( = 90 ).

beamwidths. Otherwise, a nearly circularly symmetric E and H-plane beamwidths may be achieved using superstrates or partially reflective surface (PRS) cover above the patch anof the LP patch only and the LP patch tenna [14]. The

SUTINJO et al.: A HOLOGRAPHIC ANTENNA APPROACH FOR SURFACE WAVE CONTROL

Fig. 6.

j

S

j

of LP circular patch surrounded with M periodic rings.

Fig. 8. HFSS simulated broadside gain with finite and infinite GS for M

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= 3.

Fig. 9. FDTD simulation setup.

Fig. 7. FEKO simulations for a LHCP circular patch surrounded with M at 8 GHz.

=4

surrounded by periods are shown in Fig. 6. Note that the presence of the scatterers introduces a small amount of bandwidth frequency shift and narrowing of the 10 dB to 7.91–8.3 GHz from 7.82–8.26 GHz . at The radiation pattern of the LHCP patch with 8 GHz is shown in Fig. 7 for . Unlike the LP case, the and 90 patterns are nearly the same (thus, only the is shown), since the amplitude of the circumferential illumina). However, tion of the dipoles is constant (i.e., note that the cross-polarized components in the off-broadside directions in this case are noticeably higher than in the LP case. Improving the CP performance likely requires modification to the holographic structure. The effects of finite GS was investigated using HFSS simulations. Due to the large problem size, the GS radius was limited ( at 8 GHz) which accommoto periods. Both finite and infinite GS simulations dates were performed in HFSS and the results are shown in Fig. 8 for . For this value, the broadside gain of both finite and infinite cases converge quite well, especially in the vicinity of the design frequency of 8 GHz. Similar behavior was also noted

. Such convergence suggests that the SW illuminawith tion of the board edges are indeed suppressed by the attenuation provided by the holographic surface. Radiation efficiency from the finite ground simulation results was found to be 80 to 95% in the 7 to 9 GHz range. At 8 GHz, for and for patch only. The simucase with was lated aperture efficiency for found to be 10 to 12% in the 7.7 to 8.2 GHz range, which is considered low. This, of course, is partly a consequence of the significant direct radiation from the patch and also of the circumferential SW illumination. However, we expect that the aperture efficiency may be improved when the structure is illuminated using a patch array. C. Relation to Leaky-Waves The extraction of the leaky wavenumber for the holographic structure was based on simulations in SEMCAD (FDTD) software. The essence of the simulation setup for one period of double dipoles is illustrated in Fig. 9, where and the the distance between the PMC sidewalls was height of the absorbing boundary condition (ABC) on the top . The presence of the PMC walls models surface was apart in the lateral an infinite array of dipoles spaced direction. To ensure field convergence to the SW mode, the excitation (an open circuited microstrip line) was placed away from the first cell boundary. In the approximately FDTD simulations, both the metal and dielectric were made

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^ Fig. 10. Convergence of k

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extracted from FDTD for M = 10, 12. Fig. 11. Fabricated prototype with M = 3 with LP patch excitation.

lossless. The device under test (DUT) consists of periodic field measurements at the input and cells of double dipoles. from the first and last output were taken at approximately cell boundaries, respectively. The extraction of the S-parameter from FDTD followed the method presented in [15]. To gain insights into the radiation property of the holographic were extracted from the eigenvalues of the structure, the was increased. The ABCD matrix itABCD matrix [16] as self was obtained from the corresponding S-parameter of the DUT with appropriate phase corrections to account for the field measurement distances. Since this method is approximate, was carefully monitored as was inconvergence of the creased. It was found that the convergence occurred slowly in to 12 as this case and was reached approximately at demonstrated in Fig. 10. Note that in the entire 7 to 9 GHz range, spatial harmonic is indeed fast . Thus, the radiation should occur as long as the attenuation constant is in the 7.65 to 8.6 GHz non-zero. Furthermore, the indicating radiation near broadside ( , for an end-fed LWA). Note that in the attenuation region centered around 8.3 GHz, which suggests that this crossing ocregion is a stopband. Also notice that the curs at 8.5 GHz such that the sign of is negative below this point. We found that the HFSS broadside gain (from Fig. 8) to obtained from FDTD in Fig. 10. be consistent with the , HFSS predicts a high gain region from 7.6 GHz For (15.6 dBi) to 8.2 GHz (16.9 dBi). This frequency range overlaps with the radiative attenuation region from 7.6 GHz to 8.1 GHz and also overlaps with a small portion of the stopband from plot. While the attenuation in 8.1 GHz to 8.2 GHz in the the stopband region is increasingly reactive, some amount of . radiation still exists as long as in the 7.6 to 8.1 GHz attenuation reAlthough gion, which indicates that the beam should be split into two peaks for an infinite center-fed LWA [17], a broadside maximum beam is still achieved in a center-fed finite LWA as explained and in [18]. At 7.8 GHz, [19] (where is the , is the power guided at point radius of GS,

) which is well below the beam splitting condition for finite center-fed LWA. IV. EXPERIMENTAL VALIDATION and the dipole only were The two prototypes with the fabricated. The grounded dielectric substrate material was RT/D . This radius was chosen such 6010LM with radius that the resulting board diameter of 236.25 mm is less than the smallest panel size to minimize fabrication cost. The dimensions of the holographic structure are identical to Section III-A. A patch which is nearly identical to the one in FEKO simulais used. The patch was tions in Fig. 3 with radius probe-fed from the ground plane side using a small diameter coaxial cable. The feed position was optimized using HFSS and from the center. was found to be The broadside gain measurements were performed in a 3.4 m at anechoic chamber which corresponded to and LP only) and HFSS sim7.8 GHz. The measured ( with frequency shift) are shown ulated broadside gains ( in Fig. 12. The measured gain for the case peaks at 16.04 dBi at 7.36 GHz with 1 dB gain bandwidth of approxior 8.3%). The immately 7.29–7.92 GHz ( case provement in broadside gain of at least 6 dB for the compared to the LP only case is clearly evident. Comparisons between measured results and HFSS show that the measured gain curve is shifted down in frequency by approximately 325 MHz ( 4%) compared to HFSS. Once this frequency shift is taken into account, both the gains and gain bandwidths appear quite similar as seen in the figure. The HFSS peak gain is 16.83 dBi at 8.2 GHz, which is 0.8 dB higher than the measured gain. However, the fractional gain bandwidths are nearly identical at 8%. The measured ( and LP only) and HFSS simulated (LP are reported in Fig. 13. The measured 10 dB only) bandwidth for the case is approximately 7.35–7.62 GHz or 3.6%) which is slightly narrower than the ( LP only case of 7.31–7.66 GHz ( or 4.7%). With respect to the gain bandwidths in Fig. 12, the overall bandwidths are narrower and shifted slightly to the lower frebandwidth for the quency (by 100 MHz). The measured

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Fig. 12. Measured and HFSS simulated broadside gains. The HFSS gain curve is shifted down in frequency by 325 MHz to facilitate comparison.

Fig. 13. Measured and HFSS simulated jS

j

.

patch only case (7.6–8.25 GHz) is shifted down in frequency by roughly 475 MHz ( 6%) and is slightly narrower compared to HFSS (7.31–7.67 GHz). On the whole, HFSS appears to predict the gain properties more accurately than the feed point impedance. Such behavior is generally expected in full-wave simulations since radiation has to do with an overall field behavior on the antenna structure while feed point impedance depends on field behaviors at one specific point. Measured gain patterns at the , 90 planes are shown in Figs. 14(a) and 14(b) for 7.6 GHz, which is the center of the gain bandwidth. The existence of maximum radiation at broadside is . Furthermore, the reduction of clearly demonstrated for ) and back radiation (for the radiation at the horizon (for , 90 ) is again evident. At 7.6 GHz, the radiation both at the horizon is suppressed by 10 dB compared to the LP only plane case. Again, the radiation at the horizon in the is not significantly affected by the holographic surface since the SW is not excited there. Note also that at , the presence of the holographic surface reduces the levels of the gain ripple relative to the broadside gain. Similar observations were also made at 7.45 and 7.75 GHz. These trends also hold, with some degradations, at the frequencies near the 1 dB gain band limits at 7.3 and 7.9 GHz.

Fig. 14. Gain pattern at 7.6 GHz: (a) 

=0

; jE

j

(b) 

= 90

; jE j .

V. CONCLUSION A holographic antenna based technique to improve the broadside gain and SW efficiency of a patch antenna has been presented. The holographic concept has been adapted for the microstrip patch application by introducing dual phase-shifting dipole scatterers per periodic cell. A step-by-step design method was presented and the design was validated using full-wave simulations and measured data. It was found that the holographic structure yields significant performance improvements with two or three periodic double dipole rings surrounding the patch. Prototype measurement results demonstrate broadside gain enhancement and suppression of radiation at the horizon and at the back side of the patch antenna with holographic surface consisting of three periods. Except for a few percent of frequency shift, the broadside gain and gain bandwidths were very close to the HFSS predicted values. REFERENCES [1] J. R. James, P. S. Hall, and C. Wood, Microstrip Antenna Theory and Design. London, U.K.: Peregrinus, 1981. [2] D. R. Jackson and N. G. Alexopoulos, “Simple approximate formulas for input resistance, bandwidth, and efficiency of a resonant rectangular patch,” IEEE Trans. Antennas Propag., vol. 39, no. 3, pp. 407–410, Mar. 1991.

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[3] D. Sievenpiper, L. Zhang, R. F. J. Broas, N. G. Alexopolous, and E. Yablonovitch, “High-impedance electromagnetic surfaces with a forbidden frequency band,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 11, pp. 2059–2074, Nov. 1999. [4] R. Coccioli, F. Yang, T. Itoh, and K. Ma, “Aperture-coupled patch antenna on UC-PBG surface,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 11, pp. 2123–2130, Nov. 1999. [5] N. Llombart, A. Neto, G. Gerini, and P. de Maagt, “Planar circularly symmetric EBG structures for reducing surface waves in printed antennas,” IEEE Trans. Antennas Propag., vol. 53, no. 10, pp. 3210–3218, Oct. 2005. [6] A. Neto, N. Llombart, G. Gerini, and P. de Maagt, “On the optimal radiation bandwidth of printed slot antennas surrounded by EBGs,” IEEE Trans. Antennas Propag., vol. 54, no. 4, pp. 1074–1082, Apr. 2006. [7] N. Llombart, A. Neto, G. Gerini, and P. de Maagt, “1-D scanning arrays on dense dielectrics using PCS-EBG technology,” IEEE Trans. Antennas Propag., vol. 55, no. 1, pp. 26–35, Jan. 2007. [8] M. Nannetti, F. Caminita, and S. Maci, “Leaky-wave based interpretation of the radiation from holographic surfaces,” in Proc. IEEE Antennas Propag. Symp. (APS 2007), Honolulu, HI, Jun. 2007, pp. 5813–5816. [9] M. Ettorre, S. Bruni, G. Gerini, A. Neto, N. Llombart, and S. Maci, “Sector PCS-EBG antenna for low-cost high-directivity applications,” IEEE Antennas Wireless Propag. Letters., vol. 6, pp. 537–539, 2007. [10] S. Podilchak, A. P. Freundorfer, and Y. M. M. Antar, “Planar leaky-wave antenna designs offering-conical beam scanning and broadside radiation using surface-wave launchers,” IEEE Antennas Wireless Propag. Letters., vol. 7, pp. 155–158, 2008. [11] R. G. Rojas and K. W. Lee, “Surface wave control using non-periodic parasitic strips in printed antennas,” Proc. Inst. Elect. Eng. Microw. Antennas Propag., vol. 148, no. 1, pp. 25–28, Feb. 2001. [12] A. Sutinjo, “Analysis and design of printed leaky-wave antennas for broadside radiation,” Ph.D. dissertation, Univ. Calgary, AB, Canada, May 2009. [13] W. Menzel, “A new travelling wave antenna in microstrip,” in Proc. Eur. Microwave Conf. (EuMC 1978), Paris, France, Oct. 1978, pp. 302–306. [14] R. Gardelli, M. Albani, and F. Capolino, “Array thinning by using antennas in Fabry-Perot cavity for gain enhancement,” IEEE Trans. Antennas Propag., vol. 54, no. 7, pp. 1979–1990, Jul. 2006. [15] Z. Yu, “A simple and effective method for the reflection coefficient extraction in rectangular waveguide discontinuity analysis by the FDTD,” Microw. Opt. Technol. Lett., vol. 15, no. 1, pp. 57–59, May 1997. [16] R. E. Collin, Foundations for Microwave Engineering, 2nd ed. New York: McGraw-Hill, 1992, ch. 8. [17] P. Burghignoli, G. Lovat, and D. R. Jackson, “Analysis and optimization of leaky-wave radiation at broadside from a class of 1-D periodic structures,” IEEE Trans. Antennas Propag., vol. 54, no. 9, pp. 2593–2603, Sep. 2006. [18] A. Sutinjo, M. Okoniewski, and R. H. Johnston, “Beam-splitting condition in a broadside symmetric leaky wave antenna of finite length,” IEEE Antennas Wireless Propag. Lett., vol. 7, pp. 608–611, 2008. [19] D. R. Jackson and A. A. Oliner, Antenna Engineering Handbook, 4th ed. New York: McGraw-Hill, 2007, ch. 11.

Adrian Sutinjo (M’10) received the B.S.E.E. degree from Iowa State University, Ames, in 1995, the M.S.E.E. degree from the University of Missouri-Rolla (now, Missouri S&T) in 1997, and the Ph.D. degree in electrical engineering from the University of Calgary, Calgary, AB, Canada, in 2009. From 1997 to 2004, he was an RF Engineer for Motorola in Illinois and Murandi Communications Ltd., Calgary. Currently, he is a Postdoctoral Associate at the University of Calgary. His interests include antennas, RF and microwave engineering, and electromagnetics.

Michal Okoniewski (F’09) is a Professor in the Department of Electrical and Computer Engineering, University of Calgary, Calgary, AB, Canada. He holds an endowed Libin/Ingenuity Chair in bio-engineering and Canada Research Chair in applied electromagnetics. In 2004 he co-founded Acceleware Corp. His interests range from computational electrodynamics, to tunable reflectarrays, RF MEMS and RF micro-machined devices, as well as hardware acceleration of computational methods. He is also involved in bio-electromagnetics, where he works on tissue spectroscopy and micro-imaging. Dr. Okoniewski has been an Associate Editor for the IEEE TRANSACTIONS OF ANTENNAS AND PROPAGATION for the last six years.

Ronald H. Johnston (LSM’06) was born in Drumheller, Alberta, Canada. He received the B.Sc. degree from the University of Alberta, Calgary, AB, Canada, in 1961 and the Ph.D. degree from the University of London, London, U.K. and the DIC from Imperial College, London, both in 1967. In 1970, he joined the University of Calgary and held Assistant to Full Professor positions and was the Head of the Department of Electrical and Computer Engineering from July 1997 to June 2002. He became Professor Emeritus in the Schulich School of Engineering, University of Calgary in 2006. Other professional experience includes work at Canadian General Electric (Toronto), Nortel (Ottawa), Communications Research Centre (Ottawa), Carleton University (Ottawa), University of Sydney (Australia), TRLabs (Calgary) and Nortel (Harlow UK). He has authored or co-authored about 160 journal and conference papers mostly in the areas of RF circuits, diversity antennas, small antennas, slot antennas, radiation efficiency, MIMO systems and subsurface EM wave propagation. Present research is mostly concerned with small antenna efficiency measurements, compact low Q antennas, slot array antennas and high purity circularly polarized antennas. A number of patents have been published. He has supervised (or co-supervised) over 45 postgraduate students in programs for Ph.D., M.Sc. or M.Eng. degrees.

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Receiving Polarization Agile Active Antenna Based on Injection Locked Harmonic Self Oscillating Mixers Carlos Vázquez, Samuel Ver Hoeye, Member, IEEE, Miguel Fernández, Germán León, Luis Fernando Herrán, and Fernando Las Heras, Senior Member, IEEE

Abstract—A polarization agile active antenna with phase shifter elements based on injection locked third harmonic self oscillating mixers is presented. This phase shifting topology provides the double functionality of continuous range phase shifter and downconverter. The phase shift value introduced by each circuit can be easily tuned through a DC voltage within a theoretical continuous range of 450 . The behavior of the isolated phase shifter circuit is studied, both as a function of the control voltage and versus frequency, through harmonic balance and envelope transient simulations. The polarization tuning performance of the complete active antenna is simulated, analyzing the impact of the operating parameters of the phase shifter on the overall behavior. A receiving polarization agile antenna with an input frequency band centered at 11.25 GHz and an output frequency band centered at 1.5 GHz has been manufactured for the experimental validation of the simulated results. A continuous range of polarization tuning has been observed, including two orthogonal linear polarizations along with left hand and right hand circular polarization. Index Terms—Injection locked oscillators, microstrip antennas, microwave phase shifters, polarization, receiving antennas.

I. INTRODUCTION ECONFIGURABLE antenna implementations have become widespread in recent years, owing to their capability to dynamically adjust some of their properties to the requirements of each particular scenario. In this context, polarization agility is an interesting feature for an antenna, since it simplifies the implementation of frequency reuse techniques, which can nearly double the channel capacity, and allows a good polarization matching between transmitter and receiver. Furthermore, as polarization diversity has proven to be as effective as spatial diversity to mitigate the detrimental multipath fading, both in indoor [1] and outdoor [2], [3] environments, polarization agile antenna topologies provide substantial implementation size and cost reductions.

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Manuscript received April 14, 2009; revised August 10, 2009. First published December 31, 2009; current version published March 03, 2010. This work was supported in part by the “Ministerio de Ciencia e Innovación” of Spain and “FEDER”, under projects TEC2006-12254-C02-01/TCM, TEC2008-01638/TEC (INVEMTA) and CONSOLIDER-INGENIO CSD2008-00068 (TERASENSE), by the “Gobierno del Principado de Asturias” under the “Plan de Ciencia y Tecnología (PCTI)”/“FEDER-FSE” by the Grant BP08-082, the projects EQP06-015, FC-08-EQUIP-06, PEST08-02, and IB09-081, and in part by the “Cátedra Telefónica” Universidad de Oviedo and “Fundación CTIC.” The authors are with the Department of Electrical Engineering, Universidad de Oviedo, Edificio Polivalente de Viesques, E-33203 Gijón, Spain (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2009.2039325

Several works on polarization agile antennas have been presented in the bibliography. Some passive topologies enable the selection of a discrete number of polarization states by altering the antenna layout through solid state [4], [5], or piezoelectric [6] switches. Others implement a continuous range of polarization tuning by attaching varactor loads to the radiating patch [7]. Most of the active topologies available in the literature rely on feeding two orthogonal linearly polarized radiating modes of the antenna with the same original signal, conveniently modified through phase shifting circuitry, in order to provide the polarization tuning capability. Phase shifting solutions based on a single oscillator circuit injection locked at the first harmonic component of the oscillation, enable a theoretical phase shift range limited to 180 [8]. This phase shift range can be doubled by feeding each of the radiating modes with an independent oscillator, providing any kind of coupling between the circuits is avoided [9]. Mutually coupled topologies maintain the same 180 theoretical phase shift range, which can be doubled, as in [10], by extracting the second harmonic component of the oscillation. These theoretical ranges are generally reduced in the presence of noise to about 15% smaller experimentally observable phase shift ranges. Moreover, due to the appearance of noise precursors when operating close to the limits of the synchronization range [11], the usable phase-shift range is generally further reduced to approximately 70% of the theoretical range. With regard to receiving topologies, the phase shifting functionality can be attained through a mixing operation between the conveniently phase shifted self oscillation signal, (or one of its harmonic components), and the input signal. The architecture used in this work is based on an injection locked 3rd harmonic self oscillating mixer (IL3HSOM) [12], in which the output signal is generated through mixing the input signal at frequency , with the third harmonic component at frequency of the HSOM self oscillation fundamental frequency . As a result of this operation, a downconverted signal at is obtained in a single stage [12], with the desired phase shift within a theoretical range of 540 (which assures a usable range of at least 360 ). In this paper, a receiving polarization agile microstrip antenna is presented. The wide phase shift range provided by the IL3HSOM based phase shifters, enable the antenna polarization tuning within a continuous range, comprising two orthogonal linear polarizations and both left hand and right hand circular polarization (LHCP and RHCP). Additionally, another two

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Fig. 1. (a) Topology of the polarization agile antenna. (b) Circuit diagram of the IL3HSOM based phase shifter.

linear polarizations might be produced by activating only one circuit at a time. However, these are not considered in this work since the received power would be halved, as it would be with any other of the available polarizations. The paper is structured as follows. In Section II, the proposed topology is introduced and the individual performance of its important parts is studied. In Section III, the polarization tuning capabilities of the complete active antenna solution are simulated (using Advanced Design System), for different operating conditions of the phase shifter circuits. In Section IV, the experimental results obtained with the manufactured receiving polarization agile antenna are shown. II. POLARIZATION AGILE ANTENNA BASED ON IL3HSOM A. Topology of the Polarization Agile Antenna The topology of the receiving polarization agile antenna is shown in Fig. 1(a). A two port aperture coupled patch antenna (ACPA), couples the power received in each of its two orthogonal linearly polarized fundamental modes, at GHz, onto one of its output ports, which deliver these signals to two IL3HSOM based phase shifters. Both IL3HSOM circuits are injection locked to an external GHz and phase , signal of power , frequency through a Wilkinson power divider, providing equal power and and ). phase in both branches (

Using the techniques presented in [13]–[15], the IL3HSOM circuits have been optimized to provide maximum conversion in the input frequency gain band at GHz, for the mixing opera. The phase shifts introduced by the tion IL3HSOM circuits at intermediate frequency, with respect to , the external phase reference can be separately controlled through two DC signals, enabling the polarization tuning capability. Low power samples of the output signals of both circuits are extracted through microstrip directional couplers for phase shift monitoring purposes. The sampled signals are simultaneously measured with two Agilent 89600 Vector Signal Analyzers (N8201A–N8221A). The output signal of the polarization agile antenna is obtained through a Wilkinson combiner and measured with the vector network analzer of the anechoic chamber measurement setup, which is explained in Section IV. In order to prevent detrimental reductions in the phase shift ranges of the IL3HSOM circuits and to assure their independent performance, mutual coupling between them at the harmonic components of the self oscillation frequency, , must be avoided. The required isolation levels through the input and output ports are achieved by filtering, taking advantage of the fact that none of these harmonic components falls into either the input or output frequency bands. Mutual coupling through the synchronization port at the is avoided by the harmonic components bandpass filter centred at . The Wilkinson divider is designed to feature high isolation levels, so that the synchronization signal for both circuits is exclusively determined by the external generator. B. Two Port Aperture Coupled Patch Antenna The antenna, as depicted in Fig. 2(a), consists of a square patch designed in the bottom layer of a 0.762 mm thick ARLON and at 10 GHz) and 25N substrate ( placed inverted on top of a 2.6 mm thick foam layer ( and at 10 GHz). The power received in each of the two orthogonal linearly polarized fundamental modes of the patch is electromagnetically coupled through two perpendicular slots etched in the ground plane of the distribution network, onto the microstrip transmission line connected to one of the output ports. These transmission lines are designed to have a 90 degree difference in electrical length, in order for the output signals to be in phase when the incident radiation presents right hand circular polarization. A prototype of the antenna has been manufactured and measured. An impedance matched frequency band ( dB, ) from 10.5 to 12 GHz is obtained, as shown in Fig. 2(b). High isolation levels between the ports ( dB), are observed throughout the same band. C. Performance of the IL3HSOM Based Phase Shifters The circuit topology of the IL3HSOM is shown in Fig. 1(b). The series feedback at the source terminal of the ATF36077 transistor is designed to produce negative resistance at the gate port. The component at intermediate frequency resulting from the mixing operation is selected through a bandpass filter at the

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Fig. 3. Injection locked solutions of the isolated IL3HSOM circuit for three different synchronization power levels P , as a function of the control voltage V , for (a) the conversion gain G , and (b) the phase shift at intermediate frequency . The stable and unstable regions determined through envelope transient simulations [12], are also indicated.

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Fig. 2. (a) Structure of the two port aperture coupled patch antenna. Dimensions in millimeters. (b) Measured scattering parameters of the two port ACPA.

drain port and, as a reference for the phase shifting functionis injected through ality, a signal of frequency and power the synchronization filter at the gate port. A varactor diode connected to the series feedback network provides the phase shift tuning capability. For different operation conditions, the injection locked solutions of the IL3HSOM circuit are obtained using the techniques presented in [12], based on the use of an auxiliary generator in harmonic balance and envelope transient simulations. At the GHz, the syncenter frequency of the input band , and the chronized solutions for both the conversion gain phase shift introduced by the circuit at intermediate frequency, , with respect to the external phase reference have been calculated as a function of the control voltage . The results for three different synchronization power levels are shown in Fig. 3. The traces in dashed line correspond to harmonic balance simulations and represent mathematical solutions of the system. However, only a limited range of these solutions are stable and thus, experimentally observable [12]. For values of the control voltage outside the stable ranges, which are determined using envelope transient simulations and indicated in solid line, the frequency of oscillation is no longer locked to that of the synchronization signal and it starts to vary with the control voltage.

The conversion gain is slightly dependent on the control voltage [see Fig. 3(a)] and, although this dependence increases , it is not significant for with the synchronization power dBm), its most applications since, in the worst case ( variation range remains below 0.7 dB. By tuning the varactor can be controlled within control voltage , the phase shift a stable range of about 450 degrees [Fig. 3(b)]. Even though the sensitivity to the control signal is higher for lower synchronization power levels, the stable range of variation is approximately the same for the three studied cases. Hence, the phase shift introduced at the center frequency , can be arbitrarily selected within the synchronization ranges through the varactor bias voltage. However, once is fixed at a specific point, the phase shifts introduced at other frequencies cannot be controlled, as they are determined by the frequency response of the circuit. For three synchronization power levels, this frequency response has been calculated through harmonic balance and envelope transient simulations, considering seven different phase shift values , uniformly distributed set at the center frequency throughout the corresponding stable ranges [Fig. 3(b)]. , is repThe frequency response of the conversion gain resented in Fig. 4(a). For higher synchronization power levels, a stronger variation is observed, both with frequency and with , especially when apthe operation point selected proaching the limits of the input frequency band. In order to simplify the comparison between the multiple traces represented, has been defined in (1). The first term the phase deviation of the equation is the actual frequency response of the circuit, considered. By subtracting for the different control voltages

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Fig. 5. Theoretical evolution of the antenna polarization when a phase shift component (in addition to the 90 degree delay caused by is introduced in the the feeding network layout).

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circular polarization when both components are combined in phase. The theoretical evolution of the antenna polarization compowhen an additional phase shift is introduced in the nent , is shown in Fig. 5 in terms of its axial ratio (AR), as expressed in (2), assuming both com. ponents are combined with equal voltage gain Fig. 4. Frequency response of the IL3HSOM circuit for three different synchronization power levels and for seven working points, uniformly distributed , as throughout the stable ranges (a) conversion gain. (b) Phase deviation defined in (1).

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the second term (the phase shift value established at the center , for each different trace), the offset frequency between traces is eliminated, putting them together at the center frequency.

(1) Due to the propagation throughout the circuit, the phase response presents the characteristic strong slope as a function of frequency which, in this case, conceals the small differences between the traces. Thus, for each of the considered synchronization power levels, the linear least squares fitting of the trace corresponding to the center of the stable phase shift range , is calculated to cancel out this slope. The [Fig. 3(b)], last term of (1) sets the deviation to zero at the center frequency . also increases As shown in Fig. 4(b), the phase deviation its variation, both with frequency and with the operation point , for higher synchronization power levels. selected The impact of the frequency performance of the circuit on the present application is studied in the next section. III. POLARIZATION TUNING PERFORMANCE OF THE ACTIVE ANTENNA The polarization tuning capability is attained by introducing a relative phase shift between the signals present at the output ports of the antenna. According to the antenna layout outlined in Fig. 2(a), the component corresponding to the fundamental mode polarized along the axis, , undergoes a 90 degree phase delay with respect to the other, , in order to obtain

(2) The axial ratio is defined as the relationship between the major and the minor axes of the polarization ellipse of the plane wave propagating along the direction, that results in maximum available power at the antenna global output [16]. When both components are combined in phase , the axial ratio drops to zero, corresponding to right hand circular polarization (RHCP). The axial ratio is likewise zero for , obtaining left hand circular polarization (LHCP). For and , two linear polarizations ( dB) are achieved in the and directions respectively (hereafter referred to as and ). The behavior of the complete active antenna topology, as shown in Fig. 1(a), has been simulated for three different values of the synchronization power . The control voltage of , has been swept throughout its stable range [Fig. 3(b)], while keeping working at the center point, which replicates the theoretical case studied above. In Fig. 6(a), the evolution of the axial ratio as a function of the control voltage , has been represented. As the phase shift produced by the IL3HSOM circuits is a monotonically increasing function of the control voltage [Fig. 3(b)], the peaks and minima in this case represent the same polarization states indicated in Fig. 5. For each of the synchronization power levels considered, the performance at six frequency points, uniformly distributed throughout the input band ( GHz, ), have been displayed. Both circular polarizations are successfully obtained ( dB), for all the synchronization power levels and at all the frequency points considered. With linear polarizations, on the other hand, due to their higher sensitivity to the phase shift (Fig. 5), a somewhat more frequency selective response is observed. This is especially apparent for dBm, where both linear polarizations ( dB) are only achieved at some frequencies. For the two lower synchronization power levels (

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Fig. 7. Measurement setup in the anechoic chamber.

IV. EXPERIMENTAL RESULTS

Fig. 6. (a) Variation of the axial ratio at the center of the input frequency : GHz), as the control voltage V is swept, while keeping band (f HSOM working at the center point. For each synchronization power level, the behavior at six points of the input frequency band is displayed. (b) Variation of the axial ratio over frequency for the working points specified in the legend.

= 11 25

and dBm), LP is stable for all the considered frequency values, whereas for , the axial ratio drops below the 30 dB threshold at some point. The behavior for dBm has been magnified in the first inset of Fig. 6(a), showing peaks at two successive values, which take place at different frequency points (Fig. 6(b)). This suggests the existence of an intermediate working point between those two, providing higher axial ratio levels, together with a better frequency performance. The variation of the axial ratio over frequency is shown in Fig. 6(b). Since the circular polarizations have been found to be stable in the input band [Fig. 6(a)], only the linear ones, corresponding to the working points specified in the legend, have been represented. For dBm, the behavior at the two successive values corresponding to have been included, showing linear polarization in two different frequency ranges. The most limited frequency performance is observed for dBm. The polarization is maintained along the input band for the two lower synchronization power levels, whereas for , the axial ratio drops below 30 dB at some point within this band. In conclusion, the performance of the IL3HSOM based phase shifter circuit is more stable over frequency for lower values of the synchronization power, as discussed in Section II.C, leading to potentially wider polarization bandwidths. Nevertheless, since the phase shift can be varied within a continuous range, the working point selected also has an important impact on the frequency performance, especially for linear polarizations.

In order to validate the simulated results, the proposed polarization agile active antenna has been manufactured and assembled following the topology of Fig. 1(a). Due to the fact that the signal received by the antenna is internally downconverted, obtaining its output at intermediate frequency, the anechoic chamber measurement setup shown in Fig. 7, was needed to evaluate the global performance of the system (polarization tuning capabilities, radiation patterns, ). A horn antenna with high polarization purity, fed by an external signal generator (SG), is used to transmit a tone in the input frequency band GHz. Another external signal generator provides the synchronization signal at frequency to the active antenna, which is mounted on the azimuthal positioner. The output of the system at intermediate frequency is measured by a vector network analyzer, which is triggered by the positioning control system. The sampled IF output of each IL3HSOM circuit is separately monitored by an Agilent 89600 Vector Signal Analyzer (N8201A–N8221A). All the measurement and signal generation equipment is synchronized through a 10 MHz pilot signal, which provides a common frequency reference. The polarization state of the active antenna is evaluated by registering the relative power received at the IF output, while the transmitting horn is rotated about its longitudinal axis. This procedure is repeated for different operation points as the control voltage is swept throughout the synchronization range. Since simulations showed a better performance for lower values of the synchronization power, these measurements have been carried out for and dBm. For the center frequency of the input band ( GHz), the relative IF power measured at the output of the active antenna is shown in Fig. 8, as a function of the control voltage and the polarization angle of the transmitting horn. For the sake of clarity, only a representative group of the measurements carried out has been represented in this figure. The axial ratio, measured at the center frequency of the input band, as a function of the control voltage is shown in Fig. 9(a) for and dBm. A small deviation on the performance of the manufactured HSOM circuits has been found, producing stronger variations of the conversion gain with the control voltage, especially at the upper end of the synchronization range. Although, as a consequence of this deviation,

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= 035

= 040 LP0 LP+

Fig. 9. Measurement of the axial ratio for P and P dBm. (a) As a function of the control voltage V at the center of the input frequency : GHz). (b) As a function of frequency for and band (f (upper subfigure), and for RHCP and LHCP (lower subfigure).

= 11 25

Fig. 8. Relative IF power measured at the output of the active antenna at the center frequency of the input band (f : GHz), as a function of the dBm. control voltage and the polarization angle of the probe for: (a) P (b) P dBm.

= 11 25

= 040

= 035

the second left hand circular polarization observed in simulation (corresponding to the higher value of the control signal), is not reached in measurement, all the desired polarizations have been successfully attained. Furthermore, note that, for this experimental setup the second HSOM circuit has been kept at a fixed working point and therefore, a double phase shift tuning range is available. The variation of the axial ratio over frequency is represented in Fig. 9(b) for the two orthogonal linear polarizations ( and ), and for RHCP and LHCP. Unlike in the simulation, the values of the control signal have been carefully selected to produce the best possible frequency performance. Both linear polarizations are maintained over almost the whole input frequency band for both synchronization values, although a slightly wider bandwidth is observed for dBm. The stronger variation in the conversion gain of the practical circuits mentioned earlier, has a greater impact on the circular polarizations, which present a poorer frequency performance. The polarization bandwidth is reduced with respect to the simulated results, although for dBm, the axial ratio remains below the 3 dB threshold throughout a substantial part of the band.

= 11 25

Fig. 10. Radiation pattern of the active antenna at f : GHz, for and RHCP and for two synchronization power levels, measured with the polarization of the transmitting oriented in the ' direction.

LP+

= 45

The radiation pattern of the active antenna at GHz, has been measured by rotating it about the azimuthal angle , while the polarization of the transmitting horn is oriented in the direction. The results are displayed in Fig. 10 for two different synchronization power levels. V. CONCLUSION A receiving polarization agile active antenna based on injection locked third harmonic self oscillating mixers has been presented. The multifunctional IL3HSOM circuits have proven to be an interesting phase shifting solution, providing efficient integrated signal downconversion, which enables the implementation of compact, low cost and low power consuming receiving systems. The proposed topology enables the polarization variation in a continuous range, comprising two orthogonal linear polarizations along with left hand and right hand circular polarization. The frequency performance of the complete system has

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been studied, observing a better behavior for lower synchronization power levels. The simulated results have been validated by measurements of the manufactured prototype, obtaining a good agreement. REFERENCES [1] S. Loredo, B. Manteca, and R. Torres, “Polarization diversity in indoor scenarios: An experimental study at 1.8 and 2.5 GHz,” in Proc. 13th IEEE Int. Symp. on Personal, Indoor and Mobile Radio Communications, Sep. 2002, vol. 2, pp. 896–900, Vol. 2. [2] A. Turkmani, A. Arowojolu, P. Jefford, and C. Kellett, “An experimental evaluation of the performance of two-branch space and polarization diversity schemes at 1800 MHz,” IEEE Trans. Veh. Technol., vol. 44, no. 2, pp. 318–326, May 1995. [3] F. Lotse, J.-E. Berg, U. Forssen, and P. Idahl, “Base station polarization diversity reception in macrocellular systems at 1800 mhz,” in Proc. IEEE 46th Veh. Technol. Conf. Mobile Technol. for the Human Race, Apr./May 1996, vol. 3, pp. 1643–1646, Vol. 3. [4] R.-H. Chen and J.-S. Row, “Single-fed microstrip patch antenna with switchable polarization,” IEEE Trans. Antennas Propag., vol. 56, no. 4, pp. 922–926, Apr. 2008. [5] Y. Sung, T. Jang, and Y.-S. Kim, “A reconfigurable microstrip antenna for switchable polarization,” IEEE Microw. Wireless Compon. Lett., vol. 14, no. 11, pp. 534–536, Nov. 2004. [6] S.-H. Hsu and K. Chang, “A novel reconfigurable microstrip antenna with switchable circular polarization,” IEEE Microw. Wireless Compon. Lett., vol. 6, pp. 160–162, 2007. [7] P. Haskins and J. Dahele, “Varactor-diode loaded passive polarizationagile patch antenna,” Electron. Lett., vol. 30, no. 13, pp. 1074–1075, Jun. 1994. [8] P. Hall, I. Morrow, P. Haskins, and J. Dahele, “Phase control in injection locked microstrip active antennas,” in IEEE MTT-S Int. Microw. Symp. Digest, May 1994, vol. 2, pp. 1227–1230. [9] C. Vázquez, S. Ver Hoeye, M. Fernández, L. Herrán, G. León, and F. Las Heras, “Transmitting polarisation agile antenna based on synchronised oscillators,” in Proc. IEEE Antennas Propag. Society Int. Symp. AP-S, Jul. 2009, pp. 1–4. [10] S.-C. Yen and T.-H. Chu, “A beam-scanning and polarization-agile antenna array using mutually coupled oscillating doublers,” IEEE Trans. Antennas Propag., vol. 53, no. 12, pp. 4051–4057, Dec. 2005. [11] S. Ver Hoeye, A. Suárez, and S. Sancho, “Analysis of noise effects on the nonlinear dynamics of synchronized oscillators,” IEEE Microw. Wireless Compon. Lett., vol. 11, no. 9, pp. 376–378, Sep. 2001. [12] S. Ver Hoeye, L. F. Herrán, M. Fernández, and F. Las Heras, “Design and analysis of a microwave large-range variable phase-shifter based on an injection-locked harmonic self-oscillating mixer,” IEEE Microw. Wireless Compon. Lett., vol. 16, no. 6, pp. 342–344, Jun. 2006. [13] S. Ver Hoeye, L. Zurdo, and A. Suárez, “New nonlinear design tools for self-oscillating mixers,” IEEE Microw. Wireless Compon. Lett., vol. 11, no. 8, pp. 337–339, Aug. 2001. [14] L. F. Herrán, S. Ver Hoeye, and F. Las Heras, “Nonlinear optimization tools for the design of microwave high-conversion gain harmonic selfoscillating mixers,” IEEE Microw. Wireless Compon. Lett., vol. 16, no. 1, pp. 16–18, Jan. 2006. [15] M. Fernandez, S. Ver Hoeye, L. Herran, and F. Las Heras, “Nonlinear optimization of wide-band harmonic self-oscillating mixers,” IEEE Microw. Wireless Compon. Lett., vol. 18, no. 5, pp. 347–349, May 2008. [16] C. Balanis, Antenna Theory, Analysis and Design.. New York: Wiley, 1982.

Carlos Vázquez Antuña received the M.Sc. degree in telecommunication engineering from the University of Oviedo, Gijón, Spain, in 2007, where he is currently working toward the Ph.D. degree. Since 2007, he has been working as a Research Assistant with the Signal Theory and Communications Area, University of Oviedo. His research efforts mainly focus on nonlinear analysis and the optimization of microwave circuits to be used in active antennas.

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Samuel Ver Hoeye (M’05) received the M.Sc. degree in electronic engineering from the University of Gent, Gent, Belgium, in 1999, and the Ph.D. degree from the University of Cantabria, Santander, Spain, in 2002. He is currently Associate Professor with the Department of Electrical and Electronic Engineering, University of Oviedo, Gijón, Spain. His main research is focused on nonlinear analysis and the optimization of microwave circuits and their application to active antennas.

Miguel Fernández García received the M.Sc. degree in telecommunication engineering from the University of Oviedo, Gijón, Spain, in 2006, a, where he is currently working toward the Ph.D. degree. From 2005 to 2008, he worked as a Research Assistant with the Signal Theory and Communications Area, University of Oviedo, where he is currently an Assistant Professor. His main interest are focused on nonlinear analysis and the optimization of microwave circuits to be used in active and phased antenna arrays.

Germán León Fernández was born in Alcázar de San Juan, Spain, in 1975. He received the Bachelor’s degree and the Ph.D. degree in physical science both from the University of Seville, Spain, in 1998 and in 2005, respectively. In 2005, he joined the Department of Electrical Engineering, University of Oviedo, Spain, as an Associate Professor. His research interests focus on antenna measurements, characterization and planar antennas design.

Luis Fernando Herrán Ontañón received the Telecommunication Engineering degree from the University of Cantabria, Santander, in 1999, and the Ph.D. degree from the University of Oviedo, Gijón, Spain, in 2007. In 2003, he joined the Department of Electrical Engineering, University of Oviedo. His research interest includes nonlinear optimization of phased array antennas and the design of RF/microwave subsystems.

Fernando Las Heras Andrés (M’86–SM’08) received the M.S. degree in 1987 and the Ph.D. degree in 1990, both in telecommunication engineering, from the Universidad Politécnica de Madrid (UPM), Madrid, Spain. From 1988 to 1990, he was a National Graduate Research Fellow. From 1991 to 2000, he was an Associate Professor with the Department of Signals, Systems and Radiocommunications, UPM. From 2001 to 2003, he was an Associate Professor with the Department of Electrical Engineering, University of Oviedo, Gijón, Spain, pioneering the Area of Theory of Signal and Communications. Since December 2003, he has been a Full Professor at the University of Oviedo. His main research interests include the analysis and design of antennas, electromagnetic interference (EMI), and the inverse electromagnetic problem with application to diagnostic, measurement and synthesis of antennas.

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Singly and Dual Polarized Convoluted Frequency Selective Structures Benito Sanz-Izquierdo, Edward A. “Ted” Parker, Jean-Baptiste Robertson, and John C. Batchelor, Senior Member, IEEE

Abstract—Convoluting the elements of frequency selective surfaces produces resonating structures with very small unit cell dimensions. This feature is attractive when the FSS is to be used at low frequencies, mounted on a curved surface, or when placed in the proximity of compact radiators. The characteristics of single and dual polarized convoluted FSS are analyzed and measured. The development of novel convoluted elements derived from the square loop slot is traced and their performance is examined. A novel technique of interweaving convoluted loops allows for further cell size reduction, while increasing the passband width, introducing flexibility in wideband FSS design, particularly for tailoring the Electromagnetic Architecture of buildings, and mobile communications in the built environment. Simulated transmission responses of the convoluted structures are in good agreement with the measurements. Index Terms—Built environment, convoluted elements, electromagnetic architecture, fractals, frequency selective surfaces.

I. INTRODUCTION

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HERE is an increasing interest in applying frequency selective surface to wireless communications systems with the purpose of controlling the electromagnetic architecture of buildings (EAoB) [1]–[4]. One problem that FSS encounter when applied to buildings is that the wavelengths in the bands used by most of the mobile, radio and wireless technologies employed in indoor communications are not insignificant when compared with the size of an ordinary office room. For example, the 400 MHz band employed for general mobile radio systems (GMRS) in the USA and personal mobile radio systems (PMR446) and the emergency TETRA in Europe has a corresponding wavelength of 750 mm, only 4 times smaller than the average floor-ceiling height in a building. The word convoluted was introduced in [5] as a general term to describe geometries of unit cells in FSS arrays where complex conductor or slot structures twist, turn and in some cases interweave. The main aim in that paper was to make use of their space filling properties to reduce very significantly the element size required for a fixed resonant frequency. Many such curves, though not all [6]–[9], are fractal. Fractal designs have been described specifically for FSS elements [5], [10], [11], but recent Manuscript received April 08, 2009; revised July 30, 2009. First published December 31, 2009; current version published March 03, 2010. This work was supported in part by the UK Engineering and Physical Sciences Research Council and in part by the National Policing Improvement Agency. The authors are with the Department of Electrical Engineering, University of Kent, Canterbury CT2 7NT, U.K. (e-mail: [email protected]; [email protected]; [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2009.2039321

emphasis has focused more on antenna applications, comprehensive reviews being given in [12]–[14]. Convoluting the elements of an FSS could play an important role in the future EAoB as they can considerably reduce the size of the unit cell and perform well when incorporated into curved surfaces. In addition, convoluting FSS elements improves the angular stability of the frequency responses of the surface, moving the operating bands away from the grating region determined by the periodicity of the array [5]. The Hilbert curves are a family of space filling curves that can be produced using simple mathematical formulation [15]. They offer the attractive property of being able to compact an electrically long wire within a very small space. The geometry has been applied to produce compact antennas [16]–[18], frequency selective surfaces [5] and high impedance surfaces (HIP) [19]. In [5], the effectiveness of this process was judged by the figure and the ratio , where is the corresponding of merit free space wavelength, is the lattice periodicity and is the total length of the conductor in the element. More recently, convoluted frequency surfaces [20], [21] have been proposed where the array elements extend beyond their unit cells into neighboring cells, defined as “interweaving” in [21]. Interweaving was initially applied to high impedance surfaces (HIP) in [22] using the convoluted cross dipoles of the type described in [6]. In [22], miniaturization was achieved at the expense of a reduction in the bandwidth of the HIP. This is in contrast with the bandwidth enhancement properties of these configurations when employed as FSS [20], [21]. This paper studies the miniaturization of the unit cells of frequency selective surfaces and their transmission responses. The first section adds a further iteration of the Hilbert geometry to the sequence previously published in [5]. Note, though, that in this paper, the elements are in slot form, to give bandpass transmission responses. Later sections look at novel developments of convoluted loops, and interweaving. The paper ends with a case study of the application of interwoven elements. II. HIGHLY CONVOLUTED HILBERT CURVE STRUCTURES FOR UHF APPLICATIONS A. Hilbert Geometry Fig. 1 illustrates the first, second, third and fifth generation of the Hilbert curve. The curves are generated by way of the Lindenmayer system [23] and each generation is composed of segments of length and , where the generation number , and the side of the square are related by

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Fig. 2. Transmission response of 5G Hilbert, E parallel to open side of loop: Black thick line—Measured with bi-conical antenna; Black broken line—Measured with log-periodics; Gray line—Plane wave simulation (CST Microwave Studio). Fig. 1. a) First, b) second, c) third and d) fifth generation Hilbert curves.

Frequency selective surfaces based on the first, third and four generations in [5] were able to reduce sequentially the unit cell size to below 10 percent of the free space wavelength at reso. nance, illustrated by the figure of merit A slot version of the fifth generation [see Fig. 1(d)] has now been developed in order to achieve resonant frequencies below 1 GHz, with higher figure of merit than reported in [5]. The dimensions chosen were , , and , the periodicity of the regular square lattice. The width , the influence of the slot width is of the slot was discussed later. The FSS was etched into a copper clad polyester supporting substrate 0.03 mm thick, with and loss tangent . This material was used throughout the work reported here. Simulated and measured transmission responses for an array of these fifth generation elements are shown in Figs. 2 and 3. Simulations were carried out using the frequency domain solver included in CST Microwave Studio. The very thin substrate employed has only a minor effect on the transmission response [24], [25] while complicating significantly the meshing process in the simulations, leading to increased computing time. For this reason, the thin dielectric layer was not taken into account in the simulations which were regarded only as guide to experimental implementation. B. Measurements As the Hilbert curve begins and ends at adjacent corners, the element is not symmetrical and therefore is singly polarized. The frequency selective slot structure was placed in an aperture of approximately 200 mm 200 mm, surrounded by a high frequency absorbing board of 1.52 m 1.95 m for testing purposes. At the long wavelengths employed here, measured transmission levels are likely to be perturbed by scattered signals [4]. Two sets of independent measurements were therefore carried out and compared.

In the first, two log periodic antennas, the signal source and the receiver, were each placed 1 metre from the centre of the FSS. Transmission levels were calibrated relative to that of the open aperture. Below 1 GHz multipath and leakage problems were significant. So as a trial, the log periodic transmitter were replaced by a broadband biconical dipole antenna placed in close proximity to the array, at a distance of only 5 cm, while the log-periodic receiver remained at 1 m from the FSS. Fig. 2 shows the transmission response when the electric field was aligned parallel to the open side of the element square (Fig. 1(d)). There are resonances at about 1.6 GHz and 2.6 GHz, with measurements and plane wave simulations predicting well the behavior of the FSS. There is very little difference between the two sets of measurements, although the frequencies of maximum transmission are marginally higher than predicted by the simulation. The measured losses are mentioned in Section III-A. The subsequent measurements in this paper were all carried out with the biconical antenna. Fig. 3 shows the transmission curve when the electric field was perpendicular to the open side of the square. Resonances occur at 0.75 GHz and 1.8 GHz, with a very high figure of merit at . The resonant frequency of the first resonance: convoluted elements is relatively stable to angle of incidence [6], [7] and, as is typical for slot element FSS, the width of the resonance tends to decrease in TE as the angle increases while the TM bandwidth broadens. In our case, the lowest resonant frequency drifted by less than 0.5% at TE45 and TM45 while the 10 dB bandwidth varied by about 10%. The crosspolar level in transmission was very low in simulations and comparable with that of a narrow dipole slot. Table I summarizes the characteristics of singly polarized (SP) arrays presented in [5], with the addition of the fifth generation of the Hilbert curve for MHz applications. At the time that [5] was written, we were unable to make measurements at

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Fig. 3. Transmission response, Hilbert 5G, E perpendicular to open side of loop. TABLE I FIGURES OF MERIT FOR A SELECTION OF ARRAY ELEMENTS

Fig. 4. Figure of merit 

=p and bandwidth dependence on slot width.

III. DUAL POLARIZED CONVOLUTED LOOPS A. Design and Measurements

the very long wavelengths where the presence of the low frequency resonance for the fourth generation was suspected to exist, but that problem has been overcome and plane wave measurements have now shown that the prediction was correct. The is 16.7 in Table I. is the wavecorresponding value of length of the second resonance, is the total length of the conductor or slot in the unit cell. C. Influence of Slot Width In common with less complex elements, such as the square loops, the resonance frequency is influenced by the width of the slot/conductor [26]. Fig. 4 illustrates the changes in the figure of merit and the percentage bandwidth as functions of in the fifth generation of the Hilbert curve. the slot width As can be seen from the figure, an increase in width of 0.25 , by 23%, together with an 18% increase mm decreases in the bandwidth measured between the 10 dB points in the transmission response. The results were calculated using the frequency domain solver of CST Microwave Studio and repeated with the time domain solver included in the same software package.

The “four-legged loaded element” as defined in [25] is a frequency selective structure which offers significant advantages such as bipolarization, compactness and angular stability. A convoluted version of this element has been developed and is illustrated in Fig. 5(b), together with its original structure in Fig. 5(a). Each arm has eight stubs on each side. The length , the element periodicity of the cross employed was , the width of the slot in the simulations was , the width of the stubs was , and their periodicity was . In the fabricated FSS the slot widths varied slightly across the array by approximately 0.02 mm. The measured transmission response of a slot array is shown in Fig. 6. It compares well with the plane wave simulation (grey curve) calculated using CST Microwave Studio, but again the latter curve is slightly lower in frequency. There are two clear passbands with transmission peaks at 925 MHz and 2260 MHz with 10 dB fractional bandwidths of 50% and 10% respectively. The insertion losses were approximately 2 dB at the lower band and 4 dB at the (narrower) higher band, consistent with the insertion loss-bandwidth product concept discussed in [27], much the same as reported for dipole slot arrays in Figs. 5 and 6 of [24]. The corresponding figure of merit at 925 MHz was , which is a substantial improvement on that of the element in Fig. 5(a) and the convoluted square previously reported in [5]. In general, these closed loop elements appear to present lower efficiency from the point of view of the total slot length than the open wire structures in Table I, influenced by reactive coupling within and between individual stubs. In the case of the novel convoluted element in . increased by just over 3% at TE45 Fig. 5(b), and decreased by under 2% at TM45 while the 10 dB bandwidth decreased/increased in TE/TM by about 18%. The cross-

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Fig. 5. Four-Legged element (a) and its convoluted equivalent (b).

Fig. 6. Transmission response of the convoluted 4-Legged element.

polarization levels for all the convoluted loops in this paper were similar to those of a square loop. The square loop slot [see Fig. 7(a)] is another element that can be convoluted [5]. As a comparative study, a periodic array of the slot elements in Fig. 7(b), with similar dimensions to , , the convoluted element in Fig. 5(b) ( and ) was simulated, fabricated and measured. Its transmission response had a first resonant frequency at 1 GHz and a second at about 2.75 GHz. In the simulations, the 10 dB widths were 68% and 11% respectively. The insertion losses were just below 2 dB at the lower band and around 5 dB at the higher one. The corresponding and the efficiency of the loop figure of merit is . drifted by about 2% at TE45 and TM45 while the 10 dB bandwidths again varied by about 18%. A real advantage of this structure is that adjacent elements can be interwoven, to modify the transmission response, as described in Section III-C. B. Effect of the Number of Stubs The influence of the number n of stubs present in elements of the form shown in Fig. 7(b) is illustrated in Figs. 9 and 10. The

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Fig. 7. Square loop (a) and its convoluted equivalent (b).

Fig. 8. Transmission response of convoluted square loop [see Fig. 7(b)].

increases steadily with n but slows asymptotparameter ically to about 15 when more than about 10 stubs are inserted. increases sharply up to , increasing gradSimilarly, ually after that. In Fig. 10 the addition of just one stub to the basic square produces a large reduction in the 10 dB width of the passband—the fractional bandwidth decreases by a factor of about 2, subsequently fluctuating between 50% and 60%. C. Interwoven Convoluted Loop Elements The unit cell of a convoluted square loop structure can be interwoven with its neighboring unit cell as shown in Fig. 11(a), generating the array in Fig. 11(b). Essentially, half of the cycle has been extended beyond the unit cell while the other half has been shortened to allow for the extended cycle from the adja, cent cell. An array structure with dimensions ( , and ) similar to the equivalent convoluted square loop slot configuration in Section II [Fig. 7(b)] was fabricated and measured. The element interweaving fraction was 0.85 [see Fig. 11(a), (b)], with 0.0 representing the convoluted square loop in Fig. 7(b). The interweaving fraction is defined on Fig. 11(c) by the location of the ends of the stubs between the extremes 0.0 and 1.0 on the diagram. Note that the gap between the stub ends belonging to

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Fig. 9. Effect of the number of stubs n on 

Fig. 10. Effect of the number of stubs on f

=f

=p and L=

and the

.

Fig. 11. a) two cells of an interwoven square loop, b) close up view of the interweaving method, c) two stubs completely interwoven, and the interweaving fraction scale.

010 dB bandwidth. Fig. 12. Transmission response of the interwoven square loop slot.

the two interwoven elements is preserved. The transmission response (Fig. 12) had peaks near 550 MHz and 2050 MHz. In the simulations, the 10 dB widths were 126% and 6% respectively. The measured insertion loss at the low band was about 1.8 dB, but greater than 10 dB at the narrow 2 GHz passband. Again, the transmission response simulated for plane wave illumination predicted well the behavior of the FSS. There was a 45% reduction in the lower resonant frequency with respect the original convoluted square loop, while the bandwidth increased increased by 80% to by a factor of 1.8. The figure of merit 27 and halved, to 1.14. Note that this is now almost that of the open loop, singly polarized 5th generation Hilbert curve, while this closed interwoven convoluted structure is a bipolarized element. This element is also very stable with regard to the angle of incidence. The frequency drift of was about 1% at TE45 and TM45 whilst the 10 dB bandwidths varied by about 20%.

D. Effect of Interweaving the Loop Fig. 13 illustrates the effect of interweaving on the resonant frequency and the width of the lowest passband. The fractional 10 dB bandwidth increases almost linearly from about 70% for the convoluted square loop in Fig. 7(b) (0% interwoven) to 135% for the fully interwoven element. The resonant frequency decreases by just over 50% and approximately follows a quadratic equation of the form: (2) where x is the percentage bandwidth and frequency in gigahertz.

the first resonant

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peak at around 400 MHz with 10 dB bandwidth extending from nearly 200 MHz to 700 MHz. It is followed by two very narrow band resonant modes, at 1.5 GHz and 2.1 GHz, where the measured insertion losses are 10 dB and 18 dB respectively. Simulations using CST Microwave Studio showed very acceptable angular stability. Between normal incidence, TE45 and TM45 there was no appreciable drift at 400 MHz, although the two narrow passbands drift in frequency between 1.4 GHz and 1.55 GHz, and from 2.1 GHz to 2.3 GHz, consistent with [5]. IV. CONCLUSION AND DISCUSSION

Fig. 13. Effect of interweaving on the resonant frequency and fractional bandwidth.

Fig. 14. Transmission response of the interwoven element FSS (Fig. 11), scaled by a factor of 1.35.

E. An Application of Interwoven Convoluted Loops The dimensions of the interwoven element in Section III-C were scaled by a factor of 1.35 to encompass the 400 MHz emergency band employed for emergency services in Europe, while attenuating the higher section of the radio spectrum. The fractional frequency range allocated to various forms of mobile communications is very wide: the wideband performance of FSS is probably more important here than for higher frequency applications, sometimes imposing constraints over a 10: 1 wavelength range. As pointed out in [28], in the built environment relatively small interference attenuation can result in significant improvements in the system outage probability. A 15 dB increase in the carrier-to-interference ratio can reduce the outage probability by a factor of almost 30, and with an inverse square law approximation, just 10 dB reduces the cell separation required for frequency reuse by a factor of 3, potential enhancements in the efficiency of use of the radio spectrum. Fig. 14 shows the wideband simulated and measured transmission responses. In the simulation, there is a transmission

Singly and dual polarized convoluted frequency selective structures in slot form have been characterized. The unit-cell size needed for operation at a given frequency has been reduced dramatically by using highly convoluted elements. The 5th generation of the Hilbert family of curves adds a new iteration to previous work, allowing for operation below 1 GHz with a cell size of less than 15 mm, but is singly polarized. Its transmission response is influenced by the width of the slot. The dual-polarized designs presented here are based on convoluted loops. As with FSS with simple element geometries, cascading layers is a technique for tailoring the shape of passbands in the transmission responses. The geometry of the convoluted square loop provides a further degree of flexibility in wideband design, enabling adjacent elements to be interwoven. Here, interweaving decreased the resonant frequency by over 50% and increased the 10 dB passband width by over 60%. An FSS specifically designed for mobile communications in the built environment attenuates the mobile and wireless bands between 700 MHz and 3 GHz while passing the general mobile radio systems (GMRS) in the USA, and the personal mobile radio systems (PMR446) and the emergency services TETRA band in Europe. REFERENCES [1] M. Philippakis, C. Martel, D. Kemp, R. Allan, M. Clift, S. Massey, S. Appleton, W. Damerell, C. Burton, and E. A. Parker, “Application of FSS Structures to selective control the propagation of signals into and out of buildings,” Ofcom ref AY4464A, 2003. [2] M. Hook, K. D. Ward, and C. Mias, “Project to demonstrate the ability of frequency selective surface structures to enhanced the spectral efficiency of radio systems when used within buildings,” Ofcom ref AY4462A, 2003. [3] H. H. Sung, “Frequency selective wallpaper for mitigation indoor wireless interference,” Ph.D. dissertation, Univ. Auckland, Auckland, New Zealand, 2006. [4] E. A. Parker, J.-B. Robertson, B. Sanz-Izquierdo, and J. C. Batchelor, “Minimal size FSS for long wavelength operation,” IET Electron. Lett., vol. 44, no. 6, pp. 394–395, Mar. 2008. [5] E. A. Parker and A. N. A. El Sheikh, “Convoluted array elements and reduced size unit cells for frequency-selective surfaces,” Proc. Inst. Elect. Eng. Microw., Antennas Propag., vol. 138, pt. H, pp. 19–22, Feb. 1991. [6] E. A. Parker, A. N. A. El Sheikh, and A. C. Lima, “Convoluted frequency-selective array elements derived from linear and crossed dipoles,” Proc. Inst. Elect. Eng. H, vol. 40, no. 5, pp. 378–380, 1993. [7] E. A. Parker and A. N. A. El Sheikh, “Convoluted dipole array elements,” Inst. Elect. Eng. Electron. Lett., vol. 27, no. 4, pp. 322–323, 1991. [8] D. H. Werner, D. M. Jones, and P. L. Werner, “The electromagnetic fields of elliptical torus knots,” IEEE Trans. Antennas Propag., vol. 49, pp. 980–991, Jun. 2001. [9] Z. Bayraktar, P. L. Werner, and D. H. Werner, “The design of miniature three-element stochastic Yagi-Uda arrays using particle swarm optimisation,” IEEE Antennas Wireless Propag. Lett., vol. 5, pp. 22–26, 2006.

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[10] J. Romeau and Y. Rahmat-Samii, “Dual band FSS with fractal elements,” IET Electron. Lett., vol. 35, no. 9, pp. 702–703, 1999. [11] J. Romeau and Y. Rahmat-Samii, “Fractal FSS: A novel dual-band frequency selective surface,” IEEE Trans. Antennas Propag., vol. 48, pp. 1097–1105, Jul. 2000. [12] D. H. Werner and D. Lee, “A design for dual-polarized multiband frequency selective surfaces using fractal elements,” in Proc. Antennas Propag. Soc. Int. Symp. Dig., Salt Lake City, UT, Jul. 2000, vol. 3, pp. 1692–1695. [13] D. H. Werner, W. Kuhirun, and P. L. Werner, “The Peano-Gosper fractal array,” IEEE Trans. Antennas Propag., vol. 51, pp. 2063–2072, Aug. 2003. [14] D. H. Werner, W. Kuhirun, and P. L. Werner, “Fractile arrays: A new class of tiled arrays with fractal boundaries,” IEEE Trans. Antennas Propag., vol. 52, pp. 2008–2018, Aug. 2004. [15] H. Sagan, Space-Filling Curves. New York: Springer-Verlag, 1994. [16] K. J. Vinoy, K. A. Jose, V. K. Varadan, and V. V. Varadan, “Hilbert curve fractal antenna: A small resonant antenna for VHF/UHF applications,” Microw. Opt. Tech. Lett., vol. 29, no. 4, pp. 215–219, May 2001. [17] S. R. Best, “A comparison of the performance properties of the Hilbert curve fractal and meander line monopole antennas,” Microw. Opt. Tech. Lett., vol. 35, no. 4, pp. 258–262, Nov. 20, 2002. [18] J. Zhu, A. Hoorfar, and N. Engheta, “Bandwidth, cross polarization, and feed-point characteristics of matched Hilbert antennas,” IEEE Antennas Wireless Propag. Lett., vol. 2, no. 1, pp. 2–5, 2003. [19] J. McVay, N. Engheta, and A. Hoorfar, “High impedance metamaterial surfaces using Hilbert-curve inclusions,” IEEE Microw. Wireless Compon. Lett., vol. 14, no. 3, Mar. 2004. [20] S. Barbagallo, A. Monorchio, and G. Manara, “Small periodicity FSS screens with enhanced bandwidth performance,” IET Electron. Lett., vol. 42, no. 7, pp. 382–384, 2006. [21] F. Huang, J. C. Batchelor, and E. A. Parker, “Interwoven convoluted element frequency selective surfaces with wide bandwidths,” IEE Electron. Lett., vol. 42, pp. 788–790, Jul. 14, 2006. [22] S. Tse, B. Sanz-Izquierdo, J. C. Batchelor, and R. J. Langley, “Reduced sized cells for high-impedance ground planes,” IET Electron. Lett., vol. 39, no. 24, pp. 1699–1701, 2003. [23] A. Lindenmayer, “Mathematical model for cellular interaction in development,” J. Theoretical Biol., vol. 18, pp. 280–315, 1968. [24] R. Luebbers and B. Munk, “Some effects of dielectric loading on periodic slot arrays,” IEEE Trans. Antennas Propag., vol. 26, pp. 536–542, Jul. 1978. [25] B. A. Munk, Frequency Selective Surfaces: Theory and Design. New York: Wiley, 2000, ch. 1, 2, pp. 11–42, Appendix E, pp. 393–394. [26] R. J. Langley and E. A. Parker, “Double square frequency selective surfaces and their equivalent circuit,” IET Electron. Lett., vol. 19, pp. 675–677, 1983. [27] P. Callaghan and E. A. Parker, “Loss- bandwidth product for frequency selective surfaces,” Electron. Lett., vol. 28, no. 4, p. 365, 1992. [28] A. H. Wong, M. J. Neve, and K. W. Sowerby, “Performance analysis for indoor wireless systems employing directional antennas in the presence of external interference,” in Proc. IEEE AP-S Int. Symp., Washington, DC, 2005, vol. 1A, pp. 799–802.

Benito Sanz-Izquierdo received the B.Sc. degree from the University of Las Palmas de Gran Canaria, Spain, in 1998 and the M.Sc. and Ph.D. degrees from the University of Kent, Canterbury, U.K., in 2002 and 2007, respectively. He has been working as a Research Associate in the Department of Electronics, University of Kent, since 2003. His research interests are multiband antennas, wearable electronics, RFID antennas, substrate integrated waveguides components, electromagnetic band-gap structures and frequency selective surfaces.

Edward A. “Ted” Parker received the MA degree in physics and the Ph.D. degree in radio astronomy from St. Catharine’s College, Cambridge University, U.K. He established the antennas group in the Electronics Laboratory at the University of Kent. The early work of that group focused on reflector antenna design, later on frequency selective surfaces and patch antennas. He was appointed Reader at the University of Kent in 1977, and, since 1987, has been Professor of radio communications, now Professor Emeritus Prof. Parker is a member of the IET.

Jean-Baptiste Robertson received the Master’s degree in applied mathematics from the Institut des Sciences de l’Ingenieur de Toulon et du Var, France, in 1998. He is currently working toward the Ph.D. degree at the University of Kent, Canterbury, U.K. He is studying truncated frequency selective surfaces under the supervision of Dr. John Batchelor and Prof. Ted Parker.

John C. Batchelor (S’93–M’95–SM’07) received the B.Sc. and Ph.D. degrees from the University of Kent, Canterbury, U.K., in 1991 and 1995, respectively. From 1994 to 1996, he was a Research Assistant with the Electronics Department, University of Kent, and in 1997, became a Lecturer of electronic engineering. His current research interests include printed antennas, compact multiband antennas, electromagnetic bandgap structures, and low-frequency frequency-selective surfaces.

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A Linear Rectangular Dielectric Resonator Antenna Array Fed by Dielectric Image Guide With Low Cross Polarization Asem S. Al-Zoubi, Member, IEEE, Ahmed A. Kishk, Fellow, IEEE, and Allen W. Glisson, Fellow, IEEE

Abstract—Design of a linear array of rectangular dielectric resonator antennas (DRAs) fed by dielectric image guide (DIG) is presented. Coupling between the DIG and the DRAs is predicted using the effective dielectric constant method. In order to achieve a specific power distribution, the power coupled to each DRA is controlled by changing the spacing between the DRAs and the DIG. Cross polarization reduction is achieved by wrapping a conducting strip around the middle of the DRA without affecting the co-polarized radiation pattern. The antenna is fabricated and tested. Good agreement between the measured and computed results is obtained. Index Terms—Dielectric image guide (DIG), dielectric resonator antenna (DRA), linear array, low cross polarization.

I. INTRODUCTION

D

IELECTRIC resonator antennas (DRAs) have been widely used in the microwave and millimeter frequency bands due to their attractive radiation characteristics. They offer several potential advantages such as small size, light weight, high radiation efficiency, wide bandwidth, low loss, and no excitation of surface waves [1]–[4]. Different shapes of DRAs such as cylindrical, hemispherical, elliptical, pyramidal, rectangular, and triangular have been presented in the literature. The rectangular-shaped DRAs offer practical advantages over cylindrical and hemispherical ones in that they are easier to fabricate and have more design flexibility. Different mechanisms for coupling energy to the DRA are used, such as the slot aperture [5], coaxial probe [6], microstrip line [7], and dielectric image guide. A dielectric image guide (DIG) has low losses at higher frequencies, and the DRA can be considered as a truncated DIG. The fields of the fundamental modes of the DRA and DIG are similar. Advantages of the DIG are low loss at high frequencies and ease of coupling energy to the DRAs. Since the DRAs are above a ground plane, the directivity is doubled, and the DIG can be designed to support one mode only. Recently, the aperture coupled dielectric resonator

Manuscript received January 03, 2009; revised August 08, 2009. First published December 28, 2009; current version published March 03, 2010. A. S. Al-Zoubi was with the Department of Electrical Engineering, University of Mississippi, University, MS 38677 USA. He is now with the Department of Communications Engineering, Al-Yarmouk University, Irbid 21163, Jordan (e-mail: [email protected]). A. A. Kishk and A. W. Glisson are with the Department of Electrical Engineering, University of Mississippi, University, MS 38677 USA (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2009.2039294

antenna fed by DIG has been analyzed and designed [8], [9]. Since the slot apertures radiate on both sides, such an antenna has high back radiation. A reflector was used to reduce the back radiation [10]. However, if the DRAs can be coupled to the DIG from the same side we will significantly reduce the back radiation. The array must have a specific phase and amplitude distribution in order to maximize the gain or to reduce the sidelobe levels. Several types of feeding have been used to feed a linear array of DRAs to achieve these objectives, and dielectric image guides (DIGs) can be used efficiently since they have low losses [11]–[14]. Here, a linear DRA array fed by a DIG is presented. The effective dielectric constant (EDC) [15], [16] is used to approximate the coupling between the DIG and the DRAs. A Dolph-Chebyshev amplitude distribution is used to control the sidelobe level of the array radiation pattern. From the amplitude coefficients the separation between the DIG and each DRA is obtained. The cross polarization is reduced using two methods: (a) by inserting a metal sheet at the center of the DRA normal to the propagation direction of the wave in the DIG, or (b) by wrapping a conducting strip around the DRA at the center, which is found to be simple and more practical. In Section II, the configuration of the DIG feed line will be discussed and results for measurements and simulation for the parameters are compared. Coupling theory will be studied in Section III, where the coupling length, power delivered to the DRA, and the required amplitude coefficient will be obtained. In Section IV the simulated results for the parameters and radiation patterns will be presented for 7 and 15-element DRA arrays. The results are verified experimentally for the case of the 7-element DRA array. The cross polarization suppression technique is studied in Section V. II. CONFIGURATION OF THE DIG FEED LINE The dielectric image guide (DIG) is shown in Fig. 1 with , , and . In order to excite the DIG, the DIG is tapered and connected to the rectangular waveguide as shown in Fig. 1. The dimensions used are given in the caption of Fig. 1. The guided wavelength of the DIG is obtained theoretically using the effective dielectric constant method and found to be 11.75 mm. The total length of the DIG is about 21 guided wavelengths, , (the tapered parts are not included in this length). The DIG ends are tapered to achieve smooth transition between the waveguide and the DIG. The geometry is fabricated using two X-band waveguides with wall thickness of 1.27 mm and a copper sheet ground plane of

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Fig. 1. Geometry of the DIG and the transition from rectangular waveguide : ,B : ,L : , and L to DIG with A . (a) DIG excited by a waveguide. (b) Transition from waveguide to : DIG.

24 5 mm

= 22 86 mm

= 10 16 mm

= 27 5 mm

=

thickness 0.5 mm, width 140 mm, and length of 293 mm. The material used for the DIG is the RT/duroid 6010LM. The transmission coefficient and return loss for the DIG side are simulated using HFSS commercial software [17] and shown in Fig. 2. Dielectric and conductor losses are included in the simulation. From the figure it can be seen that the system with the tranlong at 10 GHz has a total insertion sitions and a DIG 21 loss of about 1.43 dB and the reflection coefficient is less than over the entire band. III. COUPLING BETWEEN THE DRA AND THE DIG The effective dielectric constant (EDC) method is used to obtain the coupling between two identical DIGs as shown in Fig. 3. Applying the boundary conditions, the following set of equations is obtained [15]: (1) with

and (2)

with , for odd modes for even modes, where , , , and and are transverse propagation constants inside and outside the guide, respectively. The length needed for complete power transfer from guide A to B in Fig. 4 is (3) The magnitude of the coupling coefficient guides is given by

between the two

(4)

Fig. 2. and (c)

S Parameters of the feeding DIG with the transitions: (a) S S .

, (b)

S

,

For a pair of DIGs with , , and , the propagation constants were calculated for different values of spacing . The relationship between the length and the spacing is shown in Fig. 5.

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Fig. 3. Coupler configuration for (a) odd and (b) even modes. Fig. 7. Power coupled from the DIG to the DRA for different DRA lengths based on (6).

Fig. 4. Coupling section of coupled dielectric image guides.

Fig. 8. DRA array fed by DIG.

Fig. 7 shows the ratio of the power of the DRA to the power of the DIG as a function of the spacing for different DRA lengths. From the figure, as the spacing increases, the coupled power decreases. Also, as the length of the DRA increases, the coupled power increases. The losses at each transition and the insertion and transmission losses for the system are obtained from the simulation using HFSS commercial software. The Dolph-Chebyshev amplitude distribution will be used to control the sidelobe level of the array. Fig. 8 shows the array of DRAs fed by the DIG [18]. The DIG is divided into the same number of unit cells as the number of the DRAs. Therefore, each unit cell is corresponding to a DRA. The length of the cell is equal to the spacing between the elements. Each unit cell couples a fraction of the power ( ) from the DIG to the DRA and dissipates some energy due to conductor and di). The power , which is the fraction of electric losses ( the power coupled from the DIG into the DRA, and the loss can be written as

Fig. 5. Coupling length as a function of the spacing between the DIGs.

Fig. 6. Configuration of the DRA near the DIG.

The power ratio between ports 3 and 1 is (7) (5)

If the second DIG as shown in Fig. 6 (in our case this is a DRA with the same height and width as the DIG) has a length , then the power coupled to the DRA is given by (6)

(8) The formulas for the conductor and dielectric losses for the DIG can be found in [20]. is zero, this means that all the input If the output power power is radiated, which means that the last DRA element absorbs all the remaining power from the DIG, but this is impractical [18]. The required power distribution is calculated from , which is obtained using the required amplitude coefficients

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Fig. 9. Geometry of 7-element array.

TABLE I REQUIRED AMPLITUDE AND POWER DISTRIBUTION FOR 7-ELEMENT ARRAY WITH 40 dB SIDELOBE LEVEL

Dolph-Chebyshev amplitude distribution for a specific sidelobe level, which can be written as

(9)

where is the input power in the DIG and is the remaining power transmitted in the DIG at the end of the DRA array. The next step is to obtain the values of the coupled power . These values will be used to find the spacing between each DRA and the DIG. The equations used to obtain the power can be found in [18]. IV. RESULTS AND DISCUSSION A. 7-Element Array The array is designed to operate at 10 GHz using 7 DRA elements of the same height and width as the DIG. The separation between elements is 23.5 mm. The DRA with , , , and is used. The calculated dielectric and conductor losses for this DIG are obtained using the effective dielectric constant method and found to be . Table I displays the coefficients obtained using a DolphChebyshev amplitude distribution for a sidelobe level of 40 dB, the power , which is the fraction of the power coupled from the DIG into the DRA, and the required spacing between the

Fig. 10. Simulated and measured S parameters comparison for 7-element array (a) S , (b) S , and (c) S .

DRAs and the DIG. It is assumed that . The simulated geometry is shown in Fig. 9. Fig. 10 shows a comparison between the measured and simulated parameters. The transmission coefficients are almost identical, but there is a slight difference in the reflection coefficient at the input and output ports due to imperfect fabrication. It is noticed that the reflection coefficients are less than over the entire bandwidth for both the measured and simulated results. The computed and measured radiation patterns are compared with each other as shown in Fig. 11 and show good agreement. It can be noticed that the cross polarization is very high. It can

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TABLE II MEASURED AND SIMULATED GAINS AND RADIATION EFFICIENCY OF THE 7-DRA ARRAY ANTENNA

Fig. 13. Geometry of the 15-element DRA fed by DIG.

Fig. 11. Radiation patterns for the array with 40 dB sidelobe level at different frequencies. The left hand side for Co-polar patterns and the right hand side for the X-polar patterns.

Fig. 14. Simulated S parameters for the 15-element array. Fig. 12. Radiation patterns of the feed at 10 GHz: (a) Co-polar and (b) X-polar.

also be noticed that a strong field is radiated in the direction of the -axis, which can be related to the direct radiation from the waveguide feed aperture. The radiation due to the feed alone is shown in Fig. 12. The gain of the antenna is measured and compared to the simulated results as shown in Table II where good agreement is observed. The computed and measured gains are the peak values of the co-polarization. The simulated radiation efficiencies are also shown in the Table II. B. 15-Element Array As we gained experience and confidence in the software to analyze the problem and since the radiation from the feed is strong compared to the DRA array, a larger array was designed.

The array is designed to operate at 10 GHz with 15 DRA elements of the same height and width as the DIG. The separation , between elements is 23.5 mm. DRAs with , , and are used. obtained with a DolphTable III shows the coefficients Chebyshev amplitude distribution for a sidelobe level of 40 dB, between the DRAs the power , and the required spacing and the DIG. It is assumed that . The geometry shown in Fig. 13 is simulated using HFSS. The simulated parameters are shown in Fig. 14. It can be seen that the reflection . The co-polar and cross-polar coefficients are below radiation patterns for this array are shown in Fig. 15 for different frequencies. The fields are normalized by the peak value at each frequency separately. It is noticed that the cross polarization at 10 GHz is very high. The simulated gain of the antenna is shown in Table IV. At 10 GHz the gain is 12.31 dBi, while for the

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TABLE III REQUIRED AMPLITUDE AND POWER DISTRIBUTION FOR 15-ELEMENT ARRAY WITH 40 dB SIDELOBE LEVEL

TABLE IV SIMULATED GAIN AND RADIATION EFFICIENCY OF THE 15-ELEMENT ARRAY

0

Fig. 15. Radiation patterns of the 15-element DRA array for 40 dB sidelobe level. The left hand side for co-polar patterns and the right hand side for the x-polar patterns.

7-DRA array it is 7.62 dBi. The simulated radiation efficiencies of the antenna are shown in the Table III. V. CROSS POLARIZATION REDUCTION The cross polarization is radiated due to the DIG feed and the excitation of higher order modes of the DRA. The radiated and the undesired modes of the DRA are the desired modes. The mode couples well to the mode excited in the DIG. The electric fields in the DIG and the DRA above a ground plane are shown in Fig. 16. Two methods are investigated to eliminate the unwanted additional modes by which the cross polarization is reduced without changing the co-polarization patterns. Since the DRA is fed along its entire length by the DIG, inserting a metal sheet at the center of the DRA perpendicular to the propagation direction will not affect the power

Fig. 16. Electric field vectors in the (a) DRA and (b) DIG.

distribution between the two halves of the DRA. By inserting the mode and also metal sheet at the center we eliminate the the mode. The first method to suppress some of the undesired modes and to reduce cross-polarization is by inserting a metal sheet at the center of the DRA perpendicular to the propagation direction (perpendicular to the -axis) as shown in Fig. 17(a) [21].

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Fig. 17. Zoom view on the DIG and coupled DRAs with (a) shorting plates and (b) wrapped with a narrow conducting strip.

Fig. 19. Radiation patterns for the 15-element DRA array with a conducting strip around the DRA at different frequencies. The left hand side for Co-polar patterns and the right hand side for the X-polar patterns. Fig. 18. Radiation patterns for the 7-element DRA array with a conducting strip around the DRA at different frequencies. The left hand side for Co-polar patterns and the right hand side for the X-polar patterns. TABLE V MEASURED AND SIMULATED GAINS OF THE 7-DRA ARRAY ANTENNA WITH CONDUCTING STRIPS AROUND THE DRAS

two halves of the DRA. The second method, which is easier and more practical to implement, is to wrap a narrow conducting strip around the DRA at the center as shown in Fig. 17(b). The cross polarization level is reduced about 20 dB in this case. A. DRAs Wrapped With a Conducting Strip

The reduction in cross polarization is about 25 dB [20]. The results of this case are removed for brevity. This method requires splitting the DRA and gluing the conducting plate between the

Wrapping a conducting strip around the DRA is more practical and easier to implement, but it also suppresses the modes and reduces the cross polarization in a similar way as the shorting conducting plate. The following subsections show the effect of the conducting strip for the two cases discussed earlier. 1) 7-Element Array: Fig. 18 shows the simulated and measured radiation patterns at different frequencies for a 7-element

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TABLE VI SIMULATED GAIN AND RADIATION EFFICIENCY OF THE 15-ELEMENT DRA ARRAY WITH ELEMENTS WRAPPED WITH NARROW CONDUCTING STRIPS

DRA array with DRA elements wrapped by a narrow conducting strip at its center. Compared to Fig. 14, it is noticed that the cross polarization is also reduced by about 17 dB with no changes in the co-polarization, and the reduction in the cross polarization is almost the same as using the shorting conducting plates at the center of the DRAs. The computed and measured gains of the antenna are shown in Table V. The results are in good agreement. It can be noticed that the gain is low for the 7-element array because lots of power is transmitted to the second port, as can be noticed by comparing Fig. 10(c) to Fig. 14. Also, the simulated radiation efficiencies of the antenna are shown in the table. 2) 15-Element Array: For a 15-element DRA array with 40 dB sidelobe level with elements wrapped with narrow conducting strips, Fig. 19 shows the simulated radiation patterns at different frequencies. The cross polarization level is about 20 dB below the co-polarization level for this case. The simulated gain and radiation efficiencies of the antenna array with conducting strips around the DRAs are shown in Table VI. At 10 GHz the gain is 12.46 dBi, while for the 7-DRA array it was 7.61 dBi. This indicates that much less power is transmitted to the second port. VI. CONCLUSION Linear dielectric resonator antenna arrays fed by dielectric image guide were presented. The effective dielectric constant method was used to approximate the coupling between the DIG and the DRA elements. A Dolph-Chebyshev amplitude distribution was used to control the sidelobe level of the array radiation patterns. Linear arrays of 7 and 15 elements of rectangular DRAs were designed and the cross polarization was reduced by inserting a metal sheet at the center of the DRA or by wrapping the DRA at the center by narrow conducting strip. The simulated results for the 7-element arrays were verified experimentally. ACKNOWLEDGMENT The support of the Department of Electrical Engineering and the Center for Applied Electromagnetic Systems Research (CAESR) at the University of Mississippi is acknowledged. REFERENCES [1] A. A. Kishk, “Dielectric resonator antenna, a candidate for radar applications,” in Proc. IEEE Radar Conf., May 2003, pp. 258–264. [2] J. Shin, A. A. Kishk, and A. W. Glisson, “Analysis of rectangular dielectric resonator antennas excited through a slot over a finite ground plane,” in Proc. IEEE AP-S Int. Symp., Jul. 2000, vol. 4, pp. 2076–2079.

[3] I. A. Eshrah, A. A. Kishk, A. B. Yakovlev, and A. W. Glisson, “Theory and implementation of dielectric resonator antenna excited by a waveguide slot,” IEEE Trans. Antennas Propag., vol. 44, no. 53, pp. 483–494, Jan. 2005. [4] K. M. Luk and K. W. Leung, Dielectric Resonator Antennas. London, U.K.: Research Studies Press, 2004. [5] A. A. Kishk, A. Ittipiboon, Y. M. M. Antar, and M. Cuhaci, “Dielectric resonator antenna fed by a slot in the ground plane of a microstrip line,” in Proc. 8th Int. Conf. on Antennas Propag., ICAP’93, Part 1, Apr. 1993, pp. 540–543. [6] G. P. Junker, A. A. Kishk, and A. W. Glisson, “Input impedance of dielectric resonator antennas excited by a coaxial probe,” IEEE Trans. Antennas Propag., vol. 42, no. 7, pp. 960–966, Jul. 1994. [7] R. A. Kranenberg and S. A. Long, “Microstrip transmission line excitation of dielectric resonator antennas,” IEE Electron. Lett., vol. 24, pp. 1156–1157, Sep. 1988. [8] A. Al-Zoubi, A. Kishk, and A. W. Glisson, “Analysis of aperture coupled dielectric resonator antenna fed by dielectric image line,” in Proc. IEEE Antennas Propag. Society Int. Symp., Jul. 2006, pp. 2519–2522. [9] A. S. Al-Zoubi, A. A. Kishk, and A. W. Glisson, “Analysis and design of a rectangular dielectric resonator antenna fed by dielectric image line through narrow slots,” Progr. Electromagn. Res., vol. PIER 77, pp. 379–390, 2007. [10] A. Al-Zoubi, A. Kishk, and A. Glisson, “Slot-aperture-coupled linear dielectric resonator array fed by dielectric image line backed by a reflector,” in Proc. IEEE Antennas Propag. Society Int. Symp., Jul. 2008, pp. 1–4. [11] S. Shindo and T. Itanami, “Low-loss rectangular dielectric image line for millimeter-wave integrated circuits,” IEEE Trans. Antennas Propag., vol. 26, no. 10, pp. 747–751, Oct. 1978. [12] K. Solbach and I. Wolff, “The electromagnetic fields and the phase constants of dielectric image lines,” IEEE Trans. Antennas Propag., vol. 26, no. 4, pp. 266–274, Apr. 1978. [13] F. Farzaneh, P. Guillon, and Y. Garault, “Coupling between a dielectric image guide and a dielectric resonator,” in Proc. IEEE MTT-Digest, 1984, pp. 115–117. [14] M. T. Birand and R. V. Gelsthorpe, “Experimental millimetric array using dielectric radiators fed by means of dielectric waveguides,” Electron. Lett., vol. 17, pp. 633–635, Sep. 1981. [15] R. M. Knox and P. P. Toulios, “Integrated circuits for the millimeter through optical frequency range,” in Proc. Symp. Submillimeter Waves, 1970, pp. 497–516. [16] P. Bhartia and I. J. Bahl, Millimeter-Wave Engineering and Application. New York: Wiley, 1984. [17] HFSS: High Frequency Structure Simulator Based on Finite Element Method11.0.2 ed. Ansoft Corporation, 2007. [18] M. W. Wyville, “Dielectric resonator antenna arrays in the EHF band,” Master’s thesis, Carleton University, Dept. Electron., Ottawa, ON, Canada, 2005. [19] M. W. Wyville, A. Petosa, and J. S. Wight, “DIG feed for DRA arrays,” in Proc. IEEE Antennas Propag. Society Int. Symp., Jul. 2005, pp. 176–179. [20] A. S. Al-Zoubi, “Rectangular dielectric resonator antennas fed by dielectric image guides,” Ph.D. dissertation, Univ. Mississippi, Dept. Elect. Eng., University, 2008. [21] T. K. Tam and R. D. Murch, “Half volume dielectric resonator antenna,” Electron. Lett., vol. 33, no. 23, pp. 1914–1916, Nov. 1997.

Asem S. Al-Zoubi (M’09) received the B.Sc. degree from Eastern Mediterranean University, Cyprus, in 1993, the M.Sc. degree from Jordan University of Science and Technology, in 1998, and the Ph.D. degree from the University of Mississippi, University, in 2008, all in electrical engineering. Currently, he is an Assistant Professor with the Department of Communications Engineering in Al-Yarmouk University, Jordan. From October 1999 to October 2000, he worked as a Laboratory Supervisor at Princess Sumaya University for Technology (PSUT), Jordan. From October 2000 to January 2004, he worked as a Lecturer in the Telecommunication Department, Institute of Science for Telecom and Technology, Riyadh, Saudi Arabia. His current research interests include dielectric resonator antennas and microstrip antennas. Dr. Al-Zoubi is a member of the Sigma Xi Society.

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Ahmed A. Kishk (S’84–M’86–SM’90–F’98) received the B.S. degree in electronic and communication engineering from Cairo University, Cairo, Egypt, in 1977, the B.S. degree in applied mathematics from Ain-Shams University, Cairo, Egypt, in 1980, and the M.Eng. and Ph.D. degrees from the University of Manitoba, Winnipeg, MB, Canada, in 1983 and 1986, respectively. He has been a Professor of electrical engineering at the University of Mississippi, University, since 1995. He has published over 200 journal articles and book

Allen W. Glisson (S’71–M’78–SM’88–F’02) received the B.S., M.S., and Ph.D. degrees in electrical engineering from the University of Mississippi, University, in 1973, 1975, and 1978, respectively. In 1978, he joined the faculty of the University of Mississippi where he is currently Professor and Chair of the Department of Electrical Engineering. His current research interests include the development and application of numerical techniques for treating electromagnetic radiation and scattering problems, and modeling of dielectric resonators and dielectric res-

chapters. Prof. Kishk was an Editor-in-Chief of the ACES Journal from 1998 to 2001 and has been an Editor of the IEEE Antennas and Propagation Magazine since 1993. He received the 1995 and 2006 ACES Journal outstanding paper awards, and the Microwave Theory and Techniques Society Microwave Prize in 2004.

onator antennas. Dr. Glisson is a Fellow of the IEEE, a Fellow of the Applied Computational Electromagnetics Society, and a member of Commission B of the International Union of Radio Science. He received a Best Paper Award from the SUMMA Foundation and twice received a citation for Excellence in Refereeing from the American Geophysical Union. He was a recipient of the 2004 Microwave Prize awarded by the Microwave Theory and Techniques Society and received the 2006 Best Paper Award from the Applied Computational Electromagnetics Society Journal. He was selected as the Outstanding Engineering Faculty Member in 1986, 1996, and 2004. He received a Ralph R. Teetor Educational Award in 1989 and in 2002 he received the Faculty Service Award in the School of Engineering. He served as the Associate Editor for Book Reviews and Abstracts for the IEEE Antennas and Propagation Magazine from 1984 until 2006. He currently serves on the Board of Directors of the Applied Computational Electromagnetics Society, is Treasurer of the society, and is a member of the AP-S IEEE Press Liaison Committee. He has previously served as a member of the IEEE Antennas and Propagation Society Administrative Committee, as the secretary of Commission B of the U.S. National Committee of URSI, as an Associate Editor for Radio Science, as Co-Editor-in-Chief of the Applied Computational Electromagnetics Society Journal, and as the Editor-in-Chief of the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION.

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Beamforming Lens Antenna on a High Resistivity Silicon Wafer for 60 GHz WPAN Woosung Lee, Student Member, IEEE, Jaeheung Kim, Senior Member, IEEE, Choon Sik Cho, Member, IEEE, and Young Joong Yoon, Member, IEEE

Abstract—Wafer-scale beamforming lenses for future IEEE802.15.3c 60 GHz WPAN applications are presented. An on-wafer fabrication is of particular interest because a beamforming lens can be fabricated with sub-circuits in a single process. It means that the beamforming lens system would be compact, reliable, and cost-effective. The Rotman lens and the Rotman lens with antenna arrays were fabricated on a high-resistivity silicon (HRS) wafer in a semiconductor process, which is a preliminary research to check the feasibility of a Rotman lens for a chip scale packaging. In the case of the Rotman lens only, the efficiency is in the range from 50% to 70% depending on which beam port is excited. Assuming that the lens is coupled with ideal isotropic antennas, the synthesized beam patterns from the S-parameters 15 1 0 2 15 2 , shows that the beam directions are 29 3 and 29.5 , and the beam widths are 15.37 , 15.62 , 15.46 , 15.51 , and 15.63 , respectively. In the case of the Rotman lens with antenna array, the patterns were measured by using on-wafer measurement setup. It shows that the beam directions are 26 6 21 8 , 0 , 21.8 , and 26.6 . These results are in good agreement with the calculated results from ray-optic. Thus, it is verified that the lens antenna implemented on a wafer can be feasible for the system-in-package (SiP) and wafer-level package technologies. Index Terms—60 GHz WPAN, lens antenna, Rotman lens, waferscale.

I. INTRODUCTION

R

ECENTLY, many applications are being proposed in the range from millimeter-wave to terahertz frequency thanks to the developments of sources and detectors at these frequencies. The semiconductor process less than 90 nm is encouraging to realize various millimeter-wave and tera-hertz services. There are beneficial aspects such as small size, wide band, and penetration capability which are favorable for the applications of communications and imaging. The wireless personal area network (WPAN) at 60 GHz is in the process of standardization at the working group of IEEE802.15.3c and the standardization is scheduled to be completed in 2009 [1]. The service will be provided in the unlicensed band from 57 GHz to 64 GHz allocated by the Federal Communications Commission Manuscript received May 15, 2009; revised August 24, 2009. First published December 31, 2009; current version published March 03, 2010. This work was supported by the Korea Science and Engineering Foundation (KOSEF) funded by the Korean Government under Grant (R01-2007-000-11294-0). W. Lee, J. Kim, and Y. Joong Yoon are with the Department of Electrical and Electronic Engineering, Yonsei University, Seoul 120-749, Korea (e-mail: [email protected]; [email protected]; [email protected]). Choon Sik Cho is with the School of Electronics, Telecommunication and Computer Engineering, Korea Aerospace University, Gyeonggi-do 412-791, Korea (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2009.2039331

(FCC) in 2001. According to the documents proposed as a part of the standard, multiple beams are required to overcome various line-of-sight (LOS) and non-LOS situations. One of the elegant solutions to generate multiple beams is the Rotman lens because the lens has advantages of low-profile, wide bandwidth, low cost, and simple structure [2]. Since the lens was invented by Rotman and Turner in 1963 [3], many researchers have proposed several types of Rotman lenses. Archer at the Raytheon developed a microstrip type Rotman lens in 1967 [4]. Kim designed a Rotman lens on a dielectric slab to reduce conductor loss [5]. A substrate integrated waveguide was employed to design a Rotman lens [6]. Design and implementation of a three-dimensional stack of Rotman lenses feeding a planar array antenna generating beams with hexagonal grid lattice is presented in [7]. The recent advance of a semiconductor process makes it possible to implement electrical and mechanical structures on a silicon (Si) wafer [8]. There are also reports in which planar circuits and antennas were fabricated on a wafer [9], [10]. Of particular interest are millimeter-wave applications that are 60 GHz WPAN, 77 GHz car radar, and 94 GHz imaging radar because the size of a Rotman lens can be small enough to be accommodated on a wafer [11]. An on-wafer fabrication of a Rotman lens is also of interest because of the potential applications for system-in-package (SiP) and wafer-level packaging implementations. The lens can be fabricated together with sub-circuits such as filters, baluns as well as MEMS switches which are key elements in a beamforming system within a single package [12], [13]. It means that the beamforming lens system can be compact, reliable, and cost-effective. Therefore, to verify its feasibility, the implementation of Rotman lenses on a wafer has been explored in this research, which is aiming for IEEE802.15.3c (60 GHz WPAN) and potentially for 77 GHz radar and 94 GHz imaging as well. II. DESIGNS OF ROTMAN LENSES ON A SI-WAFER A. Design Equations of a Rotman Lens Since a Rotman lens was invented in 1963, the lens equation has been well explained and modified in many publications [3], [14], [15]. Its typical geometry and design parameters are shown in Fig. 1. There are two curves and one straight line. The left curve is the focal arc on which beam ports are supposed to be located. The focal arc is a semi-circle having a center at and can be modified to be slightly elliptical in order to reduce the aberration [15]. The right curve is the array curve on which array ports are supposed to be located. The straight

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Fig. 1. Geometry and design parameters of a Rotman lens.

line is the array plane on which antennas are supposed to be located as array elements. For simplicity, all parameters are normalized by the focal length . The parameters are defined as , . also A designer can calculate the length of the true time delay line and the array port coordinates P from the predeter, the scanmined design parameters that are the focal angle , the ratio of on-axis to off-axis focal length , ning angle the dielectric constant of the substrate at design frequency , . The and the effective dielectric constant of the delay line can be calculated from the following length of the delay line equations. The equations were presented by Kim but there is a was missing on typing error in the equations. We found that the third term in the equation about [5], [14]. After adding to the third term in the equation about , the equations were rearranged in the close form of the original equations presented by Rotman and Turner [3] (1)

Fig. 2. Calculated attenuation in a microstrip line on a wafer at 60 GHz.

research by considering the focal length (or the scaling factor) and the path-length error [14]. B. The Consideration of a Wafer Prior to the design, it is necessary to check the properties of a wafer as a substrate because the lens antenna is supposed to be fabricated on a wafer. The most important factor of a wafer in millimeter-wave applications is resistivity because a low resistivity less than 100 -cm can cause serious attenuation, which makes it difficult to implement a millimeter-wave device on it [16]. Therefore, the attenuation in a wafer was investigated in terms of its resistivity. The loss of a wafer results from dielectric damping and substrate conductivity. Thus, the loss tangent can be defined as (4) where is the conductivity (S/m) of a wafer, is the angular frequency (rad), and are the real and the imaginary part of electric permittivity (F/m), respectively [17]. The tan is the is the loss tangent loss tangent from dielectric loss and tan , and should from substrate conductivity. The values of of be known to estimate the losses. Then, the attenuation a microstrip line on a wafer was calculated from the following equation [18]

where

(5)

Then, with the obtained value of , the array port coordinate can be calculated from the following equations: P (2) (3) The overall size and shape of a Rotman lens can be finalized in an interactive process of the design tool developed in this

where is the guided wavelength (mm). The attenuation calculated with respect to the resistivity at 60 GHz is shown in Fig. 2. It shows that the attenuation linearly decreases as the resistivity increases in low resistivity region. According to the calcula-cm) is about tion, the attenuation of standard silicon ( 4.30 dB/mm, which is too lossy for millimeter-wave passive components to be fabricated on it. As the resistivity increases, the attenuation approaches to a certain fixed value in high resistivity region, and the converged value is determined dominantly of a by the value of tan . Because Afsar showed that tan high-resistivity silicon (HRS) is about 0.003 in millimeter wave

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Fig. 3. Cross section of an HRS.

band [19], it is apparent that the loss is small enough for the lens to be designed in this resistivity region. In Fig. 2, the attenuation in the parallel plate (the lens medium) has not been considered because it is negligibly small compared to that in microstrip lines. As a preliminary research to check the feasibility of a Rotman lens on a wafer, Rotman lenses were realized on a 6 inch HRS wafer to minimize the attenuation and increase its efficiency. The other factor to be considered is thickness of a wafer. In microstrip lines, most of the electromagnetic power propagates within a substrate under the quasi-TEM condition. As thickness of a substrate increases, it could suffer from significant loss which results from the spurious surface wave modes. Therefore, thickness of a wafer should be thin enough to support the of quasi-TEM condition at 60 GHz. The cutoff frequency the dominant quasi-TEM mode is given by [20]

Fig. 4. Layout of the proposed Rotman lens. TABLE I DESIGN PARAMETERS OF THE PROPOSED ROTMAN LENS

(6) where is the thickness in millimeters. The thickness of the HRS wafer in this work was originally 625 m which correat 60 GHz. From (6), the calculated is sponds to about 0.4 47 GHz when is 625 m. This frequency cannot support the desired application that is 60 GHz WPAN. Therefore, the thickness was reduced as thin as 300 m in the chemical-mechanical polishing (CMP) process to increase up to 102 GHz. The cross section of an HRS wafer in this research is shown in Fig. 3. The diameter of the wafer is 150 mm. Its resistivity is . The greater than 10 k -cm and its crystal orientation is thickness is 300 m and a silicon dioxide (SiO ) layer grew up to 1.5 m. The SiO layer helps to suppress surface wave slightly [21]. Gold was deposited up to 0.8 m on the both side of the Si-wafer. C. Design of a Rotman Lens Although the design equations of a Rotman lens are unique to determine its geometry, its final structure is different from designer to designer because of its variable aspects that are size, port design, treatment of termination, and so on. In addition, drawing of the design is time-consuming process as a lens has more ports. A computer design code has been developed for several years to save time in the stage of design. The outputs of the code are geometry and performance factors such as path length error, beam patterns, and lens size. The overall design procedure is interactive from the performance factors to the input parameters such as focal length, focal angle, corresponding beam angle, array size, and so on.

The designed Rotman lens is presented in Fig. 4 and its design parameters are summarized in Table I. It consists of seven array ports, five beam ports, and six dummy ports. In practice, dummy ports should be appropriately terminated not to reflect electromagnetic wave so that the beam ports and the array ports are not influenced by internal reflection from the dummy ports. At lower frequency, it has been treated by connecting 50 loads to the dummy ports. However, this method is not appropriate to the proposed lens because it is assumed to be fabricated on a Si-wafer. Instead of 50 loads, an electromagnetic absorbing layer is used for terminating the dummy ports [22]. As shown in Table I, both of the focal angle and the corresponding scanning angle are 30 and the spacing between antennas is 0.5 at 60 GHz. The diameter of the lens medium is approximately 5.41 mm and the overall size of the lens including lines to the ports is 19.6 mm 20 mm. The lens is fed by 50 microstrip lines whose corresponding width is 0.30 mm. The lens was designed to generate five beams with pointing angle of 30 , 16 , 0 , 16 , and 30 , respectively. Each beam has half power beam width of 15.37 , 14.68 , 14.53 , 14.68 , and 15.37 , respectively.

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Fig. 5. Layout of the Rotman lens with antennas.

Because the lens is supposed to be measured in a probe measurement system, a transition from microstrip to coplanar waveguide was designed for ground-signal-ground (GSG) probing contact at the ports as shown in Fig. 4 [23]. The transition was designed to operate in the band of 60 GHz WPAN (57 GHz–64 GHz) with insertion loss less than 0.9 dB as a result of the simulation in a method of moment based tool. In order to verify the performance of the transition pad, three different back-to-back structures that have different length of a microstrip line were simulated and measured. From the simulation results, it has been verified that all the three structures show the return losses more than 20 dB and insertion losses less than 2 dB in 60 GHz WPAN band. This means that the probing pad has good transition performance with insertion loss less than 1 dB loss because the back-to-back structures consist of two probing pads and a microstrip line. However, from the measurement results, it was verified that a frequency shift was occurred, thus, the probing pad works well between 58 GHz and 65 GHz. D. Design of a Rotman Lens With Microstrip Patch Antennas The layout of the lens with antennas is presented in Fig. 5. It consists of three parts: lens body, meander lines, and antennas. The geometry of the lens body is identical with the Rotman lens shown in Fig. 4. The lines between the lens body and the patch antennas were meandered to satisfy the constrained length of the delay line . The lines were designed to have constrained lengths calculated from ray-optic and then the lengths were slightly changed to tune the phase distribution condition with a finite element method based simulation tool. A microstrip patch antenna as a radiating element was designed from the basic design approach [24]. A quarter-wave transformer is required to match the patch antenna (220 ) to a 50 microstrip line. The dimension of the patch is width of 0.9 mm and length of 0.53 mm, and the spacing between the patch antennas is 0.5 . Around the patch antenna, periodic air holes are fabricated to suppress surface wave by perturbing the interface between air and silicon [25]. To verify the effects of air holes, the simulated radiation patterns of the patch antenna with and without air holes are plotted in Fig. 6. These results show that the radiation

Fig. 6. Simulated radiation patterns (H-field pattern) of the element patch antennas with and without air holes.

along the substrate surface is reduced and the antenna gain is increased slightly as a result. The simulation result shows that the antenna resonates at 60 GHz and has a gain of 5.57 dBi. The overall size of the lens with antennas is 30 mm 20 mm. The proposed lens antenna can be also designed on other substrate such as low temperature cofired ceramics (LTCC), fused silica, thin film benzocyclobutene (BCB), and alumina with microstrip structure. In the case of LTCC, the shrinkage after cofiring typically around 10 % should be considered. III. FABRICATION AND MEASUREMENT A. Fabrication Process The proposed lens antenna was fabricated on an HRS in the semiconductor process. The procedure of the fabrication includes polishing, oxidization, gold-metallization, wet-etching, and lithography. The 625 m thick HRS was thinned to be 300 m in polishing process. The HRS was oxidized to grow SiO layer of 1.5 m on the upper layer. Then, gold was deposited up to 0.8 m on both sides of the HRS in chemical vapor deposition process. The chrome masks were prepared for etching and lithography. The upper layer of gold was patterned in wet etching process. The photolithography process was followed to generate air holes around the antennas. The fabricated Rotman lens and the Rotman lens with antennas are shown in Fig. 7. B. Measurement of the Rotman Lens The measurement has been performed in the frequency range from 55 GHz to 65 GHz which is wide enough to cover the 60 GHz WPAN from 57 GHz to 64 GHz. The lens was measured in a millimeter-wave probe system that is composed of a vector network analyzer (HP8510C) and two GSG probe tips with pitch of 150 m. The calibration has been performed by using a TRL calibration substrate. As shown in Fig. 8, an absorbing layer (ECCOSORB\textregistered MFS FGM-40/SS6M) was used for terminating the dummy ports. To see how well the absorbing layer attenuates, transmission coefficient of a 50

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Fig. 7. Fabricated lenses. (a) Rotman lens and (b) Rotman lens with antennas. Fig. 9. The measured transmission coefficients of the Rotman lens from beam port 3. (a) Magnitude and (b) phase.

Fig. 8. Measurement setup of the Rotman lens.

microstrip line on the HRS was measured with and without the absorbing layer. The measurement was repeated for three different lines having different lengths. From the results, it is found that the absorbing layer attenuates electromagnetic power about 1 dB/mm. Because the lines extended from the lens to the ports are longer than 5 mm, the absorbing layer can absorb power more than 10 dB for round-trip. Thus, all the ports except two ports under test can be appropriately terminated. There are five beam ports and seven array ports so that the two-port measurement should be repeated for 35 combinations to have all the S-parameters. The absorbing layers were appropriately attached on all lines except two ports under test as shown in Fig. 8. Because all the ports are aligned on linear edge to have proper alignment with probe tips, the lengths of the lines from array ports are different from the constrained length of

delay line . Therefore, the reference plane of the measured transmission coefficient should be shifted by dembedding the length of those lines from the measurement of the microstrip lines having different lengths. From the measurements, it is verified that the return loss at the beam ports is better than 10 dB. These results indicate that most of the power transmits to the lens body with little reflection. As an example, the measured magnitude and phase of the transmission coefficients at beam port 3 are plotted in Fig. 9. The results lower than 58 GHz are not reliable because of the mismatching at the probing pad. In ideal case, the magnitudes of all ports should be the same level but there are fluctuations in the distribution of magnitude depending on the angle between a beam port and an array port. The largest variation in the peak-to-peak . The reason is that the coherent magnitude is 2.9 dB in condition in upper and lower corners can be destructively interfered by the internal reflection at the corners. On the other hand, the phase of all ports shows good agreement with the calculated . Similarly, values. The significant phase error is 5.9 in these distortions became worse when the lens is excited at the outer beam ports. The maximum magnitude variation occurred is 3.8 dB and the maximum phase error is 14.4 in in . The lens efficiencies at all the beam ports were calculated from the measured transmission coefficient. The efficiency can be defined as the ratio of the sum of the output power at all the array ports to the power fed at one of the beam ports. The

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Fig. 10. The measured efficiencies.

Fig. 11. Synthesized beam patterns from the measured S-parameters at 60 GHz.

obtained efficiencies at all beam ports are plotted in Fig. 10. The efficiencies from the measurement of the S-parameters are in the range from 50 % to 70 % depending on which beam port is excited. The efficiency at the beam port 3 (center beam port) is higher than those at the other beam ports, which can be expected from the fact that at the center beam port the power tends to transmit to all the array ports rather than to the dummy ports. The beam pattern was synthesized from the measured transmission coefficients as shown in Fig. 11. The synthesized beam at the th beam port can be computed from the patterns measurement of transmission coefficient as in [26]

Fig. 12. Measurement setup for the lens with antennas. (a) The diagram and (b) the picture.

lens is coupled with ideal antennas, beam directions are , 0.2 , 15.2 , and 29.5 . The half power beam widths are 15.37 , 15.62 , 15.46 , 15.51 , and 15.63 , respectively. The slight beam broadening is observed and caused from phase and magnitude distribution errors in the measured S-parameters. The synthesized beam patterns show that the scan angles and the beam widths are in good agreement with those in the calculated beam patterns. Compared to other works reported before, the proposed lens has small insertion loss and phase error whereas it has the widest scan angle [11], [27], [28]. C. Measurement of the Rotman Lens With Antennas

(7) total number of array ports (or where = scan angle, antennas), wave numbers of free space and the delay spacing between antennas, The length of the lines, The measured transdelay line for the th antenna, and mission coefficient from the array port to the beam port . The term cos might be replaced with antenna element pattern. The patterns are normalized to the maximum directivity of the beam at B3. The solid lines represent measured beam patterns and the dotted lines represent calculated beam patterns from ray-optic. From the measurement results, assumed that the

In addition to the measurement of the lens only, the lens coupled with antennas was measured in a millimeter-wave probe system. In practice, there are difficulties in measuring an antenna on a wafer because it requires a high performance network analyzer, flexible millimeter-wave cables, and a probe station, as well as a particularly designed rotating arm and its fixture to a probe station [29]. Fig. 12 presents the measurement setup for the lens with antenna array. As shown in Fig. 12(a), one of the ports of the network analyzer is connected to one of the beam ports by contacting with a GSG probe, and the other port of the network analyzer is connected to a standard gain horn antenna that is located 100 mm higher from the antenna

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TABLE II BEAM DIRECTIONS AND BEAM WIDTHS

performance. As a summary of this research, the simulated and measured beam directions and beam widths are summarized in Table II. Fig. 13. Measured pattern of the lens antenna at 61.5 GHz.

array. This height is long enough to guarantee the far-field pattern of the proposed lens antenna because the far-field region (15 mm) /(5 mm) begins from the distance of 2D mm. However, this distance does not satisfy the far-field condition of the horn antenna but is in the radiating near-field region. It means that the near-field pattern should be considered when conducting the probe compensation [30]. Therefore, in this work, the pattern of the horn antenna was obtained from simulation in a commercial software (Microwave Studio 2008) for the horn compensation. As the same way as in Fig. 8, the other beam ports and the dummy ports except the beam port under the test are termimm nated with the absorbing layer. The horn can shift from mm, which corresponds approximately from to to . The radiation pattern was measured by moving the horn antenna linearly in terms of two-port S-parameters. In this research, the horn was swept by manually moving the horn from mm to mm and the lens antenna was measured at 13 points with a step of 10 mm. The sweeping path is linear, instead of circular, because of the rigid waveguide connection in the measurement setup. Therefore, it is expected that because of the linear sweeping path the amplitude of pattern affected by the gain pattern of the horn. The receiving power at the lens antenna drops as the horn moves away from the center of the lens antenna. Although the lens and patch antennas were designed to operate at 60 GHz, the pattern at 61.5 GHz is more distinct which might be caused from frequency shift in matching between the lens and the antennas. Therefore, the measured beam patterns at 61.5 GHz were plotted in Fig. 13. The measured patterns are distorted from the calculated patterns. Possible reasons for the distortion include the variation in receiving power due to the linear sweeping path and the reflections from nearby struc, 0 , 16.7 , and tures. The beam directions are 26.6 . The beam widths are obtained only at beam port 2, 3, and 4 due to the limitation of the effective measurement range. The fluctuation in the beam widths might be caused by the reflection and the scattering from the metallic structure of the probing system. Despite of the limitation of the measurement setup, it is verified that the measured beam patterns show distinct beamforming

IV. CONCLUSION This paper presents a Rotman lens and a Rotman lens with an antenna array that were implemented on a wafer at 60 GHz for IEEE802.15.3c (60 GHz WPAN) applications. The lenses were fabricated in a semiconductor process that includes polishing, SiO oxidization, gold metallization, wet-etching, and lithography. Before designing the lens antenna on a wafer, the resistivity and the thickness of a wafer was considered from the attenuation and the cutoff frequency of a microstrip line on a wafer. To verify the performance of the lenses, the lens and the lens with antennas were fabricated on an HRS wafer and measured. The lens efficiency, one of the dominant factors of lens performance, is in the range from 50 % to 70 % depending on which beam port is excited. This means that 50 % to 70 % of the power from the beam port is transmitted to the array ports because the loss from the HRS substrate is negligibly small comparing to the absorption at the dummy ports. Even though the measurement resource and setup are limited, the measurement results show that the overall performance of the proposed lenses is in good agreement with the calculated performance parameters. Therefore, the implementation of the lenses on a wafer will pave the way for future millimeter-wave SiP implementations. ACKNOWLEDGMENT The authors are grateful to Prof. J.-K. Rhee at Dongguk University for allowing the authors to use the millimeter-wave probe system in his Lab, MINT (Millimeter-wave INnovation Technology) research center. REFERENCES [1] , IEEE Standard 802.15.3c [Online]. Available: http://ieee802.org/ [2] R. J. Mailloux, Phased Array Antenna Handbook. Boston, MA: Artech House, 1994, ch. 8. [3] W. Rotman and R. F. Turner, “Wide-angle microwave lens for line source applications,” IEEE Trans. Antennas Propag., vol. AP-11, no. 11, pp. 623–632, Nov. 1963. [4] D. H. Archer and M. J. Maybell, “Rotman lens development history at Raytheon electronic warfare systems 1967–1995,” in Proc. IEEE AP-S Int. Symp., Jul. 2005, vol. 2B, pp. 31–34. [5] J. Kim, C. S. Cho, and F. S. Barnes, “Dielectric slab Rotman lens for micro/millimeter-wave applications,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 8, pp. 2622–2627, Aug. 2005. [6] E. Sbarra, L. Marcaccioli, R. V. Gatti, and R. Sorrentino, “A novel Rotman lens in SIW technology,” in Proc. Eur. Radar Conf., Oct. 2007, pp. 236–239.

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[7] K. K. Chan and S. K. Rao, “Design of a Rotman lens feed network to generate a hexagonal lattice of multiple beams,” IEEE Trans. Antennas Propag., vol. 50, pp. 1099–1108, Aug. 2002. [8] J. Buechler, E. Kasper, P. Russer, and K. M. Strohm, “Silicon high-resistivity-substrate millimeter-wave technology,” IEEE Trans. Microw. Theory Tech., vol. MTT-34, no. 12, pp. 1516–1521, Dec. 1986. [9] A. Babakhani, X. Guan, A. Komijani, A. Natarajan, and A. Hajimiri, “77-GHz phased-array transceiver with on-chip antennas in silicon: Receiver and antennas,” IEEE J. Solid-State Circuits, vol. 41, no. 12, pp. 2795–2806, Dec. 2006. [10] M. R. M. Ahmadi, S. Safavi-Naeini, and L. Zhu, “An efficient CMOS on-chip antenna structure for system in package transceiver applications,” in Proc. IEEE Radio and Wireless Symp., Jan. 2007, pp. 487–490. [11] S. Lee, S. Song, Y. Kim, J. Lee, C. Cheon, K. Seo, and Y. Kwon, “A V-band beam-steering antenna on a thin-film substrate with a flip-chip interconnection,” IEEE Microw. Wireless Compon. Lett., vol. 18, no. 4, pp. 287–289, Apr. 2008. [12] M. Fernandez-Bolanosa, J. Perruisseau-Carrierb, P. Dainesia, and A. M. Ionescua, “RF MEMS capacitive switch on semi-suspended CPW using low-loss high-resistivity silicon substrate,” Microelectron. Eng., vol. 85, pp. 1039–1042, 2008. [13] K. Entesari, K. Obeidat, A. R. Brown, and G. M. Rebeiz, “A 25–75-MHz RF MEMS tunable filter,” IEEE Trans. Microw. Theory Tech., vol. 55, no. 11, pp. 2399–2405, Nov. 2007. [14] J. Kim and F. S. Barnes, “Scaling and focusing of the rotman lens,” in IEEE AP-S Int. Symp. Dig., 2001, pp. 773–776. [15] T. Katagi, “An improved design method of Rotman lens antennas,” IEEE Trans. Antennas Propag., vol. AP-32, no. 5, pp. 524–527, May 1984. [16] C. H. Doan, S. Emami, A. M. Niknejad, and R. W. Brodersen, “Millimeter-wave CMOS design,” IEEE J. Solid-State Circuits, vol. 40, no. 1, pp. 144–155, Jan. 2005. [17] D. M. Pozar, Microwave Engineering, 2nd ed. New York: Wiley, 1998, ch. 1. [18] E. J. Denlinger, “Losses of microstrip lines,” IEEE Trans. Microw. Theory Tech., vol. MTT-28, no. 6, pp. 513–522, Jun. 1980. [19] M. N. Afsar, “Dielectric measurements of millimeter-wave materials,” IEEE Trans. Microw. Theory Tech., vol. MTT-32, no. 12, pp. 1598–1609, Dec. 1984. [20] K. C. Gupta, R. Garg, I. Bahl, and P. Bhartia, Microstrip Lines and Slotlines. Norwood, MA: Artech House, 1996, ch. 2. [21] K. Singh and S. Pal, “CAD analysis of microstrip lines using micromachining techniques,” High Freq. Electron., vol. 6, no. 5, pp. 30–35, May 2007. [22] L. Musa and M. S. Smith, “Microstrip port design and sidewall absorption for printed Rotman lenses,” in Proc. Inst. Elect. Eng., Feb. 1989, vol. 136, pp. 53–58. [23] Y. C. Lee and C. S. Park, “A compact and low-radiation CPW probe pad using CBCPW-to-microstrip transitions for V-band LTCC applications,” IEEE Adv. Packag., vol. 30, no. 3, pp. 566–569, Aug. 2007. [24] C. A. Balanis, Antenna Theory: Analysis and Design, 3rd ed. New York: Wiley, 2005, ch. 14. [25] R. Gonzalo, P. Maagt, and M. Sorolla, “Enhanced patch-antenna performance by suppressing surface waves using photonic-bandgap substrates,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 11, pp. 2131–2138, Nov. 1999. [26] J. Kim, C. S. Park, and S. Min, “TM mode surface wave excited dielectric slab rotman lens,” IEEE Antennas Propag. Lett., vol. 6, pp. 584–587, 2008. [27] I. S. Song, J. Kim, D. Y. Jung, K. C. Eun, J. J. Lee, S. J. Cho, H. Y. Kim, J. Bang, I. Oh, and C. S. Park, “60 GHz Rotman lens and new compact low loss delay line using LTCC technology,” in Proc. IEEE Radio and Wireless Symp., San Diego, 2009, pp. 663–666. [28] J. Schoebel, T. Buck, M. Reimann, M. Ulm, M. Schneider, A. Jourdain, G. J. Carchon, and H. A. C. Tilmans, “Design considerations and technology assessment of phased-array antenna systems with RF MEMS for automotive radar applications,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 6, pp. 1968–1975, Jun. 2005.

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[29] S. Cheng, H. Yousef, and H. Kratz, “79 GHz slot antennas based on substrate integrated waveguides (SIW) in a flexible printed circuit board,” IEEE Trans. Antennas Propag., vol. 57, no. 1, pp. 64–71, Jan. 2009. [30] D. Slater, Near-Field Antenna Measurements. Boston, MA: Artech House, 1991, ch. 3.

Woosung Lee (S’08) received the B.S. and M.S. degrees in electrical and electronic engineering from Yonsei University, Seoul, Korea, in 2005 and 2007, respectively, where he is currently working toward the Ph.D. degree. Since 2005, he has been working as a Research Assistant involved in the projects of millimeter-wave lens antenna and packages at Yonsei University. His research interests include beamforming arrays, small antennas, and millimeter-wave antennas.

Jaeheung Kim (S’98–M’02–SM’07) received the B.S. degree in electronic engineering from Yonsei University, Seoul, Korea, in 1989, and the M.S. and Ph.D. degrees in electrical and computer engineering from the University of Colorado at Boulder, in 1998 and 2002, respectively. From 1992 to 1995, he was with the DACOM Corporation, where he was involved with wireless communication systems. From 2002 to 2006, he was with the Department of Electrical and Electronic Engineering, Kangwon National University, Chuncheon, Korea. From 2006 to 2008, he was with the Information and Communications University, Daejeon, Korea. In 2008, he joined the Department of Electrical and Electronic Engineering, Yonsei University, Seoul, Korea. His research interests include beam-forming arrays, millimeter-wave sensing and imaging, and high-efficiency active circuits.

Choon Sik Cho (S’98–M’99) received the B.S. degree in control and instrumentation engineering from Seoul National University, Korea, in 1987, the M.S. degree in electrical and computer engineering from the University of South Carolina, Durham, in 1995, and the Ph.D. degree in electrical and computer engineering from University of Colorado at Boulder, in 1998. From 1987 to 1992, he was with LG Electronics, working on communication systems. From 1999 to 2003, he was with Pantec&Curitel, where he was principally involved with the development of mobile phones. He joined the School of Electronics, Telecommunication and Computer Engineering, Korea Aerospace University, in 2004. His research interests include the design of RFIC/MMIC especially for power amplifiers, oscillators, LNAs, passive circuits, and the millimeter-wave and THz imaging.

Young Joong Yoon (M’93) received the B.S. and M.S. degrees in electronic engineering from Yonsei University, Seoul, Korea, in 1981 and 1986, respectively, and the Ph.D. degree in electrical engineering from the Georgia Institute of Technology, Atlanta, in 1991. From 1992 to 1993, he was a Senior Researcher with the Electronics and Telecommunications Research Institute (ETRI), Daejeon, Korea. In 1993, he joined the faculty of Yonsei University, where he is currently a Professor with the Department of Electrical and Electronics Engineering. And, currently, he is a Vice President at the Korean Institute of Electromagnetic Engineering & Science (KIEES). His research interests are antennas, RF devices, and radio propagations.

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High Permittivity Dielectric Rod Waveguide as an Antenna Array Element for Millimeter Waves J. Patrik Pousi, Dmitri V. Lioubtchenko, Sergey N. Dudorov, and Antti V. Räisänen, Fellow, IEEE

Abstract—Dielectric rod waveguide antennas of rectangular cross section have a number of advantages over conventional waveguide and horn antennas as an antenna array element. Dielectric rod waveguide antennas have relatively low cost, low losses, a broadband input match and a high packing potential. Additionally the radiation pattern of such antennas is almost frequency independent. In this paper the suitability of Sapphire rod waveguides for an antenna array is studied with simulations and prototype measurements at W band. Strong mutual coupling is observed when the elements are close to each other. Index Terms—Antenna array, dielectric rod waveguide, millimeter wave.

I. INTRODUCTION CTIVE and passive imaging at millimeter wavelengths have gathered a lot of interest among several research groups in recent years. Such imaging systems usually require an efficient antenna array to gather the information from the object. Also other antenna applications in the millimeter wave frequency range may require antennas with high and tunable directivity, e.g., short range communication. Dielectric rod waveguide (DRW) antennas made of relatively high permittivity materials like Sapphire or Silicon require only a small cross-sectional area, 0.5 1.0 mm at W band. Lower permittivity materials require a larger cross section , the cross section for a good matching. For area is 1.1 1.0 0.55 0.50 mm respectively. Thus high permittivity materials have a more dense concentration of the field, which leads to a denser packing of the array and low level of sidelobes. Horizontally tapered rods offer also a low polarization cross-coupling [1]. As it has been shown in previous studies, high permittivity DRW antennas of rectangular cross section have a broadband input matching and the electromagnetic field is concentrated in the rod [2]. This can provide a low mutual coupling between the elements and a high packing potential that is often difficult to achieve in antenna arrays. So far high permittivity dielectric rod waveguide antennas with a metal waveguide feed have been fabricated and measured up to

A

Manuscript received March 21, 2009; revised May 29, 2009. First published December 28, 2009; current version published March 03, 2010. This work was supported in part by the Academy of Finland through the Centre of Excellence program. The work of J. P. Pousi was supported in part by the Finnish Cultural Foundation, the Finnish Society of Electronics Engineers and in part by the Foundation of Emil Aaltone. The authors are with Department of Radio Science and Engineering and SMARAD Centre of Excellence, Helsinki University of Technology (TKK), FI-02015 TKK, Finland (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2009.2039314

Fig. 1. Illustration of a 2

2 2 DRW antenna array block.

150 GHz. The measurements showed also a nearly frequency independent radiation pattern over a large frequency band [3]. Single antenna elements with such a feed could be scalable up to 300 GHz or even higher by the fabrication of the antenna elements with smaller dimensions (0.15 0.30 mm for 220–325 GHz). In general an antenna array enables a larger gain and narrower beam than a single antenna element. By changing the element spacing and increasing the number of elements a wide variety of radiation patterns can be created. However, it has to be remembered that the larger is the number of elements, the more complex will be the element feeding network. Another way to create different radiation patterns is to change the signal phase between the antenna elements. By using electrically controlled phase shifters a very effective beam steering can be achieved. In this paper the suitability of high permittivity DRW an, for antennas, made of Sapphire tenna arrays at W band is analyzed with simulations and prototype measurements. Anisotropy of Sapphire does not cause complications to the design in this case where the waveguide is cut along the axis of Sapphire [4]. II. DRW ANTENNA ARRAYS In the case of the higher permittivity DRW a horn feed is not required and thus valuable space is saved. Fig. 1 illustrates a 2 2 DRW antenna array fed by regular metal waveguides. The radiation pattern of an antenna array is a product of the element pattern and the array factor. The element spacing larger than one wavelength leads to grating lobes. The directivity as a function of the element spacing reaches local maxima about [5], [6]. Grating lobes can be suppressed for example every by the use of a combination of subarrays and an array amplitude tapering as it has been done in [7]. Earlier dielectric rod waveguide antennas have been studied as optimized feed elements for focal plane arrays [8]. Dielectric rods were made of polyethylene and they were fed by

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Fig. 3. Two-element array feed designed with HFSS: metal waveguide splitter with an inductive post. Fig. 2. Simulated E plane radiation pattern of a two element DRW antenna array with =2;  and 3=2 element separations at 90 GHz without mutual coupling.

slotlines as the metal waveguide feeding requires a horn structure in case of low permittivity rods. Using a horn feed would require a lot of space and would lead to large element spacing. III. TWO ELEMENT ARRAY PROTOTYPE HFSS™ was used to simulate vertically ( plane) placed two-element DRW array with different element spacing. The difficulty in modeling the rectangular DRWs is the rapidly changing field near the waveguide corners. Earlier it has been studied by comparing simulations with measurements, that by , referring to the absolute difference between setting two iterations, a very accurate approximation of electromagnetic fields in rectangular DRWs can be obtained with HFSS [4]. The simulation tool allows calculating the pattern of an array by taking the radiation pattern of a single element. Fig. 2 presents the simulated radiation patterns at 90 GHz of two vertically placed Sapphire rods. The patterns obtained this way do not take into account the mutual coupling between the , a narrow main beam, about elements. If the spacing is 40 is obtained, but also high sidelobes appear. With spacing of one wavelength the main beam is about 60 and sidelobes are about 6 dB lower. It was decided to manufacture a prototype array with element spacing of one wavelength. A. Feed System One problem in DRW antenna arrays is the difficulty to design an efficient feed system with the high packing density. In [8] a metal waveguide feed and a slotline feed were considered. There the problem with the waveguide transition was the need of a horn as the rod had a low permittivity and thus the required space of the feed was too large. A slotline feed is studied more in detail in [9] and also in [10], [11]. In the proposed transition in [8] the slotline substrate is inserted in the middle of the DRW. Insertion losses between 0.5 and 1.5 dB were reported over the frequency band of 60–90 GHz. Also the mutual coupling between two adjacent transitions was studied. It was estimated to dB when the distance between the lines be less than is 10 mm. With high permittivity DRWs in W band the slotline feeds would be very difficult to realize mechanically. But as the high

Fig. 4. Simulated H field distribution in the splitter.

permittivity DRW can be well matched to a regular rectangular metal waveguide [2] also this type of DRW feed enables an antenna element separation less than . At 90 GHz mm. As it is not wanted to study separate channels with separate antenna elements, the array feed can be designed as a power splitter, where the power is equally divided among two waveguides. Such a splitter was studied and optimized in detail with HFSS. As the distance between the middle points of the two feeding waveguides is wanted to be only 3.3 mm, the matching of the junction becomes difficult. However, it was seen in simulations that when the corners near the junction are rounded the matching is significantly improved. Further matching improvement can be done with an inductive post [12], [13]. A post causes reflections that cancel out the reflection of the junction. If the amplitude of the second reflection is equal to the reflection of the junction and with an opposite phase, the sum reflection of the entire junction is zero. The shape of the object that causes second reflection is irrelevant, but it should have similar frequency dependence as the junction itself. Cylindrical post is often chosen as it is easy to manufacture. An optimal position and size of the post were studied with simulations. It was found out, that the radius of the post should be 0.25 mm, the height 0.2 mm and the distance from the junction wall 1.77 mm. The structure and schematic of the feed system is presented in Figs. 3, 4 and 5. In Figs. 6–8 the simulated and measured S-parameters are compared. The port 2 is located in the upper arm of the power splitter. From the results it can be concluded that the splitter performance agrees quite well with the simulated performance. The

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Fig. 5. Schematic structure of the feed system. Fig. 8. Simulated and measured S

Fig. 6. Simulated and measured S

of the feed system.

of the feed system. Fig. 9. Simulated mutual coupling with different distance between the elements.

Fig. 7. Simulated and measured S

of the feed system.

resonant frequency in the prototype is slightly shifted up from the desired 90 GHz, but still the dB-bandwidth is about 10 GHz. The power division is not completely equal between the ports, because of the asymmetry caused by the matching post, but near 3 dB power division is obtained, see Figs. 7 and 8. B. Mutual Coupling Mutual coupling is an important parameter in antenna array design. It can distort the adjacent channels in the array and also modify the desired radiation pattern. In simulations some effects of the mutual coupling can be seen by comparing the array radiation pattern calculated from a single antenna to a simulated

two element pattern. According to HFSS simulations, it seems that the mutual coupling is not very significant when the element separation is one wavelength. In Fig. 9 one can observe that due to a small asymmetry in the feed system the main beam . is slightly turned Mutual coupling is further studied by simulating two adjacent 56 mm long sapphire rods with different vertical distances between them. All the rod ends are tapered with 6 mm tapering section and matched to a metal waveguide. The lower rod is matched to ports 1 and 2 and the upper rod to ports 3 and 4 (Fig. 10). A vertical symmetry plane was used to reduce the simulation time. When the lower rod is fed from the port 1 the power cou. This is plotted in the pled to the upper rod can be seen in frequency range of 75–110 GHz with five different distances between the rods in Fig. 11. The results showing the are interesting and show that the coupling is very low when the distance is over 2.3 mm. So for the antenna array operation mutual coupling is not significant. With shorter distances the coupling is stronger and one can see that at some frequencies all the power is coupled to the upper rod. For is dB at example when the distance is 0.5 mm, the 80 GHz. Phenomenon is similar to the cross-talk that can occur in optical fibers, where this complete power transfer has been used in directional couplers [14]. The simulation results were

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Fig. 13. Simulated mutual coupling between two horizontally placed sapphire waveguides.

Fig. 10. Simulation setup for mutual coupling.

Fig. 14. Two element sapphire rod antenna prototype.

Fig. 11. Simulated mutual coupling with different vertical distance between the elements.

As the metal waveguides in the measurements include flanges, they were also added to the simulation setup. They were modelled as perfect electric conductor surfaces. Small deviations between the results are probably due to a slight tilting of the rods in the measurements. Also the small Teflon sheets around the rods in the feeds were not taken into account in the simulation model. In horizontal plane the mutual coupling is assumed to be stronger due to the stronger field concentration. Though, it has to be remembered that if the DRWs are fed with the metal waveguides, in practise the element separation is always over 3 mm if the elements are in parallel. Simulations were made also in this plane and the results are presented in Fig. 13. Cross-talk between the elements and the complete power transfer at some frequencies is clearly visible at the distances less than 1 mm. IV. MEASUREMENT RESULTS

Fig. 12. Comparison between simulated and measured mutual coupling between two vertically placed sapphire waveguides.

verified with measurements (Fig. 12) and the agreement between the simulations and the measurements is relatively good.

A prototype antenna was built by inserting two 20 mm long Sapphire rods into the power splitter. The separation between the rod center points is 3.3 mm. The tapering section in both ends of the rod is 6 mm. Thin Teflon sheets were used to attach the rods in the middle of the metal waveguides. Fig. 14 shows the antenna prototype.

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Fig. 15. Measured S

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of antenna prototype compared with the simulations.

Fig. 16. Simulated and measured E plane radiation pattern of the two-element antenna.

Fig. 18. Measured H plane radiation pattern of the two-element antenna.

2

Fig. 19. Simulated E and H plane radiation patterns of the 2 2 Sapphire rod array.

agrees well with the simulation. In Fig. 17 the plane pattern of the two-element antenna is compared with the measured single element pattern. As expected the single antenna pattern covers plane radiation pattern is the two-element antenna pattern. presented in Fig. 18. Finally the radiation pattern of a 2 2 Sapphire rod waveguide antenna array was simulated. The vertical and horizontal element separations are both 3.3 mm. The results are presented in Fig. 19. One can notice that the main beam shapes nearly coincide in both and planes which can be a useful property in certain applications. Fig. 17. Measured two-element antenna E plane radiation pattern compared with the measured single element antenna radiation pattern.

Reflection coefficient of the antenna was measured with HP 8510 vector network analyzer. The measurement is compared is below with the simulated values in Fig. 15. Measured dB in 82–92.5 GHz. Measurement agrees well with the simulation. and Antenna radiation pattern was measured both in plane by using the millimeter network analyzer and rotating the antenna. Measurement results are compared with the simulations made with HFSS. Measured plane radiation pattern is presented in Fig. 16. The main beam is about 60 and the pattern

V. CONCLUSION In this paper the suitability of Sapphire rod waveguide as an antenna array element for W band was studied with simulations and prototype measurements. Compared to open ended metal waveguides Sapphire waveguides can provide a good and broadband input match, lower mutual coupling and a nearly frequency independent radiation pattern over a large frequency band. Therefore they would be an interesting alternative for densely packed antenna arrays. Such arrays can also provide similar radiation pattern in both and plane. Arrays could be scalable up to 300 GHz by the fabrication of the antenna elements with smaller dimensions. The challenge is the difficulty of feeding them as the metal waveguide feeding network becomes complex with several

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elements. The mutual coupling between the antenna elements was measured and simulated not to be significant in distances used in antenna arrays. At short distances the simulations of the mutual coupling revealed a cross-talk between the elements and a complete power transfer at some frequencies. This was also verified with the measurements. This phenomenon could be used for example in frequency selective power couplers or in monitoring the propagating power in a dielectric rod waveguides. ACKNOWLEDGMENT The power splitter for the array prototype was manufactured and measured in Elmika Co., Vilnius, Lithuania. REFERENCES [1] J. Weinzierl, J. Richter, G. Rehm, and H. Brand, “Simulation and measurement of dielectric antennas at 150 GHz,” in Proc. 29th Eur. Microw. Week, Munich, Germany, Oct. 1999, vol. 2, pp. 185–188. [2] D. Lioubtchenko, S. Dudorov, J. Mallat, J. Tuovinen, and A. V. Räisänen, “Low-loss sapphire waveguide for 75–110 GHz frequency range,” IEEE Microw. Wireless Compon. Lett., vol. 11, no. 6, pp. 252–254, 2001. [3] J. P. Pousi, D. V. Lioubtchenko, S. N. Dudorov, J. A. Mallat, and A. V. Räisänen, “High permittivity dielectric rod waveguide antenna for 110–150 GHz,” presented at the Proc. 1st Eur. Conf. of Antennas Propag. (EuCAP06), Nice, France, Nov. 6–10, 2006, CD-ROM SP-262, paper 347786. [4] S. Dudorov, “Rectangular Dielectric Waveguide and Its Optimal Transition to a Metal Waveguide,” Doctoral Thesis, Radio Laboratory, Helsinki Univ. Technol., Otamedia, Finland, 2002. [5] C. A. Balanis, Antenna Theory: Analysis and Design. New York: Harper Row, 1982. [6] W. L. Stutzmann and G. A. Thiele, Antenna Theory and Design. London, U.K.: Peter Peregrius, 1984. [7] T. Sehm, A. Lehto, and A. V. Räisänen, “A high-gain 58-GHz box-horn array antenna with suppressed grating lobes,” IEEE Trans. Antennas Propag., vol. 47, no. 7, pp. 1125–1130, Jul. 1999. [8] J. Richter and L.-P. Schmidt, “Dielectric rod antennas as optimized feed elements for focal plane arrays,” in Antennas Propag. Society Int. Symp., Jul. 3–8, 2000, pp. 667–670. [9] J. Richter, Y. Yazici, C. Ziegler, and L.-P. Schmidt, “A broadband transition between dielectric and planar waveguides at millimeter wave frequencies,” in Proc. 33rd Eur. Microw. Conf., Munich, Sep. 2003, vol. 3, no. 2003, pp. 947–950. [10] H. Tehrani, M.-Y. Li, and K. Chang, “Broadband microstrip to dielectric image line transitions,” IEEE Microw. Guided Wave Lett., vol. 10, no. 10, pp. 409–411, Oct. 2000. [11] J. Tang and K. Wu, “Co-layered integration and interconnect of planar circuits and nonradiative dielectric (NRD) waveguide,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 4, pp. 519–524, Apr. 2000. [12] T. Sehm, A. Lehto, and A. V. Räisänen, “Matching of a rectangular waveguide T junction with unequal power division,” Microw. Opt. Technol. Lett., vol. 14, no. 3, pp. 141–143, Feb. 1997. [13] J. Hirokawa, K. Sakurai, M. Ando, and N. Goto, “An analysis of a waveguide T junction with an inductive post,” IEEE Trans. Microw. Theory Tech., vol. 39, no. 3, pp. 563–566, 1991. [14] A. W. Snyder, Optical Waveguide Theory. New York: Academic Press, 1983.

J. Patrik Pousi was born in Vantaa, Finland, in August 1976. He received the Master of Science (Tech.) and Licentiate of Science (Tech.) degrees in electrical engineering from Helsinki University of Technology (TKK), Espoo, Finland, in 2003 and 2006, respectively, where he is currently working towards the Doctor of Science (Tech.) degree. Since 2004, he has been a Research Engineer with the Department of Radio Science and Engineering, TKK. His current research interests include active and passive dielectric rod waveguide components for millimeter wavelengths.

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Dmitri V. Lioubtchenko was born in Gorky, Russia, in May 1971. He received the B.S. and M.S. degrees and the Ph.D. degree in applied physics and mathematics from Moscow Institute of Physics and Technology, in 1993, 1994, and 1998, respectively. From 1994 to 1997, he was a Researcher in the Institute of Radio Engineering and Electronics, Russian Academy of Sciences, Moscow. From 1997 to 1998, he was a visiting researcher at the University of Liverpool, U.K. In 1998, he joined the Department of Radio Science and Engineering, Helsinki University of Technology (TKK), Finland, where he is currently an Academy Research Fellow. His research interests and experience cover various topics including investigations of new materials for millimeter wave, microwave and optoelectronic applications particularly, on the development of active and passive dielectric waveguides for the frequency above 100 GHz.

Sergey N. Dudorov was born in the Kirov region, Russia, in May 1975. From September 1992 to June 1998, he studied at the Moscow Institute of Physics and Technology, where he received the Master of Science degree in applied physics and mathematics in June 1998. His thesis topic was “Investigation of dielectric waveguides and devices based on them.” He received the Licentiate degree and the Doctor of Science in Technology degree from the Helsinki University of Technology (TKK), Espoo, Finland, in 2001 and 2002, respectively, and the Candidate of Science degree from the Moscow Institute of Physics and Technology. His dissertation title was “Rectangular dielectric waveguide and its optimal transition to a metal waveguide”. In November 1998, he joined the Department of Radio Science and Engineering, Helsinki University of Technology (TKK), Finland, where he is currently a Postdoctoral Researcher. His research activities are focused on the dielectric property measurements in application to development of new devices for millimeter and microwave applications based on the dielectric waveguides.

Antti V. Räisänen (S’76–M’81–SM’85–F’94) received the Doctor of Science (Tech.) degree in electrical engineering from the Helsinki University of Technology (TKK), Espoo, Finland, in 1981. He was appointed to the Professor Chair of Radio Engineering at TKK in 1989, after holding the same position pro tem in 1985 and 1987–1989. He has held visiting scientist and professor positions at the Five College Radio Astronomy Observatory (FCRAO) and UMass, Amherst (1978–79, 80, 81), at Chalmers University of Technology, Göteborg, Sweden (1983), at the Department of Physics, UC Berkeley (1984–85), at JPL Caltech, Pasadena (1992–93), and at Paris Observatory and University of Paris 6 (2001–02). Currently, he is supervising research in millimeter-wave components, antennas, receivers, microwave measurements, etc. at TKK Dept. Radio Science and Engineering and MilliLab (Millimetre Wave Laboratory of Finland—ESA External Laboratory). He leads the Centre of Smart Radios and Wireless Research (SMARAD) at TKK, which obtained the national status of CoE in Research for 2002–2007 and 2008–2013. Currently he is also Head of TKK Dept. of Radio Science and Engineering. He has authored and coauthored some 400 scientific or technical papers and six books, e.g., Radio Engineering for Wireless Communication and Sensor Applications (Artech House, 2003). Dr. Räisänen is a Fellow of the IEEE since 1994 and Fellow of AMTA since 2008. He has been Conference Chairman of several international microwave and millimeter wave conferences including the European Microwave Conference in 1992. In 1997, he was elected the Vice-Rector of TKK for the period of 1997–2000. He served as an Associate Editor of the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES from 2002 to 2005. He is a member of the Board of Directors of the European Microwave Association (EuMA) for 2006–2008 and 2009–2011. He is also the Chair of the Board of Directors of MilliLab.

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Linear Sparse Array Synthesis With Minimum Number of Sensors Ling Cen, Member, IEEE, Wee Ser, Senior Member, IEEE, Zhu Liang Yu, Member, IEEE, Susanto Rahardja, Senior Member, IEEE, and Wei Cen

Abstract—The number of sensors employed in an array affects the array performance, computational load, and cost. Consequently, the minimization of the number of sensors is of great importance in practice. However, relatively fewer research works have been reported on the later. In this paper, a novel optimization method is proposed to address this issue. In the proposed method, the improved genetic algorithm that has been presented at a conference recently, is used to optimize the weight coefficients and sensor positions of the array. Sensors that contribute the least to the array performance are then removed systematically until the smallest acceptable number of sensors is obtained. Specifically, this paper reports the study on the relationship between the peak sidelobe level and the sensor weights, and uses the later to select the sensors to be removed. Through this approach, the desired beam pattern can be synthesized using the smallest number of sensors efficiently. Numerical results show that the proposed sensor removal method is able to achieve good sidelobe suppression with a smaller number of sensors compared to other existing algorithms. The computational load required by our proposed approach is about one order less than that required by other existing algorithms too. Index Terms—Beam pattern synthesis, genetic algorithms (GAs), linear arrays, peak sidelobe level (PSL), sparse arrays.

I. INTRODUCTION

A

SPARSE array is one where the inter-sensor spacing is larger than the Nyquist limit, i.e., half of the signal wavelength [1]. In order to avoid grating lobes, sparse arrays are usually designed to be unequally-spaced. Unequally-spaced arrays, or aperiodic arrays [2], have been studied for several decades. Compared with equally-spaced arrays, aperiodic arrays have the advantage of having higher spatial resolution and lower sidelobe while using a smaller number of sensors. Using a smaller number of sensors also leads to lower implementation complexity and cost. A sparse aperiodic array can be synthesized based on two main types of optimization objectives. In the first objective, the sidelobe level is minimized by adjusting the sensor positions and/or the weight coefficients. The number of sensors and the

Manuscript received November 18, 2008; revised June 22, 2009. First published December 28, 2009; current version published March 03, 2010. L. Cen and S. Rahardja are with the Institute for Infocomm Research, Singapore 138632, Singapore (e-mail: [email protected]). W. Ser is with Nanyang Technological University, Singapore 639798, Singapore. Z. L. Yu is with the College of Automation Science and Engineering, South China University of Technology, Guangzhou 510641, China. W. Cen is with Elektrotechnik GmbH, 34063 Kassel, Germany. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2009.2039292

spatial aperture are fixed a priori [3]–[9]. In [6], the inter-sensor spacings are optimized initially before the weights are adjusted to synthesize the desired beam pattern. An analytical technique is proposed in [4], [5], in which the inter-sensor spacings for a given array weight distribution are determined by performing a Legendre transformation on the array factor. Since the sensor positions are arranged as an exponential or trigonometric function, determination of the sensor positions is a nonlinear process. Recently, global optimization techniques, such as the simulated annealing (SA) algorithm [1], [7], differential evolution (DE) algorithm [3], and genetic algorithm (GA) [8]–[11], have been proposed for the synthesis of aperiodic linear array. In the second objective, the number of sensors required to produce a desired beam pattern is minimized by adjusting the sensor positions and the weight coefficients [1]. The number of sensors employed in an array affects the array performance, computational load, and cost. Consequently, not only the optimization of the beam patterns, but also the minimization of the number of sensors, is of great practical importance. However, relatively few works have been reported on this approach in the literature. One such technique is the use of a thinned array [12]–[15]. Thinned arrays are designed by selectively removing some sensors in an equally-spaced or periodic array with the aim of creating a desired gain pattern across the aperture. Some optimization algorithms such as the SA and the GA have been successfully employed to remove a certain percentage of the sensors from a fully populated half-wavelength array [12]–[14]. However, the finite set of possible sensor locations restricts the degree of freedom of the optimization process. Even when the array weights are optimized simultaneously with the sensor positions, the improvement in sidelobe suppression is still small [15]. In [16], Leahy and Jeffs presented a Simplex based algorithm. However, as the iteration stops when it reaches a local minimum of the graph formed by the Simplex tableaus and their adjacencies, there is no guarantee of a global optimum solution. A mixed integer linear programming algorithm is proposed in [17]. As is the case in most of the other methods reported, the approaches taken in [16], [17] can only be applied to symmetrical array synthesis. In [1], a SA based approach is introduced to generate the optimum array beam pattern while minimizing both the number of sensors and the spatial aperture. All the algorithm parameters, including the weights, the sensor positions, the number of sensors, and the spatial aperture, are optimized in a single process. The results show an improvement in the array beam pattern synthesis performance over other existing methods for asymmetrical array synthesis. However, there are still rooms to improve their method due to the following reasons.

0018-926X/$26.00 © 2010 IEEE

CEN et al.: LINEAR SPARSE ARRAY SYNTHESIS WITH MINIMUM NUMBER OF SENSORS

where

Fig. 1. Geometry of an aperiodic and asymmetrical linear array with sors.

N sen-

First, the method in [1] perturbs the weight coefficient and position of each sensor in turns. Since the distribution and level of the sidelobes depend on both the weights and sensor positions, it is more possible to achieve minimal sidelobe level when the weights and positions of all sensors are considered simultaneously. Second, as is the case for [16], [17], the inter-sensor spacings are confined to a finite set of candidate locations, which is likely to yield a sub-optimal solution instead. Sparse arrays with randomly spaced sensors have a high degree of freedom in lowering the sidelobe level [4], [5], [9]. Third, obtaining an optimal beam pattern as well as minimizing the number of sensors and spatial aperture is a multi-objective optimization problem. In [1], it is solved by employing a multi-term energy function where each term is assigned a weighting coefficient. Correct tuning of these weights is critical for acceptable optimization performance. These weights may be different for different design parameters, which leads to additional difficulty in the design process and in algorithm implementation. In order to address the problems encountered in realizing the second optimization objective for sparse aperiodic array synthesis, a novel iterative method is proposed in this paper, where the number of sensors is minimized in an iterative manner. The weight coefficients and the sensor positions of all sensors are jointly designed by using an improved genetic algorithm (IGA) [10], [11]. The minimization of the number of sensors is achieved by iteratively removing the sensors that are considered to have less contribution to the array factor. The fitness function (or objective function) used has only one term that aims at minimizing the sidelobe level. By not using several weighted terms, the evaluation of the fitness function is now less complex than that used in [1]. The remaining part of this paper is organized as follows. In Section II, the array synthesis formulation for sparse arrays is briefly introduced. Section III describes briefly the IGA method which will be adapted for use in the proposed iterative optimization method. The proposed iterative optimization method is described in Section IV. In order to illustrate the effectiveness of the proposed method, computer simulation results are given in Section V. Finally, Section VI presents the conclusions of the paper. II. SPARSE ARRAY SYNTHESIS FORMULATION Assume an aperiodic and asymmetrical linear array with sensors as shown in Fig. 1. The array factor AF can be expressed as [2]

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and

is the

wavelength, is the steering angle measured with respect to the axis, is the distance between the first and the th sensors is the inter-sensor spacing measured in wavelength , and between the th and the th sensors. In (1), is the weight coefficient of the th sensor. It is complex and can be expressed , where and are the amplitude and as phase of , respectively. Consequently, the array factor can be expressed as (2) . In order to guarantee the array response at the where target direction , a constraint on array factor as is set in optimization. The look direction constraint ensures the signal-to-noise ratio (SNR) gain of the synthesized arrays. The goal of array synthesis is to design the parameters of the sensor array to produce the desired beam pattern. One desired beam pattern is to have low peak sidelobe level (PSL). As the sensor positions occur as an exponential or trigonometric function, the synthesis of an aperiodic array is a non-linear process. In the following sections, a genetic algorithm based iterative optimization scheme is proposed for the synthesis of an aperiodic sparse array that uses the minimum number of sensors to achieve the desired PSL value. III. IMPROVED GENETIC ALGORITHM (IGA) FOR SIDELOBE SUPPRESSION The improved genetic algorithm (IGA) is described in details in [10]. A summary is included here for completeness. In the IGA, the sensor parameters are encoded and cascaded to form a chromosome which represents a potential solution. The sensor weights and locations are concatenated into different parameter sections. The chromosome is arranged in a multi-section way. A specified number of chromosomes are used to construct a population, which will then evolve through selection, breeding and genetic variation. With the help of such evolutionary process, the parameters of all sensors are jointly synthesized. In the IGA, a crossover process for real-value parameters is used, where three methods of crossover are randomly applied in each generation during the evolutionary process. A linear combination method is then applied to produce the cut-points. Instead of applying the usual stochastic mutation process, a self-supervised mutation is used here. The results from the previous searches are, then used to adjust the direction and step size of subsequent searches in the mutation process. The process described above will converge to a set of optimum weight coefficients and sensor positions for a given number of sensors considered. IV. PROPOSED ITERATIVE OPTIMIZATION SCHEME

(1)

In the proposed iterative optimization method, the improved genetic algorithm (IGA) [10], is used to adjust the weigh coefficients and positions of the sensor array with the aim of minimizing the sidelobe level. The minimization of the number of sensors is then achieved by iteratively removing the

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Fig. 3. Removal of the 3rd sensor from an N -sensor array to form the new array with N sensors (N = N 1).

0

Fig. 2. Relationship between  and a , where  is the removal factor and a is the amplitude of the ith sensor weight.

sensors that contribute the least to the array factor so that the removal of these sensors will lead to the smallest increase in PSL. A. Relationship Between PSL and Sensor Weights Removing sensors with non-zero weight coefficients will inevitably degrade the PSL performance. In order to analyze this effect, let us define a new term, the removal factor, , as the normalized increase in PSL due to the removal of one sensor, i.e., (3) where is the original PSL and is the PSL after the th sensor is removed from the array. Without loss of generality, let us consider a 25-sensor linear sparse array and a sidelobe . Fig. 2 shows the relationship interval of between and , where is the amplitude of the th sensor weight. The figure shows a general ascending trend in the value of as increases (the same trend has been found in other design examples). This implies that sensors with a smaller value are likely to contribute less to the value of PSL. Conseof quently, removing sensors with smaller is likely to lead to a smaller degradation in the PSL value. This is the basis of our iterative removal scheme. B. Proposed Optimization Process The optimization process starts with a relatively large value of . The sensor weights and spacings are jointly optimized (by the IGA) to minimize the PSL. A search objective, i.e., the , is defined according to the design desired PSL value ( specifications. When the PSL objective is reached or when the IGA is considered to have converged, the evolution process of the IGA is terminated. If the PSL of the resultant array is smaller than , i.e., the PSL objective has been reached, will be removed the sensors with amplitudes smaller than from the array. Here is a specified threshold value for and it for all sensors, the sensor will be discussed later. When with the smallest amplitude is removed instead. The removal process is illustrated in Fig. 3. After the removal of sensors, a new run of the IGA is re-started with the reduced number of sensors. If the IGA converges to an unsatisfied solution, i.e., , a compensation process is introduced where

sensors are inserted into the array to avoid over-removal of senwill sors. The compensation process as well as the value of be discussed later. The IGA is, then re-started with the newly inserted sensors. The iterative process is terminated when further reduction of the number of sensors does not produce a satisfactory solution. The major steps of the proposed method are summarized below: Step 1 Initialization Step 1.1 Specify the parameter values of the IGA, i.e., the population pool size, the parameters in the crossover and mutation processes, and the . , Step 1.2 Specify the initial number of sensors, and the amplitude threshold, . Step 1.3 Create an initial population pool. Step 2 Use the IGA to synthesize the array parameters. The detailed process of the IGA can be referred to in [10]. The search process is stopped if any one of the following three conditions is satisfied in the order given: 1) a satisfying solution is found, i.e., the design objective has been reached; 2) the improvement of fitness during successive generations is smaller than an acceptable level; 3) the maximum number of generations is reached. The PSL of the resultant design achieved in this step is . denoted as Step 3 We denote the numbers of sensors in the last two (previous run) and (curconsecutive IGA-runs as rent run). Note that for the first IGA-run, we have . If and , remove the sensors whose amplitudes of weights are smaller than , or remove one sensor with the smallest amplitude value when for all sensors, from the array. If and , increase the number sensors into the array. Conof sensors by inserting sequently, a new array is obtained by either removing or inserting sensors, and the resultant array is then subjected to the same optimization process described in Step 2. Step 4 The whole iteration process is terminated if one of and the following 2 conditions is met: 1) ; 2) and . For the first condition, the results obtained in the current run of the IGA are recorded as the final solution. For the second condition, take the results obtained in the previous run as the final design. C. Choice of Let us now discuss on the choice of . A larger reduces computational time but it may end up removing too many sen-

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sors leading to unacceptable performance. Based on our obis in the range of [ servations, a suitable value of ], where is the upper bound of . In the example shown in the section below, we set the constraint of , on amplitudes. The aim is to constrain the current taper ratio (CTR) [1], [17], a ratio between the maximum and minimum weight coefficient values, so as to avoid possible computational problems caused by unusually can be large weight coefficients. In the case of selected from 0.5–0.8. D. Compensation for Over-Removal of Sensors After removing some sensors, the resultant PSL may become higher than the specified PSL. Some of these sensors may then have to be put back into the array. The number of sensors to be put back is designed to be (4) where rounds down to the nearest integer. Here is the middle point of the difference between the numbers of sensors in the last two consecutive runs. It is based on the idea of the bisecsensors. tion method. A new run will then start with The initial weights and the inter-sensor spacings of the added sensors are arbitrarily selected from a given range of values. This compensation process is aimed at moderating the efon the optimization results. Without the compensafect of tion process, we may need to choose a relatively small value of to avoid over-removal of sensors. This may lead to more iteration loops and thus more computational load. Another aim for the compensation process is to know when to terminate the iterative process. V. COMPUTER SIMULATION RESULTS A. Comparison With Simulated Annealing Based Method In order to make a meaningful performance comparison, the baseline configuration used for performance evaluation in [1], [6]–[8] is employed in our simulation studies too. In [1], [6]–[8], and a the authors synthesized the array with 25 sensors over sidelobe range of , where . An iterative process to reduce the number of sensors used has been presented , and produces in [1]. The process starts with 35 sensors over . a resultant array of 24 sensors with an overall aperture of The PSL obtained (over ) is dB and the dB, denoted as dB, width of the main lobe measured at is 0.0214. In view of the above achievements made by [1], we set the objective of our proposed optimization method to be one that and a PSL attempts to find an array with fewer sensors dB over . Without loss of genlower than , and constrain erality, we set the adjacent sensor spacing to be no less than . The beam is sampled with 1024 pattern in the -region ( points. In the IGA, the population pool comprises 30 individuals. The parameters in the crossover and the mutation process

Fig. 4. Amplitude values, a , of the 25-sensor, 20-sensor and 19-sensor arrays for our proposed algorithm.

are chosen as: (crossover probability) (mutation (decreasing rate [10]) = 0.6, and (minprobability) imal acceptable improvement [10]) = 0.001. The IGA searching process is terminated when 100 generations have been evolved or the fitness improvement is less than 0.001 for 10 consecutive is generations if a satisfying solution cannot be found. The chosen to be 0.8. The computational complexity is measured in terms of the number of fitness functions evaluated, which is a common way of estimating the computational complexity of evolutionary algorithms [18]. Specifically, it involves the computation of the array factor for each when scans from 0 to degrees. If 1024 points are sampled in the interval 0- , we have to compute equation (2) 1024 times for each possible design. The time taken for the other processes incurred in the computation is relatively insignificant and can be ignored. The initial population is formed with a uniform array having , and a PSL of 25 sensors, an inter-sensor spacing of dB. The convergence condition is checked for every 10 generations. The first run of the IGA is terminated after 70 generadB. The amplitude tions when the PSL is reduced to values, , are shown in Fig. 4. As the figure shows, there are 5 sensors whose weight amplitudes are smaller than . These and , 5 sensors are marked in the Figure as respectively. Removing them from the array increases the PSL dB. The second run of the optimization process is to started and a PSL of dB is obtained after 10 generations. The result is shown in Fig. 4 too. Since all the weight amplitudes are larger than , the sensor with the smallest weight , is removed to form the initial popuamplitude, denoted as lation for the next run. In the third run, the PSL is reduced from dB to dB after 30 generations. Further removal of any sensor fails to find acceptable solutions. The results of the iterative process are summarized in Table I. The sensor weights and positions of the final design are shown in Fig. 5(a), and the beam pattern is shown in Fig. 5(b). A comparison between our results and that reported in [1] is summarized in Table II. Four observations can be made from this Table. a) Our method synthesizes the array with 19 sensors, which is 5 fewer than 24 required by the simulated annealing (SA) method reported in [1].

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TABLE III MAXIMUM SIDELOBE LEVELS AND REQUIRED MINIMUM NUMBERS OF SENSORS

Fig. 5. Performance of our proposed method using 19 sensors (a) weight ampli= 0:0203). tudes and positions (b) beam pattern (PSL = 14:49 dB, u

0

TABLE I RESULTS OF THE ITERATIVE PROCESS

c) Even with fewer sensors and smaller spatial aperture, our dB. solution is slightly better in terms of PSL and d) Although the 19-sensor array is found after performing 3 runs of the IGA, the total number of generations is only 110 and the fitness function is evaluated for only 46608 times. In [1], the energy function is evaluated for approxtimes. Based on imately this estimation, the number of function evaluations in our method is only 7.9 % of that needed in the SA method. This example has illustrated that our method can achieve a significant reduction in computational complexity compared to the method in [1]. Many other independent simulation trials have arrived at similar results too, which implies the stability of the proposed algorithm. B. Maximum Sidelobe Levels and Required Minimum Numbers of Sensors

TABLE II COMPARISON BETWEEN THE SIMULATED ANNEALING METHOD [1] AND OUR PROPOSED METHOD

As the extension of the previous example, we attempt to find the fewest sensors that are required to synthesize arrays with the PSLs no larger than a value from dB to dB with dB, over the interval . Table III a step-size of lists the minimal numbers of sensors required in the designs meeting different values of the PSLs. It can be seen from this Table that the PSL of the array with 24 sensors reaches dB that is lower than the one achieved in [1] by 4.58 dB. The size of spatial aperture in our design is , which is decompared to that in [1]. Other designs with creased by dB), i.e., 25 sensors with an aperthe same PSL value ( and 23 sensors with an aperture of , are also ture of provided in [1], which represent different tradeoff between the number of sensors and the aperture. It shows that using the same and ), we can achieve numbers of sensors (i.e., dB the designs with much smaller PSL values (i.e., and dB, respectively) compared to dB obtained in [1]. VI. CONCLUSION

b) Compared with the 24-sensor array synthesized by the SA method, our method reduces the spatial aperture size by . This improvement is due largely to the reduction in the value of used.

In this paper, an iterative optimization scheme is proposed for minimizing the number of sensors in the synthesis of linear sparse aperiodic arrays. The weight coefficients and the sensor positions are optimized simultaneously using an IGA. By

CEN et al.: LINEAR SPARSE ARRAY SYNTHESIS WITH MINIMUM NUMBER OF SENSORS

iteratively removing the sensors that have less contribution to the lowering of the PSL, a linear array satisfying a given specification can be synthesized with the minimum number of sensors. Computer simulations show that our proposed scheme is able to synthesize an aperiodic array with a smaller number of sensors and spatial aperture compared to other existing methods. The computational cost of our method has been shown to be about one order lower than that of other existing methods too. REFERENCES [1] A. Trucco and V. Murino, “Stochastic optimization of linear sparse arrays,” IEEE J. Oceanic Eng., vol. 24, no. 3, pp. 291–299, Jul. 1999. [2] A. Ishimaru, “Theory of unequally-spaced arrays,” IEEE Trans. Antennas Propag., vol. 10, no. 6, pp. 691–702, Nov. 1962. [3] D. G. Kurup, M. Himdi, and A. Rydberg, “Synthesis of uniform amplitude unequally spaced antenna arrays using the differential evolution algorithm,” IEEE Trans. Antennas Propag., vol. 51, no. 9, pp. 2210–2217, Sep. 2003. [4] B. P. Kumar and G. R. Branner, “Design of unequally spaced arrays for performance improvement,” IEEE Trans. Antennas Propag., vol. 47, no. 3, pp. 511–523, Mar. 1999. [5] B. P. Kumar and G. R. Branner, “Generalized analytical technique for the synthesis of unequally spaced arrays with linear, planar, cylindrical or spherical geometry,” IEEE Trans. Antennas Propag., vol. 53, no. 2, pp. 621–634, Feb. 2005. [6] P. Jarske, T. Saramäki, S. K. Mitra, and Y. Neuvo, “On the properties and design of nonuniformly spaced linear arrays,” IEEE Trans. Acoust., Speech, Signal Processing, vol. 36, no. 3, pp. 372–380, Mar. 1988. [7] V. Murino, A. Trucco, and C. S. Regazzoni, “Synthesis of unequally spaced arrays by simulated annealing,” IEEE Trans. Signal Processing, vol. 44, no. 1, pp. 119–123, Jan. 1996. [8] A. Lommi, A. Massa, E. Storti, and A. Trucco, “Sidelobe reduction in sparse linear arrays by genetic algorithms,” Microw. Opt. Technol. Lett., vol. 31, no. 3, pp. 194–196, Feb. 2002. [9] K. Chen, Z. S. He, and C. L. Han, “A modified real GA for the sparse linear array synthesis with multiple constraints,” IEEE Trans. Antennas Propag., vol. 54, no. 7, pp. 2169–2173, Jul. 2006. [10] L. Cen, W. Ser, Z. L. Yu, and R. Susanto, “An improved genetic algorithm for aperiodic array synthesis,” in Proc. IEEE Int. Conf. Acoust., Speech, Signal Processing, Las Vegas, Nevada, Mar. 31–Apr. 4 2008, pp. 2465–2468. [11] L. Cen, W. Ser, Z. L. Yu, R. Susanto, and W. Cen, “Pattern synthesis of linear aperiodic array using genetic algorithm,” IEEE Trans. Antennas Propag., submitted for publication. [12] R. L. Haupt, “Thinned arrays using genetic algorithms,” IEEE Trans. Antennas Propag., vol. 42, no. 7, pp. 993–999, Jul. 1994. [13] D. J. O’Neill, “Element placement in thinned arrays using genetic algorithms,” in Proc. IEEE Int. Conf. Oceans Eng.for Today’s Technol.and Tomorrows Preservation, Brest, France, Sep. 1994, vol. 2, pp. 301–306. [14] C. A. Meijer, “Simulated annealing in the design of thinned arrays having low sidelobe levels,” in Proc. IEEE South African Symp. Commun.and Signal Processing, Rondebosch, South Africa, Sep. 1998, pp. 361–366. [15] M. G. Bray, D. H. Werner, D. W. Boeringer, and D. W. Machuga, “Optimization of thinned aperiodic linear phased arrays using genetic algorithms to reduce grating lobes during scanning,” IEEE Trans. Antennas Propag., vol. 50, no. 12, pp. 1732–1742, Dec. 2002. [16] R. M. Leahy and B. D. Jeffs, “On the design of maximally sparse beamforming arrays,” IEEE Trans. Antennas Propag., vol. 39, no. 8, pp. 1178–1187, Aug. 1991. [17] S. Holm, B. Elgetun, and G. Dahl, “Properties of the beampattern of weight- and layout-optimized sparse arrays,” IEEE Trans. Ultrason., Ferroelect., and Freq. Contr., vol. 44, no. 5, pp. 983–991, Sept. 1997. [18] K. F. Man, K. S. Tang, and S. Kwong, Genetic Algorithms: Concepts and Designs. London: Springer Verlag, 1999.

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Ling Cen (S’03–M’06) received the B.Eng. degree from the University of Science and Technology of China, in 1997, the M. Eng. degree from Chinese Academy of Sciences, in 2001, and the Ph.D. degree in electrical and computer engineering, from the National University of Singapore (NUS), in 2006. She was with the General Electronic Technology Institute, China, as a project Engineer from 1997 to 1998. In 2005, she joined the Centre for Signal Processing, Nanyang Technological University (NTU), as a Research Associate, then became a Research Fellow. She is currently working as a Research fellow in the Human Language Technology Department, Institute for Infocomm Research (I2R), Singapore. Her research interests include digital filter design, array signal processing, speech synthesis, pattern recognition, and evolutionary computation. Dr. Cen has served as a Reviewer to several international journals and conferences.

Wee Ser (SM’97) received the B.Sc. (Hon.) and Ph.D. degrees, both in electrical and electronic engineering, from Loughborough University, U.K., in 1978 and 1982, respectively. He joined the Defence Science Organization (DSO), Singapore, in 1982, and became the Head of the Communications Research Division in 1993. In 1997, he joined NTU as an Associate Professor and was since appointed Director of the Centre for Signal Processing. He has published about 120 papers in refereed journals and international conferences. He holds six patents and has three pending patents. His research interests include microphone array and sensor array signal processing in general, signal classification techniques, and channel estimation and equalization techniques. Dr. Ser was a recipient of the Colombo Plan scholarship and the PSC Postgraduate Scholarship. He was awarded the IEE Prize during his studies in the U.K. While in DSO, he was a recipient of the prestigious Defence Technology (Individual) Prize in 1991 and the DSO Excellent Award in 1992. He is an Associate Editor for the IEEE Communications Letters and the Journal of Multidimensional Systems and Signal Processing (Springer). He has served in several international and national advisory and technical committees and as reviewers to several international journals.

Zhu Liang Yu (S’02–M’06) received the B.S.E.E. and M.S.E.E. degrees, both in electronic engineering, from the Nanjing University of Aeronautics and Astronautics, China, in 1995 and 1998, respectively, and the Ph.D. degree from Nanyang Technological University, Singapore, in 2006. He worked as a Software Engineer at the Shanghai BELL Company, Ltd., from 1998 to 2000. In 2000, he joined the Center for Signal Processing, Nanyang Technological University, as a Research Engineer, then became a Research Fellow. In 2008, he joined the College of Automation Science and Engineering, South China University of Technology, as an Associate Professor. His research interests include array signal processing, acoustic signal processing and adaptive signal processing.

Susanto Rahardja (SM’03) received the Ph.D. degree in electrical and electronic engineering from the Nanyang Technological University (NTU), Singapore. His research interests are in audio/video signal processing, spread spectrum and multi-user detection techniques for CDMA applications, digital signal processing algorithms and implementations, logic synthesis, of which he has more than 200 publications in internationally refereed journals and conferences. Currently, he is a Principal Scientist, Director of Personal 3D Entertainment System program and Head of Signal processing Department at I2R. He is also a Program Director of Science and Engineering Research Council of A*STAR.

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Dr. Rahardja was the recipient of the IEE Hartree Premium Award in 2002. He jointly developed scalable to lossless audio compression technology which was published as a normative international standard in ISO/IEC 14496-3:2005/ Amd.3:2006. He also contributed to audio lossless codec which was incorporated in ISO/IEC 14496-3:2005/Amd.2:2006. In recognition of his contributions to the national standardization program and digital audio signal processing and its adoption to the MPEG, he was awarded the Standards Council Merit Award by SPRING Singapore in 2006, the National Technology Award in 2007. He has served in several boards, advisory and technical committees in various ACM, IEEE and SPIE related professional activities in the areas of multimedia. He is currently serving as an Associate Editor for the IEEE TRANSACTIONS ON AUDIO, SPEECH AND LANGUAGE PROCESSING, the Elsevier Journal of Visual Communication and Image Representation and the IEEE TRANSACTIONS ON MULTIMEDIA. He served as the Conference Chair of Multimedia Systems and Applications at SPIE OpticsEast Symposium from 2006 to 2007 as well as Symposium Co-chair of the IEEE International Symposium on Multiple-Valued logic in 2006. He was the General Chair of 7th ACM SIGGRAPH VRCAI 2008. Since 2004, he was the Council member of National IT Standards Committee

in Singapore. In March 2009, he was elected as the President of SIGGRAPH Singapore Chapter (SSC) and Southeast Asia Graphics (SEAGRAPH) society in the 2009 Annual General Meeting.

Wei Cen received the B.Eng. and M.Eng. degrees in electrical engineering from the University of Science and Technology of China, and the Ph.D. degree from the Nueva Ecija University of Science and Technology, Philippines. She is currently working at Elektrotechnik GmbH, Kassel, Germany. Her research interests include effects of electromagnetic fields on biological systems, numerical methods, signal processing.

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Alternating Adaptive Projections in Antenna Synthesis Javier Leonardo Araque Quijano and Giuseppe Vecchi, Fellow, IEEE

Abstract—The projection operator is a basic building block in the application of the alternating projections method to antenna synthesis. In general it is a non-linear operator that is repeatedly applied in the course of a single synthesis process, thus having a considerable impact on the convergence properties of the resulting algorithm. A novel approach to the computation of these projections is presented which exploits a simple definition of the relevant spaces (particularly that of radiated fields). Characterization of field mask specification as scale-invariant under this definition adds a further degree of freedom, namely reference level, which impacts on the (sensitive to scaling) projector. In order to compute the optimum reference level, an iterative procedure is proposed which is simple to implement, easily integrable in standard alternating projection routines, and adds negligible computational burden. Numerical tests confirm an improved performance with respect to the fixed-scaling projection operator in terms of convergence rate and robustness against the initial guess, supporting our approach as a valid aid in overcoming the drawbacks of the alternating projections-based antenna synthesis. Index Terms—Alternating projections, antenna synthesis, intersection approach, optimization methods.

I. INTRODUCTION ORMULATION of constrained antenna synthesis problems in terms of set intersection [1], [2] is a well-established and a valuable tool in view of two key factors: generality and simplicity. The intersection approach allows tackling a wide range of synthesis problems in which realistic design goals and constraints can be specified naturally; furthermore, it affords a simple implementation, as very often results from the integration of pre-existing unconstrained synthesis tools and well-established non-linear solvers. In the set-intersection approach, the problem is formulated as that of computing the intersection between two subsets of the and the specifispace of radiated fields: the feasible set . The former contains all the fields that can be cation set effectively radiated obeying design constraints (antenna geometry, excitation dynamics, etc), while the latter contains all the fields satisfying design specifications on the pattern, i.e., exhibiting a set of desired features (side lobe level, beam width, front-to-back ratio, etc). An element belonging to both sets is a solution to the synthesis problem [1].

F

Manuscript received March 02, 2009; revised July 07, 2009. First published December 28, 2009; current version published March 03, 2010. The authors are with the Antenna and EMC Laboratory (LACE), Politecnico di Torino, Turin 10129, Italy (e-mail: [email protected]; giuseppe. [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2009.2039307

constitutes the primary challenge of the imObtaining plementation given the complexity of these sets (see Section V). A very popular way to compute this intersection is the alternating projections algorithm (APA) [3], which proceeds by proand . jecting a candidate solution alternatively on the sets Antenna synthesis based on the alternating projections algorithm has been considerably studied in the last years. The earliest application of the alternating projections algorithm to general antenna power synthesis was presented in [4], where it was noted that the algorithm is able to deal with any constraint expressible as an inequality. [5] presents the constrained synthesis of 1-D arrays through APA, with a detailed discussion on advantageous choices of the starting point. In [6] an annealing-like technique was proposed to improve the convergence properties of the alternating projections algorithm by which the excitation/field masks gradually stiffen during the process until reaching the actual specification values; the focus in this case was sensitivity to the initial guess. In [7] the space of the radiated fields is modified via an alternative field representation; considering the field square magnitude rather than the field itself allows exploiting the quasi-convexity property of the resulting quadratic variety for an improved robustness against the trapping problem; the projection operator in that case involves the minimization of a quartic functional, effected by means of a modified non-linear solver (self-scaled BFGS method). Generality, effectiveness and simplicity in the implementation explain the use of APA in recent applications [8]–[11]. On the other hand, it is worth mentioning that in particular synthesis tasks more specific algorithms may be available which may be advantageous. An alternative technique for the general problem of computation of the intersection between sets with quadratic convergence (as opposed to the linear convergence rate of alternating projections) is proposed in [12], which unfortunately is applicable only when the derivative of the projection operator is available and the sets involved are strictly convex. [13] deals with the synthesis of shaped beam arrays; starting from the assertion of feasibility criteria that provide also a plausible goal pattern, the problem is transformed into the simpler one of (possibly constrained) field synthesis. This method cannot directly handle excitation constraints, but it may be used to find sensible starting guesses when the more general Alternating Projections Algorithm is used in a second stage of the solution. In [14]–[16] the design of pencil-beam arrays with arbitrary upper bound in the side lobe region was shown to be a convex optimization problem, which both provides guarantees on the absence of false solutions and allows devising a highly efficient solution algorithm; however it cannot handle more general design goals nor deal with non-convex constraints. The previous tech-

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nique is augmented with simulated annealing in [17] to allow simultaneous synthesis of array element amplitudes and locations; having the tool in [14]–[16] as basic brick, it shares its limitations pertaining the generality of the goal and constraints that can be dealt with. [18] considers the null-steering problem with excitation dynamics, providing a feasibility criterion and a synthesis algorithm that compares favorably to APA and quadratic programming-based techniques; however it is not applicable to more general design specifications. [19] deals with geometry and weight synthesis of planar wideband arrays based on linear programming; it is limited to arrays symmetric around the origin with real excitation (thus radiated field is real everywhere) and may deal with constraints expressible as linear inequalities on the (real) variables. On the negative side, APA is known to suffer from slow convergence rate and/or trapping on local minima depending on the initial guess, which has motivated various modifications to the basic technique as described above [5]–[7]. In the present work we propose a novel approach for the application of the APA in antenna synthesis. The starting point is a simple definition of the sets involved in the intersection approach, from which it turns out that in most practical instances those are convex in the spaces where they are naturally defined. After realizing that under the definition above the conventional projection formulas [see (14)] do not conform to the definition of projector, a general procedure for the accurate and efficient computation of these projections is presented. Extensive numerical tests have shown consistent improvements on convergence rate and robustness against the initial guess with respect to the conventional application of the APA, supporting our approach as an effective aid in counteracting the main drawbacks of the APA-based synthesis technique. Preliminary results of this work were presented in [20]. This paper is organized as follows: Section II presents in detail the problem to be solved and the proposed representation for fields and antenna parameters. Section III defines and discusses the implications of the scalability of field goals which is the basis for the approach proposed. The generalities of the field projection problem and the application of scalability are described in Section V. Section VI presents three instances of the dynamic projector proposed. Finally, Section VII presents numerical tests which illustrate the performance improvements achieved, including a large-scale problem of high practical relevance.

These currents give rise to a radiated field radiation operator

via the

(2) The objective is thus finding the set of complex excitations that a) are subject to the set of (amplitude) constraints: (3) b) radiate a field with components whose amplitude is specified to lie within a given mask, i.e., to lie between a lower and an upper position-dependent value, and , respectively (4) where the superscript is used to indicate a given field component, (e.g. , , , ). In general more than one such specification is to be enforced, and this requires no modifications in the formulation. We remark that the above does not necessarily refer to far field, although the numerical tests we provide in Section VII are limited to that case. Note that in this reference problem (3) and (4) define the sets and respectively. In the far field, the radiation operator can be expressed as (5) are the centers of the radiating elements or of the where the radiation patterns of the elebasis functions, and mental antennas or of the basis functions for the considered (vector) component. This general formulation accommodates for the design of conformal sources (arrays) as in [10], [21] considering also element rotation with respect to the global coordinate system or the utilization of embedded element patterns when inter-element coupling is to be considered in array design, thus covering the relevant case of synthesis of generic circularly-polarized arrays. projective coordinates It is convenient to use the (6) with which

II. STATEMENT OF THE PROBLEM The reference problem is the synthesis of antenna source distribution with pattern (field amplitude) specifications in the form of inequalities (masks), and constraints on the excitation amplitudes. The source distribution will be described in terms of a , finite number of complex coefficients that can represent the excitations of an array, or the coefficients that represent a continuous current distribution (say, ) in terms of a chosen set of basis functions, e.g. (1)

(7) Concerning (4), we can consider enforcement of the mask inobservation points, which makes the space of equalities at all possible radiated fields finite-dimensional (since the field is ). If we describe the radicomplex, the space dimension is ated field by splitting the complex quantities into magnitude and phase instead of real and imaginary parts, the specification set in (4) looks completely different. In fact, since phase is not involved in the specifications in (4), the specification set becomes -dimensional box constrained along the magnitude-related a

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dimensions and extending infinitely along the phase-related dimensions. With this choice of space, the specification set is convex. The above applies also to design constraints in (3): denoting by the Cartesian product of all the allowed ranges of , antenna parameters, i.e., and are the limit values for the kth antenna pawhere rameter, we see that these span a convex set (hyper-rectangle) provided the intervals are simple and separable. Note however through the radiation operator that spans the set (8) which in general does not preserve the original convexity. We will be using throughout the pseudo-inverse radiation operator, defined as follows: (9) is the adjoint of . where To illustrate the above we may consider the constrained synthesis of the excitation amplitudes of a 2-element array when the field mask involves only two measurement locations. Considering complex quantities, both antenna parameters and radiated fields span . Now consider the following field specification and constraints:

(10) As discussed above we consider magnitude-phase representation of both excitations and radiated fields. Given the invariance of these sets along phase-related dimensions (no phase specifications) we can limit ourselves, for display purposes, to the magnitude-related subspaces, which are , i.e., pairs of non-negative real numbers. A graphical representation of the situation is provided in Fig. 1, where sets with equal shading are mapped to each other by when going from the antenna parameters (i.e., excitation amplitudes) to the measured fields and by when going back. Note that , i.e., the (pseudo) inverse radiation operator, provides complex array amplitudes from complex field measurements; hence, it constitutes the solution to a (linear) field synthesis problem. It is seen in Fig. 1 that even though and are convex sets in their natural domains of definition (dark rectangle and clear square at left and right spaces respectively), they span non-convex sets in the conjugate space of ). This motivates (image through of and through the widespread procedure [5], [7], [21] whereby projection is performed in the most suitable space as seen in Section IV; in particular the simple form of (14) is a consequence of the convexity of these sets under the magnitude-phase representation. Further discussion on the properties of these sets specific to the antenna synthesis problem and how to exploit them are postponed to Section V. As in every inverse problem, care should be observed in the discretization of the source and observation domains to achieve a stable solution. The choice of the number of observation points is in principle dictated by the requirement of fully characterizing the radiated field with sampled data, which for a general radiator’s support and observation domain is computed from the

Fig. 1. Sets involved in the intersection approach for the design of a 2-element array when the field is specified at two measurement locations only.

degrees of freedom of the field [21], [22]. Furthermore, we need to avoid that the field at intervening (i.e., not sampled) points be (substantially) outside the masks when sampled points are conforming to it. The latter requirement necessitates a sampling that allows reasonably accurate linear interpolation; therefore, it involves an -fold sampling refinement with respect to the depending on the desired acNyquist rate above, with curacy. The first consideration above must guide source discretization as well; in fact as the basis set representing the source in (1) grows beyond the number of degrees of freedom of the field (for fixed source dimensions), it gradually loses linear independence, making the solution both less stable and unnecessarily expensive to compute. Section VII illustrates these considerations when dealing with far-field synthesis of uniform 2-D arrays. III. SCALABILITY IN SYNTHESIS PROBLEMS In inverse problems (and in particular synthesis problems) considerable benefits in terms of computational efficiency and solution quality can be derived from a throughout characterization of the solution. In order to do so with the problem at hand, we define the properties that make a problem suitable for the application of the approach proposed. In particular we are interested on establishing the scaling invariance of goal specifications and design constraints in (3) and (4) respectively. A scalable goal is defined as that in which the specification in (4) can be extended by including a non-null scaling without modifying the synthesis problem (11) It should be noted that the design goal in many synthesis problems of practical interest fall into this category: pencil beam design with SLL specifications, shaped beam synthesis, contoured beam synthesis, etc. In all these cases the distinctive feature is that field specification is a shape stipulation, and the absolute level of the field is of no interest. On the other hand, less common synthesis problems are rather given in terms of antenna directivity/gain which do not enjoy scaling invariance (scaling in general leads to invalid directivity/gain functions in

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view of the well known normalization conditions they are subject to [23]). Analogously, a scalable constraint is defined as that in which the constraint given in (3) can be scaled without modifying the synthesis problem (12) Design constraints in many synthesis problems obey this condition, notably dynamic range constraints on array excitations and super-gain avoidance, which are very relevant in practical applications. On the other hand, constraints such as antenna physical dimensions, maximum current amplitude, total input power, etc. are absolute in nature and thus not expressible as given in (12). Nonetheless, even when such non-scalable goals are present, one may perform a partition of the design parameters such that the scalable part is dealt with separately with the technique that will be presented.

Fig. 2. Effect of goal scalability on the synthesis problem initially proposed in Fig. 1.

IV. THE ALTERNATING PROJECTIONS ALGORITHM Following the discussion above, the synthesis problem is casted into the intersection, in the space of the magnitude of and radiated fields, between a geometrically simple set . The usual practice to circumvent the a complex one projection into the intricate constraint set is, whenever is possible, perform the projection in the space where naturally defined (where it is convex whenever the convention of magnitude-only variables can be employed, as is the case of array synthesis with excitation constraints) [5]. Denoting by the set of parameters fully characterizing the antenna (in general geometry and feeding parameters), the projection operator, by the radiation operator, and by (i.e., antenna parameters at the nth iteration) is computed via the alternating projections algorithm [5] (13) In (13) the radiator operator and its (pseudo) inverse are used to perform the projection in the space better suited for each element to be projected (radiation field or antenna parameters). This procedure relies on suitable properties of the radiator operator. For instance, it is unitary in cases assimilable to the field synthesis of uniform samples of the far field of equispaced arrays, thus preserving the metric properties of the projector when moving between the conjugate spaces [2]. V. FIELD PROJECTION The conventional projection of a generic element F onto a specification set with and being its lower and upper limits is as follows [2]:

the magnitude-phase representation discussed in Section II. Even though (14) is of widespread use in alternating-projection synthesis, it may be unnecessarily restrictive as seen below is done in a similar way, so (projection on the constraint set the discussion that follows applies to that, too). The shortcomings of the direct application of (14) are better understood by referring to the actual translation of design goals/ constraints into a rigorous description by means of the sets and . For instance, with the sets in Fig. 1 one may think that the synthesis problem has no solution in view of the void intersection. However, it was discussed in Section III that in the relevant cases of scalable goal and/or constraint (c.f. Section III) these sets can be expressed in more general terms by means of an arbitrary scaling. For instance, assuming goal scalability, the sets to be intersected become those in Fig. 2, which show that the synthesis problem is actually solvable for some values of the scaling factor . This is a clear indication of the need to include any possible invariance in the synthesis process to avoid ruling out valid solutions. Fig. 2-left may lead one to think that the complexity of the inverse image of the design goal can be circumvented by the application of scaling (the set has become convex). However, this is true only in the 2-D case, while higher-dimensional setups prevent such “convexification” (the 3-D analog of Fig. 2 is a cone whose transverse section is a ring, which is non-convex). In order to integrate scaling invariance into the synthesis process we define a modified projector , which is just a generalization of that in (14) (15)

(14) where the dependence on the observation coordinates (e.g. , , matrix indices, etc.) has been omitted for clarity. According to (14), the phase of F remains unchanged, while its magnitude is clipped to stay in the allowed range, justifying

where an arbitrary norm definition has been used in the computation of . We note that this generalized projection is just the composition of the minimization procedure implied by the original projector (14) and that implied by the choice of the scaling factor that minimizes the distance to the projected version. As such, it conforms to the definition of projector onto a the union

ARAQUE QUIJANO AND VECCHI: ALTERNATING ADAPTIVE PROJECTIONS IN ANTENNA SYNTHESIS

Fig. 3. Effect of various scalings on the projection of F onto the goal set.

of a family of sets parameterized by nition in virtue of scalability)

(the extended goal defi-

(16) This in turn means that application of our approach preserves the amenable distance-reduction property of APA [1, Eq. 14]. In choosing there are alternatives to the optimum expression in (15). A possibility is that of scaling the goal set such that its image equals that of F under a suitable function. For example, , is computed as follows: to have (17) Considering the example introduced in Section II, we can see the result of various choices of in Fig. 3. Note that in virtue of scalability the goal set is not the original rectangle anymore, but the shaded triangle including any of its possible scalings. In Fig. 3 arrows indicate the result of different scaling schemes applied before using (14) on the point to be projected (X at the right bottom), where the minima and maxima equalization schemes follow from (17), while the 2-norm follows from (15) using the 2-norm of the vector of measurements : (18) In this definition, used throughout our implementation, the possibly 2-D nature of the observation domain is disregarded [e.g. in computing the 2-norm when the field is specified on the unit sphere one could choose the usual surface integral instead of (18)]. This has been done both for generality and simplicity: (18) is fully compatible with the discrete representation of fields as required in practical implementation and does not explicitly depend on the spatial distribution of these samples (the synthesis process can focus on important portions of the observation domain by simply increasing the spatial sampling therein). It is important to note, however, that there are practical constraints in the choice of those sample points, that need to be fairly well plane in order to avoid the insurgence of distributed in the super-directive sources and other spurious effects as discussed in Section II. From Fig. 3 it is seen that the results for all these approaches except the third one may lead to a point unnecessarily far away

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from the starting point, thus lacking the basic property of a projection operator and introducing excessive distortion on the original element. It should be apparent that the (repeated) application of an inadequate projector has a harmful effect on the convergence properties of the resulting synthesis algorithm, let alone the possibility of finding a solution to the synthesis problem, as seen from the apparent lack of solution (void set intersection) in Fig. 1, which is actually due to inadequate scaling. In practice such fixed scaling will lead to stagnation between the closest points in sets B and M shown in Fig. 1, while dynamic scaling overcomes this difficulty by revealing a non-empty intersection as seen in Fig. 2. It remains to determine how to compute the optimum scaling factor in (15). It should be observed that the high dimensionality of practical problems (hundreds of field samples) in general complicates the direct computation of in (15). In this work we propose a generic iterative algorithm to compute , which is thus an inner loop that runs at each step of (13), specifically when a projection step is required. A clear advantage of such approach is that in the course of the synthesis process (the outer loop, given by (13), the optimum scale factor varies smoothly, so at each external iteration the iterative projector is initialized with the previous estimate of , which is very close to the solution. As seen below, this approach is of simple implementation and high computational efficiency (minimal impact on total execution time with respect to the simplest projection operator which is not surprising in view that computational cost is mainly determined by the direct/inverse radiation operations). Our problem is closely related to that of solving a system of linear inequalities, which can be tackled with the generalized relaxation method [24], [25]. We have the additional requirement of choosing, among the possible solutions to that system of inequalities (which specify the set upon which projection is to be performed) the nearest to a given point. Specification in terms of inequalities however is too general for practical purposes: we have seen that in most cases goals are separable, thus expressible via masks. Masks dramatically reduce the number of parameters required to define the sets of interest; to have an idea, paramin n-dimensional space a mask determined by eters (two scalars specifying the box limits along each dimension plus the scaling factor) is determined by a set of inequalities parameters ( coefficients per inequality, comprising inequalities since this is the number of hyper-planes with bounding the relevant set). Furthermore, masks are explicit solutions of the system of inequalities, so we are left with the simpler problem of choosing, among known solutions, the one closest to the point given. Exploitation of these facts gives rise to a considerably simpler and more efficient algorithm as seen below. We note that more general specification sets (e.g. non-separable) may require a relaxation-like procedure. The approach proposed is summarized by the flowchart in Fig. 4, which describes the process followed at each projection step. The dynamic projector is given in input an initial guess for the scaling factor , which is either that obtained in the previous projection when available or simply one when not. The function has two implicit parameters: the maximum number of iterations and and the error threshold, which command loop break ( respectively). Concerning the loop body, functions

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Fig. 5. Depiction of the first two iterations of the 2-norm projection.

Fig. 4. Generic flowchart for adaptive mask scaling.

and are specific to the norm chosen in (15), and will be illustrated by examples in Section VI. Finally, we stress the iterative nature of our approach and the fact that the problem of computing the projection is equivalent to that of computing the optimum scaling factor in view of (15). The dynamically adaptive projection technique presented above is a fundamental building block in our proposed implementation of the APA. It is used to compute the projections required by the key operation (13), and it is therefore applicable to cases where i) the conventional APA approach is applicable, and ii) one of (11), (12) holds, covering a relevant subset of synthesis problems. In this respect the proposed approach inherits the generality of the APA in tackling a variety of synthesis problems in a unified manner, and improving convergence where applicable. On the contrary, our technique is not expected to have a bear on specific problems arising in the use of APA to synthesis tasks other than the (discrete) array problem; most notably the case of continuous sources necessitates the avoidance of super-directivity. VI. PROJECTOR INSTANCES This section illustrates three specific application instances of the general approach described in the previous section: a) 2-norm projection; b) infinity-norm projection; and c) constant 1-norm projection. These demonstrate the flexibility of the approach by obtaining an projector that either conforms to the definition of projection under given metrics [cases a) and b)] or maintains a desired property on the original field after projection [case c)]. For simplicity, the convergence of these iterative projectors was characterized through extensive numerical tests with high dimensional ( 1000) data rather than analytically; in all cases exponential or better convergence was obtained. As mentioned in Section V we disregard the geometric features of the field observation domain; fields (and masks) are simply represented as generic complex vectors that are to be projected upon a set defined through masks (19)

This implies that the original projection in (14) should be evaluated component-wise, i.e., the independent variable is simply an index into the related vectors. In the following, the vector to be projected will be denoted by , while will denote its projection through (14) with an arbitrary fixed scaling constant. Finally, the projection of through (15) will be denoted by . A. Iterative Projector in 2-Norm In this section we seek to compute the 2-norm projector [i.e., that conforming to (15) with the 2-norm definition in (18)] with the generic algorithm in Fig. 4. From Fig. 3 one can infer that the 2-norm projection of must satisfy the following modified statement of the projection theorem [26] (20) This is only a necessary condition since the goal set is not a sub-space of (in 3D it is a semi-cone with polygonal section as can be inferred from the 2D case in Fig. 3). In words it tells must lie on the boundary that the projection of a field of (a triangle contour in the 2D case in Fig. 3), and that the difference between original and projected version must be perpendicular to the boundary at that point. Based on the above, the error and update functions in Fig. 4 are computed as follows:

(21) This algorithm operates by iteratively scaling the mask in such a way that the difference between the original vector and its projection is perpendicular to the latter as seen in Fig. 5. From the two steps depicted therein, it is apparent that few iterations should suffice in approaching the 2-norm exact projection. B. Iterative Projector in Infinity-Norm A similar procedure is possible when dealing with infinityto be the infinitynorm. A necessary condition for the vector norm projection of is (22)

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This results in the following error and update functions:

(25) With these expressions, the algorithm starts by computing the 1-norm variation of the input vector upon projection and then updates the scaling such that the 1-norm of the scaled projection is the same as that of the original vector; however, given the scale change and the non-linearity of the projector, the new projection must be re-computed and the process repeated. Fig. 6. Depiction of the clipping operation with two relevant distances for infinity-norm projection: extreme up and down-clipping.

where sub-indices specify vector components. This condition follows from the definition of projector in the infinity-norm, and basically specifies that the two distances indicated in Fig. 6 must be equal (i.e., any scale increase would lead to an increase in the infinity-norm of the difference between the original vector and its projection). The relevant functions are computed as follows:

(23) With this definition the algorithm operates as follows: first these two distances (and the observation coordinates and where they occur) are computed with the current scaling value. Next, the scaling value is updated such that the new distances at these observation coordinates obey the infinity-norm projection condition in (22). This correction leads immediately to the satisfaction of the condition referring only to coordinates and , but in general leads to the condition not being satisfied elsewhere; the subsequent iteration finds the new and and performs further corrections. C. Iterative Projector for Constant 1-Norm In contrast to the previous algorithms, where it was sought to compute such that (15) holds with a given norm definition, one could define other constraints on the resulting projection. Here it is presented one such possibility, stemming from the consideration that the optimum scaling could be defined in terms of maintaining certain property of the original vector after projection. In particular, it could be required that the projection does not modify the 1-norm of the original vector, so the condition to be satisfied is (24)

VII. NUMERICAL RESULTS In this section we test the approach presented in the far-field inter-element spacing on a synthesis of planar arrays with square grid (this corresponds to the Nyquist sampling rate of the far field for a source with the given dimensions [22]). The array with , where has dimensions is the number of elements. The (u,v) plane is uniformly sampled on a grid with points, corresponding to an oversampling factor of 4 to avoid considerable deviation from specification at points not sampled as discussed in Section II. The choice of the oversampling factor is dictated by the accuracy desired, higher factors could be required in high accuracy applications since in general deviations will occur at points not sampled. Sampling rate has a definite impact on the computation time per iteration. Furthermore, it influences convergence rate, in some cases determining the realizability of a given field specification, i.e., a synthesis problem may be solvable at a given sampling rate while not at a higher one; this is particularly true for specifications presenting fast variations and/or narrow allowable ranges for some observation points. When one increases sampling rate above the factor presented here (we have tested up to 20, i.e., 40 samples per wavelength) performance remains consistently better for the technique proposed; moreover, when realizability is not affected by sampling rate, the number of iterations required for convergence is fairly constant. The maxima equalization scheme (see Fig. 3) along with the projection operators devised in Section VI were tested in 2-D array synthesis with various constraints on the excitation. Our implementation of the alternating projections algorithm follows that of [1], [5], employing the 2-D FFT in the direct/inverse radiation operators, with the only difference being the modified projection operators. In all of the following tests, the array elements are Huygens’ sources. In order to better assess the performance gain, we employ the , see definition uniform norm of the dB fields (denoted by below) as the main indicator instead of the usual 2-norm criterion. This choice provides a measure of performance more in agreement with pattern specifications (relative differences are more important than absolute ones, and the maximum of such differences throughout the observation domain should be the kept at bay). Due to its global nature, 2-norm residual practically neglects deviations in low-level zones of the pattern, which are however important in practical applications (SLL specification,

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out of coverage zones). The same applies to isolated pattern artifacts (i.e., with small support) such as nulls in the coverage -conzone and spikes in the out-of-coverage zone. Hence, vergence is a (stronger) condition that manages to better express what it is considered a compliant solution. The increased resometric is particularly useful at later lution afforded by the stages of the process since differences are usually not visible in 2-norm, yet significant. The error measures (maximum and average) are computed as follows:

otherwise (26) The alternating projections algorithm stopped when any of the following conditions was satisfied: the maximum error (convergence achieved), no significant improvement observed during 2000 iterations (stagnation) or 20000 iterations reached. It should be noted that projection technique presented can be employed in both projection steps required in an iteration of the APA as given by (13) provided the relevant spaces comply with the scale-invariance property (this is the case for array synthesis with excitation constraints). In the following tests both these steps were performed with the same projector rule for a better comparison. The iterative projector had a maximum iteration , while the threshold of the goodness criterion count . A detailed timing revealed a less than 10% is increase in the per-iteration computation time for all the tests below, confirming the efficiency of the technique. As discussed in the introduction, there are alternatives to the magnitude-phase representation of fields/sources employed here. Besides the well-known squared magnitude representation [13], we observe here yet another possible representation, that to our knowledge has not been used in literature: by resorting to the logarithm of the quantities of interest, one conveniently separates magnitude and phase, which respectively go into the real and imaginary parts of the final variable; furthermore, this (in substance) dB-magnitude/phase representation may be beneficial in view of the discussion preceding (26). We performed tests to assess the benefit of our adaptive projector with respect to the static projector also for these representations, obtaining improvements in the line of those shown here for the magnitude/phase one. On the other hand, we did not aim at determining which representation leads to the best convergence rates, since this goes beyond the scope of this work, which focuses in the projector. A. Robustness Against Initial Guess As pointed out in the introduction, the main shortcoming of APA is the marked dependence of the synthesis result on the initial guess [1], [2], [5], [7]. In this subsection we compare the

Fig. 7. Goal pattern for the robustness test.

performance of the 2-norm iterative projector to that of the conventional projector by reporting results on several synthesis runs with random initial guesses. The array considered consists of 15 15 elements, and has overall dimensions . The pattern specification is given in Fig. 7, and 800 runs were performed for both the maxima-equalization (conventional) and 2-norm iterative (proposed) projectors considering constraints on the excitation dynamics of 20 dB and 17 dB. Concerning the random initialization of the solution, a radiation pattern with real and imaginary parts uniformly distributed in the [0,1] interval was used. It is worth mentioning that the choice of these intervals, limiting all the complex amplitudes to the first quadrant of the complex plane, had a positive effect on the convergence rates observed in this test (broadside objective pattern); in fact, allowing these to vary in the interval [ 1,1] (i.e., allowing phase jumps of radians between neighboring measurements), consistently resulted in poor convergence (and thus not reported here). This observation is in the line with the minimum-phase initialization scheme proposed in [5]. For the comparison we focus on the main performance metric, as defined in (26). Histograms for the different exthat is citation constraints and projection rules are seen in Fig. 8. From the results therein it is clear that the adaptive projector proposed performs consistently better than the conventional one, with higher success rates. The improvement is more evident in the case of tighter constraints (17 dB), supporting the use of our approach in constrained synthesis tasks. Moreover, in all cases the runs using the 2-norm projector were converging when the algorithm stopped (this was asserted by checking that the best iteration was the last one and that the stop reason was reaching the maximum number of iterations), while runs with the maxima-equalization scheme were mostly interrupted due to stagnation. The results above are far from complete since only one synthesis goal was demonstrated. However, these illustrate the

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TABLE I SUMMARY OF RESULTS FOR TRIANGULAR SHAPED BEAM SYNTHESIS

Fig. 8. Histogram of the minimum per-run error  for the conventional (left) and 2-norm (right) projectors for the synthesis with excitation constraints of 20 dB (top) and 17 dB (bottom).

Fig. 10. Synthesized triangular shaped beam 15 tation constraints and 2-norm iterative projector.

2 15 array elements, no exci-

Fig. 9. Design goal for triangular shaped beam.

trend observed with many other goal patterns and constraints, i.e., that the adaptive scaling approach provides concrete advantages with very little cost considering implementation and computational efficiency. In view of this, results presented in Sections VII-B and VII-C refer to an all-ones initial guess in all cases. B. Triangular Contoured Beam Synthesis The goal for this test is the triangular shaped beam considered in [13], and shown in Fig. 9. According to the feasibility criterion in [13], at a minimum a 15 15 array is required to solve this problem in the unconstrained case. We present results for such an array and different excitation constraints in Table I. Note that the execution time per iteration was below 6 msec.; all the synthesis launches in Table I were completed in under two minutes. These results confirm the superior performance of the proposed projectors with respect to the conventional “staticscaling” approach, particularly in constrained synthesis. Figs. 10 and 11 show the pattern synthesized with different excitation constraints when using the 2-norm projector proposed.

2

Fig. 11. Synthesized triangular shaped beam 15 15 array elements, 20 dB dynamic constraint on excitation and 2-norm iterative projector.

C. Contoured Beam Synthesis for Geographical Coverage In this section we present results pertaining a large-scale synthesis problem in which the goal pattern is a contoured beam for geographical coverage from a satellite in the geostationary

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Fig. 13. Synthesized contoured beam pattern for geographical coverage with 161 161 array elements, 3 dB excitation constraints and 2-norm iterative projector.

2

Fig. 12. Design goal for geographical coverage: global view (left), main beam detail (right).

TABLE II SUMMARY OF RESULTS FOR CONTOURED-BEAM SYNTHESIS

2

Fig. 14. Synthesized array excitation for geographical coverage with 161 161 array elements, 3 dB excitation constraints and 2-norm iterative projector.

tern errors reported ( plane.

,

) include the invisible part of the

VIII. CONCLUSION

orbit, which is a case study presented in [27]. The field specification is shown in Fig. 12; it is synthesized by means of a 161 array with half-wavelength inter-element spacing 161 . Excitation constraints are given in terms of maximum amplitude dynamics; several of these were tested to better assess the performance of the technique. Pertaining the specification mask in Fig. 12, it should be noted that in the actual synthesis process the specification on the level at the out-of-coverage zone was extended to the invisible part of in order to prevent super-directive the spectrum behavior. This is to some extent equivalent to limiting excitation jumps between adjacent elements (this removes high-frequency components, hence can be seen as a simple and approximate way to limit phase difference between adjacent elements). Furinter-element spacing, we concentrate thermore, due to the due to the inherent periodicity of on the interval plane. the radiated fields in the extended Synthesis results are given in Table II. From these, it is seen that the performance of the 2-norm iterative projector is consistently superior in terms of final maximum dB-error, resulting in tighter adherence to the design specifications. The execution time was 0.7 seconds per iteration; the longest runs in Table II took about two hours to complete. As an example, the synthesized pattern and array excitations for 3 dB excitation dynamics and 2-norm projection are shown in Fig. 13 and Fig. 14. It should be noted that the radiation pat-

A novel approach for the computation of the projections required in the alternating projections algorithm has been proposed based on a scaling invariance property of design specifications and constraints which arises in synthesis problems of practical relevance. By this property, the sets to be intersected are actually an extension of the usual masks considered in the conventional approach, where a fixed reference level is used-this is an unnecessary restriction that in general reduces the measure of the solution set (if it exists) and negatively impacts convergence given the non-linearity of the projection operator, which is applied very many times in a single synthesis run. A general iterative technique was proposed to allow the adaptive computation of the optimum reference level under various criteria at each step of the synthesis process. Numerical tests on three application instances confirm a consistent improvement with respect to an extreme-scaling projector in terms of the quality of the solution obtained and robustness against the initial guess with negligible increase in the computational cost. Among the dynamic schemes tested, the 2-norm dynamic projector presented the most robust operation in terms of solution quality and convergence rate, which was to some extent expected given the preponderant role of this metric in the spaces considered. The superior performance in terms of robustness against the initial guess aids in alleviating one of the most important shortcomings of the APA-based antenna synthesis. Future work will focus on the efficient use of APA-based synthesis to compute element locations in addition to excitation amplitudes for arbitrary 3-D arrays.

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REFERENCES [1] O. M. Bucci, G. Franceschetti, G. Mazzarella, and G. Panariello, “Intersection approach to array pattern synthesis,” Proc. Inst. Elect.Eng. Microw. Antennas. Propag., vol. 137, pt. H, pp. 349–357, Dec. 1990. [2] O. M. Bucci, G. D’Elia, G. Mazzarella, and G. Panariello, “Antenna pattern synthesis: A new general approach,” Proc. IEEE, vol. 82, pp. 358–371, Mar. 1994. [3] L. Bregman, “The method of successive projection for finding a common point of convex sets,” Soviet Math. Dokl., vol. 162, no. 3, pp. 688–692, 1965. [4] G. Poulton, “Antenna power pattern synthesis using method of successive projections,” Electron. Lett., vol. 22, pp. 1042–1043, 1986, 25. [5] G. Franceschetti, G. Mazzarella, and G. Panariello, “Array synthesis with excitation constraints,” Proc. Inst. Elect. Eng. Microw. Antennas. Propag., vol. 135, pt. H, pp. 400–407, Dec. 1988. [6] D. Trincia, L. Marcaccioli, R. V. Gatti, and R. Sorrentino, “Modified projection method for array pattern synthesis,” in Proc. 34th Eur. Microw. Conf., Amsterdam, 2004, pp. 1397–1400. [7] O. M. Bucci, G. D’Elia, and G. Romito, “Power synthesis of conformal arrays by a generalised projection method,” Proc. Inst. Elect. Eng. Microw., Antennas. Propag., vol. 142, pp. 467–471, Dec. 1995. [8] F. Venneri, S. Costanzo, G. Di Massa, and G. Angiulli, “An improved synthesis algorithm for reflectarrays design,” IEE Antennas Wireless Propag. Lett., vol. 4, pp. 258–261, 2005. [9] W. Bu-Hong, G. Ying, and W. Yong-Liang, “Pattern synthesis of conformal array antenna with polarization diversity,” in Proc. 1st Asian and Pacific Conf. on Synthetic Aperture Radar, Nov. 2007, pp. 170–174. [10] B.-H. Wang, Y. Guo, Y.-L. Wang, and Y.-Z. Lin, “Frequency-invariant pattern synthesis of conformal array antenna with low cross-polarisation,” IET Microw. Antennas Propag., vol. 2, pp. 442–450, Aug. 2008. [11] M. Ciattaglia and G. Marrocco, “Time domain synthesis of pulsed arrays,” IEEE Trans. Antennas Propag., vol. 56, pp. 1928–1938, Jul. 2008. [12] S. Dharanipragada and K. Arun, “A quadratically convergent algorithm for convex-set constrained signal recovery,” IEEE Trans. Signal Processing, vol. 44, pp. 248–266, Feb. 1996. [13] T. Isernia, O. M. Bucci, and N. Fiorentino, “Shaped beam antenna synthesis problems: Feasibility criteria and new strategies,” J. Electromagn. Waves Applicat., vol. 12, pp. 103–138, 1998. [14] T. Isernia and G. Panariello, “Optimal focusing of scalar fields subject to arbitrary upper bounds,” Electron. Lett., vol. 34, pp. 162–164, Jan. 1998. [15] O. Bucci, L. Caccavale, and T. Isernia, “Optimal far-field focusing of uniformly spaced arrays subject to arbitrary upper bounds in nontarget directions,” IEEE Trans. Antennas Propag., vol. 50, pp. 1539–1554, Nov. 2002. [16] T. Isernia, P. Di Iorio, and F. Soldovieri, “An effective approach for the optimal focusing of array fields subject to arbitrary upper bounds,” IEEE Trans. Antennas Propag., vol. 48, pp. 1837–1847, Dec. 2000. [17] T. Isernia, F. Pena, O. Bucci, M. D’Urso, J. Gomez, and J. Rodriguez, “A hybrid approach for the optimal synthesis of pencil beams through array antennas,” AIEEE Trans. Antennas Propag., vol. 52, pp. 2912–2918, Nov. 2004. [18] R. Vescovo, “Consistency of constraints on nulls and on dynamic range ratio in pattern synthesis for antenna arrays,” IEEE Trans. Antennas Propag., vol. 55, pp. 2662–2670, Oct. 2007. [19] P. J. Bevelacqua and C. A. Balanis, “Geometry and weight optimization for minimizing sidelobes in wideband planar arrays,” IEEE Trans. Antennas Propag., vol. 57, pp. 1285–1289, April 2009.

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[20] J. L. Araque Quijano and G. Vecchi, “Adaptive mask scaling in alternating projections antenna synthesis,” presented at the XXIXth URSI General Assembly, Chicago, IL, Aug. 2008. [21] O. M. Bucci, C. Gennarelli, and C. Savarese, “Representation of electromagnetic fields over arbitrary surfaces by a finite and nonredundant number of samples,” IEEE Trans. Antennas Propag., vol. 46, pp. 351–359, Mar. 1998. [22] O. M. Bucci and G. Franceschetti, “On the degrees of freedom of scattered fields,” IEEE Trans. Antennas Propag., vol. 37, pp. 918–926, Jul. 1989. [23] C. A. Balanis, Antenna Theory: Analysis and Design, 3rd ed. New York: Wiley-Interscience, Apr. 2005. [24] S. Agmon, “The relaxation method for linear inequalities,” Canadian J. Math., no. 6, pp. 382–392, 1954. [25] T. S. Motzkin and I. J. Schoenberg, “The relaxation method for linear inequalities,” Canadian J. Math., no. 6, pp. 393–404, 1954. [26] D. G. Luenberger, Optimization by Vector Space Methods (Series in Decision and Control). New York: Wiley-Interscience, Jan. 1997. [27] F. Vipiana, G. Vecchi, and M. Sabbadini, “A multiresolution approach to contoured-beam antennas,” IEEE Trans. Antennas Propag., vol. 55, pp. 684–697, March 2007.

Javier Leonardo Araque Quijano was born in Bogotá, Colombia, in 1979. He received the Electronics Engineer degree from Pontificia Universidad Javeriana, Bogotá, in 2001, and the M.Sc. and Ph.D. (Dottorato di Ricerca) degrees in electronics and telecommunications engineering from the Politecnico di Torino, Turin, Italy, in 2003 and 2007, respectively, both with a grant from the ALPIP partners. In 2002, he was an Associate Professor at the Pontificia Universidad Javeriana. From 2003 to 2004, he was in the Radio Access Techniques Section, Telecom Italia Lab (TILAB). In 2005, he was a Teaching Assistant at the Politecnico di Torino, and since 2007, an Associate Researcher. His research interests are: reconfigurable antenna design, optimization techniques, computational electromagnetism, and inverse problems in electromagnetism.

Giuseppe Vecchi (M’90–SM’07–F’10) received the Laurea and Ph.D. (Dottorato di Ricerca) degrees in electronic engineering from the Politecnico di Torino, Torino, Italy, in 1985 and 1989, respectively, with doctoral research partly carried out at Polytechnic University, Farmingdale, NY. He was a Visiting Scientist with Polytechnic University from 1989 to 1990. In 1990, he joined the Department of Electronics, Politecnico di Torino, as an Assistant Professor (Ricercatore) where, from 1992 to 2000, he was an Associate Professor and, since 2000, he has been a Professor. He was a Visiting Scientist at the University of Helsinki, Finland, in 1992, and has been an Adjunct Faculty in the Department of Electrical and Computer Engineering, University of Illinois at Chicago, since 1997. His current research activities concern analytical and numerical techniques for analysis, design and diagnostics of antennas and devices, RF plasma heating, electromagnetic compatibility, and imaging.

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An Optimum Adaptive Single-Port Microwave Beamformer Based on Array Signal Vector Estimation Sadegh Farzaneh, Member, IEEE, and Abdel-Razik Sebak, Fellow, IEEE

Abstract—A single-port adaptive beamforming structure based on an optimum perturbation technique is presented. The proposed perturbation technique is based on array signal vector estimation for temporally-correlated array signals. Temporal correlation is generated by jointly reducing the receiver bandwidth and increasing the weighting rate. The error signal is generated using the estimated array signal vector that is estimated in ( + 1) perturbation cycles for an -element array. The proposed perturbation technique with the adaptive unconstrained least mean square (ULMS) algorithm achieves a lower misadjustment than previous perturbation techniques and the multi-port ULMS algorithm. With proper gradient step size variations and/or weight clipping the proposed algorithm converges with a larger maximum gradient step size than the multi-port and other single-port algorithms. The new perturbation technique can be implemented with less hardware complexity and speed than previous techniques. Index Terms—Adaptive beamforming, digital beamforming, microwave beamforming, multi-port structure, perturbation algorithms, single-port structure.

I. INTRODUCTION ANY ADAPTIVE beamforming (ABF) algorithms have been developed for the multi-port digital beamforming (DBF) that incorporates one full receiver per antenna element [1]. DBF structures allow sophisticated signal processing operations in the processor, but they suffer from high cost, size and battery power consumption [2]. In addition, the total throughput in a DBF structure is limited by the processor input/output ports speed [3]. There have been many efforts to reduce the number of receivers in the ABF structures. One structure is presented in [4], [5] which transfers the received antenna elements’ signals to the processor sequentially in a time-multiplex fashion. In [6], [7], the aerial beamforming structure is presented which uses one active branch surrounded by several reactively-loaded passive elements. Another technique is microwave beamforming (MBF) which combines signals in the microwave domain using phase shifters and variable gain attenuators/amplifiers (VGAs). In this technique, one radio frequency (RF) down-converter and analog-to-digital converter (ADC) are utilized. An MBF technique is presented in [8], [9] that integrates both high resolution phase shift and attenuation control in one simple block.

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Manuscript received February 05, 2009; revised June 09, 2009. First published December 28, 2009; current version published March 03, 2010. S. Farzaneh was with the Department of Electrical and Computer Engineering, Ecole Polytechnique de Montreal, Montreal, QC, Canada. He is now with SDP Components Inc., Montreal, QC, Canada (e-mail: [email protected]; [email protected]). A.-R. Sebak is with the Department of Electrical and Computer Engineering, Concordia University, Montreal, QC, Canada (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2009.2039303

Any single-port structure including the MBF structure is limited in signal processing due to the lack of antenna array signal vector in the processor. In [10], single-port algorithms based on the method of steepest descent and random-search have been presented. In [11], orthogonal perturbation sequences are used to estimate the gradient vector for the constrained least mean perturbations for an -elesquare (CLMS) algorithm in ment array. This technique is further developed and investigated in [12]–[15]. In [16], a coherent perturbation technique based on temporal correlation is devised which estimates the perturbations. gradient vector for the ULMS algorithm in In [17]–[19] perturbation techniques are developed that solely use phase shifters. In [20], the antenna array signal vector and as a result the gradient vector is estimated in perturbation cycles by making the consequent array signal samples temporallycorrelated. In this study, the structure in [20] is modified in different attributes. The error signal is formed using the output signal obtained by applying the weight vector to the estimated array signal vector instead of using the first perturbation output. In adinstead of dition, the array signal vector is estimated in perturbation cycles. It is observed that through these modifications the proposed single-port ULMS algorithm achieves lower misadjustment than other high performance single-port algorithms such as those in [12], [16], and [20] and even the multiport ULMS algorithm. In addition, the proposed algorithm converges with a larger gradient step size than other single-port and the multi-port ULMS algorithms. Moreover, compared to the single-port algorithm in [20], the new algorithm converges with a smaller perturbation rate which lowers the switching rate burden on the phase shifters and VGAs. Moreover, temporal correlation is generated by jointly increasing the weighting rate and reducing the receiver bandwidth. Furthermore, the proposed algorithm is studied analytically and compared with the multiport algorithm and the one in [20]. The main limitation on the proposed algorithm and those in [16], and [20] is that signals should be temporally correlated. The algorithm presented in [12]–[15] works without temporal correlation. In Section II, the new beamforming structure is introduced. In Section III, the transient analysis of the proposed adaptive algorithm is presented. Simulation results and discussions are presented in Section IV. Finally, the summary and conclusions are given in Section V. II. ADAPTIVE ULMS WITH ARRAY SIGNAL ESTIMATION A. Sequential Perturbation With

and

Perturbations

In a microwave beamforming (MBF) structure the array signal vector is not available in the processor. Fig. 1(a) shows

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Fig. 1. The proposed single-port MBF structure with the new perturbation technique. (a) Overall block diagram. (b) Schematic representation of the sequence of the operations in the beamformer processor.

the block diagram of the proposed single-port MBF structure. In adaptive ULMS algorithm the weight vector in the iteration is updated by

In (2), and

is the array signal vector in the is the error signal given by

iteration

(3) (1) and are the weight vector and the gradient where iteration, respectively, and is the gradient step vector in the size. When the array signal vector is available in the processor the gradient vector is calculated by (2)

where is the training signal. As can be seen from Fig. 1(a), the array signals are weighted using phase shifters and VGAs and combined in the microwave domain. Therefore, the gradient vector can not be computed using (2) due to lack of the array signal vector in the processor. In this work, similar to [20], the array signal vector is estimated by generating multiple beamformer outputs through applying multiple weight vectors in the microwave domain. However, in

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this work, outputs are generated instead of outputs used in [20]. In Fig. 1(a), there is a temporal correlation block that makes the array signals temporally correlated which is fundamental to the proposed technique. Temporal correlation between two samand are ples of the white noise which is band-limited to apart is given by . In [20], it was assumed that is fixed and the weighting rate or sampling rate was considered higher than the signal transmis, ) to maintain temporal corsion bandwidth ( relation. This requires using fast hardware in the structure. Alternatively, temporal correlation can be maintained by reducing . The receiver bandwidth can be rethe receiver bandwidth . It is duced by using an LPF with bandwidth noteworthy that reducing the receiver bandwidth solely modifies the temporal variation of the output signal, but does not change its spatial variation, which is important during weight calculation process. In Fig. 1(a), the temporal correlation block works in two modes: adaptation mode (AM) and normal mode (NM). In the NM, it is assumed that the weight vector is known and applied to the array. In this mode, the receiver bandwidth, controlled by LPF1, is matched to the incident signal bandwidth to maintain signal integrity. In the AM, where the algorithm calculates the optimum weight vector, the receiver bandwidth, controlled by LPF2, is reduced as shown in Fig. 1(a) to maintain temporal correlation. By lowering the LPF bandwidth, the output power is reduced. For the output signal to be in the dynamic range of the ADC, a simple baseband amplifier may be used after the LPF as shown in Fig. 1(a). It is noteworthy that the second LPF with bandwidth higher than the signal bandwidth added to the single-port structure in [20] was intended to improve the signal integrity, but the added LPF in this work (LPF2) maintains temporal correlation. Fig. 1(b) shows a schematic representation of the operations in the ABF processor. Suppose the computed weight vector iteration macrocycle is by the adaptive algorithm in the where represents a matrix transpose. In the beginning this weight vector is an arbitrary weight vector, but for the subsequent iteration macrocycles it is stored in a memory as shown in Fig. 1(b). Then, using this perturbed weight vectors are calculated weight vector by (4) where,

, and , are the perturbation vector in each perturbation cycle and is the perturbation factor. The amplitudes and phases of these weight vectors are calculated and proper control signals are sequentially applied to the phase shifters and VGAs. When weight vectors are applied, the proper outputs of the ADC are also recorded sequentially and are given by

of the ADC as shown in Fig. 1(b). Then, it can be shown that the array signal vector is estimated sequentially by (6) which is based on having temporally-correlated array signal samples. The array signal vector is formed after the serial to parallel (S/P) block shown in Fig. 1(a). In Fig. 1(a) the synchronization and timing block synchronizes different blocks. The rate that the weight vectors are applied in the microwave domain is equal to the sampling rate of the ADC. In this structure each iteration macrocycle of the adaptive algorithm is divided smaller cycles, called perturbation cycles. into The ADC, control hardware, and processor should be synchronized in each perturbation cycle. It is important that the ADC output signal samples correspond to the proper weight vectors applied to the MBF structure. were estimated sequenIn [20], the array signals for was estimated by tially by (6), but which uses and instead of , . Because , are more correlated than and , the perturbation in this work is lower than that in [20] error in estimating which strongly enhances the adaptive algorithm performance as shown later. The estimated array signal vector in (6) is used to implement the adaptive ULMS algorithm where is used instead of to estimate the gradient vector by (7) In (7), is the estimated error signal which is defined in [20] by . Another way to formulate the error signal is to apply the weight vector to the estimated array in the processor and find the error by signal vector (8) Considering the number of perturbation cycles, or , and the way the error signal is formed, there are four possibilities to implement the single-port ULMS algorithm. It is shown in this study that the single-port algorithm performance is optimum when the error signal is obtained using (8) and perturbations are used.

,

(5) In (5), the superscript denotes the Hermitian of an array, represents the array signal vector sampled at the end perturbation cycle, and refers to the output of the

B. Steady State MSE With Estimated Array Signal Vector For a multi-port structure the mean square error (MSE), de, is given by [1] fined by (9) and . For the proposed where technique in this paper which uses and , the MSE defined by may be rewritten as (10)

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TABLE I ANTENNA ARRAY AND CHANNEL INFORMATION

Fig. 2. A comparison of the steady-state MMSE for the single-port (^ multi-port ( ) algorithms.

where, that

and and

) and

. It can be shown , where

(11) and are the desired signal and the interferIn (11), ence-plus-noise covariance matrixes, respectively. Similar to a multi-port structure, the optimum solution of (10) is given by . With the reference and desired signals powers and , respectively, it can be shown that , where is the desired signal steering vector and . The optimum weight vector for the multi-port algorithm is given by or . Therefore, the optimum solution is the same as the optimum solution for the multi-port structure up to a multiplicative constant. However, the minimum mean square error (MMSE) for the proposed single-port and the multi-port algorithms are respectively given by (12) and (13) Since is a positive number, the MMSE in (12) increases monotonically with . For the multi-port MSE, is unity, but for the single-port technique can be less than unity by proper choice of and . Therefore, MMSE for the proposed single-port algorithm can be less than the multi-port algorithm. Fig. 2 shows as a function of for different values and for the channel scenario described in Table I. As can be seen, for and the minimum occurs when . It is shown in Section III-B that the gradient covariance, weight covariance, and misadjustment

are proportional to . Therefore, the reduced MMSE by the proposed technique is expected to improve the adaptive algorithm performance. For the single-port algorithm in [20] and , the MSE equals that uses given by (9). This is because and has the same spatial and statistical variation as . Therefore, the new algorithm has lower MMSE than the algorithm in [20] as well. In addition, for & or & , the MSE is the same as the multi-port algorithm. Therefore, they do and not offer the MMSE reduction that joint offers. Using is important from hardware implementaiteration is given tion. If the calculated weight vector in the , the applied perturbed weight by in subsequent perturbation cycles are vectors with , , , and . The only difference between and is the phase reversal of . Likewise, the only difference between and is the phase reversal of . Therefore, the weighting block after each antenna element can be divided into two parts. The first part is a general weighting block comphase shifter and an attenuator. This part posed of a is controlled one time in each iteration macrocycle to apply the in the beginning of weight vector each iteration macrocycle and its state remains fixed. Therefore, it does not need to be fast as its speed does not determine the temporal correlation. In practise, the speed requirement of this part is determined by the time variation of the communication phase shifter. channel. The second part is a fast one bit Different one-bit phase shifters in different elements are controlled sequentially. These phase shifters should be fast as their which conspeeds determine the maximum weighting rate trols the temporal correlation along with the LPF bandwidth. This phase shifter can be made using two fast PIN diodes. A design of this switch for microwave sampling beamformer is presented in [9] where its steady- state and transient performance are investigated. Each of these phase shifters is also switched one time in each iteration macrocycle because after the phase of each element is inverted it remains fixed during subsequent perturbation cycles. Therefore, a single control signal with an -bit shift register can control all one-bit phase shifters. Combination and phase shifters allows phase of shift control in each iteration.

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III. TRANSIENT ANALYSIS OF THE NEW SINGLE-PORT ALGORITHM A. Convergence Behaviour and Speed In this part, the convergence behaviour of the proposed single-port algorithm is studied. Averaging the weight update (1) gives (14) where

is the average weight vector and (15)

In (15),

is a diagonal matrix with diagonal elements given by Fig. 3. A comparison of the method in [20] and the method in this paper in terms of the converged weight vector.

(16) In (14), whose

is a nonlinear vector function of the weight vectors, element is given by

(17)

where

(18) In deriving (14) it is assumed that the array signals in different iteration macrocycles are independent. This is used to apply the expectation operation over terms which are functions of the weight vectors and signals separately. This issue is addressed in details in [20]. can be ignored, when where is the If largest eigenvalue of , the algorithm converges to . When elements of the diagonal matrix are the same , and the algorithm converges to , which is proportional to the multi-port weight vector. Fig. 3 studies the variations of the diagonal elements of the matrix for the proposed algorithm and the one in [20]. For this purpose the variance of the vector versus is shown for different values. As can be seen, for the proposed algorithm is much lower than the one in [20]. In particular, this term is very . Different simulasmall for the proposed technique at for , which contions show that firms the convergence of the algorithm. was ignored. In the above analysis the nonlinear term It is shown here that this term is much smaller in the proposed algorithm than in the algorithm in [20]. First, as can be seen are proportional to , from (17) and (18) the elements of

, or while that of [20] are proportional to or . Therefore, with the coefficients for the new algorithm are smaller. Fig. 4(a) shows , versus for a four-element array and with a weighting rate for both algorithms. As can be seen, , for the proposed algorithm are smaller than that of [20]. In parfor the proposed technique is much smaller due to ticular, using and instead of and as discussed estimating in Section II-A. As can be seen in Fig. 4(a), the optimum for the proposed algorithm may slightly change, but is close to . Fig. 4(b) shows the same parameters as in Fig. 4(a), . As but versus the weighting rate coefficient can be seen again, the coefficients for the proposed technique are much smaller than those based on [20]. As is shown by adaptive simulations in Section IV, this improves the convergence of the proposed technique for small values. Another reason for is that the adaptive algorithm minimizes the term ignoring in as it converges. The convergence speed is determined by the smallest and . If the maximum permissible gradient eigenvalue of is used, the convergence time is step size proportional to . Different simulations show that is smaller for the proposed single-port algorithm than for the multi-port one. However, the maximum permissible in the single-port algorithm is also limited by the nonlinear term in the beginning of iterations. If a variable with small initial values and large final values is used, the single-port algorithm converges with larger final values. In addition, by clipping the weight vectors similar to [12], the proposed single-port algorithm converges with larger than the multi-port ULMS. B. Gradient Covariance, Weight Covariance, and Misadjustment Gradient covariance affects the transient performance of the adaptive algorithm. It can be shown that the gradient covariance for the proposed single-port algorithm when algorithm converges is given by (19)

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Fig. 4. Nonlinear term coefficients in e

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. (a) Versus perturbation coefficients ( ). (b) Versus the weighting rate coefficient.

Where is given by (12). Noting that the gradient covariance [1] and shows for the multi-port ULMS is that the proposed algorithm achieves lower gradient covariance than the multi-port algorithm. For the perturbation technique in because it uses [20] the gradient covariance is close to the same error signal as the multi-port structure. and after some manipulations, By defining it can be shown that the weight covariance matrix is given by (20) which is again proportional to . Using (20), it can be shown that the misadjustment for the proposed algorithm is given by

(21) that is less than unity. which is proportional to , and , (21) is converted to the multiWhen port ULMS algorithm misadjustment. For small values, (21) reduces to (22) After some manipulations and noting that is a large number, (22) is given by

(23) After some manipulations it can be shown for and , that the factor in (23) is less than unity. Noting that for the multi-port algorithm plus the fact that and are less than unity, proves that the misadjustment for the proposed single-port

algorithm with multi-port algorithm.

and

is lower than that of the

IV. SIMULATION RESULTS AND DISCUSSIONS The same array with the scenario described in Table I is used in the simulations. Adaptive ULMS is simulated with the multiand port and single port algorithms. The gradient step size are set to 0.0005 and 2, respectively. Desired signal, noise and interference sources are modeled by spatially-independent random Gaussian sequences. These signals are generated by passing temporally independent signal samples through an ideal digital filter. Temporally-correlated samples are generated by controlling the ratio of the LPF bandwidth and the sampling rate. For the multi-port algorithm the signal samples are assumed independent, an assumption that generally made for a DBF structure. Therefore, there was no need to pass them through LPF2. However, it is assured that average signal, interference and noise powers in the receiver for both the single-port and multi-port structures are the same. The results are averaged over 1000 realizations. For both the multiport and single-port structures is a Gaussian pulse train which are correlated with the corresponding desired signals. For the multi-port structure, samples are generated to be independent, but for the multiport structure there might be a small correlation depending and the number of perturbation in on the perturbation rate . When , , samples are each iteration also independent for the single-port structure. In addition, it is assured that both the reference signals have the same average powers. Fig. 5 compares the performance of the single-port algorithm with different perturbation methods and the multi-port algorithm. For the single-port structure four cases described in Section II-B are considered which depend on the number of or ) and the way the error perturbation cycles ( refers to the error signal signal is formed. In Fig. 5, defined in [20] and refers to the error signal defined in this paper by (8). Fig. 5(a) shows the actual excess MSE normalized to the actual MMSE using the optimum single-port weight vector as a

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Fig. 5. A comparison of the adaptive multi-port and single-port ULMS with different perturbation techniques. (a) Normalized excess MSE. (b) Average power pattern.

function of the iteration numbers. This parameter is the misadjustment factor when the algorithm converges. As can be seen in , Fig. 5(a), the lowest misadjustment is achieved with and , which is the proposed algorithm. It can be seen that the performance for the two cases with ( or ) are much lower than with . In the performance with is suaddition, with case. It is noteworthy that the misadjustment perior to , and is even less than that with of the multi-port algorithm. Moreover, in Fig. 5(a) the proposed is also shown which is sualgorithm simulated with , ) with . perior to the algorithm in [20] ( This shows that the new method works with less hardware speed requirements than the method in [20]. For small iteration numbers, jumps in the curves related to the single-port algorithms . However, due to the dein Fig. 5(a) are due to the term , as algorithm converges the effect of this term is dicreasing minished. As can be seen, this effect for the proposed single-port is much less pronounced algorithm in this paper with than other single-port approaches. This allows using larger values which increases the convergence speed. By clipping the weight vectors after each iteration cycle this effect is significantly reduced. Fig. 5(b) shows the average power pattern obcalculated by tained using the converged weight vector where is the steering vector in the direction and is the last iteration number. By averaging in this manner, the effect of the weight variance is also included. The nulls depths in Fig. 5(b) follow the misadjustment trend in Fig. 5(a). The deepest nulls are achieved for and which is also superior to the multi-port structure. The proposed approach in this paper ( and ) achieves null depths deeper by around 7 dB than the approach , ). in [20] ( , with It is noteworthy that for and are uncorrelated, but for there is a small correlation. However, it should be noted that the correlation between adjacent samples are important. One problem with [20] was estimated using and which rewas that

quired them to be correlated. This increased the hardware speed requirements compared to proposed structure in this paper. Fig. 6 compares the proposed single-port algorithm with the multi-port ULMS and other single-port algorithms in the literature. Two perturbation methods that have been developed with ULMS algorithm are compared with the proposed technique. One technique which is presented in [9] and developed perturbations further in [13]–[15] for CLMS algorithm uses in each iteration. This technique is developed for ULMS algorithm in [12]. Another technique is presented in [16] which uses perturbations in each iteration and relies on temporal correlation similar to the proposed technique in this paper. Fig. 6(a) shows the normalized excess MSE for the multi-port algorithm, , 0.05, the the single-port algorithm in [12] with single-port algorithm in [16] with , 8 and , and , . The scenario the proposed technique with . As can be seen, the proin Table I is used and , , ) achieves posed single-port algorithm ( the lowest misadjustment. Fig. 6(b) shows the corresponding average power patterns with the converged weight vectors. As can be seen, the deepest nulls is also achieved with the prois the optimum value posed technique. In Fig. 6, for the technique in [12] which is obtained based on the analytical formulas developed in and [12] verified by our simulais also simulated which improves tions. In addition the convergence but increases the misadjustment. The for the algorithm in [14] is chosen through different simulation to find a value that the algorithm converges well. For this techand are shown. nique, simulations with both improves the algorithm perforAs can be seen increasing mance, but it is still lower in performance than the proposed algorithm. In all the simulations in Fig. 6 a clipping operation are used. It is observed that the proposed in [13] with clipping operation strongly improves the performance of all algorithms except the multi-port algorithm. In addition, it is observed that by using proper variable or using clipping operation the proposed technique converges with a larger maximum than all other algorithms including the multi-port algorithm.

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Fig. 6. A comparison of the proposed single-port algorithm, the multi-port algorithm and other single-port algorithms. (a) Normalized excess MSE. (b) Average power pattern.

seen the misadjustment ratio in dB is less than zero for which shows that the single-port algorithm misadjustment is lower than that of the multi-port one. The maximum difference between theory and simulation is less than 1dB. This error hapin (17) in calcupens due to ignoring the nonlinear term lating the misadjustment for the single-port algorithm. As can . Also, looking be seen in Fig. 7 this error is minimum at at Fig. 4(b), it can be seen that the nonlinear terms are minimum . Both simulations and theory show that the single-port at . The maximum error misadjustment is minimum with between (22) and (23) is 0.04 dB which verifies the approximations made in deriving (23). V. CONCLUSION Fig. 7. Misadjustment ratio using the analytical formulas and simulations with

and different perturbation rates.

= 02

For example, for the scenario in Table I, with , the achieved SINR gain after convergence with proposed algorithm, -perturbation with the multi-port algorithm, [12], and -perturbation with [16] and are 22.5 dB, 12.5 dB, 4.3 dB, and 11.3 dB, respectively. This shows that the proposed algorithm works with larger values. The proposed algorithm is compared with other algorithms in different scenarios and similar trends are observed. The effect of the quantized perturbations on the performance of the algorithm is also investigated. It is observed that a weighting resolution higher than five bits does not affect the adaptive algorithm performance considerably. In addition, the amplitude gain quantization error does not affect the algorithm performance with more than three bit resolution. Fig. 7 shows the ratio of the single-port to the multi-port misadjustment obtained using derived formulas and simulations and different values. For the single-port algowith rithm, the misadjustment is obtained using (22) and (23). The channel scenario in Table I is used in the simulations. As can be

A new perturbation technique for single-port adaptive microwave beamforming is proposed. The proposed perturbation technique is based on making the array signals temporally correlated. Temporal correlation is maintained by reducing the receiver bandwidth and/or increasing the weighting rate. The proand achieves a lower posed algorithm with misadjustment than the multi-port algorithm and other singleport algorithms. The convergence speed of the new single-port MBF algorithm is also very close to that of the corresponding multi-port algorithm with the same gradient step size. The proposed algorithm converges with larger maximum gradient step size than the multi-port and other single-port algorithms. Using is optimal from both DSP computational complexity and microwave hardware implementation point of views. REFERENCES [1] L. C. Godara, Smart Antennas. Boca Raton, FL: CRC Press, 2004. [2] T. Ohira, “Adaptive array antenna beamforming architectures as viewed by a microwave circuit designer,” in Proc. Asia-Pacific Microwave Conf., Sydney, Australia, Dec. 2000, pp. 828–833. [3] S. Jeon, Y. Wang, Y. Qian, T. Itoh, and T. Ohira, “A novel smart antenna system implementation for broadband wireless communication,” IEEE Trans. Antennas Propag., vol. 45, pp. 2324–2332, Dec. 1997. [4] J. D. Fredrick, Y. Wang, and T. Itoh, “A smart antenna receiver array using a single RF channel and digital beamforming,” IEEE Trans. Microw. Theory Tech., vol. 50, pp. 3052–3058, Dec. 2002.

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[5] J. D. Fredrick, Y. Wang, and T. Itoh, “Smart antenna based on spatial multiplexing of local elements (SMILE) for mutual coupling reduction,” IEEE Trans. Antennas Propag., vol. 52, no. 1, pp. 106–114, Jan. 2004. [6] C. Sun, A. Hirata, T. Ohira, and N. C. Karmakar, “Fast beamforming of electronically steerable parasitic array radiator antennas: Theory and experiment,” IEEE Trans. Antennas Propag., vol. 52, no. 7, pp. 1819–11832, Jul. 2004. [7] E. Taillefer et al., “Enhanced reactance-domain ESPRIT algorithm employing multiple beams and translational-invariance soft selection for direction-of-arrival estimation in the full azimuth,” IEEE Trans. Antennas Propag., vol. 56, no. 8, pp. 2514–2526, Aug. 2008. [8] S. Farzaneh and A. Sebak, “A novel amplitude-Phase weighting for analog microwave beamforming,” IEEE Trans. Antennas Propag., vol. 54, no. 7, pp. 1997–2008, Jul. 2006. [9] S. Farzaneh and A. Sebak, “Microwave sampling beamformer: Prototype verification and switch design,” IEEE Trans. Microw. Theory Tech., vol. 57, no. 1, pp. 36–44, Jan. 2009. [10] B. Widrow and J. McCool, “A comparison of adaptive algorithms based on the methods of steepest descent and random search,” IEEE Trans. Antenna. Propag., vol. 24, no. 5, pp. 615–637, Sep. 1976. [11] A. Cantoni, “Application of orthogonal perturbation sequences to adaptive beamforming,” IEEE Trans. Antennas Propag., vol. 28, pp. 191–202, Mar. 1980. [12] D. C. Farden and R. M. Davis, “Orthogonal weight perturbation algorithms in partially adaptive arrays,” IEEE Trans. Antennas Propag., vol. 33, no. 1, pp. 56–63, Jan. 1985. [13] I. Webster, R. J. Evans, and A. Cantoni, “Robust perturbation algorithms for adaptive antenna arrays,” IEEE Trans. Antennas Propag., vol. 38, pp. 195–201, Feb. 1990. [14] L. C. Godara and A. Cantoni, “Analysis of constrained LMS algorithm with application to adaptive beamforming using perturbation sequences,” IEEE Trans. Antennas Propag., vol. 34, pp. 368–379, Mar. 1986. [15] L. C. Godara and A. Cantoni, “Analysis of the performance of adaptive beamforming using perturbation sequences,” IEEE Trans. Antennas Propag., vol. 31, pp. 268–279, Mar. 1983. [16] R. M. Davis, D. C. Farden, and P. J.-S. Sher, “A coherent perturbation algorithm,” IEEE Trans. Antennas Propag., vol. 34, pp. 350–387, Mar. 1986. [17] S. Denno and T. Ohiro, “Modified constant modulus algorithm for digital signal processing adaptive antennas with microwave analog beamforming,” IEEE Trans. Antennas Propag., vol. 50, no. 6, pp. 850–857, Jun. 2002. [18] R. M. Davis, “Phase-only LMS and perturbation algorithms,” IEEE Trans. Aerosp. Electron. Syst., vol. 34, no. 1, pp. 169–178, Jan. 1998. [19] T. A. Denidni, D. McNeil, and G. Y. Delisle, “Experimental investigations of a new adaptive dual antenna array for handset applications,” IEEE Trans. Veh. Technol., vol. 52, no. 6, pp. 1417–1423, Nov. 2003. [20] S. Farzaneh and A. R. Sebak, “Fast adaptive microwave beamforming using array signal estimation,” IEEE Trans. Antennas Propag., vol. 55, no. 3, pp. 850–858, Mar. 2007.

Sadegh Farzaneh (S’05–M’08) received the B.S. and M.S. degrees (with honors) from Shiraz University, Iran, in 1996 and 1999, respectively, and the Ph.D. degree from Concordia University, Canada in 2008, all in electrical engineering. From 1997 to 2000, he worked with Iran Components Industries on the design of microwave circuits. Between 2000 and 2003, he was with the Electrical Engineering Department, Kazeroun Azad University, as a Lecturer. From 2008 to 2009, he worked at Concordia University and Ecole Polytechnique de Montreal as a Research Associate and Postdoctoral Fellow. He is currently working at SDP Components Inc., Montreal, as a Research Scientist. His research interests include smart antennas, phased array antennas, microwave circuits, and wireless communication. Dr. Farzaneh was the recipient of the Concordia University School of Graduate Studies Doctoral Teaching Assistantship, International Fee Remission Award in 2005, and the France and André Desmarais Graduate Fellowship in 2006.

Abdel-Razik Sebak (M’81–SM’91–F’09) received the B.Sc. degree (with honors) in electrical engineering from Cairo University, Egypt, in 1976, the B.Sc. degree in applied mathematics from Ein Shams University, Egypt, in 1978, and the M.Eng. and Ph.D. degrees in electrical engineering from the University of Manitoba, Winnipeg, MB, Canada, in 1982 and 1984, respectively. From 1984 to 1986, he was with the Canadian Marconi Company, Kanata, Ontario, working on the design of microstrip phased array antennas. From 1987 to 2002, he was a Professor in the Electrical and Computer Engineering Department, University of Manitoba, Winnipeg. He is currently a Professor of Electrical and Computer Engineering, Concordia University, Montreal. His current research interests include phased array antennas, computational electromagnetics, integrated antennas, electromagnetic theory, interaction of EM waves with new materials and bio-electromagnetics. Dr. Sebak is a member of the International Union of Radio Science Commission B. He received the 2003–2004 Faculty of Engineering, Concordia University double Merit Award for outstanding Teaching and Research. He has also received the 2000 and 1992 University of Manitoba Merit Award for outstanding Teaching and Research, the 1994 Rh. Award for Outstanding Contributions to Scholarship and Research in the Applied Sciences category, and the 1996 Faculty of Engineering Superior Academic Performance. He has served as Chair for the IEEE Canada Awards and Recognition Committee (2002–2004) and IEEE Canada CONAC (1999–2001). He has also served as Chair of the IEEE Winnipeg Section (1996–97). He is the Technical Program Co-Chair (2006) and served as the Treasurer (1992, 1996, and 2000) and Publicity Chair (1994) for the Symposium on Antenna Technology and Applied Electromagnetics (ANTEM). He has also served as Chair (1991–92) of the joint IEEE AP/MTT/VT Winnipeg Chapter. He received, as the Chapter Chair, the 1992 IEEE Antennas and Propagation Society Best Chapter Award.

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Decoupled 2D Direction of Arrival Estimation Using Compact Uniform Circular Arrays in the Presence of Elevation-Dependent Mutual Coupling Bu Hong Wang, Member, IEEE, Hon Tat Hui, Senior Member, IEEE, and Mook Seng Leong

Abstract—Based on the rank reduction theory (RARE), a decoupled method for 2D direction of arrival (DOA) estimation in the presence of elevation-dependent mutual coupling is proposed for compact uniform circular arrays (UCAs). Using a new formulation of the beamspace array manifold in the presence of mutual coupling, the azimuth estimates are decoupled from the elevation estimates and obtained with no need for the exact knowledge of mutual coupling. For the elevation estimation, a 1D parameter search in the elevation space for every azimuth estimate is performed with the elevation-dependent mutual coupling effect compensated efficiently. Though the computational load for the elevation estimation is increased compared to that of the original UCA-RARE algorithm, the 1D parameter search in our method overcomes most of the inherent shortcomings of the UCA-RARE algorithm. This enables unambiguous and paired 2D DOA estimation with the elevation-dependent mutual coupling effect being compensated for effectively. Numerical examples are presented to demonstrate the effectiveness of the proposed method. Index Terms—Mutual coupling, rank reduction theory (RARE), uniform circular array, 2D DOA estimation.

I. INTRODUCTION

U

NIFORM circular arrays (UCAs) are of particular importance in direction of arrival (DOA) estimation since they can provide 360 azimuthal converge and estimate both azimuth and elevation angles simultaneously. In addition, due to its circular symmetry, UCAs possess an azimuthally invariant beampattern. 2D DOA estimation with UCAs [1]–[6] is therefore very useful in practical situations. Although the generalization of 1-D classical subspace-based algorithms [7]–[9] to 2D DOA estimation is straightforward, the high computational cost due to the 2-D spectral search or iterative optimization procedures makes them not suitable for real-time applications. To use UCAs for efficient 2D DOA estimation, the beamspace transformation, based on the phase-mode excitation principle [1], is usually applied to acquire the desired Vandermode structure for the steering vector in the mode space. Some effective algorithms of this kind include UCA-RB-MUSIC [2], UCA-ESPRIT [2], and UCA-RARE [3], among which the UCA-RARE is more attractive since it decouples azimuth estimation from Manuscript received April 11, 2009; revised August 05, 2009. First published December 31, 2009; current version published March 03, 2010. This work was supported in part by the US ONR research project under the project PR: 09PR03332-01 and in part by the National Science Foundation of China under Grant 60601016. The authors are with the Department of Electrical and Computer Engineering, National University of Singapore, Singapore 117576, Singapore (e-mail: [email protected]; [email protected]; [email protected]). Digital Object Identifier 10.1109/TAP.2009.2039323

elevation estimation and relaxes the assumption of omnidirectional element patterns. However the prior knowledge of the spatial source structure needed for elevation estimation and the inherent spurious estimation problem for azimuth partly limit its scope of practical applications. All the 2D DOA estimation algorithms referred to above ignore the mutual coupling effect, which ultimately destroys the underlying model assumptions needed for their efficient implementations. Actually, mutual coupling can be rather significant in UCAs and furthermore, it is elevation-dependent. This dependence of the mutual coupling effect on elevation angle is but commonly ignored in favor of a simplified mutual coupling model or to facilitate the design of those theoretically perfect array signal algorithms. Most of the high-resolution DOA estimation methods are in fact very sensitive to the array manifold errors due to the mutual coupling effect. In practical scenarios of 2D DOA estimation, it is therefore necessary to take into account the elevation dependence of the mutual coupling effect. In [4] an excellent search-free and robust 2D DOA estimation algorithm was proposed for UCAs with the capability that mutual coupling could be taken into account. Based on a circuit model for the array and with the open-circuit voltages of the antenna elements expanded in spherical modes, the array manifold model in [4] has successfully taken the elevation-dependent mutual coupling effect (including platform effect) into account. On the other hand, the inherent ambiguous estimates associated with the original UCA_RARE algorithm, though noted clearly, have not yet been properly resolved in [4]. It will be shown in this paper that the beamspace array manifold in the presence of mutual coupling actually possesses an alternative analytical expression to that used in [4] and which can be employed to simplify the mutual coupling compensation for compact UCAs. In this paper, we first use a method proposed recently by Hui [10] to calculate the elevation-dependent receiving mutual impedances for a compact UCA. The advantage of this method is that it can take into account the change of mutual coupling with the elevation direction of the external impinging source. A decoupled 2D DOA estimation algorithm, taking into account the elevation-dependent mutual coupling, is then proposed for compact UCAs. Stemming from the UCA-RARE algorithm in [3], the azimuth estimation in our new method is decoupled from the elevation estimation and can be performed with no need for the exact knowledge of the mutual coupling. The 1D parameter search in the elevation space (for each azimuth estimate) is then employed for the paired and unambiguous 2D DOA estimation. In the 1D parameter search for elevation estimation, the

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elevation-dependent mutual coupling effect can be efficiently compensated by the elevation-dependent receiving mutual impedances. Though the computational load of our new method for the elevation estimation is higher than that of the original UCA-RARE algorithm, the 1D parameter search for elevation estimation overcomes most of the inherent shortcomings associated with the original UCA-RARE algorithm and the elevationdependent mutual coupling compensation can be performed simultaneously. Compared with the original UCA-RARE algorithm, the advantages of our new method are that the elevation-dependent mutual coupling effect can be efficiently compensated, no prior knowledge on the source structures is required for the elevation estimation, spurious estimates or signal cancellations can all be avoided, and finally the paired azimuth and elevation estimates can be obtained automatically. Computer simulations are provided to verify the effectiveness and validity of the proposed method.

(9)

(10) In the above equations, is a steering vector is a DFT matrix, and in beamspace, is the Bessel function of the first kind and order . For UCAs, phase modes can be excited at a reasonable only strength [1]. To make the residual error negligible, the number , of elements in the UCA need to satisfy the condition which equivalently means that the circumferential spacing be. This tween adjacent array elements should be less than leads to a serious mutual coupling effect for a given aperture of a UCA. steering vector In the presence of mutual coupling, the can be modeled as (11)

II. ARRAY DATA MODEL IN THE PRESENCE OF MUTUAL COUPLING identical elements, which Consider a UCA consisting of are uniformly distributed over the circumference of a circle of radius in the X-Y plane. Assume that narrowband sources, , impinge on the array from diswith a wavenumber tinct directions , , where is the elevation angle measured from the Z-axis and is the azimuth angle measured from the X-axis counter-clockwise. In the absence of mutual coupling effect, a widely used array data model can be described as (1) where is an noise-corrupted array snapshot vector, is an signal vector, and is an noise is an matrix, the vector. The array manifold matrix . columns of which are steering vectors That is, (2)

(3) (4) The mode space transformation for a UCA is generally completed via the DFT (Discrete Fourier Transformation) of the steering vector in the element space (3). That is, (5) where

(6) (7) (8)

matrix is the mutual coupling matrix where the (MCM) of the UCA. In this study, the elevation dependence of the mutual coupling effect is taken into account so that the MCM will vary with elevation angle . For a fixed elevation angle, it is well known that a complex symmetric circulant matrix provides a satisfactory model for the MCM of a UCA [11]. III. ELEVATION DEPENDENCE OF THE MUTUAL COUPLING EFFECT The open-circuit voltage method [12] is a widely used method for mutual coupling compensation. It relates the antenna terminal voltages with mutual coupling present to the so-called open-circuit voltages. A major shortcoming with the open-circuit voltage method is that the open-circuit voltages are actually not free from mutual coupling (i.e. the voltages received by isolated elements) as they were supposed to be. Except that, the open-circuit voltages method is inherently independent of the directions of the incoming source. This is because the mutual impedances are defined irrespective of the source directions [13]. There are other more accurate methods for the mutual coupling analysis [14]–[16]. However their practical applications in the receiving arrays are limited either due to the same shortcomings as the open-circuit voltage method, unrealistic approximations, or a complicated procedure for implementation. The receiving-mutual-impedance method proposed by Hui [10] can more accurately model the mutual coupling situation for a receiving array, in which the effect of the antenna terminal loads and the external signal source can all be taken into account for the calculation of mutual impedances. Different from the open-circuit voltage method, the receiving-mutual-impedance method relates the measured terminal voltages to the decoupled terminal voltages through a newly-defined receiving mu[10]. This method has been applied to tual impedance many situations [17], [18], and in which a better decoupling performance than the open-circuit voltage method has been demonstrated. In this paper, the capability of this method in taking into account of the direction change of the impinging source is employed to calculate the elevation-dependent receiving mutual

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TABLE I THE RECEIVING MUTUAL IMPEDANCES AT DIFFERENT ELEVATION ANGLES OF AN IMPINGING SOURCE

impedances of a UCA. These receiving mutual impedances are then used to decouple the elevation-dependent mutual coupling effect. Consider for example a compact UCA with monopoles tuned to . The radius of UCA is . The length of the monopole antennas is set as and the diameter of the monopole wires is 0.6 mm. This monopole array is mounted over a large conducting plane. To construct the complex circulant MCM matrix, the receiving shown in Table I (shown mutual impedances at the end of the paper) are calculated with receiving-mutual-impedance method [10] for different elevation angles of an impinging source. For comparison, the corresponding conventional mutual impedances are also calculated with the open-circuit voltage method and shown at the bottom line of Table I. Note that the conventional mutual impedances do not vary with the elevation angle of the impinging source. Although the variation of the receiving mutual impedances with elevation angle as shown in Table I seems not very significant, the change of the mutual coupling effect as indicated by these receiving mutual impedances has actually a significant effect in DOA estimations. This can be seen from the contour plots of the spatial spectra of the 2D-MUSIC algorithm shown in Fig. 1 for the detection of two impinging sources at and . It can be observed that the cases of mutual coupling compensation with the open-circuit voltage method and the single-elevation-angle ) receiving-mutual-impedance method (obtained at all fail to resolve the two sources. However, when the mutual coupling effect is compensated by the elevation-dependent receiving-mutual-impedance method, the two sources can be accurately detected. This result shows the sensitive response of the high resolution DOA estimation algorithm to the distortion of array manifold model due to the changing mutual coupling effect with elevation angle. It also tells us the necessity to use an elevation-dependent mutual coupling compensation method in the 2D DOA estimation. Practically, we can calculate the receiving mutual impedances for different elevation angles beforehand and construct

Fig. 1. The contour plots of the spatial spectra of 2D MUSIC obtained with different mutual coupling compensation methods.

a look-up table for real time applications. If the elevation-dependence of mutual coupling is not particularly evident, we can compress the look-up table data with one receiving mutual impedance for an elevation angle range, such as a 10 interval, over which the variation of the mutual coupling is assumed negligible. By virtue of this well-established look-up table for the receiving mutual impedances at different elevation angles, we can compensate for the mutual coupling effect more efficiently and accurately. When this is combined with the new 2D DOA estimation algorithm developed in the next section, more accurate and efficient 2D DOA estimation can be achieved, especially with compact antenna arrays. IV. DECOUPLED AZIMUTH ESTIMATION The algorithm proposed in this section stems from the original UCA-RARE algorithm [3]. Despite its attractive ability to decouple the azimuth estimation from the elevation estimation, improvements with respect to the following points

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are still required: (1) mutual coupling, especially the elevation-dependent mutual coupling, was not considered in the original UCA-RARE algorithms for 2D DOA estimation; (2) the inherent spurious DOA estimates or signal cancellation phenomenon and the corresponding measures to resolve them have not been analyzed in details; and (3) the a priori knowledge on the sources structure, i.e., the number of sources with the same azimuth angle, is needed for the elevation estimation. The new algorithm proposed below can overcome the above shortcomings associated with the original UCA-RARE algorithm. in the presence of We start from the steering vector mutual coupling. Due to the circulant MCM for a UCA and (i.e. the the symmetry structure of the circulant vector first row of the MCM), an interesting and useful observation is is essentially the periodic convolution of the ideal that with the circulant vector , i.e., steering vector (12)

where “ ” denotes the Hadamard product (i.e., element-wise product) of vectors. Recall the formulation for the steering vector used in [3], i.e., (15) where (16) (17) (18) (19) is a anti-diagonal unit matrix and in (16), In (17), is assumed for omnidirectional elements. In the presence of the mutual coupling, (15) can be rewritten as

where “ ” denotes the periodic convolution of discrete sequences. This interesting property facilitates the formulation considerably. The of the mode space transformation of is essentially the DFT mode space transformation matrix or IDFT (inverse discrete Fourier transformation) and can be reformulated as

(20) Due to the special structure of

.. .

.. .

, we notice that (21)

.. . which means

(22) (23)

.. .

(13) denotes the elewhere denotes the ments of the DFT and elements of the IDFT. According to the discrete convolution theory, the mode space transformation of can be written as

is a centro-symmetry vector which results That is, vector given by in a more concise expression for

(24) with (25) Now we can account for the elevation-dependent mutual couof the original UCA-RARE pling effect just by revising , where for difalgorithm with ferent source elevation angles can be calculated using the receiving-mutual-impedance method in [10]. Based on the RARE theory, we can achieve the azimuth estimates without the exact knowledge of the mutual coupling. This is due to the orthogonality between the noise subspace and signal subspace of the array covariance matrix , i.e.,

(14)

(26)

WANG et al.: DECOUPLED 2D DOA ESTIMATION USING COMPACT UNIFORM CIRCULAR ARRAYS

Similar to the UCA-RARE, a spatial spectrum function for the azimuth estimation can be constructed as

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(see [3]) cannot be deduced and the recursive relationship can no longer be employed to construct the UCA-ESPRIT-like algorithm.

(27) V. ELEVATION ESTIMATION WITH MUTUAL COUPLING COMPENSATION

or (28) (29) where is the determinant of a matrix. The denominator of (28) is suggested in [3] to eliminate the spurious azimuth estifor the case of impinging sources with the mates at . Due to the Vandermonde strucsame elevation angle in (18), the polynomial-rooting-based method deture of veloped in the original UCA-RARE method can still be used to obtain the azimuth estimates in the presence of mutual coupling, but no knowledge of the mutual coupling is needed for its imfor and plementation. Furthermore, we found that for are also spurious azimuth estimates from the spectrum function of (27), which have not been reported in the development of the original UCA-RARE algorithm and which cannot be cancelled by the denominator of the (28). To obtain the elevation estimation for a given azimuth estimate, a specifically-designed closed-form algorithm similar to UCA-ESPRIT was proposed in the original UCA-RARE algorithm. Despite its search-free implementation, there are some shortcomings that make it somewhat unsuitable for practical applications. (1) In our intensive and numerous simulations using the original UCA-RARE algorithm under noisy conditions or with some special sources structures, such as sources with azpresent simultaneously (the rank of the imuths and matrix will be smaller than the number of the sources with the same azimuth angle), the multiplicity of the roots on the unit circle of (27) (i.e., the number in [3]) does not always equal to the number of sources with the same azimuth angle. Consequently, in order to acquire the final elevation estimates accurately, the exact a priori knowledge of the source structure, i.e., the number of sources with the same azimuth, is usually required to construct the linear relationship of the subspace fitting used in the subsequent UCA-ESPRIT-like algorithm. This is actually unrealistic in the practical 2D DOA estimation. (2) The spurious azimuth estimate ( for or for ) has not been taken into account in the subsequent elevation estimation. This will result in errors in the final paired elevation estimates. (3) If the denominator of (28) is used for the cancellation of the spurious azimuth , a signal cancellation would appear at estimate at when there is a source exactly at . All these spurious azimuth estimates will be demonstrated in our simulation examples. It should be noted that due to the elevation-dependence , a closed-form UCA-ESPRIT-like algorithm for of elevation estimation proposed in the original UCA-RARE algorithm cannot be applied any more. This is because a to the Bessel function matrix constant matrix relating

To obtain the elevation estimation in the presence of elevation-dependent mutual coupling effect, we suggest a searchbased algorithm based on the null space analysis of the matrix . Though this will require a greater computational load than the original UCA-RARE algorithm, the inherent shortcomings associated with the original UCA-RARE algorithm as described in Section IV can all be overcome. The following criteria establish the theoretical basis for our new elevation estimation. Under the assumption of unambiguous array manifold in the presence of mutual coupling: can be found belonging to null space of matrix (i) no for spurious azimuth estimates; (ii) only associated with the true elevation angles befor true azimuth long to the null space of the matrix angles. With the assumption that the array manifold including mutual ambiguity, the proof of the above coupling is free of rank criteria can be completed readily by contradiction and is omitted here for brevity. Criterion (I) guarantees that the spurious azfor and for imuth estimates, such as can all be cancelled off in the subsequent elevation estimation since no elevation estimates can be found to pair with these spurious azimuth estimates. Criterion (II) guarantees that all elevation estimates which pair with the same azimuth angle can be estimated accurately. Interestingly, the measure (i.e. the denominator of (28)) used in the original UCA-RARE for the spurious estimation associated with the scenario that the sources possess the same elevation angle is not necessary any more. This is because in the searching procedure for elevation estimates, the will be canceled off from the spurious azimuths at ultimate azimuth estimates due to the fact that there is no elevation estimate paired with them. At the same time, this also makes for avoiding the signal cancellation phenomenon as mentioned above. Due to the relatively smaller searching range for elevacompared with for azimuth, the additional tion, computational load due to parameter search is still acceptable in practical applications, after considering its robustness to the spurious azimuth estimates. With the above considerations, the steps for the proposed decoupled 2D DOA estimation algorithm can be summarized as follows: for (i) Calculate the circulant mutual coupling vector different elevations by the receiving-mutual-impedance as method [10] and construct a look-up table for (30) where (31) (32)

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(ii) Obtain the azimuth estimates with a similar method as UCA-RARE from the spatial spectrum function (33) The above spatial spectrum function is used, rather than (28), in case that cancellation of the sources at would appear. In this step, all possible spurious azimuth estimates can still exist but will be eliminated in the final paired 2D DOA estimation by the elevation parameter search in the next step. (iii) For every azimuth estimate, including the spurious estiand construct the basis mamates, calculate matrix for its null-space, for example, by the Matlab trix function . (iv) Using the 1D elevation parameter search in the range , find all elevation angles such that the belongs to the corresponding null-space of . These are the corresponding elevation estimates for the given azimuth estimates. The dependence between two subspaces can be evaluated, for exIf ample, by the Matlab function there is no which belongs to the null-space of , the given azimuth estimate is then regarded as a spurious estimate and will be removed from the final paired 2D DOA estimation. The paired 2D DOA estimation found during the 1D elevation parameter search will serve as the final 2D DOA estimation. Of particular importance is that in this step, the elevation-dependent mutual coupling ef. fect can be efficiently compensated by It should be further noted that in the above formulation of the decoupled 2D DOA estimation algorithm, omnidirectional . Straightforward extenelement patterns are assumed for sion of this algorithm to directional element patterns can be with the corresponding Fourier coachieved by revising efficients of the directional element patterns as suggested in the original UCA-RARE algorithm.

Fig. 2. The root distribution for the numerator of (28) with two impinging sources at [ = 30  = 20 ] and [ = 30  = 60 ].

VI. SIMULATION RESULTS ON DOA ESTIMATION A. The Spurious Azimuth Estimates of the Original UCA-RARE Algorithm A UCA composed of elements with a radius is considered. This array has been studied in [3] and the elements are assumed to be ideal sensors with no mutual coupling between them. The ambiguous scenarios associated with the original UCA-RARE algorithm are demonstrated in the following cases of 2D DOA estimation. (i) For the case of two impinging sources at and (with same elevation angle), Figs. 2 and 3 depict the root distributions nearby the unit circle for the numerator and denominator of (28), , respectively. A total of four spurious roots at , , can be seen in Fig. 2. But and here are only two corresponding signal roots at and on the unit circle for the denominator of (28) as shown in Fig. 3. This results in

Fig. 3. The corresponding root distribution for the denominator of (28) for the case in Fig. 2.

and a final spurious azimuth estimations at in the original UCA-RARE. This means that the denominator of (28) cannot eliminate the spurious azand . imuth estimates at (ii) For the case of three sources at , , and , the elevation angles of the first and second sources are the same but different from that of the third one. The azimuth angle of the third source is exactly at . Similarly, in this case, a total of three spurious roots , , at appear in the root distriand bution for the numerator of (28) and there are signal roots

WANG et al.: DECOUPLED 2D DOA ESTIMATION USING COMPACT UNIFORM CIRCULAR ARRAYS

Fig. 4. 2D DOA estimation using the new algorithm with mutual coupling compensated by the elevation-dependent receiving-mutual-impedance method.

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Fig. 5. 2D DOA estimation using the new algorithm with mutual coupling compensated by the single-elevation-angle receiving-mutual-impedances method with receiving mutual impedances obtained at an elevation angle of 45 .

at and on the unit circle for the denominator of (28). This leads to a final signal cancellation . Furthermore, the spurious azat the azimuth and still exist. imuth estimates at (iii) Another scenario that will lead to erroneous elevation estimates in the original UCA-RARE algorithm is when the azimuth angles of the impinging sources satisfy the . This results in a more serious relation and disables the subsequent elrank reduction of evation estimation in the original UCA-RARE algorithm. The above-mentioned spurious estimates and the signal cancellation problem can all be avoided by the 1D search procedure in the elevation space. B. Monte-Carlo Simulation for 2D DOA Estimation With the New Algorithm We now consider a compact UCA with a radius of and 16 monopoles elements . For this compact array, the full electromagnetic response including the mutual coupling effect was obtained by using the well-known method of moments (MoM) [19]. Two uncorrelated impinging sources and were at also modeled in MoM as two plane wave sources. Figs. 4 to 6 show the results of 2D DOA estimation using the new and the number of snapshots algorithm for . These results represent a 100 times implementation of the new algorithm by independent Monte-Carlo simulations. The signals and noise in our simulation are assumed to be stationary, zero mean, and uncorrelated Gaussian random processes. Noise is both spatially and temporally white. In each Monte-Carlo simulation, two normally distributed random numbers with zero mean are generated in Matlab. These two random numbers are used to represent the two signals. The terminal voltages on the antenna elements are then calculated independent by MoM using two plane wave excitations. and normally distributed random numbers are also generated to represent the random noise at the antenna terminals. The received signals at the antenna terminals are then obtained by in (28) is calculated the signal model in (1). The vector

Fig. 6. 2D DOA estimation using the new algorithm with mutual coupling compensated by the open-circuit voltage method.

from the estimated covariance matrix . Fig. 4 shows the results when mutual coupling is compensated by the elevation-dependent receiving-mutual-impedance method. Fig. 5 shows the results when mutual coupling is compensated by the single-elevation-angle receiving-mutual-impedance method with receiving mutual impedances obtained at an elevation angle of 45 . Fig. 6 is for the results with mutual coupling compensated by the open-circuit voltage method. From these results, it can be observed that even at such a low SNR, the azimuth estimates still distribute around the true values of 30 and 60 . However, for the elevation estimates, the results of the elevation-dependent receiving-mutual-impedance method in Fig. 4 give the most accurate estimation as compared to the other two compensation methods. The worst case is in Fig. 6 which corresponds to the open-circuit voltage method. To further verify the statistical efficiency and accuracy of the new method, 500 times independent Monte-Carlo simulations are performed with SNR varied from 0 dB to 35 dB. The biases and variances of the azimuth estimates of the two sources are shown in Figs. 7 and 8. The corresponding Cramer Rao bounds (CRBs) are also depicted in Fig. 8. From Fig. 8, it can be seen

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Fig. 7. The bias of the azimuth estimate versus SNR by using the new algorithm.

Fig. 10. 2D DOA estimation using the new algorithm with a platform effect (a shorted element at the centre of the UCA).

Fig. 8. The variance of the azimuth estimate versus SNR by using the new algorithm.

accurate elevation estimation. Another interesting finding in Fig. 9 is that the elevation estimation is rather robust to the change in SNR. This, when combined with its robustness to the errors in the corresponding azimuth estimation as shown in Figs. 4to 6, is another attractive feature of the new algorithm. However, as also observed from Fig. 9, the use of an accurate mutual coupling compensation method is very importance for the elevation estimation in the new method. Lastly, it should be noted that the elevation estimation methods in the original UCA-RARE and the method in [4] are all search-free procedures. Elevation estimates in our 2D DOA estimation algorithm are acquired with the spectrum search of in the null-space of over the range of . The CPU time for this spectrum search with a searching step of 0.1 on a personal computer with a Pentium IV processor running at 2400 MHz is about 0.3268 s (with the “tic” and “toc” run of Matlab), while the CPU time reported in [4] for a traditional Root-MUSIC-like algorithm is about 0.11 s. Hence the price paid for the elevation-dependent mutual coupling compensation and the ambiguity-free azimuth estimations is some loss of computational efficiency. C. Performance in the Presence of a Platform Effect

Fig. 9. The averaged bias of elevation estimates versus SNR by using the new algorithm.

that the variances of the azimuth estimates obtained by the new algorithm follow the trend of the corresponding CRBs. The average biases for the elevation estimates of the two sources corresponding to the above simulations are depicted in the Fig. 9, in which comparisons between the results obtained by the three different mutual coupling compensation methods are presented. It can be clearly noticed that the elevation-dependent receiving-mutual-impedance method again provides the most

In [4], it was reported that platform effects due to near-field scattering from proximity objects affect the performance of a UCA with dipole elements. To demonstrate the performance of the new algorithm in the presence of a platform effect, an additional shorted monopole is added to the center of the UCA considered above. The results for a 100 times implementation of the new algorithm is depicted in Fig. 10 with the same simulation conditions as in Section VI-A. It can be seen that the biases of the 2D DOA estimation are now somewhat increased. This may be due to the fact that the scattering effect introduced by the central monopole element bears an azimuth variation component which has not been modeled accurately by the mutual in (11). This example shows that a further coupling matrix modification of the array data model is needed for the new algorithm in order to account for the possible happening of platform effects.

WANG et al.: DECOUPLED 2D DOA ESTIMATION USING COMPACT UNIFORM CIRCULAR ARRAYS

VII. CONCLUSION A decoupled 2D DOA estimation method in the presence of elevation-dependent mutual coupling effect is proposed for compact UCAs. The azimuth estimates are decoupled from the elevation estimates and can be realized without the exact knowledge of mutual coupling. The elevation-dependent mutual coupling effect is compensated efficiently in the search-based elevation estimation procedure. Although the computational load of the elevation estimation in the new method is higher than that of the original UCA-RARE method, the advantages of the new method, such as efficient compensation of the elevation-dependent mutual coupling effect, no requirement on the a priori knowledge of the spatial source structure, and robustness to the spurious 2D DOA estimates, far exceed its shortcomings. The new method manifests itself as a suitable method for practical 2D DOA estimation for compact UCAs. REFERENCES [1] D. Longstaff, P. E. K. Chow, and D. E. N. Davies, “Directional properties of circular arrays,” Proc. Inst. Elec. Eng., vol. 114, Jun. 1967. [2] C. P. Mathews and M. D. Zoltowski, “Eigenstructure techniques for 2-Dangle estimation with uniform circular arrays,” IEEE Trans. Signal Process., vol. 42, pp. 2395–2407, 1994. [3] M. Pesavento and J. F. Böhme, “Direction of arrival estimation in uniform circular arrays composed of directional elements,” in Proc. Sensor Array and Multichannel Signal Processing Workshop, Aug. 2002, pp. 503–507. [4] R. Goossens and H. Rogier, “A hybrid UCA-RARE/Root-MUSIC approach for 2-D direction of arrival estimation in uniform circular arrays in the presence of mutual coupling,” IEEE Trans. Antennas Propag., vol. 43, pp. 841–849, 2007. [5] R. Goossens, H. Rogier, and S. Werbrouck, “UCA Root-MUSIC with sparse uniform circular arrays,” IEEE Trans. Signal Process., vol. 56, pp. 4095–4099, 2008. [6] T. T. Zhang, Y. L. Lu, and H. T. Hui, “Compensation for the mutual coupling effect in uniform circular arrays for 2D DOA estimations using a new searching algorithm,” IEEE Trans. Aerosp. Electron. Sys., vol. 44, pp. 1215–1221, Jul. 2008. [7] R. O. Schmidt, “Multiple emitter location and signal parameter estimation,” IEEE Trans. Antenn. Propag., vol. 34, no. 3, pp. 276–280, 1986. [8] P. Stoica and K. C. Sharman, “Maximum likelihood methods for direction-of-arrival estimation,” IEEE Trans. Acoust., Speech, Signal Processing, vol. 38, no. 7, pp. 1132–1143, 1990. [9] M. Viberg, B. Ottersten, and T. Kailath, “Detection and estimation in sensor arrays using weighted subspace fitting,” IEEE Trans. Signal Process., vol. 39, no. 11, pp. 2436–2449, 1991. [10] H. T. Hui, “Improved compensation for the mutual coupling effect in a dipole array for direction finding,” IEEE Trans. Antennas Propag., vol. 51, no. 9, pp. 2498–2503, Sep. 2003. [11] B. Friedlander and A. J. Weiss, “Direction finding in the presence of mutual coupling,” IEEE Trans. Antennas Propag., vol. 39, pp. 273–284, Mar. 1991. [12] I. J. Gupta and A. A. Ksienski, “Effect of mutual coupling on the performance of adaptive arrays,” IEEE Trans. Antennas Propag., vol. 31, no. 9, pp. 785–791, Sep. 1983. [13] E. C. Jordan, Electromagnetic Waves and Radiating Systems. Englewood Cliffs, N. J.: Prentice-Hall, 1968, ch. 11. [14] H. Steyskal and J. S. Herd, “Mutual coupling compensation in small array antennas,” IEEE Trans. Antennas Propag., vol. 38, no. 12, pp. 1971–1975, 1990. [15] C. K. E. Lau, R. S. Adve, and T. K. Sarkar, “Minimum norm mutual coupling compensation with applications in direction of arrival estimation,” IEEE Trans. Antennas Propag., vol. 52, no. 8, pp. 2034–2040, 2004. [16] J. W. Wallace and M. A. Jensen, “Mutual coupling in MIMO wireless systems: A rigorous network theory analysis,” IEEE Trans Wireless Commun., vol. 3, no. 4, pp. 1317–1325, 2004. [17] H. T. Hui, “A practical approach to compensate for the mutual coupling effect of an adaptive dipole array,” IEEE Trans. Antennas Propag., vol. 52, no. 5, pp. 1262–1269, 2004. [18] H. T. Hui, B. K. Li, and S. Crozier, “A new decoupling method for quadrature coils in magnetic resonance imaging,” IEEE Trans. Biomed. Eng., vol. 53, no. 10, pp. 2114–2116, 2006. [19] R. F. Harrington, Field Computation by Moment Methods. New York: IEEE Press, 1993.

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Bu Hong Wang (M’06) received the M.S. and Ph.D. degrees in signal and information processing, all from the Xidian University, Xi’an, China, in 2000 and 2003, respectively. From 2003 to 2005, he was a Postdoctoral Fellow in the Postdoctoral Technical Innovation Centre, Nanjing Research Institute of Electronics Technology, Nanjing, China. From 2006 to 2008, he was an Associate Professor with the School of Electronic Engineering of Xidian University. Since January 2009, he has been a Research Fellow with the Department of Electrical and Computer Engineering, National University of Singapore, Singapore. His research interest is mainly in the area of array signal processing and its application in radar and communications. Prof. Wang was honored with the Excellent Doctoral Dissertation of Shaanxi Province Award for his Ph.D. dissertation, “On Some Crucial Aspects of High Resolution Direction of Arrival Estimation.” His postdoctoral studies were supported by the National Postdoctoral Science Foundation.

Hon Tat Hui (SM’04) received the B.Eng. degree (with first class honors) and the Ph.D. degree both from the City University of Hong Kong, in 1994 and 1998, respectively. From 1998 to 2001, he worked at the City University of Hong Kong as a Research Fellow. From 2001 to 2004, he was an Assistant Professor at Nanyang Technological University in Singapore. From 2004 to 2007, he was a Lecturer in the School of Information Technology and Electrical Engineering, University of Queensland. He is currently an Assistant Professor with the Department of Electrical and Computer Engineering, National University of Singapore. His main research interests are in antennas and wireless communications. His recent research interest also includes quantum computing, quantum information processing, and biomedical imaging (especially MRI). He has published more than 50 papers in internationally referred journals and presented many papers at international conferences. Dr. Hui has been a project reviewer of various industrial and government organizations. He was an exceptional performance reviewer for the IEEE Antennas and Propagation Society in 2008. He served as an editorial board member in various international journals. He has helped organize many local and international conferences.

Mook Seng Leong received the B.Sc.Eng. (Hons I) degree in electrical engineering and the Ph.D. degree in microwave engineering, from the University of London, London, U.K., in 1968 and 1972, respectively. After a two year Postdoctoral Research Fellowship attachment to Andrew Antennas under the UK SERC, he joined the then University of Singapore in October 1973 and has been a Professor of electrical engineering at the National University of Singapore since 1989. His main research interests include advanced electromagnetics, analysis and design of microwave antennas and waveguides, and EMC control and management. Has published over 300 peer reviewed research papers and is a coauthor of the book entitled Spherical Wave Functions in Electromagnetic Theory (Wiley). Prof. Leong is a Fellow of the Institution of Electrical Engineers (now IET), London. He is the founding Chairman of MTT/AP/EMC Chapter, Singapore IEEE Section. He received the MINDEF-NUS Joint R&D Award from DSO National Laboratories, Singapore, in 1966, for his contributions to their collaborative research projects. He received the Teaching Excellence Award (1996/97, 1999/2000) and was listed among NUS’s Top 100 teachers, in 2002. He is an Associate Editor of the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION (2004–2007, 2007–2010), and a member of the Editorial Boards for IET Microwaves, Antennas and Propagation (2007–2009), and Microwave and Optical Technology Letters (since 1998). He has also been in the Organizing Chairman of the international conferences, hosted by Singapore, such as APMC 1999, PIERS 2003 and APMC 2009.

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Ultrawideband Aperiodic Antenna Arrays Based on Optimized Raised Power Series Representations Micah D. Gregory, Student Member, IEEE, and Douglas H. Werner, Fellow, IEEE

Abstract—Past research has shown that application of mathematical and geometrical concepts such as fractals, aperiodic tilings, and special polynomials can provide elegant solutions to difficult antenna array design problems. For example, design issues such as beam shaping and control, sidelobe levels, bandwidth and many others have been addressed with such concepts. In this paper, mathematical constructs based on the raised power series (RPS) are utilized to provide easily controlled aperiodicity to a linear array of antenna elements in order to achieve wideband performance. In addition, recursive application of raised power series subarrays and implementation of an optimization technique based on the genetic algorithm is demonstrated to realize impressive ultrawideband performance. The technique introduced here is shown to offer bandwidths of many octaves with excellent sidelobe suppression and no grating lobes. Moreover, the ultrawideband performance for one of the optimized RPS array examples is verified through full-wave simulations which take into account the coupling environment experienced by realistic radiating elements (in this case half-wave dipole antennas for three different operating frequencies). Index Terms—Aperiodic arrays, genetic algorithm, raised power series, ultrawideband arrays.

I. INTRODUCTION

M

UCH research has been invested into developing antenna array configurations that are usable over an extended frequency range. That is, they do not exhibit grating lobes when the electrical distance between elements becomes large. Early methods for creating wideband arrays involved statistical processes, where predictions of patterns were made that offered a certain probability of achieving a targeted sidelobe level [1]. In this case, uniformly excited elements are randomly distributed over a specified aperture. A large aperture would emulate a wide bandwidth, however, this method offers no guarantee that elements are adequately spaced or that the sidelobe level goal is indeed met. Other designs using fully determined patterns from small arrays over limited bandwidths were examined in [2]. Later, designs based on fractal concepts have produced multi-band arrays as well as array configurations that are fairly insensitive to bandwidth, but with limited sidelobe level suppression [3]–[6]. Many array design methods approach the issue of sidelobe level reduction or pattern control only for narrowband operation or over a limited scanning range [7]–[11]. Optimizations with simple element perturbations using scanned Manuscript received March 30, 2009; revised July 09, 2009. First published December 28, 2009; current version published March 03, 2010. The authors are with The Pennsylvania State University, Electrical Engineering, University Park, PA USA (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2009.2039315

arrays with a relatively small number of elements (e.g., arrays with 8 or 16 elements) of realistic radiators have also been performed in [12]. Beam scanning, however, can be considered analogous to an increase in bandwidth. Beam steering shifts the array factor across the visible region, whereas increasing bandwidth compresses the array factor into the visible region; both actions can cause grating lobes to appear in narrowband array designs [13], [14]. Recently, there have been several new design methodologies introduced for aperiodic antenna arrays that are capable of producing low sidelobe level performance over very large bandwidths. The linear polyfractal arrays introduced in [13], [15], and [16] are one such example, yielding very low sidelobes over bandwidths of many octaves. In addition to linear array design techniques, application of aperiodic tiling theory to array design has yielded planar array configurations with ultrawideband performance [17]. As in [8]–[13], [15], and [16], a genetic algorithm was applied to optimize the arrays for best performance, in this case by evolving the optimal aperiodic tiling patterns. The RPS method presented here has been developed as an aperiodic way of defining the locations of elements in an antenna array using a minimal set of defining parameters. Two classes of arrays are introduced here; one is a small configuration designed to be a simple building block, whereas the other class uses these building blocks to construct more complicated arrays which are capable of exhibiting ultrawide bandwidth performance. Similar to [13], [15], and [16], the array design method presented here assumes uniformly excited elements. The design technique for the simple building blocks (subarrays) is introduced in Section II and their performance is examined in Section III. The complex arrays, which are optimized with the use of a genetic algorithm, are constructed and evaluated in Section IV. Several examples of optimized arrays are presented which demonstrate their ultrawideband performance capabilities. Some differentiating characteristics of this new ultrawideband array design methodology compared to other previously reported techniques include having the ability to specify a predefined number of elements and a targeted aperture size, design features which can facilitate practical array construction, and ease of integration into a variety of binary or real-valued evolutionary optimization algorithms. Additionally, fewer parameters are typically required using this new method to describe complex array layouts as compared to alternative linear array design techniques such as [13], [15], and [16], which makes it particularly attractive in the context of an optimization problem. Of practical importance in any antenna array is the method of excitation. Conventional periodic and early aperiodic array

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of “raised power series”, i.e., conventional power series where power each of the exponents are raised to the

(4) (5)

Fig. 1. Array geometry and coordinate system.

designs utilize corporate feed networks, however, these can be unattractive because of feed losses and radiation [18], [19]. They also typically have a narrow operating bandwidth. Alternatively, each element (in a T/R module) can contain all necessary electronics such as amplifiers and phase shifters, allowing excitation via coaxial cable, for example. These monolithic arrays have the advantages of simplicity in construction as well as increased design flexibility and improved performance through self-contained feed networks [18], [20]–[22]. When used with ultrawideband array methodologies, monolithic elements would greatly reduce fabrication complexity; UWB elements would simply be placed at the locations specified by the RPS array design technique. Additionally, the use of uniformly excited elements in the array design allows for easier monolithic element construction compared to amplitude-tapered arrays.

The new array now has an aperiodic arrangement of elements. Instead of elements located periodically at a distance apart, the distance between the elements varies depending on their lo, the array recation; however, in the special case when . Typically, elements in an array mains periodic with apart to avoid strong mutual coupling are placed at least which can have a detrimental affect on the driving point impedances [23]. In order to preserve this property in the RPS array, the scale factor must be defined according to (6). The scaling since with , the two factors are different about most closely spaced elements are at the ends of the array while , the two closest elements are near the origin. The with positions of the elements for the RPS array can be determined element is located at . Here, from (7), where the for and for (6)

II. ARRAY CONFIGURATION The raised power series (RPS) arrays are based on a modification of the conventional power series expression to accommodate a linear set of antenna elements. For example, it is well known that the array factor for a standard periodic array of uniformly excited and equally spaced antenna elements can be expressed in the form of a power series (see (1) below). In this case, all elements are assumed to lie on the z-axis with the angle measured from the x-axis as illustrated in Fig. 1. Here, is the spacing between elements and is the free-space wave number. The power series representation for the array factor given in (1) can also be expressed as (2), which reduces to the convenient closed form of (3) [23] for the unsteered array

(7)

(1)

Aperture size is an important consideration when designing an array. For a linear array, this is simply the distance between the two farthest elements. Since the RPS array is mirrored about . For a typical RPS the origin, the aperture size is simply , the size of the array is . With array with , the smallest array the same design constraint of allowable is the periodic variety, which has an aperture size of for elements. It becomes clear that for and , the aperture size of the RPS array can become very large, several times the size of a periodic array of a similar number of elements. For example, a 101-element array with and has an aperture size of , about seven times that of the corresponding periodic array. For this reason, the parameter will be typically limited to the range . This avoids the creation of extremely sparse arrays which are undesirable in many practical applications.

(2)

III. BASIC RPS ARRAY PERFORMANCE

(3)

power Modifying (1) and (2) by raising to the and introducing a scale factor (required for maintaining a ) yields the new array desired minimum element spacing, factor expressions given in (4) and (5). Note that (4) is in terms

In this section the broadband characteristics of RPS arrays will be examined, focusing on sidelobe performance at minto . These minimum element spacings ranging from to imum spacings correspond to operating frequencies of , with designated as the lowest operating frequency of the array. Fig. 2 shows sidelobe levels of variously sized, uni. formly excited RPS arrays with operating at As expected, for a periodic array with , the sidelobe level corresponds to the well know value of

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Fig. 2. Sidelobe levels of various sized RPS arrays with r varied from 0.75 to 1.25 and 0:5 d 5:0. The scaling constant  is appropriately adjusted for each value of r . Arrays are periodic and exhibit grating lobes when r = 1. For arrays of fixed size, the peak sidelobe levels are nearly identical at frequencies = 1). Elements are uniformly excited and steered to broadside. above 2f (d

13.2 dB [23]. This lobe is adjacent to the main beam and although it is not a grating lobe, any reduction is generally desirable. Fig. 2 indicates that suppression can be achieved with values of slightly greater than 1. When increasing the operwith a minimum element spacing of , ating frequency to a grating lobe appears for the periodic array structure but not . Moreover, they continue to exfor the RPS arrays when hibit grating lobe suppression for wider bandwidths, with up to a 10:1 bandwidth for the arrays considered in Fig. 2. Increasing ) genthe number of elements in the RPS array (with erally lends itself to a lower peak sidelobe level; and for fixed values of slightly away from unity, sidelobe levels generally . remain constant with increasing frequency, especially for This feature is a particularly useful attribute of the RPS arrays, which enables significant sidelobe suppression over wide operating bandwidths (e.g., at least a 10:1 bandwidth for the cases considered in Fig. 2). (101 eleFig. 2 demonstrates that for the case where , the peak sidelobe level is low and indepenments) and dent of operating frequency (minimum spacing) over the range . This lends itself to a suitable wideband array design, therefore the properties of the array factor over an extended bandwidth are investigated. Array factors at several operating frequencies are shown plotted in Fig. 3. A compar-





Fig. 3. Visible half-regions (0 sin  1) of normalized array factors at various minimum spacings for a RPS array with N = 50 and r = 0:81 and a periodic array. The arrays are comprised of uniformly excited isotropic radiators with their main beam steered to broadside (sin  = 0).

ison of the sidelobe level and directivity with a periodic array of an equivalent element count is included in Fig. 4. An expression for the directivity of RPS arrays can be found by following the procedure outlined in [24], which results in (8), shown at the bottom of the page. Since the array factors are symmetrical when the main beam , it is only necessary to is steered to broadside compute and plot the radiation patterns over half of the visible region. Close inspection of the array factor as the operating frequency is increased reveals that the pattern becomes compressed towards broadside. This occurs for all linear arrays

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Fig. 5. Element locations of the 101-element periodic (top), 123-element periodic (middle) and 101-element RPS (bottom) arrays.

Fig. 6. RPS method with subarray construction. A sample two stage, 5 by 5 array is shown here; more stages can be applied for increasingly complex and larger structures, allowing for better bandwidth performance.

0

Fig. 4. Peak sidelobe levels, 3 dB beamwidths, and maximum directivity for a 101-element periodic array, a 123-element periodic array, and a RPS array with N = 50 and r = 0:81 (101 elements). The 123-element periodic array has an aperture size approximately equivalent to the RPS array. All arrays are comprised of uniformly excited isotropic radiators with their main beam steered to broadside.

and implies that the maximum sidelobe level of an array can only monotonically increase as the operating frequency (and minimum spacing) increases, and vice versa [14], [25]. When optimizing arrays for low sidelobe levels, this property becomes particularly useful. For example, the array factor is generally computed (optimized) at the highest intended operating frequency (or largest minimum element spacing) to have the lowest possible sidelobe level. This guarantees that the sidelobe level for the array will not exceed this value at any lower , frequency of operation. Alternatively, and represents the peak sidelobe where level of the array relative to the main beam at frequency . In addition, the scanning performance can be predicted by using a ) because of the aforementioned higher frequency (larger similarities between scanning and frequency. In fact, the scanning performance can be exactly determined by the relationship that is scanned given in (9) [13]. An array with spacing from broadside has an equivalent unsteered spacing of . For these reasons, the scanned and broadside performance of an array is generally critiqued by determining its peak sidelobe level at the highest intended operating frequency (or largest minimum element spacing) at broadside only (9) The RPS array presented gives stable, low sidelobe level performance over a remarkably wide 80:1 bandwidth, starting out with 9.4 dB at , and only jumping up to 8.6 dB at about . The periodic array by comparison develops a grating lobe minimum spacing . Directivity for the RPS array is at approximately that of the periodic array but fluctuates less near and then remaining nearly constant thereafter. Ripple in the directivity of the periodic array is due to repeated introduction of grating lobes into the visible region of the array factor. RPS arrays (and in general, most broadband arrays) mitigate the ripple

effect by suppressing the grating lobes and spreading of the energy across the pattern [26]. and , the average For the RPS array with at . It requires about 25% element spacing is more space than the respective 101-element periodic array, but at a great advantage in terms of bandwidth. Element locations for the 101-element periodic array, the 101-element RPS array, and the periodic array with an aperture size equal to the RPS array (123-elements) are depicted in Fig. 5. IV. OPTIMIZED RPS ARRAYS For a fixed number of elements in an array, the RPS design method has only a single parameter, , that can be adjusted to control the radiation characteristics such as sidelobe level and bandwidth. Hence, the design flexibility of RPS arrays is limited. One way to configure an array with more customizable radiation properties is by employing a system of subarrays. Similar subarray-based design methods for planar arrays have been employed with success in [27], [28], and later with amplitude tapered linear arrays in [29] as it is a simple way to reduce the dimensionality of the optimization problem, allowing large structures to be created without requiring an intractable number of controlling parameters. Here, the same RPS method will be applied recursively to obtain aperiodically located subarrays, each consisting of an RPS array. Fig. 6 shows an example where each parameter controls the structure of the RPS subarray parameter controls the locations of the subarrays and the RPS subarrays is formed). This (i.e., an RPS array of method can be applied recursively for an unlimited number of branches or stages, however only a two stage array is illustrated in Fig. 5. The element locations and the array factor expression for this array are given in (10) and (11) respectively. This type of generalized RPS array configuration is well suited for optimization using techniques such as the genetic algorithm [30]. Careful , and must be made selection of the values for , to avoid overlapping subarrays. Additionally, constraints can be placed on these parameters to enforce a specific aperture size, or allow extra room between subarrays for practical reasons such as space for a structure that would physically support the subarrays. The branched-subarray method is similar to the fractal structures considered in [3] and [15] but element locations are determined in a different manner, where the tree structures are

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TABLE I OPTIMIZATION PARAMETERS FOR THE 55-ELEMENT RPS ARRAY

not necessarily self-similar, and the number of elements in the array is predetermined

(10)

Fig. 7. Element locations for the 55-element optimized RPS array. Minimum spacing is 0:5 and average spacing is 0:69 at f = f .

(11)

Several examples of the generalized RPS arrays will be presented here that demonstrate their ability to exhibit ultrawideband performance. Each array is optimized using a parallelized binary genetic algorithm scheme [30]. The first example that will be considered is comprised of five subarrays with 11 elements each (55 elements total). The global scaling parameter is a function of the nominal scaling parameter, i.e., with , which leaves enough space for the 11 elements in each subarray. The optimization parameters for this array design can be found listed in Table I. The binary GA encodes each parameter in an efficient manner as a string of 1s and 0s (the chromosome), thus allowing them to have a number of possible values equal to a power of two. For this array there are six -parameters, namely and through . Given six bits per parameter, the (greater than 50 billion) six-dimensional solution space has possible outcomes, and therefore represents a problem that is a good candidate for optimization using a technique like the genetic algorithm. Although binary representation of each -parameter limits the solution space through quantization, it has the benefit that subarrays will possess some amount of homogeneity, making physical construction of large arrays more practical. The array fitness (12) is evaluated based solely on the peak relative sidelobe level (SLL) at a single value for the minimum at the highest intended operating frequency of spacing the array. Since the GA utilizes a scaling-invariant tournament selection process, any monotonic scaling of the fitness function will result in the same evolution [30]. Additionally, elitism is used to preserve the best member of the population at each generation. The first array was optimized for best performance minimum element spacing, which guarantees that it at a will have at least a 20:1 frequency bandwidth at the resulting sidelobe level. Computational time was less than 3 minutes on a 3.0 GHz quad-core Intel processor machine running approximately 92 generations of 64 population members. The final set of optimized values is shown in Table I, resulting

Fig. 8. Normalized array factor of the optimized 55-element RPS array at f = . All elements are uniformly excited and the main beam is steered to broadside (sin  = 0). The peak relative sidelobe level is 14.1 dB. The directivity of the array is 18.1 dB.

f

0

Fig. 9. Normalized array factor of the optimized 55-element RPS array at f = 20f . All elements are uniformly excited and main beam is steered to broadside (sin  = 0). The peak relative sidelobe level is 09.55 dB. The directivity of the array is 17.4 dB.

in an array with an average element spacing of

with

(12) The optimized parameter values in Table I produce the array with element locations illustrated in Fig. 7 and the array factors for minimum element spacings of and shown in Figs. 8 and 9, respectively. The maximum sidelobe level at is slightly lower than a standard periodic array. However, unlike a periodic array, the low sidelobe properties are still maintained . The peak SLL at is lower than the basic even at RPS array presented in Figs. 2 through 4 yet there are nearly half as many elements in this array. The next example utilizes 21 subarrays of 21 elements each, yielding a 441-element array. The optimization parameters and

GREGORY AND WERNER: UWB APERIODIC ANTENNA ARRAYS BASED ON OPTIMIZED RPS REPRESENTATIONS

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TABLE II OPTIMIZED PARAMETERS FOR THE 441-ELEMENT ARRAY. EACH LOCAL r -PARAMETER HAD A 5-BIT DEFINITION IN THE GA CHROMOSOME WHILE THE GLOBAL PARAMETER HAD A 7-BIT DEFINITION

=

Fig. 10. Normalized array factor of the optimized, 441-element array at f f . All elements are uniformly excited and the main beam is steered to broadside  . The peak relative sidelobe level of the array is 14.92 dB and the directivity is 27.2 dB.

(sin = 0)

0

Fig. 12. Sidelobe levels for the optimized RPS arrays and a periodic array from f . Construction of the 55-element array is a 2-stage, 5 by f f to f 11 element design. Construction of the 441-element array is a 2-stage, 21 by 21 element design. Construction of the 425-element array is a 3-stage, 5 by 5 by 17 element design. Construction of the 1785-element array is a 3-stage, 5 by 7 by 51 element design. All arrays are uniformly excited and steered to broadside.

=

=

Fig. 11. Normalized array factor of the optimized, 441-element array at f . All elements are uniformly excited and the main beam is steered to broadside  . The peak relative sidelobe level of the array is 14.92 dB and the directivity of the array is 26.4 dB.

20f

(sin = 0)

= 80

0

their resulting values for this design are shown in Table II. The value of the global scaling parameter in this case is with . The local -parameters were limited to the range in the genetic algorithm while the global -parameter was limited . Optimization with 128 population members to over 20 generations took approximately 10 minutes on a computer cluster utilizing 40 processor cores at 2.4 GHz each. Final and minimum spacing average spacing of the array was at the lowest operating frequency of . Elitism was and the fitness function defined in (12) were employed in the optimization of this array. Again, a single point optimization was minimum element spacing, which guarancarried out at a tees that the array will have at least a 20:1 bandwidth at the resulting sidelobe level. Array factors for the optimized 441-eleand ment array are shown plotted in Figs. 10 and 11 for , respectively. These radiation pattern plots demonstrate excellent sidelobe suppression and no grating lobes over the desired operating bandwidth of 20:1 targeted for the array. In fact, as shown in Fig. 12, the optimized array produces sidelobe levels of about 13.5 dB or below over an 80:1 bandwidth. Lastly, two 3-stage arrays were optimized in a manner similar to the 2-stage designs. One was comprised of a 5 by 5 by 17 element RPS subarray structure, yielding 425 elements. The

Fig. 13. Relative element locations for the 441-element, 2-stage design (bottom row), 425-element, 3-stage design (middle row), and 1785-element, 3-stage design (top row). Vertical lines represent the location of an array element. Each array is scaled for visual purposes (all arrays are not of the same aperture size). It can be easily seen how elements tend to be lumped closely together in groups with the subarray method.

other example had a 5 by 7 by 51 element layout with 1785 total elements. Additional stages allow more customizability in the array layout, but at the expense of a more difficult optimization problem since there is typically an increased number of controlling parameters. Although longer optimization times will usually be required, the more customizable array has the potential for increased bandwidth performance over the less complicated arrays. For instance, the 425-element array requires 31 parameters, 9 more than the similarly sized 2-stage design. The 1785-element array requires 41 optimizable parameters. All parameters . As in the for both arrays lie in the range previous example, the arrays were optimized for best sidelobe . performance at a minimum element spacing of The average element spacings, 3 dB beamwidths, and directivities for these arrays can be found in Table III; the element locations are shown in Fig. 13. The sidelobe suppression capability of these arrays are even more remarkable than the previous two

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TABLE III PERFORMANCE SUMMARIES FOR THE FOUR OPTIMIZED RPS ARRAYS. THE 55 AND 441 ELEMENT ARRAYS ARE 2-STAGE DESIGNS WHILE THE 425 AND 1785  ELEMENT ARRAYS USE A 3-STAGE CONSTRUCTION. ALL ARRAYS WERE OPTIMIZED AT d

= 10

TABLE IV GENETIC ALGORITHM OPTIMIZATION SPECIFICS FOR THE FOUR EXAMPLE ARRAYS

TABLE V PERFORMANCE SUMMARIES FOR THE 3-STAGE, 425-ELEMENT OPTIMIZED RPS ARRAY. SOURCE IMPEDANCES FOR ALL DIPOLES ARE 75

examples, with a nearly 16 dB SLL maintained over an 80:1 bandwidth for the 1785-element design as seen from Fig. 12. It is also observed that even though the 425-element, 3-stage design contains fewer elements than the previous 2-stage array, the GA was able to find an array configuration with a lower sidelobe level at the optimization frequency, although more time was required for convergence as shown in Table IV. In addition to the examples presented here, the same design methodology can be applied to generate a variety of different array configurations to meet specific SLL, bandwidth and directivity specifications as well as practical size constraints. V. SIMULATIONS In an ideal ultrawideband array, each antenna element would operate efficiently over the entire bandwidth that the array supports. Additionally, elements would not couple to one another, which can have a detrimental impact on input impedances, especially if UWB antennas are employed [23]. Typically, most issues with the realization of UWB arrays arise in the lower range of operating frequencies where elements are electrically close and coupling is strongest. Above a minimum spacing of a few wavelengths, elements begin to behave as if they were in isolation [31]. Fortunately, with the optimized RPS arrays shown in Figs. 12 and 13, the system can be designed such that the minimum operating frequency corresponds with an element spacing of several wavelengths (or more), reducing any coupling concerns while still offering good bandwidth performance. In order to validate the UWB performance of the presented arrays, the 425-element case will be chosen for full-wave simulation , in FEKO [32]. Simulations will be performed at , and with dipole antenna elements designed for each



Fig. 14. Dipole antenna element locations for the simulated, 425-element RPS array. Minimum element spacing is 0.5 meters, average element spacing is 0.87 meters.

respective operating frequency. Computation feasibility will be maintained by driving each element at the input rather than implementing a realistic feed network such as the corporate feed. This method of excitation at the element also closely mimics the aforementioned monolithic arrays’ behavior. The lowest operating frequency of the array is chosen to be in the VHF range at 300 MHz, which implies a fixed minimum element spacing of 0.5 meters for the array. Dipole lengths and electrical distances vary as the operating frequency changes, however, the physical element locations shown in Fig. 14 are retained. The dipole antennas are arranged as shown in Fig. 1 and are parallel to the y-axis such that their plane of uniform radiation aligns with the array factor (xz-plane). Therefore, no element pattern compensation will need to be considered (other than normalization) for comparison to the basic array factor. The simulated radiation pattern for the array at (300 MHz) is shown in Fig. 15, along with the VSWR for each element in Fig. 16. The radiation patterns and VSWRs for the and are shown in Figs. 17, array at 18, and 19, 20, respectively. No grating lobes appear in any of the patterns, even with a minimum element spacing of up to . The array delivers excellent bandwidth performance and all simulated sidelobe levels are near their predicted values. As expected, mutual coupling between elements is strongest at

GREGORY AND WERNER: UWB APERIODIC ANTENNA ARRAYS BASED ON OPTIMIZED RPS REPRESENTATIONS

Fig. 15. Radiation pattern at 300 MHz (d = =2) for the simulated array of dipoles compared to the computed array factor. The peak sidelobe level of the simulated array is 15.11 dB. Excellent agreement is observed between the theoretical and simulated radiation patterns.

0

Fig. 16. Simulated VSWR for each dipole antenna of the 425-element, op= =2). Peak and average VSWR is timized RPS array at 300 MHz (d 2.50:1 and 1.77:1, respectively. Source impedances for the dipoles are 75 .

Fig. 17. Radiation pattern at 1.2 GHz (d = 2) for the simulated array of dipoles compared to the computed array factor. The peak sidelobe level of the simulated array is 14.76 dB. Again, excellent agreement is observed between the theoretical and simulated radiation patterns.

0

Fig. 18. Simulated VSWR for each dipole antenna of the 425-element, opti= 2). Peak and average VSWR is 1.95:1 mized RPS array at 1.2 GHz (d and 1.33:1, respectively. Source impedances for the dipoles are 75 .

Fig. 19. Radiation pattern at 6.0 GHz (d = 10) for the simulated array of dipoles compared to the computed array factor. The peak sidelobe level of the simulated array is 15.10 dB. Again, excellent agreement is observed between the theoretical and simulated radiation patterns.

0

Fig. 20. Simulated VSWR for each dipole antenna of the 425-element, opti= 10). Peak and average VSWR is 1.29:1 mized RPS array at 6.0 GHz (d and 1.19:1, respectively. Source impedances for the dipoles are 75 .

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as evidenced by the large VSWRs in Fig. 16. At the higher operating frequencies, elements begin to behave as if they were in isolation; no elements exhibit VSWR above 2:1. If corredesired, the array could be designed such that sponds to the lowest operating frequency and VSWRs would be well below 2:1 even when the elements are electrically closest. The array would still exhibit a 5:1 bandwidth with a peak sidelobe level of no more than about 15 dB. VI. CONCLUSION RPS arrays have shown themselves to be a fruitful and useful addition to the toolbox of ultrawideband array design techniques. They possess attributes that can make them particularly appealing in certain design scenarios compared to other wideband array optimization techniques. The RPS arrays can be designed with a predetermined number of elements and offer a high degree of control over the final aperture size. Additionally, they utilize small subarrays that serve as building blocks to construct much larger arrays, offering some structural similarities and design flexibility including the choice of subarray spacing. These properties can be exploited to simplify the design and construction of RPS arrays compared to other ultrawideband array design approaches that have been reported (e.g., [13], [17]). Several RPS-based array design examples have been presented that successfully illustrate their capacity for structured randomness and easy integration into robust optimization techniques such as the genetic algorithm. Excellent wideband performance was achieved at minimum element spacings of from examples featuring 55, 441, 425, and 1785 elements with peak sidelobe levels of 9.5 dB, 14.9 dB, 15.2 dB, and 15.9 dB, respectively. In fact, the sidelobe suppression capability of the optimized 1785-element array was shown to remain at nearly 16 dB over a remarkably wide 80:1 bandwidth. Full-wave simulations have shown that the RPS arrays behave as expected even with realistic radiating elements. REFERENCES [1] Y. T. Lo, “A mathematical theory of antenna arrays with randomly spaced elements,” IEEE Trans. Antennas Propag., vol. 12, no. 3, pp. 257–268, May 1964. [2] D. King, R. Packard, and R. Thomas, “Unequally-spaced, broad-band antenna arrays,” IEEE Trans. Antennas Propag., vol. 8, no. 4, pp. 380–384, Jul. 1960. [3] Y. Kim and D. L. Jaggard, “The fractal random array,” Proc. IEEE, vol. 74, no. 9, pp. 1278–1280, Sep. 1986. [4] C. Puente-Baliarda and R. Pous, “Fractal design of multiband and low side-lobe arrays,” IEEE Trans. Antennas Propag., vol. 44, no. 5, pp. 730–739, May 1996. [5] D. H. Werner, M. A. Gingrich, and P. L. Werner, “A self-similar fractal radiation pattern synthesis technique for reconfigurable multi-band arrays,” IEEE Trans. Antennas Propag., vol. 51, no. 7, pp. 1486–1498, Jul. 2003. [6] D. H. Werner and S. Ganguly, “An overview of fractal antenna engineering research,” IEEE Antennas Propag. Mag., vol. 45, no. 1, pp. 38–57, Feb. 2003. [7] N. Jin and Y. Rahmat-Samii, “Advances in particle swarm optimization for antenna designs: Real-number, binary, single-objective and multiobjective implementations,” IEEE Trans. Antennas Propag., vol. 55, no. 3, pp. 556–567, Mar. 2007. [8] R. L. Haupt, “Thinned arrays using genetic algorithms,” IEEE Trans. Antennas Propag., vol. 42, no. 7, pp. 993–999, Jul. 1994. [9] D. S. Weile and E. Michielssen, “Integer coded pareto genetic algorithm design of constrained antenna arrays,” Electron. Lett., vol. 32, no. 19, pp. 1744–1745, Sept. 1996.

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[10] F. J. Ares-Pena, J. A. Rodriguez-Gonzalez, E. Villanueva-Lopez, and S. R. Rengarajan, “Genetic algorithms in the design and optimization of antenna array patterns,” IEEE Trans. Antennas Propag., vol. 47, no. 3, pp. 506–510, Mar. 1999. [11] S. Caorsi, A. Lommi, A. Massa, and M. Pastorino, “Peak sidelobe level reduction with a hybrid approach based on GAs and difference sets,” IEEE Trans. Antennas Propag., vol. 52, no. 4, pp. 1116–1121, Apr. 2004. [12] M. G. Bray, D. H. Werner, D. W. Boeringer, and D. W. Machuga, “Optimization of thinned aperiodic linear phased arrays using genetic algorithms to reduce grating lobes during scanning,” IEEE Trans. Antennas Propag., vol. 50, no. 12, pp. 1732–1742, Dec. 2002. [13] J. S. Petko and D. H. Werner, “The pareto optimization of ultrawideband polyfractal arrays,” IEEE Trans. Antennas Propag., vol. 56, no. 1, pp. 97–107, Jan. 2008. [14] J. H. Doles, III and F. D. Benedict, “Broad-band array design using the asymptotic theory of unequally spaced arrays,” IEEE Trans. Antennas Propag., vol. 36, no. 1, pp. 27–33, Jan. 1988. [15] J. S. Petko and D. H. Werner, “The evolution of optimal linear polyfractal arrays using genetic algorithms,” IEEE Trans. Antennas Propag., vol. 53, no. 11, pp. 3604–3615, Nov. 2005. [16] J. S. Petko and D. H. Werner, “An autopolyploidy-based genetic algorithm for enhanced evolution of linear polyfractal arrays,” IEEE Trans. Antennas Propag., vol. 55, no. 3, pp. 583–593, Mar. 2007. [17] T. G. Spence and D. H. Werner, “Design of broadband planar arrays based on the optimization of aperiodic tilings,” IEEE Trans. Antennas Propag., vol. 56, no. 1, pp. 76–86, Jan. 2008. [18] D. M. Pozar, “Microstrip antennas,” Proc. IEEE, vol. 80, no. 1, Jan. 1992. [19] M. D. Gregory, J. S. Petko, and D. H. Werner, “Design validation of prototype ultra-wideband linear polyfractal antenna arrays,” presented at the Proc. IEEE Int. Symp. on Antennas and Propagation and USNC/ URSI National Radio Science Meeting, San Diego, CA, Jul. 5–11, 2008. [20] H. An, B. Nauwelaers, and A. Van De Capelle, “Broadband active microstrip array elements,” IEEE Electron. Lett., vol. 27, no. 25, pp. 2378–2379, Dec. 1991. [21] , C. A. Balanis, Ed., Modern Antenna Handbook. Hoboken, NJ: Wiley, 2008. [22] , R. C. Johnson, Ed., Antenna Engineering Handbook. New York: McGraw-Hill, 1993. [23] C. A. Balanis, Antenna Theory, Analysis and Design, 3rd ed. Hoboken, NJ: Wiley, 2005. [24] D. H. Werner, W. Kuhirun, and P. L. Werner, “The Peano-Gosper fractal array,” IEEE Trans. Antennas Propag., vol. 51, no. 8, pp. 2063–2072, Aug. 2003. [25] A. Ishimaru and Y.-S. Chen, “Thinning and broadbanding antenna arrays by unequal spacings,” IEEE Trans. Antennas Propag., vol. 13, no. 1, pp. 34–42, Jan. 1965. [26] D. H. Werner and R. Mittra, Frontiers in Electromagnetics. Piscataway, NJ: IEEE Press, 2000. [27] N. Toyama, “Aperiodic array consisting of subarrays for use in small mobile earth stations,” IEEE Trans. Antennas Propag., vol. 53, no. 6, pp. 2004–2010, Jun. 2005. [28] K. C. Kerby and J. T. Bernhard, “Sidelobe level and wideband behavior of arrays of random subarrays,” IEEE Trans. Antennas Propag., vol. 54, no. 8, pp. 2253–2262, Aug. 2006. [29] R. L. Haupt, “Optimized weighting of uniform subarrays of unequal sizes,” IEEE Trans. Antennas Propag., vol. 55, no. 4, pp. 1207–1210, Apr. 2007. [30] R. L. Haupt and D. H. Werner, Genetic Algorithms in Electromagnetics. Hoboken/Piscataway, NJ: Wiley/IEEE Press, 2007. [31] T. G. Spence and D. H. Werner, “Full-wave mutual coupling analysis of wideband arrays based on aperiodic tilings,” presented at the IEEE Int. Symp. on Antennas and Propagation and USNC/URSI National Radio Science Meeting, Charleston, SC, Jun. 1–5, 2009. [32] EM Software and Systems—S.A ver. 5.4, FEKO [Online]. Available: www.emssusa.com

Micah D. Gregory (S’06) was born in Williamsport, PA, in 1984. He received the B.S. degree in electrical engineering from Bucknell University, Lewisburg, PA, in 2006. He is currently working toward the M.S. degree in electrical engineering at the Pennsylvania State University (Penn State), University Park. He is currently a Research Assistant for the Computational Electromagnetics and Antennas Research Lab (CEARL), Penn State. His research interests include ultrawideband and phased array antenna design, evolutionary strategies, frequency selective surfaces, and horn antennas. Other interests include parallel and high performance computer programming. Mr. Gregory was the recipient of the 2005 MTT-S undergraduate/pregraduate scholarship award and the 2009 A. J. Ferraro award for research excellence in the field of antenna engineering.

Douglas H. Werner (S’81–M’89–SM’94–F’05) received the B.S., M.S., and Ph.D. degrees in electrical engineering and the M.A. degree in mathematics from The Pennsylvania State University (Penn State), University Park, in 1983, 1985, 1989, and 1986, respectively. He is a Professor in the Pennsylvania State University Department of Electrical Engineering. He is the Director of the Computational Electromagnetics and Antennas Research Lab (CEARL) http://labs.ee.psu. edu/labs/dwernergroup/ as well as a member of the Communications and Space Sciences Lab (CSSL). He is also a Senior Scientist in the Computational Electromagnetics Department of the Applied Research Laboratory and a faculty member of the Materials Research Institute (MRI) at Penn State. His research interests include theoretical and computational electromagnetics with applications to antenna theory and design, phased arrays, microwave devices, wireless and personal communication systems, wearable and e-textile antennas, conformal antennas, frequency selective surfaces, electromagnetic wave interactions with complex media, metamaterials, electromagnetic bandgap materials, zero and negative index materials, fractal and knot electrodynamics, tiling theory, neural networks, genetic algorithms and particle swarm optimization. Dr. Werner was presented with the 1993 Applied Computational Electromagnetics Society (ACES) Best Paper Award and was also the recipient of a 1993 International Union of Radio Science (URSI) Young Scientist Award. In 1994, he received the Pennsylvania State University Applied Research Laboratory Outstanding Publication Award. He was a coauthor (with one of his graduate students) of a paper published in the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION which received the 2006 R. W. P. King Award. He has also received several Letters of Commendation from the Pennsylvania State University Department of Electrical Engineering for outstanding teaching and research. He is a former Associate Editor of Radio Science, an Editor of the IEEE ANTENNAS AND PROPAGATION MAGAZINE, a member of the American Geophysical Union (AGU), URSI Commissions B and G, the Applied Computational Electromagnetics Society (ACES), Eta Kappa Nu, Tau Beta Pi and Sigma Xi. He has published over 375 technical papers and proceedings articles and is the author of eight book chapters. He edited the book, Frontiers in Electromagnetics (Piscataway, NJ: IEEE Press, 2000). He has also contributed a chapter for Electromagnetic Optimization by Genetic Algorithms (New York: Wiley Interscience, 1999) as well as for Soft Computing in Communications (New York: Springer, 2004). He coauthored Genetic Algorithms in Electromagnetics (Hoboken, NJ: Wiley/IEEE, 2007). He has also contributed an invited chapter on “Fractal Antennas” for the 2007 edition of the popular Antenna Engineering Handbook (New York: McGraw-Hill). He was the recipient of a College of Engineering PSES Outstanding Research Award and Outstanding Teaching Award in March 2000 and March 2002 respectively. He was also presented with an IEEE Central Pennsylvania Section Millennium Medal. In March 2009, he received the PSES Premier Research Award. He is a Fellow of the IEEE, the IET, and the ACES.

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Original and Modified Kernels in Method-ofMoments Analyses of Resonant Circular Arrays of Cylindrical Dipoles George Fikioris, Senior Member, IEEE, Dimitrios Tsamitros, Savvas Chalkidis, and Panagiotis J. Papakanellos, Member, IEEE

Abstract—Properly dimensioned circular arrays of cylindrical dipoles are known to possess very narrow resonances. It is also known that analyzing such arrays using moment methods presents unique and particular difficulties, as application of such methods to the usual Hallén-type integral equations can yield meaningless results from which no further conclusions should be drawn. In the present paper, we apply moment methods to properly modified integral equations and obtain much more reliable results. We also observe that certain difficulties still remain, and discuss them in detail. Index Terms—Antenna arrays, Galerkin method, moment methods, resonance.

N = 20

-element circular array. The dipoles are identical, Fig. 1. Top view of a parallel, and non-staggered, with their centers lying on the plane.

I. INTRODUCTION

A

number of theoretical and experimental studies [1]–[7] have shown that properly designed circular arrays of cylindrical dipoles possess a series of very narrow resonances, called “phase-sequence resonances.” The dipoles are identical, parallel and non-staggered. The array is large, both in the sense of many elements and in the sense of many wavelengths in dipoles is center-driven dimension. Only one of the by a voltage ; the rest are unloaded and non-driven. A top -element array is shown in Fig. 1. For view of a resonances to occur, it is necessary to properly select the and inter-element spacing , as well as the dipoles’ length radius ; many conventional choices yield no resonances at all. Resonance means a narrow peak when the driving-point is plotted vs. the frequency conductance , where is the current at the center of the driven dipole; at resonance, also, the driving-point suscepis zero (to be more precise, is very tance close to its zero). Each resonance can be characterized by a large, integer parameter , where peaks that are successive and increasing in Manuscript received April 01, 2009; revised July 08, 2009. First published December 31, 2009; current version published March 03, 2010. G. Fikioris is with the School of Electrical and Computer Engineering, National Technical University, GR 157-73 Zografou, Athens, Greece (e-mail: [email protected]). D. Tsamitros and S. Chalkidis are with the Hellenic Air Force Academy, Dekelia Air Force Base, GR 1010 Dekelia, Attiki, Greece (e-mail: [email protected]; [email protected]). P. J. Papakanellos is with the School of Electrical and Computer Engineering, National Technical University, GR 157-73 Zografou, Athens, Greece and also with the Hellenic Air Force Academy, Dekelia Air Force Base, GR 1010 Dekelia, Attiki, Greece (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2009.2039317

z=0

frequency correspond to successive and increasing values of . Throughout this paper, we assume for simplicity that is even. ; The last and most narrow resonance corresponds to case of [2], [4, Ch. 11] the values of are there, for the predicted to be extremely large, at least if one assumes that the dipoles are lossless. The narrowness and height of the peaks decrease rapidly as decreases and eventually the peaks become unnoticeable. For lossless dipoles, the narrowness and height of fixed the peaks also decreases rapidly as decreases with [2], [4, Ch. 11]. Furthermore, the extremely large values greatly decrease when one takes the dipoles’ finite conductivity into account [3], [4, Ch. 12]. Many further properties of resonant circular arrays, including radiation patterns and the surface-wave nature of near fields, are discussed in detail in [1]–[8] (see also [9], [10]). From a practical point of view, an application as a microwave beacon for the coastal navigation of ships and airplanes is proposed in [11] and [12]. An application as a surface-wave generator at 30 MHz is described in [6] (this application is based on the resonances of an array with two driven elements; similarities and differences with the case of a single driven element are fully discussed in [6]). On the other hand, in the directional borehole-radar application of [13], circular-array resonances are undesirable phenomena that should be avoided. By far the most important related potential application seems to be the resonant closed-loop array of a more general shape (i.e., not necessarily circular) [14], [15] (see also [16]). A study of two coupled resonant dipole arrays and analogies with certain optical resonators can be found in [17]. Resonant dipole arrays are also related, more-or-less, to structures studied by other research groups; such structures include optical

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resonators, plasmonic devices, electroacoustic transducers, and periodic gratings [18]–[34]. The analysis method in [1]–[7] (see also [35]) is the “twoterm theory” (originally developed by R. W. P. King in the 1950s—see [4] for original references), which is an approxcoupled integral equations for the imate solution to the currents along the cylindrical dipoles that consist the array. Narrow resonances in large circular arrays were initially discovered in 1990 using the two-term theory in its original form [1]. After the publication of [1], it was found that the original for some . This occurred two-term theory gave negative for certain parameters , , , different from those in [1], equals see [4, §11.8] for a specific example. Since the radiated power (plus the power dissipated on the dipoles must always—for any when their conductivity is finite), array with a single driven element, and at any frequency—be a positive quantity. Thus, the aforementioned negative were clearly meaningless results from which no further conclusions should be drawn. This difficulty (as well as other considerations) motivated D. K. Freeman and T. T. Wu [36], [37] to re-examine the kernels of the integral equations, leading to the replacement of the “original kernels” by the “modified kernels,” where our terminology for the kernels is consistent with [2] and [4], especially [4, §11.2, §11.8]. These modified kernels were successfully incorporated in [2]–[7]. In [38], an attempt is made to analyze a circular array—with -element experimental parameters the same as the array of [2]–[4]—via the method of moments (MoM) and, specifically, by straightforward application of the Numerical Electromagnetics Code (NEC) [39]. The essential result of [38] is a plot of vs. [38, Fig. 1], in which narrow “peaks” are shown, most of which are negative—and thus meaningless. Therefore a MoM analysis of resonant circular arrays is an especially demanding problem, presenting unique difficulties. [38] attributes these difficulties to the particular complications of the integral equations themselves and, secondarily, to roundoff-error effects; these latter effects become increasingly severe as increases and the sought-after peaks become higher and narrower [4, §13.5]. As proposed in [38], “the first step in obtaining accurate results via MoM would be to decouple the integral equations via the method of symmetrical components [4, §4.1] and, more importantly, to use the ‘modified kernels.’” The purpose of the present paper is to carry this first step out and to discuss the results in detail: We first briefly review the relevant integral equations, the method of symmetrical components, and the original/modified kernels in Section II; this background material is collected from previous works, mainly [4]. The remaining material in the present paper is original: We apply MoM to the integral equations using each set of kernels and compare the results. We also compare to corresponding results obtained using the two-term theory. Although use of the modified kernels does seem to eliminate the problem of “negative peaks,” we run into certain new difficulties not encountered with the two-term theory. We explain why such difficulties remain, and suggest probable remedies, by invoking results from [1]–[7], as well as conclusions regarding the application of MoM to Hallén’s equation with the delta-function generator [40].

Our specific MoM is Galerkin’s method with pulse functions, the same as the method applied in [40] to Hallén’s equation. We believe that results obtained using MoM with other basis and testing functions would be similar in nature. It is important to isolate the effects we wish to study from other effects. Accordingly, when we apply MoM, (i) we only consider one of the decoupled integral equations, which is predicted to have a single peak. (We consider all decoupled integral equations, however, when we calculate the field.) (ii) To isolate from effects due to roundoff and numerical integration, we use a not-too-large value (as in Fig. 1). On the other hand, the parameters of , , , remain the same as in the experimental array of [2]–[4] and in the NEC analysis of [38]. Finally, we assume lossless dipoles, but briefly indicate extensions to the lossy case. II. BACKGROUND: PHASE-SEQUENCE INTEGRAL EQUATION; ORIGINAL AND MODIFIED KERNELS denote the current distribution along array element Let no. when only element no. 1 is center-driven by a delta-function generator maintaining a voltage . can be written as aca superposition of the “phase-sequence currents” cording to

(1) In (1), the first expression for is [4, Eq. (4.18b)]. The second expression follows because the symmetries for all of the circular array imply . The aforementioned symmetries are discussed in detail in [5, Ch. 3]. The second expression shows (rather than ) different phase-sequence that there are satisfies the “ th phase-sequence currents integral equation”

(2) Equation (2), as well as (3)–(5) below, are immediate consequences of [4, Eqs. (4.3a)–(4.3g), (4.20a)]. In (2) (3) , condition

is a constant to be determined from the , and the “ th phase-sequence kernel”

FIKIORIS et al.: ORIGINAL AND MODIFIED KERNELS IN MoM ANALYSES OF RESONANT CIRCULAR ARRAYS

is the following superposition of the self- and mutualinteraction kernels

(4) In (4), the mutual-interaction kernels by

are given

(5)

in which is the distance from the axis of dipole 1 to the axis of dipole . The expression in (4), which involves cosines, comes from an expression involving exponentials [see [4, Eq. (4.3d)]; the said expression is similar to the first expression in (1)]. Once again, the equality of the two expressions is a consequence of the symmetries of the circular array, discussed in [5, Ch. 3]. Depending on whether the original or the modified kernels are , see (6a)–(6b) at the used, the self-interaction kernel is bottom of the page. The original kernel (6a) is in [4, Eq. (4.3d)], while (6b) is in [4, Eq. (11.13)]. Equations (1)–(6) are the result of applying the well-known method of symmetrical components (described, for example, in [4, Ch. 4]) to the coupled integral equations for the current distributions . The said coupled equations are explicitly written in [4, Eq. (10.1)]. Thus for a circular array, instead of solving coupled integral equations, integral (2) for , and then obtain we solve the from the second expression (1). is physically As discussed in [4, p. 388 and p. 396], the current on element no. 1 when the array is driven at its th phase-sequence, meaning that all elements are voltage-driven between adjacent voltages, with a constant phase shift being the voltage driving element no. 1. When and with the array is driven at its th phase-sequence, all elements are geometrically and electrically in the same environment—this is why the integral equations decouple. The “ th phase-sequence is defined as admittance” (7) When the array is driven at its th phase-sequence, is the driving-point admittance of any array element. Note that the

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is proportional to the total phase-sequence conductance must always (for radiated power. Thus, as is true for , any circular array, at any frequency) be a positive quantity. Obviously, the term “ th phase-sequence resonance” for an array with a single driven element reflects the fact that each peak in the driving-point conductance is due to a peak in the plot of for a certain value of [2], [4, §11.3]. At the th phase-sequence resonance, only the th term in the right-hand side of (1) is important and—as a consequence of (3) and (7)—we have and . The original kernels (5) with (6a) have been successfully used in circular-array analyses for many years (see, e.g., [41], [42]) and are instantly recognizable: The self-term (6a) is what commonly is called the “approximate kernel” in Hallén- and Pocklington-type integral equations [40]. The mutual terms (5) assume an axis-to-axis interaction; this assumption is logical (because the dipoles are thin) and perfectly adequate for conventional arrays. The modified kernels (5) with (6b) result from (5) with (6a) if one simply sets in the imaginary part of the self-term—but not in the real part of the self-term, which remains the same. Loosely speaking, (6b) treats the imaginary part of the self-term on an equal footing as the imaginary part of the mutual terms—but treating the real part in such a manner would result in a kernel with a non-integrable singularity. A more rigorous justification of using (6b) can be found in [2], [4, §11.8] (see also [7], [36], [37]). The consequences of using (6b)(6b) within the two-term theory are fully discussed in [2], [4, Ch. 11]. The specific MoM that we apply to (2) is Galerkin’s method with pulse basis functions, exactly as applied in [40] to Hallen’s integral equation (in [40], the symbol is used in place of our ). Note from [40, Eq. (8)] that the Toeplitz-matrix elements, which are originally double integrals, can be reduced to single integrals; this reduces the effect of numerical-integration errors. III. NUMERICAL RESULTS AND COMPARISONS We first show MoM results obtained with the original kernels: as a function of freThe solid lines in Fig. 2 are , corresponding to the last quency (we choose -element array (Fig. 1) phase-sequence resonance), for a , , and (see the with Introduction for the origin of these parameters). The leftmost peak corresponds to , the next to , and so on , which is the rightmost peak, so that increasing until the number of basis functions always shifts the resonance leftwards in frequency. The dotted line is the corresponding result

(6a)

(6b)

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G 20 m = 10 2h = 4:36 cm a = 3:17 mm M = 26 M = 25

f

N=

Fig. 2. Phase-sequence conductance as function of frequency for , , , , and . All results obtained with the original kernels (5) and (6a). Solid lines are MoM results, with in the leftmost peak, in the next, and so on until , which is the rightmost peak. Dotted line is two-term theory result.

d = 3:546 cm

M=5

obtained using the two-term theory with the same (original) kernels.1 It is seen that all peaks are “negative peaks,” so that the results in Fig. 2 are meaningless, from which no further conclusions should be drawn. This is true for all MoM results in Fig. 2 of basis functions), but also for (whatever the number the corresponding two-term theory result. It is also seen that the ; two-term theory result is closest to the MoM result with this is true both for the “resonant frequency” and the “negative peak height” , where the meaning of our notation is that attains its maximum value when . The existence of a negative peak for a particular value of reminds us of the NEC results and discussions of [38]. In particular, the negative peak does not tend to disappear when one uses and for an increasing number of basis functions. For as in Fig. 2, we found that only the peak values of is negative. But other values of give rise to more than one and , the and negative peak, e.g., when peaks are positive, while the last two peaks ( , ) are negative. Fig. 3 is just like Fig. 2, but with the modified kernels. The simple change from (6a) to (6b) eliminates all negative peaks, whatever the value of . Here, the two-term theory result is and . Note that closer to the MoM results with of Fig. 3 are very close to those of Fig. 2. At least for the the two-term theory, this was expected, because the obtained [2], via the two-term theory depend on the real part of [4, §11.3], whereas the change from (6a) to (6b) only affects . Fig. 4 is just like Fig. 3, but with the driving-point susceptance shown in place of . As one might expect beforehand—and as explained in [4, §12.4] via the two-term theory behaves like the input susceptance of a high- se—ries RLC circuit: vanishes, roughly when and , and the peak values of are approximately 1The relevant two-term theory formulas with the modified kernels can be found in [2] or [4, §11.2]. To obtain such formulas with the original kernels, by in [2, Eq. (B9)] or [2, simply replace Eq. (B9)] or [4, Eq. (11.13)]. We stress that there is no parameter similar to in the two-term theory, which is a closed-form approximate solution.

sin kz=z sin kpz + a =pz + a

Fig. 3. Like Fig. 2, but with the modified kernels (5) and (6b) in place of the original kernels.

M

Fig. 4. Like Fig. 3, but phase-sequence susceptance .

G

B

shown instead of

. (Curves behaving in this manner are often encountered in electrical engineering problems, e.g., [43] describes a similar situation occurring in highly mismatched transmission lines). The assertion above is true at least for the MoM peaks . The leftmost peaks on the right in Fig. 4, up to about appear “elevated,” and this trend continues if is increased a in Fig. 4. bit further than its largest value We had expected an unfamiliar and unnatural behavior when is sufficiently large: Fig. 5 shows the imaginary part of the current along dipole no. 1 (normalized to ), for the resonant case when . This MoM result is seen to exhibit unnatural oscillations near the dipole’s center. (Additional oscillations occur near the dipole’s end; we focus on those near the center because of their relevance to the bein Fig. 4.) We found that oscillations always havior of the is sufficiently large, and that they become more occur when rapid and intense as increases. Therefore, one should not trust values in Fig. 4 when is large. Similar oscillations the also occur in the case of Hallén’s equation with the approximate exceeds the value [40], compare kernel, roughly when Fig. 5 especially to [40, Fig. 2]. Behavior as in Fig. 5 was expected because the integral (2) shares the relevant properties of Hallén’s equation that cause the unnatural oscillations: namely, that is a smooth function of and a right-hand a kernel side with a discontinuous derivative (the discontinuity is a direct consequence of the delta-function generator) [40], [44]. Note

FIKIORIS et al.: ORIGINAL AND MODIFIED KERNELS IN MoM ANALYSES OF RESONANT CIRCULAR ARRAYS

Fig. 5. Imaginary part of the normalized current ImfI (z )=V g = ImfI (z )=V g as a function of distance z= to center of dipole 1; N = 20, m = 10, 2h = 4:36 cm, a = 3:17 mm, d = 3:546 cm, and M = 20. The frequency f is the resonant frequency f when M = 20, = 2:576 GHz. (The horizontal coordinate of the nth so that f = f dot coincides with that of center of the nth basis function, while the vertical coordinate equals the imaginary part of the nth basis-function coefficient.).

Fig. 7. Peak (resonant) values G

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of Fig. 3 as function of 1=M .

Fig. 8. Normalized power pattern jrE j for array with a single driven element obtained from the two-term theory with f = 2:767 GHz (solid curve) and from the MoM currents for M = 7 and f = 2:748 GHz (dots) as function of  for  = =2. Fig. 6. Resonant frequencies f

of Fig. 3 as function of 1=M .

especially that with the chosen parameters, has the value , a value much smaller than the encountered in conventional arrays. One might have hoped for better agreement with the two-term increases, but Figs. 3 and 4 reveal that this theory result as not the case. Even if one did not have available the two-term theory result, one might have expected the MoM results to “apis made large (but, at least for the case pear convergent” if of , not much larger than ). In particular, one might have anticipated convergence of and , which are shown in Figs. 6 and 7, respectively, as a function of . decreases, Fig. 6 does allow one to very roughly estiAs by extrapolation: appears to be between 2.4 and mate 2.5 GHz. On the other hand, it is much more difficult to make , which does not appear convergent. such an estimation for Finally, for the case of a resonant array with a single driven element, Figs. 8 and 9 compare the normalized power pattern obtained from the two-term theory (specifically from [4, , which is the two-term theory Eq. (11.30)], with resonant frequency) to corresponding results obtained from the MoM of the present paper. The MoM results were specifically inobtained by applying our Galerkin’s method to the tegral equations in (2), and then using (1) to obtain the currents

Fig. 9. Like Fig. 8, but pattern shown as function of  for 

= 0.

when one element is driven. Because our basis functions thus obtained are staircase-type functions; are pulses, the thus, the total far field can be found as a superposition of the far Hertzian dipoles. We took and fields of . Fig. 8 the corresponding resonant frequency shows the pattern as function of the polar angle in the plane of the dipoles’ centers, with each of the two curves normalized to the maximum value. Similarly, Fig. 9 shows the . The results obnormalized pattern as function of for tained by the two theories are seen to agree very well. The field

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consists of beams, in accordance with the discussions in [4, §11.5]. Narrower resonances tend to yield beams that are equal to one another. Such patterns are useful for the microwave-beacon application alluded to in Section I. IV. POSSIBLE EXTENSIONS The results of Section III point to a number of possible extensions which we list here. of this paper (1) As already mentioned, the value is unusually small, leading to problems even for relatively small values of . The specific values of , , and were taken from the -element experimental circular array of [2]–[4] (the considerations for choosing the parameters of the experiment can be found in [5, §8.2] and, in less detail, [3]). The two-term theory results of [2, Table 1] and [4, Table 11.1] indicate that there exist , , and that would (i) still yield narrow resonances, and (ii) lead to a somewhat larger value of . This would allow a MoM analysis with correspondingly larger values of . In particular, one could find , , and that would lead to slightly smaller values of which, in the present paper, is unusually large (its value is nearly 0.03). Let us re-stress that many conventional parameter choices yield no resonances at all. (2) For the case of an isolated dipole, [45] proposes a possible remedy for oscillations such as those in Fig. 5. Briefly, one considers that the oscillating MoM current lies on the -axis, calculates the magnetic field due to this current at a distance , and defines a new current by . This new current appears in [45] to be smooth. Such a technique might also “work” for the present case of a circular array. Even for the isolated dipole, however, the proximity of this new current to “correct” results is not well understood and is currently under investigation. (3) Difficulties encountered with MoM analyses of an isolated dipole driven by a frill generator have been discussed in [46]. If one used a frill generator in place of the delta-function generator, the discussions in [46] indicate that one would not encounter oscillations near (similar to those in Fig. 4), something that would lead to more trustworthy values of . On the other hand, it is most likely that one would still encounter oscillations near in both the real and the imaginary parts (as in [46, Figs 1 and 2] (4) [2] and [4, §11.8] discuss the “refined modified kernels.” One still uses the mutual interaction kernels (5), but appropriately modifies the real part of the self-interaction kernel (6b). Importantly, the imaginary part of (6b) remains the same; the imaginary part of (6a) should not be used. We believe that MoM results obtained with such kernels will no longer present the main difficulties that we encountered with the modified kernels in Section III. Work is in progress to obtain such MoM results and to compare with the two-term theory. To make a fair comparison, one should incorporate what [2] and [4, §11.6]] call the “refinements for numerical

calculations” into the version of the two-term theory used in the present paper. (5) Actual dipoles have a finite (but large) conductivity. This very important effect has been incorporated into the various versions of the two-term theory [4, Ch. 12]. It would be of interest to apply MoM to the relevant th phase sequence integral equation in [4, Ch. 12] (specifically, to [4, Eq. (12.12)], an equation that seems to have first appeared in [5]; related equations can be found in the early work [47] and have been used recently in [48], [49]). V. CONCLUSION Resonant circular arrays of electrically short and thick cylindrical dipoles have been studied extensively in previous works using the two-term theory, which is an approximate closed-form solution of the integral equations for the currents along the dipoles. Obtaining accurate results by applying moment methods to the usual integral equations is known from [38] to present unique and particular difficulties. In the present paper, we make a first step towards obtaining such accurate results using moment methods. Our main conclusion is the crucial importance of using the “modified” kernels in place of the “original” ones. (The modified kernels were proposed in the early 1990s after detailed studies by D. K. Freeman and T. T. Wu, and their crucial importance for two-term theory solutions of resonant arrays has already been documented; their unimportance for conventional array problems has also been noted.) Even with the modified kernels, difficulties remain. In particular, the integral equations dealt with herein have certain common features with Hallén’s equation, the difficulties of which have been studied in detail [40]; this allowed us to explain why the values of phase-sequence susceptance are not trustworthy when the number of basis functions becomes too large. Finally, based mainly on extensive previous studies on resonant circular arrays [1]–[7], we proposed several ways of obtaining more reliable moment-method solutions. ACKNOWLEDGMENT The work of D. Tsamitros and S. Chalkidis comes from their Senior Thesis at the Hellenic Air Force Academy. The authors thank I. Chremmos for useful discussions. REFERENCES [1] G. Fikioris, R. W. P. King, and T. T. Wu, “The resonant circular array of electrically short elements,” J. Appl. Phys., vol. 68, no. 2, pp. 431–439, Jul. 1990. [2] G. Fikioris, R. W. P. King, and T. T. Wu, “A novel resonant circular array: Improved analysis,” in Progress in Electromagnetics Research, J. A. Kong, Ed. Cambridge, MA: EMW Publishing, 1994, vol. 8, ch. 1, pp. 1–30. [3] G. Fikioris, “Experimental study of novel resonant circular arrays,” Proc. Inst. Elect. Eng. Microw. Antennas Propag., vol. 145, no. 1, pp. 92–98, Feb. 1998. [4] R. W. P. King, G. Fikioris, and R. B. Mack, Cylindrical Antennas and Arrays. Cambridge, U.K.: Cambridge University Press, 2002. [5] G. Fikioris, “Resonant arrays of cylindrical dipoles: Theory and experiment,” Ph.D. dissertation, Harvard Univ., , 1993. [6] G. Fikioris, R. W. P. King, and T. T. Wu, “Novel surface-wave antenna,” Proc. Inst. Elect. Eng. Microw. Antennas Propag., vol. 143, no. 1, pp. 1–6, Feb. 1996.

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[7] D. K. Freeman and T. T. Wu, “Variational-principle formulation of the two-term theory for arrays of cylindrical dipoles,” IEEE Trans. Antennas Propag., vol. 43, pp. 340–349, Apr. 1995. [8] G. Fikioris and K. Matos, “Near fields of resonant circular arrays of cylindrical dipoles,” IEEE Antennas Propag. Mag., vol. 50, no. 1, pp. 97–107, Feb. 2008. [9] D. Margetis, G. Fikioris, J. M. Myers, and T. T. Wu, “Highly directive current distributions: General theory,” Phys. Rev. E, vol. 58, no. 2, pp. 2531–2547, Aug. 1998. [10] D. Margetis and G. Fikioris, “Two-dimensional, highly directive currents on large circular loops,” J. Math. Phys., vol. 41, no. 9, pp. 6130–6172, Sep. 2000. [11] R. W. P. King, M. Owens, and T. T. Wu, “Properties and applications of the large circular resonant dipole array,” IEEE Trans. Antennas Propag., vol. 51, pp. 103–109, Jan. 2003. [12] R. W. P. King, “A microwave beacon and guiding signals for airports and their approaches,” IEEE Trans. Antennas Propag., vol. 51, pp. 110–114, Jan. 2003. [13] S. Ebihara and T. Yamamoto, “Resonance analysis of a circular dipole array antenna in cylindrically layered media for directional borehole radar,” IEEE Trans. Geoscie. Remote Sens., vol. 44, no. 1, pp. 22–31, Jan. 2006. [14] G. Fikioris, “Field patterns of resonant noncircular closed-loop arrays,” J. Electromagn. Waves Appl., vol. 10, no. 3, pp. 307–327, 1996. [15] G. Fikioris, S. D. Zaharopoulos, and P. D. Apostolidis, “Field patterns of resonant noncircular closed-loop arrays: Further analysis,” IEEE Trans. Antennas Propag., vol. 53, pp. 3906–3914, Dec. 2005. [16] R. C. Hansen, Electrically Small, Superdirective, and Superconducting Antennas. New York: Wiley, 2008, sec. 2.5. [17] I. Psarros and I. Chremmos, “Resonance splitting in two coupled circular closed-loop arrays and investigation of analogy to traveling-wave optical resonators,” in Progr. Electromagn. Res. PIER 87, 2008, pp. 197–214. [18] V. Veremey, “Superdirective antennas with passive reflectors,” IEEE Antennas Propag. Mag., vol. 37, no. 2, pp. 16–27, Apr. 1995. [19] S. A. Maier, M. L. Brongersma, and H. A. Atwater, “Electromagnetic energy transport along arrays of closely spaced metal rods as an analogue to plasmonic devices,” Appl. Phys. Lett., vol. 78, no. 1, pp. 16–18, Jan. 2001. [20] A. N. Fantino, S. I. Grosz, and D. C. Skigin, “Resonant effects in periodic gratings comprising a finite number of grooves in each period,” Phys. Rev. E, vol. 64, no. 016605, Jun. 2001. [21] S. A. Maier, M. L. Brongersma, and H. A. Atwater, “Electromagnetic energy transport along Yagi arrays,” Mater. Sci. Eng. C, vol. 19, pp. 291–294, 2002. [22] A. D. Yaghjian, “Scattering-matrix analysis of linear periodic arrays,” IEEE Trans. Antennas Propag., vol. 50, pp. 1050–1064, Aug. 2002. [23] D. C. Skigin, A. N. Fantino, and S. I. Grosz, “Phase resonances in compound metallic gratings,” J. Optics A: Pure Appl. Opt., vol. 5, pp. S129–S135, 2003. [24] J. Poon, J. Scheuer, and A. Yariv, “Wavelength-selective reflector based on a circular array of coupled microring resonators,” IEEE Photon. Technol. Lett., vol. 16, pp. 1331–1333, May 2004. [25] W. K. Kahn, “Tolerance study for a range limited sensor utilizing the extended near-field of a circular array,” in Proc. URSI EMTS, Pisa, Italy, May 23–27, 2004, pp. 179–181. [26] S. V. Boriskina, “Theoretical prediction of a dramatic -factor enhancement and degeneracy removal of whispering gallery modes in symmetrical photonic molecules,” Opt. Lett., vol. 31, no. 3, pp. 338–340, Feb. 2006. [27] J. Wiersig and M. Hentschel, “Unidirectional light emission from high- modes in optical microcavities,” Phys. Rev. A—Atomic, Molecular, Opt. Phys., vol. 73, no. 3, pp. 1–4, 2006. [28] E. I. Smotrova, A. I. Nosich, T. M. Benson, and P. Sewell, “Threshold reduction in a cyclic photonic molecule laser composed of identical microdisks with whispering-gallery modes,” Opt. Lett., vol. 31, no. 7, pp. 921–923, Apr. 2006. [29] A. L. Burin, “Bound whispering gallery modes in circular arrays of dielectric spherical particles,” Phys. Rev. E, vol. 73, no. 066614, Jun. 2006. [30] S. V. Boriskina, “Spectrally engineered photonic molecules as optical sensors with enhanced sensitivity: A proposal and numerical analysis,” J. Opt. Soc. Amer. B: Opt. Phys., vol. 23, no. 8, pp. 1565–1573, 2006. [31] S. V. Boriskina, T. M. Benson, P. D. Sewell, and A. I. Nosich, “Directional emission, increased free spectral range and mode -factors in 2-D wavelength-scale optical microcavity structures,” IEEE J. Sel. Topics Quant. Electron., vol. 12, no. 6, pp. 1175–1182, Nov./Dec. 2006.

Q

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[32] I. D. Chremmos and N. K. Uzunoglu, “Properties of regular polygons of coupled microring resonators,” Appl. Opt., vol. 46, no. 30, pp. 7730–7738, Oct. 2007. [33] G. S. Blaustein, M. I. Gozman, O. Samoylova, I. Y. Polishchuk, and A. L. Burin, “Guiding optical modes in chains of dielectric particles,” Opt. Express, vol. 15, no. 25, pp. 17380–17391, Dec. 2007. [34] I. Y. Polishchuk, M. I. Gozman, O. M. Samoylova, and A. L. Burin, “Interference of guiding modes in ’traffic’ circle waveguides composed of dielectric spherical particles,” Phys. Lett. A, vol. 373, pp. 1396–1400, Feb. 2009. [35] I. Psarros and G. Fikioris, “Two-term theory for infinite linear array and application to study of resonances,” J. Electromagn. Waves Applicat., vol. 20, no. 5, pp. 623–645, Apr. 2006. [36] D. K. Freeman and T. T. Wu, “An improved kernel for arrays of cylindrical dipoles,” presented at the IEEE/AP-S Int. Symp., London, Ontario, Jun. 24–28, 1990. [37] D. K. Freeman, “Extremely narrow resonances in closed-loop arrays of quantum mechanical and electromagnetic interactions,” Ph.D. dissertation, Harvard Univ., , 1992. [38] G. Fikioris, P. J. Papakanellos, J. D. Koundouros, and A. K. Patsiotis, “Difficulties in MoM analyses of resonant circular arrays of cylindrical dipoles,” Electron. Lett., vol. 41, no. 2, pp. 54–55, Jan. 2005. [39] NEC-WIN Pro User’s Manual. Antenna Analysis Software Version 1.1. Hollister, CA: Nittany Scientific, Inc., 1997. [40] G. Fikioris and T. T. Wu, “On the application of numerical methods to Hallén’s equation,” IEEE Trans. Antennas Propag., vol. 49, pp. 383–392, Mar. 2001. [41] J. D. Tillman, Jr, The Theory and Design of Circular Antenna Arrays. Knoxville: Univ. Tennessee Eng. Experiment Station, 1966. [42] R. W. P. King, R. B. Mack, and S. S. Sandler, Arrays of Cylindrical Dipoles. Cambridge, U.K.: Cambridge Univ. Press, 1968, ch. 4. [43] G. Fikioris, “Analytical studies supplementing the smith chart,” IEEE Trans. Education, vol. 47, no. 2, pp. 261–268, May 2004. [44] T. T. Wu, Introduction to Linear Antennas, R. E. Collin and F. J. Zucker, Eds. New York: McGraw-Hill, 1969, ch. 8, pt. I. [45] P. J. Papakanellos and G. Fikioris, “A possible remedy for the oscillations occurring in thin-wire MoM analysis of cylindrical antennas,” Progre. Electromagne. Res. PIER 69, pp. 77–92, 2007. [46] G. Fikioris, J. Lionas, and C. G. Lioutas, “The use of the frill generator in thin-wire integral equations,” IEEE Trans. Antennas Propag., vol. 51, pp. 1847–1854, Aug. 2003. [47] R. W. P. King and T. T. Wu, “The imperfectly conducting cylindrical transmitting antenna,” IEEE Trans. Antennas Propag., vol. 14, pp. 524–534, Sep. 1966. [48] G. W. Hanson, “Fundamental transmitting properties of carbon nanotube antennas,” IEEE Trans. Antennas Propag., vol. 53, pp. 3426–3435, Nov. 2005. [49] G. W. Hanson, “Radiation efficiency of nano-radius dipole antennas in the microwave and far-infrared regimes,” IEEE Antennas Propag. Mag., vol. 50, no. 3, pp. 66–77, Jun. 2008.

George Fikioris (S’90–M’94–SM’05) received the Diploma of Electrical Engineering from the National Technical University of Athens, Greece (NTUA), in 1986, and the S.M. and Ph.D. degrees in engineering sciences from Harvard University, Cambridge, MA, in 1987 and 1993, respectively. From 1993 to 1998, he was an Electronics Engineer with the Air Force Research Laboratory, Hanscom AFB, MA. From 1999 to 2002, he was a Researcher with the Institute of Communication and Computer Systems, NTUA. From 2002 to 2007, he was a Lecturer at the School of Electrical and Computer Engineering, NTUA, where, since February 2007, he has been an Assistant Professor. He is the author or coauthor of over 30 papers in technical journals, and a similar number of conference presentations. He has coauthored Cylindrical Antennas and Arrays (Cambridge University Press, 2002). He is the author of Mellin-transform Method for Integral Evaluation: Introduction and Applications to Electromagnetics (Morgan & Claypool Publishers, 2007). His research interests include electromagnetics, antennas, and applied mathematics. Dr. Fikioris is a member of the American Mathematical Society and of the Technical Chamber of Greece.

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Dimitrios Tsamitros was born in Veria, Greece, in February 1987. He received the degree in telecommunications and electronics from the Hellenic Air Force Academy, Athens, Greece, in July 2009. He is now a Second Lieutenant in the Hellenic Air Force. He is concurrently studying at the School of Electrical and Computer Engineering, National Technical University of Athens, Greece.

Savvas Chalkidis was born in Drama, Greece, in June 1987. He received the degree in telecommunications and electronics from the Hellenic Air Force Academy, Athens, Greece, in July 2009. He is now a Second Lieutenant in the Hellenic Air Force. He is concurrently studying at the School of Informatics, Aristotle University of Thessaloniki, Greece.

Panagiotis J. Papakanellos (S’99–M’05) was born in Athens, Greece, in 1976. He received the Diploma and Ph.D. degree in electrical and computer engineering from the National Technical University of Athens (NTUA), Athens, Greece, in 1999 and 2004, respectively. During his postgraduate studies, he was supported by a scholarship from the Onassis Foundation. During 2007, he was a post-doctorate Researcher, supported by the State Scholarships Foundation of Greece. He is currently an Adjunct Lecturer with the Hellenic Air Force Academy. His main research interests are in the areas of antennas, wireless communications, computational electromagnetics and electromagnetic compatibility. Dr. Papakanellos is a member of the Technical Chamber of Greece.

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A New Fast Physical Optics for Smooth Surfaces by Means of a Numerical Theory of Diffraction Felipe Vico-Bondia, Miguel Ferrando-Bataller, Member, IEEE, and Alejandro Valero-Nogueira, Member, IEEE

Abstract—A new technique to compute the physical optics (PO) integral is presented. The technique consists of a blind code that computes the different contributions (stationary phase points, end points, etc.) numerically. This technique is based on a decomposition of the surface into small triangles and a fast evaluation of each triangle by means of a deformation of the integration path in the complex plane. This algorithm permits a fast and accurate evaluation of the PO integral for smooth large surfaces. The CPU time is almost independent of frequency. Index Terms—Asymptotic expansion, highly oscillatory integrals, path deformation, physical optics (PO), uniform theory of diffraction (UTD).

I. INTRODUCTION

T

HE physical optics (PO) approximation is a well-known technique used to analyze very large conducting structures. In scattering problems and radiation of large reflectors, the PO technique provides acceptable accuracy [1], [2] for some applications. This technique allows avoiding the hard solution of the method of moments (MoM) linear system by approximating this solution by the explicit PO current (1) The PO current is proportional to the tangent magnetic field in the lit region, and zero in the shadow region. The radiated field can be computed by a direct integration of these currents. When the scatterer has a very large electrical size, the nature of the integrals is highly oscillatory; therefore, the numerical evaluation can be very time-consuming. To speed up this process a number of techniques have been proposed in the past. The Gaussian Beam technique [3] is based on a decomposition of the incident field in terms of a relatively small number of beams with Gaussian shape. The diffraction of each beam is computed analytically. Manuscript received March 03, 2009; revised July 23, 2009. First published December 28, 2009; current version published March 03, 2010. This work was supported by the Spanish Ministerio de Educación y Ciencia under the project CSD2008-00068. F. Vico-Bondia is with the Grupo de Radiación Electromagnética GRE, Departamento de Comunicaciones, Instituto de aplicaciones científicas y técnicas ITEAM, Universidad Politécnica de Valencia, Valencia, Spain (e-mail: felipe. [email protected]). M. Ferrando-Bataller and A. Valero-Nogueira are with the Departamento de Comunicaciones, Universidad Politécnica de Valencia, Valencia, Spain (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2009.2039308

The technique proposed by A. Boag in [4], [5] is based on a subdomain decomposition of the scatterer and a subsequent aggregation of the scattering patterns associated to the subdomains. This technique is very fast when computing the full radiation pattern. Other techniques make use of analytical expressions to compute the PO integral. The most classical one was developed by Gordon [6] and Ludwig [7] and permits a linear amplitude variation and a linear phase variation. The solution is obtained in a closed form. Different generalizations of the Gordon and Ludwig’s work were performed first by Glenn Crabtree in [8] and later by F. Cátedra in [9]. They allow a quadratic phase and amplitude variation. Different approximated closed forms are proposed. The purpose of this paper is to introduce a numerical method to compute the PO integral with a low computation time and high accuracy. Another important aspect of this method is the versatility and robustness. A large number of problems can be treated with the same algorithm changing only the input file. In that sense the algorithm is blind. The method that we propose is explained in the next two steps. First (Section II) we obtain a quadratic fast physical optics (in a similar way as in [8] and [9]). The radiated field is obtained dividing the surface in small triangles. For each triangle, a quadratic approximation for the phase and amplitude function is performed. The contribution of each triangle is computed rigorously by using a path deformation technique. Second (Section III) we speed up the previous technique by emptying the triangles with a negligible contribution to the total radiated field. At the end, only a few of the triangles are computed and contribute to the total radiated field (those containing stationary phase points, endpoints and groups of them). Neglecting triangles introduces nonexistent boundaries at the frontier between these and the computed ones. To avoid it, the contribution of artificial edges is computed separately and extracted. The result is a robust blind code with a CPU time almost independent of frequency and with a physical interpretation that directly corresponds to diffraction theory. This algorithm can be applied to compute efficiently any kind of highly oscillatory integral (2) where is a large parameter. Such integrals are quite common in different fields of physics: nuclear physics [10], quantum mechanics [11], acoustic scattering [12] and also in the electromagnetic wave theory (equivalent edge currents [13], incremental currents [14], physical op-

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Fig. 1. Triangular quadratic element.

Fig. 2. Decomposition of the triangle into different trapezoidal regions.

tics currents [1], [2]). Nevertheless, to the authors knowledge, the surface PO integral is the most challenging to evaluate in terms of CPU time and accuracy. II. QUADRATIC FAST PHYSICAL OPTICS Next we develop a quadratic fast physical optics algorithm. The first step is to divide the full surface in small triangles (this is performed in the parametric domain). Next we find a quadratic approximation for the amplitude and phase functions on each triangle; that is (3)

asymptotic expansion, but the result of different rigorous analytical steps like path deformation, integration by parts and others. Therefore the result will be exact for all frequencies and all positions of the possible stationary phase point inside the triangle. The first step consists on using a special coordinate system to is a find the canonical form of the function . As we said second order polynomial function, therefore (6) Applying a particular linear change of variable it is always possible (in the non-degenerate case) to obtain the following expression:

where (7) (8)

(4) The approximation can be computed using the Taylor polynomial or the Lagrange interpolation polynomial. In that case the total number of triangles is strongly reduced (with respect to the linear approximation [6], [7]) due to the higher order of the functions. Nevertheless, as the Louville’s theory shows, the arising integrals cannot be solved analytically. (See [15]). The shadow-lit region problem is not faced in this paper. Standard -buffer algorithms can be applied to select and discard the lit and the shadow triangles respectively (see [16]). The goal now is to obtain a fast algorithm to compute the contribution of each triangle. The integral will be decomposed into elementary functions and slow varying line integrals. For each triangle we have (see Fig. 1) (5) and are second order polynomial functions, is the wave number, and is a triangular integration region in the plane. Next, different analytical techniques will be used to obtain an exact decomposition of (5) into different contributions: edges, vertex and stationary phase point. Yet, it is important to notice that the decomposition that will be obtained here is not an

and are the eigenvectors of the where quadratic form (6). Constants and are just a shift to move the minimum of the quadratic form to the origin of coordinates . See [17] for more details. The signs appearing on (8) depend on the sign of the eigenvalues of the quadratic form . Using (7) on (5) the following expression is derived: (9) where (10) and is the image of through the affine transformation (7) (linear transformation followed by a translation), which is again a triangle. with is not written exTo simplify, the dependency of plicitly in (10), that is, from now to the end of this section, we will assume to be in a generic triangle of the surface in the decomposition performed in (3). For the shake of simplicity also, in (8) are positive, therefore let us assume that both signs of we focus our attention on the following integral: (11)

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Fig. 4. Decomposition of the triangle into different trapezoidal regions. Contribution of each trapezoidal region in red.

Fig. 3. High frequency interpretation of the decomposition.

The other possibilities will be studied later. The integration region is triangular; therefore, (11) can be decomposed into several trapezoidal pieces as (12). See Fig. 2 (12) A particular procedure combining analytical transformations and path deformation techniques is proposed and described in Appendix II to split (12) into the following terms: (13) where, see (14) at the bottom of the page.

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Fig. 3 shows that we can interpret each element in (14) as the contribution of stationary phase points and endpoints. This claim can be proved by applying the asymptotic expansion of (14) and comparing with the non-uniform asymptotic expansion of (12). In order to compute the integral in the full triangle, the previous process is applied to each trapezoidal piece of the decomposition. Fig. 4 illustrates the points that contribute to the result. Most of the contributions of different trapezoidal regions cancel. The only contributions that remain are the vertex contributions, the edge contributions and the stationary phase contribution, as Fig. 5 shows.

otherwise

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Fig. 5. (a) High frequency interpretation of the decomposition for the whole triangle: (b) Notation for the equation of the triangle.

In summary, the following decomposition is obtained for each triangle:

(15) where, see (16) and (17) at the bottom of the following page. , are the intersection points of Fig. 23. In (17), Now (11) is written as a sum of slowly varying integrals and analytic expressions, therefore, the evaluation of (11) will be very fast and the computation time will not be -dependent. It is important to notice that there is no approximation in the process, neither functional nor asymptotic approximation (each new path of integration is obtained analytically, see (62)–(64)

in Appendix II). Therefore the result is uniformly approximated for all frequencies and positions of the stationary phase point. In this decomposition it is also very important to have access separately to each of the 7 contributions: , , etc. which, for high frequencies, can be identified as the classical contributions (stationary phase point, end point and vertex contribution) of the asymptotic approximation. This decomposition (and identification) will be the key to speed up this numerical process. This will be the topic of the next section. The previous development successfully solves the positive .” For the negative case ’ ’ a similar procecase “ dure could be applied. A similar decomposition as in (15) would be obtained. Nevertheless, a devastating difference arises. In the negative case, the 7 contributions could not be identified as the classical contributions of the asymptotic approximation. Therefore, the decomposition that would be obtained for the negative case using the same procedure could not be used in the next section for the improved algorithm that we will call numerical theory of diffraction. Fortunately a slightly different approach can be applied suc.” The trick consists cessfully for the negative case “ of a 45 degree rotation in the axes

(18) where (19)

(16)

(17)

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and

,

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are given by (24)

and see (25) at the bottom of the page, where (26)

Fig. 6. (a) High frequency interpretation of the decomposition for the whole triangle. (b) Notation for the equation of the triangle.

As in the previous case, decomposition into trapezoidal regions can be performed (see Fig. 2). The following integral is obtained: (20) A particular procedure combining analytical transformations and path deformation techniques is proposed and described in Appendix III to split (20) into the following terms: (21) where

(22)

(23)

equals 0 or 1 depending on the situation and described in Fig. 25. Using a similar reasoning as the previous one in Fig. 4, the integral over the whole triangle can be written as

(27) where we have (28) at the bottom of the page. For this case it is also important to notice that this decomposition of in 7 contributions is again exact (no approximation process has been applied) and for high frequencies, they can be identified as the classical contributions (stationary phase point, end point and vertex contribution) of the asymptotic approximation. Once we can analyze each parabolic patch efficiently, a quadratic fast physical optics can be successfully implemented. As Fig. 7 shows, the surface is divided into small triangles. The phase and amplitude function on each triangle is approximated by a second order Lagrange interpolatory polynomial, and the fast algorithm described above is applied. To conclude this section it is important to emphasize that the integral of each triangle has been decomposed in terms of closed forms and slow-varying line integrals. Therefore the computation time (for each triangle) is independent of frequency. Nevertheless, for a given accuracy, the total number of triangles required obviously increases with frequency and so does the total computation time (for the quadratic fast physical optics). This

(25)

(28)

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Fig. 7. Quadratic fast physical optics. Fig. 8. Naïve NTD. Spurious end point contributions.

drawback will be solved in the next section where we will introduce a substantial improvement which we have called the numerical theory of diffraction (NTD). NTD will speed up the previous method. In the end, the total CPU time will be almost frequency independent. III. NUMERICAL THEORY OF DIFFRACTION In the previous section, a fast technique to compute the diffraction of quadratic patches was introduced. The total field of each patch is split into different contributions , , etc. that can be obtained in a very short computation time. The decomposition is exact (no approximation or asymptotic process is applied). Moreover, the decomposition has an additional property: For high frequency, each contribution , , etc. tends to the classical asymptotic contribution (stationary phase, end point, etc.). In this section, this key property will be fully exploited to develop a new algorithm which indeed can be considered a new theory. A whole new approach to diffraction problems which we will call the numerical theory of diffraction (NTD). The classical asymptotic evaluation of the PO integral [18], approximates the total value of the integral in terms of different contributions as stationary phase points, stationary phase lines, end points, etc. Therefore, classical asymptotic techniques suggest that the main contribution of the PO integral is concentrated on the vicinities of the contributing points (stationary phase points, end points,…). A naïve NTD implementation would extend the domain of the PO integral only over the contributing patches as Fig. 8 shows

Fig. 9. NTD. No spurious contributions.

This approximation would be obviously wrong. Taking the integral over the reduced area, extra non-existing end point contributions would be included as Fig. 8 shows. In order to improve the naïve NTD above mentioned and make a technique that really works we have to remove the extra non-existing end point contributions (see Fig. 8). This is now possible by using the special decomposition developed in Section II. For each triangle in the selected integration region the contribution of the fictitious edges are removed from the decomposition (15) and (27). Fig. 9 shows the choice. Fig. 10 shows another example to clarify the idea. The final contribution of the complete triangles is given by

(29) where only the contributing triangles are included.

(30)

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Fig. 11. CP for the calculation of the phase variation along the edge (a) inner triangles; (b) triangles on the border. Fig. 10. (a) PO extended over the contributive triangles (spp, end points and edges) removing the non-existing edges. (b) Zoom of the Integration Region.

As it was shown in previous sections. For incomplete triangles, the contribution is as follows. In case of absence of one edge (edge number 1)

(31) In case of absence of two edges (edge number 1 and 2) Fig. 12. Nurbs surface.

(32) That is, the contributions corresponding to the sides of the non-existing edges have been removed. This simple technique (NTD) allows to obtain the total contribution of complex phenomena (two near stationary phase points, three stationary phase points and end point, etc.) numerically, just computing the integral over the reduced region that contains the contributions and remove the extra non-existing edges. This algorithm is particularly interesting when a uniform asymptotic contribution of the complex phenomena is difficult to find analytically [19]. This algorithm (NTD) is based on the selection of the contributing regions. In order to automate this process, a heuristic technique is used. The total variation of the phase is computed for each triangular patch. The selection of the contributing triangles is performed by comparison with a threshold

Fig. 13. RCS diagram. Comparison between different methods.

Finally, the triangles of the vertex are always selected (the end point contribution).

(33) where

(34) and is a set of control points in the triangle. For is the set of interpolation points the inner triangles, in the standard Lagrange interpois the subset of 3 lation. For the triangles in the edges, interpolation points along the edge (see Fig. 11).

IV. NUMERICAL EXPERIMENTS (NTD) In this section, we validate the methods presented above testing them with different geometries. We also show the internal process of the NTD and how the algorithm computes the contribution of certain regions on the surface. The first example is the surface of Fig. 12. The RCS diagram and . is computed for an angle Fig. 13 shows a perfect agreement between the NTD method and previous methods.

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Fig. 16. Nurbs surface.

Fig. 17. RCS diagram. Comparison between different methods. Fig. 14. NTD algorithm. (a), (c), (e): Phase function for different angles of and ' (b)  incidence. (b), (d), (f): Integration region. (a)  and ' (c)  and ' (d)  and ' (e)  and ' (f)  and ' .

=0 = 90

= 57 = 57

= 45 = 90

= 57 = 57

=0 = 45

= 57 = 57

Fig. 18. NTD for a surface with two separated saddle points.

Fig. 15. NTD for a surface with saddle point near wedge.

Fig. 14 shows the phase function in the parametric domain and the triangles selected automatically by the NTD algorithm for different angles of incidence. As we can see, the diagram contains a transition between no stationary phase point (SPP) and a stationary phase point near a wedge . The NTD algorithm selects the integration region around the SPP. The accuracy is uniform. Fig. 15 shows the triangularization of the surface in the space domain , the selected triangles and the removed artificial edges.

The next example shows a transition between one SPP and two SPPs. The surface analyzed is illustrated in Fig. 16. In that case the diagram is computed for an incidence of and . Fig. 17. shows perfect agreement among different methods. Fig. 18 shows the triangularization of the surface in the space , the selected triangles and the removed artifidomain cial edges. The next sequence of figures shows the phase function and integration region for different angles. In the sequence we can see the transition between one and two stationary phase points. We can also see how the integration region is divided into two disconnected regions when the two SPP move away. As we can see in the previous two examples, the integration region is strongly reduced and only contains regions of the surface that effectively contribute to the total diffracted field. At

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Fig. 21. CPU time of Fig. 22 (Full diagram, 10 directions).

V. CONCLUSION

Fig. 19. NTD algorithm. (a), (c), (e): Phase function for different incidences. and ' (b)  and (b), (d), (f): Integration region. (a)  ' (c)  and ' (d)  and ' (e)  and ' (f)  and ' .

= 45 = 060 = 45 = 57

= 065 = 45 = 57 = 45

= 45 = 45

= 57 = 040

Fig. 20. CPU time of Fig. 18 (Full diagram, 10 directions).

the same time, the contribution of each region is computed numerically without requiring special functions for each particular case and transition as UTD does. The next figures show the CPU time comparison between NTD, quadratic FPO and Linear PO. As shown in Figs. 20 and 21, the CPU time is strongly reduced by using the NTD algorithm. Moreover, the complexity is nearly constant with frequency; that is, the CPU time is . This is a completely new property for a numerical technique, which was reserved only for asymptotic approaches like GTD or UTD.

In this paper, we develop a new fast physical optics technique (FPO) called the numerical theory of diffraction (NTD). The technique is based on a new approach for the quadratic fast physical optics developed by F. Catedra in [9]. The new algorithm allows to compute fast and efficiently the radiated field by curved surfaces of arbitrary shape. The algorithm was tested successfully with different geometries. The NTD is based on a decomposition of each quadratic patch into different contributions that extends the asymptotic contribution for high frequencies (stationary phase point, end point, etc.) This condition allows neglecting all the triangles where no high frequency phenomena take place. The algorithm can also be applied to speed up the computation time of a single direction of observation or any number of points in the radiation pattern. In that sense, the algorithm is better that other algorithms based on domain decomposition and aggregation [4], [5] that only improves the CPU time of the full radiation pattern. As far as the scalability of the method is concerned, the ; that is, it does not incomputation time is crease with frequency. This is a property that previously only the purely asymptotic methods like GTD or UTD had. Therefore we introduce a numerical technique (NTD) (blind for different smooth surfaces) with uniform accuracy for different incidences and constant CPU time with frequency. This technique is applied to compute the radiation of the PO current, as this current is the most time consuming in the PO algorithms, nevertheless the same algorithm could be extended to compute the radiation of different equivalent edge currents (or any highly oscillatory integral), so that a full numerical theory of diffraction could be completed. APPENDIX I FRESNEL REPEATED FUNCTIONS The definition of Fresnel repeated function of order is given by the following expression: (35)

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where (36)

where Appendix I). Therefore

is the

and

-repeated Fresnel integral (see

(42) (37)

is the complex continuation of the The function complementary error function (see [20]). The most efficient way to evaluate these functions for is by using the following recursive formula:

where

(43)

(38)

APPENDIX II INTEGRAL ON A TRAPEZOIDAL REGION. THE POSITIVE CASE In this section, an efficient technique to compute the integral (39) on trapezoidal regions is developed for the positive case (39) Now has the same sign (positive or negative). To sim. For the other case , plify let us assume that is performed. a simple change of variable The integral with is solved analytically (using integration by parts)

(40)

(44) is a highly oscillatory line integral that can be solved analytically (again in terms of the Fresnel function). Nevertheless it is important (for the development of the NTD) to obtain sepa, and rately the contribution of the stationary phase point the end point contributions

(45) where, see (46)–(48) at the bottom of the page. is a highly oscillatory line integral that cannot be solved analytically. Therefore a path deformation technique will be used. The integrand is the product of a complex exponential and a Fresnel function, that is, two oscillatory functions. In order to find the path of integration it is important to find the phase behavior of the full integrand in the whole complex plane. First the behavior of the Fresnel functions is studied. The oscillatory behavior of the Fresnel function of order 0 is plotted in Fig. 22. The figures suggest that we can write the following:

(49)

(41)

(46)

(47) else (48)

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Multiplying by

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the decomposition (55)

(57)

Fig. 22. Oscillatory behavior of the Fresnel function of order 0.

where and are slow varying functions, and a partition of the complex plane given by

is

where , and are slow varying functions. Usually, steepest descent technique is used on functions , nevertheless (57) with only one phasefront, that is shows two different phase fronts (depending on the region of the complex plane). The idea is to split the integral into two parts and solve them independently. Let us define the following discontinuous Fresnel complementary integrals:

(50) This decomposition can be generalized for the rest of the repeated Fresnel integrals (51)

(58) Obviously we have the following phase behavior for (58)

(59) (52) Analogously we introduce the following piecewise polynomial function:

(60) Both functions are discontinuous, and verify the following trivial functional relationship: (53) (61) where (54) Therefore, a decomposition of the form

This decomposition allows applying the steepest descent technique separately on each addend. Let us define the following steepest descent path for the integral:

(55) (62) is obtained where (63) (56) (64)

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of the new path, but they will be important in the next decomposition. The indexes in are used only to stress the dependency of the path with these variables , and . Nevertheless, they will not be used if the context suggests these values. These paths verify the following steepest descent condition for :

Fig. 23. Path deformation.

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where (65) (66) Points A and B are the intersection of and with the frontier between and . They are irrelevant in the definition

The new integration path (62)–(64) is used directly over the complex continuation of the integral(44). The expressions for the integral over the new path is shown in (68) at the bottom of the page. Now applying the decomposition (61), see (69) at the bottom of the page, where we have (70) at the bottom of the page. The non-analytic part is an integral over the steepest descent path, therefore it is a slowly varying integral and can be solved

(68)

(69)

(70)

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efficiently using a numerical quadrature rule shown in (71)–(74) at the bottom of the page. The integrand of the analytic part is zero on , therefore the integration path can be reduced to the A-B segment. Therefore, we have (75) at the bottom of the page. Using integration by parts, see (76) at the bottom of the page, where

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To summarize we have

(79) where, see (80) at the bottom of the following page. Each contribution can be interpreted as the effect of an end point or a stationary phase point, as illustrated in Fig. 3. APPENDIX III INTEGRAL ON A TRAPEZOIDAL REGION. THE NEGATIVE CASE

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In this section, an efficient technique to compute the integral (81) on trapezoidal regions is developed for the negative case

Note that is a second order polynomial function (the is only to stress the dependency on and ). index

(81)

(71) (72) (73) (74)

(75)

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Applying integration by parts for the first integral

(82) As (82) shows, the integrand contains two different phasefronts; that is (83) In order to apply path deformation it would be desirable to split this “multi-phasefront” integral into two “single- phasefront” integrals

(84) Unfortunately this simple step introduces much more complexity than desirable.

is smooth, thereThe function fore integrable in any reasonable integration theory (Riemann, ). Nevertheless, the functions and Lebesgue separately contain poles at up to third de, , . Therefore no standard integragree, that is: tion theory can be applied to cover this situation. Notice that Cauchy’s principal value does not work for the second order poles. Therefore, the step (84) is not allowed in a standard sense. The idea for the next step is to apply path deformation, thereand must be fore both phasefronts somehow separated. In order to do that, a nonstandard integration theory is used. The one more useful for our purpose is the Abel’s summation technique (see [21]). This integration technique will allow us to split both phasefronts into two finite quantities, and make finite the integral over the singularities. In a more abstract sense, the Abel’s summation technique is an extension of the ordinary integral (Lebesgue) that extends the operator on a larger space of non-integrable functions. This extension verifies some regularity properties (linearity, addition, etc.). See [21] for more details. Applying Abel’s definition: see (85) at the bottom of the page, where, see (86) at the bottom of the following page.

else

(80)

(85)

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Fig. 24. Path deformation for > 0.

It is remarkable the fact that both expressions and are written in terms of “single phasefront” functions. It is also imboth integrals are standard portant to notice that for any (they exist in the sense of Riemann/Lebesgue, without singularities and high order poles on the real line) and the limit also exists. Another important fact to understand the whole piccannot be interchanged with the ture is that the limit integral because the hypothesis of the Dominate Convergence Theorem is not verified (the functions are not dominated by any Lebesgue integrable function (see [22]). This point shows explicitly why Abel’s summation technique is a true extension of the standard Lebesgue integral theory. Thus, see (87) at the bottom of the page. the path deformation technique can be Now, for each applied. The new integration path is showed in Fig. 24. but It is important to notice that the pole is no more in in . As the singularity is in the upper semiplane, at an infinitely small distance from the origin. Therefore, deand , pending on the sign of and the position of , the residue of the singularity must be taken into account. Fig. 25 shows all the possibilities.

Fig. 25. Path deformation for different values of a and the position of L , L and ' (0) (a) and (c) the residue must be included (b) and (d) the residue must be not included. (a) a > 0 (b) a > 0 (c) a < 0 (d) a < 0.

To summarize, if the pole is inside the grey region and small enough, then the residue must be taken into account. It is important to notice that, once the path deformation is can be interchanged with the integral applied, the limit again (In the new path, the Lebesgue dominate convergence theorem is verified). The equations for the new integration path are ) the following: (in the limit

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(89)

(86)

(87)

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(96)

(90) where

REFERENCES (91)

These paths verify the following steepest descent condition for :

(92) Therefore, the integral

equals 0 or 1 depending on the sitWhere uation described in Fig. 25.

can finally be spit into

(93) where

(94)

(95) , is the contribution of the pole at And in the limit when ; that is, the pole at . The residue of each integrand is given by (96), shown at the top of the page, where (97) Therefore,

(98)

[1] P. H. Pathak, “Techniques for high frequency problems,” in Antenna Handbook Theory, Application and Design, Y. T. Lo and S. W. Lee, Eds. New York: Van Nostrand Reinhold, 1988. [2] L. Infante and M. Stefano, “Near-field line-integral representation of the Kirchhoff-type aperture radiation for parabolic reflector,” IEEE Antenna Wireless Propag Lett., vol. 2, pp. 273–276, 2003. [3] P. H. Pathak and R. Burkholder, “Novel Gaussian beam method for the rapid analysis of large reflector antennas,” IEEE Trans. Antennas Propag., Jun. 2001. [4] A. Boag, “A fast physical optics (FPO) algorithm for high frequency scattering,” IEEE Trans. Antennas Propag., vol. 52, no. 1, pp. 197–204, Jan. 2004. [5] A. Boag and E. Michielssen, “A fast physical optics (FPO) algorithm for double-bounce scattering,” IEEE Trans. Antennas Propag., vol. 52, no. 1, pp. 205–212, Jan. 2004. [6] W. B. Gordon, “Far-field approximations to the Kirchhoff-Helmholtz representation of scattered fields,” IEEE Trans. Antennas Propag., pp. 590–592, Jul. 1975. [7] A. C. Ludwig, “Computation of radiation patterns involving numerical double integration,” IEEE Trans. Antennas Propag., pp. 766–769, Nov. 1968. [8] G. Crabtree, “A numerical quadrature technique for physical optics scattering analysis,” IEEE Trans. Magn., vol. 27, no. 5, pp. 4291–4294, Sep. 1991. [9] C. Delgado, J. Gónzalez, and F. Cátedra, “Analytic field calculation involving current modes and quadratic phase expressions,” IEEE Trans. Antennas Propag., vol. 55, no. 1, pp. 233–240, Jan. 2007. [10] K. T. R. Davies, “Complex-plane methods for evaluating highly oscillatory integrals in nuclear physics: II,” J. Phys. G: Nucl. Phys., no. 14, pp. 973–993, Jan. 1988. [11] V. Kucherenko, “Quasiclassical asymptotics of point-source functions for the stationary Schrodinger equation,” Theor. Mat., no. 3, pp. 384–406. [12] M. Ganesh, S. Langdon, and I. H. Sloan, “Efficient evaluation of highly oscillatory acoustic scattering surface integrals,” J. Comput. Appl. Math., vol. 204, no. 2, pp. 363–374, Jul. 2007. [13] C. E. Ryan and L. Peters Jr., “Evaluation of edge diffracted fields including equivalent currents for the caustic regions,” IEEE Trans. Antennas Propag., vol. AP-17, pp. 292–299, 1969. [14] R. Tiberio and S. Maci, “An incremental theory of diffraction: Scalar formulation,” IEEE Trans. Antennas Propag., vol. 42, May 1994. [15] J. F. Ritt, Integration in Finite Terms. Liouville’s Theory of Elementary Methods. New York: Columbia Univ. Press, 1948. [16] C. Qinfeng and X. Penggen, “Application of improved Z-buffer technique to RCS computation,” J. Natural Sci., vol. 3, no. 1, pp. 53–55, Mar. 1998. [17] T. M. Apostol, Calculus. New York: Wiley, 1969, vol. 2, pp. 128–130. [18] O. M. Conde, J. Perez, and M. F. Catedra, “Stationary phase method application for the analysis of radiation of complex 3-D conducting structures,” IEEE Trans. Antennas Propag., vol. 49, no. 5, pp. 724–731, May 2001. [19] A. Borovikov, “Stationary phase method for a saddle point near the boundary of the domain of integration,” in Mathematical Notes. New York: Springer, 2005, pp. 91–98. [20] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions. Norwood, MA: Dover, 1972. [21] G. H. Hardy, Divergent Series. Providence, RI: AMS Bookstore, 2000. [22] W. Rudin, Real and Complex Analysis. New York: McGraw-Hill, 1987.

VICO-BONDIA et al.: A NEW FAST PHYSICAL OPTICS FOR SMOOTH SURFACES BY MEANS OF A NTD

Felipe Vico-Bondia was born in Valencia, Spain, in 1981. He received the M.S. and Ph.D. degrees in electrical engineering from the Universidad Politécnica de Valencia, in 2004 and 2009, respectively, and the M.S. degree in mathematics from the Universidad Nacional de Educación a Distancia, Valencia, in 2009. From 2004 to 2005, he was with the Institute of Telecommunications and Multimedia Applications (iTEAM) as a Research Assistant. During 2006 to 2007, he was on leave at the Instituto Superiore Mario Boella, Politecnico di Torino, Italy, where he developed hybrid techniques for high frequency EM scattering problems. Since 2007, he has been an Assistant Professor in the Communications Engineering Department, Universidad Politécnica de Valencia, where he is currently working. His research interests include numerical methods applied to scattering and radiation problems, asymptotic techniques, hybridization of high frequency and numerically rigorous methods and fast computational techniques applied to electromagnetics. Dr. Vico-Bondia was awarded a Research Fellowship by the Spanish Ministry of Culture for 2005 to 2006. In 2006 he received the “Best Conference Paper Award” at the European Conference on Antennas and Propagation.

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Miguel Ferrando-Bataller (S’81–M’83) was born in Alcoy, Spain, in 1954. He received the M.S. and Ph.D. degrees in telecommunication engineering from the Universitat Politecnica de Catalunya, Barcelona, Spain, in 1977 and 1982, respectively. From 1977 to 1982, he was a Teaching Assistant in the Antennas, Microwave and Radar Group, Universitat Politecnica de Catalunya, and in 1982, became an Associate Professor. In 1990, he joined the Universidad Politecnica de Valencia, Spain, where he was Director of the Telecommunication Engineering School and Vice-Chancellor. He is currently Director of Long-life learning Office and Professor of antennas and satellite communications. His current research interest includes numerical methods, antenna design and e-learning activities.

Alejandro Valero-Nogueira (M’09) was born in Madrid, Spain on July 19, 1965. He received the M.S. degree in electrical engineering from the Universidad Politécnica de Madrid, Madrid, Spain, in 1991 and the Ph.D. degree in electrical engineering from Universidad Politécnica de Valencia, Valencia, Spain in 1997. In 1992, he joined the Departamento de Comunicaciones, Universidad Politécnica de Valencia, where he is currently an Associate Professor. During 1999, he was on leave at the ElectroScience Laboratory, The Ohio State University, Columbus, where he was involved in fast solution methods in electromagnetics and conformal antenna arrays. His current research interests include computational electromagnetics, Green’s functions, waveguide slot arrays and gap waveguides.

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Planar Electromagnetic Bandgap Structures Based on Polar Curves and Mapping Functions Charity B. Mulenga and James A. Flint, Senior Member, IEEE

Abstract—A type of electromagnetic bandgap structure is described that is easily parameterized and can produce a range of square and spiral geometries. Individual electromagnetic bandgap (EBG) geometries are defined on a cell-by-cell basis in terms of their convolution factor , which defines the extent to which the elements are interleaved and controls the coupling slot length between adjacent elements. Polar equations are used to define the slot locus which also incorporate a transformation which ensures the slot extends into the corners of the square unit cell and hence extends the maximum slot length achievable. The electromagnetic properties of the so-called polar EBG are evaluated by means of numerical simulation and measurements and dispersion diagrams are presented. Finally, the performance is compared with other similar miniaturized EBG cell geometries. It is shown that the polar EBG has better angular stability than the equivalent square patch design and is comparable in terms of performance to other low frequency EBG elements. At the same time it retains the ability to fine tune the response by adjusting . Index Terms—Electromagnetic bandgap (EBG), mapping functions, metamaterials, polar curves.

circuits [18]–[20] and improvement of signal isolation in RF mixed signal systems [20]. The properties of the EBG only exist within a specific band and thus careful design is necessary in order to select the resand the bandwidth (BW) of a particular onant frequency array geometry. For a given array the center frequency is given by [21] (1) where and are the per unit area inductance and capacitance respectively. In the simplest element geometry, which consists of an array of square patches shorted to ground at their centers, the capacitance is mainly due to fringing fields between elements and the inductance is mainly due to the current loop length between adjacent vias [1]. The bandwidth is also an important consideration and is given by [21] (2)

I. INTRODUCTION VER the past decade periodic structures such as electromagnetic bandgap (EBG) materials have attracted extensive research interest in the microwave and millimeter wave domains. The major characteristic of EBG surfaces is the existence of one or more bands whereby propagating surface waves are effectively suppressed. In addition, many of these structures also have the property that they reflect normally-incident plane within certain waves with a reflection coefficient of frequency bands. In this case the surface behaves as a wall of magnetic symmetry and hence is often termed an artificial magnetic conductor (AMC) [1], [2]. There are numerous applications of these periodic metamaterials in the field of antenna design [3]–[9]. Specific uses for these materials include producing improved ground planes for individual antennas. [10], and reducing mutual coupling for patch antennas or arrays [7], [11]–[14]. Additionally, EBG structures have been applied to coplanar waveguides (CPW) to reduce leaky waves. [15], [16] and to realize directive base station antennas [17]. Other applications include the mitigation of switching noise in high-speed

O

Manuscript received May 06, 2009; revised July 16, 2009. First published December 31, 2009; current version published March 03, 2010. This work was supported in part by Metamorphose NoE and in part by Loughborough University. The authors are with the Department of Electrical and Electronic Engineering, Loughborough University, Loughborough LE11 3TU, U.K. (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2009.2039319

where is the free space impedance. In low frequency applications, it is often the case that the array becomes physically very large for a given number of cell periods and thus it is necessary to apply special methods and geometries in order to increase and . The most desirable approach would be to increase the inductance as it can be seen in (2). This leads to a bandwidth improvement (e.g., by introducing meandering vias [22], [23], slanting vias [24], or high permeability materials [25]). However there are disadvantages to this approach such as introducing more difficult manufacturing processes or increasing losses. Another approach which can be taken in addition to increasing inductance, is increasing the capacitance between adjacent elements and that is the focus of the current paper. Capacitance increase can be achieved by using high permittivity substrates [23], interleaving and convoluting elements [26], introducing inter-digitated edges between adjacent patches [6], [27] and by reducing the gap width between adjacent elements. It is also possible to produce thick conducting elements at the surface which exhibit both fringing and parallel-plate type capacitance. Frequency-reducing geometries which have been proposed often make use of fractal space filling methods such as Hilbert and Peano curves. As the iteration order of the curve increases, the footprint is preserved while the length of the curve increases. The Hilbert curve geometry has been applied successfully to produce AMC structures with low resonant frequencies. [2], [28]. One issue with these element types is the necessity to select an integer order number which means that fine tuning of the

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MULENGA AND FLINT: PLANAR EBG STRUCTURES BASED ON POLAR CURVES AND MAPPING FUNCTIONS

resonant frequency has to be achieved by adjusting other parameters such as the array periodicity, substrate thickness and gap widths. It is an objective of the current paper to demonstrate an array element geometry which can be tuned without changing the substrate thickness or gap width, which are often inconvenient to change. In contrast, it is typically simple to modify the slot geometry on a substrate via photolithographic-etching or routing methods. If the array periodicity can be fixed between elements it also allows the frequency of different regions to be adjusted independently, potentially allowing geometrical perturbations in the surface to be corrected for. This paper is organized as follows. In Section II, the concept of using polar curves and mapping functions in the design of low frequency EBG structures is introduced along with its resultant bandgap and AMC features. Section III discusses a method by which the surface can be tuned. The angular stability of polar-EBG structures is discussed in Section IV. Section V presents comparisons of the proposed geometry to well-known low frequency EBG structures, followed by conclusions in Section VI. II. GEOMETRY AND PERFORMANCE OF POLAR EBGS A. Polar EBG Geometry A diagram showing the upper surface of the proposed spirallike polar-EBG structure is shown in Fig. 1. The area in the center has been darkened in order to indicate the unit cell with the darkest areas representing the metallic areas in the array and the lighter parts representing the slots cut into the upper conductor. Below the upper surface is a layer of dielectric and finally a layer of conducting material below this to form a continuous ground plane. The ground plane is connected to the center of the metal patches in the array by a set of thin via pins which are labeled in Fig. 1. If it is desired to increase the surface capacitance per unit area, then it is possible to decrease the width of the slot, but arguably a better way and an approach more conducive to easy manufacture is to increase the slot length. A straightforward method of defining the slot geometry is to make use of a polar equation which defines the locus of the slot (shown dotted in Fig. 1). We define a non-integer factor, , termed the convolution factor, which controls the extent to which the spiral geometry is modified. Higher convolution factors yields a geometry which tends towards the standard “mushroom” patch surface analyzed by Sievenpiper [1], lower numbers achieve spiral geometries of varying levels of convolution. The polar equation is shown in (3) (3) where describes the locus of the spiral slot shown dotted in Fig. 1 and other geometrical dimensions are as labeled. Equa-

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Fig. 1. Top view of the polar EBG structure (dark gray—copper). The locus of the slot (shown dotted in the unit cell) is defined by a polar equation.

tion (3) is derived by assuming that the center point of each individual slot within the unit cell of the original patch remains fixed, as do the extremities. This simple equation ensures that more convoluted slots (with lower values) will always remain equidistant as they cross the line of the original slot locus. The other slots in the unit cell are then determined by rotational symis a mapping function which further modifies the metry. polar function in order to better utilize the surface area available in the unit cell by extending the spiral-slot length. For generates the plain circular spiral-slots shown on the left hand side of Fig. 2. For a square unit cell, can be chosen as the periodic function, as shown in (4) at the bottom . The function, shown plotted of the page, where in Fig. 3, produces warped spiral-slot patterns on the right hand side of Fig. 2. Note that these patterns are no longer idealized spirals but benefit from longer slot lengths due to their longer path within the unit cell. The geometrical effects of changing the convolution factor and applying warping are clearly evident in Fig. 2. It can be seen that a variety of spiral and square-type EBG materials can generated by this technique which have varying properties, dependent on the factor . The effect of varying on the slot length is shown in Fig. 4. The slot lengths shown are normalized to the ), case. In theory it is possquare unit cell length, (i.e., sible to use extremely small values of with their corresponding long slots, however practical constraints due to the finite width of the slot, , eventually prevent further reduction of . B. Computer Simulation Periodic structures of the type described can be thought of as having two distinct properties—the AMC properties where the

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Fig. 4. Variation of slot length with convolution factor, k .

2

Fig. 2. Schematic of 2 2 Polar-EBG array showing the effects of changing the curve order on the length of the spiral. The lines indicate the slots cut into the upper conductive layer.

Fig. 5. Reflection phase comparison between the conventional EBG and Polar-EBG with equal periodicity a = 7:5 mm, k = 0:5 and g = 0:5 mm.

Fig. 3. Mapping function that provides a warping transformation.

reflected wave at the surface is considered to be more in phase with the incident wave than out of phase and the EBG property in which surface waves are suppressed. In this section the two properties are characterized for the spiral-slot based polar-EBG structure. To verify the AMC property a polar-EBG and a conventional square patch EBG structure having equal periodicity, substrate material, and slot width were compared. The EBG structures were analyzed using the TLM method and a periodic boundary condition [29]. The cells were modeled on a dielectric slab 3.18 mm thick with relative permittivity of 2.95. The length of the used was 7 mm. In both cases conventional square patch

the array periodicity was mm and the slot width, mm. The polar-EBG structure was designed using a convolution which gave the shape seen already in Fig. 2. factor The simulated reflection phase results are shown in Fig. 5. The frequency band where the EBG surface had a reflection phase was taken to be the useable bandwidth. The in the range reflection phase variation with frequency indicates a similar behavior in both cases, however the EBG has a 30% frequency reduction compared to the conventional EBG. An expected reduction is also seen in the bandwidth in case of the polar-EBG. The lower resonant frequency is primarily due to first resonance of the structure and can be related directly to the spiral-slot length. Additionally, Fig. 5 demonstrates the viability of the polar-EBG as an artificial magnetic conductor. The stop-band and pass-band frequencies of an EBG structure can be more fully expressed by using dispersion diagrams. Dispersion analysis has the main advantage of estimating the stop-bands of EBG structures without considering the entire structure. An infinite structure is simulated by imposing periodic boundary conditions with appropriate phase shifts onto the unit cell in a suitable eigenmode solver. The wave propagation in the structure can be represented by certain vectors in the unit cell

MULENGA AND FLINT: PLANAR EBG STRUCTURES BASED ON POLAR CURVES AND MAPPING FUNCTIONS

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Fig. 7. TE Surface wave transmission on an EBG surface (a) polar-EBG k =

Fig. 6. Dispersion diagram for the Polar-EBG structures (a) Polar-EBG k = 0:5 (b) Conventional EBG.

that constitute a boundary region of propagation, often referred to as the irreducible Brillouin zone. Deriving the propagating modes in this zone suffices to cover all the possible directions of propagation within the unit cell. The Polar-EBG structure has its irreducible Brillouin zone as the entire unit cell given that it has rotational symmetry and not reflective symmetry. dispersion diagram Fig. 6 shows the full of the Polar-EBG structure and a conventional EBG structure. The dispersion diagrams have been calculated using an eigenmode solver based on the Finite Integration Method [15]. The Polar-EBG exhibits a bandgap from 2.8–3.9 GHz while the conventional EBG shows a bandgap from 4–6.2 GHz. The stopbands predicted by the dispersion diagrams show very good agreement with those predicted through reflection phase analysis. The suppression of surface waves plays a useful role in improving the radiation efficiency of antennas and also allows the control of unwanted side and back lobes in the pattern. This capability of EBG structures can be evaluated experimentally using surface wave measurements.

0:5 (b) conventional EBG.

To perform these surface wave measurements a 12 12 array mm was etched on a Taconic subwith periodicity strate (TLE-95) of thickness 3.18 mm and relative permittivity . The EBG property of the fabricated prototype was measured using the coaxial monopole probe methodology from [1]. The measured responses for the polar-EBG and conventional EBG are shown in Fig. 7. The area shaded gray shows the predicted stop-band edges as calculated using (1) and (2). The measured reflection coefficient obtained by means of the coaxial probes shows good agreement with the theoretical stop-band values. Additionally, these results clearly show the presence of a bandgap for the fabricated prototypes. The conventional EBG shows a bandgap between 4.13 and demonstrates a 6.0 GHz while the polar-EBG with bandgap with band edges at 3.06 GHz and 3.8 GHz (Fig. 7). The stop-bands derived from the measurement results agree well with those in the dispersion diagram determined via simulations. The minor discrepancies found can be partly attributed to fabrication tolerances and due to the finite array used in the measurements, as compared to the infinite array simulated.

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Fig. 8. Effect of convolution factor on the fundamental stop-band of the polarEBG.

Fig. 9. Reflection phase variation with curve order.

IV. ANGULAR STABILITY III. BANDGAP TUNING BY VARYING THE CONVOLUTION FACTOR A convenient means of controlling the resonant frequency and bandgap of polar-EBG structures is through the use of the convolution factor . This convolution factor offers an additional degree of freedom over the conventional EBG and can be used to design structures for arbitrary frequencies. As discussed in Section II, the convolution factor controls the length of the embedded spiral-slot which contributes proportionally to the capacitance. In this case we consider a polar-EBG on identical substrate to that described in the previous section and again with periodicity 7.5 mm. The spiral-slot width selected was 0.5 was varied from 0.25 to 4. mm and the convolution factor The effect of the convolution factor on the resonant frequency and bandwidth was investigated via simulation using the TLM technique with periodic boundaries at the edges of the unit cell. the simulation results are almost indistinNote that for guishable from the results for the standard square patch element. This can be confirmed by noting the slot length in Fig. 4 is ap. Values of were not simulated as proximately this would have removed an excessive amount of metal from the array. By considering the reflection phase and measuring the center frequency and bandwidth the bandgaps for different convolution factors were obtained. The results of these simulations are shown plotted in Fig. 8. It is clearly shown that as is reduced the resonant frequency decreases in a nonlinear fashion. The greatest rate of change so this could be considered a lower bound occurs for for useful modification. At the lowest values of k the resonant frequency is still reducing, however this corresponds to increasingly spiraled structures which become difficult to manufacture and the slot width becomes physically unobtainable. Another important consideration at the left hand side of the plot is the decrease in fractional bandwidth which falls off even more rapidly than the frequency. This reduction in fractional . Fig. 9 illusbandwidth is particularly pronounced for trates this effect, demonstrating in addition that when , a second stop-band appears. This is an indication that the next stop band is also reduced in frequency.

EBG surfaces do not exhibit uniform surface impedance with respect to different spatial harmonics radiated by antennas [17]. The resonant frequency at which PMC effects are observed depends on the incidence angle, and therefore the resultant interaction of the EBG surface and antenna will be a summation of in-phase and out-of-phase effects [30]. Angular dependence is consequently an important characteristic to be determined for an EBG structure. The angular dependence of the surface impedance can be considered separately for both E-plane (transverse electric, TE) and H-plane (transverse magnetic, TM) polarized waves. For the polar EBG the angular dependence has been determined by applying the methodology in [31]. A frequency domain solver is used to apply incident waves at several incident angles in the range 0–60 for both TE and TM and the resonant frequency of the EBG is determined with the objective of determining the change in frequency and the total amount of deviation observed for a range of incident angles. Table II presents the absolute deviation and relative deviation of the resonant frequency for the polar EBG for various values. The reference case is the case which corresponds to the standard mushroom EBG. These reference results are in good agreement with those presented by Simovski et al. [31]. In the case of the polar EBG, it is notable that in the case of both TE and TM, the polar spiral pattern has the desirable effect of reducing the angular dependence. The effect is a general trend to reduce the deviation for decreasing . Also, the percentage deviation in the resonant for the case frequency for TM and TE is very similar and hence this is a useful design. In comparison to the spiral designs in [30], the polar-EBGs demonstrated greater angular stability for both , and in both TE and TM incident fields. In addition, the polar designs have better angular stability for the . The angular stabilization effect observed in case for both the TE and TM instances in the polar surface is consistent with the observations in [31] where the excitation of the vias causing, electric currents (TM) and magnetic currents (TE) was produced.

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TABLE I BANDGAP CHARACTERISTICS ANALYSIS THROUGH REFLECTION PHASE ANALYSIS

TABLE II RELATIVE AND ABSOLUTE DEVIATION OF RESONANT FREQUENCY FOR DIFFERENT VALUES OF THE CONVOLUTION FACTOR, k

TABLE III DESIGN SPECIFICATIONS FOR EACH EBG UNIT CELL STRUCTURE

Fig. 10. Schematics of artificial magnetic conductors (AMC) surfaces: (a) Conventional mushroom structure, (b) spiral-like EBG, (c) edge-located vias, (d) second order Hilbert curve, (e) polar-EBG.

V. COMPARISON WITH KNOWN LOW-FREQUENCY EBG STRUCTURES Fig. 10 shows schematic diagrams of five different types of EBG, which were designed on substrates with thickness

mm and relative permittivity of 2.65. The radius of the via pins, where present, was kept constant at 0.4 mm. The lattice in Fig. 10(a) is the conventional mushroom structure (as in [3]) which consists of square metal patches connected to the ground plane by metal via pins in the center of the patch. The lattice in Fig. 10(b) is the spiral-like EBG structure proposed by Zheng et al. [33], while the lattice in Fig. 10(c) consists of metal patches with offset metallic vias [34]. Additionally, Fig. 10(d) is the Hilbert curve based high impedance metamaterial surface of iteration order 2 as in [2]. These were compared to the polar-EBG with warped spirals having . The EBG structures were all designed to operate at 3.25 GHz and were modeled and optimized using a full-field TLM technique. The final design parameters for each of these surfaces are summarized in Table III. The different geometries were evaluated for plane wave incidence by determining the reflection phase. The bandwidth was calculated from the simulation refrom sults by observing the convention of the phase being the phase at the center frequency. The results are summarized in Table IV. The size reduction figures quoted are the percentage of the square unit cell for the geometry measured relative to the conventional mushroom EBG. It is worth noting that the Hilbert curve and edge-located via structures are asymmetrical and therefore necessitate two different orthogonal polarizations making the resonant frequency polarization dependent. In this study, for both the Hilbert curve and edge-located via structures, only the polarization that yields a resonant frequency of 3.25 GHz was considered. The results of the parametric study show that each of the four modified EBG structures has a much lower fractional bandwidth than the conventional mushroom type. The second-order Hilbert curve demonstrates

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TABLE IV SIMULATED BANDWIDTHS AND CELL SIZE REDUCTION OF EBG STRUCTURES

ACKNOWLEDGMENT The authors gratefully thank Dr. A. Chauraya for help with the prototypes and measurements. They also wish to thank Taconic who kindly donated the laminate samples for the designs and CST for the simulation software. REFERENCES

the least fractional bandwidth at 3.6%. The edge-located via show remarkably similar fractional and Polar-EBG at bandwidths but with the Polar-EBG having a higher cell size reduction at nearly 56%. The spiral-like EBG exhibits the greatest % while the Hilbert-curve shows the cell size reduction least cell size reduction at 24%. Although the edge-located via EBG marginally performs better than the polar-EBG structure in terms of bandwidth, the performance of the Polar-EBG is still better than that of the spiral-like EBG and Hilbert curve in terms of the fractional bandwidth. In the case of the Hilbert-curve, the polar-EBG performs better, because of the greater utilization of the patch area to increase surface capacitance while the Hilbert-curve relies more on its inductive nature. The polar-EBG also has the benefit of being symmetrical and is therefore not polarization dependent while the Hilbert-curve and edge-located via EBGs are polarization dependent. This characteristic may be important to some applications. The difference in performance of the Spiral-like EBG and polar-EBG can be understood by analyzing each individual structure. The polar-EBG is a direct derivate of the conven. The gaps between tional mushroom structure when the patches have been tessellated, and warped in order to increase the capacitance by making better use of the patch area. In the case of the spiral-like EBG, apart from increasing the lateral length of the patch element, interleaving has also been incorporated to a greater degree into the structure to increase the overall capacitance. VI. CONCLUSION In this paper, a novel EBG structure designed using polar curves and mapping functions has been presented and evaluated. Simulations and experiments have been performed which demonstrate that as is reduced, the angular stability of the surface improves and the resonant frequency reduces, however, as found in other designs, this comes at the cost of reduced bandwidth. In this regard, the polar EBG performs at least as well as the others studied and in some cases better. One of the major advantages in this method of defining the EBG geometry is the ability to adjust the frequency by changing only the slot length via a simple, single parameter, . It is of course possible to adjust the slot width and substrate thickness however the availability of substrate and manufacturing tolerances then become more important when tuning the surface.

[1] D. F. Sievenpiper, “High-Impedance Electromagnetic Surfaces,” Ph.D. dissertation, Univ. California, Los Angeles, 1999. [2] J. McVay, N. Engheta, and A. Hoorfar, “High impedance metamaterial surfaces using Hilbert-curve inclusions,” IEEE Microw. Wireless Compon. Lett., vol. 14, pp. 130–132, 2004. [3] D. Sievenpiper, L. Zhang, R. F. J. Broas, N. G. Alexopolous, and E. Yablonovitch, “High-impedance electromagnetic surfaces with a forbidden frequency band,” IEEE Trans. Microw. Theory Tech., vol. 47, pp. 2059–2074, 1999. [4] S. Wang, A. P. Feresidis, J. C. Vardaxoglou, and G. Goussetis, “Artificial magnetic conductors for low-profile resonant cavity antennas,” in AP-S Int. Symp. (Digest) IEEE Antennas Propag. Society, 2004, vol. 2, pp. 1423–1426. [5] F. Yang and Y. Rahmat-Samii, “Reflection phase characterization of an electromagnetic band-gap (EBG) surface,” in Proc. IEEE Int. Symp. Antennas Propag., 2002, pp. 744–747. [6] L. Yang, M. Fan, F. Chen, J. She, and Z. Feng, “A novel compact electromagnetic-bandgap (EBG) structure and its applications for microwave circuits,” IEEE Trans. Microw. Theory Tech., vol. 53, pp. 183–190, 2005. [7] Y. Yao, X. Wang, and Z. Feng, “A novel dual-band compact electromagnetic bandgap (EBG) structure and its application in multi-antennas,” in Proc. IEEE Int. Symp. Antennas and Propag. Society, 2006, pp. 1943–1946. [8] L. Inclan-Sanchez, E. Rajo-Iglesias, J. L. Vazquez-Roy, and V. Gonzalez-Posadas, “Design of periodic metallo-dielectric structure for broadband multilayer patch antenna,” Microw. Opt. Technol. Lett., vol. 44, pp. 418–421, 2005. [9] R. F. J. Broas, D. F. Sievenpiper, and E. Yablonovitch, “A high-impedance ground plane applied to a cellphone handset geometry,” IEEE Trans. Microw. Theory Tech., vol. 49, pp. 1262–1265, 2001. [10] J. R. Sohn, H. Tae, J. Lee, and J. Lee, “Comparative analysis of four types of high-impedance surfaces for low profile antenna applications,” in AP-S IEEE Int. Symp. (Digest) Antennas Propag. Society, 2005, vol. 1A, pp. 758–761. [11] W. Zhang, J. Mao, and X. Sun, “Patch antenna array embedded on a high-impedance ground plane,” J. Electromagn. Waves Applicat., vol. 19, pp. 2007–2014, 2005. [12] F. Yang and Y. Rahmat-Samii, “Microstrip antennas integrated with electromagnetic band-gap (EBG) structures: A low mutual coupling design for array applications,” IEEE Trans. Antennas Propag., vol. 51, pp. 2936–2946, 2003. [13] R. Gonzalo, P. De Maagt, and M. Sorolla, “Enhanced patch-antenna performance by suppressing surface waves using photonic-bandgap substrates,” IEEE Trans. Microw. Theory Tech., vol. 47, pp. 2131–2138, 1999. [14] F. Yang and Y. Rahmat-Samii, “Microstrip antennas integrated with electromagnetic band-gap (EBG) structures: A low mutual coupling design for array applications,” IEEE Trans. Antennas Propag., vol. 51, pp. 2936–2946, 2003. [15] F. Yang, K. Ma, Y. Qian, and T. Itoh, “A novel TEM waveguide using uniplanar compact photonic-bandgap (UC-PBG) structure,” IEEE Trans. Microw. Theory Tech., vol. 47, pp. 2092–2098, 1999. [16] F. Yang, K. Ma, Y. Qian, and T. Itoh, “A uniplanar compact photonic-bandgap (UC-PBG) structure and its applications for microwave circuit,” IEEE Trans. Microw. Theory Tech., vol. 47, pp. 1509–1514, 1999. [17] G. K. Palikaras, A. P. Feresidis, and J. C. Vardaxoglou, “Cylindrical electromagnetic bandgap structures for directive base station antennas,” IEEE Antennas Wireless Propag. Lett., vol. 3, pp. 87–89, 2004. [18] T. Kamgaing and O. M. Ramahi, “Design and modeling of high-impedance electromagnetic surfaces for switching noise suppression in power planes,” IEEE Trans. Electromagn. Compat., vol. 47, pp. 479–489, 2005.

MULENGA AND FLINT: PLANAR EBG STRUCTURES BASED ON POLAR CURVES AND MAPPING FUNCTIONS

[19] T. Kamgaing and O. M. Ramahi, “A novel power plane with integrated simultaneous switching noise mitigation capability using high impedance surface,” IEEE Microw. Wireless Compon. Lett., vol. 13, pp. 21–23, 2003. [20] R. Abhari and G. V. Eleftheriades, “Metallo-dielectric electromagnetic bandgap structures for suppression and isolation of the parallel-plate noise in high-speed circuits,” IEEE Trans. Microw. Theory Tech., vol. 51, pp. 1629–1639, 2003. [21] D. Sievenpiper, “Review of theory, fabrication, and applications of high-impedance ground planes,” in Metamaterials, Physics and Engineering Explorations, N. Engheta and R. W. Ziolkowski, Eds. : IEEE Press, 2006, pp. 295–297. [22] M. F. Abedin, M. Z. Azad, and M. Ali, “Wideband smaller unit-cell planar EBG structures and their application,” IEEE Trans. Antennas Propag., vol. 56, pp. 903–908, 2008. [23] D. J. Kern, D. H. Werner, M. J. Wilhelm, L. Lanuzza, and A. Monorchio, “The design synthesis of multiband artificial magnetic conductors using high impedance frequency selective surfaces,” IEEE Trans. Antennas Propag., vol. 53, pp. 8–17, 2005. [24] S. Tse, Y. Hao, and C. Parini, “Mushroom-like high-impedance surface (HIS) with slanted vias,” in Proc. LAPC Antennas and Propag. Conf., Loughborough, 2007, pp. 309–312. [25] D. J. Kern, D. H. Werner, and M. J. Wilhelm, “Active negative impedance loaded EBG structures for the realization of ultra-wideband artificial magnetic conductors,” in IEEE AP-S Int. Symp. (Digest) Antennas Propag. Society, 2003, vol. 2, pp. 427–430. [26] G. S. A. Shaker and S. Safavi-Naeini, “Reduced size electromagnetic bandgap (EBG) structures for antenna applications,” in Proc. Canadian Conf. Elect. and Comput. Eng., 2005, pp. 1198–1201. [27] D. J. Kern, D. H. Werner, K. H. Church, and M. J. Wilhelm, “Genetically engineered multiband high-impedance frequency selective surfaces,” Microw. Opt. Technol. Lett., vol. 38, pp. 400–403, 2003. [28] J. McVay, A. Hoorfar, and N. Engheta, “Radiation characteristics of microstrip dipole antennas over a high-impedance metamaterial surface made of Hilbert inclusions,” in IEEE MTT S Int. Microw. Symp. Dig., 2003, vol. 1, pp. 587–590. [29] P. B. Johns, “A symmetrical condensed node for the TLM method,” IEEE Trans. Microw. Theory Tech., vol. 35, pp. 370–377, 1987. [30] M. Hosseini, A. Pirhadi, and M. Hakkak, “Design of an AMC with little sensitivity to angle of incidence using an optimized Jerusalem cross FSS,” in IEEE Int. Workshop on Antenna Technology: Small Antennas and Novel Metamaterials, 2006, pp. 245–248. [31] C. R. Simovski, P. De Maagt, A. A. Sochava, M. Paquay, and S. A. Tretyakov, “Angular stabilisation of resonant frequency of artificial magnetic conductors for TE-incidence,” Electron. Lett., vol. 40, pp. 92–93, 2004.

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[32] C. R. Simovski, P. De Maagt, and I. V. Melchakova, “High-impedance surfaces having stable resonance with respect to polarization and incidence angle,” IEEE Trans. Antennas Propag., vol. 53, pp. 908–914, 2005. [33] Q.-R. Zheng, Y.-Q. Fu, and N.-C. Yuan, “A novel compact spiral electromagnetic band-gap (EBG) structure,” IEEE Trans. Antennas Propag., vol. 56, pp. 1656–1660, 2008. [34] E. Rajo-Iglesias, L. Inclan-Sanchez, J. Vazquez-Roy, and E. GarciaMuoz, “Size reduction of mushroom-type EBG surfaces by using edgelocated vias,” IEEE Microw. Wireless Components. Lett., vol. 17, pp. 670–672, 2007.

Charity B. Mulenga received the B.Sc. degree in electrical engineering from Makerere University, Uganda, in 2001 and the M.Sc. degree in digital communication systems from Loughborough University, Loughborough, U.K., in 2004, where, since October 2006, she has been working toward the Ph.D. degree. Her research interests focus on the design of broadband antennas, periodic structures, computational electromagnetics and frequency selective surfaces.

James A. Flint (M’96–SM’07) was born in Holbrook, Derbyshire, U.K., in 1973. He received the M.Eng. and Ph.D. degrees in electronic and electrical engineering from Loughborough University, Loughborough, U.K., in 1996 and 2000, respectively. He worked in the automotive industry as a Project Engineer and later returned to Loughborough when he was appointed Lecturer in Wireless Systems Engineering following in 2001. He was promoted to Senior Lecturer in 2006 and is currently the Head of the Communications Research Division, Department of Electrical and Electronic Engineering. His research interests span fault-tolerant signal processing, novel acoustic and electromagnetic transducers, metamaterials and electromagnetic compatibility. Dr. Flint is a Chartered Engineer in the U.K., a member of the Institution of Engineering and Technology, and has acted as a consultant to numerous highprofile companies and government agencies.

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Active Phase Conjugating Lens With Sub-Wavelength Resolution Capability Vincent F. Fusco, Fellow, IEEE, Neil B. Buchanan, and Oleksandr Malyuskin, Member, IEEE

Abstract—Experimental results are presented for the focusing capability of an active phase conjugating lens for a single and a dipole source pair and these are compared with predictions. In addition for a single source we illustrate the ability of the lens to project a null at the lens focus instead of a peak. A scheme is also presented such that when a source or pair of sources is imaged through an identical pair of passive scatterers located symmetrically about the lens that imaging with sub-wavelength resolution is possible. The rationale for the operation of the lens and aberrations observed due to its finite array size is discussed and is supported throughout by means of numerical simulation. Index Terms—Array, diffraction limit, microwave imaging, phase conjugation, subwavelength resolution.

I. INTRODUCTION

R

ECENTLY, the theoretical possibility for microwave focusing based on plane wave phase conjugation was reported in [1] for a conceptual surface wherein the tangential components of an incident electromagnetic plane wave were phase conjugated. In [2] analysis of a structure based on two back-to-back wire frequency selective surfaces (FSS) interconnected through hypothetical phase conjugating mixers was conducted, however the structure was not constructed. In [3] basic experimental results were produced which demonstrated that negative refraction through phase conjugation would lead to focussing, no comparison was made to theory. Source imaging with sub-wavelength resolution is of value in applications such as non-destructive testing, biological and medical diagnostics, [4]. In the past few years several techniques have been proposed for near field sub-wavelength imaging, e.g., the left-handed material superlens in [5], the transmission-line metamaterial lens [6] and wire media lens in [7]. To date for far field imaging an auxiliary lensing device is used [8] in conjunction with the techniques above. Diffraction limited source localization in free space using time reversal techniques has been previously demonstrated in optics and acoustics [9]. For monochromatic signals time reversal and phase conjugation are equivalent. Thus, in general, if Manuscript received May 21, 2009; revised August 20, 2009. First published December 31, 2009; current version published March 03, 2010. This work was supported by the UK Engineering and Physical Science Research Council under contracts EP/D045835/1, EP/E01707X/1, and in part by the Department of Education for Northern Ireland Strengthening all Island Grant, Mobile Wireless Futures. The authors are with The Institute of Electronics, Communications and Information Technology (ECIT), Queen’s University Belfast, Queen’s Island, Belfast BT3 9DT, Northern Ireland (e-mail: [email protected]; n.buchanan@ee. qub.ac.uk; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2009.2039332

a time reversal or phase conjugation lens is positioned in the far field zone of a source in a homogeneous space the resolution of this lens is always diffraction limited because it is illuminated only by propagating spectrum as evanescent spectrum dies before reaching the lens. Thus only propagating waves reach the phase conjugation surface where they are negatively refracted, however scattering off in-homogeneities can in certain circumstances result in evanescent-to-propagating mode as well as reciprocal conversion, [10], [11] thus sub-wavelength imaging becomes a possibility. In this paper we present simulated and experimental results for a practical phase conjugating lens prototype which deploys a precision active phase conjugating architecture with dc tuneable phase offset, Fig. 1(a). The operation of the phase conjugating unit is described in detail in [12]. The lens is able to receive and process power levels down to 110 dBm and retransmits with constant output power even when received power level varies. A further feature of this architecture, relevant to lens applications, is that the phase delay in each conjugation path is readily adjustable via VCO frequency pushing. This property is used to demonstrate, lens null focusing capability and image displacement control. Additionally the paper demonstrates a simple means by which the lens when can be operated with sources augmented with scatterers in order to provide sub-wavelength resolution of closely spaced sources. II. PHASE CONJUGATING LENS DESCRIPTION The phase conjugating lens (PC) is comprised of two back to back ten element patch arrays, interconnected pair wise via an IQ modulator phase conjugating circuit, A schematic of a single cell is shown in Fig. 1(a), its operation is described in [12]. A photograph of the lens, mounted inside an anechoic chamber, is shown in Fig. 1(b). All the antennas were identical linearly , polarized microstrip patches with , , the spacing between patches . Vertical linear polarization was used on both source was and image sides of the lens. III. LENS PERFORMANCE PREDICTION In order to calculate the transmitted field of the ten element patch PC lens we adopt the following procedure. First we position a half-wavelength dipole source at broadside to the 10 1 patch element array. Then we calculate using the FEKO EM simulator, [13], the voltage received by each source side patch element. This voltage is then phase conjugated and applied to its paired image side patch radiator. To facilitate simulation each of the source side patch antennas are terminated in a lumped load impedance corresponding to the input impedance of the phase

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is the incident field due to dipole excitation, is a point source dyadic Green’s function, is the outward normal , and index numbers patches to the antenna’s surface, from 1 to 10, is the surface area of the j-th antenna. Having solved (1) one can find a scalar potential at each patch terminal point as (3) The current distribution in each of the lumped loads connected to each of the patches is then defined by the potential difference between the terminal point and PEC screen. The voltages across the load terminals can be found as (4)

Fig. 1. Phase conjugating lens. (a) Single phase conjugating cell schematic. (b) Photograph of phase conjugating lens.

Fig. 2. PC array equivalent model. (a) Receive (source) side. (b) Retransmit (image) side.

conjugating circuit, Fig. 2(a), in this case a parallel RC load, , . The field calculation proceeds as follows; The magnetic vector potential integral equation representing the arrangement in Fig. 2 is given by (1) (1) where the total magnetic vector potential count mutual coupling in the structure

takes into ac(2)

the load at the th patches and is the th terminal where current. In the next step these voltages are phase conjugated and reapplied to each of the corresponding patch antenna terminals on the image side of the array, Fig. 2(b), and the field radiated by the image side patch array is calculated using a standard Green’s function method [14]. The simulated transmitted PC field is shown Fig. 3. The PC lens is illuminated by a z-oriented half-wavelength voltage generator fed wire antenna (diameter 1 mm) with voltage source 1 V and zero degree phase radiating at frequency 2.4 GHz. The separation between the source and PC lens is 30 cm (2.4 @2.4 GHz). The image side field of the ten element patch array is shown in Fig. 3(a), and focussing action is clearly evident. The x and y orientations on the graph are the same as in Fig. 4. The amplitude profile of the source side signal across the central portion of the array is as shown in Fig. 3(b) and is preserved at the lens image side when making the field predictions in Fig. 3(a). In practice with the arrangement in Fig. 1(a) all of the phase conjugation units produce identical output power levels, thus the image side power distribution across the array is constant. This effect was modelled and the image side field distribution recalculated, as shown in Fig. 3(c). Examination of Fig. 3(a) and Fig. 3(c) show that signal re-transmission with uniform distribution alters the structure of the imaged focal spot. In general with input amplitude variation removed at the lens output side the focal spot tends to be slightly enlarged with respect to the case when input amplitude variation is preserved across the array. It is also interesting to note that in both cases the most intense portion of the spot lies , which is closer that the distance associated at with the 30 cm separation distance. There are two reasons for this. The first one is the finiteness of the array leads to larger mutual coupling variations on the edge elements than would otherwise be the case in a longer or infinite array, this aspect will be described further in Section V. Secondly since the array operates close to the resonant frequency of its patches and due to its finite length and its excitation by a packet of plane waves the array mutual impedance has an inductive part. The impact of this on focal spot position is as follows. The magnetic vector potential defined from (1) contains a , defines the plane of PC phase factor lens, at each antenna terminal due to the incident field excited

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Fig. 4. Lens setup definitions peak focusing capability.

Since the transmitted field is calculated for , the Green’s function contains the factor . Convolution of the Green’s function and current density will give in the expresrise to an exponential term sion for the transmitted field. So we expect the transmitted field . However the inductive nature of the maxima to lie at at array leads to an additional phase contribution of the transmitting side. Therefore the transmitted field expression , will now contain the multiplier thus the focal spot maximum position will be determined by the , i.e. maximum will be shifted towards the relation PC array plane. Also it is interesting to note that along the y-direction that . This effect is due to the 1D the 3 dB focal spot width is nature of the PC lens resulting in the angular spectrum loss and consequent focal spot enlargement [15]. Fig. 3. Simulated transmitted field magnitude for ten element patch array. (a) Simulated image side E field with receive distribution preserved at array output. (b) Simulated incident field and induced received voltage distribution across the centre portion of the array. (c) Simulated image side E field retransmit distribution amplitude limited at array output.

by the half-wavelength antenna. This factor is encoded into a scalar potential (3) and after incident voltage phase conjugation the re-transmit voltages are entered into the expression for the transmitted field, i.e. they are embedded into the current density on the retransmit side of the lens with inverse (phase conjugated) sign, i.e.,

(5)

IV. EXPERIMENTAL SETUP The experimental setup, shown in Fig. 4, employed a half wavelength dipole placed on either side of the active phase conjugating lens. One dipole is used as source and the other is used as an E field probe. These dipoles can be varied in position relative to their distance from the array (y direction), and across the array (x direction). The IQ phase conjugating circuit, Fig. 1(a) allows us to programme the transmit side, TX, (source) and receive side, RX, (image) frequencies. For the experiments presented below we used split frequency operation, 2.4 GHz at the TX dipole and 2.41 GHz at the RX dipole. This means that any stray fringing signal from the TX dipole does not interfere with the results at the RX side. In addition input and output polarizations were both vertical linear. The value of the phase delays in each

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Fig. 6. Measured bistatic focused beam peak characteristics. (a) Measured focused peak beam intensity for various TX dipole x positions, dotted line shows ideal linear behaviour.

Fig. 5. Field distributions associated with focused image of source dipole. (a) Measured E field around imaged location and (b) measured near field bistatic responses for various single source positions.

of the conjugation path, , were set equal. This was done by placing a TX source in far field of the lens (7 m) and RX dipole 30 cm from lens. The phase of each of the phase conjugating units was set to yield 0 degrees at the RX dipole when it was ) each corresponding placed directly in front of (at element. Investigation of the field intensity around the imaged region was carried out for the situation when the source was located , , the RX dipole was scanned across the at , ). The measured results shown in Fig. 5(a) lens (varying show that in accordance with Fig. 3 that the maximum intensity i.e. position of the focal spot is located closer to the PC than at , c.f. , simulated in Fig. 3. Next a bistatic measurement , Fig. 5(b), was made by on the image side of the lens at varying with fixed. This shows that phase conjugation is leading to far field signal collimation. Fig. 6 shows that by tracking the peak amplitude positions of the bistatic patterns in Fig. 5(b) we obtain, over the region 20 cm about , an almost linear variation of the transversal displacement of the image of the source dipole from the lens boresight direction. Some deviation of the image displacement curve from the linear variation is due to the finite size of the array. These results indicate that we should be able

(in the absence of mutual coupling between sources) to determine source positions to a 3 dB level which are displaced by and to 0.5 dB level to . In order to demonstrate the imaging resolution for two half wavelength sources, (mutual coupling present), the setup of Fig. 4 was employed. Here two dipoles were used, each placed distance of 30 cm from the lens. The separation at a distance between these was varied in the direction, while . The dipoles were fed with their centre was held at equiphase and amplitude signals. This arrangement shown in Fig. 7(a) was simulated using the procedure elaborated in Section III. Fig. 7(b) displays the simulated field distribution . It is evident that the due to two antennas separated by two sources cannot be resolved. Fig. 7(c) demonstrates the predicted transmitted field for the case of two antennas separated . As expected the sources can now be resolved. by The arrangement in Fig. 7(a) was experimentally characterized and the results of this experiment shown in Fig. 7(d). These illustrate that it was possible to obtain two distinct ( 3 dB) peak images at the image side of the lens for dipole separations of . Also as seen from Fig. 7(d) that as dipole separation is the images begin to smear and below reduced to separation the image formed has a single peak. Thus the two individual sources can no longer be resolved, i.e. mutual coupling between the source dipoles is affecting resolution. V. LENS FOCUS CONTROL A. Image Displacement By varying the distributions of in Fig. 4 we can focus the lens to various locations on the lens image side. For example slightly from their far field caliby varying the values of brated assignments by using the d.c. phase assist control shown in Fig. 1(b) we can compensate for the image offset aberrations

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Fig. 8. Measurement of image displacement capability: to position spot at 2:4 (b) y = 6:4, y moved, x calibrated at 0 .

Fig. 7. Two mutually coupled sources separated in the transverse plane. (a) Two =2 sources separated by distance D in the transversal plane. The distance from the sources to the PC lens is 2:4. (b) Simulated image field distribution due to two sources separated by =2. (c) Simulated image field due to PC surface excited by two antennas separated by distance D = 1:04 in the transversal plane. (d) Normalized experimental image field obtained using two dipole sources, for various dipole spacings (y = 30 cm, 2:4).

re-calibrated

= 0 cm,

described above and deliver the image to exactly the 30 cm position when the source is located at 30 cm. Fig. 8(a) shows that as required. As expected the focal spot is now centred on , while focusing at the 3 dB points in the x direction spans in the y direction it is , c.f. numerical simulation prediction of in the x direction and in the y direction. In this way, a source at any position on the TX side of the lens can be brought to a focus at any convenient position on the image, RX (image), side of the lens. Hence the location of the PC lens with respect to the source is not critical. Thus a fixed PC lens and image dipole or dipoles could be used to locate the spatial position of a source or sources as it/they reach a certain prescribed location. are Fig. 8(b) shows the situation when the values of held at their initial far field calibration values and the source is 0 cm, and moved to . It is observed position of that is not located at but occurs at . Additionally if we reconsider Fig. 5(a) we see that in both cases while the respectively the peak intensity of the focal spot lies at , , and i.e. the best peak to sidelobe responses occur for actual spatial location of the source. Fig. 8(a) shows that after phase correction has been applied this effect is also true. The reasons underpinning this property of the lens are due to the form of the transmitted field as a superposition of modulated spherical waves. Consequently the focal spot maximum position

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will be determined both by the phases and amplitudes which depend not only on excitation but also on the mutual coupling in the array. Mutual coupling produces the amplitude and phase variation which is different from an ideal infinite lens case thus focal spot displacement occurs. For larger arrays these mutual coupling induced effects occur less rapidly than at the ends of short finite size arrays thus the composite overall effect is that focal spot position lies closer to what we would expect to observe in the ideal case, i.e. the image lies in the mirror symmetric position with respect to the source and the peak to side lobe response is aligned with this position. In detail the far field generated by the PC lens is obtained from (1), (2) upon taking into account the far field component of the Greens function

(6)

Fig. 9. PC transmitted field due to a PC lens illuminated by a single dipole source. The separation between the source and the PC surface is 2:4. Maximum and the centre of the spot at half maximum are located at 2:4. PC lens is a 40 40 dipole array with =4 spacing between elements.

2

, . Using the far field where one can get an approximate approximation expression for the field in the image plane as

(7)

, is a displacement across the array where is the polarization vector, is the rain the image plane, is the radiation pattern phase diation pattern amplitude and of the th transducer of a PC lens. The radiation pattern is described by the expression following from (6) as Fig. 10. Measured null focusing varying TX dipole position from the lens, y = 30 cm (2:4).

(8) The nulls of the transmitted field are formed in the vicinity of , for the real part of the points the field and , for the imaginary part. In the vicinity of nulls the amplitudes of partial modulated spherical waves (7) are small and are therefore do not dominate. At the location of the focal spot maxima the phase is small, but the amplitudes are large so the spot position and shape critically depends on the amplitude variation. For larger arrays mutual coupling induced end effects have less impact on focal spot position than for a small finite size array. Fig. 9 illustrates this by means of a simulation of a 2D 40 40 PC array excited by a single source, here it can be seen that beam focussing in both x and in the y direction is occurring due to the 2D nature of the array and that the focal spot and maximum side lobe to main lobe positions are well defined and position with respect to the source occurring in the mirror location.

B. Null Focusing Capability With the phase conjugation arrangement in Fig. 1(a) it is possible, that for a single source, instead of projecting a peak image from the source dipole, we can vary the values of in Fig. 4 in order to position a sharp null on the image side. This was carried out by setting both the RX and TX dipoles, at , ). Referring to Fig. 4, the far ( field calibration values of at alternate elements were set to be in antiphase. The measured image field results, Fig. 10, show that much higher focal point positional location than can be obtained when a null is formed, and that image displacement effects are being self-cancelled by the applied phase offsets across the array. Here the 3 dB points of the null cover 4 cm in the and 6 cm in the y direction. This x direction gives an overall image resolution at 3 dB points of . It should be noted that multiple sources cannot be null imaged in this way since their null images merge together so there is no

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advantage in null focusing for multiple sources located closer than a wavelength. VI. SUBWAVELENGTH SOURCE RESOLUTION IN THE FAR FIELD USING AN AUXILIARY SCATTERING ARRANGEMENT In the previous sections the far field resolution of a PC lens in free space is shown to be diffraction limited, i.e. two sources separated closer than wavelength cannot be distinguished. This is explained by the fact that the evanescent spectrum of the source carrying subwavelength information does not reach the conjugating surface due to its fast decay from the source. It is known in acoustics [9] that in an inhomogeneous medium phase conjugating techniques can be used for subwavelength source localization. Here scattering off heterogeneities can result in evanescent-to-propagating mode conversion and during back-propagation the retrodirected, by a time reversal array, propagating to evanescent wave recovery can occur due scattered off inhomogeneities. The reciprocity between propagating and evanescent waves in the scattering process and especially the crucial role of phase conjugation has been established in [10]. Application of this scheme to electromagnetic waves has been studied recently, in experimental paper showing subwavelength resolution in an entirely closed reverberation time reversal chamber [11] and in a theoretical paper discussing subwavelength resolution using PC lens with scattering arrangement in free space [16]. It is important to note that the use of PC lens in subwavelength imaging assisted with evanescent-to-propagating modes conversion is crucial for source reconstruction due to the PC lens autocorrecting properties, namely all the propagating waves of the source angular spectrum (including those carrying subwavelength information due to scattering) arrive at the image plane with corrected amplitudes and phases [17], an operation which cannot be performed by a conventional lens. A. Directivity Enhancement of a Single Source Produced by Adding Lumped Loaded Scatterers Direct sensing of the evanescent field of the source in the far field is not possible due to its fast decay from the source. The fine features of the source can be however encoded into the propagating spectrum by the way of scattering [11], [16], [18]. Practically this can be realized by inserting a resonant scatterer into the near field of the source [16], [18]. In a free space imaging setup it is important that the radiating wave from the source should be directed towards the PC lens and not screened or reflected from the PC lens by the auxiliary scatterer. By suitable loading we can ensure that the energy will be directed towards the PC surface, as shown in Fig. 11. This can be achieved by inclusion of a parallel LC, , , load into a half wavelength long scatterer which provides the desired result. Fig. 11 also shows the measured response of the arrangement. The separation between the loaded source dipole and the scattering dipole is . Without the load present in the scatterer the source energy is directed away from the PC lens.

Fig. 11. Measured and simulated directivity of a half-wavelength antenna accompanied by a parallel LC loaded half wavelength scatterer. The separation between the antenna and scatterer is 0:1 and wire diameter is 1 mm.

Fig. 12. Simulated normalized field magnitude generated by a PC patch array when excited by a dipole source 1 Am located 2:4 away from the array which is parallel LC loaded. Amplitude variation of voltage is constant on the output side of the array.

In Fig. 12 the consequence of positioning a loaded scatterer close to the source on PC lens array focusing capability is computed. From this it is evident that on the image side of the lens that the resultant beamwidth is narrowed with respect to Fig. 3(c) but that is still diffraction limited, i.e. the spot size is . B. Demonstration of the Role of the PC Lens and Scattering Arrangement We will now demonstrate the general principal for optimal arranging the scattering elements. Fig. 13(a) shows that the presence of a scatterer with no PC lens will lead to a scattered response which is always y directed irrespective of the relative . x position of the source and scatterer, in this case Fig. 13(b) shows the case with a PC lens present together with a scatterer at the image side and no scatterer at source side. Here the scattered field distribution pattern points towards the

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Fig. 13. Effect of source and image scatterers on focal spot size. (a) No PC lens, scatterer at image side, no scatterer at source side. Scattered field distribution pattern is aligned along y direction at x position irrespective of scatterer displacement x . (b) PC lens, scatterer at image side, no scatterer at source side. Scattered field distribution points more towards the mirror location of the source at image side than Fig. 13(a) due to focusing properties of the PC lens. (c) PC lens, scatterer at image side, scatterer at source side. Scattered field pattern points towards the mirror location of the source at image side. Also due to the focusing properties of the PC lens and additional propagating-to-evanescent spectrum conversion the localized source position x measured as a centre of the field distribution at half-maximum is less than x and x . (d) PC lens, scatterer at image side, scatterer at source side. Solid line—PC lens which preserves both amplitude and phase information; dotted line—PC lens with constant output amplitude profile. Scattered field distribution pattern points towards the mirror location of the source at image side but focusing properties of the PC lens with constant power output are deteriorated with respect to those obtained for lens with power profile preserved.

mirror location of the source at image side due to the focusing even properties of PC lens, it is now pointing to

though the scatterer is located at . With the PC lens, scatterer at image side, and scatterer at the source side

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Fig. 15. Measured resolution of two mutually coupled dipole sources using auxiliary scatters at source and image positions.

C. Experimental Results

Fig. 14. Experimental focusing response for symmetrical scattering arrange. (b) Source ments with PC lens in position. (a) Source and scatterer at x and scatterer :  (left and right).

x =01

=0

both in position, Fig. 13(c) the scattered field distribution pattern again points towards the mirror location of the source at image side. However in this case the focal spot size is more localized than it was for either Fig. 13(a), or Fig. 13(b). This effect is due to the focusing properties of the PC lens and propagating-to-evanescent spectrum encoding on the source side and decoding on the receive side, [10]. Fig. 13(d) shows, as suggested earlier, that the effect of limiting the output amplitude of the array is to broaden the focal spot size response of the lens.

The experimental results shown in Fig. 14(a) for a single source show that the responses expected in Fig. 13 do indeed occur, namely beam narrowing due to presence of loaded scatfrom the dipole source. Fig. 14(a) illustrates that terers at the presence of the scatterer only on the RX side gives rise to y directed scattering. Thus the source and scatterer located on the the TX and the TX/RX scatterer responses are y axis, indistinguishable. Sympathetic lateral displacement of the scatterer on both source and image side with respect to the source will cause which is held at a fixed location on the y axis, the imaged scattered signal to tilt towards the source location as shown in Fig. 14(b). The effect that symmetrical source image scatterer pairs have on improving resolution between two sources was measured and the results are presented in Fig. 15. This shows the results for two dipole sources with aligned scatterers displaced about to 31.25 mm spacing, and positioned at from lens. Loaded scatterers were present in mirror locations on both TX and RX sides, at from dipoles. In Fig. 15 the dotted line represents the case when no scatterer is present on the RX side. This result shows that the presence of the symmetrical scattering arrangement is leading to improved resolution for mutuspacing, spacing a 1.5 dB dip ally coupled dipoles at spacing a 4 dB dip is observed. In conoccurs, while trast Fig. 7(d) shows that without scatterers present individual source resolution was not possible at this separation distance. Next we investigated experimentally the effect of moving two dipole sources at across the lens with each dipole having from it. a scatterer present at The dual dipole, scatterer arrangement we moved in symx-direction on the TX (source) side, while pathy along the and on the RX (image) side, they were kept fixed at about position. Fig. 16 shows the result of the experiment. the It can be seen that an asymmetry between the two peaks occurs. This is about 3 dB for a dipole movement of around 12 mm.

FUSCO et al.: ACTIVE PHASE CONJUGATING LENS WITH SUB-WAVELENGTH RESOLUTION CAPABILITY

Fig. 16. Two laterally displaced Dipole Imaged Results.

The asymmetry noted in these responses is due to the scatterers trying to preserve pointing to the actual source position, as was the case in Fig. 14(b) for a scatterer augmented single dipole source. For the two dipole source arrangement the scatter at the position furthest from the image of the actual source location is partly obscured by the scatterer nearest the source image hence it amplitude response is highest. VII. CONCLUSION This paper has shown practical verification of a phase conjugating lens using a precision analogue IQ modulator phase conjugation unit. Agreement has been obtained with numerical simulation predictions for the structure of the imaged spot size obtained. In its basic form the lens has been shown to provide classical diffraction limited focusing. In addition it has been shown that with suitable phase adjustments the lens can project a null for improved single source location on the image side. It has also been shown that the focal length on both sides of the lens, normally and automatically set by the PC source position to be identical to the separation between the source and the lens, can be electronically set to different values. Further, the ability of the lens to resolve two dipole sources below the diffraction limit has been shown to be possible by augmenting the lens source arrangement with auxiliary scatterers positioned on both the source and image side of the lens. ACKNOWLEDGMENT The authors would also like to acknowledge the assistance of Mr. M. Major for the fabrication of the phase conjugation circuits used in this paper. REFERENCES [1] S. Maslovski and S. Tretyakov, “Phase conjugation and perfect lensing,” J. Appl. Phys., vol. 94, no. 7, pp. 4241–4243, Oct. 2003. [2] O. Malyuskin, V. Fusco, and A. Schuchinsky, “Phase conjugating wire FSS lens,” IEEE Trans. Antennas Propag., vol. 54, no. 5, pp. 1399–1404, 2006.

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[3] C. A. Allen, K. M. K. H. Leong, and T. Itoh, “A negative reflective/refractive “meta-interface” using a bi-directional phase-conjugating array,” in IEEE Int. Microw. Theory Tech. Symp. Dig., Jun. 8–13, 2003, vol. 3, pp. 1875–1878. [4] K. P. Gaikovich, “Subsurface near-field scanning tomography,” Phys. Rev. Lett., vol. 98, p. 183902, May 2007. [5] V. Veselago, “Electrodynamics of substances with simultaneously negative electrical and magnetic permeabilities,” Sov. Phys. Uspekhi, vol. 92, no. 3, pp. 517–526, 1967. [6] A. K. Iyer and G. V. Eleftheriades, “Free-space imaging beyond the diffraction limit using a Veselago-Pendry transmission-line metamaterial superlens,” IEEE Trans Antennas Propag., vol. 57, no. 6, pp. 1720–1727, Jun. 2009. [7] P. A. Belov, Y. Hao, and S. Sudhakaran, “Subwavelength microwave imaging using an array of parallel conducting wires as a lens,” Phy. Rev. B, vol. 73, p. 033108, 2006. [8] M. Tsang and D. Psaltis, “Theory of resonantly enhanced near-field imaging,” Optics Express, vol. 15, no. 19, pp. 11959–11970, 2007. [9] M. Fink, “Time reversal of ultrasonic fields,” IEEE Trans. Ultrason., Ferroelect., Freq. Control, vol. 39, no. 5, pp. 555–566, Sep. 1992. [10] R. Carminati, J. J. Saenz, J. J. Greffet, and M. Nieto-Vesperinas, “Reciprocity, unitarity, and time reversal symmetry of the S matrix of fields containing evanescent components,” Phys. Rev. A, vol. 62, pp. 012712/ 1–012712/7, 2000. [11] G. Lerosey et al., “Focusing beyond the diffraction limit with far-field time reversal,” Science, vol. 315, p. 1120, 2007. [12] V. F. Fusco and N. B. Buchanan, “Dual mode retrodirective/phased array,” IET Electron. Lett., vol. 45, no. 3, pp. 139–141, Jan. 2009. [13] [Online]. Available: www.feko.info [14] H. Chen, Theory of Electromagnetic Waves. A Coordinate Free Approach. New York: McGraw-Hill, 1983. [15] O. Malyuskin and V. Fusco, “Negative refraction lensing and signal modulation using a tuneable phase conjugating frequency selective surface,” in Proc. IEEE Antennas Propag. Int Symp., July 2008, pp. 1–4. [16] O. Malyuskin and V. Fusco, “Far field subwavelength source resolution using phase conjugating lens assisted with evanescent-to-propagating spectrum conversion,” IEEE Antennas Propag., accepted for publication. [17] , R. A. Fisher, Ed., Optical Phase Conjugation. New York: Academic, 1983. [18] F. Simonetti, M. Fleming, and E. Marengo, “Illustration of the role of multiply scattering in subwavelength imaging from far-field measurements,” J. Opt. Soc. Am. A, vol. 25, no. 2, pp. 292–303, Feb. 2008.

Vincent F. Fusco (S’82–M’82–SM’96–F’04) received the Bachelors degree (1st class honors) in electrical and electronic engineering, the Ph.D. degree in microwave electronics, and the D.Sc. degree, for his work on advanced front end architectures with enhanced functionality, from The Queens University of Belfast (QUB), Belfast, Northern Ireland, in 1979, 1982, and 2000, respectively. His research interests include nonlinear microwave circuit design, and active and passive antenna techniques. He is the Technical Director of the High Frequency Laboratories, Queens University of Belfast, and is also Director of the International Centre for Research for System on Chip and Advanced MicroWireless Integration, SoCaM. He has published over 420 scientific papers in major journals and international conferences, and is the author of two textbooks. He holds several patents on active and retrodirective antennas and has contributed invited chapters to books in the fields of active antenna design and EM field computation. Prof. Fusco is a Fellow of the Royal Academy of Engineering and a Member of the Royal Irish Academy. In 1986, he was awarded a British Telecommunications Fellowship and 1997 he was awarded the NI Engineering Federation Trophy for outstanding industrially relevant research.

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Neil B. Buchanan received the Bachelors degree in electrical and electronic engineering and his Ph.D. degree in microwave electronics from The Queens University of Belfast (QUB), Belfast, Northern Ireland, in 1993 and 2000, respectively. His research interests include novel phase conjugating architectures for self steered antennas, mm-wave MMIC and antenna design, specialized mm-wave measurement techniques and innovative mm-wave receivers. At present he is employed by the High Frequency Laboratories at ECIT, as a Senior Engineer. He has published over 30 scientific papers in major journals and in refereed international conferences, and has acted as referee for IET/IEEE publications/conferences.

Oleksandr Malyuskin (M’04) received the M.Sc. degree in radiophysics and electronics and the Ph.D. degree in electrical engineering from Kharkiv National University, Ukraine, in 1997 and 2001 respectively. He joined the Institute of Electronics, Communications and Information Technology, The Queens University of Belfast (QUB), Belfast, Northern Ireland, in March 2004 as a Postdoctoral Research Fellow involved in the development of novel composite materials for advanced EM applications. His research interests include analytic and numerical methods in electromagnetic wave theory, characterization and application of complex and nonlinear materials, antenna arrays and time reversal techniques.

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A Geometrical Optics Model of Three Dimensional Scattering From a Rough Layer With Two Rough Surfaces Nicolas Pinel, Associate Member, IEEE, Joel T. Johnson, Fellow, IEEE, and Christophe Bourlier, Associate Member, IEEE

Abstract—An asymptotic method is described for predicting the bistatic normalized radar cross section of a rough homogeneous layer made up of two rough surfaces. The model is based on iteration of the Kirchhoff approximation to calculate the fields scattered by the rough layer, and is reduced to the high-frequency limit in order to obtain numerical results rapidly. Shadowing effects, significant for large incidence or scattering angles, are taken into account through the use of shadowing functions. The model is applicable for moderate to large surface roughnesses having small to moderate slopes, and for both lossless and lossy inner media. It was validated for a rough layer with a rough surface over a perfectly flat surface in a preceding contribution. Here, the extension of the model to a rough layer with two rough surfaces is developed, and results are presented to validate the asymptotic model by comparison with a numerical reference method. Index Terms—Electromagnetic scattering by rough surfaces, multilayered media, physical optics, multistatic scattering.

Fig. 1. Multiple scattering from a rough layer with two rough interfaces, repx; z^). The points on the upper surface 6 are denoted as resented in the plane (^ fA ; A ; . . . ; A g, whereas the points on the lower surface 6 are denoted as g.  is the elevation incidence angle, and  is the elevafB ; B ; . . . ; B ^. tion scattering angle in reflection, measured with respect to the vertical axis z The positive sense is defined as clockwise.

I. INTRODUCTION II. CALCULATION OF THE SCATTERED FIELDS DERIVED WITH THE KA AND THE MSP

T

A. Problem Presentation

Manuscript received February 26, 2009; revised June 09, 2009. First published December 28, 2009; current version published March 03, 2010. N. Pinel and C. Bourlier are with the IREENA (Institut de Recherche en Electrotechnique et Electronique de Nantes Atlantique) Laboratory, Université Nantes Angers Le Mans, Polytech’Nantes, 44306 Nantes Cedex 3, France (e-mail: [email protected]; [email protected]). J. T. Johnson is with the Department of Electrical and Computer Engineering and ElectroScience Laboratory, The Ohio State University, Columbus, OH 43210 USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2009.2039306

The studied system (see Fig. 1) is made up of a rough layer with two rough interfaces (with an upper interface and a ), separating homogeneous media , with lower interface . Each of the three media , with relative permit, is assumed to be non magnetic (relative permeability tivity ). The same notations as in [3] are used. As for a flat lower interface [3], in order to calculate the fields and , the Kirchhoff approximation (KA) (which is sometimes also called physical optics approximation, or tangent plane approximation) is iterated for each scattering inside the rough layer, i.e., on the interaction among interfaces (and not for multiple scattering from the same rough interface). The formulation is further simplified by applying the method of stationary phase (MSP) for each scattering point inside the rough layer. Using these two approximations in the calculation of scattered fields (note the MSP could have been applied only when calculating the NRCS, but applying the MSP for the scattered fields allows simpler calculations), simplified expressions for and can be obtained: in the case of 1-D surfaces, with , and 5 fold numerical integra1 numerical integration for tions for [4]. Here, for the case of 2-D surfaces, the number of numerical integrations is doubled as shown in what follows.

HE aim of this paper is to extend the Kirchhoff approximation to the case of a rough layer with two rough interfaces, and to obtain a formulation of the bistatic normalized radar cross section (NRCS) in the high-frequency limit in the case where the upper and lower surfaces of the layer are uncorrelated. The model, which takes shadowing effects into account [1], [2], has been described in a recent publication [3] for a rough layer with a rough surface over a perfectly flat surface. Here, the extension of the model to a rough layer with two rough surfaces is developed. Numerical results are presented and compared with a reference numerical method to validate the model, for lossless as well as for lossy inner media.

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B. Fields Scattered by the Rough Layer Under the KA, the derivation of the first-order scattered field , corresponding to the scattering from a single rough surface, is well-known and developed in [3]. The higher-order scat, etc.) are obtained by the same way as tered fields ( for the flat lower interface case [3], i.e., by iterating the KA at each scattering among interfaces. The main difference with the flat case concerns the scattering from the lower interface, where the KA must be applied to model lower surface roughness scattering. Thus, the second-order scattered field in reflection is given under the MSP by the relation

(2)

, and

with given by

(3) (4) (5)

(1) and indicate surface illuminawith then implies 2 5 fold tion functions. The calculation of numerical integrations. By using the same principle for the higher orders, i.e., by iterating the KA at each scattering among interfaces, and using the MSP, it is possible to obtain the expressions of the scattered at any order . Nevertheless, their fields in reflection expression is long and is consequently not given here. III. NRCS IN THE HIGH-FREQUENCY LIMIT A general description of the NRCS in the high-frequency limit can be found in Section III of [3]. A. Expression of the Second-Order Contribution The first-order NRCS in reflection corresponds to the NRCS in reflection from a single rough interface. Under the KA and the MSP, and by using the Geometric Optics Apis well-known [5], [6], and proximation (GOA), expressed by [3, Eq. (7)]. , the principle For the second-order contribution is the same as for 1-D surfaces (see [4, Subsection 3.1.1]), and the expression is similar to the flat lower interface case. The main difference comes from the scattering in reflection from the can be rough lower interface. Thus, the second-order NRCS written as

refers to the rough surface slope probability denHere for the upper or lower insity function; use of the terface is apparent from the argument of the function. The bistatic average shadowing function in reflection is given by [3, Eq. (17)], and the bistatic average shadowing functions in transmission are given by , and , where is the Beta function (also called the Eulerian integral of the , and first kind). The terms are polarization square matrices of dimension 2, given by [3, Eq. (22)], their components being given by [3, Eqs. (23) and (24)] for reflection and transmission, , and are respectively. The wave vectors given by , and , with and the wavenumbers inside and , respectively. The normalized wave vectors , and are given by [3, Eqs. (1a)–(1d)]. The terms , and are the vertical components (i.e., the projections with respect to the axis ) of , and , respectively. It can be noticed that the computation of implies four fold numerical integrations, which should in general require rather long computing time. Nevertheless, the method being geometric-based, it is possible to optimize the numerical integrations, as described in [7, Section 3.A]. Thus, a reasonable computing time can be achieved, as described in Section IV-A. Higher-order NRCS quantities can be calculated in a similar manner, but are not considered here. B. Model Validity Domains and Properties As for the flat lower interface case [3], the validity domains of the model are similar to those in the 2-D problem (with 1-D surfaces): see [7, Section 2.C] and [3, Section III.B]. The main difference with [3] comes from the fact that the constraints also concern the lower rough interface. Based on the iteration of the Kirchhoff approximation (KA) to compute scattering interactions among the interfaces of the rough layer and the high frequency approximation, the overall approach has the validity domain of the geometric optics approximation (GOA). That is why

PINEL et al.: A GEOMETRICAL OPTICS MODEL OF 3-D SCATTERING FROM A ROUGH LAYER

this method is called the geometric optics approximation for a rough layer and is denoted GOA. The model has the same general properties as for the flat lower interface case [3], see second paragraph of Subsection III-B. In particular, the model in itself, as based on the GOA, cannot deal with lossy media a priori. Still, by using exactly the same approach as for the 2-D case (see [4, Section 7]), as it will be shown here in the following numerical results, lossy media can be taken into account a posteriori and give satisfying results. To do so, the power propagation losses can be evaluated by considering . Then, in the particular case of normal flat interfaces, , the losses correspond to a back-and-forth of incidence the wave inside the lossy layer of mean thickness . The assois given by ciated imaginary part of the phase (6) the wavenumber inside the vacuum, the complex with relative permittivity of the lossy layer, and the imaginary part operator. As a result, the power propagation losses are given by (7) In the next section, asymptotic model (GOA) predictions of the first two order contributions of the NRCS, and , are compared with a reference numerical method for validation. The validation concerns the second-order , correcontribution, as the first-order contribution sponding to the scattering in reflection from a single rough surface, is well-known. IV. NUMERICAL RESULTS: GOA VALIDATION A. Numerical Reference Method The same numerical reference method as in [3] (see [8]–[13] for additional information) is used, as this code is capable of treating both flat and rough lower layer boundaries. The results to be illustrated consider a layer of relative permittivity or or , above a rough perfectly conducting boundary . The surface profiles were generated as independent realizations of a Gaussian stochastic process with an isotropic Gaussian correlation function, and the cases considered used identical surface statistics (but independent surface realizations) for the upper and lower interfaces. Three roughness cases are considered: rms height or or and correlation length , corresponding to rms slope 0.15 or 0.2 or 0.3, respectively. A mean distance of 2.41 free space wavelengths between the layers was used in the numerical method. The reference numerical method used surface sizes of 24 by 24 free space wavelengths, discretized into 256 by 256 points for a total of 393216 unknowns in the matrix equation (4 or 2 unknowns for each point on the upper and lower interfaces, respectively). A total of 32 surface realizations (sufficient to achieve mean NRCS estimates accurate to within approximately 2 dB) were used in each simulation, with the required computations performed on parallel computing resources at the Maui High Performance Computing Center. Results for a single surface realization used 8 processors as described in [3], and required approximately 8 hours of CPU time. By comparison, the typical

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CPU time to compute the GOA is of the order of 25 seconds on a standard office computer using MATLAB for each obser), for a total of 15 minutes for each vation configuration ( simulation presented in the following. In the comparisons to be shown, the incident wave is linearly polarized with an incident elevation angle of 0 (normal incidence), 20 or 40 , and the incident azimuth angle is always . The numerical results present the -th order total NRCS in reflection for HH, HV, VH, VV polarizations (the first term representing the polarization of the scattered wave, and the second term the polarization of the incident wave) and in eior in scattering planes rother the plane of incidence tated azimuthally with respect to the plane of incidence ( not zero). More precisely, about the GOA, the first two order contributions (reported as “1” in the legend of the figures) and (reported as ‘ ’ in the legend of the figures) of are computed for the asymptotic model. For the numerical reference method, and are computed, together with the total contribution which takes all the significant contributions into account (reported as “Tot” in the legend of the figures). It must be noted that under the numerical reference method, it was checked that for all cases (and especially when the rms height is ), the coherent contribution is negligible by comparison with the incoherent contribution, which is a necessary condition for the applicability of the GOA. B. Simulation Results For all configurations, it is recalled that the layer mean thickness , and the lower medium is perfectly con. Moreover, for the first configuration to ducting, be simulated, for both surfaces the rms height , which makes the rms slope , and the layer is of relative permittivity . Fig. 2 presents numerical results for scattering in the plane of incidence (azimuth angle ) for an incidence angle . For the GOA, the first-order contribution is plotted as a black line with circles. The second-order contribution is plotted as a dotted line with plus signs. For the numerical simulations of the reference numerical method, results are presented in all polarizations for normal incidence, but were computed only in HH and VH polarizations for this case. Then, to plot VH and VV polarizations, the following symmetries of the bistatic NRCS for a normally incident plane wave were used (8) (9) The contribution from the upper interface alone obtained from the numerical method, corresponding to , is plotted as a solid line. The result after one iteration of the method, corresponding to , is plotted as a dash-dot line, and the result after many iterations, corresponding to , is plotted as a dashed line. In co-polarizations HH and VV, the first two order contributions of the total NRCS and of the GOA have the same basic properties as in the 2-D problem (see [4, Section 6] and [7, Section 3.C]). The second-order total NRCS contributes for all scattering angles, and is much larger than that of the first-order NRCS . Similarly as for the 2-D problem, the

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Fig. 2. Simulation of the first two total NRCSs   and   in dB scale, with respect to the observation angle  in the plane of incidence (azimuth angle  = 0 ), for an incidence elevation angle  = 0 . The rough surfaces have  = 2:41 with same rms slope  = 0:15, the layer is of mean thickness H relative permittivity  = 3, and the lower medium is perfectly conducting ( = i ).

1

case without shadowing effects (which is not represented here for the sake of clarity of the figure) diverges for observation angles : this highlights the relevance of taking shadowing into account for grazing angles. In cross-polarizations VH and HV, as expected by the first-order KA, the first-order NRCS of the GOA has a negligible contribution compared to the second-order contribution . For HH polarization, the comparison with the reference numerical method shows a very good agreement for , which corresponds to the scattering from the upper surface when no lower layer is present. The differences that appear for grazing angles, , are likely impacted by the finite surface size as well as the limitations of the GOA (more precisely, the neglect of the multiple scattering from the same interface effect) for this region. Very good agreement is also observed for the second-order contribution ; significant differences are observed only for grazing angles, , also likely impacted by the finite surface size as well as limitations of the GOA. Indeed, the observed differences in co-polarizations are very similar to the ones obtained for a 2-D problem: for instance, see [7, Fig. 2] for a similar configuration. The result of the numerical method for many iterations highlights that for all scattering angles for these surfaces, there is no significant difference with the first iteration , which means that is sufficient to quantify the scattering process. This result is in agreement with observations made for a 2-D problem (see [7, Section 3.B]). Thus, in co-polarization, the second-order contribution of the GOA model can correctly quantify the scattering process for moderate observation angles, (the difference being likely attributed to the neglect of multiple scattering effects). The same general comments and conclusions can be drawn for VV polarization. For cross-polarization VH, the comparison of the GOA with the reference method highlights an underestimation for the

Fig. 3. Same simulations as in Fig. 2, but with an azimuth angle  = 15 .

for moderate observation angles second-order contribution . This may be attributed to multiple scattering from the same interface effect or possibly to finite surface size effects, although such effects would likely not be major contributors for angles within 30 degrees scattering angle. The total scattering coefficient computed from the reference method shows larger contributions for all . The same general comments and conclusions can be drawn for HV polarization, where the GOA for all angles . underestimates Nevertheless, as for the flat lower interface case, for this rms moves slope, as soon as the azimuthal scattering direction away from 0 degrees, the agreement of the GOA with the reference numerical method is good both in co- and cross-polar: for the izations. This is illustrated in Fig. 3 for , in VH and HV polarizations the second-order contribution GOA is in very good agreement with the numerical method, for . The agreement of the GOA with scattering angles the reference method is improved in this comparison compared to that for the plane-of-incidence, since first-order scattering effects are more important in both polarizations, leading to a lower relative contribution of the multiple scattering effects. Moreover, the difference between the total scattering coefficient and in the numerical method becomes weak here, and is significant only for scattering angles for both cross-polarizations. In HH and VV co-polarizations, the same general are applicable, with the remarks and conclusions as for agreement of the GOA being very good for moderate scattering angles. Fig. 4 presents results for the same parameters as in Fig. 2, but . As is a 90 azimuthal for azimuth angle rotation from the results of Fig. 3 where , the numerical results for co-polarizations are similar to those for cross-polarization obtained in Fig. 3 and vice-versa. Therefore, co-pol and cross-pol results here can largely be interpreted in the same manner as used for cross-pol and co-pol results, respectively. Other comparisons (not presented here) for various rotations of the scattering plane (i.e., values) investigated whether both

PINEL et al.: A GEOMETRICAL OPTICS MODEL OF 3-D SCATTERING FROM A ROUGH LAYER

Fig. 4. Same simulations as in Fig. 2, but with an azimuth angle 

= 105

.

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Fig. 6. Same simulation parameters as in Fig. 5, but only the GOA is represented here, for which a comparison is made between the cases of a rough and a flat lower interface (second-order contribution  ).



A comparison (not presented here) between the in-plane co-polarized 3-D results and the 2-D results of the GOA model was made for the same roughness statistics and layer dielectric properties. The comparison showed that for the first-order contribution , the ratio of the 3-D case to the 2-D case is weak (less than 1 dB) for moderate incidence angles and moderate scattering angles. More generally, it can be shown that this ratio is given for a Gaussian height pdf and without shadowing effects by the general relation (10)

Fig. 5. Same simulations as in Fig. 2, but with surfaces of rms slope 

= 0:2.

the GOA and the reference numerical method captured the appropriate symmetries of the bistatic NRCS for a normally incident plane wave, given by (8) and (9). Both methods were found to achieve these symmetries. Fig. 5 presents comparisons for the parameters of Fig. 2, but with surfaces of rms slope . While the surface correlation length remains unchanged, the reference numerical method used a surface RMS height to achieve . The numerical results are similar to those of Fig. 2, the agreement of the GOA with the reference numerical method being at least as good as in the previous configuration. The results for azimuthal rotations of lead to the same general results and comments. Thus, the same general conclusions can be drawn. Moreover, a similar scenario, but with a layer mean thickness was tested, leading to very similar quantitative results on the numerical method, which confirms the general property of the GOA of being independent of the layer mean thickness for lossless inner media .

, the ratio cannot be For the second-order contribution expressed from a simple mathematical formula. Nevertheless, by comparing the numerical results, it can be noticed that the two curves have the same general behavior. For this typical configuration with , the ratio is practically nearly constant and of the order of dB in this case. However, this ratio decreases for increasing rms slope , and significantly varies for . with Fig. 6 presents comparisons for the parameters of Fig. 5, but the results make comparisons of the GOA model for flat and rough lower interfaces, as plotted in [3, Fig. 2]. Then, for the second-order contribution , the GOA model with the case of a flat lower interface is plotted in dash-dot line with crosses. Similarly as for the 2D case (see [4, Section 6]), significant differences between a rough and a flat lower interface appear. For a flat lower interface, the second-order contribution is concentrated around the specular direction . Indeed, as the lower interface is flat, the energy incident on the lower interface is not scattered in all directions like for the rough case but reflected in the specular direction. On the contrary, for a rough lower interface, the second-order contribution is lower in and around the specular direction , and is more uniformly distributed in all scattering angles. Indeed, the rough lower interface scatters energy in all directions.

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= 105 .

Fig. 7. Same simulations as in Fig. 5, but with an incidence elevation angle  and an inner relative permittivity  i : .

Fig. 8. Same simulations as in Fig. 7, but with an azimuth angle 

Fig. 7 presents comparisons with the numerical method for the parameters of Fig. 5, but with an incidence angle and an inner relative permittivity . The model itself cannot deal with lossy media a priori, but the effect of lossy media can be taken into account a posteriori. Then, the lossy inner medium is taken into account in the GOA as described in [4, Section 7 ]. The results show the same general behavior of the GOA as in the preceding configuration. The results for the case without shadowing effects (not presented here for the sake of clarity of the figure) again diverge for grazing . The results of the GOA are compared with the reference numerical method for all polarizations (HH, VH, HV, VV). In co-polarizations HH and VV, the first-order contribution highlights a good agreement of the GOA with the reference method. Again differences that appear for larger values, and in particular for , can be attributed to the limitations of the GOA as well as tapered wave effects at the larger angles. The second-order contribution highlights a good agreement of the GOA with the reference method for . As in Figs. 2 and 5, results from the reference numerical method highlight that for all scattering angles , the higher orders are negligible: the second-order is enough to quantify the scattering phenomenon. In cross-polarizations VH and HV, results from the GOA confirm that has a negligible contribution, while has a relatively low contribution. The reference method again shows appreciable contributions for that are impacted by the tapered wave and are likely to be overestimates of the true scattering. Once again, the GOA underestimates cross-polarized scattering, especially for moderate . This is likely due to the impact of multiple scattering on the upper interface, which plays a significant role generally in cross-polarized scattering. Still, a rather good agreement is found in VH polarization for relatively high scattering angles, . The results of the reference method for higher orders show that the second-order contribution when computed exactly underestimates the total scattering only slightly for these surface statistics and incidence angles.

Fig. 8 presents numerical results for the same parameters as in Fig. 7, but for a rotated scattering plane at azimuth angle . Again, the main changes from appear in co-polarization. Overall, a good agreement is found with the reference method because multiple scattering effects are less important when compared to first-order scattering processes. Thus, there is a good agreement of the GOA with the reference method for moderate . for Other comparisons of the second-order contribution all values of (i.e., over the whole upper scattered hemisphere) are plotted in Fig. 9 for incidence polarization and in Fig. 10 for incidence polarization, for the same simulations parameters as in Fig. 7. The discretization is 15 for and 5 for . The plus sign indicates the specular direction. It must be noted that the data are not interpolated, even if interpolating should be more relevant to get a more realistic overall representation (especially around and ). The three sub-figures at the top of the figure show the co-polarization, and at the bottom the cross-polarization. Results of from the numerical reference method are plotted on the left, from the GOA in the middle, and the difference of the numerical method with the GOA (differences taken in terms of NRCS values in decibels) on the right. The comparisons confirm the good agreement of the GOA with the reference numerical method for moderate scattering angles . Fig. 11 presents comparisons for the parameters of Fig. 5, but with an incidence elevation angle , an inner relative permittivity , and surfaces with rms slope . For this configuration, the surfaces have higher rms slopes, which corresponds to the qualitative rms slopes limits of the validity domain of the GOA. Then, the results should highlight limitations of the GOA by comparison with the numerical reference method. It can indeed be seen that contrary to the previous configurations where the cross-polarizations highlight rather good agreement of the GOA for high scattering angles (especially for VH polarization), in this case the NRCS is always significantly underestimated by the GOA. This can easily be understood: by significantly increasing the rms slopes, the relative

= 20

= 3 + 0 05

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Fig. 9. Same simulations as in Fig. 7, but with an azimuth angle  ranging [0; 180] , with 15 discretization: Numerical results of   for H incidence polarization.

Fig. 12. Same simulations as in Fig. 11, but with an azimuth angle  ranging [0; 180] , with 2 discretization: Numerical results of  for H incidence  polarization.

Fig. 10. Same simulations as in Fig. 9, but with V incidence polarization.

Fig. 13. Same simulations as in Fig. 12, but with V incidence polarization.

Fig. 11. Same simulations as in Fig. 5, but with an incidence elevation angle  = 40 , an inner relative permittivity  = 3 + i0:01, and surfaces with rms slope  = 0:3.

tion by the GOA is a bit more important for grazing than for the previous cases, mainly owing to the higher rms slopes . Other comparisons of the second-order contribution for various values of are plotted in Fig. 12 for incidence polarization and in Fig. 13 for incidence polarization. The discretization is this time 2 for and 2 for , providing a higher resolution image of the bistatic scattering pattern. The comparisons confirm the general good agreement of the GOA with the reference numerical method for moderate scattering angles . Moreover, limitations of the model are highlighted mainly for grazing , in and for cross-polarizaaround the in-plane configuration tions, and in and around the cross-plane configuration for co-polarizations. It can be noticed that the underestimation by the GOA with shadowing effects is more significant for HH polarization than for VV polarization. Thus, all these comparisons validate the GOA in its validity domain, and help to quantify limitations of the approach for grazing scattering angles and for cross-polarized predictions in the plane of incidence. V. CONCLUSION

contribution of the multiple scattering from the same interface effect is increased. For the co-polarizations, the agreement remains good, except for larger angles. Here, the underestima-

The GOA with shadowing effects for a rough layer has been extended to a general 3-D problem with 2-D surfaces, allowing it to model more realistic problems and to study the influence

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of cross-polarizations. Comparisons with a reference numerical method validated the GOA in its validity domain. The different configurations used confirmed that the model is independent of the layer mean thickness for lossless inner media, and that the model can deal with lossy inner media as well with good predictions. Results showed that for moderate scattering angles, the is generally sufficient to quansecond-order contribution tify the scattering process. Observed differences of the GOA with the reference method can be attributed primarily to multiple scattering effects on the same interface. As a prospect of this paper, the GOA model for a rough layer could be improved by incorporating multiple scattering effects on the same rough interface, similarly as done by several authors for the double scattering in reflection from a single rough interface [14]–[16]. ACKNOWLEDGMENT The authors would like to thank the anonymous reviewers for their helpful, relevant, and constructive comments. REFERENCES [1] C. Bourlier, G. Berginc, and J. Saillard, “Monostatic and bistatic statistical shadowing functions from a one-dimensional stationary randomly rough surface according to the observation length: I. Single scattering,” Waves Random Media, vol. 12, no. 2, pp. 145–73, 2002. [2] N. Pinel, C. Bourlier, and J. Saillard, “Energy conservation of the scattering from rough surfaces in the high-frequency limit,” Opt. Lett., vol. 30, no. 15, pp. 2007–2009, Aug. 2005. [3] N. Pinel, J. Johnson, and C. Bourlier, “A geometrical optics model of three dimensional scattering from a rough surface over a planar surface,” IEEE Trans. Antennas Propag., vol. 57, no. 2, pp. 546–554, Feb. 2009. [4] N. Pinel, N. Déchamps, C. Bourlier, and J. Saillard, “Bistatic scattering from one-dimensional random rough homogeneous layers in the high-frequency limit with shadowing effect,” Waves Random Complex Media, vol. 17, no. 3, pp. 283–303, Aug. 2007. [5] A. Fung, Microwave Scattering and Emission Models and Their Applications. Boston, MA: Artech House, 1994. [6] L. Tsang and J. Kong, Scattering of Electromagnetic Waves, Volume III: Advanced Topics. New York: Wiley, 2001. [7] N. Pinel and C. Bourlier, “Scattering from very rough layers under the geometric optics approximation: Further investigation,” J. Opt. Society Amer. A, vol. 25, no. 6, pp. 1293–1306, Jun. 2008. [8] K. Pak, L. Tsang, and J. T. Johnson, “Numerical simulations and backscattering enhancement of electromagnetic waves from two dimensional dielectric random rough surfaces with sparse matrix canonical grid method,” J. Opt. Society Amer. A, vol. 14, no. 7, pp. 1515–1529, 1997. [9] J. T. Johnson, R. T. Shin, J. A. Kong, L. Tsang, and K. Pak, “A numerical study of the composite surface model for ocean scattering,” IEEE Trans. Geosci. Remote Sensing, vol. 36, no. 1, pp. 72–83, 1998. [10] J. T. Johnson, R. T. Shin, J. A. Kong, L. Tsang, and K. Pak, “A numerical study of ocean polarimetric thermal emission,” IEEE Trans. Geosci. Remote Sensing, vol. 37, no. 1, pt. I, pp. 8–20, 1999. [11] J. T. Johnson and R. J. Burkholder, “Coupled canonical grid/discrete dipole approach for computing scattering from objects above or below a rough interface,” IEEE Trans. Geosci. Remote Sensing, vol. 39, pp. 1214–1220, 2001. [12] J. T. Johnson, “A numerical study of scattering from an object above a rough surface,” IEEE Trans. Antennas Propag., vol. 40, pp. 1361–1367, 2002.

[13] J. T. Johnson and R. J. Burkholder, “A study of scattering from an object below a rough interface,” IEEE Trans. Geosci. Remote Sensing, vol. 42, pp. 59–66, 2004. [14] A. Ishimaru, C. Le, Y. Kuga, L. Sengers, and T. Chan, “Polarimetric scattering theory for high slope rough surfaces,” Progr. Electromagn. Res., vol. 14, pp. 1–36, 1996. [15] E. Bahar and M. El-Shenawee, “Double-scatter cross sections for twodimensional random rough surfaces that exhibit backscatter enhancement,” J. Opt. Society Amer. A, vol. 18, no. 1, pp. 108–16, Jan. 2001. [16] C. Bourlier and G. Berginc, “Multiple scattering in the high-frequency limit with second-order shadowing function from 2D anisotropic rough dielectric surfaces: I. Theoretical study,” Waves Random Media, vol. 14, no. 3, pp. 229–52, 2004.

Nicolas Pinel (A’09) was born in Saint-Brieuc, France, in 1980. He received the Engineering degree and the M.S. degree in electronics and electrical engineering both from Polytech’Nantes (Ecole polytechnique de l’université de Nantes), Nantes, France, in 2003 and the Ph.D. degree from the University of Nantes, in 2006. He is currently working as a contract research Engineer at IREENA Laboratory (Institut de Recherche en Electrotechnique et Electronique de Nantes Atlantique), Nantes. His research interests are in the areas of microwave and optical remote sensing, propagation. In particular, he works on asymptotic methods of electromagnetic wave scattering from rough surfaces and layers, and its application to oil slicks on sea surfaces at moderate and grazing incidence angles.

Joel T. Johnson (F’08) received the Bachelor of Electrical Engineering degree from the Georgia Institute of Technology, Atlanta, in 1991 and the S.M. and Ph.D. degrees from the Massachusetts Institute of Technology, Cambridge, in 1993 and 1996, respectively. He is currently a Professor in the ElectroScience Laboratory, Department of Electrical and Computer Engineering, The Ohio State University, Columbus. His research interests are in the areas of microwave remote sensing, propagation, and electromagnetic wave theory. Dr. Johnson is a member of Tau Beta Pi, Eta Kappa Nu, and Phi Kappa Phi and Commissions B and F of the International URSI. He received the 1993 Best Paper Award from the IEEE Geoscience and Remote Sensing Society, was named an Office of Naval Research Young Investigator, received the National Science Foundation Career Award, the Presidential Early Career Award for Scientists and Engineers in 1997, and was recognized by the U.S. National Committee of Union of Radio Science (URSI) as a Booker Fellow in 2002.

Christophe Bourlier (A’08) was born in La Flèche, France, on July 6, 1971. He received the M.S. degree in electronics and the Ph.D. from the University of Rennes, Rennes, France, in 1995 and 1999, respectively. While at the University of Rennes, he was with the Laboratory of Radiocommunication where he worked on antennas coupling in the VHF-HF band. He was also with the SEI (Système Electronique et Informatique) Laboratory. Now, he is with IREENA Laboratory (Institut de Recherche en Electrotechnique et Electronique de Nantes Atlantique, France) on the Radar team at Polytech’Nantes (University of Nantes, France). He works as an Assistant Researcher of National Center for Scientific Research on electromagnetic wave scattering from rough surfaces and objects for remote sensing applications. He is author of more than 90 journal articles and conference papers.

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Dual-Grid Finite-Difference Frequency-Domain Method for Modeling Chiral Medium Erdogan Alkan, Member, IEEE, Veysel Demir, Member, IEEE, Atef Z. Elsherbeni, Fellow, IEEE, and Ercument Arvas, Fellow, IEEE

Abstract—A dual-grid finite-difference frequency-domain (DG-FDFD) method is introduced to solve for scattering of electromagnetic waves from bianisotropic objects. The formulations are based on a dual-grid scheme in which a traditional Yee grid and a transverse Yee grid are combined to achieve coupling of electric and magnetic fields that is imposed by the bianisotropy. Thus the underlying grid naturally supports the presented formulations. Introduction of a dual-grid scheme doubles the number of electromagnetic field components to be solved, which in turn implies increased time and memory of the computational resources for solution of the resulting matrix equation. As a remedy to this problem, an efficient iterative solution technique is presented that effectively reduces the solution time and memory. The presented formulations can solve problems including bianisotropic objects. The validity of the formulations is verified by calculating bistatic radar cross-sections of three-dimensional chiral objects. The results are compared with those obtained from analytical and other numerical solutions. Index Terms—Chiral media, electromagnetic scattering, finite difference methods, iterative methods.

I. INTRODUCTION

T

HE analysis of chiral materials has been an important topic in computational electromagnetics especially after artificial chiral materials have been manufactured in the microwave range in the last decade. Numerical analysis of chiral materials has been carried out using a variety of numerical methods, such as the method of moments (MoM) [1]–[4], the finite-difference time-domain (FDTD) method [5]–[11], finite-difference frequency-domain (FDFD) method [12], generalized multipole technique (GMT) method [13], surface integral equation method (SIEM) [14], and hybrid finite element method (FEM) [15]. Every one of these methods has its own unique strengths and weaknesses, depending on the problem considered. One of the strengths of FDFD scheme is that it has no analytical load, such Manuscript received January 21, 2009; revised August 16, 2009. First published December 28, 2009; current version published March 03, 2010. E. Alkan was with the Department of Electrical Engineering and Computer Science, Syracuse University, Syracuse, NY 13244 USA. He is now with PPC Syracuse, East Syracuse, NY 13057-4010 USA (e-mail: [email protected]; [email protected]). E. Arvas is with the Department of Electrical Engineering and Computer Science, Syracuse University, Syracuse, NY 13244 USA (e-mail: [email protected]). V. Demir is with the Department of Electrical Engineering, Northern Illinois University, DeKalb, IL 60115 USA (e-mail: [email protected]). A. Z. Elsherbeni is with the Department of Electrical Engineering, The University of Mississippi, University, MS 38677 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2009.2039297

as derivation of structure dependent Green’s functions, and thus it is easy to understand and implement. Furthermore, it is easy to model complex materials such as inhomogeneous, anisotropic, or dispersive materials. Therefore, FDFD can simulate structures with no known analytical solution. The technique is also very robust and does not suffer from stability problems often encountered in time domain methods such as FDTD [16]. Despite these advantages, the memory required by FDFD method is very high and it requires multiple runs to obtain the broadband response of a given problem. For example, FDTD supplies a very broadband data with a single simulation run with much less memory requirements. In [12], the classical Yee [17] cells are used to compose the computational space. On the traditional Yee grid the electric field components and magnetic field components are not located at the same spatial positions. On the other hand, the electric and magnetic field components in chiral media are coupled by the chirality parameter. This coupling requires that the electric and magnetic field components are positioned at the same node. This problem is overcome in [12] by averaging the known field components for the field positions where they are needed. An alternative method, in which two transverse Yee grids are overlapping so that the same components of electric and magnetic fields coexist at the same locations, is proposed in [7] and [8] to solve one-, two-, and three-dimensional problems using FDTD method. Three-dimensional FDTD formulations employing this dual-grid scheme are presented in [11]. The FDFD method, on the other hand, has not yet benefited from the advantages of this approach. No results are published for one-, two- and three-dimensional scattering problems. In this paper, the general 3D frequency domain numerical method based on dual-grid (DG-FDFD) approach has been formulated for general bianisotropic materials and results for chiral objects—a subclass of bianisotropic materials—are presented. In order to show the validity of the derived formulations, bi-static radar cross-sections (RCS) of arbitrary shaped objects have been obtained and the results are compared to exact solution and other numerical solutions. As a drawback, the introduction of a dual-grid scheme doubles the number of electromagnetic field components to be solved, which in turn implies increased time and memory in solution of the resulting matrix equation. As a remedy to this problem, an efficient iterative solution algorithm is developed. The new algorithm effectively reduces the time and memory requirements, thus compensates for the implied overhead.

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II. SCATTERED FIELD FORMULATION The presented method is based on the scattered field formulation, such that (1) (2) and are total fields, scattered fields are where denoted by and , whereas the incident fields are denoted and . “a” and “b” represents the index numbers by of the field components on the first grid and the second grid, respecitively. Using the bi-isotropic constitutive relations in Maxwell’s equations in frequency domain and using the relations in (1) and (2) for the total fields, one can obtain (3) (4)

Fig. 1. The field components on the modified Yee cell for DG-FDFD method.

TABLE I ACTUAL SPATIAL LOCATIONS OF THE FIELD COMPONENTS

where (5) (6) Here, is permittivity, is permeability, the measure of nonreciprocity, and is the measure of chirality of the material. As can be seen from (3) and (4), the electric and magnetic field components are coupled by means of and parameters. This coupling requires the coexistence of the same-polarized components of electric and magnetic fields at the same spatial poand , and , and and . However, sitions, i.e. on a traditional Yee grid same-polarized field components are not defined at the same positions. The chiral FDFD formulation presented in [12] is based on the traditional Yee grid and this problem is overcome by averaging the field components for the positions where they are needed. The dual-grid approach employs two Yee grids: one traditional Yee grid, and the other a transverse Yee grid in which the electric and magnetic field positions are swapped. Dual-grid is the combination of these two grids such that same-polarized components of electric and magnetic fields coexist at the same spatial positions. The approach is named as dual-grid finite-difference frequency-domain (DG-FDFD) method for this reason. In this case, there is no need to perform averaging during the calculations as in [12], and formulation is more compatible with the underlying spatial discretization. The spatial positions of the field components on the modified Yee cell used for dual-grid approach are shown in Fig. 1. Additionally, actual index numbers of the field components are depicted in Table I. Decomposing the vector equations in (3) and (4) in Cartesian coordinates for both defined grids, representing the space derivatives by central-difference approximations and modifying the equations to model the absorbing boundaries as PML layers with the notations being used in [18], one can obtain the following for the -components.

For the first grid

(7)

(8) where in the non-PML region (9)

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and in the PML region

(10) For the second grid

cell and 2 grids, the total number of scattered field components, . The coefficients thus the number of unknowns in (15) is is a non-singular, highly sparse large matrix. Using matrix direct solution methods solving large sparse matrices, such as Gaussian elimination, Cramer’s rule and inverse matrix method, is very costly in terms of memory and CPU time. Therefore an iterative method is preferred to solve sparse linear set of equations to save time and memory. Results in this paper are obtained based on the iterative method called improved “vanilla” BiCGStab(l) iterative method written by Botchev based on [19]. The BiCGStab (bi-conjugate gradient stabilized) method has been introduced in [20]. The method is enhanced in [21]–[23]. BiCGStab algorithm starts with an initial solution vector . As the iterations proceed, the solution vector converges to the actual solution. The iteration is terminated when the relative error of the solution is less than a given tolerance. IV. TIME AND MEMORY REQUIREMENTS

(11)

(12) where in the non-PML region (13) and in the PML region

The amount of memory required for the solution of a sparse matrix equation is mainly determined by the size of the coefficients matrix—the number of non-zero coefficients. Examining one of the equations, for instance (7), one can verify that there . Thus the number of are 6 non-zero coefficients in a row of , and number of non-zero coefficients in unknowns is is for the DG-FDFD method. The FDFD method in [12], solves for 6 field components per . The number of cell, thus the number of unknowns is non-zero coefficients in a row of coefficients matrix is 13, which as makes the total number of non-zero coefficients in for the FDFD method. Basically, there is no significant difference between the memory requirements of DG-FDFD and FDFD. However, when the problem given in Fig. 2 is run using DG-FDFD and FDFD, a significant difference has been observed between the solution times of these two methods; the simulation takes approximately 312 minutes for the DG-FDFD method, and 210 minutes for the FDFD method. The following section discusses the enhancement of the solution algorithm to improve the computation time of the DG-FDFD method. V. DG-FDFD METHOD WITH IMPROVED SOLUTION ALGORITHM

(14)

III. SOLUTION OF THE DERIVED EQUATIONS After all the equations are obtained for both grids, and for , , and components, they are combined to form a matrix equation for a computation space composed of cells as (15) where is the coefficients matrix, is the unknown vector containing scattered electric and magnetic field components of is the excitation vector, due to incident field, repboth grids, resenting the right hand sides of (7), (8), (11), (12), and the rest of the derived equations. Since there are 6 field components per

As discussed in the previous section, although DG-FDFD requires less memory, it is inefficient in the solution time. One unknowns, while this can notice that FDFD solves for number is for DG-FDFD. Following is an algorithm that reduces the number of unknowns for DG-FDFD and reduces the simulation time. The BiCGStab algorithm requires the computation of the right-hand side of the matrix equation for a given solution at every time step. The algorithm then uses this vector temporary right-hand side internally to check for the conver. gence of the solution and generate a new solution vector For the DG-FDFD, the temporary right hand side is calculated by due to (15). As the number of by multiplying unknowns increase, the solution of matrix-vector multiplications in BiCGStab takes longer times, resulting in longer total simulation times as it is observed from initial experiment.

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Therefore, in order to increase the method efficiency, number of unknowns should be reduced. Actually, the equations relating the electric and magnetic fields [(7), (8), (11), (12), and other equations] can be put in the following form: (16) (17) Here, the electric and magnetic field vectors are (18a) (18b) Fig. 2. A sketch of the problem setup for the chiral sphere.

and are coefficients matrices and they can be ex. and are excitation vectors and tracted from vector. they are upper half and lower half subsections of Equation (17) can be rearranged as

TABLE II FDFD VERSUS EFFICIENT DG-FDFD

(19) Then electric field vector can be obtained substituting (19) in (16) as

(20) where

is identity matrix. As one can observe from (20), is a new coefficients matrix and is the vector for which the solution is sought. Thus, while solving the matrix equation, we account for only the electric fields; only the electric field components are treated as unknowns. Then the BiCGStab algorithm can be employed to solve (20) rather than (15). It should be noted that a new coefficients matrix is not con; it is found that such a coefstructed for ficients matrix will include 17 nonzero coefficients in each row, thus increasing the memory requirement significantly. Instead, (20) is kept as is and the following described algorithm is employed. The right-hand side of (20) is calculated before calling BiCGStab. Inside the BICGStab at every iteration the temporary right-hand side is calculated in three steps as follows: ; Step 1) ; Step 2) . Step 3) is a temporary vector which stores the result in the Here intermediate steps. and has Each of the new coefficients matrices rows and 5 non-zero coefficients in a row. Thus their . sizes will be VI. RESULTS The computer being used for the simulations has Intel Quad 2.5 GHz processor and 4 GB DDR RAM. The program is written and compiled in 64 bit compatible Intel Fortran v10.1.21.

In this section, first simulation time and memory requirements of the efficient DG-FDFD and the FDFD [12] methods are compared based on an example problem given in Fig. 2. Then, the radar cross-sections of different objects are obtained using the efficient DG-FDFD method and compared to exact and other numerical methods. A. Comparing the Methods The sketch of the problem is given in Fig. 2. The problem space includes a sphere of radius 7.2 cm, which is illuminated by an -polarized -traveling incident plane wave at a frequency of 1 GHz. The relative permittivity of the sphere material is 4, while the chirality is 0.5. The computational space is composed of one million cells (on a single grid), each cell with size of 0.25 cm on a side. For the example problem, DG-FDFD method with improved solution algorithm results in 60 million unknowns (5 nonzero coeffients and 12 field components). The number of operations to calculate the above temporary right-hand side in BiCGStab multiplications and is additions, every time the right-hand side is calculated. For the FDFD method [12], number of operations additions and multiplications, every time the right-hand side is calculated. When DG-FDFD with the improved solution algorithm is run to solve the example problem, the solution time is recorded as 208 minutes, which is a very significant reduction from 312 minutes. With the improved solution algorithm, the solution time of DG-FDFD becomes comparable to that of the FDFD method. The comparison of these two methods is given in Table II.

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Fig. 3. Co-polarized RCS of the chiral sphere. Fig. 5. A sketch of the problem setup for the chiral cube.

Fig. 4. Cross-polarized RCS of the chiral sphere.

Fig. 6. Co-polarized bistatic RCS of the chiral cube.

B. Scattering from a Chiral Sphere The sphere problem in Fig. 2 is run using the DG-FDFD method and the calculated bistatic radar cross-sections, and , are compared with exact solutions obtained from the program in [24]. The co- and cross-polarized components are plotted in Fig. 3 and Fig. 4, respectively. As observed from these figures, increasing chirality from 0 to 0.5, gave rise to a cross-polarized field, which is the same order as the co-polarized field and that exhibits the optical activity property of the chiral media. Fig. 7. Cross-polarized bistatic RCS of the chiral cube.

C. Scattering From an Inhomogeneous Chiral Cube A cubic structure is illustrated in Fig. 5. The scatterer is 14 cm long on a side and it has two equal halves separated by the plane. The dielectric constants of both halves are the same as , meanwhile the chirality values are different. The chirality of the left half is zero. The right half has a chirality value . The scatterer is illuminated by an -polarized, of -traveling wave at 0.75 GHz. The computational space is comcubic cells. Each cell has a length of posed of 0.4 cm on a side. The simulation results for co- and cross-polarized radar crosssections of the scatterer are shown in Fig. 6 and Fig. 7, respectively. The results are compared to those obtained using method of moment (MoM) [25] and FDFD method [12]. The DG-FDFD results agree with the MoM and FDFD results.

D. Scattering from a Finite Chiral Cylinder Here, a finite chiral cylinder is simulated. The co-polarized and cross-polarized bistatic radar cross sections of the structure are obtained and compared with the results of MoM, finite-difference time-domain (FDTD) [26] and FDFD method. The computation space is excited by an x-polarized, z-traveling plane wave at 1 GHz. The height of the cylinder is 24 cm and the radius is 12 cm. For this simulation, parameters of the Drude-Born-Fedorov (DBF) constitutive relations are used. The conversions among these parameters are given in [27]. , Namely, the material parameters of the cylinder are and . The converted values are , and . The computation space is divided into

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ponents is much smaller than the magnitude of the co-polarized wave component for the cylinder. VII. CONCLUSION

Fig. 8. A sketch of the problem setup for the chiral cylinder.

In this work, the dual-grid finite-difference frequency-domain (DG-FDFD) scheme for the scattering analysis of three-dimensional objects is proposed. The PML boundary condition and the scattered field approach are successfully implemented into the DG-FDFD formulation. The formulations are based on a dual-grid scheme in which a traditional Yee grid and a transverse Yee grid are combined to achieve coupling of electric and magnetic fields that is imposed by the bianisotropy. Since the coefficent matrix of the method includes all the non-zero coefficients of the electric and magnetic field components, it yields a very long simulation time. For this reason, a more efficient solution algorithm is introduced to increase the simulation speed. Since both electric and magnetic field components are defined at every node on the Yee cell, there is no need to perform averaging as in FDFD method. Therefore, formulation is more compatible with the underlying spatial discretization. In addition to these features, it is shown that, DG-FDFD requires less memory than that of the FDFD, while maintaining comparable simulation time. Finally, the proposed method is validated by analyzing spherical, cubical, and cylindrical chiral scatterers. The results from DG-FDFD are in good agreement with exact, MoM, FDTD, and FDFD results. REFERENCES

Fig. 9. Co-polarized bistatic RCS of the chiral cylinder.

Fig. 10. Cross-polarized bistatic RCS of the chiral cylinder.

cubic cells with 9-cell thick PML layers. The computation space is sketched in Fig. 8. The simulated radar cross-sections of the finite chiral cylinder are given in Fig. 9 and Fig. 10, respectively. As seen from these figures, the obtained results by the DG-FDFD method show a good agreement with the results obtained from other numerical methods. Note that, the magnitude of the cross-polarized com-

[1] F. Altunkilic, “Transmission through an arbitrary aperture in an arbitrary three-dimensional conducting surface enclosing chiral material,” Ph.D. dissertation, Syracuse University, Syracuse, NY, Dec. 2007. [2] M. Hasanovic, C. Mei, J. R. Mautz, and E. Arvas, “Scattering from 3D inhomogeneous chiral bodies of arbitrary shape by the method of moments,” IEEE Trans. Antennas Propag., vol. 55, p. 1817, 2007. [3] D. Worasawate, J. R. Mautz, and E. Arvas, “Electromagnetic scattering from an arbitrarily shaped three-dimensional homogeneous chiral body,” IEEE Trans. Antennas Propag., vol. 51, p. 1077, 2003. [4] M. Yuceer, J. R. Mautz, and E. Arvas, “Method of moments solution for the radar cross section of a chiral body of revolution,” IEEE Trans. Antennas Propag., vol. 53, p. 1163, 2005. [5] M. G. Bray and D. H. Werner, “A simple dispersive Chiral FDTD formulation implemented on a Yee grid,” in Proc. IEEE Antennas Propag. Society Int. Symp. and URSI National Radio Science Meeting, 2005, vol. 1, p. 126. [6] V. Demir, A. Z. Elsherbeni, and E. Arvas, “FDTD formulation for dispersive chiral media using the Z transform method,” IEEE Trans. Antennas Propag., vol. 53, p. 3374, 2005. [7] A. Grande, I. Barba, A. C. L. Cabeceira, J. Represa, P. P. M. So, and W. J. R. Hoefer, “FDTD modeling of transient microwave signals in dispersive and lossy bi-isotropic media,” IEEE Trans. Microw. Theory Tech., vol. 52, p. 773, 2004. [8] A. Grande, I. Barba, A. C. L. Cabeceira, J. Represa, K. Karkkainen, and A. H. Sihvola, “Two-dimensional extension of a novel FDTD technique for modeling dispersive lossy bi-isotropic media using the auxiliary differential equation method,” IEEE Microw. Wireless Compon. Lett., vol. 15, p. 375, 2005. [9] L. D. S. Alcantara, “An unconditionally stable FDTD method for electromagnetic wave propagation analysis in bi-isotropic media,” in Proc. SBMO/IEEE MTT-S Int. Microw. Optoelectron. Conf., 2005, p. 661. [10] A. Akyurtlu and D. H. Werner, “BI-FDTD: A novel finite-difference time-domain formulation for modeling wave propagation in bi-isotropic media,” IEEE Trans. Antennas Propag., vol. 52, p. 416, 2004. [11] M. G. Bray, “Finite-difference time-domain simulation of electromagnetic bandgap and bi-anisotropic metamaterials,” Ph.D. dissertation, The Pennsylvania State University, University Park, Dec. 2005.

ALKAN et al.: DG-FDFD METHOD FOR MODELING CHIRAL MEDIUM

[12] L. Kuzu, V. Demir, A. Z. Elsherbeni, and E. Arvas, “Electromagnetic scattering from arbitrarily shaped chiral objects using the finite difference frequency domain method,” in proc. Progr. Electromagn. Res., PIER, 2007, vol. 67, pp. 1–24. [13] Z. Ming and W. X. Zhang, “Scattering of electromagnetic waves from a chiral cylinder of arbitrary cross section-GMT approach,” Microw. Opt. Technol. Lett., vol. 10, p. 22, 1995. [14] A. I. Fedorenko, “Solution of the problem of electromagnetic wave scattering by a homogeneous chiral cylinder using the surface integral equation method,” J. Commun. Technol. Electron., vol. 40, p. 134, 1995. [15] Y. J. Zhang and E. P. Li, “Scattering of three-dimensional chiral objects above a perfect conducting plane by hybrid finite element method,” J. Electromagn. Waves Applicat., vol. 19, p. 1535, 2005. [16] M. Gokten, “New frequency domain electromagnetic solvers based on multiresolution analysis,” Ph.D. Dissertation, Syracuse Univ., Syracuse, NY, Dec. 2007. [17] K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag., vol. 14, p. 302, 1966. [18] M. Al Sharkawy, V. Demir, and A. Z. Elsherbeni, “Iterative multiregion technique for large-scale electromagnetic scattering problems: Twodimensional case,” Radio Sci, vol. 40, Sep. 2005. [19] D. R. Fokkema, Enhanced Implementation of BiCGStab (l) for Solving Linear Systems of Equations Preprint 976, Dept. Math., Utrecht University, The Netherlands. [20] H. A. Van der Vorst, “Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems,” SIAM J. Sci. Stat. Comput., vol. 13, pp. 631–644, 1992. [21] G. L. G. Sleijpen and D. R. Fokkema, “BiCGstab(l) for linear equations involving unsymmetric matrices with complex spectrum,” Electron. Trans. Numer. Analy. (ETNA), vol. 1, pp. 11–32, 1993. [22] G. L. G. Sleijpen and H. A. van der Vorst, “Maintaining convergence properties of BiCGstab methods in finite precision arithmetic,” Numer. Algorithms, vol. 10, p. 203, 1995. [23] G. L. G. Sleijpen and H. A. van der Vorst, “Reliable updated residuals in hybrid Bi-CG methods,” Computing, vol. 56, p. 141, 1996. [24] V. Demir, A. Elsherbeni, D. Worasawate, and E. Arvas, “A graphical user interface (GUI) for plane-wave scattering from a conducting, dielectric, or chiral sphere,” IEEE Antennas Propag. Mag., vol. 46, p. 94, 2004. [25] M. Hasanovic, “Electromagnetic scattering from an arbitrarily shaped three-dimensional inhomogeneous chiral body,” Ph.D. dissertation, Syracuse University, Syracuse, NY, May 2006. [26] V. Demir, “Electromagnetic scattering from three-dimensional chiral objects using the FDTD method,” Ph.D. Dissertation, Syracuse Univ., Syracuse, NY, Jun. 2004. [27] I. V. Lindell, A. H. Sihvola, S. A. Tretyakov, and A. J. Viitanen, Electromagnetic Waves in Chiral and Bi-Isotropic Media. Boston, MA: Artech House, 1994.

Erdogan Alkan (M’08) was born in Nigde, Turkey, in 1978. He received the B.S.E.E. and Ph.D. degrees from Syracuse University, Syracuse, NY, in 2003 and 2009, respectively. He was a Research Assistant at Syracuse University, from 2001 to 2005, where he worked on LNAs, high power amplifiers, pulse circuits, and RF and microwave passive circuits such as filters, and couplers. His research interests are in electromagnetic scattering and finite difference frequency domain method. He is currently working at PPC, Syracuse, NY, as an RF Design Engineer.

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Veysel Demir (S’00–M’05) was born in Batman, Turkey, in 1974. He received the B.S.E.E. degree from Middle East Technical University, Ankara, Turkey, in 1997 and the M.S.E.E. and Ph.D. degrees from Syracuse University, Syracuse, NY, in 2002 and 2004, respectively. He was with Sonnet Software, Inc., Liverpool, NY, from 2000 to 2004. He worked as Visiting Research Scholar at the Electrical Engineering Department, University of Mississippi, University, from 2004 to 2007. He is now with the Department of Electrical Engineering, Northern Illionis University, Dekalb, working as an Assistant Professor. His research interests include computational electromagnetics, numerical techniques such as FDTD, FDFD, MoM, RF and microwave circuits, and antenna design.

Atef Z. Elsherbeni (S’84–M’86–SM’91–F’07) is a Professor of electrical engineering, Associate Dean for Research and Graduate programs, Director of The School of Engineering CAD Lab, and the Associate Director of The Center for Applied Electromagnetic Systems Research (CAESR), The University of Mississippi, University. He is also an adjunct Professor in the Department of Electrical Engineering and Computer Science, L.C. Smith College of Engineering and Computer Science, Syracuse University. He has conducted research dealing with scattering and diffraction by dielectric and metal objects, finite difference time domain analysis of passive and active microwave devices including planar transmission lines, field visualization and software development for EM education, interactions of electromagnetic waves with human body, sensors development for monitoring soil moisture, airports noise levels, air quality including haze and humidity, reflector and printed antennas and antenna arrays for radars, UAV, and personal communication systems, antennas for wideband applications, and antenna and material properties measurements. He is the coauthor of the following books The Finite Difference Time Domain Method for Electromagnetics With MATLAB Simulations (Scitech 2009), Antenna Design and Visualization Using Matlab (Scitech, 2006), MATLAB Simulations for Radar Systems Design (CRC Press, 2003), Electromagnetic Scattering Using the Iterative Multiregion Technique (Morgan & Claypool, 2007), and Electromagnetics and Antenna Optimization using Taguchi’s Method (Morgan & Claypool, 2007). He is the main author of the chapters “Handheld Antennas” and “The Finite Difference Time Domain Technique for Microstrip Antennas” in the Handbook of Antennas in Wireless Communications (CRC Press, 2001). Dr. Elsherbeni is a Fellow of the Institute of Electrical and Electronics Engineers (IEEE) and a Fellow of the Applied Computational Electromagnetic Society (ACES). He is the Editor-in-Chief of the ACES Journal and an Associate Editor of the Radio Science Journal.

Ercument Arvas (M’85–SM’89–F’03) received the B.S. and M.S. degrees from the Middle East Technical University, Ankara, Turkey, in 1976 and 1979, respectively, and the Ph.D. degree from Syracuse University, Syracuse, NY, in 1983, all in electrical engineering. From 1984 to 1987, he was with the Electrical Engineering Department, Rochester Institute of Technology, Rochester, NY. In 1987, he joined the Electrical Engineering and Computer Science Department, Syracuse University, where he is currently a Professor in the Electrical Engineering and Computer Science Department. His research and teaching interests are in electromagnetic scattering and microwave devices. Prof. Arvas is a Member of the Applied Computational Electromagnetics Society (ACES).

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Optimized Analytic Field Propagator (O-AFP) for Plane Wave Injection in FDTD Simulations Tengmeng Tan, Student Member, IEEE, and Mike Potter, Member, IEEE

Abstract—Optimizations are proposed for the original frequency domain analytic field propagator (AFP) to source plane waves in a 1D/2D/3D total-field scattered-field (TF/SF) FDTD formulation. The proposed technique essentially produces no field leakage error (limited by 300 dB machine precision error) with finite sampling points dictated by temporal aliasing constraints. By taking advantage of the inherent 1D nature of a plane wave, the memory requirements for the original AFP are reduced significantly. The proposed optimized solution (O-AFP) greatly reduces the computational complexity by calculating a number of spatial fields of ( ) rather than ( 2 ), which then makes the technique practical enough to include fully 3D simulations. Index Terms—Finite difference time domain (FDTD) methods.

I. INTRODUCTION

T

HE premise of the plane wave total-field/scattered-field (TF/SF) formulation as outlined in Taflove [1] is to introduce volume sources into the finite difference time domain (FDTD) in an efficient manner. This is accomplished by separating the computational domain into a total field (TF) and a scattered field (SF) region, separated by a boundary which is often called the Huygens’ surface. Fields that are excited tangentially on this surface serve to generate an incident field in the TF region. If perfect cancellation of the unwanted outward incident wave is achieved then the SF region (where the scattered fields are observed) contains only the scattered fields whereas the TF region contains both the scattered and incident fields. For example, if no scatterers exist in the TF region then the fields in SF region will be identically zero for all time whereas the fields in the TF region will be only the incident wave itself. Computationally, the advantage of this premise is that it is no longer required to calculate source fields everywhere—they need only be calculated on the Huygens’ surfaces to produce an identical effect. Because a plane wave is inherently a 1D object, one would expect that this computational burden could be reduced further. The first FDTD TF/SF implementation appeared in [2] to excite the Huygens’ surface, where the analytic incident field solution was used. However, the discrete equivalent of this on a point-by-point basis in the FDTD grid does not propagate identically, because the FDTD dispersion relation does not equal Manuscript received March 11, 2009; revised July 10, 2009. First published December 28, 2009; current version published March 03, 2010. This work was supported by the Natural Science and Engineering Research Council (NSERC) of Canada. The authors are with the Department of Electrical and Computer Engineering, University of Calgary, Calgary, AB T2N 1N4, Canada (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2009.2039310

its continuous counterpart. That is to say, what propagates in the discrete grid is not just a sampled version of the analytic solution. Due to the mismatch between the intended (analytic) source and the propagating wave in the grid, non-physical reflections as large as 10% of the incident plane wave were observed in [3] in the scattered field region. In short, the key for minimizing leakage error lies on the construction of a plane wave source which exhibits the same numerical characteristics as the FDTD grid. The first truly 1D method to source plane waves is the incident field array (IFA) technique developed in [1], [4], which evaluates the plane wave source via interpolation from an auxiliary 1D-FDTD grid. Although a significant improvement over the technique of Merewether et al., the IFA technique can never perfectly annihilate the outward field for plane waves propagating at arbitrary angles over all frequencies, as shown in [5]–[8]. This is because in the IFA technique the auxiliary 1D FDTD grid does not have the numerical anisotropy as exhibited in the main 2D/3D FDTD grid. Consequently, the dispersion relations for the main and auxiliary grids cannot match, except at a single frequency or at the magic angles where the dispersion relation reduces to a single term (perfect cancellation is possible at these angles). One can further improve the leakage error of the IFA technique by modifying the 1D auxiliary grid parameters so that the difference in their dispersion relations is minimized over a specific frequency range. The matched numerical dispersion (MND) developed in [9], and the technique reported in [10], [11] effectively minimize the dispersion mismatch error over all frequencies by optimizing the 1D grid parameters. However, as stated in [11], polarization projections can complicate 3D implementations and therefore only a 2D problem is reported, where the reflected fields are indeed suppressed by aldB. Other techniques use higher order interpolation most [12] or a smoothing window function [13] to limit the source spectrum and improve the error due to mismatch. Typically, for a bandlimited sinusoidal incident wave source the IFA technique can only suppress the incident field propagating in the SF region by about dB. Combined with the MND, the techniques in [12] or [13] (which are valid for 3D) can add an approximately dB improvement, which amounts to approximately dB incident field isolation in general. In [7], [8], [14] we reported on a 1D multipoint auxiliary propagator (1D-MAP), which uses a 1D field update equation derived from the main grid FDTD dispersion relation, which is valid for a plane wave propagating at rational angles over all frequencies. We demonstrated in 2D that the incident field in dB (subject to double the SF region can be suppressed by precision implementation) even with a square wave function as

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TAN AND POTTER: O-AFP FOR PLANE WAVE INJECTION IN FDTD SIMULATIONS

its incident wave. The 1D-MAP method has not been extended to 3D because of complications caused by the frequency dependent polarization projection, which is fundamentally caused by the non-transverse numerical electromagnetic waves [15], [16]. Specifically, the FDTD wavenumber vector, the electric field and the magnetic field are no longer orthogonal and this nonorthogonality is, in general, frequency dependent. Another track has been to tackle the problem in the frequency domain. In [16], [18], the numerical plane wave source function is constructed from the 1D plane wave Green’s function . In that work, is evaluated from the FDTD dispersion relation, which typically must be solved numerically since no closed form exists. In addition, the frequency dependent projection as discussed earlier needs to be included in the Green’s function to properly account for nonorthogonality (see [16] for details). The importance of this correction can be seen dB of leakage error can be achieved in [16] and [18] where dB is obtainable ([18] with polarization correction and only mostly focuses on multi-layer applications) without the correction. Both papers reported the procedure for 2D problems, and the method became known as the Analytic Field Propagator (AFP) in the sense that the propagator is analytically known and interpolation errors are virtually nonexistent. Although the AFP technique produces a leakage error much superior than IFA for the same bandwidth spectrum, the AFP scheme as reported is also limited to 2D applications. The AFP requires all incident field values for every point on the Huygens’ surface to be calculated before runtime, since the time domain fields are found from an inverse FFT. Perhaps largely due to this prodigious storage requirement, a 3D TF/SF implemented by an AFP methodology has never been reported even though a relatively complete procedure to establish the 3D polarization vector has been discussed in [16]. To summarize, in 3D the current TF/SF formulations are undB. Alable to achieve incident field isolation better than though this isolation may be satisfactory for some applications, many other applications—e.g., biomedical imaging and RCS dB. Furtherstudies—can demand dynamic ranges near more, one should be able to create a plane wave propagator in the frequency domain that is indeed 1D, and hence mitigate the storage requirement of the AFP. In this paper, we propose an optimized AFP (O-AFP) solution that combines the ideas of the AFP and 1D-MAP methods and that extends its functionality to 3D. The two specific objectives of this hybridization are i) to ensure that the proposed solution can produce mismatch errors similar to the values demonstrated in the 1D-MAP; and ii) to make the AFP algorithm much more practical for larger simulations by reducing the Huygen’s surface from computing spatial points to only computing points, effectively producing a truly 1D propagator in the frequency domain. In what follows we provide the details of the method that was briefly presented in [17]. In particular, we describe how the AFP technique can be optimized to make the technique practical for application to 3D problems. Numerical examples are then presented to demonstrate the accuracy of the new methodology, and its efficiency is compared with the AFP. Finally, a discussion on implementation issues and practical considerations is given.

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Fig. 1. Pictorial representation of the AFP method, where the incident field E at a location x is found from the field at a reference location E via a propagator. The propagator must account for the proper phase delay, but the points of calculation could be anywhere.

II. O-AFP FORMULATION The premise of the original AFP method [16] is that the amon a particular plitude and phase of the incident field wavefront can be found from the field at a reference location through a proper transformation. Because it is a plane wave, the transformation is a simple propagation operator in where the frequency domain, i.e., (see Fig. 1), where the operator accounts for the phase delay in traveling from a reference point to the calculated point; in FDTD this phase delay includes compensating for numerical dispersion in the grid. The resultant field in the frequency domain is then transformed back to the time domain via an inverse FFT. As previously mentioned, one of the problems in extending the AFP method to 3D is associated with the memory burden. Incident fields need to be calculated and stored in a preprocessing stage for every point on the Huygens’ surface. Specif, the Huygens’ surface for a ically, for a TF region of size 3D problem will occupy discrete grid points, time steps then there will be and if the simulation is run for source points to be stored in memory. For electrically large problems—e.g., if is a few hundred points and is run for a few thousand time steps, then the AFP scheme can easily consume over tens of billions of data points. What follows will be a discussion of an optimized AFP method (O-AFP) to bring this storage requirement to more practical levels while maintaining the accuracy. Since we are concerned only with plane waves, the frequency domain source values only need to be calculated along a line/map (1D) which lies in a particular direction with respect to the main computational grid. Incident field values for the Huygens’ surface at known locations share a common wavefront (perpendicular to the 1D map) at a distance from along the 1D map. The field along some reference location this 1D map is also polarized in some arbitrary direction, so a projection operation must also take place to find the required field component. For instance, for an arbitrary electric field component, the frequency domain source field is given by (1) where . The term accounts for the projection operation, and will be explained in a later subsection. First we will explain how the original AFP is optimized to reduce the number of required computations. A. Optimizing the AFP Grid A plane wave by definition only propagates in a single direction with a plane as its wavefront, which means that

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can be calculated on a 1D grid. One can show that the number of points that need to be evaluated are then , which is obviously more desirable unless , which for large translates to extremely fine angular resolution. B. Angles and Projection of Field Components

Fig. 2. An example of how the 1D auxiliary grid is related to points of the main 2D/3D grid which lie on the Huygens’ surface.

it can always be described in 1D. However, when setting (the propagator), none of the up the Green’s function existing AFP schemes [16], [18], [19] make use of this reduction in dimension. The goal of the 1D auxiliary grid is to which by design must equal the compute field values at , which main grid field evaluated at and must lie on the same wavefront. The means that plane wave is propagating in a fixed direction specified by . As we recently showed in [20], a plane wave in a 3D FDTD grid can propagate at a countably infinite set of discretized angles related by rational numbers, and the can be taken to have uniform associated 1D grid in . The rational angle condition imposes that spacing of hold are integers that effectively count the number of where gridcells in a particular direction. With these conditions, the , where is an spacing of the 1D grid is such that integer and (2) Thus every point in the main grid (specifically the Huygens’ surface for this application) indexed by has an associated integer which indexes the 1D grid. Effectively, this integer equation maps a plane wave in the general 3D discrete domain into a 1D discrete domain. The practical implication is that one point computed on the AFP grid can be used to populate many points along the Huygens’ surface of the 2D/3D grids. Because of dimensional reduction, we will refer to this AFP grid as an optimized AFP or O-AFP for short. Fig. 2 shows an example of how the points on the 1D grid are related to those of the 3D grid. Note that any angle of propagation could be represented large enough. here by taking The exact computational saving is now explored. With no loss of generality, the total field region is assumed to be occupied by a square box and hence one Huygens’ surface contains discrete points. Consider that Fig. 2 represents the -plane of spatial a Huygens’ surface. The original AFP requires points to be calculated in preprocessing. However, as discussed in [20], many of these points actually share a wavefront and

We now address the projection term in (1). In order to determine the correct form for this term, we must consider the nature of numerical dispersion in the main grid. An incident field of arbitrary polarization must be decomposed into Cartesian components via field projections, but these projections must also take into account the numerical anistropy of the grid. The original AFP method is developed by directly translating the 2D TM FDTD equations into the frequency domain, and part of this projection is actually done implicitly in the resulting equations. As shown in [20], the equivalent operation in 3D can be derived directly from the FDTD dispersion relation. Specifically, we define the FDTD projections to be (3) . Again, the tilde symbol signifies a nuwhere . Notice that these projection merical variable and for values by construction satisfy all , which means that any standard trigonometric relation can be used to identify the associated FDTD numerical angles and . For example (4) Having established the numerical propagation angles allows us to now deal with the projection term in (1). An electromagnetic plane wave has a specific polarization vector. Furthermore, the orientation of this vector also has a fixed direction since the direction of was defined to be independent of freis parallel to ; adopting quency, i.e., ), the the operators established in [1] (which we will label variable measures the polarization angle from the direction . The scalar electric variable is the polarization field at the reference location. Therefore, obtaining its Cartesian requires a polarization projection operation components which will be done by . However, because the FDTD space is anisotropic, ought to account for this. If one makes use of the FDTD projection or the complex angles and , then the complete functional form for is

(5) In other words, one simply replaces and with and . Equation (5) also show that the polarization projection in general depends on frequency, and its value is also purely real below the cutoff and becomes complex after the cutoff point. at which the This cutoff point is defined as the frequency

TAN AND POTTER: O-AFP FOR PLANE WAVE INJECTION IN FDTD SIMULATIONS

wavenumber changes from purely real to complex-valued. is dependent on angle, and is therefore dependent Note that . As stated earlier, occurs only at magic anon the gles, which implies that holds only at the magic angles. C. Phase Accumulation Term We now address the third term in (1), which accounts for the nonlinear phase and magnitude distortions accumulating along the FDTD grids as the plane wave propagates away from the reference point. In particular, when is unity but the value of is real then the magnitude it decays exponentially when turns complex (beyond cutoff). In particular, if is large then the exponential term dominates over any practical input . In fact, it is well known that frequency components beyond cutoff are highly attenuated, and the FDTD algorithm is effectively a low-pass filter. One other nuance needs to be mentioned to tie this method together. The 1D grid has implicitly assumed that field components are collocated. However, following the convention in [1] for the geometry of the voxels, fields in the main grid are found at the midway point of voxel edges and faces, so an offset must be introduced; thus must be used. This offset takes a simple form based on the geometry of the 1D and 3D grids (6) where the addition in the subscript is understood to be a cyclic permutation: i.e., , and so forth. holds for every Specifically, component provided one uses the evaluated by (6). In short, (1) constitutes the heart of the proposed O-AFP scheme which is capable of solving a 3D TF/SF formulation over a full spectrum and at any rational angles. The factorization is distorted also shows that the simple Green’s function by additional magnitude and phase distortions caused by the , except at the magic angles where anisotropic projection reduces to . The correct FDTD plane wave transfer function is given by (7) The key to achieving a TF/SF formulation that has no leakage simplifies to evaluating the inverse transform —or equivalently (1)—correctly. D. Infinite Impulse Response and Temporal Aliasing Error This section develops a practical criterion for the AFP or O-AFP scheme to mitigate numerical integration error. To this end, the discrete-time sequence for the incident field in (1) is given by the DTFT (discrete-time Fourier transform) relation (8) where is normalized to . This integral equation must be evaluated numerically since no closed form solution is availis obtained by root-finding). If it is approximated by able ( a DTFS (Discrete-Time Fourier Series) defined by

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, then discrete signal processing theory states that (9) where the second term represents temporal aliasing which has is not time-limited a non-zero value when the sequence . Unfortunately, due to the recursive nature beyond of the FDTD update equations, (8) in general does not form a time-limited sequence. For example, if the source function is exsuch that its , cited by an impulse function one can numerically verify (by running a 1D-FDTD grid with ) that the output does not form a time-limited sequence even though its input sequence is time-limited. In other words, the FDTD update equations form an IIR (infinite impulse response) low pass filter. This means that padding zero to the source sequence as suggested in [19] may not be an efficient way for improving the DTFS approximation. An effective alternative to zero-padding is to use a source that has a low pass characteristic. This confunction dition is obtained from the fact that ideally what we want is for within the desired temporal limit . We now show that a necessary condition that can ensure which has no this equality is to have an excitation . To this end, define significant energy above , where the unity window function for and zero otherwise is designed to remove the temporal aliasing term of (9). can be analytically obtained from Next, the spectrum periodic convolution (10) is the DTFT of , which has a where . Because of the convolution operation, the bandwidth of since maximum bandwidth of (10) is bounded by at some large as stated earlier the bandwidth for cannot exceed . On the other hand, recall that by defini, which clearly does not equal tion over all . Particular trouble occurs at frequensince cannot have signifcies above icant energy content above cutoff. In other words, (10) states that in the DTFS approximation of (9) it is impossible to avoid or the source function temporal aliasing unless either is spectrally bounded below . Hence, a much more effective way of mitigating the temporal aliasing problem since is to employ a source function bandlimited to (zero-padding) taxes memory storage and comincreasing putational cost. A good DTFS approximation of (8) requires that the error at every discrete point must be small. It can be numerically verified [through numerical integration of (10)] that lower frequency points produce almost no spectral error. For example, holds true even for a full spectral excitation , and significant numerical error does not start until approaches . Hence the band

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limitedness in our discussion specifically refers to the error assowhich for larger ciated with frequencies . than a few hundred points practically reduces to In particular, if the numerical value for over all is at the machine precision level, then there is no practical difference between the DTFT or DTFS implementation of (8). This condition is automatically satisfied when the source function already has energy near the machine precision . In other words, banlimit for frequencies dlimiting the source provides better control over the leakage error that propagates through the SF region than zero-padding does. It should be noted that the use of bandlimiting to control source errors is not necessary for the full O-AFP formulation to work, but it shows that one can control the accuracy to whatever level one requires, depending on the application. Notice also that (10) can be used to construct an exact DTFS evaluation of (1) over a full spectrum. This formulation will not be considered in the present paper since it involves a numerical integration before the DTFS is performed. E. Computing the

Field

Recall that the time-domain electric field for the computational domain has already been comentire puted by since the integer relation maps every unique to some . This observation suggests that the fields can be computed by the standard FDTD update equations, which only store the present values. to be the x compoFor example, if one defines nent of the magnetic field which is not spatially collocated to any other FDTD components, then this component can be computed by

(11) The other y and z components can be evaluated in the same way. III. NUMERICAL EXAMPLES In all numerical examples, we use a frequency modulated Gaussian source where the parameters and are used to vary the centre frequency, spectral bandwidth, and delay respectively. Although complex is not required, we will also include complex to make our examples general enough to demonstrate incident fields with a band. To emphasize that the frewidth extending beyond quency content affects the leakage errors, we plot the spectrum (top figure) and the time signature (bottom two figures) as one unit. For convenience, the spectral plots are organized as a doubled horizontal plot with the frequency on the bottom axis and on the top axis, and the points per wavelength signifies the cutoff point. The inset plot provides a close-up view

near cutoff to compare the magnitudes of the excitation spec(dashed black line) and the output (solid blue trum line). The incident wave starts at one corner of the TF box and radiates into the TF region at some rational angle defined by . Four probes record the -component of the electric field TF and the SF time sequences with the first pair (laand ) located two cells inside and two cells outbeled side the TF region near the corner where the O-AFP grid starts. and ) are also two cells inThe second pair (labeled side and two cells outside the TF region, but at the furthest diagonal corner away from the first pair. All simulations emwith polarized at ploy Courant stability values of 20 and unity aspect ratio, but terminated with a PEC. The 1D is evaluated, and for a better FDTD transfer function contrast the source and are renormalized to their maximum values, but all the time-domain plots are true , specifically values. Frequency scales are normalized with means . The Matlab floating-point relative accuracy on our machine is , hence any relative accuracy approaching this limit is effectively machine noise. The 3D-FDTD grids for the following three simulations contain 45 45 45 cells with a 30 30 30 square-box TF resurrounded by 15 cells of SF region gion at the center on all sides. The O-AFP grid is propagating at or rational angles of and and has a cutoff of or points per wavelength. Therefore, the proposed O-AFP scheme only comspatial points whereas the existing putes spatial points. AFP scheme would compute Fig. 3 shows a simulation result with source function with pawith a delay and a peak frequency rameters ( points per wavelength). The discrete of points and the simulaversion of (8) is sampled at tion is run for time steps. As shown in the inset plot, since the source function is bandlimited for , the magnitude is nearly down from its at . This means that unity peak value which occurs at the FDTD filter has not filtered out any useful energy outside . The time-domain incident field spectrum of and clearly show that inside the TF region plotted by these two fields are delayed versions of each other (containing small nonlinear phase errors). On the other hand, the time-doand show that these signals have been main plotted by smaller than and . Therefore, a suppressed by perfect cancellation of the unwanted incident fields propagating in the SF region is demonstrated. Fig. 4 investigates a wider source spectrum with (same delay ) and modulated with a lower frequency source, ( points per wavelength). Furthermore, time sampling is doubled compared to the first example because time-steps. As shown in the inset plot, the source is no longer bandlimited at the cutoff bespectrum down from its peak value. Uncause its magnitude is only like the previous example, the FDTD transfer function will filter out some useful energy contained in to produce . Points and clearly suggest that the incident

TAN AND POTTER: O-AFP FOR PLANE WAVE INJECTION IN FDTD SIMULATIONS

Fig. 3. Example 1—Demonstrating that a perfect match can be achieved if the source spectrum is kept below cutoff. In the top figure, the solid line is the source spectrum, the dashed line is the transfer function of the grid. The other figures show the total field (middle) and scattered field (bottom) at two locations in the domain.

Fig. 4. Example 2—Uses the same simulation domain as Example 1, but with a source that is no longer bandlimited. As a result, the incident field isolation from the SF domain is worse.

wave suffers larger nonlinear phase and magnitude distortions. and show that the incident field Consequently, points waves in the SF region are only suppressed by from and . Therefore dB incident field isolation error is obtained.

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Fig. 5. Example 3—Demonstrates that a higher frequency source for a domain with lower number of sampling points per wavelength still results in perfect isolation of the source.

or Fig. 5 pushes the peak sampling to points per wavelength but with a narrower Gaussian profile to and ), and limit its spectral bandwidth ( . Though the incident wave is at a higher frequency, the inset plot still shows that the source spectrum is bandlimited . Consequently, the leakage errors propagating in to and approach floating point error. An the SF region dB is demonstrated; essentially there is no isolation of practical leakage error even with such low sample points per wavelength. Our last example simulates a problem where storage can become an issue for the existing AFP method. This more realistic (surrounded by a 10 cell example has a TF region of SF region), using the same angles as in the previous examples, time-steps. Therefore, the AFP scheme and is run for spatial points to populate the Huywould require gens’ surface, whereas the proposed O-AFP scheme would only spatial points (we save a more detailed discusrequire sion on memory savings for this and the other examples for the or next section). The source function is at points per wavelength but with a wider Gaussian spectrum than (but ). Fig. 6 shows the simFig. 3 since is bandlimited to ulation result without the spectral plot ( ). Although the incident wave does not suffer magnitude distortion (bandlimited below ), the incident waves at and demonstrate distortion due to numerical points dispersion: the plane wave effectively has traveled nearly six away from its source, and phase error surely acthousand cumulates over such a large distance. On the other hand, points and demonstrate that effectively the leakage errors are still limited only by numerical precision.

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Fig. 6. Example 4—Demonstrates a simulation domain where the source travels over several wavelengths before reaching the far side of the domain. Therefore there is a large accumulation of phase distortion (compared with an analytic plane wave). Nevertheless, there is still perfect isolation from the SF region. The solid line is for the sensor point nearest the source, and the dahsed line is for a sensor point in the corner furthest from the source.

TABLE I SPATIAL POINTS AND MEMORY STORAGE FOR AFP AND O-AFP

IV. DISCUSSION We now examine the exact memory saving available from the proposed O-AFP scheme, and the final results are summarized in Table I. Without the integer mapping reduction, the AFP spatial points for the first grid needs to evaluate three examples, whereas the proposed O-AFP grid only evalspatial points. Even with such a small TF box uates size , a memory reduction factor of 50 has been achieved by the O-AFP grid. The total data points is the product of and if we assume then the total storage requirement for the AFP and O-AFP are respectively and data points. The memory storage is not a major and are small for these simulations. concern since both These are included here mainly to show that (9) can be samand yet no temporal aliasing occurs. pled at much smaller points (with a different In fact, we even tested with bandlimited source spectrum) and yet similar leakage errors are value observed. We excluded its result because such a small has a little practical value for FDTD simulations. On the other hand, our last example if run by an AFP grid spatial points but the O-AFP grid would compute spatial points, which amounts to a reduconly requires tion factor of nearly 254. It is clear that for a practical problem ) alwhere is large, judicious choice of angles (smaller lows the O-AFP to always compute fewer spatial points than rethat of the AFP grids. As a result, the memory storage quired for the O-AFP grid will be reduced by the same factor.

time-steps, the last example only For instance, with pieces of data whereas the AFP grid needs to store data points. Without the O-AFP grid, it would store increases, then memory storage will become is clear that if an issue. The above examples demonstrate that a perfect match (down to machine precision level) for the AFP formulation can be accomplished by evaluating the discrete summation of (8). In particular, the contrast of examples Figs. 3 and 4 suggest that the plays a very critical role for minispectral bandwidth of does not necesmizing the temporal aliasing error—larger sarily produce better leakage error. Another indirect conclusion can also be said about the validity of the FDTD dispersion relation, at least up to the cutoff frequency . The fact that cancellation of the unwanted reflection—down to machine level—was achieved implies that the numerical plane wave constructed by (1) must be identical to that of an FDTD plane wave (though only the bandlimited source was demonstrated). By the same token, (1) only utilizes information taken directly from the FDTD dispersion relation. It is therefore easy to conclude that the FDTD dispersion relation must be valid plane wave dispersion relation, otherwise any of these AFP methods would not be so successful. The proposed solution presented here is compatible with solving TF/SF simulations involving planar half-spaces such as in [19]. However, one must be aware that the efficiencies gained from the rational angle condition may not carry over into the transmitted field region. Specifically, one has the freedom to choose the angle of incidence in the region from which the incident wave comes, and hence the reflected wave is just the mirror image of that. However, once the incident wave parameters have been chosen, along with the material parameters in the transmitted region, there are no longer any degrees of freedom. In general, the transmitted angle may require large integer ratios ; in other words, accuracy can always be achieved but memory saving may not always hold in general. Although the proposed method is valid for 1D and 2D simulations, it is anticipated that its main application will be for 3D problems. This is because the 1D-MAP method reported in [8] is just as accurate and more efficient than the O-AFP for 2D simulations, or 3D simulations that are on a principal grid plane. The main contribution of the O-AFP method is that it allows for simulations with extremely accurate and efficient plane wave propagation at arbitrary angles, which was not possible before with the IFA or AFP techniques, nor with the 1D-MAP. V. CONCLUSION The proposed solution has addressed two challenging issues faced by the existing TF/SF formulation: accuracy and memory storage. We demonstrated that efficient plane wave sources in a 1D/2D/3D FDTD TF/SF formulation without any practical dB benchmark) can be solved by a leakage error (a hybrid of AFP and 1D-MAP schema. First, we identified that the root cause of the existing AFP error originates in the temporal aliasing term, and hence established a practical condition to equate the DTFT signal to the DTFS representation. We also made use of the rational angle propagation condition to construct a one-to-many mapping that can further save both

TAN AND POTTER: O-AFP FOR PLANE WAVE INJECTION IN FDTD SIMULATIONS

memory storage and computational points. This mapping allows one to reduce the Huygens’ source computations from to . As a result, the memory storage for the O-AFP is no longer cubic but quadratic . Nevertheless, it is important to also state that the storage issue required by the O-AFP scheme may still be are large. challenging if indeed both and REFERENCES [1] A. Taflove and S. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. Norwood, MA: Artech House, 2005. [2] D. E. Merewether, R. Fisher, and F. W. Smith, “On implementing a numeric Huygen’s source scheme in a finite difference program to illuminate scattering bodies,” IEEE Trans. Nucl. Sci., vol. 14, no. 3, pp. 1829–1833, Dec. 1980. [3] D. E. Merewether and R. Fisher, “An application of the equivalence principle to the finite-difference analysis of EM fields inside complex cavities driven by large apertures,” in Proc. IEEE Antennas Propag. Soc. Int. Symp., May 1982, vol. 20, pp. 495–498. [4] K. Umashankar and A. Taflove, “A novel method to analyze electromagnetic scattering of complex objects,” IEEE Trans. Electromagn. Compat., vol. EMC-24, no. 4, pp. 397–405, Nov. 1982. [5] T. Tan and M. Potter, “Perfectly matched plane wave generated by a time domain multipoint 1D propagator for total field/scattered-field finite difference time domain (FDTD) formulations,” in Proc. Antenna Technol. and Appl. Electromagn. Conf., Montreal, Jul. 16–19, 2006, pp. 605–608. [6] T. Tan and M. E. Potter, “Perfect plane wave source for total-field/ scattered-field formulation in FDTD using time domain multipoint 1D auxiliary grid,” in Proc. IEEE Antennas Propag. Soc. Int. Symp., Jul. 9–14, 2006, pp. 3777–3780. [7] T. Tan and M. E. Potter, “A 1D multipoint auxiliary propagator (1DMAP) for sourcing plane waves for FDTD,” in Proc. IEEE Antennas Propag. Soc. Int. Symp., Jun. 9–15, 2007, pp. 1669–1672. [8] T. Tan and M. E. Potter, “1-D multipoint auxiliary source propagator for the total-field/scattered-field FDTD formulation,” IEEE Antennas Wireless Propag. Lett., vol. 6, pp. 144–148, 2007. [9] C. Guiffaut and K. Mahdjoubi, “Perfect wideband plane wave injector for FDTD method,” in Proc. IEEE Antennas Propag. Soc. Int. Symp., Salt Lake City, UT, 2000, vol. 1, pp. 236–239. [10] S. C. Winton, P. Kosmas, and C. M. Rappaport, “FDTD simulation of TE and TM plane waves at nonzero incidence in arbitrary layered media,” IEEE Trans. Antennas. Propag., vol. 53, no. 5, pp. 1721–1728, May 2005. [11] P. Kosmas, “Review of an FDTD TF/SF formulation based on auxiliary 1-D grids,” presented at the Workshop on computational electromagnetics in time-domain, CEM-TD, 2007. [12] U. Oguz and L. Gurel, “Interpolation technique to improve the accuracy of the plane-wave excitations in the FDTD method,” Radio Sci., vol. 32, no. 6, pp. 2189–2199, Nov./Dec. 1997.

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[13] U. Oguz and L. Gurel, “An efficient and accurate technique for the incident-wave excitations in the FDTD method,” IEEE Trans. Microw. Theory Tech., vol. 46, no. 6, pp. 869–882, Jun. 1998. [14] T. Tan and M. Potter, “1D Multipoint auxiliary propagator (1D-MAP) for perfectly matching plane wave Huygen’s sources in FDTD,” presented at the Appl. Computational Electromagn. Society Conf., Verona, Italy, Mar. 19–23, 2007. [15] M. Celuch-Marcysiak and W. K. Gwarek, “On the nature of solutions produced by finite difference schemes in time domain,” Int. J. of Numer. Model., vol. 12, pp. 23–40, 1999. [16] J. B. Schneider, “Plane waves in FDTD simulations and a nearly perfect total-field/scattered-field boundary,” IEEE Trans. Antennas Propag., vol. 52, no. 12, pp. 3280–3287, Dec. 2004. [17] T. Tan and M. E. Potter, “An optimized AFP scheme for the total/ scattered field FDTD formulation,” presented at the IEEE Antennas Propag. Soc. Int. Symp., San Diego, CA, 2008. [18] C. D. Moss, F. L. Teixeira, and J. A. Kong, “Analysis and compensation of numerical dispersion in the FDTD method for layered, anisotropic media,” IEEE Trans. Antennas Propag., vol. 9, no. 50, pp. 54–56, Sep. 2002. [19] J. B. Schneider and K. Abdijalilov, “Analytic field propagation TFSF boundary for FDTD problems involving planar interfaces: PECs, TE, and TM,” IEEE Trans. Antennas Propag., vol. 54, no. 9, pp. 2531–2542, Sept. 2006. [20] T. Tan and M. E. Potter, “On the nature of numerical plane waves in FDTD,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 505–508, 2009.

Tengmeng Tan (S’09) received the B.Sc. degree in electrical and computer engineering with a minor equivalence in applied mathematics from the University of Calgary, AB, Canada, where he is currently in the process of completing his doctoral dissertation. His main research interest is in the applied computational electromagnetics and much of his work involves time-domain techniques. Mr. Tan was awarded an excellent Teaching Assistants Award in 2005, and twice received Honorable Mention at the AP-S/URSI, in 2008 and 2009, respectively.

Mike Potter (M’94) received the B.Eng. degree in engineering physics (electrical) from the Royal Military College of Canada, Kingston, ON, Canada, in 1992 and the Ph.D. degree in electrical engineering from the University of Victoria, Victoria, BC, Canada, in 2001. From 1992 to 199,7 he served as an Officer in the Canadian Navy as a Combat Systems Engineer. After completing his service and attaining the rank of Lieutenant (Navy), he completed his doctoral work in Victoria, British Columbia, Canada. He was then a Postdoctoral Fellow at the University of Arizona, Tucson, from 2001 to 2002. He currently holds the position of Associate Professor in the Department of Electrical and Computer Engineering, University of Calgary, Calgary, AB, Canada. His research interests include computational electromagnetics and the FDTD method. Dr. Potter is a member of the Optical Society of America, and serves as a member of the APS Education Committee.

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Development of the CPML for Three-Dimensional Unconditionally Stable LOD-FDTD Method Iftikhar Ahmed, Member, IEEE, Eng Huat Khoo, and Erping Li, Fellow, IEEE

Abstract—A convolutional perfectly matched layer (CPML) is developed for three-dimensional unconditionally stable, locally one dimensional (LOD)-finite-difference time-domain (FDTD) method. The formulation of the LOD-FDTD CPML is derived and numerical results are demonstrated at different positions for different Courant Friedrich Levy numbers in the simulation domain. The method is validated numerically with FDTD-CPML. Index Terms—Alternating direction implicit-finite-difference time-domain (ADI-FDTD), convolutional perfectly matched layer (CPML), Courant Friedrich Levy (CFL) limit, finite-difference time-domain (FDTD), locally one dimensional (LOD)-FDTD, perfectly matched layer (PML).

[8]. In addition, the CPML permits an easy implementation of the complex frequency shifted (CFS) stretching factor that allows the reflection of the low frequency evanescent waves to be significantly reduced [9]. Due to generality and simplicity of the CPML, it is implemented with FDTD method on graphical processing unit (GPU) [10] to accelerate the simulation speed. For three dimensional ADI-FDTD and two-dimensional LOD-FDTD methods CPML is developed in [11], [12]. Due to significance of the CPML absorbing boundary condition, in this paper it is developed for the three dimensional LOD-FDTD method. It is abbreviated as LOD-CPML. The formulation and numerical results are discussed in the following sections.

I. INTRODUCTION

II. FORMULATION OF LOD-CPML

T IS KNOWN that Courant Friedrich Levy (CFL) limit reduces computational efficiency of the finite-difference timedomain (FDTD) method [1] for the simulation of structures where fine mesh is needed. The CFL limit was removed with the development of unconditionally stable alternating direction implicit (ADI)-FDTD method [2], [3]. Nevertheless, recently another unconditionally stable technique, known as the locally one-dimensional (LOD)-FDTD has been introduced [4] for two dimensional structures. Later on, it has been extended to the three-dimensional LOD-FDTD method [5] and compared with the ADI-FDTD method. It is found that the LOD-FDTD method is efficient than the ADI-FDTD method due to the requirement of less number of arithmetic operations [5]. The FDTD and ADI-FDTD methods have been applied to open structures, where to truncate the open boundaries numerous absorbing boundary conditions (ABCs) have been developed [1]. Among absorbing boundary conditions perfectly matched layer (PML) [6], uniaxial perfectly matched layer (UPML) [7] and CPML [8] are most commonly used boundary conditions. The PML and the UPML approaches show absorption errors at low frequencies and evanescent waves. In addition, for these both approaches formulation modification is needed if the material of a structure changes. On the other hand, the CPML is completely independent of the host medium and there is no need of modifications in formulation, when apply to lossy, dispersive, anisotropic, nonlinear, and inhomogeneous media. The use of CPML provides significant saving in memory

I

Maxwell’s equations for an isotropic and lossless media are

For the three dimensional LOD-FDTD method, these equations are written as follows. Step 1 (1a) (1b) (1c) (1d) Step2 (2a) (2b)

Manuscript received June 09, 2009; revised August 26, 2009. First published December 31, 2009; current version published March 03, 2010. The authors are with the Department of Computational Electromagnetics and Photonics, Institute of High Performance Computing, Singapore 138632, Singapore (e-mail: iahmed; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2009.2039334 0018-926X/$26.00 © 2010 IEEE

(2c) (2d)

AHMED et al.: DEVELOPMENT OF THE CPML FOR THREE-DIMENSIONAL UNCONDITIONALLY STABLE LOD-FDTD METHOD

Step3 (3a) (3b) (3c) (3d) For the implementation of CPML, we propose to replace the above 3-step LOD-FDTD equations with the following ones where quantities are the auxiliary variables defined in [8]. Step 1

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of the regions. It should be noted that in FDTD-CPML and ADI-CPML [8], [12], auxiliary variable is needed for each spatial differential term in a scalar Maxwell’s equation (6 scalar Maxwell’s equations required for FDTD, 12 for ADI-FDTD and 12 for LOD-FDTD). In the case of LOD-CPML only one auxiliary variable is needed in each equation (12 auxiliary variables for LOD-CPML, 12 auxiliary variables for FDTD-CPML and 24 for ADI-CPML). Therefore, in LOD-CPML the number of auxiliary variables is same as in FDTD-CPML, while half of that of the ADI-CPML. Why only one variable is enough for LOD-CPML in each equation is discussed in paragraph below. For elucidation, (4a) is considered with two auxiliary variables, similar to FDTD-CPML and ADI-CPML and is written as

(4a)

(7)

(4b)

contains the same time In this equation auxiliary term at which field to be calculated (left side). Similarly other (4b)–(6d) can also be modified. However, from numerical experiments with the method it is found that this additional does not affect numerical results signifivariable at cantly (either use or ) except additional complexities in simulation. Therefore, for simplicity both auxiliary variables are considered at same time “ ” and as a result (7) reduces to (4a). Due to implicit nature of the LOD-CPML (4a)–(6d), in their current form, it is computationally expensive to simulate. Therefore, for efficient simulation these equations are further simplified and are given in the following. Step 1 The resultant equation after putting (4d) into (4a) is

(4c) (4d) Step 2 (5a) (5b) (5c) (5d) Step 3 (6a) (6b) (6c) (6d) From (4a)–(6d), it is clear that each equation contains one extra (auxiliary) variable compared to the equations of conventional LOD-FDTD method (1a)–(3d). These auxiliary variables are used in the CPML region only and are zero in rest

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(8a) (8d)

The resultant equation after putting (4c) into (4b) is

Similar to (8a)–(8d), equations of step 2 and step 3 can be simplified. In (8a)–(8d) all electric field components have tridiagonal matrices on the left hand side of equations. This simplification in the form of tridiagonal matrix provides faster solution of the algorithm compared to that without tridiagonal matrix. In this paper tridiagonal matrices are formed for electric field only, however these matrices can be obtained for other way around also (i.e., for magnetic fields, in that case no need to make tridiagonal matrices for electric fields). In above equations the variable “ ,” is considered for each field and direction. This additional variable is responsible for wave absorption in the absorbing boundary regions. As an example one of the auxiliary equations is written as

(9) where

(9a) (9b) where (8b) There is no need to simplify (4c) and (4d) because to calculate and , all the parameters are known

(8c)

(9c) (9d) (9e) where is polynomial grading and is considered equal to 4, , Ncpml is total number of CPML layers, are number of CPML in and directions. In (8a)–(8d) and are calculated from (9d). During provariables gramming the variable should be used in such way, that the (9c) and (9d) should be 0 and 1 respectively, at value of the interface of vacuum and CPML, while maximum at outer boundaries. If compared to the ADI-FDTD method, both LOD-FDTD and the ADI-FDTD method have same number of equation (12), but LOD-FDTD has less number of arithmetic operations [5].

AHMED et al.: DEVELOPMENT OF THE CPML FOR THREE-DIMENSIONAL UNCONDITIONALLY STABLE LOD-FDTD METHOD

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Fig. 2. Reflection error of the LOD-CPML and FDTD-CPML for CFLN = 1 at position B.

Fig. 1. Structure layout (free space truncated by CPML).

The LOD-FDTD with CPML also has less number of auxiliary variables compared to the ADI-FDTD with CPML. Therefore, lower number of auxiliary variables and less arithmetic operations make the LOD-CPML efficient as compared to the ADI-CPML [12]. III. NUMERICAL RESULTS For numerical results the structure under study is shown in Fig. 1 with dimensions 44 44 44 cells including 10 layers of CPML in each direction. The cell size in each direction is mm). same and uniform ( In Fig. 1 point A at location (22, 22, 29) and point B at location (29, 29, 22) are observation points. Gaussian pulse is used as a source. For more robust analysis three field compoand are exnents (center of pulse) and cited with same (width of pulse), where and are center point of and directions respectively. For CPML following parameters are considered:

and in (9c) to (9d). In general parameter is variable and has minimum value at the interface of CPML and simulation domain, and gradually moves to maximum at the ends. In this paper is considered constant (0.08 S/m) throughout the CPML domain. For validation the LOD-CPML is compared with the FDTDCPML. Fig. 2 shows the reflection error of both methods ob. Here the term CFLN is defined as the served at ratio between the time step taken and the maximum CFL limit of the conventional FDTD method. The reflection error is calculated with the following formula:

(10)

Fig. 3. Reflection error with respect to time for different CFLN at position B.

is the field value observed at a point in the where is obtained by extending simulation domain, while dimensions of the simulation domain (to minimize the reflecis the maximum value tion effect from boundaries). observed in the extended domain. Fig. 2 demonstrates that both FDTD-CPML and LOD-CPML have almost same reflection error. For further analysis of the method, Fig. 3 shows reflection error with respect to time for different CFLN at point “B” as mentioned in Fig. 1. Point B is at a diagonal direction from source and 5 cells away from CPML boundary. From Fig. 3 it can be observed that the reflection error decreases with the increase in CFLN. This decrease in reflection error is actually an increase in absolute reflection error which varies towards lower values compared to the ADI-CPML [11], in which error varies towards higher values with increase in CFLN. The reason observed for decrease in LOD-CPML reflection that increases with the increase in error is value of CFLN due to numerical dispersion. The numerical dispersion effect of the three dimensional LOD-FDTD method is explained in [13]. It is also mentioned in [13] that dispersion error may vary upward or downward with increase in the CFLN from a reference point. For calculation of reflection error the value at is used as a reference. This is because the proposed method and the conventional FDTD-CPML have same error at .

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Fig. 4. Relative error with respect to different CFLN. Fig. 6. Reflection error with respect to time for different CFLN at position A.

IV. CONCLUSION In this paper, CPML is developed for the three dimensional unconditionally stable LOD-FDTD method. Formulation of the method is derived. Reflection and relative errors are studied at different CFLN. The reflection error of the proposed method is compared with the conventional FDTD method at and are in good agreement. However, the advantage of the LODCPML is its applicability at higher CFLN. This development of CPML for the three dimensional LOD-FDTD method will enhance applications of the method. ACKNOWLEDGMENT Fig. 5. Normalized electric field intensity in simulation domain versus time at different CFLN.

For additional clarifications, Fig. 4 is plotted that shows the relative error with respect to different CFLN. It also shows that with the increase in CFLN the relative error increases and vice versa. The relative error is calculated with the following formula: (11) is the field value calculated with FDTD-CPML at and is the field value calculated with LOD-CPML at different CFLN. Fig. 5 shows normalized electric field intensity versus time at different CFLN, calculated at point A. From this figure it can be observed that the field intensity increases with the increase in CFLN. This is the reason why the reflection error of the LODCPML decreases with the increase in CFLN. For reflection error analysis, other than at point “B” Fig. 6 is plotted between reflection error and time for different CFLN at point “A” as shown in Fig. 1. The point “A” is located straight from source and is 5 cells away from the CPML boundary. In this case error also increases with the increase in CFLN similar to Fig. 3. However, the reflection error at point “A” is less than that of the error at point “B”, due to less reflection from the absorbing boundary. The reason of more reflection error at point “B” is its diagonal location, where the effect of reflected waves is more as compared to that at point A. where

The authors wish to express their sincerest gratitude to Prof. R. Mittra, Pennsylvania State University, University Park, for his useful suggestions and discussions. REFERENCES [1] A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method. Boston, MA: Artech House, 2005. [2] F. Zheng, Z. Chen, and J. Zhang, “A finite-difference time-domain method without the Courant stability conditions,” IEEE Microw. Guided Wave Lett., vol. 9, no. 11, pp. 441–443, 1999. [3] T. Namiki, “A new FDTD algorithm based on alternating-direction implicit method,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 10, pp. 2003–2007, 1999. [4] J. Shibayama, M. Muraki, J. Yamauchi, and H. Nakano, “Efficient implicit FDTD algorithm based on locally one-dimensional scheme,” Electron. Lett., vol. 41, no. 19, pp. 1046–1047, 2005. [5] I. Ahmed, E. K. Chua, E. P. Li, and Z. Chen, “Development of the three-dimensional unconditionally stable LOD-FDTD method,” IEEE Trans. Antennas Propag., vol. 56, no. 11, pp. 3596–3600, 2008. [6] J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys., vol. 114, pp. 195–200, 1994. [7] Z. S. Sacks, D. M. Kingsland, R. Lee, and J. F. Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Antennas Propag., vol. 43, pp. 1460–1463, 1995. [8] J. A. Roden and S. D. Gedney, “Convolutional PML (CPML): An efficient FDTD implementation of the CFS-PML for arbitrary media,” Microw. Opt. Technol. Lett., vol. 27, pp. 334–339, 2000. [9] J. P. Berenger, “Numerical reflection from FDTD-PMLs: A comparison of the split PML with the unsplit and CFS PMLs,” IEEE Trans. Antennas Propag., vol. 50, no. 3, pp. 258–265, 2002. [10] M. J. Inman, A. Z. Elsherbeni, J. G. Maloney, and B. N. Baker, “GPU based FDTD solver with CPML boundaries,” in IEEE Antennas Propag. Society Symp., Jun. 2007, pp. 5255–5258. [11] I. Ahmed, E. P. Li, and K. Krohne, “Convolutional perfectly matched layer for an unconditionally stable LOD-FDTD method,” IEEE Microw. Wireless Compon. Lett., vol. 17, no. 2, pp. 816–818, 2007. [12] S. D. Gedney, G. Liu, A. Roden, and A. Zhu, “Perfectly matched layer media with CFS for an unconditionally stable ADI-FDTD method,” IEEE Trans. Antennas Propag., vol. 49, no. 11, pp. 1554–1559, 2001.

AHMED et al.: DEVELOPMENT OF THE CPML FOR THREE-DIMENSIONAL UNCONDITIONALLY STABLE LOD-FDTD METHOD

[13] I. Ahmed, C. E. Kee, and E. P. Li, “Numerical dispersion analysis of the unconditionally stable three-dimensional LOD-FDTD method,” IEEE Trans. Antennas Propag., submitted for publication.

Iftikhar Ahmed (S’02–M’05), received B.Sc. degree electrical engineering (electronics and communications) degree from the University of Engineering and Technology ,Taxila, Pakistan, in 1995, the M.Sc. electrical engineering (electronics and communications) degree from the University of Engineering and Technology Lahore, Pakistan, in 1999, and the Ph.D. degree in electrical engineering from Dalhousie University, Halifax, NS, Canada, in 2006. In 2006, he joined Institute of High Performance Computing (IHPC), A*STAR, Singapore, where he is currently a Senior Research Engineer. He has authored and coauthored over 30 journal and conference papers. His current research interests include computational electromagnetics, computational plasmonics and nanophotonics, modeling and simulation of nano devices using time domain methods (FDTD, ADIFDTD and LOD-FDTD).

E. H. Khoo received the B.Eng. degree (with honours) and Ph.D. degree in electrical and electronics engineering from the Nanyang Technological University, Singapore, in 2003 and 2008, respectively. He is currently working as a Research Engineer at the Institute of High Performance Computing (IHPC), A*STAR, Singapore. His current research interest include plasmonics, photonic crystal devices, passive and active photonic, quantum mechanics and solid state physics.

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Erping Li (S’91–M’92–SM’01–F’08) received the Ph.D. degree in electrical engineering from Sheffield Hallam University, Sheffield, U.K., in 1992. From 1989 to 1992, he was a Research Associate/Fellow in the School of Electronic and Information Technology at Sheffield Hallam University. Between 1993 and 1999, he was a Senior Research Fellow, Principal Research Engineer, and the Technical Director at the Singapore Research Institute and Industry. Since 2000, he has been with the Singapore National Research Institute of High Performance Computing, where he is currently Head of the Advanced Electronic Systems and Electromagnetics Department. He is also a Guest Professor at Xi’an Jiaotong University, Xi’an, China, and a Guest Professor of Peking University, Beijing, China. He authored or coauthored over 150 papers published in the referred international journals and conferences, and five book chapters. He holds and has filed a number of patents at the US patent office. His research interests include fast and efficient computational electromagnetics, micro/nano-scale integrated circuits and electronic package, electromagnetic compatibility, signal integrity and nanotechnology. Dr. Li is a Fellow of IEEE, and a Fellow of the Electromagnetics Academy. He was the recipient of 2006 IEEE EMC Technical Achievement Award, the 2007 Singapore IES Prestigious Engineering Achievement Award, and the prestigious Changjiang (Yangtze) Chair Professorship Award from the Ministry of Education in China in 2007. He is an elected IEEE EMC Distinguished Lecturer for 2007 to 2008. He is currently an Associate Editor for the IEEE MICROWAVE AND WIRELESS COMPONENTS LETTERS and IEEE TRANSACTIONS ON EMC. He has been a Technical Chair, Session Chair for many international conferences. He was the President for the International Zurich Symposium on EMC held in 2006 and 2008 in Singapore, the General Chair for the 2008 Asia-Pacific EMC Symposium and the Chairman of the IEEE EMC Singapore Chapter for 2005–2006. He has been invited to give numerous invited talks and keynote speeches at various international conferences and forums.

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An Auxiliary Differential Equation Formulation for the Complex-Frequency Shifted PML Stephen D. Gedney, Fellow, IEEE, and Bo Zhao, Student Member, IEEE

Abstract—An efficient auxiliary-differential equation (ADE) form of the complex frequency shifted perfectly matched layer (CPML) absorbing media derived from a stretched coordinate PML formulation is presented. It is shown that a unit step response of the ADE-CPML equations leads to a discrete form that is identical to Roden’s convolutional PML method for FDTD implementations. The derivation of discrete difference operators for the ADE-CPML equations for FDTD is also presented. The ADE-CPML method is also extended in a compact form to a multiple-pole PML formulation. The advantage of the ADE-CPML method is that it provides a more flexible representation that can be extended to higher-order methods. In this paper, it is applied to the discontinuous Galerkin finite element time-domain (DGFETD) method. It is demonstrated that the ADE-CPML maintains the exponential convergence of the DGFETD method. Index Terms—Absorbing boundary conditions, finite-difference time-domain (FDTD) methods, finite element methods, perfectly matched layer.

I. INTRODUCTION HE perfectly matched layer (PML) absorbing boundary condition [1] has revolutionized the termination of unbounded domains for differential equation based solvers such as the finite-difference time-domain (FDTD). The significant advantage of the PML absorbing media is that it provides a mesh truncation algorithm that is independent of frequency, wave polarization, and angle of incidence, and has extremely small reflection errors. It has also been shown that the PML is inherently “material-independent”, and can terminate domains with inhomogeneous, dispersive, and non-linear materials [2]–[4]. Berenger’s original method is now referred to as the “splitfield” PML. Other variants of the PML have also been introduced, and are now referred to as the un-split, or anisotropic PML [2], [5], and the stretched coordinate PML [6]. While each of these techniques offers different mathematical representations of the PML, the formulations will lead to equivalent reflection properties [7]. As a consequence, more recent research on improving the performance of the PML has focused on modifying the choice of the constitutive parameters [8]–[12]. The most accurate and robust choice for the constitutive parameters currently in use is the complex-frequency shifted (CFS) PML parameters [7], [8], [10]–[12]. An efficient im-

T

plementation of the CFS-PML for FDTD methods is Roden’s stretched coordinate formulation implemented with the discrete recursive-convolution method [12]. The discrete recursive-convolution method is second-order accurate and is efficient in terms of memory and computational cost. With the proper choice of scaling the constitutive parameters, this approach provides excellent absorption of both propagating and evanescent waves, and the PML boundary can be placed extremely close to the device under test [7], [10]. The method can also be applied to terminate domains with arbitrary media without any specialization of the implementation [7], [12]. Recently, there has been a thrust of developing high-order solution methods, such as the pseudo-spectral time-domain method [13], or the discontinuous Galerkin method [14]–[16]. Such techniques also have benefited from PML truncations of the mesh. However, in order to utilize the CFS-PML constitutive parameters, a formulation that allows high-order implementations of the CFS-PML must be developed. In this paper, an alternate form of the CFS-PML expressed via an auxiliary differential equation (ADE) form is presented. The ADE-CPML for a multiple-pole PML is also introduced. The methods are validated based on a second-order accurate FDTD formulation. They are also validated based on a high-order discontinuous Galerkin finite-element time-domain (DGFETD) formulation [17]. II. ADE FORM OF THE CFS-PML Consider the time-harmonic Maxwell’s equations in a lossless, source-free media. In the PML region, the equations can be expressed in complex stretched coordinates. For example, the -projection of Faraday’s and Ampere’s laws are expressed as [6]: (1) (2) where are the stretched coordinate metric coefficients. Initially, these coefficients are chosen via the CFS-PML parameters [10]–[12] (3)

Manuscript received October 13, 2008; revised July 07, 2009. First published December 04, 2009; current version published March 03, 2010. The authors are with the Department of Electrical and Computer Engineering, University of Kentucky, Lexington, KY 40506-0046 USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2009.2037765

and are assumed to be positive real, and can be where one dimensional functions along the -axis. Transforming (1) or (2) into the time domain leads to a convolution between the inverse of the stretched coordinate parameters and the partial derivatives [12]. Alternatively, an auxiliary

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variable can be introduced which is constrained via the introduction of the appropriate ADE based on the identity [18]

non-TEM waveguides. Employing a multiple-pole PML, (1) can be written as

(4)

(15)

where (5)

where, and are CFS-PML parameters, as in (3). A characterization of the absorption (and reflection) properties of the double-pole PML is presented in Appendix B. From (A.6) in Appendix A, (15) can be expressed as

Combining (4), the derivative on the right-hand side of (1) can be expressed as (16)

(6) where the new auxiliary parameter

where and are defined in (A.7) and (A.8). This is simplified through the introduction of auxiliary variables

satisfies (7)

Applying (5), (7) is transformed into the time-domain, leading to the auxiliary differential equation (ADE)

(17) The entire expression is transformed into the time-domain leading to

(8) In conclusion, (1) and (2) can be rewritten as

(18) (9)

where,

satisfy the differential equations

and (10) (19) Combining all coordinate directions, Maxwell’s equations can be written in the PML as (11)

where, and . and . This can be completed for all the projections of Maxwell’s curl equations, which can then be expressed as

(12)

(20) (21)

where (13)

where, (22)

and

(and similarly

) satisfy the ADE The two auxiliary vectors

and

, each satisfies the ADE

(14) where,

is the unit normal to the PML interface, and

is the

normal axis of the PML media. In a single PML region, represents a pair of auxiliary variables. In overlap regions, there is a pair of auxiliary variables for each . III. MULTIPLE-POLE PML TENSOR When using a PML to terminate waves with a significant amount of energy in both evanescent and propagating modes over a broadband, it can be advantageous to employ a PML with stretched coordinate parameters that have multiple-poles [9]–[20]. Example applications would be periodic structures or

(23) and satisfy similar where, is the normal to the PML. equations. Higher-order multiple-pole PML forms can be derived via a recursive expression. Thus, an -pole PML will require auxiliary variables per spatial derivative term. While this leads to additional memory, there may be some applications that demand such a higher-order PML to enable the reduction of the overall mesh dimension so that the total degrees of freedom can be reduced.

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IV. ADE-CPML FOR FDTD Equations (11)–(14) can be discretized to second-order accuracy via the finite-difference time-domain (FDTD) approxiis co-located mation [21]. To this end, the auxiliary fields in space and time with the fields and respectively. Maxwell’s equations in (11) and (12) are discretized using the standard Yee-algorithm. Subsequently, a discrete form for (14) must be derived. Two methods are presented here. The first is a unit step response. The second a finite-difference approximation.

Grouping terms, this can be posed in the form of a recursive update scheme as (30) where,

(31) A. Unit Step Response Equation (8) can be viewed as a first-order ordinary-differen, driven by the forcing function . tial equation for For second-order accuracy, it is sufficient to approximate the forcing function as a piecewise constant function. Then, each can be represented via time interval a unit-step response solution, with source , where is the unit-step function, and initial condition . From (8), it is seen that has the steady-state value

It is noted that the second-term in (29) held at time step , rather than using a linear average in time, as is typically done with the loss term in FDTD [21]. In fact, if this term is averaged in ), this leads to an instability in the time (i.e., combined ADE-FDTD formulation. One can show that for the update equations in (27) or (30), the combined ADE and FDTD equations will be stable within the Courant Stability limit if the coefficients fall within the range

(24)

(32) and

Given these conditions, the unit-step response to (8) is expressed as

(25) where, (26) Finally, letting

then from (25) (27)

(33) The former inequality, (32), guarantees the stability of the difference operator in (30). The latter inequality (33), ensures that positive loss is injected back into Maxwell’s equations, and thus ensuring attenuation of the fields within the PML region. If , and are positive real numbers the coefficients in (28) and (31) satisfy this stability relation. Thus, both explicit update schemes in (30)–(31) and (27)–(28) are stable and well posed. It is shown in Section VI that both maintain second-order accuracy of the standard FDTD discretization and have the same reflection properties. C. Discrete Multiple-Pole Formulation

where, (28) and, is given by (26). Comparing the recursive update in (27) to Roden’s CPML formulation [12], [7], it is observed that the expression is identical to that derived using the recursive convolution method (cf., [7, p. 306, Eq. (7.110a)]). Thus, it is concluded that the recursive convolution approximation is identical to the unit-step response of the auxiliary differential equation (8). B. Finite-Difference Approximation The auxiliary differential equation (8) can alternatively be discretized by following a finite difference procedure. To this end, (8) is approximated as (29)

The multiple-pole PML can also be formulated within the FDTD method. To this end, (20) and (21) are discretized using the Yee-algorithm. The discretization is identical to the singlepole PML, with the exception of the additional auxiliary variables. Equation (23) is discretized for each auxiliary field in an identical manner as that presented in either Sections IV-A or IV-B, leading to a second-order convergent formulation for the FDTD method. V. APPLICATION TO THE DGFETD METHOD The ADE-CPML representation of Maxwell’s equations in (11)–(14) can be applied to more general solution techniques. Here, it is applied to a Discontinuous Galerkin finite-element time-domain (DGFETD) formulation [17]. The DGFETD problem domain is spatially decomposed into sub-domains, where each sub-domain supports an independent finite-element mesh. Within each sub-domain, a Galerkin FEM formulation of Maxwell’s coupled curl equations is posed. Neighboring

GEDNEY AND ZHAO: AN AUXILIARY DIFFERENTIAL EQUATION FORMULATION FOR THE CFS PML

sub-domains then couple through their shared boundaries by weakly constraining the continuity of the tangential fields across shared boundaries. A high-order time-integration, such as Runge-Kutta (RK) methods, is applied leading to high-order time-dependent solutions [17]. Following the procedure outlined in [17], the weak-form of Maxwell’s curl equations based on (11)–(14) can be expressed as

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and are The field intensities and the test-vectors discretized via -curl conforming basis functions [22]. The auxiliary vectors are instead expanded as (38) where is the -th -curl conforming basis function representing the E or H-field. It is noted that basis functions that are parallel to are thus neglected from the auxiliary vector function space. The test vectors space of

are expanded via the dual

such that (39)

(34)

is the Kronecker delta function. With this choice of where field and test spaces, (34)–(37) can be re-cast in a discrete form as (40) (41) (42)

(35)

(36)

(37)

where,

is the sub-domain volume,

is the surface bounding

the sub-domain volume, , and represent the , and auxiliary field test spaces, and and represent the tangential fields in the neighboring subdomain on boundary [17]. Equations (36) and (37) represent the weak-form equations of the ADE for each PML overlap region with unit normal . The vector in (34) and (35) is defined such that when . Otherwise, .

(43) where the matrices are defined with one-to-one correspondence with the volume and surface integrals in (34)–(37) with the vector field and test functions expanded into their respective function spaces, indicates matrix transpose, and are the coefficient vectors representing the discrete H and E-fields, and are the unknown coefficient vectors of the neighboring and are the coefficient vectors of unsubdomains, and knowns representing the auxiliary fields in the PML regions. The -matrices are diagonal matrices, and are a function of the PML-tensor coefficients. Outside the PML regions, the and matrices are zero, and the formulation reduces to the standard DGFETD method. The system of equations is solved via a high-order time-integration technique. In this work, a fourthorder Runge-Kutta method has been applied. The proposed ADE-PML formulation offers a number of advantages over the PML formulation presented in [17]. Notably, the matrices of the auxiliary equations are diagonalized, thus reducing storage and computational cost. Secondly, the ADECPML formulation is independent of the host medium material type. The ADE-CPML formulation also naturally incorporates the CFS-PML tensor. It is further noted that the multiple-pole ADE-CPML can also be easily incorporated into the DGFETD method. The one tradeoff of the proposed method is that it requires the PML mesh to extend normally off the PML interface boundary. While this is difficult for automated mesh generators, this can be realized with a simple extrusion of the mesh off of the exterior boundary.

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Fig. 1. Two dimensional 40 mm 40 mm working volume excited with a TM polarized current source at the center of the domain. The field is sampled 2 mm from the corner of the PML boundary.

VI. VALIDATION A. FDTD Validation The ADE-CPML method is initially validated for the proposed FDTD implementations. Polynomial scaling is applied to the parameters [7]. It is noted that and are scaled such that they are zero at the interior PML interface, and maximum at the terminating boundary. Conversely, is scaled such that it is a maximum at the FDTD/PML interface, and scales to zero at the terminating boundary. This is necessary for the CFS-PML to absorb both evanescent and propagating modes [10]. It has been found that linear scaling for is optimal [7], and is done so in all cases presented herein. Consider a two-dimensional free space excited by a TM -poplaced at the center of the larized electric current source domain, as illustrated in Fig. 1. The source has a time-signature described as a differentiated Gaussian pulse, with a 3 GHz bandwidth. The 2D space has a dimension of 40 40 mm, and mm. is discretized with a uniform cell spacing The domain is padded with an additional ten cells of PML on all four sides with the same cell spacing. The field is probed at in the corner of the domain that is 2 mm from both a point sides of the lower corner of the domain (cf., Fig. 1). The error in the field relative to a reference problem, with the side-walls extended sufficiently far so that no reflection from the exterior boundary would be present for over 1000 time steps. The error in the vertical electric field was then computed via the form (44) where, represents the maximum value over all time steps represents the field computed using the original problem represents via the FDTD method bounded by the PML, and the field computed using the reference problem. The fields were simulated using both the original convolutional PML (CPML) form (27)–(28) and the ADE-CPML form presented in Section IV (30)–(31). The data in Fig. 2 represents a and were varied. Both set of simulations for which and were scaled with third-order polynomial scaling . was fixed at 0.24, and was scaled linearly. Note that is normalized by , which is the optimal value of predicted by [7, p. 294, Eq. (7.67)]. Observing Figs. 2(a) and (b),

Fig. 2. Contour plot of the maximum relative error in the electric field at point = and  (with m = 3), with a = 0:24 and B as a function of  linear scaling for the (a) CPML, and (b) ADE-CPML.

the ADE-CPML may show some advantage. However, overall, the results are quite comparable. The data in Fig. 3 represents and were varied, and a set of simulations for which was fixed at 15. and were again scaled with third-order polynomial scaling , and was scaled linearly. Once again, the ADE-CPML performs comparably to the CPML. Next, we shift to a three-dimensional example, which is the interaction of a transient wave with a long, thin PEC plate [7]. The geometry is illustrated in Fig. 4. The wave is excited by a small vertical dipole placed just above one corner of the plate . The horizontal field is probed at the opposite end of the plate. The discrete FDTD model was constructed with a unimm. The PML inform mesh with terface was placed 3 cells away from the plate on all sides. Thus, the global mesh (excluding the PML region), consisted of 31 106 6 cells. The mesh was then extended in all 6 directions by PML media. A much larger reference problem was also simulated in order to predict the reflection error due to the PML boundary. The error in the probed electric field was computed again via (44).

GEDNEY AND ZHAO: AN AUXILIARY DIFFERENTIAL EQUATION FORMULATION FOR THE CFS PML

Fig. 3. Contour plot of the maximum relative error in the electric field at point (linearly scaled) and  = (with m ), with B as a function of a  , for the (a) CPML and (b) ADE-CPML.

=3

= 15

( ) ( )

Fig. 4. Thin PEC plate illuminated by a vertical current dipole J placed just above one corner of the plate. The horizontal electric field E is probed at the opposite corner of the plate.

Fig. 5 illustrates the error relative to the maximum reference field as a function of time for various PML thicknesses as computed via the CPML and the ADE-CPML methods. These (both results were obtained with ), and with linear polynomial scaled with scaling. The CPML and ADE-CPML results are quite comparable. A 10 cell thick PML can provide nearly 4 digits of accuracy from the absorbing boundary. Next, the PML constitutive parameters were swept to find optimal ranges for the parameters. A sampling of this study is presented in Figs. 6 and 7. In Fig. 6, both and are swept, with

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Fig. 5. Relative error versus time for the thin plate problem for PML of various thicknesses as a function of time as computed via the (a) CPML and (b) ADECPML.

being held fixed. In Fig. 7, and were swept with fixed at 15. Again, and were polynomial scaled , and was linearly scaled. This was found to be with the near optimal scaling for a 10 cell thick PML. Also, in all , or cases, the minimum error is recorded around is optimal in the range of perhaps a little larger. Also, is optimal in the range of 0.1 to 0.25. Note 10–20, and that we are sensing a cross-polarized wave. For the co-polar, which is the dominant field, the optimal value ized wave , and and are slightly lower. It for is noted that the ADE-CPML form is observed to be slightly better in terms of reflection error. However, for all practical purposes, the two formulations have comparable accuracy. The final example is a rectangular waveguide terminated with PML. The example geometry is illustrated in Fig. 8. A -directed electric dipole was used to excite both propagating and evanescent modes in the waveguide. The waveguide was discretized with a uniform discretization with cell sizes mm. A reference problem that was much larger than the original problem was also run to provide a

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Fig. 6. Contour plot of the relative error versus  = and  , polynomial scaled with m , and a : and linearly scaled, for the thin plate problem resulting from the (a) CPML and (b) ADE-CPML formulations (10 cell thick PML).

Fig. 7. Contour plot of the maximum relative error s versus a and  = , with  and  and  polynomial scaled with m , and a is linearly scaled, for the thin plate problem resulting from the (a) CPML and (b) ADE-CPML formulations (10 cell thick PML).

reference for the relative error of the PML for the reduced domain. The reflection error of the PML versus frequency is presented in Fig. 9(a) for a 7-cell thick PML, and Fig. 9(b) for a 10-cell thick PML. In both figures, the CFS-PML is compared to the double-pole PML. For the CFS-PML, (both polynomial scaled with ), and (linearly scaled). For the double-pole PML, (all were scaled with ), , and was scaled according to (B.9), with . In both cases, the double-pole PML provides dB improvement in the reflection error. It is noted that the largest error occurs near the cutoff frequencies of the two lowest order modes of the waveguide. Near the cutoff frequency, the wave is highly evanescent (just below cutoff), or very slowly propagating (just above cutoff), and is very difficult for the PML to absorb [19]. The double-pole PML dramatically improves the ability of the PML to absorb these waves over a very broad band. This is consistent with that found by [9] and [17], [18].

Fig. 8. Air-filled rectangular waveguide excited by a vertical current dipole. The waveguide is terminated via PML absorbers.

=3

= 0 24

= 15

=3

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Fig. 10. Reflection error versus  of a 2 cell thick PML termination for the DGFETD method for various basis function orders.

Fig. 9. Reflection error versus frequency due to a first order and a second order CFS-PML termination of the rectangular waveguide for (a) 7 cell and (b) 10 cell thick PML regions.

B. DGFETD Validation The ADE-CPML formulation for the DGFETD method is next validated via a canonical test case. To this end, the dominant mode parallel-plate waveguide is excited. The waveguide is discretized with hexahedron that are 0.1 m on a side. The fundamental TEM mode in the guide is excited with a broadband pulse. The guide is terminated with a 2 cell thick PML with a constant material profile. Since the mode is propagating and . The maximum reflection error over 2500 time steps as a function of the PML conductivity is illustrated in Fig. 10 for , and 5. Also plotted is the theoretical rebasis orders flection error based on the attenuation of the wave encountered via a full round trip through an ideal PEC-backed PML slab. The measured reflection error matches the theoretical reflection error until gets sufficiently large so that discretization error dominates the measured reflection error. It is observed that as the basis order is increased by two, the reflection error decreases by more than 30 dB.

VII. CONCLUSION In this paper an auxiliary differential form of the complex-frequency shifted PML was derived based on a stretched-coordinate PML formulation. This is referred to as the ADE-CPML. It was shown that a unit-step response solution of the ADE-CPML equations leads to identical update coefficients as Roden’s original CPML formulation [12]. Well-posed finite-difference operators for the ADE-CPML formulation were also derived. Limits for the coefficients of the discrete difference operators to ensure stability were also discussed. This method is then generalized to a multiple-pole form of the PML, derived by cascading PML constitutive parameters. Through numerical simulations with the FDTD method, it was validated that the ADE-CPML has comparable reflection error as Roden’s CPML for the same constitutive parameters. It was also demonstrated that a multiple-pole ADE-CPML can be highly effective at simultaneously absorbing purely evanescent and purely propagating waves over a broad frequency band, such as found in a bounded waveguide analysis. The ADE-CPML formulation was extended to the discontinuous DGFETD method. This has several advantages over previous methods. The first is a reduction in memory as compared to the original PML formulation posed for the DGFETD method, since the auxiliary differential equations can be diagonalized. Secondly, the ADE-CPML formulation is material independent. Finally, it provides a means to incorporate the CFS-PML and a multiple-pole PML into a higher-order method such as the DGFETD method, without having to rely on a complicated high-order approximation of the discrete convolution. APPENDIX A Applying (4), it can be shown that (A.1) The right-hand side can be expanded as

(A.2)

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The last term is expanded via a partial fraction expansion (A.3) where, (A.4) and (A.5)

amplifies the attenuation of the Therefore, the real part attenuevanescent part of the wave, and the imaginary part ates the propagating part of the wave. From (B.3) it is seen that at very low frequencies, (i.e., when and ), . As a consequence, there will be no attenuation of a propagating wave. This would be catastrophic for a TEM wave, as the PML would offer no attenuaor to tion at all. This can be mitigated by choosing either . Consequently, from (B.3) be 0. Here, we will choose

Combining (A.2) and (A.4), it can be shown that (B.5) (A.6) where, (B.6)

(A.7) (A.8)

Thus, at low frequencies approximated as

, the imaginary term is

(B.7)

(A.9) Similarly, at high frequencies, where

(B.8)

APPENDIX B The reflection and absorption properties of the PML with are studied in this ApPML tensor coefficient pendix. The theoretical reflection error resulting from an ideal PML slab of depth backed by a PEC wall can be expressed as [6], [19][20] (B.1) is the complex wave-number along the normal-axis where, of the PML. For simplicity, but without losing generality, assume that the stretched coordinate parameters are constant within the PML. Thus,

in (B.4), it is As a consequence, assuming observed that, the double-pole form of the PML-tensor provides broad band absorption of propagating waves, including down to zero frequency. It is further observed from (B.8) that, when and , we expect typical maximum choosing values for values for these parameters to be and . can potentially be negative. It is observed in (B.5) that This would lead to an unconditional instability. To avoid such an instability, it can be required that [17] (B.9)

(B.2) It is thus fruitful to expand into its real and imaginary parts, as shown in (B.3) at the bottom of the page. From (B.2), the propagation is thus expressed as

where is a constant that can be specified by the user. Consequently, is expressed as (B.10) At high frequencies (where

(B.4)

) (B.11)

(B.3)

GEDNEY AND ZHAO: AN AUXILIARY DIFFERENTIAL EQUATION FORMULATION FOR THE CFS PML

At low frequencies (where

) (B.12)

If the conductivity is scaled, then near the interface boundary and , and where (B.13) Thus, should be chosen in a similar form to the CFS-PML to reduce reflection error off the front boundary interface. In summary, it has been shown in this Appendix that the double-pole PML provides broad band absorption of both evanescent and propagating waves, including at very low frequencies in a superior manner as compared to the standard single-pole CFS-PML formulation. REFERENCES [1] J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys., vol. 114, pp. 195–200, 1994. [2] S. D. Gedney, “An anisotropic perfectly matched layer-absorbing medium for the truncation of FDTD lattices,” IEEE Trans. Antennas Propag., vol. 44, pp. 1630–1639, Dec. 1996. [3] S. D. Gedney, “An anisotropic PML absorbing media for the FDTD simulation of fields in lossy and dispersive media,” Electromagnetics, vol. 16, pp. 399–415, 1996. [4] A. P. Zhao, “Application of the material-independent PML absorbers to the FDTD analysis of electromagnetic waves in nonlinear media,” Microw. Opt. Technol. Lett., vol. 17, pp. 164–168, 1998. [5] Z. S. Sacks, D. M. Kingsland, R. Lee, and J. F. Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Antennas Propag., vol. 43, pp. 1460–1463, 1995. [6] W. C. Chew and W. H. Weedon, “A 3D perfectly matched medium from modified Maxwells equations with stretched coordinates,” Microw. Opt. Technol. Lett., vol. 7, pp. 599–604, 1994. [7] S. D. Gedney, “Perfectly matched layer absorbing boundary conditions,” in Computational Electrodynamics: The Finite-Difference TimeDomain Method, A. Taflove and S. B. Hagness, Eds., 3rd ed. Boston: Artech House, 2005. [8] J. P. Berenger, “Numerical reflection from FDTD-PMLs: A comparison of the split PML with the unsplit and CFSPMLs,” IEEE Trans. Antennas Propag., vol. 50, pp. 258–265, Mar. 2002. [9] M. W. Chevalier and U. S. Inan, “A PML using a convolutional curl operator and a numerical reflection coefficient for general linear media,” IEEE Trans. Antennas Propag., vol. 52, pp. 1647–1657, Jul. 2004. [10] S. D. Gedney, “Scaled CFS-PML: It is more robust, more accurate, more efficient, and simple to implement. Why aren’t you using it?,” in Proc. IEEE Antennas Propag. Society Int. Symp., Washington, DC, 2005, pp. 364–367. [11] M. Kuzuoglu and R. Mittra, “Frequency dependence of the constitutive parameters of causal perfectly matched anisotropic absorbers,” IEEE Microw. Guided Wave Lett., vol. 6, pp. 447–449, 1996. [12] J. A. Roden and S. D. Gedney, “Convolutional PML (CPML): An efficient FDTD implementation of the CFS-PML for arbitrary media,” Microw. Opt. Technol. Lett., vol. 27, pp. 334–339, Dec. 5, 2000. [13] Q. H. Liu, “The PSTD algorithm: A time-domain method requiring only two cells per wavelength,” Microw. Opt. Technol. Lett., vol. 15, pp. 158–165, June 20, 1997. [14] S. Gedney, C. Luo, B. Guernsey, J. A. Roden, R. Crawford, and J. A. Miller, “The discontinuous Galerkin finite-element time-domain method (DGFETD): A high order, globally-explicit method for parallel computation,” presented at the IEEE Int. Symp. on Electromagnetic Compatibility, Honolulu, HI, 2007.

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[15] J. S. Hesthaven and T. Warburton, “High-order accurate methods for time-domain electromagnetics,” CMES-Comput. Modeling Eng. Sci., vol. 5, pp. 395–407, May 2004. [16] T. Xiao and Q. H. Liu, “Three-dimensional unstructured-grid discontinuous Galerkin method for Maxwell’s equations with well-posed perfectly matched layer,” Microw. Opt. Technol. Lett., vol. 46, pp. 459–463, Sept. 2005. [17] S. D. Gedney, C. Luo, J. A. Roden, R. D. Crawford, B. Guernsey, J. A. Miller, T. Kramer, and E. W. Lucas, “The discontinuous galerkin finite-element time-domain method solution of Maxwell’s equations,” Appl. Comput. Electromagn. Society J., vol. 24, pp. 129–142, Apr. 2009. [18] L. N. Wang and C. H. Liang, “A new implementation of CFS-PML for ADI-FDTD method,” Microw. Opt. Technol. Lett., vol. 48, pp. 1924–1928, Oct. 2006. [19] D. Correia and J. M. Jin, “On the development of a higher-order PML,” IEEE Trans. Antennas Propag., vol. 53, pp. 4157–4163, Dec. 2005. [20] D. Correia and J. M. Jin, “Performance of regular PML, CFS-PML, and second-order PML for waveguide problems,” Microw. Opt. Technol. Lett., vol. 48, pp. 2121–2126, Oct. 2006. [21] A. Taflove and S. B. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain, 3rd ed. Boston, MA: Artech House, 2005. [22] J. P. Webb, “Hierarchical vector basis functions of arbitrary order for triangular and tetrahedral finite elements,” IEEE Trans. Antennas Propag., vol. 47, pp. 1244–1253, 1999.

Stephen D. Gedney (F’04) received the B.Eng.Honors degree from McGill University, Montreal, QC, Canada, in 1985, and the M.S. and Ph.D. degrees in electrical engineering from the University of Illinois, Urbana-Champaign, in 1987 and 1991, respectively. He is currently a Professor with the Department of Electrical and Computer Engineering, University of Kentucky, Lexington, where he has been since 1991. From 1985 to 1987, he worked for the U.S. Army Corps of Engineers, Champaign, IL. In summers 1992 and 1993, he was a NASA/ASEE Faculty Fellow at the Jet Propulsion Laboratory, Pasadena, CA. In 1996, he was a Visiting Professor at the Hughes Research Labs (now HRL laboratories), Malibu, CA. His research is in the area of computational electromagnetics with focus in high-order solution techniques, fast solver technology, advanced time-domain methods, and parallel algorithms. His research has focused on applications in the areas of electromagnetic scattering and microwave circuit modeling and design. He has published over 150 articles in peer reviewed journals and conference proceedings and has contributed to a number of books in his field. Prof. Gedney received the Tau Beta Pi Outstanding Teacher Award in 1995. In 2002, he was named as the Reese Terry Professor of Electrical and Computer Engineering at the University of Kentucky. He is also a Fellow of the IEEE.

Bo Zhao (S’07) was born in the city of Shenyang, China, in July 1983. He received the B.S. degree from Zhejiang University, Hangzhou, China, in 2006 and the M.S. degree from the University of Kentucky, Lexington, in 2007. During his undergraduate studies, he was a member of the State Key Laboratory of Millimeter Waves at Southeast University where he performed research on metamaterial simulation and modeling under the guidance of Prof. Tie Jun Cui. Currently, he is a Graduate Research Assistant with the Electromagnetics Lab, University of Kentucky. His research interests include advanced techniques for electromagnetic simulation.

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Integral Equation Analysis of Scattering From Multilayered Periodic Array Using Equivalence Principle and Connection Scheme Fu-Gang Hu, Member, IEEE, and Jiming Song, Senior Member, IEEE

Abstract—An integral equation approach is presented to analyze the two-dimensional (2-D) scattering from multilayered periodic array. The proposed approach is capable of handling scattering from the array filled with different media in different layers. Combining the equivalence principle algorithm (EPA) and connection scheme (EPACS), it can be avoided to find and evaluate the multilayered periodic Green’s functions. For 2 identical layers, the elimination of the unknowns between top and bottom surfaces can be accelerated using the logarithm algorithm. More importantly, based on EPACS, an approach is proposed to effectively handle the semi-infinitely layered case in which a unit consisting of several layers is repeated infinitely in one direction. Index Terms—Domain decomposition, equivalence theorem, integral equations, method of moments, multilayered media, periodic structures.

I. INTRODUCTION

P

ERIODIC structures can provide a variety of applications in the area of electromagnetics, including microwave or quasi-optical filters, polarizers, radomes, and artificial dielectrics. Furthermore, the periodic structures provide novel characteristics if they are multilayered or cascaded. For instance, the two-dimensional photonic band-gap structure, which consists of multilayered periodic structure, finds the potential application to narrow-band filters, substrates for antennas, and so on. The cascaded periodic strip gratings can serve as a functional frequency-selective surface to provide varieties of filter characteristics. In the last decade, the electromagnetic scattering from periodic structures has been investigated extensively by various analytical and numerical methods. Among them, T-matrix method [1], [2], the method of moments (MoM) [3]–[6], and finite element method (FEM) [7]–[9] play important roles. Recently, the domain decomposition method (DDM) has attracted much attention from the society of computational electromagnetics because of its potential capability and advantages in solving electrically large problems with complex structures. It is natural to apply the DDM to differential equaManuscript received March 20, 2009; revised June 26, 2009. First published December 28, 2009; current version published March 03, 2010. This work was supported in part by the National Science Foundation CAREER Grant ECS0547161. The authors are with the Department of Electrical and Computer Engineering, Iowa State University, Ames, IA 50011 USA (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2009.2039313

tion methods, such as the finite element (FE) [10]–[12] and finite-difference (FD) methods [13]. In addition, DDM also can be employed via the integral equation (IE) method with the aid of the equivalence principle [14]–[18]. Specially, the integral equation method using periodic boundary condition and a connection scheme is used for modeling of multilayered lossy periodic structures in [3]. This approach can handle the case of the metallic patches at the interface between layers or on the periodic boundary. In this paper, an IE approach is developed to investigate the frequency response of a singly-periodic array. By the application of the equivalence principle algorithm and connection scheme (EPACS), the case of the periodic array filled with different media in different layers can be treated [3]. The computational domain first is restricted to one period of the multilayered array. In one period, each layer can be treated as an individual cell. Then, the equivalence principle can be applied separately to each individual cell to obtain the integral equations for equivalent currents on the outside boundary of the cell and the perfect electric conductor (PEC) surface. In general, when the cells are not overlapping or connected with each other, it is necessary to build up the relationship between them by applying IE to their outside boundary [17]. However, for the multilayered periodic structure, two neighboring cells are connected via the same interface. Thus, by combining the periodic boundary condition (PBC) with the connection scheme, the relationship can be established between the currents or fields on the topmost and bottommost surfaces. Finally, two integral equations on the topmost and bottommost surfaces, which involve the periodic Green’s function, are required to establish the complete equation system. After seeking the solution of equivalent currents on the topmost and bottommost surfaces, the proper way is proposed to correctly calculate the current on the surface of PEC object. Moreover, based on EPACS, an effective approach is proposed to handle the semi-infinitely layered case in which one unit consisting of several layers is repeated infinitely along one direction. Each unit can be regarded as a two-port network. For this semi-infinitely layered array, the impedance matrix representing the relationship between the equivalent magnetic and electric currents on the top surface of each unit should be identical because the network is infinitely extended when one looks into it from the top surface of any unit. Based on this fact, the equation for this impedance matrix can be established by EPACS and solved using a proper iterative method. After obtaining the impedance matrix, the integral equation on the top-

0018-926X/$26.00 © 2010 IEEE

HU AND SONG: INTEGRAL EQUATION ANALYSIS OF SCATTERING FROM MULTILAYERED PERIODIC ARRAY

Fig. 1. Multilayered infinitely periodic PEC array with a period of x-direction.

P

along

most surface is required to construct the complete system of equations for solving the fields or currents on the top surfaces. Then the reflection coefficients of Floquet’s harmonics can be found. It should be mentioned that the direct IE approach definitely cannot handle the case of semi-infinitely layered array since the number of unknowns for the direct IE approach will be infinite. Finally, numerical results are given to verify the proposed approaches. II. FORMULATION In this paper, a 2-D scattering problem is considered. The scatterers are assumed to be infinitely long along -direction. Fig. 1 shows the -layer infinite periodic PEC cylinder along -direction. This array is array, whose period is located in free space. It is filled with different media with and permeability in different layers relative permittivity . and can be complex numbers. The PEC objects are buried inside the media and their cross section may vary from layer to layer. The incident plane impinges on the array from free wave , where space. Its wave vector is given by , , , and are the incident angles.

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Fig. 2. Cells in each layer.

Fig. 3. Unknown field and current on the interior side of the outer surface C and inner surface C of Cell 1.

, ) must exist on . Hence, the electric ( field integral equations (EFIE) can be found

(1) where

A. Multilayered Array of PEC Because the array is infinitely periodic along -direction, one may consider just one cell in each layer shown in Fig. 2. In this work, the TMz case is considered. Without loss of generality, the integral equation is applied to Cell 1 shown in Fig. 3. Assume and are the electric field and current on the interior side includes , , , of outer surface , respectively. . includes , , and . is the electric and current on the inner surface (i.e., PEC surface), as shown in Fig. 3. A problem equivalent to the original problem internal to can be set up as follows. Let the original field exists internal with the original medium, and null field exists external to to with the same medium as the original one internal to . To support this field, the equivalent magnetic and electric currents

Because the relative permittivity and permeability in the Green’s function can be complex numbers, the present approach can handle the case where the media in each layer are lossy. Then, discretization of the integral equations yields (2) on

and (3)

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on ,

Assume the following relationship has been found between Layer 1 and Layer

. Assume the dimension of , , and to be , and , respectively. , , , , , and are matrices with the dimensions , , , , , and , on can be eliminated respectively. Then, the unknowns and the information on can be transferred to the outer surface . From (3), one can get

(11) where

(4) Substituting (4) into (2) yields (5)

Combining (9) and (11) and using the boundary condition of continuity of the tangential field (12)

where one can get the relationship between Layer 1 and Layer

(13) In order to obtain the relationship between the top and bottom surfaces, the unknowns on the left and right sides of the cell have to be eliminated by applying the periodic boundary condition

where see (14) at the bottom of the page. Thus, one can recurby sively find the relationship between Layer 1 and Layer using (14) (15)

(6) Through the matrix manipulation, one can obtain (7) where

can be rewritten as (8)

It should be noted that if each layer is identical, the logarithm algorithm can be applied to speed up the procedure for finding . For , the -time process of applying the contimes by replacing nection scheme can be reduced to that of with in (14). In addition to (15), two more conditions are required to solve the scattering problem. On the top surface of Layer 1, the following equation holds

The details for derivation of are given in Appendix. In a , one can obtain similar manner, for layer (9)

(16) where

where and

is found by replacing and with . If each layer is identical, it holds that , which indicates that one can save . CPU time for creating In the next step, the connection scheme will be applied to eliminate the unknowns between Layer 1 and Layer . From (7), one gets the initial relationship

since for the flat surface, can be dropped. is the periodic Green’s function in free space, and can be efficiently and accurately calculated by using Veysoglu’s transformation or the other transformations [4], [5]. Discretization of (16) yields

(10)

(17)

where

And on the bottom surface of Layer

, (18)

(14)

HU AND SONG: INTEGRAL EQUATION ANALYSIS OF SCATTERING FROM MULTILAYERED PERIODIC ARRAY

Discretization of (18) yields (19) Actually, (16) and (18) are obtained by setting up the problems equivalent to the original problem above the topmost surface and below the bottommost surface, respectively. Because there is no source exciting the incident field in the region below the bottom surface of Layer , no incident field contributes to the total field in (18). Combining (15), (17), and (19) yields

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other units. However, it is unnecessary for each layer to be the same in one unit. Each unit can be regarded as a two-port network and the semi-infinitely layered array is equivalent to the connection of infinite number of two-port networks. For this semi-infinitely layered array, the impedance matrix representing the relationship between the equivalent magnetic and electric currents on the top surface of each unit should be identical because the network is infinitely extended when one looks into it denote each of these from the top surface of any unit. Let impedance matrices. Thus, by using the tangential continuity condition one obtains

(20) (24)

where

On the other hand, as discussed in the above subsection, by using EPACS one gets (25) Moreover, the formulations are applicable to the case in which there is no PEC object inside the cell through just replacing (5) by (21) After solving (20), the other fields and currents of all the cells can be calculated. There are two ways to attempt this. The first and by using (7) since and have one is to get been found. Similarly one can obtain and layer by layer from the top to bottom with the aid of (12) and is extremely (9). Unfortunately, the condition number of large so that one cannot accurately calculate the inverse of which is required to solve (7) for and . Thus, one should abandon the first way and consider the other way. In fact, during the procedure of achieving (13), one can find (22) and where, see (23) at the bottom of the page. Because are known from (20), one can obtain recursively and from the bottom to top by using (22). For Cell 1, , , , are obtained through (39) (in Appendix) and (6). Then and one can find the current on the PEC by using (4). In the same manner, the current on the PEC of the other cells can be found. Numerical results will be shown to verify the second way.

can be obtained by making use of (14) or (10). where Using the second set of equations from both (25) and (24), one can get (26) Substitution of (26) into the first set of equations from (25) gives (27) Comparing (27) with the first set of equations from (24), one can achieve (28) Equation (28) is an equation for unknown matrix . It is impossible to find the explicit solution for . However, it can be solved by using the iterative procedure (29) is a matrix to be deterwhere is a relaxation factor, and . Combining (17) and the first mined, which satisfies set of (24) gives the complete system of equations (30)

B. Semi-Infinitely Layered Array of PEC layers is repeated Assume a unit consisting of infinitely along direction. Each unit must be identical to the

and , one can find and by After obtaining using (24) and (26). Following the procedure similar to that for

(23)

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multilayered array, , , , and are obtained and then can be found. In the same manner, one can find the current on PEC surface of the other cells starting from top and going downwards. Similarly, this approach is applicable to the semi-infinitely layered array without PEC objects inside each layer by means of (21). C. Floquet’s Harmonics of Scattered Fields The reflection coefficient of Floquet’s harmonics can be found after seeking the solution to the integral equation. In this subsection, the expression for the reflection coefficient will be given. The scattered field can be expressed as a superposition of Floquet’s harmonics [4]

Fig. 4. Power reflection coefficient of the zeroth Floquet’s harmonic of 1-layer : P , , and  . array of circular PEC cylinder. R

= 0 15

=1

=1

(31) . Here is the -coordinate of the where top surface of Cell 1. The Floquet’s harmonics are orthogonal to each other over one period along the direction. Assume the reference plane is located at . Thus, (32) On the other hand, surface of Cell 1

is given in terms of

and

on the top (33) Fig. 5. Power reflection coefficient of the zeroth Floquet’s harmonic of 4-layer , and  i ; ; . : P , array. R

= 0 15

From (32) and (33), one gets

(34) In terms of the definition of scattering matrix in [4], the reflecpropagating harmonic can be extion coefficient of the pressed by (35) When there are no PEC objects inside each layer, the scattering problem reduces to that of reflection of incident wave from multilayered media. For this case, only zeroth harmonics is the reflection coefficient of the incident wave exists, and from the planar boundary of infinitely extended media. and in (34) have been found through (20) and (30) for cases of multilayered and semi-infinitely layered array, respectively. III. NUMERICAL RESULTS A. Multilayered Periodic Array In the following examples, each cell is assumed to be square. In the first example, we consider a PEC object that is a circular . All layers are identical cylinder with the radius

=1

= 1 ( = 1 . . . 4)

with medium of free space. Here the point matching and pulse basis function are applied to obtain the numerical results. The TM wave is incident normally. The numerical results can be found alternatively by directly solving the EFIE applied on the PEC surface [4], which does not involve the equivalence principle and connection scheme. The direct IE approach employs the periodic Green’s function instead of the free-space Green’s function. The number of elements are 80 and 40 for outside boundary and the PEC surface, respectively. In the direct IE approach, 40 elements are applied to each PEC object surface. Figs. 4 and 5 show the power reflection coefficient of the zeroth Floquet’s harmonic for the one-layer and four-layer cases, respectively. The Fortran program, which is used to calculate the power reflection coefficient, is run on a PC with 3.2 GHz Pentium IV processor. The CPU time is about 1.5 seconds for each frequency point. There is a good agreement between results from the present approach, and the direct IE and the T-matrix approaches for the one-layer case. Also, the good agreement can be observed between results from the present approach and the direct IE approach for the four-layer case. Figs. 6 and 7 show the current distribution on PEC surface of and , respectively. the four-layer array at is the polar angle about the respective circle center. The results agree well with that of direct IE approach. As shown in Fig. 5,

HU AND SONG: INTEGRAL EQUATION ANALYSIS OF SCATTERING FROM MULTILAYERED PERIODIC ARRAY

Fig. 6. Current distribution on circular PEC cylinder of four-layer array. R :  . i ; ; . : P and P , and 

0 15

=04

=1

= 1 ( = 1 . . . 4)

853

=

Fig. 8. Reflection coefficient R of three-layer media.  = 1 0 j 0:2,  = 2 0 j 0:2,  = 4 0 j 0:2,  = 1,  = 2, and  = 1. (a) Real part. (b) Imaginary part.

Fig. 7. Current distribution on circular PEC cylinder of four-layer array. R :  . , and  i ; ; . : P and P

0 15

=09

=1

= 1 ( = 1 . . . 4)

=

most of energy is reflected at , and thus wave can hardly pass through the array. Therefore, the induced current on PEC decreases as the layer index increases. By contrast, the current does not decrease as the layer index increasing at since this frequency point is within the passband. The second example is the three-layer array filled with lossy medium but without PEC objects inside the cell. Fig. 8 shows the reflection coefficient of the zeroth Floquet’s harmonic

Fig. 9. Power reflection coefficient circular PEC array. R : P , , , and  j : ,

40 02

= 0 15 =1 =2

jR j of three-layer media and periodic = 1 0 j 0:2,  = 2 0 j 0:2,  = = 1.

for the three-layer array without PEC object inside. The medium , , in each layer is lossy. , , , and . The CPU time is about 2.0 seconds for each frequency point. There is a good agreement between results from the proposed approach and the anafor the same array with and lytical solutions. Fig. 9 shows without the circular PEC object of . Compared with

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Fig. 11. Average CPU time per frequency point versus number of layers. R j : and  . : P ,

0 15

Fig. 10. Reflection coefficient R of PEC array with 64 identical layers. R : P , j : and  . (a) Real part. (b) Imaginary part.

0 15

=20 02

=2

= 20 02

=

the array without PEC, the array with PEC has the larger reflection at lower frequencies. The PEC objects inside the array play important roles in changing the reflection at lower frequencies. Fig. 10 shows the reflection coefficient of a PEC array with and . 64 identical layers. Each layer has As shown in this figure, using logarithm algorithm leads to little compared to the scheme without the logarithm alchange in gorithm. Thus, it is stable to use the logarithm algorithm for the . Fig. 11 gives a whole picture for the average solution of CPU time per frequency point versus the number of layers. The CPU time for each connection is short because the number of unknowns for each cell is small. Hence, CPU time does not change too much as the number of layers increases for both schemes with and without the logarithm algorithm. However, for the three-dimensional (3-D) case, the number of unknowns for each cell will increase significantly. In this situation, the logarithm algorithm is expected to play an important role in reducing the CPU time when the number of layers is large. The integral equation is locally applied to each cell. When is large, the total CPU time is mainly for constructing ( is from 1 to ) since the part of CPU time for connection of matrix is negligible. Thus, the CPU time increases linearly with the number of layers. If each layer is identical (indicating ), the CPU time is just for constructing and so independent of . Thus, one can significantly save the CPU time. Fur-

Fig. 12. Convergence of

4 0 j 0:2 and  = 2.

=2

P

with and without circular PEC cylinder.

=

 =

thermore, the times of connection will be reduced from to . However, if the integral equation is applied directly to PEC objects, the computational complexity is without application of the fast algorithm. Thus, the present approach has computational advantages over the direct IE approach when is large. In addition, when is large, the memory requirement for the present approach is also much smaller than that for direct IE approach. B. Semi-Infinitely Layered Periodical Array Before a solution of reflection coefficient of semi-infinitely layered array can be found, it is interesting and necessary to investigate the convergence of . In the following example, the unit being repeated infinitely consists of one layer, namely, . The cell corresponding to each layer is square. The is located at the center of the PEC object with radius cell. The media inside each cell has and . The initial value of elements in is set to 1.0, and the relaxation . Fig. 12 shows a good convergence of factor is set to for . The relative error is defined as (36)

HU AND SONG: INTEGRAL EQUATION ANALYSIS OF SCATTERING FROM MULTILAYERED PERIODIC ARRAY

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the analytical solution. Finally, the reflection coefficient for the semi-infinitely layered array with PEC objects is calculated and compared with that for one-layer, two-layer, and four-layer circular PEC cylinder array. The average CPU time for each frequency point is about 2.2 and 2.4 seconds for the cases with and without PEC objects, respectively. The memory requirement is about 6 MB. Because the media are lossy, when the number of layer increases, the solution should converge to that of the semi-infinitely layered array. Fig. 14 shows the convergence of the reflection coefficient. When the array has four layers, the results are very close to that of the semi-infinitely layered array. IV. CONCLUSION Fig. 13. Reflection coefficient R of semi-infinitely layered array without PEC j : and  . object. 

= 40 02

=2

In this paper, a DDM approach, which is based on equivalence principle and connection scheme (EPACS), is developed to calculate the scattering from multilayered periodic arrays. This approach does not involve the multilayered periodic Green’s function. More importantly, based on EPACS, an effective approach is proposed to handle the case of semi-infinitely layered periodic arrays. The numerical results are provided to verify the proposed method. The efficiency of the proposed method is also demonstrated in this paper. This method can be extended readily to calculate the scattering from 3-D doubly periodic structures with multiple and semi-infinite layers. APPENDIX In what follows, the derivation of Rewriting (5) as

in (7) is given in detail.

and employing (6), one will obtain

(37)

Fig. 14. Reflection coefficient R of semi-infinitely layered and multilayered j : , and  . (a) array of circular PEC cylinder. R : P , Real part. (b) Imaginary part.

= 0 15

= 40 0 2

=2

It is worth pointing out that the convergence of may suffer from the resonance problem of integral equation if the media filled in the cells are lossless. The promising remedy is the application of combined field integral equation (CFIE) [19], which will be investigated in our future work. In addition, Fig. 13 shows the reflection coefficient of semiinfinitely layered array without PEC objects. There is a good agreement between the results from the proposed methods and

where and ( ,2,3,4). Reorganizing the last two sets of equations in (37) yields

(38) Thus,

(39)

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Reorganizing the first two sets of equations in (37) yields

(40) Combining (39) and (40), one can get (41) where

From (41) (42)

[11] Y.-J. Li and J.-M. Jin, “A new dual-primal domain decomposition approach for finite element simulation of 3-D large-scale electromagnetic problems,” IEEE Trans. Antennas Propag., vol. 55, pp. 2803–2810, Oct. 2007. [12] B. Stupfel and M. Mognot, “A domain decomposition method for the vector wave equation,” IEEE Trans. Antennas Propag., vol. 48, pp. 653–660, May 2000. [13] Y. Lu and C. Y. Shen, “A domain decomposition finite-difference method for parallel numerical implementation of time-dependent Maxwell’s equations,” IEEE Trans. Antennas Propag., vol. 45, pp. 556–562, Mar. 1997. [14] W. C. Chew and C. C. Lu, “The use of Huygens’ equivalence principle for solving the volume integral equation of scattering,” IEEE Trans. Antennas Propag., vol. 41, pp. 897–904, Jul. 1993. [15] S. Chakraborty and V. Jandhyala, “A surface equivalence-based method to enable rapid design and layout interations of coupled electromagnetic components in integrated packages,” in Proc. IEEE 13th Topical Meeting on Electrical Performance of Electronic Packaging, Portland, OR, 2004, pp. 45–48. [16] M.-K. Li and W. C. Chew, “Multiscale simulation of complex structures using equivalence principle algorithm with high-order field point sampling scheme,” IEEE Trans. Antennas Propag., vol. 56, pp. 2389–2397, Aug. 2008. [17] M.-K. Li and W. C. Chew, “Wave-field interaction with complex structures using equivalence principle algorithm,” IEEE Trans. Antennas Propag., vol. 55, pp. 130–138, Jan. 2007. [18] T.-M. Wang and H. Ling, “Electromagnetic scattering from three-dimensional cavities via a connection scheme,” IEEE Trans. Antennas Propag., vol. 39, pp. 1505–1513, Oct. 1991. [19] J. R. Mautz and R. F. Harrington, “ -field, -field, and combinedfield solutions for conducting body of revolution,” Arch. Elektr. Ubertragung, vol. 32, pp. 157–164, 1978.

H

E

where (43) REFERENCES [1] T. Kushta and K. Yasumoto, “Electromagnetic scattering from periodic arrays of two circular cylinders per unit cell,” in Progr. Electromagn. Res.. Philadelphia, PA: PIER, 2000, vol. 29, PIER, pp. 69–85. [2] K. Yasumoto, H. Toyama, and T. Kushta, “Accurate analysis of two-dimensional electromagentic scattering from multilayered periodic arrays of circular cylinders using lattices sums technique,” IEEE Trans. Antennas Propag., vol. 52, no. 10, pp. 2603–2611, Oct. 2004. [3] L. C. Trintinalia and H. Ling, “Integral equation modeling of multilayered doubly-periodic lossy structures using periodic boundary condition and a connection scheme,” IEEE Trans. Antennas Propag., vol. 52, no. 9, pp. 2253–2261, Sep. 2004. [4] A. F. Peterson, S. L. Ray, and R. Mittra, Computational Methods for Electromagnetics. Oxford, U.K.: Oxford Univ. Press, 1998. [5] L. Tsang, J. A. Kong, K.-H. Ding, and C. O. Ao, Scattering of Electromagnetic Waves. Numerical Simulations. New York: Wiley, c2001. [6] I. Stevanovic´, P. Crespo-Valero, K. Blagovic´, F. Bongard, and J. R. Mosig, “Integral-equation analysis of 3-D metallic objects arranged in 2-D lattices using the Ewald transformation,” IEEE Trans. Microw. Theory Tech., vol. 54, pp. 3688–3697, Oct. 2006. [7] L. E. R. Petersson and J. M. Jin, “A three-dimensional time-domain finite-element formulation for periodic structures,” IEEE Trans. Antennas Propag., vol. 54, no. 1, pp. 12–19, Jan. 2006. [8] T. F. Eibert, J. L. Volakis, D. R. Wilton, and D. R. Jackson, “Hybrid FE/BI modeling of 3-D doubly periodic structures utilizing triangular prismatic elements and an MPIE formulation accelereted by the Ewald transformation,” IEEE Trans. Antennas Propag., vol. 47, no. 5, pp. 843–850, May 1999. [9] G. Pelosi, A. Cocchi, and A. Monorchio, “A hybrid FEM-based procedure for the scattering from photonic crystals illuminated by a Gaussian beam,” IEEE Trans. Antennas Propag., vol. 48, no. 6, pp. 973–980, Jun. 2000. [10] S.-C. Lee, M. N. Vouvakis, and J.-F. Lee, “A non-overlapping domain decomposition method with non-matching grids for modeling large finite antenna arrays,” J. Comput. Phys., vol. 203, pp. 1–21, Feb. 2005.

Fu-Gang Hu (M’06) was born in 1977 in Jiangxi, China. He received the B.Eng. and M.Eng. degrees from Xidian University, Xi’an, China, in 1999 and 2002, respectively. He is currently working toward the Ph.D. degree at Iowa State University, Ames. He was an Associate Scientist with Temasek Laboratories, National University of Singapore, Singapore, from 2002 to 2007. His current research interest includes electromagnetic modeling using numerical techniques.

Jiming Song (S’92–M’95–SM’99) received the B.S. and M.S. degrees in physics, from Nanjing University, China, in 1983 and 1988, respectively, and the Ph.D. degree in electrical engineering from Michigan State University, in 1993. From 1993 to 2000, he worked as a Postdoctoral Research Associate, a Research Scientist and Visiting Assistant Professor at the University of Illinois at Urbana-Champaign. From 1996 to 2000, he worked as a Research Scientist at SAIC-DEMACO. He was the principal author of the Fast Illinois Solver Code (FISC), which has been distributed to more 400 government and industrial users. From 2000 to 2002, he was a Principal Staff Engineer/Scientist at Digital DNA Research Lab., Semiconductor Products Sector of Motorola, Tempe, AZ. In 2002, he joined Department of Electrical and Computer Engineering, Iowa State University, as an Assistant Professor and is currently an Associate Professor. His research has dealt with modeling and simulations of interconnects on lossy silicon and RF components, the wave scattering using fast algorithms, the wave propagation in metamaterials, and transient electromagnetic field. He has co-edited one book and published seven book chapters, 40 journal papers and 110 conference papers. Dr. Song received the NSF Career Award in 2006 and the Excellent Academic Award from Michigan State University in 1992 and was selected as a National Research Council/Air Force Summer Faculty Fellow in 2004 and 2005. Dr. Song is a senior member of IEEE.

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857

EFIE Analysis of Low-Frequency Problems With Loop-Star Decomposition and Calderón Multiplicative Preconditioner Su Yan, Student Member, IEEE, Jian-Ming Jin, Fellow, IEEE, and Zaiping Nie, Senior Member, IEEE

Abstract—Low-frequency electromagnetic problems are analyzed using the electric field integral equation (EFIE) with loop-star basis functions to alleviate the low-frequency breakdown problem. By constructing the loop-star basis functions with the curvilinear RWG (CRWG) basis and the Buffa-Christiansen (BC) basis, respectively, the recently proposed Calderón multiplicative preconditioner (CMP) is improved to become applicable at low frequencies. The Gram matrix arisen from CRWG loop-star basis and BC loop-star basis is studied in detail. A direct solution approach is introduced to solve the Gram matrix equation. The proposed Calderón preconditioner improves the condition of the EFIE operator at low frequencies, which results in a fast convergence of the preconditioned EFIE system. Several numerical examples demonstrate the fast and mesh-independent convergence of the preconditioned system. Index Terms—Buffa-Christiansen basis functions, Calderón multiplicative preconditioner, electric field integral equation (EFIE), loop-star decomposition, low-frequency problems, method of moments.

I. INTRODUCTION NALYSIS of low-frequency electromagnetic problems has received more attention recently because of its importance in the simulation of high-speed/high-frequency circuits. When the analysis is performed using an integral equation based method, the electric field integral equation (EFIE) is often preferred because of its excellent accuracy. However, the EFIE suffers from the so-called “low-frequency breakdown” due to the decoupling of the fields produced by electric currents and charges at very low frequencies. Furthermore, the EFIE operator has an undesired spectrum where the eigenvalues are clustered around the origin and at infinity as the mesh density

A

Manuscript received July 02, 2009; revised August 06, 2009. First published December 31, 2009; current version published March 03, 2010. This work was supported in part by the China Scholarship Council (CSC), the National Science Foundation of China (NSFC) under Contract 60728101, and in part by the 111 project under Contract B07046. S. Yan is with the Department of Microwave Engineering, University of Electronic Science and Technology of China, Chengdu, Sichuan, 610054, China and also with the Center for Computational Electromagnetics, Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801-2991 USA (e-mail: [email protected]). J. Jin is with the Center for Computational Electromagnetics, Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801-2991 USA (e-mail: [email protected]). Z. Nie is with the Department of Microwave Engineering, University of Electronic Science and Technology of China, Chengdu, Sichuan 610054, China (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2009.2039336

increases. Consequently, the EFIE matrix is highly ill-conditioned and hence is difficult to be solved accurately and efficiently. Unfortunately, a dense mesh is often unavoidable in the circuit analysis due to the need to discretize fine geometries. Over the past few years, much effort has been devoted to the solution of the EFIE at low frequencies. Following the Helmholtz decomposition, loop-tree and loop-star basis functions have been proposed to overcome the low-frequency breakdown problem [1]. However, the impedance matrices discretized from EFIE are still ill-conditioned. Therefore, robust preconditioners, either based on a basis rearrangement [2] or incomplete factorization [3], are always required to deal with these problems. Although these preconditioners succeed in some cases, they do not change the spectrum property of the integral equation operator. Recently, a Calderón multiplicative preconditioner (CMP) has been introduced to precondition the EFIE so that its eigenvalues could cluster at a certain point far away from the origin and infinity [4]. Different from the previous methods which are also based on the Calderón identity [5]–[8], the use of the Buffa-Christiansen (BC) basis functions [9] makes the CMP purely multiplicative, avoiding some complicated operator manipulations. In this paper, we employ the CMP to improve the condition of the EFIE matrix system discretized with a loop-star decomposition of curvilinear Rao-Wilton-Glison (CRWG) basis functions [10], [11] at low frequencies. The study of the loop-star decomposition of the BC basis functions paves a way to discretize the dual EFIE operator and yields a non-singular Gram matrix which links the domain and range of the EFIE operators. Unfortunately, although non-singular, the Gram matrix is not well conditioned due to the overlapping of the large definition domains of loop and star basis functions based on CRWG basis and BC basis, which leads to a significant increase in the non-zero off-diagonal entries. The poor condition of the Gram matrix makes it inefficient to solve with an iterative solver. Here we employ a direct sparse solver called UMFPACK [12] to solve this Gram matrix, whose analytical expression can be derived [13]. II. LOW-FREQUENCY BREAKDOWN OF EFIE Consider the problem of electromagnetic wave scattering by a conducting surface whose normal is denoted by . The scattered field is related to the incident field on the conducting surface by

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(1)

858

in which the operator

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is defined as

The matrices in (10) are given by (2)

(11)

where denotes the induced surface current, and denote the free-space wave number and impedance, and denotes the free-space Green’s function. Reformulating (1) into

(12)

(13) (3) and using a set of basis functions to expand the induced sur, where is the number face current density as the testing of degrees of freedom, and further using function to discretize (3), we obtain a matrix equation (4)

and the vectors and have the same expression as (7). Equation (10) can be scaled as (14) or more compactly as (15)

where (5) (6) (7) is the vector of unknowns. Note that and and are independent of frequency, and the amplitude of is the excitation vector in . , (4) degenerates to At DC limit, (8) At very low frequencies, , the matrix equation resulting , is singular. from the EFIE is very similar to (8). Since As a result, the matrix (4) is very difficult to solve. From a physical point of view, the induced current density can be decomposed into two components [2], an irrotational and a solenoidal component in . component in Although the surface current density is dominated by , the two components are equally important for the scattered field component due to the cancellation of the scattered field pro[14]. Therefore, if we rewrite duced by (9) where and , and expand and using a set of solenoidal basis functions and a set of non-solenoidal basis functions , respectively, EFIE (3) can be converted into and as the testing functions: the matrix equation using (10)

Throughout this paper, we will use to represent the impedance matrix after frequency scaling, and term it as the “normalized impedance matrix” since the magnitude of all elements in the are used to represent diagonal blocks are the same. and the unknown vector and the excitation vector after frequency normalization, respectively. With the treatment described above, the problem of low-frequency breakdown of EFIE has been overcome, and the matrix system (14) is now solvable. However, the block matrix is still very ill-conditioned, which compromises the condition of the entire matrix. When an iterative solver is used to solve (14), it converges very slowly, or even diverges in some cases. Therefore, it is critical to find an effective and robust preconditioner in order to solve (14) with a high efficiency and a good accuracy. III. CALDERÓN MULTIPLICATIVE PRECONDITIONER In recent decades, different kinds of preconditioners have been introduced to accelerate the convergence of the iterative solution of matrix equations. Many of them are very successful in some applications, such as the preconditioners based on the sparse approximate inverse (SAI) [15] and the incomplete LU factorization (ILU) [16]. In order to accelerate the convergence of EFIE at low frequencies, a basis rearrangement preconditioner [2], which is based on the observation that the electrostatic problem based on pulse basis converges rapidly, has been proposed when loop-tree or loop-star basis functions are employed. In addition, the preconditioner using the incomplete factorization with a heuristic drop strategy has also been proposed [3] to accelerate the convergence of EFIE discretized with loop-star basis functions. Although these preconditioners have achieved a certain success, they do not change the spectrum property of the EFIE operator. However, the spectrum property of EFIE can be improved by using the Calderón identity.

YAN et al.: EFIE ANALYSIS OF LOW-FREQUENCY PROBLEMS WITH LOOP-STAR DECOMPOSITION

A. Calderón Identity The Calderón identity is [4]–[8] (16) where

is the magnetic field integral equation (MFIE) operator (17)

and the

and

operators are defined as

859

From the above analysis, it can be observed that in order to in practice, three requirements need to be satisfied discretize [17]. and have to be divergence-conforming finite ele1) ment spaces in order to expand the unknown current denand ; sity of and have to be curl-conforming finite element 2) spaces in order to fully test and operator defined in (2); has to be non-singular in order to be 3) The Gram matrix inverted accurately and efficiently. B. CMP in Mid-Frequencies

(18) (19) It is suggested from (16) that is actually a second kind in. Therefore, tegral operator with the spectrum clustered at the EFIE operator can be used as a very effective preconditioner for itself, and this is called “self-regularizing property” of the EFIE operator. Based on the observation above, the solution of (1) can be obtained more efficiently by solving the preconditioned system: (20) is not Unfortunately, the discretization of the operator does not have a closed form. One straightforward since is to use the dual finite element space to way to discretize discretize the inner and outer operator

are very At mid-frequencies, CRWG basis functions widely chosen as the basis functions in , and can be used as the basis functions in . In order to satisfy the third requirement mentioned above, the Buffa-Christiansen [9] and can be used as (BC) vector functions the basis functions in and , respectively. Therefore, the at mid-frequencies can be discretized as operator (26) yielding the CMP matrix system (27) In (27), (28) is the impedance matrix obtained by using BC basis functions to discretize the outer operator,

(21)

(29)

and to denote the domain space of and Here we use , and to denote the range space of and , respectively. Apparently, we need to project an arbitrary field in into a field in . This can be realized by defining and inverting . For simplicity, we first define the inner the Gram matrix product

is the impedance matrix obtained by using CRWG basis funcoperator, and tions to discretize the inner

(22) (23) where and defined as

is the Gram matrix linking with . in (27) denotes the unknown vector reFurthermore, is the sulting from the CRWG expansion of , and . known excitation vector due to the incident field To make (30) easy to implement, the Gram matrix can be further written as (31)

are two vectors. Then the Gram matrix can be where the elements of (24)

In (24), and respectively. As a result, the

are the basis in and , operator can be discretized as (25)

where respectively.

(30)

and

are the basis in

and

,

are given by (32)

and and stand for the transformation matrices mapping the BC basis functions and the CRWG basis functions in the original mesh to the CRWG basis functions in the barycentric and [4]. All the matrices , , and mesh denoted by are sparse matrices which have only nonzero entries. If curvilinear triangular patches are used to discretize the surface

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of the object [18], [19], the Gram matrix is normalized due to the normalization of CRWG basis functions in terms of the interior edge length and hence has the same explicit expression in both uniformly and nonuniformly discretized cases. In order to solve the matrix (27), the matrix vector product need to be calculated. Since the Gram matrix is (MVP) very sparse and is well-conditioned, this MVP can be accomiteratively, in plished by solving the matrix equation only constant iteration counts. C. CMP at Low Frequencies doesn’t suffer from the low-frequency breakAlthough down as shown in (16), it will still fail at the low frequencies due to the numerical discretization error. As a result, neither CRWG basis functions nor BC basis functions can be used to expand the induced current density directly. Instead, the induced current density should be decomposed into a solenoidal and a non-solenoidal part as described in Section II. Practically, loop-tree and loop-star basis functions are commonly used to decompose the induced current density into a solenoidal and a non-solenoidal part, which will be discussed in detail in Section IV. Although the impedance matrix generated by loop-tree basis functions are reported to have a much better condition [20] and hence have a much faster convergence rate, the loop-star basis functions are preferred when we intend to use the Calderón preconditioner because the CRWG loop-star and BC loop-star basis functions constitute the dual finite element spaces. According to the definition of the BC basis, it is easy to see that (33)

where , , and are the same as (31), and and stand for the transformation matrices mapping the BC loop-star basis and the CRWG loop-star basis to the BC basis and the CRWG basis, are sparse respectively. All the matrices , , , , and nonzero entries. matrices which have only can be discretized in the same Theoretically, the operator way as in the mid-frequency case. However, since the overlapping of the large definition domains of the CRWG loop-star basis and the BC loop-star basis, the Gram matrix is no longer well-conditioned, but has many nonzero off-diagonal entries. Therefore, the straightforward use of the discretization expression (26) cannot guarantee the fast convergence anymore. In this paper, we use (36) instead, because this form has an eigenvalue distribution same and hence ensures a small condition number for as the discretized matrix. is the inverse of the transpose of the Gram maIn (36), . and are frequency normalized impedance trix matrices generated by using BC loop-star basis and CRWG loop-star basis, respectively (37) (38) As a result, the preconditioned EFIE (20) can be written in a matrix form as (39)

when three BC basis are associated with one triangle and oriented counterclockwise. This property is dual with respect to the star basis based on CRWG. As a result, when loop-star basis functions are used for discretizing both inner and outer operators, the Gram matrix that links and is well defined. and star In this paper, the CRWG based loop basis are used as the basis functions in , basis and are used as the basis functions in . Their dual and loop basis are used basis, the BC based star basis , and are used as the basis functions in . These choices result in the Gram as the basis functions in matrix as

denotes the normalized unknown vector resulting where is the from the CRWG loop-star discretization of , and . normalized excitation vector due to the incident field and Similar to the mid-frequency case, two MVPs need to be calculated in order to solve the preconditioned is matrix (39) at low frequencies. Unfortunately, although nonsingular, it is not well-conditioned because of the reason mentioned above. Therefore, it is not feasible to calculate these and two MVPs by solving the matrix equations iteratively. Instead, we use a direct solver to solve these two matrix equations, and this is described in detail in Section V. In Section IV, the loop-star decomposition based on both CRWG basis and BC basis are studied in order to have a better understanding of this low-frequency CMP. IV. LOOP-STAR DECOMPOSITION

(34) Similar to the mid-frequency CMP, the Gram matrix can be further written as (35)

The loop-tree and loop-star decompositions are commonly used to split the induced current density into a solenoidal part and a non-solenoidal part [1]–[3]. Although the impedance matrix obtained from the loop-tree basis has a better condition number, which is about one order of magnitude smaller than that of the impedance matrix obtained from the loop-star basis [20], the loop-star basis functions are preferred for discretizing

YAN et al.: EFIE ANALYSIS OF LOW-FREQUENCY PROBLEMS WITH LOOP-STAR DECOMPOSITION

the low-frequency CMP operator since the CRWG loop-star and BC loop-star basis functions can form the dual finite element subspaces. In this section, the CRWG based loop-star basis and BC based loop-star basis are studied. A. Loop-Star Decomposition Based on CRWG Basis The loop and star basis based on CRWG basis functions can be defined as the combination of the CRWG basis

861

B. Loop-Star Decomposition Based on BC Basis The Buffa-Christiansen basis functions [9] describe the current flows along an interior edge in the mesh grid. The loop and star basis functions using BC basis can be expressed as (46)

(47) (40) (41) In (40), is the th CRWG loop basis function, which is the summation of all the CRWG basis functions associated with denotes the th internal node on the surface of the scatterer. the number of edges (also the number of facets) associated with the th node. It is easy to point out from the definition (40) because the divergence of the that adjacent CRWG basis on the same patch cancels each other. In (41), is the th CRWG star basis function, which is the summation of all the CRWG basis functions associated with the th facet. Apparently, there are always three CRWG basis associated with one facet in closed cases. The CRWG loop basis function, which is used to expand the solenoidal component of the induced surface current, represents the current that flows around a certain point on the surface of an object. The CRWG star basis function is used to expand the current that flows out of a triangle. Although is not curl free, the star basis contains a subspace of the irrotational component of the induced surface current. This is called “incomplete Helmholtz decomposition.” Hence, (42) (43)

The BC loop basis function defined in (46), which expands the solenoidal component of the induced surface current, represents the current that flows around a certain facet on the surface after the reguof an object. It is easy to see that lation in (33). The BC star basis function represents the current is the same as in (40). Simthat flows out of a node. In (47), ilar to the CRWG loop-star basis, the BC loop-star basis are also the “incomplete Helmholtz decomposition” of the induced surface current density. 1) BC Loop Basis: Fig. 1 illustrates the detailed definition of a BC loop basis function. Fig. 1(a) shows the definition domain of the BC loop basis function. The bold dashed line (in blue) is the boundary of the definition domain of a BC loop basis. Three internal nodes are denoted by , , and . Three internal , , and edges are denoted by . This BC loop basis is the summation of the three BC basis functions which are associated with , , and . The ,2,3, barycentric triangles that have one vertex touching , , where is the number are denoted by , of such barycentric triangles and is the number of initial triangles associated with the node . In this figure, for example, , , and . The entire definition domain can . be expressed as The loop basis functions based on the BC basis can also be defined similarly to (44), with a much more complicated potential function . By examining the definition of the BC basis and BC loop basis carefully, we can rewrite the BC loop basis in terms of the potential functions

The loop basis function based on CRWG basis can also be defined by

(48)

(44)

(49)

where can be considered as a “potential function” which can control the amplitude and direction of the loop current. When curvilinear triangular patches are used to discretize the object’s surface, is defined as

(50)

(45)

(52)

in which is the th coordinate ( , 2, or 3) in the parametric coordinate system of the th triangular facet [11]. Apparently, is satisfied after the definition of (44), . since

There are four components in one BC loop basis, as shown , 2, 3, 4, their directions in Fig. 1(b). Denoted by , are shown in this figure. In (48), stands for the number of barycentric triangles that comprise the definition domain of the

(51)

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Fig. 1. Detailed definition of a loop basis based on BC basis. (a) The definition domain of a BC loop basis function. The bold dashed line (in blue) is the boundary fn ; n g, e of the definition domain of the BC loop basis. Three internal nodes are denoted by n , n , and n . Three internal edges are denoted by e fn ; n g. This BC loop basis is the summation of the three BC basis functions which are associated with e , e , and e . The barycentric fn ; n g, and e ,2,3, are denoted by T , j ; ; N , where N is the number of such barycentric triangles and N is triangles that have one vertex touching n , i ,N , and N . The entire definition domain can be expressed as the number of initial triangles associated with the node n . In this figure, N T . (b) The components of the BC loop basis function. There are four components in one BC loop basis. Denoted by f , p , 2, 3, 4, their directions are shown in this figure. The definition domain of f consists of the barycentric triangles T . The definition domain of f consists of

=

=



= 1 ... 2 =6 =6

=

the barycentric triangles

T

=1

=T

T

T

, where T

=T

=5

=1

. The definition domain of f consists of the barycentric triangles

. The definition domain of f consists of the entire domain



th current component. From the definitions (49)–(52), we can , , , and see that . The definition domain of consists of the barycen. The definition domain of contric triangles sists of the barycentric triangles

2

=

(53) The current component defined by (54) and flows in the clockwise direction if has an amplitude of and in the counterclockwise direction if . In (53), is used to adjust the distribution of to make sure that it is a continuous function at the entire definition domain of . 2) BC Star Basis: The detailed definition of BC star basis is shown in Fig. 2. The BC star current flows out from the internal and into the nodes , , where is the node number of internal edges associated with . In this figure, for example, . We use to denote the domain that consists of the barycentric triangles with one vertex touching , i.e., the domain inside the bold solid line (in red) in this figure. Similarly, , to denote the domain that consists of we use ,

, where

the barycentric triangles with one vertex touching , i.e., the domain inside the bold dashed line (in blue) in this figure. The . entire definition domain can be expressed as The BC star basis defined on has the property

, where

. The definition domain of consists of the , where . The barycentric triangles definition domain of consists of the entire domain . Generally, in (49)–(52), the potential functions are in a form of

T

.

(55) (56) In (55) and (56), stands for the Jacobian of the barycentric triangles, stands for the Jacobian of the initial triangles, and . In (56), stands for the number of internal edges . In Fig. 2 for example, associated with , , and . With the definitions (48)–(52), (55) and (56), we can finally , , derive the explicit expressions of the Gram matrix and (see [13] for detail). They are needed in the direct solution of the Gram matrix which is discussed in Section V. V. DIRECT SOLUTION OF THE GRAM MATRIX Linking the domain and range of the inner and outer EFIE operators, the Gram matrix is defined as the inner product of and the basis functions in . Therethe basis functions in fore, it is a highly sparse real-valued matrix with only nonzero entries. In order to carry out the MVP and in the preconditioned matrix (39), we need to solve the Gram matrix equations (57) (58)

YAN et al.: EFIE ANALYSIS OF LOW-FREQUENCY PROBLEMS WITH LOOP-STAR DECOMPOSITION

863

can be carried out as

Hence the MVP

(61)

Similarly, (58) can be further expressed as

(62)

Therefore, the MVP

can be carried out as

Fig. 2. Detailed definition of star basis based on BC basis. The BC star current ; ;N , flows out from the internal node n and into the nodes n , i where N is the number of internal edges associated with n . In this figure, . We use to denote the domain consists of the barycentric triangles N with one vertex touching n , i.e., the domain inside the bold solid line (in red) ,i ; ; N , to denote the domain consists of in this figure, and use the barycentric triangles with one vertex touching n , i.e., the domain inside the bold dashed line (in blue) in this figure. The entire definition domain can be . expressed as

= 1 ...

=6







=

= 1 ...



In the mid-frequency CMP, since the spectrum property of is very good, (57) and (58) can be solved iteratively, requiring only a constant number of iteration counts. However, at low freis no longer very well quencies, although not singular, the conditioned. As a result, an iterative solution of (57) and (58) is not practical anymore. In this paper, we employ a direct sparse solver called UMFPACK to solve this Gram matrix. UMFPACK is a package that uses the unsymmetric multifrontal method and direct sparse LU factorization to solve the unsymmetric sparse linear systems [12]. To this end, (57) is further expressed as

(59)

Since

(see [13] for details), we have

(63) From (61) and (63), it is clear that, instead of solving the original Gram matrix equations, we only need to solve two smaller matrix equations, which are obviously faster and memory efficient. The analytical expressions needed in the direct solution of the Gram matrix are given in [13]. Using these analytical expressions, the time requirement of constructing the Gram matrix . is only VI. NUMERICAL RESULTS In this section, the performance of the proposed preconditioner is demonstrated through several numerical examples. Using the method of moments [21], these computation are carried out and compared by using different basis functions with and without preconditioners: the loop-star basis without any preconditioner (LS), the loop-tree basis without any preconditioner (LT), the loop-tree basis with the basis rearrangement preconditioner (LT-BR) [2], and the loop-star basis with the proposed low-frequency CMP preconditioner (LS-CMP). A. Sphere

(60)

We first analyze the low-frequency scattering from a PEC sphere with a radius of 0.05 m. Fig. 3(a) displays the number of MVPs needed in an iterative solution in order to achieve a relative residual error (RSS) of , with respect to different mesh densities. In this comparison, the sphere is under the excitation of a 30-Hz and HH-polarized incident wave. The BiCGstab(1)

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for different mesh densities remains at 20, while the numbers needed by the LT-BR method increase rapidly as the mesh density increases. When the number of unknowns increases to 3528, it took LT-BR 436513 MVPs to achieve a RSS of 0.0014. The performances of LT basis and LS basis without using a preconditioner were even worse. It took the LT basis 629762 MVPs at the coarsest mesh with 288 unto achieve the RSS of knowns, while it took the LS basis 3 million MVPs to reach 0.0032. Obviously, the performance of the proposed preconditioner is much better than the other three. The preconditioned EFIE converges rapidly and independently with respect to mesh density, which is very important in a low-frequency analysis because a very dense mesh is always needed to have a good modeling of geometry details. Fig. 3(b) and (c) show the far-field RCS calculated by the proposed method. Fig. 3(b) shows the bistatic RCS at . The results obtained from LT-BR and LS-CMP 3 agree very well, and are practically identical to the Mie series solution. Fig. 3(c) shows the back-scattering RCS over a very to 300 MHz. The rewide frequency range, from 3 sults from the LS-CMP method and Mie series agree with each other very well. In fact, in mid-frequency cases, the accuracy of the CMP preconditioned EFIE is one order better than the EFIE without any preconditioner [18] because of the dramatic improvement of the condition of the preconditioned system. The same is also observed at low frequencies. B. Cone Sphere With a Gap A benchmark model, a cone sphere with a gap, is analyzed in this section. As shown in Fig. 4(a), this object is 0.689 m in length and orients in the direction. A 1.0 Hz incident wave is incident from the tip of the cone (the incident angle is and ), and is HH polarized. In order to discretize the sharp tip and the gap, a nonuniform mesh is used and 7509 unknowns are generated. Fig. 4(b) shows the convergence history for solving this problem. The iterative method used is GMRes(30) [23], [24]. As is clearly seen in this figure, the in 66 proposed LS-CMP method achieved the RSS of MVPs. However, the LS, LT, and LT-BR could only reach the RSS of 0.073, 0.020, and 0.00016, respectively, after 53000 MVPs. This example demonstrates the excellent convergence of the proposed preconditioner when dealing with a highly nonuniform mesh. Fig. 4(c) shows the real part of the current density induced on the surface of the scatterer in two different view angles. The current distribution is presented here because in a low-frequency analysis, it is usually more important to calculate and compare the current distribution than the far-field distribution. Fig. 3. Analysis of low-frequency scattering from a PEC sphere. The radius of this sphere is 0.05 m. (a) Number of MVPs needed at different mesh densities (different number of unknowns) to achieve a relative residual error of 10 , under the excitation of a 30 Hz, HH polarized incident wave. (b) Comparison of the bistatic RCS at 3 10 Hz. The RCS data are calculated by the loop-tree basis with the basis rearrangement technique and the loop-star basis using CMP as a preconditioner. Mie series is used as the reference data. (c) Backward scattering RCS over a very wide frequency range, from 3 10 Hz to 300 MHz.

2

2

[22], [23] is used as the iterative solver. From Fig. 3(a), it can be seen that the number of MVPs needed by the LS-CMP method

C. Bent Spring In this section, a more realistic object is considered. Shown in Fig. 5(a) is a spring which is bent at the center point. The total length of this bent spring is 2.98 cm, and the straight part lies in the direction. This object is discretized into 2181 degrees of freedom. A 1.0 Hz incident wave comes from the and and is HH polarized. GMRes(30) is angle again used as the iterative solver. Fig. 5(b) shows the convergence history with the targeted RSS setting to . The LS

YAN et al.: EFIE ANALYSIS OF LOW-FREQUENCY PROBLEMS WITH LOOP-STAR DECOMPOSITION

Fig. 4. Analysis of low-frequency scattering from a cone sphere with a gap. The total length of this object is 0.689 m. A 1.0 Hz plane wave is incident from the angle  and  and is HH polarized. (a) Problem description. (b) Convergence history. (c) Real part of the current density induced on the surface of the scatterer from two different view angles (in linear scale).

=0

=0

basis with the proposed low-frequency CMP preconditioner needed only 538 MVPs to converge, while the other three methods could not converge to the desired RSS even after 218100 MVPs, which is 100 times of the number of unknowns. The LS, LT, and LT-BR could only achieve the RSS of 0.0043, , respectively, at 218100 MVPs. The 0.00094, and 9.15 real part of the current density is shown in Fig. 5(c) from three different view angles. From the bottom view and the top view, we can see that the current distributions at the bottom side and the top side are almost the same. This is due to the diffraction of the low-frequency wave, and is different from the mid- and high-frequency cases, where the lit and shadow regions can be distinguished clearly.

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Fig. 5. Analysis of low-frequency scattering from a bent spring. The total length of this object is 2.98 cm. A 1.0 Hz plane wave is incident from the and  and is HH polarized. (a) Problem description. (b) angle  Convergence history. (c) Real part of the current density induced on the surface of the bent spring from three different view angles (in linear scale).

=0

=0

D. Computer Chip As the last example, the low-frequency behavior of a computer chip is analyzed. As shown in Fig. 6(a), the chip lies in the plane and has one corner cut. There are 11 legs at each side of the chip. The length of each side is 1.3 cm. Since the legs are very thin and the chip is flat, a nonuniform mesh has to be used. This object is discretized into 2564 triangular facets, leading to 3846 unknowns. Two different excitation cases are considered, and GMRes(30) is also used as the iterative solver to solve the preconditioned low-frequency EFIE system in both cases. 1) Plane Wave Excitation: In the first case, the chip is illuminated by a VV polarized, 50 Hz incident wave coming from the

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achieve the RSS of 0.0020, 0.00094, and 0.0032, respectively, after 384600 MVPs. The loop-star basis without using any preconditioner even began to diverge at the last part of iteration. It is interesting to see from this figure that the LT-BR method has a worse performance than the loop-tree basis without using any preconditioner, suggesting that the basis rearrangement technique may counteract in some cases and hence is not robust and reliable. 2) Voltage Source Excitation: In the second case, a 50 Hz 1 volt source is placed at the end of one leg. Fig. 6(c) shows . The the convergence history with the targeted RSS set to LS basis with the proposed low-frequency CMP preconditioner converged within 94 MVPs, whereas the other three methods could not converge to the desired RSS. The LS and LT stopped converging after 4 and 10 MVPs, only achieving the RSS of 0.70537 and 0.09717, respectively. The LT-BR reduced the RSS to 0.00207, after 384600 MVPs. VII. CONCLUSION

Fig. 6. Low-frequency analysis of a computer chip. There are 11 legs at each side of the chip. The total length of each side is 1.3 cm. Case 1: a 50 Hz plane wave is incident from the angle  and  and is VV polarized. Case 2: a 50 Hz 1 volt source is placed at the end of one leg. (a) Problem description. (b) Convergence history of the case 1. (c) Convergence history of the case 2.

= 45

= 45

A Calderón multiplicative preconditioner for the EFIE analysis of low-frequency problems based on the loop-star decomposition is presented in this paper. By applying the Calderón identity and the loop-star decomposition of both CRWG and BC basis functions, the preconditioned EFIE has an excellent spectrum property at low frequencies. With the aid of the proposed preconditioner, the EFIE converges at an arbitrarily low frequency rapidly and independently with respect to the mesh density, and is immune from the convergence deterioration caused by nonuniform meshes. The Gram matrix encountered in the implementation of the Calderón multiplicative preconditioner is investigated in detail and a direct solver is used for its solution. All the numerical examples demonstrate the excellent performance of the proposed method. Finally, we note that during the preparation of this manuscript, a paper has been published on a similar subject [25]. As an independent research effort, our work is different from [25] in the following aspects. First, our algorithm does not require the application of the frequency scaling to the Gram matrix. In fact, the frequency scaling may counteract the performance of the algorithm as it makes more difficult to solve the Gram matrix . Second, with the use of a direct sparse equation when algorithm to solve the Gram matrix, we have successfully overcome the conditioning problem of the Gram matrix obtained by using RWG loop-star basis and BC loop-star basis, a problem noted in [25] for future research. Third, this paper presented the very detailed definition of the BC loop basis, which can be used to construct the explicit expression of the Gram matrix. Finally, we have presented many examples to demonstrate the excellent performance of our algorithm and to support all our observations. REFERENCES

angle and . Fig. 6(b) shows the convergence . The LS basis with history with the targeted RSS set to the proposed low-frequency CMP preconditioner needed 120 MVPs to converge, whereas the other three methods could not converge to the desired RSS. The LS, LT, and LT-BR could only

[1] D. R. Wilton and A. W. Glisson, “On improving the electric field integral equation at low frequencies,” in Proc. URSI Radio Sci. Meet. Dig., Los Angeles, CA, Jun. 1981, pp. 24–24. [2] J. S. Zhao and W. C. Chew, “Integral equation solution of Maxwell’s equations from zero frequency to microwave frequencies,” IEEE Trans. Antennas Propag., vol. 48, pp. 1635–1645, Oct. 2000.

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[3] J. F. Lee, R. Lee, and R. J. Burkholder, “Loop star basis functions and a robust preconditioner for EFIE scattering problems,” IEEE Trans. Antennas Propag., vol. 51, pp. 1855–1863, Aug. 2003. [4] F. P. Andriulli, K. Cools, H. Ba˘gci, F. Olyslager, A. Buffa, S. Christiansen, and E. Michielssen, “A multiplicative Calderón preconditioner for the electric field integral equation,” IEEE Trans. Antennas Propag., vol. 56, pp. 2398–2412, Aug. 2008. [5] G. C. Hsiao and R. E. Kleinman, “Mathematical foundations for error estimation in numerical solutions of integral equations in electromagnetics,” IEEE Trans. Antennas Propag., vol. 45, pp. 316–328, Mar. 1997. [6] R. J. Adams, “Physical and analytical properties of a stabilized electric field integral equation,” IEEE Trans. Antennas Propag., vol. 52, pp. 362–372, Feb. 2004. [7] R. J. Adams and N. J. Champagne, “A numerical implementation of a modified form of the electric field integral equation,” IEEE Trans. Antennas Propag., vol. 52, pp. 2262–2266, Sep. 2004. [8] S. H. Christiansen and J. C. Nedelec, “A preconditioner for the electric field integral equation based on Calderón formulas,” SIAM J. Numer. Anal., vol. 40, no. 3, pp. 1100–1135, Aug. 2002. [9] A. Buffa and S. H. Christiansen, “A dual finite element complex on the barycentric refinement,” Mathematics of Computation, vol. 76, no. 260, pp. 1743–1769, Oct. 2007. [10] S. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag., vol. 30, pp. 409–418, May 1982. [11] R. D. Graglia, D. R. Wilton, and A. F. Peterson, “Higher order interpolatory vector bases for computational electromagnetics,” IEEE Trans. Antennas Propag., vol. 45, pp. 329–342, Mar. 1997. [12] T. A. Davis, UMFPACK: Unsymmetric Multifrontal Sparse LU Factorization Package 2007 [Online]. Available: http://www.cise.ufl.edu/research/sparse/umfpack/ [13] S. Yan, J.-M. Jin, and Z. Nie, “EFIE analysis of low-frequency problems with loop-star decomposition and Calderón multiplicative preconditioner,” University of Illinois at Urbana-Champaign, 2009, Tech. Rep. CCEML 1-09. [14] J. S. Zhao, W. C. Chew, T. J. Cui, and Y. H. Zhang, “Cancellations of surface loop basis functions,” in Proc. IEEE Antennas Propag. Symp., 2002, pp. 58–61. [15] G. Alléon, M. Benzi, and L. Giraud, “Sparse approximate inverse preconditioning for dense linear systems arising in computational electromagnetics,” Numer. Algorithms, vol. 16, pp. 1–15, 1997. [16] B. Carpentieri, I. S. Duff, and L. Giraud, “Experiments with sparse preconditioning of dense problems from electromagnetic applications,” CERFACS, Toulouse, France, 2000, Tech. Rep. TR/PA/00/04. [17] K. Cools, F. P. Andriulli, F. Olyslager, and E. Michielssen, “On Calderón techniques in time-domain integral equation solvers,” in Proc. Int. Conf. on Electromagnetics in Advanced Applications, Sep. 2007, pp. 360–363. [18] S. Yan, J.-M. Jin, and Z. Nie, “Implementation of the Calderón multiplicative preconditioner for the EFIE solution with curvilinear triangular patches,” in Proc. IEEE Antennas Propag. Symp., North Charleston, SC, Jun. 2009. [19] F. Valdés, F. P. Andriulli, K. Cools, and E. Michielssen, “High-order quasi-curl conforming functions for multiplicative Calderón preconditioning of the EFIE,” in Proc. IEEE Antennas Propag. Symp., North Charleston, SC, Jun. 2009. [20] T. F. Eibert, “Interative-solver convergence for loop-star and loop-tree decompositions in method-of-moments solutions of the electric-field integral equation,” IEEE Antennas Propag. Mag., vol. 46, no. 3, pp. 80–85, Jun. 2004. [21] R. F. Harrington, Field Computation by Moment Methods. New York: MacMillan, 1968. [22] G. L. G. Sleijpen and D. R. Fokkema, “BiCGstab(1) for linear equations involving unsymmetric matrices with complex spectrum,” Electro. Transa. Numer. Analys., vol. 1, pp. 11–32, Sep. 1993. [23] R. Barrett, M. Berry, T. F. Chan, J. Demmel, J. M. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and H. V. der Vorst, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods. Philadelphia, PA: SIAM, 1994. [24] Y. Saad and M. H. Schultz, “GMRes: A generalized minimal residual algorithm for solving nonsymmetric linear systems,” SIAM J. Sci. Stat. Comput., vol. 7, no. 3, pp. 856–869, Jul. 1986. [25] M. B. Stephanson and J. F. Lee, “Preconditioned electric field integral equation using Calderón identities and dual loop/star basis functions,” IEEE Trans. Antennas Propag., vol. 57, pp. 1274–1279, Apr. 2009.

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Su Yan (S’08) was born in Chengdu, China, in 1983. He received the B.S. degree in electromagnetics and microwave technology from the University of Electronic Science and Technology of China, Chengdu, in 2005, where he is currently working toward his Ph.D. degree. Since September 2008, he has been a Visiting Researcher in the Center for Computational Electromagnetics, Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, under the financial support from the China Scholarship Council. His research interests include numerical methods in computational electromagnetics, especially integral equation based methods and fast algorithms. Jian-Ming Jin (S’87–M’89–SM’94–F’01) received the B.S. and M.S. degrees in applied physics from Nanjing University, Nanjing, China, in 1982 and 1984, respectively, and the Ph.D. degree in electrical engineering from The University of Michigan at Ann Arbor, in 1989. He is currently a Professor of electrical and computer engineering and Director of the Electromagnetics Laboratory and Center for Computational Electromagnetics with the University of Illinois at Urbana-Champaign. He was appointed as the first Henry Magnuski Outstanding Young Scholar in the Department of Electrical and Computer Engineering in 1998 and later as a Sony Scholar in 2005. He was appointed as a Distinguished Visiting Professor in the Air Force Research Laboratory in 1999 and was an Adjunct, Visiting, or Guest Professor with the City University of Hong Kong, University of Hong Kong, Anhui University, Beijing Institute of Technology, Peking University, Southeast University, Nanjing University, and Shanghai Jiao Tong University. He has authored or coauthored over 190 papers in refereed journals and 20 book chapters. He has also authored the book The Finite Element Method in Electromagnetics (Wiley, 1st ed, 1993, 2nd ed, 2002) and Electromagnetic Analysis and Design in Magnetic Resonance Imaging (CRC, 1998), coauthored Computation of Special Functions (Wiley, 1996) and Finite Element Analysis of Antennas and Arrays (Wiley, 2008), and coedited Fast and Efficient Algorithms in Computational Electromagnetics (Artech, 2001). He has been an Associate Editor for Radio Science and is also on the Editorial Board for Electromagnetics and Microwave and Optical Technology Letters. His current research interests include computational electromagnetics, scattering and antenna analysis, electromagnetic compatibility, high-frequency circuit modeling and analysis, bioelectromagnetics, and magnetic resonance imaging. Dr. Jin is a member of Commission B of USNC/URSI and Tau Beta Pi. He was an Associate Editor of the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION. He was a recipient of the 1994 National Science Foundation Young Investigator Award, the 1995 Office of Naval Research Young Investigator Award, and the 1999 Applied Computational Electromagnetics Society Valued Service Award. He was also the recipient of the 1997 Xerox Junior Research Award and the 2000 Xerox Senior Research Award presented by the College of Engineering, University of Illinois at Urbana-Champaign. He was the Co-Chairman and Technical Program Chairman of the Annual Review of Progress in Applied Computational Electromagnetics Symposium in 1997 and 1998, respectively. His name often appears in the University of Illinois at Urbana-Champaign’s List of Excellent Instructors. He was elected by ISI as one of the World’s Most Cited Authors in 2002. Zaiping Nie (SM’96) was born in Xi’an, China, in 1946. He received the B.S. degree in radio engineering and the M.S. degree in electromagnetic field and microwave technology from the Chengdu Institute of Radio Engineering (now UESTC: University of Electronic Science and Technology of China), Chengdu, China, in 1968 and 1981, respectively. From 1987 to 1989, he was a Visiting Scholar with the Electromagnetics Laboratory, University of Illinois, Urbana. Currently, he is a Professor with the Department of Microwave Engineering, University of Electronic Science and Technology of China, Chengdu. He has published more than 300 journal papers. His research interests include antenna theory and techniques, fields and waves in inhomogeneous media, computational electromagnetics, electromagnetic scattering and inverse scattering, new techniques for antenna in mobile communications, transient electromagnetic theory and applications.

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A Newly Developed Formulation Suitable for Matrix Manipulation of Layered Medium Green’s Functions Jie L. Xiong and Weng Cho Chew, Fellow, IEEE Abstract—Based on a recently developed formulation of the dyadic Green’s function for layered media (DGLM), this work gives the matrix element representation of the DGLM for general RWG basis functions, including basis functions straddling across different layers and half-RWG basis function attached to an infinite ground plane. It has been rigorously proved that this representation is absent of any undesired line integrals, the same as the popular Michalski-Zheng’s formulation of type C. This work also gives the analytical solution for some typical cylindrically symmetric problems, which can be used to validate any formulation for layered medium problems. Index Terms—Green’s function, layered-medium (LM), method of moments (MoM).

I. INTRODUCTION

S

OMMERFELD first solved the problem of a vertical electric dipole on top of a half space in 1909 using Hertzian potentials [1]. In 1972, Kong used the components of the electromagnetic field instead of Hertzian potentials and extended its use to layered media [2]. The dyadic Green’s function for layered medium (DGLM) had been derived in various forms by different authors [3]–[13]. As we know, in the free space case, if the fields are expressed in terms of vector and scalar potentials, the kernel of the integral equation has a weaker singularity. The electric field integral equation (EFIE) based on vector and scalar potentials is called the mixed-potential integral equation (MPIE). It is even more important for layered media, because the spectral Sommerfeld integrals in the MPIE formulations converge more rapidly and save considerable computation resources. In the MPIE formulations, part of the coordinate-space differential operators is moved to the current using integration by parts. Since the differential operators in coordinate-space translate into higher spectral components in the Fourier space, the convergence of the Sommerfeld integrals would be more rapid after these operators being moved. The MPIE for arbitrarily shaped objects in a layered medium was first published in [14] and various forms of MPIE have been derived [15], [8], [16], Manuscript received April 02, 2009; revised August 02, 2009. First published December 31, 2009; current version published March 03, 2010. J. L. Xiong is with the Center for Computational Electromagntics and Electromagnetic Laboratory, Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801 USA and also with the Faculty of Engineering, University of Hong Kong, Hong Kong. W. C. Chew is with the Center for Computational Electromagntics and Electromagnetic Laboratory, Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801 USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2009.2039318

[17] later. Among them, the Michalski-Zheng formulation [8] is vastly popular and has been adopted by many workers1 [10], [18]–[24]. The formulation is suitable for analyzing objects of arbitrary shapes that penetrate the interface of two or more layers and the Sommerfeld integrals have better convergence properties. In the previous published work [25], we have used a totally different approach other than MPIE and arrived at a new form of DGLM. Nevertheless, the new form of DGLM implemented using method of moments (MoM) has similar advantages compared to the popular Michalski-Zheng’s formulation. It starts with the vector wave decomposition of the free space dyadic Green’s function, and is generalized for the layered medium. After exchanging the order of integrals and derivatives, the DGLM could be expressed in terms of two Sommerfeld integrals involving zeroth order Bessel function, one for TE waves and the other for TM waves. The new form of DGLM would be amenable for designing acceleration techniques. The matrix representation of the DGLM using MoM is manipulated so that the coordinate-space singularities in the integrand are as weak as possible. As a result, the associated Sommerfeld integrals have less high frequency components and converge rapidly. In this paper, we would generate the matrix element representation of this DGLM for basis functions that straddles across different layers and half basis functions that are attached to an infinite ground plane, so it is suitable for analyzing penetrating objects. We would rigorously prove that the undesired line integrals due to straddling basis functions would vanish in this formulation due to boundary conditions. In Section IV, we would give a brief comparison between this formulation and Michalski-Zheng formulation. In Section V, some typical cylindrically symmetric problems are solved analytically and the solutions would be used to validate our numerical results at Section VI. II. MATRIX REPRESENTATION FOR GENERAL BASIS FUNCTIONS In this section, we will give the final expression of the dyadic Green’s function which has been developed in [25]. It starts with a vector wave function decomposition of the dyadic Green’s function in free space [11], then is generalized for layered medium. The dyadic Green’s function for a layered medium can be derived in terms of generalized vector wave functions as

(1) 1This

list is by no means complete.

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where

. The expressions for and can be referred to as the scalar Green’s function for layered medium

(2) Fig. 1. (a) An RWG basis function in homogeneous medium. (b) An RWG basis function that straddles across layers.

(3) and consist of the primary (direct) Here, field terms and the secondary (reflected) field terms. Since it is more expedient to express the primary field in its closed form in coordinate space, we only consider the secondary field in this in the integrands of and paper. The pole at is pseudo, because the pole disappears when (2) and (3) are substituted in (1). In our earlier work [25], we have given the matrix element representation (4) of the dyadic Green’s function when both testing and source basis are embedded [Fig. 1(a)]. For implementation of (4), there are different options: one may want to evaluate the derivatives exactly by bringing them inside the spectral integrals of the scalar Green’s function. In this case, the Green’s functions sandwiched between the source basis and the test basis can be combined into one integral, and tabulated for efficiency. Quasi-static extractions should be used to accelerate the convergence of the Sommerfeld integral in the first term when the source basis function and the testing basis function reside in the same layer. Alternatively, one can compute and only. Their tabulate the two scalar Green’s functions, derivatives can be approximated by finite difference on tabulated values. This way of implementation would save both computational resource and memory for tabulation. In this paper, we have chosen the former approach for better accuracy

function. For example, if the th basis function straddles, we have

(5) As shown in Fig. 1(a), the RWG basis function is a vector basis function defined on a pair of triangles and , which are associated with a common edge [26]

(6) When moves on the common edge, we have the following identity: (7)

(4) In this paper, we will prove that this expression still holds for straddling basis [see Fig. 1(b)]. In addition, we will also discuss the case when half of the basis function is attached to an infinitely conducting ground. As shown in Fig. 1, straddling basis functions exist when the object penetrates two or more layers. For these basis functions, line integrals associated with the scalar potentials would occur in some formulations as Michalski has discussed [8]. The line integral appears when we use integration by parts to move the operator from the kernel to the basis function and testing

The above is zero for non-common edges. If an RWG basis function resides entirely in one layer, the line integrals on the positive patch and negative patch have the same magnitude but are of opposite signs, thus they cancel each other. But when a basis function straddles across layers like in Fig. 1(b), the expression on each patch is different. They will of the integrand is continuous with cancel each other only if the function respect to . Similarly, line integrals occur when we move the operator to the testing functions and they will cancel only if is continuous with respect to . Before starting the derivation, we first define an operator as

(8)

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And the scalar Green’s function can be simply written as

continuity condition of the propagation factor with respect to . The symmetry relation of the propagation factor [27] is given as (9)

1) Case 1: Only the Testing Basis Function Straddles: Here, we assume the positive patch of testing basis function lies in . The source basis function layer and the negative patch resides entirely in layer . There are two terms we need to consider. • • To study the properties of the line integral, the boundary conditions of the propagation factors are needed. From the ordinary differential equation the propagation factor satisfies, the across a discontinuity could boundary conditions for be derived ([11, p. 47]). Assuming the source is fixed in layer , and is right at the interface of layer and , we have

(14) Again, stands for for TM wave and for TE wave. After using this relation, the boundary condition (10) and (11) becomes

(15) (16) The two terms that contain •

operator are

• The line integral from the TM wave is proportional to

.

(10) (11) The contribution of the line integral from the TM wave is proportional to

(17) This term goes to zero when we use boundary condition (15). The contribution from the TE wave is proportional to

(12) is the common edge of th basis function and it lies at the . The integral vanishes when interface between layer and we substitute the boundary condition (10) into it. The TE part is proportional to

(13) From the boundary condition (11), we know is continuous with respect to at the interface and thus the integral also vanishes. 2) Case 2: Only the Source Basis Function Straddles: Next, we assume the positive patch of source basis function lies in . The testing basis function layer and the negative patch resides in layer . To study this case, we need to derive the

(18) Similarly, this is obtained by using (16). So far, we have shown that the line integral vanishes in this case too. 3) Case 3: Both the Testing Basis Function and Source Basis Function Straddle: The interaction of two RWG basis functions can always be split into the interactions of four triangle pairs. From the previous derivation, we know that all the line integrals eventually cancel each other. Thus, the expression of the matrix elements given by (4) is correct not only for the basis functions residing in a single layer, but for the straddling basis functions as well. This is because of the continuity property of the propagation factor at the interface of the layers. 4) Case 4: Basis Functions Attached to the Conducting Ground: When we model microstrip structures, the coaxial feed is often modeled as a metallic via connecting the patch and the ground plane. A delta voltage source on the via serves as the excitation. In the layered medium Green’s function approach, we would treat the dielectric substrate and the ground plane

XIONG AND CHEW: A NEWLY DEVELOPED FORMULATION SUITABLE FOR MATRIX MANIPULATION

Fig. 2. Current flow at the feed of a microstrip patch antenna.

as a layered medium, and only discretize the patch and the via. The effect of the ground plane is included in the DGLM. Since the via is physically attached to the ground plane (Fig. 2), the continuity condition of the current at the junction of the via and the ground plane should be satisfied. If we model the via with full RWG functions only, then the normal current is forced to zero at the boundary edges of the via and it would be an incorrect representation of the physics of the problem. To establish a correct model, we need to take into account the non-zero normal current at the edge of the via. The correct solution is to include half-RWG basis functions at the boundary of the via which is attached to the ground plane. Then normal current is allowed to flow from the via to the ground. In addition, after performing integration by parts, the line integrals associated with the half-RWG basis (5) need to be discarded. The reason is that each line integral has a negative image with respect to the perfect conducting ground plane, and they cancel each other as the line integrals associated with the half-RWG basis reside right on the ground plane. As a conclusion, when metallic objects are attached to the conducting ground plane, half-RWG basis functions should be constructed at the junction on the via and the associated line integrals can be discarded.

III. DISCUSSIONS Michalski-Zheng’s type C formulation of MPIE has been very popular in layered medium problems. So a brief comparison between Michalski-Zheng’s formulation and this newly developed formulation is given here. First, the derivation is different. After defining vector and scalar potentials, Michalski constructed the dyadic Green’s function and scalar Green’s function for each potential respectively. A correction term had been introduced to satisfy the continuity equation of current and charge. Overall, it was an equation-driven mathematical approach. On the other hand, the derivation presented in this paper is generated naturally from the vector wave decomposition of the free space dyadic Green’s function. The TM and TE waves are naturally decomposed. It is a physics-driven approach. Michalski has pointed out that the scalar potential kernels have different continuity properties in different mixed-potential formulations [8]. As a result, contour integrals exist in some formulations (Formulation B in [8]), while not in others (Formulation A and C). One key advantage of Michalski-Zheng’s of the formulation C is that the scalar potential kernel formulation is continuous at the interface with respect to both and , and this leads to the cancellation of the undesired

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line integrals at the interface [8], [28]. Similar to formulation C, the matrix element representation of this newly developed dyadic Green’s function enjoys the absence of line integrals when the object penetrates one or more interfaces, which has been proved in the last section. In this aspect, it is as convenient as Michalski-Zheng’s formulation C. Another point we would like to discuss is the convergent rate of the Sommerfeld integrals involved in this formulation. The reason that MPIE formulations are preferable is because they only involve the potential forms of the Green’s functions, which are less singular than their derivatives needed in the other variants of EFIE, and this leads to faster convergent Sommerfeld integrals in the layered medium case. This is the major advantage of MPIE over field formulations since Sommerfeld integrals are laborious to evaluate. If we examine the expressions of Sommerfeld integrals in [28], two of the integrands decay as and the third one decays as . In this paper, we work with the dyadic Green’s function instead of the vector and scalar potentials, so it is completely different from the MPIE formulation. We would like to investigate the convergence rate of the Sommerfeld integrals in this formulation. According to (4), there are five Sommerfeld integrals involved if the derivatives are brought inside the spectral integrals. The TM part of the integral in the first term conoperator enhances the verges most slowly because the order of the singularity. After quasi-static image extraction, the would remaining propagation factor decay on the order of since it goes to zero at . After taking all the factors into consideration, the integrand deoverall. Similarly, the integrands of the rest cays as as well. Thus, in terms of conof the terms decay as vergent rate of the Sommerfeld integrals, this new formulation is at least as good as the Michalski’s formulations. This is because we have used integration by parts as much as possible to manipulate the matrix element. Another advantage of this DGLM is that it is amenable for incorporating fast algorithms. We will discuss it in detail in our later works. IV. ANALYTICAL SOLUTION FOR SOME CYLINDRICALLY SYMMETRIC PROBLEM In order to validate the layered medium problem, it is desirable to have the analytical solution for some special cases. Here, we would solve some cylindrically symmetric problems analytically. Later, the solutions can be used to verify the numerical results. A. Current Loop Inside A Cylindrical Waveguide Assuming the electric current loop is of radius and located at inside a cylindrical waveguide as in Fig. 3(a), the field it generates inside the waveguide is a TE to field. We define a potential to be

(19)

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Fig. 3. (a) Geometry of a current loop inside a cylindrical waveguide. (b) Geometry of a current loop in the vicinity of an impenetrable cylinder. Fig. 4. (a) A cylinder straddles across two infinite half-space. (b) A cylinder straddles across layered medium within two infinitely large PEC plate.

The electric and magnetic field components of the TE field can be expressed by this potential as

(20)

(21) and parameters The unknown coefficients solved from the boundary conditions

could be

(22) (23) If we let and change the summation into an integral, we could get the radiated field generated by a current loop in free space

(24) If the media inside the cylinder is a layered medium, then we with . only need to replace B. Current Loop in the Vicinity of a Cylinder The geometry is shown in Fig. 3(b). First, we assume that there exists another cylindrical waveguide of radius . Later, we can let go to infinity, and convert the summation into an integration. We define the potential as [29] (25) The associated TE to

field is

Fig. 5. The field component E along z and  directions in the geometry shown in Fig. 4(a). (TE case). (a) Near field along ^, (b) Near field along z^.

The unknown parameters boundary conditions

and

can be solved from the

(27) (26)

(28)

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Fig. 7. Cylindrical cavity resides in layered medium.

If we would like to generate TM to field, we can replace the electric current loop with a magnetic current loop. After similar derivation, the magnetic field component is given by

(31) in this case. If the curwhere rent loop and the cylinder is located in a layered medium, with or we only need to replace . V. NUMERICAL RESULTS In this section, some numerical results are presented and compared with either analytical results or results from other formulation. The good agreement between them shows the validity of the new formulation of layered medium Green’s function. A. A Cylinder Straddles Across Two Half-Space

Fig. 6. The field component E along z and  directions in the geometry shown in Fig. 4(b). (TE case). (a) Near field along ^, (b) Near field along z^.

And the unknown coefficients current continuity condition

can be solved from the

(29) , When , where the expression of the electric field becomes an integral from the asymptotic analysis [29]

(30)

In this part, two examples are constructed and compared with the analytical results from Section IV. The first one [see Fig. 4(a)] is a conducting cylinder straddling across two lossy half-space, excited by an electric current loop. The radius of the cylinder is 0.1 m and its height is 1.2 m. Half of it is in and the other half is buried the upper space in the lower space . The radius of the current loop is 0.2 m and it is 0.3 m above the interface of the two space. In the analytical model, the cylinder is infinitely long. Here we choose two highly lossy half-spaces so that we can use a truncated cylinder to approximate the infinitely long one. The near field distributions along both and directions obtained from numerical methods agree well with the analytical results (Fig. 5). It validates the matrix expression of this new formulation for straddling objects. The second example is similar to the first one, except that we have perfect electric conductor (PEC) both as the ground and the ceiling in the structure. Here, the conducting cylinder is terminated by both the ground and the ceiling [Fig. 4(b)], and the model used in numerical analysis is the same as the one used in theoretical analysis in Fig. 3(b). The medium between the two and a lossless dielecPEC plate is an air substrate tric substrate . The thickness of both substrates is 1 m. The cylinder has a radius of 0.1 m and a height of 2 m. The radius of the current loop is 0.2 m and it is 0.3 m above the interface of the two substrates. The near field distributions obtained from the numerical method again agree well with the analytical results (Fig. 6). This example proves the validity of treating the boundary basis functions as half RWG basis functions when objects are connected to the ground or the ceiling.

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VI. CONCLUSION In this paper, we have provided some important supplements for the MoM implementation of a newly developed dyadic Green’s function for layered medium, i.e., the treatment of the straddling basis functions and half-RWG basis functions attached to ground planes. The undesired line integrals associated with straddling basis functions have been proven to disappear. Although its derivation is totally different from the MPIE approach, the final Sommerfeld integrals in the matrix element representations have nice convergence property similar to MPIE. We also call it a matrix-friendly formulation. Some cylindrically symmetric problems are solved analytically and serve as benchmarks and validation for general layered media problems. REFERENCES

Fig. 8. The field component E along z and  directions inside a cylindrical cavity. (a) Near field along ^, (b) Near field along z^.

From the current distribution, we found that the current is not terminated at the top and bottom edge of the cylinder. B. A Cylindrical Cavity in Layered Medium In this example, the electric current loop is placed inside a cylindrical cavity (Fig. 7). It could be analyzed by the model of a current loop inside a cylindrical waveguide with PEC ground and ceiling [Fig. 3(a)]. In our numerical model, only the top and the wall of the cavity are discretized. The top part lies on top of the dielectric substrate. The wall penetrates the substrate and is attached to the ground plane. The thickness of the substrate is 0.5 m and its relative permittivity is 2.6. The height of the cavity is 0.5 m and its radius is 2.5 m. The radius of the current loop is 2.0 m and it is 0.25 m above the ground plane. This structure contains both horizontal and vertical structures which straddle across two layers, thus it serves as a good test case. The near field distributions obtained from the numerical method agree well with the analytical results (Fig. 8) in this case.

[1] A. Sommerfeld, “Uber die Fortpflanzung des Lichtes in dispergierenden Medien,” Ann. Physik, vol. 28, pp. 665–737, 1909. [2] J. A. Kong, “Electromagnetic fields due to dipole antennas over stratified anisotropic media,” Geophys., vol. 37, p. 985, 1972. [3] E. Arbel and L. B. Felsen, “Theory of radiation from sources in anisotropic media, part 1: General sources in stratified media,” Electromagn. Theory Antennas, vol. 1, pt. 1, pp. 391–420, 1963. [4] V. G. Daniele and R. S. Zich, “Radiation by arbitrary sources in anisotropic stratified media,” Radio Sci., vol. 8, pp. 63–70, Jan. 1973. [5] S. Ali and S. Mahmoud, “Electromagnetic fields of buried sources in stratified anisotropic media,” IEEE Trans. Antennas Propag., vol. AP-27, no. 5, pp. 671–678, Sep. 1979. [6] J. K. Lee and J. A. Kong, “Dyadic Green’s functions for layered anisotropic medium,” Electromagnetics, vol. 3, no. 2, pp. 111–130, 1983. [7] L. Tsang, J. A. Kong, and R. T. Shin, Theory of Microwave Remote Sensing. New York: Wiley Interscience, 1985. [8] K. A. Michalski and D. Zheng, “Electromagnetic scattering and radiation by surface of arbitrary shape in layered media-Part I: Theory,” IEEE Trans. Antennas Propag., vol. 38, no. 3, pp. 335–344, Mar. 1990. [9] A. Dreher, “A new approach to dyadic Green’s function in spectral domain,” IEEE Trans. Antennas Propag., vol. 43, no. 11, pp. 1297–1302, Nov. 1995. [10] G. Dural and M. I. Aksun, “Closed-form Green’s functions for general sources and stratified media,” IEEE Trans. Microw. Theory Tech., vol. 43, pp. 1545–1552, Jul. 1995. [11] W. C. Chew, Waves and Fields in Inhomogeneous Media, ser. IEEE Electromagnetic Series, 2nd ed. New York: IEEE Press, 1995. [12] K. Michalski and J. Mosig, “Multilayered media Green’s functions in integral equation formulations,” IEEE Trans. Antennas Propag., vol. 45, no. 3, pp. 508–519, Mar. 1997. [13] W. C. Chew, J. S. Zhao, and T. J. Cui, “The layered medium Green’s function—A new look,” Microw. Opt. Technol. Lett., vol. 31, no. 4, pp. 252–255, 2001. [14] K. A. Michalski, “The mixed-potential electric field integral equation for objects in layered media,” Archiv Fuer Elektronik und Uebertragungstechnik, vol. 39, pp. 317–322, Sep./Oct. 1985. [15] L. Barlately, J. R. Mosig, and T. Sphicopoulos, “Analysis of stacked microstrip patches with a mixed potential integral equation,” IEEE Trans. Antennas Propag., vol. 38, no. 5, pp. 608–615, May 1990. [16] G. A. E. Vandenbosch and A. R. Van De Capelle, “Mixed-potential integral expression formulation of the electricfield in a stratified dielectric medium-application to the case of a probe current source,” IEEE Trans. Antennas Propag., vol. 40, no. 7, pp. 806–817, July 1992. [17] J. Sercu, N. Fache, F. Libbrecht, and P. Lagasse, “Mixed potential integral equation technique for hybridmicrostrip-slotline multilayered circuits using a mixed rectangular-triangular mesh,” IEEE Trans. Microw. Theory Tech., vol. 43, no. 5, pp. 1162–1172, May 1995. [18] Y. L. Chow, N. Hojjat, and S. Safavi-Naeini, “Spectral Green’s functions for multilayer media in a convenient computational form,” IEE Proc.-H., vol. 145, pp. 85–91, Feb. 1998. [19] K. A. Michalski, “Extrapolation methods for Sommerfeld integral tails (invited review paper),” IEEE Trans. Antennas Propag., vol. 46, pp. 1405–1418, Oct. 1998.

XIONG AND CHEW: A NEWLY DEVELOPED FORMULATION SUITABLE FOR MATRIX MANIPULATION

[20] W. Cai and T. Yu, “Fast calculation of dyadic Green’s functions for electromagnetic scattering in a multi-layered medium,” J. Comput. Phys., vol. 165, pp. 1–21, 2000. [21] F. Ling and J. M. Jin, “Discrete complex image method for Green’s functions of general multilayer media,” IEEE Microw. Guided Wave Lett., vol. 10, no. 10, pp. 400–402, 2000. [22] J. Y. Chen, A. A. Kishk, and A. W. Glisson, “Application of new MPIE formulation to the analysis of a dielectric resonator embedded in a multilayered medium coupled to a microstrip circuit,” IEEE Trans. Microw. Theory Tech., vol. 49, pp. 263–279, Feb. 2001. [23] T. M. Grzegorczyk and J. R. Mosig, “Full-wave analysis of antennas containing horizontal and vertical metallizations embedded in planar multilayered media,” IEEE Trans. Antennas Propag., vol. 51, no. 11, pp. 3047–3054, Nov. 2003. [24] E. Simsek, Q. H. Liu, and B. Wei, “Singularity subtraction for evaluation of Green’s functions for multilayer media,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 1, pp. 216–225, Jan. 2006. [25] W. C. Chew, J. L. Xiong, and M. A. Saville, “A matrix friendly formulation of layered medium Green’s function,” IEEE Antennas Wireless Propag. Lett., vol. 5, no. 1, pp. 490–494, Dec. 2006. [26] S. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by surface of arbitrary shape,” IEEE Trans. Antennas Propag., vol. 30, no. 3, pp. 409–418, May 1982. [27] W. C. Chew and S. Y. Chen, “Response of a point source embedded in a layered medium,” IEEE Antennas Wireless Propag. Lett., vol. 2, no. 14, pp. 254–258, 2003. [28] K. A. Michalski and D. Zheng, “Electromagnetic scattering and radiation by surface of arbitrary shape in layered media-Part II: Implementation and results for contiguous half-spaces,” IEEE Trans. Antennas Propag., vol. 38, no. 3, pp. 345–352, Mar. 1990. [29] W. C. Chew and B. Anderson, “Propagation of electromagnetic waves through geological beds in a geophysical probing environment,” Radio Science, vol. 20, no. 3, pp. 611–621, May–Jun. 1985. Jie L. Xiong was born in Hubei, China, in 1981. She received the B.S. degree from Tsinghua University, Beijing, China, in 2002 and the M.S. degree from the University of Massachusetts, Amherst, in 2004. She is currently working toward the Ph.D. degree at the University of Illinois at Urbana-Champaign. Currently, she is also a Research Associate with Professor Chew, in the Faculty of Engineering, University of Hong Kong. Her research interests include the fast algorithms for layered medium Green’s function and the numerical methods for the evaluation of the Casimir force.

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Weng Cho Chew (S’79–M’80–SM’86–F’93) was born in Malaysia, on June 9, 1953. He received the B.E., M.S. Engineer, and Ph.D. degrees, all in electrical engineering, from the Massachusetts Institute of Technology, Cambridge, in 1976, 1978, and 1980, respectively. Previously, he was a Department Manager and a Program Leader at Schlumberger-Doll Research, Ridgefield, CT. Currently, he is a Professor at the University of Illinois at Urbana-Champaign, where he is also the Director of the Center for Computational Electromagnetics and Electromagnetics Laboratory (CCEML). His research interest is in the areas of waves in inhomogeneous media for various sensing applications, integrated circuits, microstrip antenna applications, and fast algorithms for solving wave scattering and radiation problems. He is the originator of several fast algorithms for solving electromagnetic scattering and inverse problems. He has led a research group that has developed parallel codes solving tens of millions of unknowns for integral equations of scattering. He authored Waves and Fields in Inhomogeneous Media (New York: Van Nostrand Reinhold, 1990; reprint by Piscataway, NJ: IEEE Press, 1995), coauthored Fast and Efficient Methods in Computational Electromagnetics (Norwood, MA: Artech House, 2001), and authored and coauthored over 300 journal publications, over 400 conference publications, and over ten book chapters. Dr. Chew is an OSA Fellow, an IOP Fellow, and was an NSF Presidential Young Investigator. He received the Schelkunoff Best Paper Award from the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, the IEEE Graduate Teaching Award, Campus Wide Teaching Award, and was a Founder Professor of the College of Engineering, and currently, a Y. T. Lo Endowed Chair Professor in the Department of Electrical and Computer Engineering at the University of Illinois. Since 2005, he serves as an IEEE Distinguished Lecturer. Recently, ISI Citation elected him to the category of Most-Highly Cited Authors (top 0.5%). He served on the IEEE Adcom for Antennas and Propagation Society as well as Geoscience and Remote Sensing Society. He has been active with various journals and societies.

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Analysis of Frequency Selective Surfaces on Periodic Substrates Using Entire Domain Basis Functions Alireza Yahaghi, Arya Fallahi, Habibollah Abiri, Mahmoud Shahabadi, Christian Hafner, and Rüdiger Vahldieck, Fellow, IEEE

Abstract—The paper presents an efficient procedure to utilize the existing method of moments/boundary integral-resonant mode expansion approach for the analysis of frequency selective surfaces printed on periodic and inhomogeneous substrates. The Green’s function matrix of the structure is obtained using a multiconductor transmission line model. The resulting series equations are solved by using the periodic method of moments with entire domain functions as both basis and test functions. The required entire domain functions are extracted using boundary integral-resonant mode expansion. The Fourier coefficients of the basis functions are evaluated from the coupling integrals by applying a coordinate transformation. Some examples are analyzed and compared to measurement data. To illustrate the excellent convergence of the method, a comparison between using entire domain and sub-domain functions is made. It is shown that by using the proposed approach not only the number of required basis functions reduces but also lower truncation orders in the Fourier expansions need to be considered. This leads to a drastic reduction of the computation time. Index Terms—Boundary integral resonant mode expansion, entire domain basis functions, frequency selective surfaces, method of moments, multiconductor transmission lines method, periodic structures, spectral domain analysis.

I. INTRODUCTION INCE the introduction of frequency selective surfaces (FSSs) in the late 1960s, there has been a remarkable effort in the study of FSS performances in electromagnetic systems. In these studies, this group of metamaterials is investigated from the viewpoint of theory, measurement, and fabrication [1], [2]. FSSs found first a variety of applications in microwave systems such as in reflector antennas [3], [4] antenna radomes [5], [6] and polarizers [7]. With the development of the concept of metamaterials in the recent years [8], FSS were also considered as a kind of planar metamaterials

S

Manuscript received April 21, 2009; revised July 04, 2009. First published December 31, 2009; current version published March 03, 2010. This work was supported by the ETH Zürich. A. Yahaghi* is with the Laboratory for Electromagnetic Fields and Microwave Electronics, ETH Zürich, Zürich CH-8092, Switzerland on leave from the Department of Electrical Engineering, Shiraz University, Shiraz, Iran (e-mail: [email protected]). A. Fallahi*, C. Hafner, and R. Vahldieck are with the Laboratory for Electromagnetic Fields and Microwave Electronics, ETH Zürich, Zürich CH-8092, Switzerland (e-mail: [email protected]). H. Abiri is with the Department of Electrical Engineering, Shiraz University, Shiraz, Iran. M. Shahabadi is with the School of Electrical and Computer Engineering, Faculty of Engineering, University of Tehran, Tehran 14395-15, Iran. *The authors equally contributed. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2009.2039327

to realize some fascinating properties. New applications like planar absorbers [9], [10] and artificial magnetic conductors (AMC) [11] were introduced. The progress of FSS structures caused the need for efficiently simulating and designing such structures. There has been an extensive research and consequently a large number of publications on the FSS analysis. Different methods such as periodic method of moments (PMoM) [1], finite difference time domain (FDTD) [12], finite element method (FEM) [13] were applied to simulate them. Various schemes were introduced to design FSS for specific applications [2]. In addition, some studies have been also done on the optimization of FSS [11], [14]. In order to engineer the electromagnetic properties of an FSS, there are two main possibilities, namely designing the shape of the patches and using substrates with particular properties. To exploit the first opportunity, the analysis method should be able to analyze an FSS with an arbitrary shape for the patches. Concerning the second one, multilayered substrates are widely utilized. In addition, a few reports were published on using special materials like ferrite [15], liquid [16] and chiral [17] media as the substrate. However, these substrates are rarely used because of manufacturing difficulties. Recently another idea for engineering substrates based on introducing periodic inhomogeneity was proposed [18], [19] which is easily manufacturable. For example, drilling holes in the substrate and/or filling them with other materials changes the effective permittivity or permeability of the substrate. Furthermore a periodic substrate itself is a resonating structure, therefore this kind of substrates could be used in applications, which are based on resonance effects [19]. In [18] method of moment/transmission line (MoM/TL) procedure is introduced for the analysis of FSS with so-called periodic substrates. The method took advantage of sub-domain rooftop basis and test functions to expand the excited currents on the patches. Applying this kind of basis functions has two main drawbacks: First exact modeling of patches with curved boundaries is not possible, Second to have a reasonable model for complicated patches with or without curved boundaries, one needs to use very fine meshes. Since in PMoM we deal with Fourier expansions of the basis functions, having very fine meshes necessitates a very large set of Fourier coefficients. This issue does not cause remarkable problem in analyzing FSS with homogenous substrates. However, analysis of periodic substrates by the multiconductor transmission line method includes treating some matrices whose dimensions are directly determined by the number of Fourier components. Therefore, having very fine meshes strongly increases the computation cost of the method. In some cases, obtaining an acceptable

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determining the Green’s function matrix. The detailed explanations of each part is found in the literature. However, for better understanding of the procedure, the main steps in each part are summarized in the following. A. Periodic Method of Moments (PMoM)

Fig. 1. A typical geometry of a multilayered FSS with periodic substrate.

result even with investing considerable time and memory is impossible. These difficulties can be overcome by implementing entiredomain basis functions. Recently, a method of moments/ boundary integral-resonant mode expansion (MoM/BI-RME) approach was introduced to analyze FSS with arbitrary unit cell configurations based on entire-domain basis and test functions [20], [21]. The basis functions are assumed to be the modes of a waveguide with cross section similar to the patch shape. Hence, the very efficient BI-RME method [22] was used to find the modes and use them as basis functions. This paper presents how one can take advantage of the MoM/BI-RME approach along with TL method to analyze FSS on periodic substrates. It is well known that the number of entire-domain basis functions needed to obtain an almost exact result is usually much less than the corresponding number of rooftop basis functions [1]. In this study, it is shown that the required Fourier coefficients are less and the computation cost is much lower when entire-domain basis functions are applied. In Section II the main parts of the numerical procedure are explained. In order to demonstrate the validity of the method, some example problems are solved in Section III and their results are compared with measured data. The convergence efficiency of the proposed method compared to its rooftop version is also discussed. II. FORMULATION OF THE PROBLEM Fig. 1 illustrates a typical geometry of the problem. The structure consists of arrays of arbitrary shaped metallic patches arranged in a two dimensional lattice and printed on a substrate which may be periodic. The periodic substrate shown in Fig. 1 is obtained by drilling holes in a homogenous substrate. The goal of the analysis is the determination of the reflected and transmitted electromagnetic fields when the structure is illuminated by a plane wave with a given angular frequency , and direction ( , ) (Fig. 1). Other sources can be treated as a summation of plane waves. The numerical procedure is actually constituted of three methods: PMoM [1], [23] as the core of the procedure, BI-RME [22] for extracting the entire-domain basis functions and multiconductor transmission line method [24]–[27] for

One of the main approaches for the analysis of FSS is based on formulation of the electric field in terms of an integral equation which is then solved by the MoM. Because of the periodicity of the structure, Floquet’s theorem leads to a discrete spectrum for every quantity in the spectral domain. Therefore, the integral equation reduces to a series equation consisting of Fourier components of currents [1]. This method is usually referred as the PMoM. The resulting equation can be written in matrix form as

(1) where (2) (3) is the wave vector of the incident plane wave are lattice constants of the periodic structure in and , directions, respectively. is a row matrix containing the exponential terms ( and are the coordinates is a zero matrix with for a point on the metallic patch) and . In this paper the transthe same size as pose sign ( ) is used to distinguish between row and column and are column vectors representing the Fourier vectors. coefficients and in basis. Note that the indices and should vary in all the vectors and matrices with is a diagonal matrix with diagthe same order. The matrix onal elements equal to . In the case of homogeneous substrates the Green’s function matrix contains four diagonal matrices, including the values ( ). Therefore, each Fourier component of the field only depends on the Fourier component of the induced electric currents of the same order. In this case the calculation of the Green’s function matrix is straightforward [1]. When periodic substrates are involved, the different Fourier components of the fields are coupled throughout the substrate. In other words, the -th Fourier order of the electric current on the patch can affect the -th Fourier order of the diffracted electric field. Hence the four submatrices in are no longer diagonal. To evaluate this matrix, the multiconductor transmission line model is applied [26], [27]. For solving (1) using the concept of the MoM, electric currents excited on the patches should be expanded by some basis functions (4)

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where and are row vectors containing the basis functions and respectively. The unknown coefused for expanding ficients of these functions are arranged in the column vector . Using Galerkin’s method and after some algebraic operations the following system of equations is obtained: Fig. 2. An example of a unit cell containing a single patch. S and @S demonstrate the surface and boundary of the patch, respectively.

(5)

and is the incident elecwhere and are matrices whose ’th columns tric field vector. are Fourier coefficients of ’th corresponding basis Functions. The signs and stand for the complex and Hermitian conjuand in (1) are related to and gate respectively. through the following equation:

(6)

Using the obtained coefficients , all the desired quantities such as reflection and transmission coefficients could easily be calculated [1]. is considered as a phase factor in The term all basis functions. Therefore, the Fourier coefficients are calcu, ) basis. This avoids multiple calculalated in ( tion of Fourier coefficients for each and leads to higher efficiency in the numerical computations. In addition, it enhances the convergence of the procedure [1]. Using the presented formulation, the performance of the FSS can be simulated. Choosing a proper set of basis functions affects the efficiency of the method drastically. In this work, the BI-RME method is applied to extract the entire domain basis ( ). This was functions done previously for FSS on homogeneous substrates [20], [21]. However, to fulfill the same task for periodic substrates needs taking some points into account. These are explained in the following subsection.

B. Entire-Domain Basis Functions The basis functions in (4) should be chosen properly to be able to expand the excited current correctly. For this purpose they should not only be a complete set of functions but also have the same boundary conditions as the electric current on the patch. It means that they should be a complete set of functions with zero normal component on the boundary. From the waveguide theory, transverse magnetic fields of different guided modes of a metallic waveguide satisfy these requirements. So, the problem of finding entire domain basis functions is reduced to finding modes of a waveguide with the same cross-section as the metallic patch. The BI-RME method fulfills this task very efficiently [22].

Fig. 2 illustrates an example of a unit cell containing a single patch. Transverse magnetic field vectors of different modes of a metallic waveguide are given by (7) (8) (9) and are the eigenfunctions where the pairs of the homogeneous Helmholtz equation in the domain with Dirichelet and Neumann boundary conditions, respectively. When the patch is an -times connected surface, we have TEM basis functions which are obtained by solving the Laplace equation for with the boundary condition on an internal contour and elsewhere. By further investigation of (7)–(9), one can gain some interesting physical insight into the basis functions (10) (11) (12) Therefore only TE-type basis functions are responsible for producing electric charges on patches and TM and TEM-type of basis functions can only generate solenoidal currents. Which kind of basis functions (curl-less or divergence-less) has the main contribution in the current expansion, depends on the shape of the patches and specially on the incident wave specifications. The primitive outputs of the BI-RME are eigenvalues , and over the boundary ( is ). One can use these the outward normal derivative on boundary values to calculate and in the whole domain , and consequently the basis functions are obtained through (7) and (8). In the case of multiply connected domains, the standard Boundary Integral Method (BIM) can be implemented to obtain on , and finally the TEM basis functions (9) are computed for the whole cross section [28]. As it is clear from (5), only Fourier coefficients of basis functions are involved in the formulation of the problem. For calculating these Fourier coefficients we need to evaluate a large number of surface integrals numerically. In [20], [21], this problem is overcome by considering the propagating fields in different layers as the summation of different TE and TM modes. In that case, one encounters coupling integrals between the basis functions and TE/TM modes. Using the Green’s identity and the Helmholtz equations these integrals are transformed

YAHAGHI et al.: ANALYSIS OF FSSs ON PERIODIC SUBSTRATES USING ENTIRE DOMAIN BASIS FUNCTIONS

to line integrals of the fields or their normal derivatives. These values are the original products of the BI-RME method. Thus, the evaluation of coupling integrals are done efficiently. As mentioned previously, because of the existing coupling between different Fourier components in the periodic substrates, there is no pure TE or TM propagating mode. Therefore, one needs to find the real coefficients in the Fourier expansion of the basis functions. By applying the usual coordinate transformations [1], the Fourier integrals can be written in terms of the coupling integrals in [20]. The obtained equations are as follows:

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Fig. 3. A multicoductor transmission line model of the structure is used to determine the spectral Green’s function matrix. Input admittance matrices from different interfaces are involved in the calculation.

and are input admittance matrices looking upwhere ward and downward respectively. The admittance matrix is defined as (18) (13)

(14)

(15)

where, for example is a vector consisting of Fourier coefficients of in basis. For a homogeneous medium, the spectral domain immittance approach can easily be applied to compute the input admittance matrix [29]. For periand , the multiconductor odic substrates, after obtaining transmission line model is used to transform along the pe. In order to figure out how this riodic region and compute transformation should be accomplished, one needs to model the field propagation inside the periodic region. Electromagnetic fields within a periodic medium can be formulated by the multiconductor transmission line model which leads to a set of telegraph equations [26], [27]

(19)

(16)

where represents the transverse magnetic field vectors of different TM, TE and TEM modes of the other form a metallic waveguide. In the case of of the integrals reported in [21] are used. All above integrals can be calculated analytically in terms of the BI-RME outputs. Therefore, it leads to a great computational advantage. The only remaining step of the numerical procedure which is devoted to calculation of Green’s function matrix by multiconductor transmission line is explained in the next subsection. C. Green’s Function Matrix A side view of a typical structure is shown in Fig. 3. In spectral domain, the Green’s function matrix is indeed an impedance matrix which can be calculated using the following equation [18]: (17)

and are the voltage and current vectors of a multiconductor transmission line, respectively. They are defined as (20) The formulas to calculate the induction matrices and for isotropic, anisotropic, electrically or magnetically periodic media are reported in [27] and [18]. The solution of the coupled differential (19) can be written as (21) where is a matrix whose columns are eigenvectors of . as its diagonal The diagonal matrix consists of elements where is an eigenvalue of . in which is a diagonal matrix with diagonal elements equal and are constant complex vectors which are into . terpreted as the amplitudes of different space harmonics propagating along the -axis in opposite directions and they depend on the excitation source.

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Fig. 4. An FSS with homogeneous substrate. (a) A curved-boundary patch which is printed in each unit cell of the structure. (b) The side-view of the FSS. (c) Photo of the fabricated structure with printed copper patches on a homogeneous RO3010 substrate.

Fig. 6. Transmitted power versus frequency for the structure shown in Fig. 4 is simulated in the case of normally incident plane wave. The FSS contains a homogeneous substrate. The simulated result is compared with measured data.

Fig. 7. Transmitted power versus frequency for the structure shown in Fig. 4 is simulated in the case of obliquely incident TE-polarized plane wave. The FSS contains a homogeneous substrate. Comparison between simulation and and  . measurement is done for incidence angle 

= 30

Fig. 5. (a) Schematic of the measurement setup. The fabricated FSS is placed in a holder frame and is put between two broadband horn antennas. (b) a photo of the setup.

This solution can be used for transforming resulting matrix is

to

. The

(22) where

= 90

III. NUMERICAL RESULTS Two example problems are solved using the prescribed method. The goal in the first example is to validate the results of analyzing an FSS structure whose analysis using rooftop basis functions may be cumbersome or time-consuming. Measurement data are used for this purpose. The effect of existing periodic inhomogeneities in the substrate is also investigated in this example. In the second example, the convergence efficiency of the method compared to its rooftop version is examined. A. Validation of the Method

(23) and is an identity matrix. After solving (5), the induced current on the patch is obtained which yields the scattered electric field. The total diffracted fields are calculated as a superposition of scattered fields from the patches and reflected or transmitted fields from the substrates.

The FSS which is considered in the first example consists of a 2D lattice of patches with curved boundaries printed on a low loss RO3010 1.27 mm (50 mil) substrate. The unit cell configuration and the dimensions of the patches are demonstrated in Fig. 4. Different values are reported for dielectric constant of RO3010 substrates [30]–[32]. According to [31] the dielectric constant is about 11.5 at 3.36 GHz, however because of the dispersive property of this substrate its dielectric constant increases

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Fig. 10. High gain antennas are used to eliminate the effect of large half-power beamwidth of the broadband antennas in low frequencies which is shown in Fig. 9.

Fig. 8. An FSS with periodic substrate. (a) A curved-boundary patch which is printed in each unit cell of the structure. (b) The unit cell of the periodic substrate. (c) Photo of the fabricated structure with printed copper patches on the periodic substrate.

Fig. 11. Comparison between simulation results of transmitted power versus frequency for the two considered FSSs shown in Figs. 4 and 8, in the case of a normally incident plane wave.

Fig. 9. Transmitted power versus frequency for normal incidence of a plane wave on the FSS shown in Fig. 8 is simulated and compared with measurement. The FSS is contains a periodic substrate.

at higher frequencies [30]. In this paper we have supposed a dielectric constant which linearly varies from 11.5 at 3 GHz to 12 at 26 GHz. A photo of the FSS printed on a homogenous substrate is shown in Fig. 4(c). To measure the electromagnetic characteristics of the FSS, an experiment as shown in Fig. 5 is set up. The considered FSS is placed in a holder frame whose dielectric constant is close to air and therefore has a negligible influence on the FSS performance. The structure is placed between two broadband horn antennas to measure the transmission coefficient. The FSS is designed in such a way that only the zeroth diffracted orders are able to propagate in air. Therefore, the transmitted energy can be computed from (24)

and are the amplitudes of the zeroth diffraction where orders for transverse incident and transmitted electric fields, respectively. The entire measurement is performed with setups including and excluding the FSS. The transmission coefficient is then found from the difference of the obtained results. The simulation result for normally incident waves is compared to measured data in Fig. 6. A very good agreement between simulation and measurements is observed which verifies the validity of the method. The difference between simulation and measured data in the low frequency part of the diagram is due to the large half-power beamwidth of the broadband antenna in these frequencies. A MATLAB code is written based on the explained method and is run on a 2 AMD Opteron 254, 2.8 GHz CPU to obtain the simulation results. In the simulation, the Fourier series are . The computation time for extracting truncated at 41 entire domain basis functions for the present example was 28 second, and then it took 0.11 second for each frequency to evaluate the transmission coefficient. Therefore, it takes about 1 minute to obtain the complete curve with a resolution of 0.1

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Fig. 13. An FSS with periodic substrate. (a) A crossed shaped patch which forms the unit cell of the patch lattice. (b) The unit cell of the periodic substrate. (c) The side-view of the FSS.

Fig. 14. The absolute reflection coefficient of a normally incident plane wave is calculated by both rooftop and entire domain basis functions for the structure shown in Fig. 13. The FSS contains a periodic substrate. Four frequency points are chosen for convergency comparison.

Fig. 12. Transmitted power versus frequency for the structure shown in Fig. 8 is simulated in the case of obliquely incident TE-polarized plane wave. The FSS contains a periodic substrate. Comparisons between simulations and measurements are done for different incident angles equal to (a)  and  , (b)  and  , (c)  and  .

= 30

= 90

= 40

= 90

= 20

= 90

GHz. For further investigations, the measurement is repeated for oblique incidence of the plane wave. The incidence angles are and (Fig. 7). To prevent surrounding absorbers from blocking the incident and transmitted energy, in this experiment the FSS itself was the only turning part of the setup. Rotating the FSS opens up some gaps between it and the absorbers and results in differences between simulation and measurement.

However a good agreement for the resonance frequencies is still observed (Fig. 7). To explore the effect of having periodic inhomogeneities, holes are drilled in the substrate (Fig. 8). As shown in Fig. 9 for normal illumination of the plane wave, the simulation results agree with the measured ones. The experiment is repeated using high gain antennas in the low frequency part of the diagram to exclude the effect of large half-power beamwidth of the broadband antennas in these frequencies (Fig. 10). In the case of periodic substrates, matrix inversions and eigenvalue computations which are unavoidable steps in the multiconductor transmission line method, considerably increase the computation time. This reflects another credit side of using entire domain basis functions. Due to these matrix operations, the computation cost increases rapidly with the dimensions of the involved matrices. This factor is in turn directly determined by the number of Fourier components. The entire domain functions require usually much less Fourier coefficients than rooftop basis

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Fig. 16. The reflection coefficient of an obliquely incident plane wave is calculated by both rooftop and entire domain basis functions for the structure shown in Fig. 13. The FSS contains a periodic substrate. Three frequency points are chosen for convergency comparison.

to engineer the electromagnetic properties of an FSS by implementing this manufacturing technique. Results for other incident angles are also shown in Fig. 12. Again the observed difference is due to the gaps between the sample and surrounding absorbers. B. Convergence Efficiency of the Method

M = 13 GHz

Fig. 15. Reflection coefficient of a normally incident plane wave versus truncation order ( ) as a measure of convergency for the structure shown in Fig. 13 at different frequencies. The FSS contains a periodic substrate. (a) f : , , (c) f : , (d) f ,. (b) f

= 24 6 GHz

= 27 GHz

= 11 3GHz

functions to gain an acceptable accuracy. After extracting basis ; functions and with truncating the Fourier series at the computation time for evaluating transmission coefficient in each frequency was about 10 seconds. Note that in this study, the symmetries of the unit cell is not considered. Taking the symmetries into account enables one to further decrease the computation time. A simple comparison between transmission coefficients of the two FSS types (Fig. 11) reveals that it is possible

Convergence properties of the MoM/BI-RME method for non-periodic substrates is discussed in [33]. In order to compare the convergence efficiency of the proposed method with its rooftop version, an FSS consisting of crossed shaped patches that can be modeled by rooftop basis functions, is chosen (Fig. 13). The periodic substrate is obtained by drilling holes according to Fig. 13. The dielectric constant of the substrate medium equals 6.15. Since for periodic substrates the computation time is mainly determined by the number of Fourier components, here only the effect of different truncation orders of Fourier series is studied. The magnitude of the reflection coefficient is calculated utilizing both rooftop and entire domain basis functions. The in both presented results are obtained by setting cases. There is a small frequency shift between two results (Fig. 14). This may be due to the singularity of the tangential component of the current on the patch edge. The entire domain basis functions can form this abrupt change better than the rooftop functions. Some sample frequencies are chosen to use for convergency comparison. Variation of reflection coefficient ) is a proper versus truncation order of Fourier series ( measure to compare the convergence of the two methods. This is shown in Fig. 15 for four different frequencies. The mentioned small frequency shift can result in noticeable differences between the converged value of reflection coefficient particularly near resonance frequencies (Fig. 15(c)). It is observed in the figure that the number of Fourier coefficients needed for obtaining accurate results, is much smaller in the case of entire domain basis functions than for the rooftop ones. This issue which is not a critical point in FSS with non-periodic

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lead to an almost similar convergency in , the average convergence efficiency is still much better when entire domain basis functions are used instead of rooftops. IV. CONCLUSION Three methods, PMoM, BI-RME and multiconductor transmission line method, are incorporated and an efficient versatile scheme is developed for the analysis of FSS with periodic substrates. Comparison with measurement results demonstrates the validity of the method. Furthermore, the convergence of the method is studied and compared with the case when subdomain basis functions are used. It was shown that using entire domain functions allows one to reach good accuracy with lower number of Fourier coefficients. This leads to a better efficiency in the simulation of the problem. Utilizing periodic substrate in FSS offers a new and easily applicable way for engineering the electromagnetic characteristic of these structures. The method can be modified for anisotropic periodic substrates. REFERENCES [1] T. K. Wu, Ed., Frequency Selective Surface and Grid Array. New York, Wiley, 1995. [2] B. A. Munk, Frequency Selective Surfaces Theory and Design. New York: Wiley, 2000. [3] F. O’Nians and J. Matson, “Antenna feed system utilizing polarization independent frequency selective intermediate reflector,” U.S. 3-231892, Jan. 25, 1966. [4] J. Encinar, “Design of two-layer printed reflectarrays using patches of variable size,” IEEE Trans. Antennas Propag., vol. 49, no. 10, pp. 1403–1410, Oct. 2001. [5] B. A. Munk et al., “Transmission through a two-layer array of loaded slots,” IEEE Trans. Antennas Propag., vol. 22, no. 6, pp. 804–809, Nov. 1974. [6] “High Bandpass structure for the selective transmission and reflection of high frequency radio signals,” U.S. 5 103 241, Jul. 1992. [7] R. Ulrich, “Far-infrared properties of metallic mesh and its complementary structure,” Infrared Phys., vol. 7, no. 1, pp. 37–50, 1967. [8] N. Engheta and R. W. Ziolkowski, Eds., Metamaterials Physics and Engineering Explorations Hoboken, NJ, Wiley, 2006. [9] N. Engheta, “Thin absorbing screens using metamaterial surfaces,” in Proc. IEEE Antennas and Propagation Society Int. Symp., San Antonio, TX, 2002, pp. 392–395. [10] G. Kiani, K. Ford, K. Esselle, A. Weily, and C. Panagamuwa, “Oblique incidence performance of a novel frequency selective surface absorber,” IEEE Trans. Antennas Propag., vol. 55, no. 10, pp. 2931–2934, Oct. 2007. [11] D. Kern, D. Werner, A. Monorchio, L. Lanuzza, and M. Wilhelm, “The design synthesis of multiband artificial magnetic conductors using high impedance frequency selective surfaces,” IEEE Trans. Antennas Propag., vol. 53, no. 1, pp. 8–17, Jan. 2005. [12] W. Ko and R. Mittra, “Implementation of Floquet boundary condition in FDTD for FSS analysis,” in Antennas and Propagation Society Int. Symp. Digest, Jun. 1993, pp. 14–17. [13] I. Bardi, R. Remski, D. Perry, and Z. Cendes, “Plane wave scattering from frequency-selective surfaces by the finite-element method,” IEEE Trans. Magn., vol. 38, no. 2, pp. 641–644, 2002. [14] A. Fallahi, M. Mishrikey, C. Hafner, and R. Vahldieck, “Efficient procedures for the optimization of frequency selective surfaces,” IEEE Trans. Antennas Propag., vol. 56, no. 5, pp. 1340–1349, May 2008. [15] T. Chang, R. Langley, and E. A. Parker, “Frequency selective surfaces on biased ferrite substrates,” Electron. Lett., vol. 30, no. 15, pp. 1193–1194, Jul. 1994. [16] A. C. d. Lima, E. A. Parker, and R. J. Langley, “Tunable frequency selective surface using liquid substrates,” Electron. Lett., vol. 30, no. 4, pp. 281–282, Feb. 1994. [17] T. Ege, “Scattering by a two dimensional periodic array of conducting rings on a chiral slab,” in Antennas and Propagation Society Int. Symp. Digest, Jun. 1995, vol. 3, pp. 1667–1670. [18] A. Fallahi, M. Mishrikey, C. Hafner, and R. Vahldieck, “Analysis of multilayer frequency selective surfaces on periodic and anisotropic substrates,” Metamaterials, 2009.

Q

M = 18 6 GHz

Fig. 17. Reflection coefficient of an obliquely incident plane wave versus truncation order ( ) as a measure of convergency for the structure shown in Fig. 13 , at different frequencies. The FSS contains a periodic substrate. (a) f (b) f : , (c) f : .

= 24 8 GHz

= 11GHz

substrates has a considerable impact on the computation time of the problem. For example, obtaining an almost accurate result in 291 frequency points using 32 entire domain basis takes about 290 seconds, however functions with to achieve a similar result with rooftop basis functions one at least to 9, for which the computation time should set is about 5680 seconds. In this case, each rooftop basis function is defined on a 1 mm 1 mm rectangle. The same study is done for an obliquely incident plane wave and . The results shown in Fig. 16 are with in both BI-RME and rooftop obtained by setting cases. Three frequencies are chosen for convergency comparison. The magnitude of reflection coefficient versus the number of considered Floquet modes in these frequencies are depicted in Fig. 17. As observed in the figure, although both methods

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[19] A. Fallahi, M. Mishrikey, C. Hafner, and R. Vahldieck, “Radar absorbers based on frequency selective surfaces on perforated substrates,” J. Computat. Theoretical Nanosci., vol. 5, no. 4, pp. 704–710, Mar. 2008. [20] M. Bozzi, L. Perregrini, J. Weinzierl, and C. Winnewisser, “Efficient analysis of quasi-optical filters by a hybrid MoM/BI-RME method,” IEEE Trans. Antennas Propag., vol. 49, no. 7, pp. 1054–1064, Jul. 2001. [21] M. Bozzi and L. Perregrini, “Analysis of multilayered printed frequency selective surfaces by the MoM/BI-RME method,” IEEE Trans. Antennas Propag., vol. 51, no. 10, pp. 2830–2836, Oct. 2003. [22] M. G. G. Conciauro and R. Sorrentino, Advanced Modal Analysis. New York: Wiley, 2000. [23] R. Mittra, C. H. Chan, and T. Cwik, “Techniques for analyzing frequency selective surfaces-a review,” Proc. IEEE, vol. 76, no. 12, pp. 1593–1615, Dec. 1988. [24] M. Shahabadi, K. Schünemann, and H.-G. Unger, “Modelling of diffraction at dielectric biperiodic objects using an equivalent network,” in Digest of 20th Int. Conf. on Infrared Millimeter Waves, Orlando, FL, Dec. 1995, pp. 397–398. [25] T. Tamir and S. Zhang, “Modal transmission-line theory of multilayered grating structures,” J. Lightwave Technol., vol. 14, pp. 914–927, May 1996. [26] M. Shahabadi, S. Atakaramians, and N. Hojjat, “Transmission line formulation for the full-wave analysis of two-dimensional dielectric photonic crystals,” Inst. Elect. Eng. Sci., Meas. Technol., vol. 151, no. 5, pp. 327–334, 2004. [27] A. Fallahi, K. Z. Aghaie, A. Enayati, and M. Shahabadi, “Diffraction analysis of periodic structures using a transmission-line formulation: Principles and applications,” J. Comput. Theoretical Nanosci., vol. 4, no. 3, pp. 649–666, May 2007. [28] G. Conciauro, M. Bressan, and C. Zuffada, “Waveguide modes via an integral equation leading to a linear matrix eigenvalue problem,” IEEE Trans. Microw. Theory Tech., vol. 32, no. 11, pp. 1495–1504, Nov. 1984. [29] T. Itoh, “Spectral domain immitance approach for dispersion characteristics of generalized printed transmission lines,” IEEE Trans. Microw. Theory Tech.., vol. 28, pp. 733–736, Jul. 1980. [30] RO3000 Series High Frequency Circuit Materials. Chandler, AZ, Rogers Corporation advanced circuit materials, 1993. [31] J. B. Jarvis, M. D. Janezic, B. Riddle, C. L. Holloway, N. G. Paulter, and J. E. Blendell, “Dielectric and Conductor-Loss Characterization and Measurements on Electronic Packaging Materials,” NIST, 2001, Tech. Rep. 1520. [32] X. Y. Fang, D. Linton, C. Walker, and B. Collins, “Non-destructive characterization for dielectric loss of low permittivity substrate materials,” Meas. Sci. Technol., vol. 15, pp. 747–754, 2004. [33] M. Montagna, M. Bozzi, and L. Perregrini, “Convergence properties of the MoM/BI-RME method in the modeling of frequency selective surfaces,” in Proc. Eur. Microwave Conf., 2007, pp. 162–165.

Alireza Yahaghi received the B.S. degree in electrical engineering from the University of Tehran, Tehran, Iran, in 2002 and the M.Sc. degree in fields and waves communication engineering from the Iran University of Science and Technology, Tehran, in 2004. He is currently working toward the Ph.D. degree at Shiraz University, Shiraz, Iran. Since June 2008, he is with the Laboratory for Electromagnetic Fields and Microwave Electronics (IFH), ETH Zürich, Zürich, Switzerland, as an academic guest. His research interests are numerical methods in Electromagnetics and optics. His current research is about applications of planar metamaterials in microwave frequencies.

Arya Fallahi received the B.S. degree in electrical and electronics engineering from the Sharif University of Technology, Tehran, Iran, in 2004, the M.Sc. degree in fields and waves communication engineering from the University of Tehran, Tehran, in 2006. He is currently working toward the Ph.D. degree at the Swiss Federal Institute of Technology (ETH) Zürich, Switzerland. For his Master’s degree he has done theoretical studies on analysis of photonic crystal structures using a transmission line formulation. From

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September 2004 to March 2005, he was working with Iran Telecommunication Research Center on characterization of optical fibers. He was involved in a microwave project at University of Tehran from November 2005 to June 2006, in which he designed microstrip antennas with SIW feeds. In November 2006, he joined the ETH Zürich, where he is currently with the Laboratory for Electromagnetic Field Theory and Microwave Electronics (IFH). His research interests are design and numerical simulation of metamaterials. His current research is about metamaterials in microwave frequencies with focus on radar absorbers, artificial magnetic conductors and electromagnetic band-gap surfaces.

Habibollah Abiri (M’05) received the B.S. degree in electrical engineering from Shiraz University, Shiraz, Iran, and the D.E.A. and Doctor d’ Ingenieur degrees from National Polytechnique Institute of Grenoble (INPG), Grenoble, France, in 1978, 1981, and 1984, respectively. In 1985, he was an Assistant Professor with the University of Savoie, Chambery, France. Since 1985, he has been with the Electrical Engineering Department, Shiraz University, where he is now a Professor. In 1995, he was on sabbatical leave at the Electrical Engineering Department, Colorado State University, Fort Collins. His research interests include numerical methods in electromagnetic theory, microwave circuits, and integrated optics. Dr. Abiri is a member of the Iranian Association of Electrical and Electronics Engineers.

Mahmoud Shahabadi received the B.Sc. and M.Sc. degrees from the University of Tehran, Tehran, Iran, and the Ph.D. degree from Technische Universitaet Hamburg-Harburg, Germany, in 1988, 1991, and 1998, respectively, all in electrical engineering. Since 1998, he has been an Assistant Professor and then an Associate Professor with the School of Electrical and Computer Engineering, University of Tehran. From 2001 to 2004, he was with the Department of Electrical and Computer Engineering, University of Waterloo, Canada, as a Visiting Professor. Additionally he is a co-founder and CTO of MASSolutions Inc., a Waterloo-based company with a focus on advanced low-profile antenna array systems. His research interests and activities encompass various areas of microwave and millimeter-wave engineering as well as photonics. Computational electromagnetics for microwave engineering and photonics are his special interest. He is currently conducting research and industrial projects in the field of antenna engineering, THz engineering, photonic crystals, plasmonics, left-handed materials, and holography. Dr. Shahabadi was awarded the 1998/1999 Prize of the German Metal and Electrical Industries, Nordmetall, for his contribution to the field of millimeterwave holography and spatial power combining.

Christian Hafner was born in Zurich, Switzerland, in 1952. He received the diploma and Ph.D. degree in electrical engineering from the ETH Zurich, in 1975 and 1980, respectively. In 1987 he received the Venia Legendi for analytical and numerical calculations of electromagnetic fields from the ETH Zurich. Since 1976, he was a Scientific Assistant and Lecturer at the ETH Zurich, where he studied different topics of electricity and magnetism. Since 1980, he developed the MMP code for numerical computations of dynamic fields for a large range of applications (electrostatics, guided waves on different structures, scattering, EMC). In addition he works on philosophical and historical concepts of physics and engineering, evolutionary and genetic strategies for optimization, computer graphics and animation, and on “strange” theories (Chaos theory, fractals in classical electromagnetics, cellular automata for electromagnetics, irregular grid worlds, etc.). In 1999 he was given the title of Professor. He currently is the head of the Computational Optics Group (COG) at the IFH, ETH Zurich.

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Rüdiger Vahldieck (F’00) received the Dipl.-Ing. and Dr.-Ing. degrees in electrical engineering from the University of Bremen, Bremen, Germany, in 1980 and 1983, respectively. From 1984 to 1986, he was a Postdoctoral Fellow with the University of Ottawa, Ottawa, ON, Canada. In 1986, he joined the Department of Electrical and Computer Engineering, University of Victoria, Victoria, BC, Canada, where he became a Full Professor in 1991. During the fall of 1992 and the spring of 1993, he was a Visiting Scientist with the Ferdinand-Braun-Institute für Hochfrequenztechnik, Berlin, Germany. In 1997, he became a Professor of EM-field theory with the Swiss Federal Institute of Technology (ETH) Zürich, Zürich, Switzerland, and Head of the Laboratory for Electromagnetic Fields and Microwave Electronics (IFH) in 2003. His research interests include computational electromagnetics in the general area of electromagnetic compatibility (EMC) and, in particular, for computer-aided design of microwave, millimeter-wave, and opto-electronic integrated circuits. Since 1981, he has authored or coauthored over 300 technical papers in books, journals, and conferences, mainly in the field of microwave computer-aided design.

Prof. Vahldieck is the past president of the IEEE 2000 International Zürich Seminar on Broadband Communications (IZS2000). Since 2003, he has been president and general chairman of the International Zürich Symposium on Electromagnetic Compatibility. He is a member of the Editorial Board of the (EEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES . From 2000 to 2003, he was an associate editor for the IEEE MICROWAVE AND WIRELESS COMPONENTS LETTERS, and from July 2003 until the end of 2005, he was the Editor-in-Chief. Since 1992, he has served on the Technical Program Committee (TPC) of the IEEE Microwave Theory and Techniques Society (IEEE MTT-S) International Microwave Symposium (IMS), the IEEE MTT-S Technical Committee on Microwave Field Theory, and in 1999, on the TPC of the European Microwave Conference. From 1998 to 2003, he was the Chapter Chairman of the IEEE Swiss Joint Chapter on Microwave Theory and Techniques, Antennas and Propagation, and EMC. Since 2005, he has been President of the Swiss Research Foundation on Mobile Communications. He was the recipient of the J. K. Mitra Award of the Institution of Electronics and Telecommunication Engineers (IETE) (in 1996) for the best research paper in 1995 and was corecipient of the Outstanding Publication Award of the Institution of Electronic and Radio Engineers in 1983. He was the corecipient of the 2004 Applied Computational Electromagnetic Society (ACES) Outstanding Paper Award.

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Further Development of Vector Generalized Finite Element Method and Its Hybridization With Boundary Integrals O. Tuncer, Student Member, IEEE, Chuan Lu, Member, IEEE, N. V. Nair, Student Member, IEEE, B. Shanker, Fellow, IEEE, and L. C. Kempel, Fellow, IEEE

Abstract—Recently, vector generalized finite element method (VGFEM) was introduced for the solution of the vector Helmholtz equation, and its applicability was validated for canonical problems. VGFEM uses a local Helmholtz decomposition to construct basis functions in overlapping local domains of some canonical shape. While using a canonical shape for local domains adds flexibility to the method, one needs to provide information regarding boundaries of domains/inhomogeneities. The need for surface information proves to be a bottleneck in using the method for a larger class of problems. This paper is targeted towards overcoming these deficiencies; here, we will introduce the modifications to this method that permit interfacing with arbitrarily shaped local domains (to facilitate interfacing with existing meshing software), integrate this method with boundary integrals and provide a framework for studying dispersion. As will be apparent, the hybridization of the method with boundary integrals is not a simple adaptation of existing methods onto the VGFEM framework. Likewise, dispersion analysis is nontrivial due to the overlapping nature of VGFEM basis functions. A range of practical problems has been analyzed within the presented framework and results are compared either against measurements or existing FEM data to validate the presented methodology. Index Terms—Boundary integrals, generalized finite element methods (GFEM), numerical dispersion, partition of unity methods, RCS.

I. INTRODUCTION

T

HE finite element method (FEM) has been successfully used for the solution of various scalar and vector electromagnetic problems [1]–[3]. FEM is typically based on the space of functions that are defined on an underlying tessellation. These basis functions are typically interpolatory polynomial functions. Both the basis functions and their derivatives satisfy specific conditions at boundary interfaces [1]. For instance, Whitney elements, which are tangentially continuous across inter-element Manuscript received April 11, 2009; revised August 12, 2009. First published December 31, 2009; current version published March 03, 2010. This work was supported in part by the National Science Foundation (NSF) under DMS: 0811197, in part by AFOSR under Grant FA9550-06-1-0023, and in part by the High Performance Computing Center at Michigan State University. O. Tuncer, N. Nair, B. Shanker, and L. C. Kempel are with the Department of Electrical and Computer Engineering, Michigan State University, East Lansing, MI 48824 USA (e-mail: [email protected]). C. Lu is with Ansoft, LLC, Pittsburgh, PA 15219 USA. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2009.2039322

boundaries, are commonly used in computational electromagnetics. FEM has seen continuous development in order to analyze/solve practical problems both accurately and efficiently. In addition, there has always been an interest in developing a framework to enrich the underlying approximation space. Developing the means to do so can be useful in many different scenarios. They may be tailored to better capture electromagnetic fields in non-Lipschitzian domains; they can possibly permit numerical basis functions; and, they can enable a smaller transition regime in open domain problems and possibly provide a more seamless transition in domain decomposition approaches. Over the years, many methods have been developed including -cloud method [4], element-free Galerkin method [5], and generalized finite element method (GFEM) [6]–[12] which is based on the partition of unity (PU) approach [13]. While these methods have largely been used for elliptic, parabolic, and scalar Helmholtz equations, extension to the vector electromagnetic problems has been a recent endeavor [14]–[17]. The methodology presented in [16] has been used for the analysis of problems in both frequency and time domains [16], [18]. The work presented in this paper builds upon complete vector function spaces that were introduced by some of the authors, and constitutes an attempt to transit this methodology from nascent stages of development to a more robust one. Typically, VGFEM proceeds as follows. The problem domain is covered by a union of canonically shaped local domains/patches, on which basis functions are defined. One then uses an appropriate weak form to create the matrix elements. Since a union of patches is used to cover the domain of interest, one needs to prescribe the boundaries or properly define surfaces that partition the domain into piecewise homogeneous regions. Given the definition of these surfaces, it is then necessary to determine its intersection with these local domains/patches. Thus, using a union of canonical shapes/domains to describe the computational domain poses challenges when analyzing more realistic objects. Similar arguments hold if VGFEM is to be integrated with boundary integral techniques. Thus, this paper seeks to address three related problems: (i) it presents the formulation necessary to use VGFEM for arbitrarily shaped patches; (ii) it shows the steps necessary to integrate the method with boundary integrals, and (iii) it presents a method to help understand dispersion characteristics of this method. Addressing each of these issues is nontrivial. For instance, while it is relatively trivial to define basis functions in patches

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that can be transformed into canonical shapes such as cuboid, tetrahedra, hexahedra, spheroids, etc., it is not clear how the same process can be carried out in the patches that cannot be transformed onto canonical shapes. Furthermore, in a generic VGFEM procedure, two sets of functions need to be defined; one that forms a partition of unity and another that provides higher order approximation within the patch. Likewise, developing a boundary integral framework is not a straightforward extension of the method used in classical FEM. This is a consequence of the Babuska-Bressi condition, which does not permit the definition of an auxiliary basis function space on the patch surfaces. Finally, it is necessary to understand dispersion characteristics of the proposed method. However, contrary to classical FEM, the presence of overlapping domains precludes simple analysis. This paper presents a resolution to those problems. The outline of the paper is as follows. Section II presents the problem to be solved, a definition of partition of unity domains, and a subsequent redefinition of basis functions. A brief discussion on dispersion is also presented in this section. Section III presents the technique necessary for integrating the boundary integrals (BI) with VGFEM, and Section IV presents a plethora of results that demonstrate the applicability of the proposed method. Finally, some conclusions and future work are presented in Section V. II. FORMULATION A. Statement of the Problem whose Consider a linear and homogenous domain boundary is denoted by . Interior to the satisfies the vector Helmholtz equation domain, function

Fig. 1. An illustration of GFEM overlapping patches covering the domain .

where is the maximum number of patches covering the point . On each domain , a partition of unity (PU) function is defined such that (3) A space of vector basis function can then be defined as [16] (4) where denotes local approximation space, is the pilot vector, and and are used to identify the function space for each type of mode appearing in local Helmholtz decomposition. Details of this basis set, its convergence properties and completeness, and the manner in which this can be modified to avoid spurious modes can be found in [16], [19]. In addition, a technique by which one can obtain an independent set is also presented in [16] as well as other candidate function spaces (introduced by the authors and others). The PU function can be constructed using any Lipschitz continuous function by (5)

(1) where is used to denote either the electric field or the magnetic field, is the impressed source, is the wavenumber, is a differential operator, and is the function imposed on . In the above equation, it is assumed that . To solve this problem using VGFEM, we need to develop (i) an appropriate partitioning of the problem domain, and (ii) basis functions that are defined on these domains. These are elucidated next. B. Definition of Basis Functions Consider the domain depicted in Fig. 1. The domain is covered by sub-domains defined around nodes such that they satisfy the point-wise overlap condition (2)

and smoothly vanishes at providing that the patch boundaries, where denotes the index of the domain covering the point , [13]. A simple function that can be used to construct PU function in one dimension (1D) is a hat function, (6) where denotes the center of the patch , and represents the size of the patch. It is apparent that hat-based PU function has discontinuous derivatives. Alternatively, PU functions that have higher order continuous derivatives can be constructed using piecewise polynomials such as in (7), shown at the bottom of the page. Construction of three-dimensional (3D) PU function

(7) .

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is simply a product of 1D PU functions. The effects of using different types of PU functions on dispersion will be investigated later. Local approximation functions are defined on each local domain , and chosen from a space of functions , where is the order of local approximation function. Thus, the approximation to the unknown in (1) can be written as

(8) where and are unknown coefficients, and and are the number of basis functions for each type on the patch . Total number of basis functions on is . The unknown coefficients are solved from the linear system of equations that is obtained by applying Galerkin’s method with the given boundary conditions

(9) where

is the unit outward vector normal to the boundary and . Specifically, the bilinear form for (1) is written as

(10) where the constants gence [16].

, and

contribute uniform conver-

C. Partition of Unity Domains and Associated Basis Functions The definition of basis function, in (VG)FEM, is intimately tied to the support of the function. As was apparent from the description thus far, classical VGFEM relies on using an open cover of the domain , i.e., . This is advantageous from a couple of perspectives; (i) it enables a meshless description of the computational domain, and (ii) it permits the use of canonical elements (such as spheres, cubes, etc.) to form the open cover [16], [20]. The latter permits the use of simple functions to construct the PU function. However, the principal drawbacks of this construction are the need to define exterior

Fig. 2. Two different GFEM PU domain schemes. (a) Mesh A: Nodes are uniformly distributed. Each patch is associated with a node, (b) Mesh B: Patches—are constructed from the geometry mesh. Each patch is associated with a node, and patch domain comprises of the subdomains connected to this node.

surface and surfaces that separate piecewise homogeneous domains, and to determine their intersection with canonical PU domains. This is illustrated in Mesh-A in Fig. 2(a). This is possible by appropriately identifying these surfaces within each PU domain, expressing it using a set of polynomials, and then fitting this surface using a least square algorithm. This can be a significant bottleneck on two counts. It is difficult to utilize this sequence for geometrically complex problems, and it is difficult to enable analysis of practical problems due to lack of availability of such GFEM meshing schemes. The development here is intended to make this transition easier. In this section, we seek to develop a technique such that existing meshing information (such as that illustrated in Fig. 2(b)) can be used to extract the underlying GFEM mesh. Thus, we will develop a framework for construction of PU domains using readily available quadrilateral meshes as illustrated in Fig. 2(b). Note, a 2D quadrilateral mesh is shown in the figure to better understand the framework. Extension to forming of PU domains with 3D elements and other types of elements is rather trivial. Specifically, the PU domain is constructed from the subdomains that are connected to a node, (11) denotes the PU domain of node denotes the dowhere main of the subdomain , and is an array having the indexes of nodes of the subdomain . The definition of PU function goes hand-in-hand with the redefinition of the PU domains. The definition of PU function on an arbitrary shape PU domain is difficult due to the condition that the PU function must vanish at the

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local approximation functions. Consequently, the domain of the approximation function does not need to be identical to that of the PU domain. In fact, it is necessary for the domain of the PU at any node to be a subset of the approximation function at that node, i.e., . This is due function domain , which ensures to the fact that . For inthe basis function stance, if Legendre polynomials are used, a rectangular domain is needed for the definition of the functions. Thus, we construct the rectangular patch by centering it at the patch node position and finding the size of the patch such that it encloses the physical patch domain. For example, the size of rectangle for linear quadrilateral elements can be easily determined by and

(14)

Fig. 3. Definition of the domains of GFEM basis functions. (a) Geometry map, (b) PU hat function.

boundaries of the domain. In order to overcome this problem, each patch, which is defined around a node and composed of four subdomains sharing this node, is mapped onto a canonical as shown in Fig. 3(a). Each subdodomain of main belongs to the set

(12) The PU function then can be defined in this reference domain as depicted in Fig. 3(b). The mapping of the PU function to physical domain is performed using the transformation

(13) and denote node positions in where the physical and reference subdomains respectively, belong to the set and shifting constants that corresponds to the . If the PU function is continuous and differentiable set in this domain, one can readily prove that those properties translate to the physical domain as well. Next, the local approximation functions can be defined in the physical domain as illustrated in Fig. 3. As shown in this figure, it is important to realize that what we are interested in is the defas a product of the PU and the inition of the basis function

where and are used to denote the th node of th subdomain. Likewise, if exponential functions are used, it makes with a radius of sense to center them in a circle about , where and are given in (14). Next, it is important to understand the computational complexity of the method presented here. It is assumed that all examples are in 3D, a uniform brick meshing is used, and the number of basis is identical for each patch (reduction to 2D is trivial). To this end, assuming that the approximation is of order , then and the local approximation functions in (8) are of order respectively. It follows that the upper bound on the total . number of vector basis functions is Of course, this number is reduced considerably once the dependent vector basis functions are removed from the approximation space; in our experience, by approximately a factor of two. For instance, if the Legendre polynomials are used for local approximation function, the total number of vector basis functions in for , and for . Then, each patch is , where is the number the total number of unknowns is of patches. Further, in the scheme described above, each patch overlaps number of patches, where the maximum value of is 27 for a non-boundary patch. Consequently, it can be inferred , that the number of non-zero matrix entries per row is and the maximum number of non-zero matrix entries in a row is . As a point of comparison, assuming that a cuboid is discretized uniformly with nodes forming brick elements, then the number unknowns for higher order FEM is , where . Likewise, it can be easily proven that the upper bound on the number of non-zero matrix entries per row . for FEM is D. Dispersion Analysis In this subsection, dispersion characteristics of GFEM will be studied. The motivation herein is based on the fact that the numerical solution to the Helmholtz equation with a high wave number can notably deviate from the exact solution despite a moderate resolution of the geometry. This is largely due to phase

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Fig. 4. Definition of 1D PU function on patches separated by nodal distance h for  : .

= 15

error that is endemic in differential equation solvers. Since the phase error is progressive throughout the domain, it especially plays a key role on the accurate solution of electrically large and complex geometries. While dispersion analysis has been performed for classical FEM [21]–[23], it is a little more difficult for GFEM due to overlapping patches. To this end, we develop a semi-analytic technique to analyze phase error in GFEM. This technique presented herein is applicable to the dispersion analysis of both scalar and vector problems. Consider an infinite, linear, homogenous, isotropic, and source free region. In order to better understand dispersion analysis within the framework of GFEM, we first perform analysis for the wave propagation in 1D. This will then be followed by dispersion analysis in two-dimensional (2D) scalar and vector problems. An exact solution to the 1D scalar Helmholtz equation (15) in this region is a plane wave propagating along the -axis, , where is the wavenumber, and unity amplitude is assumed. To solve the problem using GFEM, the comsurputation domain is first covered by the PU domains rounding nodes i as shown in Fig. 4. Then, the scalar basis funcon the domains with the size of tions are defined by , which is related to the nodal distance by a constant , and . The PU function is constructed using either of the weighting functions given in (6) and (7), and are chosen from Lethe local approximation functions gendre polynomials. number of basis functions in a PU Assume that there are domain ; the same set of basis functions is assumed to exist in all PU domains that extend to infinity to the left and right of this parent PU domain. Since the PU domain overlaps only with the adjacent PU domains and , a sparse and diagonal matrix is constructed applying Galerkin’s method. The system conand related sists of square sub-block matrices to the intersection domains, and thus elements are given by

(16)

Fig. 5. Overlapping patches separated by a nodal distance h in 2D.

the numerical wavenumber. Then, the coefficients of the basis functions in the neighboring patches are (18) Setting the weighted residue integral to 0 for the PU domain a block matrix equation is obtained

,

(19) A nontrivial solution to (19) exists if , which has roots. However, there is only one valid root that satisfies the and . Then, the phase error conditions per wavelength in degrees is computed using the conventional definition (20) We should note that the phase error is a function of and . Since the dispersion analysis of GFEM is considerably more complicated unlike FEM, it is difficult to get an analytic dispersion expression as a function of and by solving the determinant. Generalization of the technique to two-dimension is straightforward but results in more complicated analytical expressions; therefore, we briefly describe the extension of the technique here. Assume a polarized plane wave with an incident angle from the x-axis propagates over an infinite 2D domain depicted , where additional index is used in Fig. 5. The PU domain to denote the discretization along -axis, now overlaps eight adjacent PU domains. The block coefficient vectors of the adjacent domains are the of the domain . shifted form of the coefficient vector The amounts of shifts along the -axis and -axis are respecand with the corresponding tively given by and components of the numerical wavenumber . Then, is the block matrix equation for

(17) where

and are the indexes of the basis and testing functions respectively. Since the system is infinitely periodic, the coefficients are the phase shifted version of each other. Thus, be the block vector representing the let unknown coefficients of the basis functions in , and be

(21)

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Assume that numerical wave propagates along the incident wave. Then, (21) is reduced to a one variable block matrix and . As in 1D, equation by using is determined by finding the appropriate root of the equation for each and values. Incident angle-phase error for each incident relation is also determined by extracting angle. This technique can also be applied to vector problems. Again, consider a 2D domain that is identical to the one considered earlier. An exact solution for the vector Helmholtz equation in this domain is a plane wave (22) where is the incident angle, and unity amplitude is assumed. For the solution of the problem with VGFEM, the PU domains are defined as in the scalar case as shown in Fig. 5 and vector . Lebasis functions are defined by gendre polynomials are used for local approximation function. Following the same procedure in scalar dispersion analysis, the block matrix equation identical to (21) is established. However, the elements of the block matrices for non-self and self-patch interactions are now respectively given by

(23) and

field by the ground plane without the aperture with surface area . The free space dyadic Green function in (25) is (26) is the free where is the free space wavenumber, and . The aperture space Green’s function, and magnetic field in Region I can be obtained by taking the curl of (25). The aperture fields in both regions are related by the continuity of the tangential electric and magnetic fields, and respectively. Thus, after taking the curl of (25) and applying the , the electric field in Region II is continuity equation at written as

(27) is the incident magnetic field, and is the where free space impedance. Equation (27) is one of the Neumann boundary conditions imposed at the aperture as the vector Helmholtz equation in Region II is solved using the VGFEM with Galerkin’s method. Additionally, although the normal components of the electric fields are discontinuous at the interface between two different media, they are continuous for this problem at the aperture due to domain homogeneity. Thus, imposition of the boundary condition ensures smoother convergence of VGFEM-BI, and it can be readily shown as

(24) where and are the indexes of vector basis and testing functions respectively, and denotes any combination of the indexes of the overlapping domains given in (21) providing that . is determined following the same Numerical wavenumber procedure in the scalar analysis. Using the valid root, convergence of VGFEM with Legendre polynomials can be analyzed for various incident angles. The proposed semi-analytic technique can be easily extended to 3D scalar and vector dispersion analysis of GFEM.

(28) As these boundary conditions are plugged into (10), bilinear form of the equations can be written as

III. HYBRID VECTOR GFEM-BOUNDARY INTEGRAL Next, we introduce Hybrid VGFEM-Boundary Integral technique for the analysis of scattering from cavity-backed aperture in an infinite ground plane as illustrated in Fig. 6. The problem and Redomain is divided into two regions: Region I . In Region I, the electric field satisfies gion II the vector wave equation given in (1), where now denotes the electric field. Applying equivalence principle and image theory, the fields in both regions are decoupled, and the electric field in Region I is given by [1]

(25) where source

is the incident electric field radiated by the current in the free space, and is the reflected electric

(29)

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Fig. 7. Illustration of integration steps for singular integrals.

Fig. 6. Illustration of scattering problem from a cavity-backed aperture.

where the vector denotes the approximation to the electric represents the testing function, is field in Region II, the incident electric field, is the volume of the support domain of the testing function, and are the surfaces of the domain of testing function on the PEC and aperture respectively, and is the outward unit normal vector of the support domain. The term with constant in (29) is used to force divergence of the field to vanish. As stated in [16], the constants , and contribute uniform convergence. Nitsche’s method is used to impose the Dirichlet boundary condition on the PEC walls. A. Differences Between VGFEM-BI and FEM-BI At this stage, we should point out a key difference between implementation of VGFEM-BI and that of FEM-BI: In FEM-BI, an auxiliary trace space is defined on the boundary to represent the magnetic field that is related to the electric field via . However, it is well known that the definition of auxiliary space is not possible in VGFEM as the Babuska-Brezzi condition is not satisfied with this auxiliary space. As a result, the formulation has to be done in terms of fields [8]. However, unlike the approach in [8], the definition of patches in this paper helps to avoid the use of auxiliary Kirchoff surfaces. Since the PU domain is constructed from the geometry brick elements, there exists boundary patches that are defined around each boundary node. The vector basis functions defined on these boundary patches are forced to satisfy the boundary conditions. The integral with dyadic term in (29) can be rewritten using the vector identities and divergence theorem as

Evaluation of the integrals in (30) is not straightforward as the PU function is piecewise continuous and the kernel is singular. The integrations are performed on each subdomain using Gauss-Legendre rule for the prescribed accuracy. To elucidate evaluation of the integrals, consider the subdomain shown in Fig. 7 and let us assume that it is on the aperture of an arbitrary cavity. For non-singular VGFEM terms, the integrals are evaluated on each subdomain in the reference domain that is shown in Fig. 3(a). More specifically, for an integral (31) where denotes the subdomain of patch , represents in in the physical domain. the portion of PU function This integral is performed numerically in the reference domain as (32) where and are the number of quadrature points, are the quadrature weights corresponding to the quadrature points , and is the determinant of the jacobian matrix. The transformation of the PU function that is defined in the reference domain, , is performed by using (13). Evaluation of VGFEM-BI terms in (30) is even more challenging due to basis function definitions of VGFEM. The surface-surface integral terms in (30) have singularity for the interaction of the patches that share this subdomain. Specifically, consider an integral form (33) where and are the source and observation points respectively. Evaluation of this integral is performed carefully in three steps as shown in Fig. 7. First, the quadratic subdomain is divided into two triangles in physical domain, and then the rule presented in [24] is applied by dividing each triangle into three sub-triangles around the observation point . The integration in each subtriangle is done numerically using [24] (34)

(30) is the outward normal vector of the contour . Conwhere trary to the FEM-BI formulation, the line integrals along the boundary are included in the formulation as the derivatives of the basis functions may not be continuous across the boundaries.

where is the jacobian of transformation and are the weights corresponding to the area coordinate sample points. The reader is referred to [24] for details. In order to evaluate (34), we need , at . However, to know the value of the PU function, we know the PU function is defined in the reference domain.

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Thus, we find the value of the PU function at the corresponding point in the reference domain by doing the transforms and respectively. The former is a simplex coordinate transform (35) and are the areas of sub-triangles 1 and 2 respecwhere tively, is the area of the original triangle as shown in Fig. 7, . The latter transform is done by and first scaling the domain , then shifting it to the corresponding defined reference subdomain that is an element of the set in (12). For the line-line integrals in (30), if an edge is shared by source and observation subpatches, singularity is removed using Cauchy’s integral; otherwise, the integration is evaluated following the procedure for non-singular integral evaluation that has been mentioned before. IV. NUMERICAL RESULTS In this Section, we will present a plethora of results to demonstrate the contribution of this paper. The method has been applied to three class of problems: (i) dispersion analysis, (ii) wave propagation in S-bend waveguide, and (iii) scattering from cavity-baked apertures. A. Dispersion Results In this subsection, convergence of the dispersion and its dependence on incidence angle are presented for both scalar GFEM and VGFEM via a set of simulations. First dispersion results for scalar GFEM and then the results for VGFEM are presented. Legendre polynomials and hat-PU function that is given by (6) are used unless otherwise is stated. The parameter re, is set to 1.5 lating element size to patch size , for all simulations. Fig. 8(a) shows the 1D dispersion results for scalar GFEM. For the first order polynomials, both scalar FEM and GFEM show approximately the same convergence behavior, whereas GFEM exhibits considerably better performance for the higher orders. It is well-known that the order of phase error in [23]; therefore, we conclude that the phase FEM is error in GFEM with Legendre polynomials is of the same order with an additional dependent constant coefficient, , which shifts the GFEM results down as compared to FEM results. Next, 2D dispersion results of scalar GFEM are shown in Fig. 8(b) for axially and diagonally incident wave. The numbers of scalar basis functions (per patch) used for the simulations are 3, 6, and 10 for the approximation orders of 1, 2, and 3, respectively. The axial results are identical to those in 1D, as expected. The phase error of diagonally incident wave becomes larger than the phase error of axially incidence wave as the polynomial order increases, and this relation does not change as the mesh size increases as seen in Fig. 8(b). Next, incident angle dependency of the phase error is further investigated for various and polynomial orders in Fig. 9. Note that the results for are scaled by the factors of and , respectively, to plot them on the same figure. For and , the phase error for the axially incident wave is about 2 more than that for the diagonally incident wave as in FEM with quadrilateral mesh structure [23]. In FEM, the phase error for the diagonally incident wave is always smaller and the shape of the plot

0

Fig. 8. h p convergence of the phase error per wavelength in degrees for scalar GFEM using Legendre polynomials and hat-PU functions. (a) FEM vs. GFEM, 1D, (b) h; p convergence for axial and diagonal incidences, 2D.

0 0

Fig. 9. Incident angle dependency of the phase error per wavelength in degrees for scalar GFEM with hat-PU function, h = .

= 10

is preserved as the polynomial order increases (it is only scaled) [23]. On the other hand, the phase error of GFEM for the diagonally incident wave gets larger as the polynomial order increases and each polynomial order exhibits different incident angle dependency plot as seen in Fig. 9. Next, we investigate the dispersion characteristics of scalar GFEM using higher-order (HO) PU function that is obtained

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0 0

Fig. 11. h; p convergence of the phase error per wavelength in degrees for VGFEM with Legendre polynomials and hat-PU function.

0 0

Fig. 10. h; p convergence and incident angle dependency of the phase error per wavelength in degrees for scalar GFEM with HO-PU function. (a) Hat-PU vs. HO-PU, axial incidence, (b) incidence angle dependency, h = .

= 10

using (7). Fig. 10(a) shows that the -h convergence plots of both , whereas HO-PU and hat-PU functions are identical for HO-PU exhibits better phase error performance for higher orders of Legendre polynomials. In addition, we investigate the incident angle dependency of the phase error for HO-PU function. As HO-PU results in Fig. 10(b) are compared with hat-PU results in Fig. 9, it is observed that the phase error for the diagonally incident wave gets larger as the polynomial order increases for both PU functions. However, HO-PU function suppresses the phase error for axially incident wave much more than hat-PU function. Thereafter, the dispersion results for VGFEM is presented and they are compared with those of scalar GFEM. The numbers of vector basis functions per patch used for the simulations are 5, 9, and 14 for the polynomials orders of 1, 2, and 3, respecconvergence of the phase error of VGFEM with tively. hat-PU function shown in Fig. 11 is exactly the same as that of scalar GFEM for axial incidence. Interestingly, the slope of the convergence is different for diagonal incidence and it is reduced . Next, the incident angle by a factor that is dominant for and dependency is shown in detail in Fig. 12(b) for with . The shape of dispersion curve of VGFEM for is very similar to that of scalar GFEM given in Fig. 9, where maximum phase error occurs for axially incidence wave. appears for diagHowever, the maximum dispersion for onal incidence unlike that of scalar GFEM.

0 0

Fig. 12. h; p convergence and incident angle dependency of the phase error per wavelength in degrees for VGFEM. (a) h; p convergence for axial inci= . dence, (b) incidence angle dependency, h

0 0 = 10

Finally, we investigate the dispersion in VGFEM using the convergence HO-PU function. Fig. 12(a) illustrates the of the VGFEM for both PU functions. convergence plots , of both HO-PU and hat-PU functions are identical for whereas HO-PU considerably suppresses the phase error for . Fig. 12(b) compares the incident angle dependency of the dispersion for HO-PU function against that for hat-PU function. for HO-PU function suppresses the phase error more for near diagonal incidences, but overall the shapes of the curves are , the shapes of the curves are, however, comsimilar. For pletely different. While the maximum error is seen at diagonal

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Fig. 13. Geometry of WR-90 waveguide, a .

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= 22:9 mm, b = 10:2 mm, r =

incidence for HO-PU function, it is observed at axial incidence for hat-PU function. Overall, HO-PU function considerably reduces dispersion. B. Application of VGFEM to Waveguide Problems In this subsection, we analyze the wave propagation in an S-bend rectangular hollow waveguide using VGFEM. Fig. 13 shows the geometry of cascaded H-plane bends in a WR-90 waveguide with mm, mm, mm . Note, the length of the straight section connecting and the cascaded curved H-plane bends is variable, and the geometry of the curved sections is constructed with linear elements, which obviously contributes to some error. While fine meshing is performed to reduce the error from geometry representation at the curved regions, coarse meshing is done for the straight waveguide sections, the lengths of which are chosen to be . The boundary conditions imposed at the input and output ports of the waveguide are respectively given by [1]

Fig. 14. S of WR-90 waveguide is simulated using different local approximation functions and compared against MoM data [25] for L mm and L mm, (b) mm. (a) mm.

= 25

L=0

=0

L = 25

(36) (37) where

and . Dirichlet boundary condition is imposed at the metallic boundaries using Nitsche’s method. Transmission characteristics of WR-90 waveguide have been well studied with the method of moment and mode matching technique in [25]. Thus, we will compare our results against theirs mm and mm cases computing parameter for mode by of

= 25

Fig. 15. S of WR-90 waveguide with L mm is simulated using different PU functions and compared against MoM data [25].

(38) where represents the total electric field computed numerically, and denotes the surface of the input port. We investigate the wave propagation in the WR-90 waveguide using two different local approximation functions: plane wave basis and Legendre polynomials. The space of plane wave , where is basis functions is the patch center and is the direction of the plane wave basis functions given by , is chosen as the roots of , and Legendre polynomials of order ,. Thus, the total number of basis . Fig. 14 compares obfunctions per patch is tained using VGFEM against MoM and the measurement data

for a range of frequencies. Note, two different VGFEM simulations have been performed by using Legendre polynomials of for one simulation, , and 21 plane waves order . VGFEM with plane waves per patch for the other one, gives better result than VGFEM with Legendre polynomials as shown Fig. 14(a). As the length of intermediate straight section gets longer, the results get much closer to MoM as seen in Fig. 14(b). Small deviation between VGFEM and MoM results can be explained with the geometry approximation of the curved sections and boundary conditions that are used to truncate the computational domain. Finally, we examine the contribution of higher order PU function on the accuracy. In Fig. 15, it is observed that the HO-PU function better captures the resonance.

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Fig. 16. Backscattered RCS of cavity-backed apertures compared against FEM-BI results given in [3] and [26] (a) small aperture: a : ; b : , : , (b) large aperture: a : ; b : , and c : . and c

= 1 73

=15

=15

=07 =01 =06

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Fig. 17. Backscattered RCS of the cavity-backed aperture with a = 2:5; b = 0:25, and c = 0:25 compared against FEM-BI results [27] (a)  = 90 , (b)  = 0 .

C. Hybrid VGFEM-BI Next, we will investigate the capability of the hybrid VGFEM-BI by simulating scattering from cavity backed apertures. The accuracy of the method is validated by comparing the RCS results with FEM-BI data or measurement results. For this purpose, far zone scattered field is first calculated by using [1]

(39) where are the spherical coordinates of the observaare the source points at the aperture surface tion point, . Then, co-polarized and cross-polarized RCS of cavities are computed by using

and (40) Legendre polynomials and hat-PU function are used for VGFEM-BI simulations.

First, VGFEM-BI is applied for computing RCS of a deep cavity with a small aperture. The dimensions of the cavity are . A polarized plane wave, with is incident to the empty cavity as illustrated in are used for Fig. 6. The vector basis functions of order VGFEM-BI. Fig. 16(a) compares the co-polarized and cross-polarized RCS of the cavity computed by VGFEM-BI and FEM-BI [3]. As it is seen, the agreement is excellent. Then, RCS of a , is simulated. cavity with larger aperture size, is PU domain with an approximate edge length of used. Fig. 16(b) shows the convergence of VGFEM-BI. The results with higher orders match with FEM-BI data [26] very well. cavity is simulated for Next, RCS of both E-polarized and H-polarized incident fields. The approximate edge length of cubic patch and the order of polynomials and respectively. VGFEM-BI and used are agree very well as shown FEM-BI results [27] for in Fig. 17(a). Fig. 17(b) shows the computed RCS for using the first and second order vector basis functions. The RCS’s of the cavity due to both polarization for different orders of the basis functions agree well with each other. Finally, RCS of a long cavity with the dimensions of is simulated at 12 GHz using cubic patches with and Legendre polyedge length of approximately . Fig. 18 shows the RCS results of nomials of order

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Fig. 18. Backscattered RCS of the cavity-backed aperture with a = 16:26; b = 0:2, and c = 0:85 compared against measurement result at 12 GHz [2].

VGFEM-BI against the measurement results provided in [2]. As seen in Fig. 18, VGFEM-BI results are very close to the measurement results like those of FEM-BI obtained in [2]. V. CONCLUSION In this paper, we have introduced several modifications that lays the foundation for using VGFEM to analyze more complex practical problems. Principally, the mathematics necessary to use arbitrary non-canonical partition of unity domains have been developed together with the manner in which the PU and local approximation functions can be defined on these domains. VGFEM has been hybridized with boundary integral by developing necessary mathematics to analyze cavity backed apertures. In addition, a semi-analytic technique has been developed to analyze dispersion characteristics of the scalar and vector GFEM. We have validated the proposed method for a range of practical problems against existing data, and demonstrated excellent agreement. Finally, the improvements made in this paper permit VGFEM to operate in either a “meshless” environment or a “meshed” environment, or a mixture of both. The framework proposed in this paper can permit an easy integration of the method with classical -adaptive FEM. We have earlier shown that this method has excellent approximation properties [16] by permitting mixtures of different kinds of approximation spaces. Some of these benefits, especially integration with classical FEM to create a highly flexible method, is a topic of current research and will be presented elsewhere. ACKNOWLEDGMENT The authors would like to thank Prof. J.-Fa. Lee at Ohio State University for commenting on their work. REFERENCES [1] J. M. Jin, The Finite Element Method in Electromagnetics. New York: Wiley, 2002. [2] J. M. Jin, J. L. Volakis, and J. D. Collins, “A finite-element-boundaryintegral method for scattering and radiation by two- and three-dimensional structures,” IEEE Antennas Propag. Mag., vol. 33, pp. 22–32, 1991.

[3] J. M. Jin and L. Volakis, “A finite element- boundary integral fromulation for scattering by three-dimensional-cavity-backed apertures,” IEEE Trans. Antennas Propag., vol. 39, pp. 97–104, 1991. [4] C. A. M. Duarte and J. T. Oden, “Hp Clouds—A meshless method to solve boundary value problems,” Univ. Texas at Austin, TICAM Tech. Rep. 95-05, 1995. [5] T. Belytschko, Y. Y. Lu, and L. Gu, “Element-free Galerkin methods,” Int. J. Numer. Methods Eng., vol. 37, pp. 229–256, 1994. [6] T. Strouboulis, I. Babuska, and R. Hidajat, “The generalized finite element method for Helmholtz equation: Theory, computation, and open problems,” Compt. Methods Appl. Mech. Eng., vol. 195, pp. 4711–4731, 2006. [7] T. Strouboulis and R. Hidajat, “Partition of unity method for Helmholtz equation: Q-covergence for plane-wave and wave-band local bases,” Applicat. Math., vol. 51, pp. 181–204, 2006. [8] C. Lu and B. Shanker, “Hybrid boundary integral-generalized (partition of unity) finite-element solvers for the scalar Helmholtz equation,” IEEE Trans. Magn., vol. 43, pp. 1002–1012, 2007. [9] C. Lu and B. Shanker, “Solving boundary value problems using the generalized (partition of unity) finite element method,” in Proc. IEEE Antennas Propag. Society Int. Symp., 2005, vol. 1B, pp. 125–128. [10] C. Lu and B. Shanker, “Development of hybrid boundary integral-generalized (partition of unity) finite element solvers for scalar problems,” in Proc. IEEE Antennas Propag. Society Int. Symp., 2005, vol. 1B, pp. 129–132. [11] C. Lu, J. Villa, B. Shanker, and L. C. Kempel, “Generalized finite element method for analyzing scattering from non-Lipschitzian domains,” in Proc. IEEE Antennas Propag. Society Int. Symp., 2006, pp. 1783–1786. [12] O. Tuncer, N. Nair, B. Shanker, and L. C. Kempel, “Dispersion analysis in scalar generalized finite element method,” in Proc. IEEE Antennas Propag. Society Int. Symp., 2008, pp. 1–4. [13] I. Babuska and J. M. Melenk, “The partition of unity method,” Int. J. Numer. Methods Eng., vol. 40, pp. 727–758, 1997. [14] I. Tsukerman, “General tangentially continuous vector elements,” IEEE Trans. Magn., vol. 39, pp. 1215–1218, 2003. [15] L. Proekt and I. Tsukerman, “Method of overlapping patches for electromagnetic computation,” IEEE Trans. Magn., vol. 38, pp. 741–744, 2002. [16] C. Lu and B. Shanker, “Generalized finite element method for vector electromagnetic problems,” IEEE Trans. Antennas Propag., vol. 55, pp. 1369–1381, 2007. [17] O. Tuncer, C. Lu, N. Nair, B. Shanker, and L. Kempel, “Analysis of error propagation in vector generalized finite element method,” ICEAA, pp. 822–825, 2007. [18] C. Lu, B. Shanker, and E. Michielssen, “Development of generalized finite element method for vector electromagnetic problems,” in Proc. IEEE Antennas Propag. Society Int. Symp., 2006, pp. 2813–2816. [19] O. Tuncer, B. Shanker, and L. C. Kempel, “Hybrid generalized finite element-boundary integral method for aperture design,” in Proc. EuCAP 3rd Eur. Conf. on Antennas and Propag., Mar. 23–27, 2009, pp. 2499–2502. [20] M. Griebel and M. A. Schweitzer, “A partical-partition of unity method (part ii: Efficient cover construction and reliable integration),” SIAM J. Sci. Comp., vol. 23, pp. 1655–1682, 2002. [21] A. Deraemaeker, I. Babuska, and P. Bouillard, “Dispersion and pollution of the FEM solution for the Helmholtz equation in one, two and three dimensions,” Int. J. Numer. Methods Eng., vol. 46, pp. 471–499, 1999. [22] R. Lee and A. C. Cangellaris, “A study of discretization error in the finite element approximation of wave solutions,” IEEE Trans. Antennas Propag., vol. 40, pp. 542–549, 1992. [23] G. S. Warren and W. R. Scott, “Numerical dispersion of higher order nodal elements in the finite element method,” IEEE Trans. Antennas Propag., vol. 44, pp. 317–320, 1996. [24] M. A. Khayat and D. R. Wilton, “Numerical evaluation of singular and near-singular potential integrals,” IEEE Trans. Antennas Propag., vol. 53, pp. 3180–3190, 2005. [25] A. Weisshaar, S. M. Goodnick, and V. K. Tripathi, “A rigorous and efficient method of moment solutions for curved waveguides bends,” IEEE Trans. Antennas Propag., vol. 40, pp. 2200–2206, 1992. [26] J. Liu and J. M. Jin, “A special higher order finite-element method for scattering by deep cavities,” IEEE Trans. Antennas Propag., vol. 48, pp. 609–703, 2000. [27] K. Barkeshli and L. Volakis, “Scattering from narrow rectangular filled grooves,” IEEE Trans. Antennas Propag., vol. 39, pp. 804–810, 1991.

TUNCER et al.: FURTHER DEVELOPMENT OF VGFEM AND ITS HYBRIDIZATION WITH BOUNDARY INTEGRALS

Ozgur Tuncer (S’01) received the B.Sc. degree in electrical and electronics engineering from the Middle East Technical University (METU), Ankara, Turkey, in 2003 and the M.Sc. degree in electrical engineering and information technology from the Technische Universität München, Munich, Germany, in 2005. He is currently working toward the Ph.D. degree at Michigan State University, East Lansing. In 2006, he worked as a System Engineer in the Communication Department, Infineon Technologies, Munich, on the modeling of 3G mobile communication products for HSDPA/HSUPA. Since 2007, he has been a Research Assistant at Michigan State University. His research interests include all aspects of theoretical and computational electromagnetics, specially vector generalized finite elements and integral equations.

Chuan Lu (S’03–M’07) was born in Tianjin, China, on September 16, 1979. He received the B.S. degree in electrical engineering from Tsinghua University, Beijing, China, in 2002 and the Ph.D. degree from Michigan State University, East Lansing, in 2007. Since then, he is a Development Engineer in Ansoft, LLC, where he has developed the electromagnetic simulation software for both high and low frequencies. His research interests are in the area of computational electromagnetics, especially differential equation-based numerical solver such as generalized finite element method and curvilinear element.

Naveen Nair (S’03) received the B.Tech. and M.Tech. degrees in mechanical engineering from the Indian Institute of Technology, Madras, India. in 2003. He is currently working toward the Ph.D. degree at Michigan State University, East Lansing. His research interests include computational and numerical techniques with specific applications to electromagnetics, inverse problems and nondestructive evaluation.

Balasubramaniam Shanker (SM’03–F’09) received the B.Tech. degree from the Indian Institute of Technology, Madras, India, in 1989, and the M.S. and Ph.D. degrees from the Pennsylvania State University, in 1992 and 1993, respectively. From 1993 to 1996, he was a Research Associate in the Department of Biochemistry and Biophysics, Iowa State University, where he worked on the molecular theory of optical activity. From 1996 to 1999, he was with the Center for Computational Electromagnetics, University of Illinois at

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Urbana-Champaign, as a Visiting Assistant Professor, and from 1999 to 2002, with the Department of Electrical and Computer Engineering, Iowa State University, as an Assistant Professor. Currently, he is an Professor in the Department of Electrical and Computer Engineering, Michigan State University, East Lansing. He has authored/coauthored over 250 journal and conferences papers and presented a number of invited talks. His research interest include all aspects of computational electromagnetics (frequency and time domain integral equation based methods, multiscale fast multipole methods, fast transient methods, higher order finite element and integral equation methods), propagation in complex media, mesoscale electromagnetics, and particle and molecular dynamics as applied to multiphysics and multiscale problems. Prof. Shanker was an Associate Editor for the IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS (AWPL), is a fellow of the IEEE and a full member of the USNC-URSI Commission B. He has also been awarded the Withrow Distinguished Junior scholar (in 2003) and the Withrow Teaching Award (in 2007).

Leo C. Kempel (S’89–M’94–SM’99–F’09) was born in Akron, OH, in October 1965. He received the B.S.E.E. degree from the University of Cincinnati, Cincinnati, OH, in 1989 and the M.S.E.E. and Ph.D. degrees from the University of Michigan, East Lansing, in 1990 and 1994, respectively. After a brief Postdoctoral appointment at the University of Michigan, he joined Mission Research Corporation in 1994 as a Senior Research Engineer. He led several projects involving the design of conformal antennas, computational electromagnetics, scattering analysis, and high power/ultrawideband microwaves. He joined Michigan State University in 1998. His current research interests include computational electromagnetics, conformal antennas, microwave/millimeter wave materials, mixedsignal electromagnetic interference techniques, and measurement techniques. Prof. Kempel has been awarded a CAREER award by the National Science Foundation and the Teacher-Scholar award by Michigan State University in 2002. He also received the MSU College of Engineering’s Withrow Distinguished Scholar (Junior Faculty) Award in 2001. He was elected as a Fellow of the Applied Computational Electromagnetics Society (ACES) in 2009. He served as an IPA with the Air Force Research Laboratory’s Sensors Directorate from 2004–2005 and 2006–2008. He was the inaugural director of the Michigan State University High Performance Computing Center and was the Associate Dean for Special Initiatives in the College of Engineering at Michigan State. He now serves as the Associate Dean for Research in the College of Engineering. He served as the technical chairperson for the 2001 Applied Computational Electromagnetics Society Conference and technical co-chair for the Finite Element Workshop held in Chios, GREECE in 2002. He was a member of the Antennas and Propagation Society’s Administrative Committee and a member of the ACES Board of Directors. He currently is the Chairperson of the Chapter Activity Committee for the IEEE Antennas and Propagation Society. He served as an Associate Editor for the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION. He is an active reviewer for several IEEE publications as well as JEWA and Radio Science. He coauthored The Finite Element Method for Electromagnetics (IEEE Press). He is a member of Tau Beta Pi, Eta Kappa Nu, and Commission B of URSI.

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Multiple Loaded Scatterer Method for E-Field Mapping Applications Mohamed A. Abou-Khousa, Member, IEEE, and Reza Zoughi, Fellow, IEEE

Abstract—Spatial mapping of electric field distribution is an important objective in many applications including antenna pattern, radar cross-section, and specific absorption rate measurements. Modulated scatterer technique (MST) based on small loaded dipoles has been successfully used for these purposes for many years. However, MST suffers from several inherent limitations. These limitations become more severe as the frequency increases. This paper introduces a new and efficient method based on a multiple loaded scatterer (MLS) approach. The proposed method allows for accurate recovery of an unknown incident electric field with measurements conducted at a single observation point. The formulation of the MLS method along with several key simulation results illustrating its efficacy for electric field distribution measurement is presented. In addition, the results of several comparisons with conventional MST are also provided. Index Terms—Electric field mapping, modulated scatterer technique (MST), multiple loaded scatterer (MLS), PIN diode.

I. INTRODUCTION EASURING electric field distributions at microwave and millimeter wave frequencies is of great importance for design verification, optimization and diagnostics purposes in a wide variety of applications including antenna design, electromagnetic compatibility (EMC), nondestructive testing (NDT) and imaging. Basically, a probe (a single antenna or array of antennas) sensitive to the electric field component of interest is used to measure the electric field in a specified spatial domain, i.e., mapping the field. A conventional probe used for this purpose is based on modulated scatterer technique (MST) which was first proposed in [1]. Since its introduction, the MST with scanned single modulated scatterer, i.e., loaded dipole, and array of scatterers, has been widely used for electric field mapping and imaging applications [2]–[4]. Modulating the scatterer enhances the measurement sensitivity, i.e., reduces the effect of unmodulated noise, through coherent averaging over many modulation cycles. Furthermore, modulation allows for spatial signal tagging when array of scatterers are used to map the electric field, and consequently, a

M

Manuscript received January 30, 2009; revised August 29, 2009. First published December 28, 2009; current version published March 03, 2010. Portions of this paper were published in the Proc. Int. Symp. on Antennas and Propagation (ISAP08), Taipei, Taiwan, Oct. 27-30, 2008, pp. 565-568, and used here with permission from the Chinese Microwave Association Publisher. M. A. Abou-Khousa is with the Imaging Research Laboratories, Robarts Research Institute, University of Western Ontario, London, ON N6A 5K8, Canada (e-mail: [email protected]). R. Zoughi is with the Applied Microwave Nondestructive Testing Laboratory (amntl), Electrical and Computer Engineering Department, Missouri University of Science and Technology, Rolla, MO 65409 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2009.2039302

single receiver may be used rendering low overall system cost and complexity [2]. A new electric field distribution measurement method employing multiple loaded linear scatterer (MLS) instead of the conventional MST arrays has been recently introduced by the authors in [5]. The MLS method was conceived to overcome many of the practical limitations associated with the conventional MST, as will be described later. To this end, the MLS method utilizes a linear scatterer loaded with PIN diodes, at multiple discrete locations over its length, while placed in the field of interest. Using multiple loads allows for loading the scatterer structure with distinct modulation states, or loading conditions. As an intermediate step towards recovering the field of interest, the MLS method reconstructs the current distribution induced on the scatterer when all of its loads are short-circuited. This is founded on the fact that the scattered electric field from a loaded scatterer is the superposition of the field scattered when the load is short-circuited and the radiated field form the scatterer when it is fed at the load location with an equivalent load voltage [6]. The equivalent load voltage itself is a function of the current passing through the load location when it is short-circuited. This short-circuit current, while related to the incident electric field of interest, is independent of the loading conditions [6]. Hence, by measuring the scattered electric field at a single observation point under each loading condition, it is possible to form a system of linear equations which can be solved to give the current over the length of the scatterer when the loads are all short-circuited. This current is subsequently used to recover the electric field of interest. Multiple and continuous loading of linear wire scatterers and antennas are well-established concepts used for antenna pattern synthesis [7], radar cross-section (RCS) control [8], and bandwidth extension purposes [9]. However, the utility of multiple loaded scatterer for the purpose of measuring unknown electric field distributions is yet to be fully investigated. In this paper, we extend the preliminary investigation and results reported in [5] by providing a comprehensive analysis of the MLS method and comparisons of its performance to the conventional MST. We start by providing the pertinent background on the conventional MST electric field measuring arrays. Then, the MLS method is presented and its unique aspects compared to the conventional MST are discussed. Finally, the performance of the MLS method is demonstrated and compared to the conventional MST via numerical simulations. II. CONVENTIONAL MST In the broad sense, the common MST-based electric field probes can be regarded as single or array probes. Typically, the

0018-926X/$26.00 © 2010 IEEE

ABOU-KHOUSA AND ZOUGHI: MULTIPLE LOADED SCATTERER METHOD FOR E-FIELD MAPPING APPLICATIONS

Fig. 1. Typical MST bistatic electric field measurement configuration using a 1D array of PIN diode-loaded short dipoles.

single probe uses a short dipole scatterer loaded with a PIN diode. The spatial distribution of the electric field of interest is measured by scanning the probe in that field while switching the PIN diode between ON and OFF states, i.e., which in turn modulates the scattered field [1], [10]. With a single scatterer, it takes a long time to complete the mapping process, which is undesirable in many applications where real-/pseudo-real time mapping is desired. To speed up this process, one- or two-dimensional arrays of small loaded scatterers may be used [2]. In most applications, a compact array of closely-spaced scatterers is needed in order to construct a representative map of the field of interest. In this configuration, the interaction between the array elements not only limits the system sensitivity and dynamic range, but it must also be compensated for in order to obtain an accurate and representative field distribution (see [11] and the references therein). In general, the MST can be applied in monostatic or bistatic configurations depending on the requirements of the application. The development in this work is focused on the general bistatic configuration. A typical MST-based bistatic electric field measurement setup using a 1D array of PIN diode-loaded dipoles each of length is depicted in Fig. 1. The loaded dipoles are interspaced by and the total length of the array is . The array is used to measure an unknown incident electric field, , radiated by a source under test, by measuring the scattered field, , at a point , while modulating the dipoles. Modulation is performed by turning a single given PIN diode ON and OFF while all remaining diodes are turned OFF. This process is repeated for all diodes sequentially.1 Ideally, the ON and OFF states of the PIN diode approximate open- and short-circuit loads, respectively. When the dipole length, , is small relative to the wavelength, , the scattered field from the shorted-dipole 1Parallel

modulation using orthogonal codes is also possible in some cases.

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is proportional to the incident field at the dipole location. Hence, modulation allows for tagging the scattered field as a function of the location of the modulated dipole while the measurements are taken at a single receiving point, i.e., one receiver is needed. The dipole interspacing, , should be at most in order to achieve proper sampling of the incident field [2]. Applying the MST involves calibrating the system, e.g., the array and receiver, with a known field distribution such as a uniform plane-wave [2]. The performance of the MST-based electric field measurement system depends on many factors. As stated in [11]; “Six basic factors that can have an impact on measurement system performances are as follows: 1) dynamic range, 2) interelement mutual coupling, 3) interaction between the MST array and test antenna, 4) parasitic signals (modulated and/or unmodulated), 5) dispersion of element scattering characteristics, and 6) probe correction.” These factors impose many trade-off considerations that need to be resolved on per-application basis. For instance, the interelement mutual coupling combined with the fact that the utilized elements are commonly small scatterers, i.e., short dipoles, can adversely impact the system sensitivity, and hence reduce the overall system dynamic range. On the other hand, using stronger scatterers, e.g., resonant scatterers as suggested in [12], can enhance the sensitivity when effective probe correction and compensation techniques are employed. Such a trade-off was investigated in [13] where it was articulated that “For practical reasons, instead of trying to build noninterfering (and therefore more complicated and perhaps less sensitive) sensors, one can use common sensors (simpler, inexpensive, and possibly more sensitive) and include them in the reconstruction formalism itself.” At high frequencies such as those in or near the millimeter wave region, the implementation of conventional MST-based systems, using an array of dipoles, becomes more challenging due the factors mentioned above. At these frequencies, the factors pertaining to parasitic loads which are difficult to characterize, and element dispersion become a significant issue in the system design. Furthermore, the measured response of the weakly modulated array dipole to small variations in the incident field is likely to be masked by the signals arising from the interaction among the array dipoles. The multiple loaded scatterer (MLS) method presented next is expected to effectively overcome most of the abovementioned issues and challenges. III. MULTIPLE LOADED SCATTERER (MLS) METHOD A. Preliminaries The current induced on a linear wire scatterer of length and radius aligned with the z-axis due to an incident electric , can be found using Pocklington’s integral equation field, [14, pp. 720] using the method-of-moments (MoM) [15], where the wire is modeled as interconnected segments each with a . The induced length and centered at current is given by (1)

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MLS considered here can be analyzed by finding the current induced along its length using linear network theory. Assuming the loads are linear or can be linearized, the scatterer can be -port network ( load terminals plus the treated as a illuminating source terminals). In this treatment, each load is represented by an equivalent voltage source at its location along the scatterer. An expression for the load equivalent voltages can be obtained from the short-circuit parameters relating the port currents and voltages in addition to the load boundary conditions. The port currents and voltages are related by (4) is the transfer admittance matrix of size and are the port current and voltage vectors, respectively. Here, the voltage across the source terminals . The load conditions on the port voltis represented by ages are where

(5) Fig. 2. MLS-based electric field distribution measurement configuration.

Using (4) and (5), the following set of equations is formulated to solve for the load equivalent voltages

where is the “short-ciris the scatcuit” current induced on the scatterer, terer admittance matrix obtained after applying the MoM, is the excitation vector, is the wavenumber, and is the intrinsic impedance of the medium. The z-component of the scattered electric field can be written as (2)

(6) where lent voltage vector, trix of size

is the load equivais the transfer admittance madefined between the load ports, is the load matrix, and is a vector consisting of the transfer admittances defined between the illuminating source terminals and the load ports on the MLS. The are given by elements of (7)

where

such that

where is the current passing through the th load location when that load is short-circuited. Using (7) into (6), the load equivalent voltage vector is found as (3) (8)

and are given in [14, p. 284]. The functions The MLS method is based on using a single linear scatterer controllable loads, e.g., lumped active elements loaded with such as PIN diodes, with general admittances of , located at , as shown in Fig. 2. The scatterer is illuminated by an unknown incident , produced by a radiation source. The objective is field, , over to measure the incident electric field distribution, the length of the scatterer using the measured scattered field, , obtained at an arbitrary observation point, . The development of the MLS method starts with explicitly expressing the scattered electric field in terms of the load admittances, as outlined next. The general analysis of straight wire scatterers loaded with lumped linear elements can be found in [6]. An equivalent approach based on Maxwellian circuits theory was also recently proposed in [16]. Following the approach outlined in [6], the

where is the represents samples short-circuit current vector. Note that taken at the load locations. of The total current along the scatterer is the superposition of the short-circuit current induced when all loads are short-circuited, as given in (1), and the current induced due to the equivalent be a vector of dimension with all load voltages. Let load locations where it is populated with zeros except at the the corresponding elements of , then the solution for the total current can be found as (9) Thus, the total scattered field at point

is now given by (10)

ABOU-KHOUSA AND ZOUGHI: MULTIPLE LOADED SCATTERER METHOD FOR E-FIELD MAPPING APPLICATIONS

Note that the first term in (10), i.e., the scattered field when all ports are short-circuited, is not a function of the load matrix, . This fact will be used later in the MLS formulation.

2 be the interpolated short-circuit current obtained from The incident electric field over the length of the scatterer is found as

(14)

B. Field Mapping Using MLS The premise of this method is based on the ability to obtain the short-circuit current, , induced along the scatterer by , e.g., realusing different sets of load values izing different loading conditions, and measuring the scattered field under each loading condition at the observation point, . , can be uniquely Given this current, the field of interest, determined from (1). For this purpose, distinct loading conditions are realized by electronically changing the load values such that the th loading condition corresponds to a unique , and results in load voltage vectors load matrix denoted by and (evaluated from (8) using in lieu of ). Let’s designate one of the loading conditions as the reference . Under condition, for example the 1st loading condition . Using (10), the this condition, let the measured field be difference between the field scattered due to the reference and the th loading conditions is written as

(11) and and have non-zero components Since the vectors positions only only at the load locations, the corresponding vector contribute to . Define a vector in the which is populated with these components. The field difference in (11) can now be written as

(12) Given the scatterer geometry, frequency of operation, and the load values for the different loading conditions, the above equation has everything known except the load short-circuit current vector which represents the unknown loads short-circuit currents. To solve for this current, equations are formed using loading conditions. Thus, by measuring the scattered field for all loading conditions and using the first loading condition as the reference, the load short-circuit current vector is determined as (13) is the vector of scattered such that . The loading conditions should result in linearly independent current distributions over the length of the scatterer in order to , as per (13). Simulation results have uniquely determine is always of rank , i.e., full rank, when such shown that current distributions are realized. In Section IV, PIN diode loads are used to obtain the required loading conditions. Before the incident electric field can be recovered using the computed short-circuit current, this current must be interpolated over the length of the scatterer, e.g., approximating . Let where field differences, and

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The functional procedure of the MLS method can now be summarized as follows: 1) activate the reference loading condition and measure ; 2) activate the rest of the loading conditions and measure ; 3) from the obtained measurements, compute the field differ(off-line); ence vector (no measurement is required); 4) compute and interpolate it; 5) solve for 6) finally, solve for the incident electric field using (14). The salient feature of the MLS method is the inherent differential field measurements, i.e., subtracting reference measurement from all subsequent measurements, which are used to construct the incident electric field. With this feature, the effects of clutter and distributed parasitics are significantly reduced. Unintentional parasitics and other “static” loads needed for biasing, such as decoupling capacitors, do not change with modulation and they result in a constant term being added to (10). Since the MLS method is not based on discrete elements, the limiting factors associated with conventional MST which are related to array element dispersion and interelement coupling do not limit its performance. Furthermore, the modulated scattered signal level in the MLS implementation, which is based on modulating a strong scatterer, is expected to be much higher than that in the conventional MST case. In practice, this fact translates to higher measurement sensitivity. In addition, no probe correction is needed with the MLS method since the presence of the scatterer is accounted for in recovering the incident field. Various attributes of the MLS method will be examined further in the next section. IV. NUMERICAL RESULTS AND DISCUSSION A linear wire scatterer representing an MLS of radius was considered to illustrate the method. Uniformly interspaced ideal PIN diodes were used to load the MLS, as shown in Fig. 2. PIN diodes, there are different possible loading With loading conditions needed in the MLS conditions. The method are a subset of the total possible loading conditions and should result in linearly independent current distributions over the length of the scatterer. The loading condition which consists of all diodes being turned ON is always used as the reference condition. Turning one diode OFF at time while the rest loading conditions. This are ON represents the remaining set of loading conditions was verified to meet the requirement mentioned above. For example, consider an MLS with loaded with PIN diodes. Fig. 3 shows the magnitude and phase of the current induced on the MLS due to an incident uniform plane-wave for the selected 4 loading conditions. Namely; 2When there are no loads placed close to the ends of the scatterer, the boundary conditions, i.e., vanishing current at both ends of the scatterer in this case, can be enforced to obtain a better estimate of the short-circuit current.

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Fig. 3. Magnitude and phase of the current induced on the MLS (L = ; M = 3) due to an incident uniform plane-wave for the selected 4 loading conditions.

Fig. 4. Example of linearly dependent current distributions induced on the MLS (L = ; M = 3).

all diodes are ON, only the first diode is turned OFF, only the second diode is turned OFF, and only the third diode is turned OFF. The current distributions shown in Fig. 3 are linearly inis of rank 3. dependent and the matrix When the loading conditions result in linearly dependent curbecomes rank-deficient. For inrent distributions, the matrix stance, let’s replace the third loading condition from the selected set, i.e., only the third diode is turned OFF, with the condition where the first and second diodes are turned OFF at the same time while the third diode is turned ON. Fig. 4 shows the magnitude and phase of the current induced on the MLS for the new set of loading conditions. In this case, the obtained current distributions are linearly dependent (cf. the current distributions in in Fig. 4) and the matrix is now the interval uniquely. of rank 2, i.e., it can not be used to determine A. Short-Circuit Current Estimation The procedure outlined in the previous section using an MLS of length and PIN diodes was used to solve for the shortcircuit current induced on the scatterer due to an incident uniform

Fig. 5. Real and imaginary parts of the short-circuit current induced on the MLS due to an incident uniform plane-wave compared to recovered currents using the MLS method with M = 20.

plane-wave while the scattered electric field observations were . Fig. 5 shows the induced short-circuit made at current distribution (real and imaginary) of the actual computed found using the current , the load short-circuit current MLS, and the interpolated short-circuit current . Cubic spline from . Other interpointerpolation was used to find lation methods such as ideal low-pass filtering, i.e., using sink functions, were also considered. The cubic spline interpolation, however, provided the best results especially near both ends of the MLS. It is evident from Fig. 5 that the recovered current using the MLS closely matches the actual induced current. In general, the accuracy associated with recovering the load short-circuit current is not a function of the number of the loads per wavelength.3 The overall accuracy of the interpolated current , however, depends on this number. For accurate interpolation, the sampling theorem dictates that the loads, i.e., the , should be less than apart. For instance, samples in Fig. 6 shows the recovered short-circuit current using the MLS method with 5 loads only. Although the load short-circuits were recovered accurately, the interpolated current was not representative of the actual short-circuit current since the sampling criteria was not satisfied in this case. Fig. 7 shows the root-mean-squared error (RMSE) in estinormalized to current dynamic range as a function mating , the sampling theorem is of the number of loads. With . As shown in Fig. 7, the interpolation satisfied with error decreases rapidly as increases beyond 10. The error is attributed to the numerical interpofloor shown for lation error. B. Electric Field Mapping To demonstrate the efficacy of the MLS method for recovering an unknown electric field distribution, the following incident electric field was considered as an example (15) 3The formulation used to obtain the short-circuit at the load locations as given in (13) is exact.

ABOU-KHOUSA AND ZOUGHI: MULTIPLE LOADED SCATTERER METHOD FOR E-FIELD MAPPING APPLICATIONS

Fig. 6. Real and imaginary parts of the short-circuit current induced on the MLS due to an incident uniform plane-wave compared to recovered currents . using the MLS method with

M =5

L = 5

Fig. 7. Normalized RMSE in estimating the short-circuit current induced on . the MLS as a function of the number of loads,

The incident field in this case is a plane-wave with angle of incidence, , and it represents more than 100 dB of electric field magnitude dynamic range and more than two phase cycles over the length of the wire. The MLS method as described above was used to recover the incident electric field from the scattered field . Fig. 8 illustrates the efobservations computed at for measuring fectiveness of the MLS method (with the incident electric field distribution (real and imaginary) by comparing its result with the actual field given in (15). The MLS method yielded very accurate results in recovering the electric field over the whole length of the scatterer except toward the ends. The errors at the end of the wire were encountered in other simulations performed using different incident field distributions. This is thought to be due to the numerical errors associated with applying the MoM. C. Comparison With Conventional MST Since the MLS method is based on modulating long scatterer as opposed to several electrically small dipoles, the corre-

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M = 20

Fig. 8. Real and imaginary parts of the actual incident field (15) and the recov. ered field using the MLS method with

L = 10

Fig. 9. RCS of 1D MST array and the corresponding MLS of the same length with 10 PIN diode loads as a function of the loading/modulation conditions.

sponding scattered signal level and consequently the signal-tonoise ratio at the receiver is higher compared to the conventional MST. To compare these two methods, one may use the RCS as an appropriate figure-of-merit which is directly related to the scattered signal level. RCS of an MST array of PIN diodeloaded dipoles, similar to the one shown in Fig. 1, was computed and compared to the RCS of an MLS of the same length and long consisting of 10 number of loads. The MST array was interspaced by . identical loaded dipoles of length Fig. 9 shows the RCS (normalized to ) of the MST array and the MLS as a function of the loading/modulation conditions. For the MST, the first modulation condition is when all diodes are OFF and the remaining ten conditions, i.e., conditions 2–11, consist of one diode being turned ON at a time while the rest are turned OFF. Fig. 9 shows that the RCS of the MLS for all loading conditions is at least two orders of magnitude higher than the conventional MST. To highlight the practical implication of the difference in scattered signal levels between the conventional MST and the MLS

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Fig. 10. Magnitude of the estimated electric field distribution in the presence of measurement uncertainty obtained using conventional MST array and the MLS  and 10 PIN diodes. of the same length L

= 10

method on the accuracy of the estimated electric field distribution, both methods were used to recover the incident field (15) with the consideration of the receiver noise presence, i.e., un. Comcertainty in the measured scattered field at plex Gaussian uncertainty with zero mean and standard deviper dimension was added to the computed ation of scattered field observations before they were processed by both methods. In this experiment, the MST method was given the advantage of perfect calibration using a uniform plane-wave. Fig. 10 shows the magnitude of the actual incident field and the estimated magnitude distributions obtained using the MST and the MLS method after coherent averaging over 100 modulation cycles. The results in Fig. 10 show that the MLS method is more immune to measurement uncertainty due to the its relatively high RCS as expected. As mentioned earlier, the effects of static loads, i.e., fixed loads, that are not controlled dynamically in the measurement process such as dc-block capacitors and load biasing structures do not impact the accuracy of the MLS method since the differential scattered field is used in recovering the field of interest. This point was verified via simulation and is considered as an advantage of the MLS method since the effect of any fixed unintentional parasitic loads, e.g., due to component mounting, is nulled out in the process. In the simulation results presented here, the PIN diodes were assumed to have ideal load values in both ON and OFF states, i.e., short and open, respectively. The sensitivity of the MLS method to non-ideal load values was studied via simulations as well by considering practical PIN diode loads. Theoretically, using non-ideal load values do not affect the performance of MLS method as long as these values are known. In practice, however, the non-ideal loads might decrease the sensitivity of the MLS method in the presence measurement uncertainty. With a non-ideal PIN diode which represents relatively high forwardand low reverse-biase impedances, the change in the MLS RCS between the reference condition and some of the remaining conditions might be reduced. Typical high frequency PIN diodes

Fig. 11. Modulated RCS as a function of the loading condition for ideal and . typical non-ideal load values with M

= 10

have forward-biase resistance less than 6 Ohm and reverse-biase capacitance in the order of 0.02 pF [17]. Consequently, the nonideal PIN diode admittance in the ON and OFF states can be and around represented by 10 GHz, respectively. Fig. 11 shows the modulated RCS of the as a function of the loading condition when MLS with ideal and practical load values are used. The modulated RCS was computed as the absolute difference between RCS of the first loading condition and all other conditions. The reduction in the modulated MLS RCS due to the non-ideal loads is apparent in Fig. 11. It is remarked that the modulated RCS of the MLS with non-ideal loads considered here is still higher than that of the conventional MST with ideal loads as shown in Fig. 11. It is emphasized that the load values should be known a priori in order to implement the MLS method. Furthermore, the scatterer structure should be modeled accurately. These are not hard-limiting factors since the load values can be measured experimentally and the structure can be modeled using available numerical techniques with high accuracy over a wide range of frequencies. V. CONCLUSION To improve upon the limitations associated with the conventional modulated scatterer technique (MST), a new electric field distribution measurement method based on modulated multiple loaded linear scatterer (MLS) was developed and analyzed. The MLS method is shown to overcome most of the challenges encountered when using the conventional MST-based array of dipoles. The developed MLS method estimates the short-circuit current distribution induced over the length of the scatterer from multiple scattered field observations taken at a single point using different modulation conditions. Consequently, the estimated current distribution is used to recover the incident electric field distribution of interest. The operation of the MLS method was verified via numerical simulations and its performance was compared to that of the conventional MST. It was shown that the MLS method offers distinct advantage over the

ABOU-KHOUSA AND ZOUGHI: MULTIPLE LOADED SCATTERER METHOD FOR E-FIELD MAPPING APPLICATIONS

conventional MST as it relates to the accuracy of the recovered field distributions from noisy scattered field observations. It is remarked that the proposed method should provide an efficient alternative to the conventional and well-established MST arrays in applications where higher sensitivity is required such as in imaging applications. The proposed method can be applied for 2D field mapping by combining many MLS elements, i.e., to form modulated surfaces, provided that the mutual coupling between these elements is properly accounted for. Future work will consider constructing a prototype MLS-based electric field measurement system. Other MLS configurations and geometries will also be subject of future research. REFERENCES [1] J. H. Richmond, “A modulated scattering technique for measurement of field distributions,” IEEE Trans. Microw. Theory Tech., vol. 3, no. 4, pp. 13–15, Jul. 1955. [2] J.-C. Bolomey and G. E. Gardiol, Engineering Applications of the Modulated Scatterer Technique. Norwood, MA: Artech House, 2001. [3] T. P. Budka, S. D. Waclawik, and G. M. Rebeiz, “A coaxial 0.5–18 GHz near electric field measurement system for planar microwave circuits using integrated probes,” IEEE Trans. Microw. Theory Tech., vol. 44, no. 12, pt. 1, pp. 2174–2184, Dec. 1996. [4] S. Caorsi, M. Donelli, and M. Pastorino, “A passive antenna system for data acquisition in scattering applications,” IEEE Antennas Wireless Propag. Lett., vol. 1, no. 1, pp. 203–206, 2002. [5] M. A. Abou-Khousa and R. Zoughi, “Electric field measurement using multiple loaded linear scatterers,” in Proc. Int. Symp. on Antennas and Propag. ISAP’08, Taipei, Taiwan, Oct. 2008, pp. 565–568. [6] R. Harrington and J. Mautz, “Straight wires with arbitrary excitation and loading,” IEEE Trans. Antennas Propag., vol. 15, no. 4, pp. 502–515, Jul. 1967. [7] R. Harrington and J. Mautz, “Pattern synthesis for loaded N-port scatterers,” IIEEE Trans. Antennas Propag., vol. 22, no. 2, pp. 184–190, Mar. 1974. [8] R. Harrington and J. Mautz, “Optimization of radar cross section of N-port loaded scatterers,” IEEE Trans. Antennas Propag., vol. 22, no. 5, pp. 697–701, Sep. 1974. [9] A. Poggio and P. Mayes, “Bandwidth extension for dipole antennas by conjugate reactance loading,” IEEE Trans. Antennas Propag., vol. 19, no. 4, pp. 544–547, Jul. 1971. [10] M.-K. Hu, “On measurements of microwave E and H field distributions by using modulated scattering methods,” IRE Trans. Microw. Theory Tech., vol. 8, no. 3, pp. 295–300, May 1960. [11] J.-C. Bolomey et al., “Rapid near-field antenna testing via arrays of modulated scattering probes,” IEEE Trans. Antennas Propag., vol. 36, no. 6, pp. 804–814, Jun. 1988. [12] R. F. Harrington, “Small resonant scatterers and their use for field measurements,” IRE Trans. Microw. Theory Tech., vol. MTT-10, pp. 165–174, May 1962. [13] O. Franza, N. Joachimowicz, and J.-C. Bolomey, “SICS: A sensor interaction compensation scheme for microwave imaging,” IEEE Trans. Antennas Propag., vol. 50, no. 2, pp. 211–216, Feb. 2002. [14] C. A. Balanis, Advanced Engineering Electromagnetic. New York: Wiley, 1989. [15] R. Harrington, “Matrix methods for field problems,” Proc. IEEE, vol. 55, no. 2, pp. 136–149, Jan. 1967.

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[16] M. Abou-Khousa and R. Zoughi, “Maxwellian circuits-based analysis of loaded wire antennas and scatterers,” IEEE Antennas Wireless Propag. Lett., vol. 7, pp. 318–320, 2008. [17] Tyco Electronics, M/A-COM Wirless Components, [Online]. Available: http://www.macom.com/DataSheets/MA4AGP907_FCP910.pdf

Mohamed A. Abou-Khousa (M’09) was born in Al-Ain, UAE, in 1980. He received the B.S. E.E. degree (magna cum laude) from the American University of Sharjah (AUS), Sharjah, UAE, in 2003, the M.S. E.E. degree from Concordia University, Montreal, QC, Canada, in 2004, and the Ph.D. degree in electrical engineering from Missouri University of Science and Technology (Missouri S&T), Rolla, in 2009. Currently, he is a Research Engineer (RF) with the Imaging Research Laboratories, Robarts Research Institute, London, ON, Canada. His current research interests include millimeter wave and microwave instrumentation, numerical electromagnetic analysis, modulated antennas, RF design for high field MRI systems, and wideband wireless communication systems. Dr. Abou-Khousa frequently serves as a reviewer for various IEEE technical publications.

R. Zoughi (S’85–M’86–SM’93–F’06) received the B.S.E.E., M.S.E.E., and Ph.D. degrees in electrical engineering (radar remote sensing, radar systems, and microwaves) from the University of Kansas, Lawrence. From 1981 until 1987, he was at the Radar Systems and Remote Sensing Laboratory (RSL), University of Kansas. Currently he is the Schlumberger Endowed Professor of Electrical and Computer Engineering at the Missouri University of Science and Technology (Missouri S&T), Rolla, formerly the University of Missouri-Rolla (UMR). Prior to joining Missouri S&T in January 2001 and since 1987, he was with the Electrical and Computer Engineering Department, Colorado State University (CSU), where he was a Professor and established the Applied Microwave Nondestructive Testing Laboratory (amntl). He held the position of Business Challenge Endowed Professor of Electrical and Computer Engineering from 1995 to 1997 while at CSU. He is the author of the textbook Microwave Nondestructive Testing and Evaluation Principles (Kluwer Academic Publishers, 2000) and the coauthor of a chapter on Microwave Techniques in an undergraduate introductory textbook entitled Nondestructive Evaluation: Theory, Techniques, and Applications (Marcel and Dekker, Inc., 2002). He is the coauthor of over 100 journal papers, 240 conference proceedings and presentations and 85 technical reports. He has nine patents to his credit all in the field of microwave nondestructive testing and evaluation. Dr. Zoughi has been the recipient of numerous teaching awards both at CSU and Missouri S&T. He was the recipient of the 2009 American Society for Nondestructive Testing (ASNT) Research Award for Sustained Excellence and the 2007 recipient of the IEEE Instrumentation and Measurement Society Distinguished Service Award. He is a Fellow of the American Society for Nondestructive Testing (ASNT), and the Editor-in-Chief of the IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT. For more information please see http://amntl.mst.edu/people/zoughi.html.

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Deterministic Approach for Spatial Diversity Analysis of Radar Systems Using Near-Field Radar Cross Section of a Metallic Plate Ramin Deban, Halim Boutayeb, Member, IEEE, Ke Wu, Fellow, IEEE, and Jean Conan

Abstract—A deterministic analysis of spatial diversity is presented in connection with radar systems. A numerical technique based on physical optics is used for our analysis. Contrary to statistical models, the proposed technique takes into account accurate near-field radar cross section of the target, and radiation characteristics of transmitting and receiving antennas. The power scattered by the target and received by multiple antennas as a function of the target aspect angle and distance is analyzed. Two combining methods of received powers are tested and statistical analysis is performed showing that, using spatial diversity, the angular range can be increased significantly and the standard deviation of the target response can be reduced. In order to validate our analysis and proposed scheme, experimental measurements were carried out using a metallic plate and a car as targets. This work has potential applications in automotive collision warning/avoidance radar systems. Index Terms—Automotive collision warning, near-field radar cross section (RCS), physical optics (PO), radar, spatial diversity, target fading.

I. INTRODUCTION N telecommunications, spatial diversity and other types of diversity scheme (polarization diversity, frequency diversity and beam-switching) used in multiple input multiple output (MIMO) systems [1] are well known for reducing Rayleigh fading and decreasing the error rate. In radar systems, spatial diversity is a concept that has been recently proposed to improve the detection performance [4]. This concept is different from beam-forming technique used in radar composed of an antenna array [5]–[7]. The main idea of spatial diversity, as proposed in [4], is to exploit the target angular spread to combat scintillation effect. Different architectures of phased arrays suitable for automotive collision avoidance radar, remote sensing, tactile missile and communication applications have been developed [5]–[7]. They allow rapid non-mechanical beam scanning which reduces multi-path fading. However, another type of fading due to the

I

Manuscript received April 28, 2009; revised July 13, 2009. First published December 31, 2009; current version published March 03, 2010. This work was supported by the National Science Engineering Research Council of Canada (NSERC) and Centre de Recherche En Electronique Radiofrequence (CREER). R Deban was with the Electrical Engineering Department, Ecole Polytechnique, Montreal, QC H3T 1J4, Canada. He is now with SCP Science, Montreal, QC H9X 4B6, Canada (e-mail: [email protected]). H. Boutayeb, K. Wu, and J. Conan are with the Electrical Engineering Department, Ecole Polytechnique, Montreal, QC H3T 1J4, Canada. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2009.2039328

scintillation of target radar cross section (RCS) is not resolved with an agile antenna system. Most targets present RCSs that vary strongly with the aspect angle and target position, implying that a rotation of the target of a fraction of degree or its lateral motion of a few centimeters could introduce fluctuations of more than 10 dB in the received signal power [8]–[11]. Target scintillations can have two effects. For a moving object they can introduce an amplitude modulation of the received power which leads to an additional multiplicative noise [9]. For a stationary target, scintillations can also reduce the probability of detection when the received power falls below the threshold level of the receiver. Since the amplitude of transmitted power is limited due to the regulatory constraints, it is necessary to reduce the target scintillations in order to increase the probability of detection. A method to reduce the target fluctuations and widen the angular range consists in using multiple and enough spaced receiving antennas. If the receiving antennas are sufficiently spaced, different aspects of the target can be observed by the receiver. In [4], using a statistical model, the authors have analyzed the effect of spatial diversity in radars, and they have shown that better performances in terms of detection probability are obtained with this type of radar compared to the performance of the phased array radars. The main purpose of this paper is to show the effect of spatial diversity in radar systems using a realistic geometrical model, which takes into account the accurate radar cross section of targets as well as the radiation characteristics of the transmitting and receiving antennas. A numerical technique based on physical optics (PO) and near field RCS is used to analyze the effects of rotation or lateral motion of a target on the received power, when one or multiple receiving antennas are used. Car crashes are mostly caused by human errors. An approach for reducing accidents is to make cars smarter, enabling them to sense potential collision automatically and possibly forcing them to take control of braking and steering systems. Automobile collision avoidance radar has become a recent feature in automotive design [12]–[15]. In order to be most effective, a combination of medium-range detection (in front of the car) and short-range detection (in all directions) is required in automotive radars. Ultrawideband (UWB) has been allocated for short-range detection at 24 GHz in USA, and temporally in Europe [12]. Consequently, we selected 24 GHz as the operating frequency. A potential application of the analysis proposed in this paper would be in an automotive collision-warning radar

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Fig. 1. Collision warning/avoidance radar scenario, showing how the target can turn.

system that determines the distance between two cars on the same lane. A problem in such an application is that the target can turn at any time, in any direction, as illustrated in Fig. 1. It can also move laterally. As a result, due to the scintillations in the radar cross section of the target, important fluctuations in the received power are observed. As demonstrated in this paper, spatial diversity can allow reducing these scintillations. Similarly to mobile communications, different combining methods capitalizing on uncorrelated fading of separate antennas in a space diversity array can be considered [16]. Selection diversity combining (SDC) is probably the simplest method. This method consists in connecting the receiver having the highest signal-to-noise ratio (SNR) to the output. Another technique, called “maximal ratio combining” (MRC) was proposed by Kahn [17]. In this method, the signals are weighed proportionately to their power to noise ratios and then summed. The inconvenience of this method is that the signal must be co-phased before combining. Furthermore, it might be difficult to provide variable weighting of signals. Equal gain combining (EGC) is a simpler method where the gains are all set to unity. In this paper, we restrict ourselves to use SDC and EGC techniques to evaluate the performance of spatial diversity in radars systems. The remainder of the paper is organized as follows. Section II describes the numerical technique that is used for the analysis, based on a semi-analytical model formula for the radar cross section of a metallic plate. In Section III, the numerical analysis of received and combined powers for two receiving antennas is presented. Section IV presents the experimental results for the two types of targets: a metallic plate and a car. Finally, the concluding remarks are given in Section V.

Fig. 2. Model of radar with spatial diversity: one antenna is transmitting and multiple antennas are receiving the response of the target.

where is the free-space wavelength, and represent the eland are the transmitevation and azimuth aspect angles, ting and receiving antennas gains, is the radar cross section of denote the distances between the target the target, and and and the transmitting and receiving antennas, respectively. We assume that the target can only turn along the azimuth , which is suitable for collision warning/avoidance angle radar applications. Using the coordinate system shown in , the Fig. 2, the transmitting antenna is at the position , for receiving antennas are at the positions and the target is at the position . B. Far-Field Radar Cross Section of a Metallic Plate Most targets have radar cross sections that vary strongly when the target is turned by a few degrees. Let us consider a perfectly . For our conducting rectangular sheet plate of dimension application, the target is at the same level as transmitting and . Moreover, receiving antennas so the elevation angle we are interested in calculating the received power for small . From this, the simple PO method aspect angles which is accurate only for angles near specular direction can be used [18]. Using PO, radar cross section of the plate is given by (2)

II. NUMERICAL APPROACH

(3)

In this section, we present the technique that we use for analyzing spatial diversity in radar systems. To simplify the analysis, a metallic object with a canonical rectangular shape is used as a model of the target. A. Physical Model and Computation of the Received Power The model of radar with multiple receiving antennas to be considered in this paper is schematized in Fig. 2. Neglecting losses in receiving and transmitting antennas, and using raytracing analysis, the power at the receiving antenna can be written as (1)

where is the free-space wave-number. Fig. 3 depicts the RCS of a metallic sheet using (2)–(3) as a function of the aspect angle , at 24 GHz, with m, which corresponds to the dimensions of a metallic plate available at our laboratory. From this curve, the RCS can vary by more than 10 dB over a small angle variation. C. Near-Field RCS of a Metallic Plate Equations (2)–(3) give the far-field RCS of a rectangular metallic plate using PO. The limit between near-field and far-field is usually set to be at a distance where the high-order terms of scattered magnetic field can be neglected [18]. This occurs at where is the greater dimension between

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Fig. 3. Far-field RCS of a 0.6 based on PO.

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2 0.6 m

metallic sheet, at 24 GHz, calculated Fig. 5. Near-field RCS of a 0.6 range, at normal aspect angle.

2 0.6 m

metallic sheet, at 24 GHz versus

metallic plate versus aspect angle at 24 GHz and different ranges: 3, 5 and 10 m. From these results, it is important to note that the RCSs are quite different at 3 m and 10 m. The calculated RCS tends to the far-field RCS (Fig. 3) when the target range is increased. Fig. 5 depicts the RCS of the metallic plate at normal aspect angle versus range. This curve is quite similar to the result presented in [19], Fig. 16. Our calculation method is also successfully validated experimentally in Section IV. III. NUMERICAL ANALYSIS

Fig. 4. Near-field RCS of a 0.6 24 GHz and different distances.

2 0.6 m

metallic sheet versus aspect angle, at

those of antennas and scatterer. For a 0.6 0.6 m metallic sheet at 24 GHz, the far-field starts at 58 m. Therefore, in our application where the target is usually at a few meters, it is more appropriate to use near-field RCS rather than far-field RCS. The computation of the RCS of a metallic plate in near-field has been studied by several authors using PO Integration [19]–[21]. However, to our knowledge, the near- field RCS of a metallic plate for different aspect angles has not been reported yet. The method proposed in [19]–[21] consists in dividing the plate into a number of small cells such that far-field RCS can be applied for each cell. The near-field RCS of the plate is obtained by integrating the radar cross section contributions over the entire plate while taking into account the phase factors. For example, in [19], Pouliguen et al. have computed the near-field RCS of a 1 1 m metallic plate at normal aspect angle versus ranges. In this work, we use the near-field RCS of a metallic plate for different aspect angles using PO integration. Fig. 4 shows the numerical results for the near-field RCS of a 0.6 0.6 m

In this section, based on the proposed metallic sheet model, we present the analysis of the received power in the case of two receiving antennas. The simulation includes a metallic sheet having the dimensions 0.6 0.6 m . All the antennas are identical 15 dB gain pyramidal horns. The antenna pattern has been obtained using full-wave finite element method (Ansoft HFSS) and incorporated in ADS (Agilent Technologies) schematic design. Note that, in this work, we use bistatic radar cross section of the target by calculating bistatic bisector aspect angles relative to transmitting and receiving antennas [18]. The configuration as illustrated in Fig. 6 is considered. The transmitting antenna is located at (0, 0, 1 m) and the two receiving antennas at positions ( m, 1 m) and (0, 0.2 m, 1 m) respectively. Different distances (3–40 m) between the conducting plate and transmitting antenna were tested. At this point, one remarks that the half-power beamwitdh (HPBW) of a 15 dB gain antenna is around 36 . At m this angle corresponds to a segment of about 2 m. Knowing that the metallic plate has a 0.6 m width, we can conclude that, at this distance or farther, the illumination is quasi-uniform on the metallic plate. When viewed from behind the horn antennas, we call the received power from the left/right receiving antenna . This convention is used throughout the whole paper. The source provides 30 dBm power signal at 24 GHz. The ADS simulated received signal power versus aspect angle

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TABLE I STANDARD DEVIATION AND ANGULAR RANGE

Fig. 6. Schematic for the analysis of the effect of the metallic sheet rotation on received power.

Fig. 7. Simulations results of received and combined signal powers for two receiving antennas positioned at 0.2 m, at 24 GHz, r m. Received powers , (b) at , (c) after applying SDC (d) after applying EGC. (a) at

Rx

Rx

=3

is shown in Fig. 7(a) and (b) for the two receiving antennas, at m.

Fig. 5(c) and (d) shows the results obtained by combining the two received powers using, respectively, selection diversity combining (SDC) and equal gain combining (EGC) techniques. From these curves, one can see qualitatively that by applying anyone of the combining methods power variations are decreased and angular range is improved. To quantify scintillation mitigation, standard deviations of individual received powers and those obtained after applying SDC or EGC techniques have been calculated for [0 –10 ] region and are reported in Table I. This angular range limit is enough for our application (automotive collision warning/avoidance), especially in the highway. In the Table I, we present also the angular range defined as the distance between the points corresponding to half-power signal. To evaluate the standard deviation reduction, we use the greater standard deviation between those of powers and , because the target can turn at negative as well as positive angles. Hence from Table I, it can be seen that at 3 m standard deviation is reduced by 3.8 dB in the [0 –10 ] region and angular range is increased by 3.8 using SDC or EGC methods. This corresponds to more than 40% improvement in scintillation mitigation and angular range enhancement for [0 –10 ] region. It should also be noted that when the received power is too low for [10 –20 ] region, the reduction of standard deviation is less than 0.5 dB. However, simulations show that this could be increased to 1 dB if the receiving antennas are located at 0.25 m. The reduction of scintillation effect versus position of receiving antennas is investigated later in this section. As mentioned previously, for our application we limit the angular range to [0 –10 ] region. It is convenient at this point to present mitigation in scintillation effects using SDC or EGC methods as a function of distance . Fig. 8 presents the reduction of standard deviation for ranging from 3 to 40 m, in [0 –10 ] angular region. From Fig. 8, it is worth noting that reduction of standard deviation is more than 2 dB for distance less than 4 m, more than 1 dB in 12–15 m range and around 2.5 dB for ranges near 30 m. Furthermore, from Fig. 8, one can note that SDC and EGC techniques give essentially similar results. Hence in this case and in practice, SDC method is advantageous over EGC because of ease in implementation. The reduction of scintillation versus range and position of receiving antennas has also been investigated numerically and are depicted in Fig. 9. For the target at a specific range, bright areas show reduction of 2 dB or more that can be obtained by locating the receiving antennas appropriately. From this figure, it is clear that for the target at 3–4 m, standard deviation reduction of more than 2 dB is easily achievable choosing any position of receiving antennas. For other ranges, although more than one solution may be available, the position of receiving antennas must be chosen according to the numerical analysis results.

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Fig. 8. Simulations results of standard deviation reduction obtained using SDC or EGC techniques versus distance r , for receiving antennas positioned at 0.2 m from transmitting antenna.

Fig. 9. Simulations results of standard deviation reduction obtained using EGC technique versus distance r and position of receiving antennas.

To investigate the angular range improvement, combined rem for two posiceived powers versus aspect angle at tions of receiving antennas (at m and m) are depicted in Fig. 10. In this figure, the received power for an antenna placed at the same position as the transmitting antenna is also presented as the reference for evaluating the angular range enhancement. From Fig. 10, we observe about 4 of angular range enhancement for receiving antennas at m. In addition if the distance between transmitting and receiving antennas is too long, as shown in Fig. 10 with receiving antennas at m, significant power decay appears at normal aspect angle. Fig. 11 presents angular range enhancement versus position of receiving antennas at different ranges (3, 5, and 10 m). In these curves, for each range, the positions of receiving antennas are limited because of the power decay at normal aspect angle (see Fig. 10). From Fig. 11, increasing the distance between receiving antennas improves the angular range. It is also

Fig. 10. Simulations results of combined powers (using SDC) for receiving m. Result for receiving antennas at 0 antennas at 0 m, 0.2 m and 0.6 m at r m (center) is used as a reference for calculating the angular range enhancement.

=3

Fig. 11. Angular range enhancement versus position of receiving antennas at ; and 10 m, applying SDC. ranges r

:35

seen in this figure that smaller angular range enhancement is observed for long ranges. Furthermore, for long ranges, increasing the distance of receiving antennas has less effect on the angular range. To conclude this section, the position of the receiving antennas should be chosen adequately to mitigate the scintillation effect and to increase the angular range. The best improvements are observed at short ranges (less than 4 m) for any position of receiving antennas. At long range, significant angular range enhancement cannot be achieved but important standard deviation reduction can be obtained. These observations are validated in the next section using experimental results. IV. EXPERIMENTAL RESULTS To validate the proposed analysis, measurements were carried out at 24 GHz. In Subsection A we present the experimental

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Fig. 12. Measurement setup in anechoic chamber. (a) Transmitting and receiving antennas. (b) Metallic plate.

results obtained for a metallic plate in anechoic chamber and Subsection B is dedicated to outdoor measurements using a car as a target. A. Measurement Set-Up and Data for the Metallic Plate Fig. 12(a) shows the set-up used in the compact range anechoic chamber of our research center for the measurements. A directive transmitting antenna and two directive receiving antennas are considered, consisting of 15 dB gain pyramidal horn antennas. A 0.6 0.6 m metallic sheet, as shown in Fig. 12(b), is positioned at m from the transmitting antenna. The receiving antennas are 0.2 m apart from the transmitting antenna, and on the same level as the metallic sheet center. Only relative powers at the two receiving antennas can be measured in the anechoic chamber. Consequently, all power values are normalized and expressed in dB. A 0.1 resolution angle has been used. The normalized measured and simulated powers versus aspect angle are shown in Fig. 13. It can be noted that in spite of a small misalignment (less than 0.5 ), the measured signal variations compare favorably with the simulated ones up to 10 , which validate our near-field numerical calculation method. We also carried out measurements where the metallic sheet was replaced by absorbers for evaluating the noise floor in the anechoic chamber. According to the results, the relative noise dB. This mainly explains the discrepancies floor is around between simulated and measured results for angles beyond 10 (see Fig. 13). As in the previous section, we analyze standard deviations and angular range of received and combined powers for aspect angles ranging from 0 to 10 . The results are depicted in

Fig. 13. Measured and simulated received powers for two receiving antennas m. Received powers (a) at and (b) at located at 0.2 m, at 24 GHz, r .

Rx

=3

Rx

TABLE II STANDARD DEVIATION AND ANGULAR RANGE

Table II, showing that standard deviation can be reduced by 3.3 dB and angular range can be improved by 4 . These results are in good accordance with the simulations (see Table I). B. Measurement Set-Up and Results for a Car In order to link our analysis, which uses a metallic plate as a target, to collision avoidance/warning radar application, shortrange measurements were carried out at 24 GHz with a real car. Fig. 14 shows the power detection system used for these measurements. The experimental data was collected at PolyGRAMES parking lot using a Toyota Corolla, which is one of the most popular cars in North America.

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Fig. 14. Power detection system at 24 GHz. One emitting antenna and two receiving antennas are at 10 m from the car (Toyota Corolla).

As previously, a directive transmitting antenna and two directive receiving antennas are considered, consisting of 15 dB gain pyramidal horn antennas. All antennas are positioned at 10 m from the car. The receiving antennas are at 0.6 m from the transmitting antenna. This distance has been chosen because it is possible to incorporate two antennas spaced by 1.2 m inside front bumper of most popular cars. The transmitting source provides 30 dBm power signal at 24 GHz. For different aspect angles the measurement set up has been moved with a step of 8.7 cm corresponding to 0.5 , from 0 to 20 . In the experiment, we observed the noise floor at about 20 dB below the maximum measured peak. The measured results of received and combined powers are plotted in Fig. 15. As noted in previous section, it is difficult to evaluate the angular range enhancement at this range (10 m). However, from Fig. 15, it can be noted that the combined signals have less scintillations with respect to the signals received by each antenna. Using these results, Table III presents the statistical measures of the four signals ranging from 0 to 10 . This table shows that the second combining method performs better. It can reduce standard deviation of the received signal by 2.2 dB for [0 –10 ] region. One can note that this reduction is significant compared to the maximum standard deviations of received powers before applying the combining methods (4.9 dB for and 4.3 dB for ).

Fig. 15. Measured received and combined signal powers for two receiving antennas. Received powers (a) at Rx , (b) at Rx , (c) after applying SDC, (d) after applying EGC.

V. CONCLUSION

TABLE III STANDARD DEVIATION

Using two receiving antennas, an analysis of spatial diversity in radar systems has been proposed, taking into account the near-field RCS of the target and the radiation characteristics of the transmitting and receiving antennas. A parametrical study of the received power for different positions of receiving antennas and aspect angles of the target consisting of a rectangular metallic plate has been presented. Applying combining methods, numerical and experimental results show that scintillations of the received power at 24 GHz from a 0.6 0.6 m metallic plate can be mitigated and the angular range can be improved. At ranges less than 4 m, we observed a reduction of the standard deviation of more than 2.5 dB in ] region and an enhancement of the angular range of more than 4 . Our near-field calculation method was successfully validated by experimental measurements carried out in an anechoic chamber using 0.1 resolution angle.

At long ranges (10–40 m), numerical results show that significant angular range enhancement cannot be achieved but important standard deviation can be obtained (more than 2 dB). Furthermore, measurement results at 24 GHz of received powers from a vehicle at 10 m have been presented to link our work to automotive collision warning/avoidance radar applications. These measurements show that, using two receiving antennas, one can reduce standard deviation of the received signal power by 2.2 dB for aspect angle in [0 –10 ] region. As a conclusion, we have analyzed and demonstrated quantitatively the advantage of using multiple receiving antennas in radar systems.

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REFERENCES [1] A. Goldsmith, Wireless Communications. New York: Cambridge Univ. Press, 2005. [2] C. B. Dietrich, K. Dietze, J. R. Nealy, and W. L. Stutzman, “Spatial, polarization and pattern diversity for wireless handheld terminals,” IEEE Trans. Antennas Propag., vol. 49. [3] A. S. Konanur, K. Gosalia, S. H. Krishnamurthy, B. Hughes, and G. Lazzi, “Increasing wireless channel capacity through MIMO systems employing co-located antennas,” IEEE Trans. Microw. Theory Techn., vol. 53, pp. 1837–1844, Jun. 2005. [4] E. Fishler, A. Haimovich, R. S. Blum, L. J. Cimini, D. Jr. Chizhik, and R. A. Valenzuela, “Spatial diversity in radars: Models and detection performance,” IEEE Trans. Sign. Proc, vol. 54, pp. 823–838, Mar. 2006. [5] S. Haykin, J. Litva, and T. J. Shepherd, Radar Array Processing, 1st ed. New York: Springer-Verlag, 1993. [6] D. Parker and D. C. Zimmermann, “Phased array—Part II: Implementations, applications, and future trends,” IEEE Trans. Microw. Theory Tech., vol. 50, pp. 688–698, Mar. 2002. [7] L. Schulwitz and A. Mortazawi, “A compact dual-polarized multibeam phased-array architecture for millimeter-wave radar,” IEEE Trans. Microw. Theory Tech., vol. 53, pp. 3588–3594, Nov. 2005. [8] N. Levanon, Radar Principles, 1st ed. New York: Wiley, 1998. [9] J. H. Dunn, D. D. Howard, and A. M. King, “Phenomena of scintillation noise in radar-tracking systems,” in Proc. IRE, May 1959, vol. 43, pp. 855–863. [10] H. L. V. Trees, Detection, Estimations, and Modulation Theory. New York: Wiley, 1968, vol. III. [11] M. Skolnik, Introduction to Radar Systems, 3rd ed. New York: McGraw-Hill, 2002. [12] P. Bell, “Impact iminent [automative radar],” IEE Rev., vol. 50, pp. 42–45, May 2004. [13] M. E. Russell, A. Crain, A. Curran, R. A. Campbell, C. A. Drubin, and W. F. Miccioli, “Millimeter-wave radar sensor for automotive intelligent cruise control,” IEEE Trans. Microw. Theory Tech., vol. 45, pp. 2444–2453, Dec. 1997. [14] C. Eckersten and B.-O. As, “A high performance automotive radar for adaptive cruise control and collision warning/avoidance,” in Proc. IEEE Conf. on Intelligent Transportation System, Boston, MA, Nov. 1997, pp. 446–451. [15] M. E. Russell, C. A. Drubin, A. S. Narinilli, W. G. Woodington, and M. J. Del Checcolo, “Integrated automotive sensors,” IEEE Trans. Microw. Theory Tech., vol. 50, pp. 674–677, Mar. 2002. [16] W. C. Jakes Jr., Microwave Mobile Communications. New York: Wiley, May 1974. [17] L. R. Kahn, “Ratio squarer,” in Proc. IRE, Nov. 1954, vol. 42, pp. 1704–. [18] G. T. Ruck, D. E. Barrick, W. D. Stuart, and C. K. Krichbaum, Radar Cross Section Handbook. New York: Plenum Press, 1970. [19] P. Pouliguen, J. F. Damiens, R. Hemon, and J. Saillard, “RCS computation in near field,” in Proc. Int. Conf. Days on Diffraction, May–Jun. 2006, pp. 252–265. [20] W. B. Gordon, “Near field calculations with far field formulas,” in IEEE Antennas Propag. Symp. Dig., July 1996, vol. 2, pp. 950–953. [21] S. R. Legault, “Refining physical optics for near-field computations,” Electron. Lett., vol. 40, no. 1, pp. 71–72, Jan. 2004.

Ramin Deban received the B.Sc. degree in telecommunications from Sharif University, Teheran, Iran, in 1985 and the Diplome d’Ingenieur in electronics from Ecole Nationale Superieure d’Electrotechnique d’Electronique d’Informatique et d’Hydraulique de Toulouse (E.N.S.E.E.I.H.T), France, in 1889. He was working in France and Canada for more than ten years before starting the research program in Poly-Grames in 2003. From January 2003, he was with the Ecole Polytechnique de Montreal in M.S. Eng. and Ph.D. degrees. His main fields of interest are antennas, radars, local positioning systems, microwaves circuits and RF technologies. Since September 2009, he has been with SCP Science, Montreal, as an R&D Manager.

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Halim Boutayeb (M’03) received the Diplome d’Ingenieur (B.Sc.) degree in signal processing and telecommunication from the Institut de Formation Superieure en Informatique et Communication, Rennes, France, in 2000 and the D.E.A. (M.Sc.) and Ph.D. degrees in electrical engineering from the University of Rennes, France, in 2000 and 2003, respectively. From March 2004 to December 2006, he was with the Institut National de la Recherche Scientifique, Montreal, Quebec, Canada. Since January 2007, he has been a Researcher with Ecole Polytechnique de Montreal. He has also been Coordinator and a member of the Centre de Recherche en Electronique Radiofrequence (CREER), which is a strategic cluster that provides a unique platform for putting together 35 professors/researchers of Quebec in the field of applied electromagnetics and RF technologies. His main fields of interest are antennas, microwaves circuits, FDTD method, artificial materials, radars, local positioning systems, and phased arrays. He has authored or coauthored more than 55 journal and conference papers and has received one patent. He has been a Reviewer for a number of scientific journals and conferences. Dr. Boutayeb received the NSERC Postdoctoral Fellowship (2004–2006). He was a co-recipient of the Best Paper Award at the JINA international conference in November 2004. He was a Technical Program Committee member of the IEEE Vehicular Technology Conference 2006.

Ke Wu (M’87–SM’92–F’01) received the B.Sc. degree (with distinction) in radio engineering from Nanjing Institute of Technology (now Southeast University), China, in 1982 and the D.E.A. and Ph.D. degrees in optics, optoelectronics, and microwave engineering (with distinction) from Institut National Polytechnique de Grenoble (INPG) and University of Grenoble, France, in 1984 and 1987, respectively. He is a Professor of electrical engineering, and Tier-I Canada Research Chair in RF and millimeter-wave engineering at Ecole Polytechnique (University of Montreal). He also holds a number of visiting (guest) and honorary professorships at various universities including the first Cheung Kong Endowed Chair Professorship at Southeast University, the first Sir Yue-Kong Pao Chair Professorship at Ningbo University, and Honorary Professorship at Nanjing University of Science and Technology and City University of Hong Kong. He has been the Director of the Poly-Grames Research Center and has recently become the Founding Director of the “Centre de recherche en électronique radiofréquence” (CREER) of Quebec. He has (co)authored over 680 referred papers, a number of books/book chapters and patents. His current research interests involve substrate integrated circuits (SICs), antenna arrays, advanced CAD and modeling techniques, and development of low-cost RF and millimeter-wave transceivers. He is also interested in the modeling and design of microwave photonic circuits and systems. He serves on the Editorial Board of the Microwave Journal, Microwave and Optical Technology Letters, and Wiley’s Encyclopedia of RF and Microwave Engineering. He is an Associate Editor of the International Journal of RF and Microwave Computer-Aided Engineering (RFMiCAE). Dr. Wu is a member of Electromagnetics Academy, the Sigma Xi Honorary Society, and the URSI. He has held many positions in and has served on various international committees, including Co-Chair of the Technical Program Committee (TPC) for 1997 and 2008 Asia-Pacific Microwave Conferences (APMC), General Co-Chair of 1999 and 2000 SPIE’s Inter. Symposia on Terahertz and Gigahertz Electronics and Photonics, General Chair of 8th Inter. Microwave and Optical Technology (ISMOT’2001), TPC Chair of 2003 IEEE Radio and Wireless Conference (RAWCON’2003), General Co-Chair of RAWCON’2004, Co-Chair of 2005 APMC Inter. Steering Committee, General Chair of 2007 URSI Inter. Symp. on Signals, Systems and Electronics (ISSSE), and General Co-Chair of 2008 and 2009 Global Symposia on Millimeter-Waves, and Inter. Steering Committee Chair of 2008 Int. Conference on Microwave and Millimeter-Wave Technology. In particular, he will be General Chair of 2012 IEEE MTT-S International Microwave Symposium (IMS). He has served on Editorial or Review Boards of various technical journals, including the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, and the IEEE MICROWAVE AND WIRELESS COMPONENTS LETTERS. He served on the Steering Committee for the 1997 joint IEEE AP-S/URSI Inter. Symp. and the TPC for the IEEE MTT-S Inter. Microwave Symp. He is currently Chair of the joint IEEE chapters of MTTS/APS/LEOS in Montreal. He is an elected MTT-S AdCom

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member for 2006–2012 and was Chair of the IEEE MTT-S Transnational Committee. He is Chair of the newly formed IEEE MTT-S Member and Geographic Activities (MGA) Committee. He was the recipient of a URSI Young Scientist Award, IEE Oliver Lodge Premium Award, Asia-Pacific Microwave Prize, IEEE CCECE Best Paper Award, University Research Award “Prix Poly 1873 pour l’Excellence en Recherche” presented by the Ecole Polytechnique on the occasion of its 125th anniversary, Urgel-Archambault Prize (the highest honor) in the field of physical sciences, mathematics and engineering from ACFAS, and 2004 Fessenden Medal of IEEE Canada. In 2002, he became the first recipient of the IEEE MTT-S Outstanding Young Engineer Award. He isa Fellow of the Canadian Academy of Engineering (CAE) and a Fellow of the Royal Society of Canada (The Canadian Academy of the Sciences and Humanities).

Jean Conan received the Radio Electronics and Nuclear Engineering degrees from the Institut Polytechnique de Grenoble, France, in 1964 and 1965, respectively, the M.S.Eng. degree from the University of Michigan, Ann Arbor, in 1971, and the Ph.D. degree from McGill University, Montréal, Quebec, Canada, in 1981. He is currently a Full Professor of electrical engineering at Ecole Polytechnique de Montréal. His current interests in research include information theory, MIMO wireless communication systems as well as problems pertaining to the analysis and design of wireless communication networks.

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Effect of Human Presence on UWB Radiowave Propagation Within the Passenger Cabin of a Midsize Airliner Simon Chiu and David G. Michelson, Senior Member, IEEE

Abstract—We have characterized the effect of human presence on path gain and time dispersion over ultrawideband (UWB) channels within the passenger cabin of a typical midsize airliner. We measured a few hundred channel frequency responses over the range 3.1–6.1 GHz between various locations within a Boeing 737-200 aircraft, with and without volunteers occupying the passenger seats. The links were deployed in a point-to-multipoint configuration with the transmitting antenna along the centre-line of the forward part of the cabin at either the ceiling or headrest level and the receiving antenna at the headrest or armrest level at selected locations throughout the rest of the cabin. As the density of occupancy increased from empty to full, path gain dropped by no more than a few dB on the ceiling-to-headrest paths but dropped by up to 10 dB on the ceiling-to-armrest and headrest-to-armrest paths. The gain reduction reached its maximum at the mid-point of the cabin and decreased thereafter. In all cases, increasing the density of occupancy caused the distance dependence of the rms delay spread to decrease greatly, the decay rate of the scattered components in the power delay profile (PDP) to almost double and the number of significant paths to drop by almost half. The results suggest that human presence substantially affects both path gain and time dispersion within the aircraft and should therefore be considered when assessing the performance of in-cabin wireless systems. Index Terms—Aircraft, propagation measurements.

I. INTRODUCTION

H

UMAN presence in the vicinity of a short-range, lowpower wireless link often leads to shadowing and scattering that affect both the path gain and time dispersion experienced by the link [1], [2]. Concern for the effect of human presence on short-range wireless links has motivated both measurement- and simulation-based studies of: (1) the depth and duration of shadow fading due to pedestrians moving in the vicinity of such links [3]–[5], (2) the effect of human presence on wireless personal area networks (WPANs), i.e., where one end of the link is located either close to or on a person [6]–[9], and (3) the effect of human presence on wireless body area networks Manuscript received April 16, 2009; revised July 29, 2009. First published December 31, 2009; current version published March 03, 2010. This work was supported in part by grants from Nokia Products, Bell Canada (through its Bell University Labs program) and in part by Western Economic Diversification Canada. The authors are with the Radio Science Laboratory, Department of Electrical and Computer Engineering, University of British Columbia, Vancouver, BC V6T 1Z4, Canada (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2009.2039326

(WBANs), i.e., where both ends of the link are located either close to or on a person [10]–[12]. In recent years, airlines and aircraft manufacturers have expressed much interest in deploying short-range wireless links within the passenger cabins of airliners in order to: (1) permit deployment of in-flight entertainment (IFE) and network access services and (2) facilitate operations and maintenance through deployment of sensor networks [13]–[17]. Although various wireless technologies have been considered and evaluated, ultrawideband (UWB) wireless technology that operates within the frequency band between 3.1 and 10.6 GHz have attracted particular interest for future systems because it: (1) can support very high data rates (up to 480 Mbps) over short distances, (2) occupies a particularly small footprint, radiates little RF energy, and consumes little power, and (3) can support precise positioning capabilities. With its cylindrical structure, its confined volume, the regular layout of its seating, and its high density of occupancy, an airliner passenger cabin is fundamentally different from the residential, commercial and industrial indoor environments considered previously by UWB researchers [18], [19]. The confined volume and high density of occupancy suggest that human presence will affect the performance of wireless systems in aircraft passenger cabins more than it will in other environments. Two previous studies presented characterizations of the UWB wireless channel within aircraft passenger cabins [20], [21], but disclosed only limited information concerning the effect of human presence on UWB wireless propagation in such environments. In other previous work, assessments of the excess pathloss introduced by human presence and internal components in passenger cabins were presented based upon: (1) narrowband measurements collected using CDMA handsets onboard a Boeing MD-90 with up to 17 passengers in the cabin [22] and (2) simulations of the effect of passengers and internal components on electromagnetic field strength inside Boeing B747, B767 and B777 aircraft passenger cabins [23]. Other previous work has yielded estimates of the manner in which the presence of windows, people and furnishings affect the field statistics and spoil the Q-factor of an enclosed space that functions as a multimode cavity [25]. However, designers require a more complete description of the effect of human presence on propagation in aircraft passenger cabins that account for the different types of paths within such environments and which are based upon larger data sets. A very recent study, conducted about the same time, as ours, considered the effect of human presence on UWB propagation within a large wide-body aircraft [26]. Here, we present

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the results of a complementary study conducted within a smaller narrow-body aircraft. After completing a pair of rigorous research ethics reviews and recruiting almost 100 volunteers to occupy passenger seats, we collected a few hundred UWB channel frequency responses (CFRs) over the frequency range of 3.1–6.1 GHz in a point-tomultipoint configuration within the passenger cabin of a Boeing 737-200 aircraft. We mounted the transmitting antenna at either the cabin ceiling or headrest level along the centerline of the forward part of the cabin and collected channel frequency response data with the receiving antenna mounted at headrest or armrest level at selected locations throughout the cabin with three degrees of occupancy: empty, partially filled and completely filled. We processed the result to determine the manner in which human presence affects the distance and frequency dependence of path gain, the form of the channel impulse response, the distance and frequency dependence of rms delay spread, and the number of significant paths below a given threshold within the passenger cabin of a typical mid-size airliner. We selected the frequency range 3.1–6.1 GHz, which corresponds closely to Band Groups 1 and 2 as defined by the WiMedia Alliance, because it is more likely that the lower portion of the UWB band will be used for point-to-multipoint coverage over large portions of the aircraft passenger cabin while the higher portions of the band are used to implement short-range peer-to-peer links [27]. The remainder of this paper is organized as follows. In Section II, we describe our VNA-based UWB channel sounder, our procedure for calibrating it, our data collection procedure and our measurement database. In Section III, we present the results of our investigation of path gain. In Section IV, we present the results of our investigation of time dispersion. Finally, in Section V, we summarize our key findings and their implications. II. MEASUREMENT SETUP A. UWB Channel Sounder Our UWB channel sounder consists of an Agilent E8362B vector network analyzer (VNA), 4-m FLL-400 SuperFlex and 15-m LMR-400 UltraFlex coaxial cables, a pair of Electro-metrics 6865 omnidirectional UWB biconical antennas, tripods and fixtures suitable for mounting the antennas at various locations throughout the aircraft, and a laptop-based instrument controller equipped with a GPIB interface. During data collection, a MATLAB script running on the laptop controlled the VNA and logged the received data. We recruited volunteers to occupy passenger seats during the measurement session. In order to meet RF emission limits imposed upon us by the Research Ethics Boards at the University of British Columbia and the British Columbia Institute of Technology, we set the transmit power to 5 dBm. We set the intermediate frequency bandwidth of the VNA to 3 kHz which reduced the resulting displayed average noise level (DANL) to 107.2 dBm. The minimum sweep time was automatically set to 2 seconds. As configured, the channel sounder can resolve at channel impulse responses (CIRs) with an transmitter-receiver separation distances of up to 15 m assuming

a distance exponent of 2.2, based on the worst case observed in our previous work [20], and average transmit and receive antenna gains of 0 dBi over all angles and directions. During data collection, the VNA was configured to sweep from 3.1 to 6.1 GHz over 2560 frequency points. The frequency sampling interval of 1.1718 MHz corresponds to a maximum unambiguous excess delay of 853 ns or a maximum observable distance of 256 m. The frequency span of 3 GHz gives us a temporal resolution of 333 ps or a spatial resolution of 100 mm. B. Channel Sounder Calibration Before measurement data can be collected, the channel sounder must be calibrated so that systematic variations in the amplitude and phase of the measured frequency response due to factors other than the propagation channel can be removed. The process involves two steps. The first step is to use the VNA’s built-in calibration routines, which are based upon a standard 12-term error model, to compensate for amplitude and phase distortions up to the point where the cables attach to the transmitting and receiving antennas. Care must be taken to ensure that the distortions for which the error correction model is compensating do not change appreciably during the measurement session, e.g., due to significant cable flexion and torsion, so that the error correction process will not introduce its own distortions. Appropriate cable handling and management techniques are the most effective way to avoid such problems. The second step, which is much more difficult, is to compensate for the distortions introduced by the antennas themselves. Because the radiation patterns of practical UWB antennas vary with both direction and frequency, individual multipath components (MPCs) arriving at the receiving antenna from different directions will be distorted in different ways. The measured channel response includes elements of the response of both: (1) the propagation channel and (2) the transmitting and receiving antennas. The result is often referred to as the response of the radio channel. In order to perfectly de-embed the propagation channel response from the radio channel response, one would need to measure the frequency-dependent double-directional channel response that accounts for the angle-of-departure (AoD) and angle-of-arrival (AoA) of each ray and the frequency-dependent three-dimensional radiation pattern of each antenna [28]. Implementing the required measurement setup within the confines of the aircraft passenger cabin would be problematic, however. The antenna calibration problem is simplified considerably if we can assume that the environment is rich with scatterers so that the physical MPCs arrive from all possible directions and each resolvable MPC includes many physical MPCs. Because the directivity of any antenna averaged over all directions is unity for all frequencies, the measured CFR will be independent of the radiation patterns of the transmitting and receiving antennas. In such cases, after appropriate account has been taken for the return loss of the antennas and the amplitude of any line-of-sight (LOS) components, the measured channel response will be equivalent to the propagation channel response. The dense single cluster form of the CIRs that we observed within that environment suggests that the density of scatterers

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within the cabin is very high. Moreover, previous work in conventional indoor environments has shown that the AoA distribution in the vertical plane broadens considerably as the size of the enclosed space becomes smaller [29]. Accordingly, it is not unreasonable to assume that the scattering is sufficiently broad that the effective gain of the transmitting and receiving antennas over all directions and frequencies is unity. Thus, while our results strictly characterize the radio channel, it seems likely that the measured channel is a useful approximation to the propagation channel. C. Data Collection We collected the CFR measurements within the passenger cabin of a Boeing 737-200 aircraft. The cabin, which can seat 130 passengers, is 3.54 m in width, 2.2 m in height and 21 m in length of which 18 m actually includes passenger seating. So that we could assess the effect of human presence on RF propagation aboard the passenger cabin, we collected measurement data with three levels of occupancy: empty, partially full and completely full. When the cabin was partially full, volunteer passengers sat in alternating seats from rows 4 through 19. When the cabin was full, volunteer passengers sat in every seat from row 4 through 19. During data collection, all of the passengers were asked to engage in quiet activities such as talking or reading while seated rather than standing in the aisle or moving about the aircraft. Before we collected production data, we verified that we could exploit the bilateral and translational symmetry inherent in the cabin layout to dramatically reduce the number of measurements needed to characterize propagation within the aircraft. We mounted the transmitting antenna along the centerline of the cabin at row 2 at either ceiling or headrest height, as appropriate, in the manner of an access point. We considered three different path types: ceiling-to-headrest (C-to-H), ceiling-to-armrest (C-to-A) and headrest-to-armrest (H-to-A). For both the C-to-H and C-to-A path types, we mounted the transmitting antenna at the ceiling level and used a custom-designed mount to place the receiving antenna at the headrest or armrest level of passenger seats in a reproducible manner on the port side of the aircraft from rows 4 to 19. For the C-to-H path type, the receiving antenna was placed on alternating aisle, middle and window seats, while for the C-to-A path type, the receiving antenna was placed only on alternating middle and window seats. For the H-to-A path type, we mounted the transmitting antenna at the headrest level and placed the receiving antenna at the armrest level of alternating middle seats on the port side of the aircraft from rows 4 to 18. The two different receiving antenna mounting positions not only represent typical use cases such as using a cell phone (at headrest level) or a laptop (at armrest level) but also represent both LOS (at the headrest) and NLOS (at the armrest) channels. A cross-section view of the cabin that shows the various antenna mounting positions is given in Fig. 1. A plan view of the cabin is shown in Fig. 2. D. Measurement Database During the development phase, we considered three transmitter locations at rows 2, 11 and 16 and over 50 receiver locations in the empty passenger cabin. For selected paths, we took

Fig. 1. Cross-sectional view of the passenger cabin showing the positions at which the transmitting and receiving antennas were deployed in the ceiling-toheadrest and ceiling-to-armrest configurations. The transmitting antenna is lowered to headrest level for the headrest-to-armrest configuration.

multiple sweeps to verify the static nature of the channel and the reproducibility of our measurements. This yielded over 200 CFRs in the development phase. During the production phase, we used only one transmitter location and collected data only on the port side of the aircraft. For each of the three levels of occupancy, i.e., empty, partially filled and completely filled, we collected CFRs at 24 and 16 different receiver locations along the port side of the aircraft for the C-to-H and C-to-A path types, respectively. For the empty and full aircraft cases, we also collected CFRs at 8 selected receiver locations for the H-to-A path type. This yielded 152 CFRs in the production phase. In total, we collected over 360 CFRs. III. EFFECT OF HUMAN PRESENCE ON PATH GAIN IN THE AIRCRAFT ENVIRONMENT The manner in which path gain decreases with distance determines the maximum range that can be achieved by a wireless link. For UWB-based wireless systems, path gain is an especially important consideration given the relatively low power levels that such systems are permitted to radiate. Within the passenger cabin, path gain decreases with increasing transmitter-receiver separation due to the combined effects of spatial spreading and obstruction by cabin fixtures, seating and passengers. Assessing the effect of human presence on path gain within the aircraft environment allows system designers to more accurately predict the coverage and reliability of UWB-based point-to-multipoint wireless systems deployed within such environments. We modeled the path gain within the passenger cabin environment as follows. First, we divided the 3.1–6.1 GHz frequency , each of which is 1.5 range into two band groups GHz wide. Over each band group, we verified that the envelope of the frequency response was effectively flat. We obtained the by taking the average of distance-dependent path gain

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Fig. 2. Locations of the transmitting antenna (.), receiving antenna ( = transmitting antenna at ceiling, O = transmitting antenna at headrest level) and volunteers ( ) within (a) the empty, (b) the partially filled and (c) the completely filled Boeing 737-200 aircraft.

the magnitude of the measured complex CFRs, each band group, yielding

, across

(1) is the where is the transmitter-receiver separation distance, number of frequency steps in each band group, and is the th frequency step. At each location, we estimated the path gain when the cabin was empty, and then estimated the reduction in , when the cabin was partially and fully occupath gain, pied. The configuration of the transmitting and receiving antennas and their antenna patterns remained constant as the level of occupancy increased. Thus, any variation in antenna gain due to changes in the path geometry with distance would have cancelled out when the difference in the estimated path gains was calculated. , observed in band In Fig. 3, the reduction in path gain, group 1 is presented as a function of distance, , for different path types and, within each plot, for different levels of occupancy. Although we had anticipated that the reduction in path gain would generally increase with distance over the length of the cabin, the actual relationship was more complicated. Ini-

tially, path gain decreases as the distance between the transmitter and receiver increases. Beyond the mid-point in the cabin (a distance of between 7 and 9 meters), however, the trend reverses. The time dispersion results presented in the next section do not reveal a similar breakpoint at the mid-point of the cabin so it seems likely that AoA effects are responsible. Although our measurement data are insufficient to reveal such effects, ray tracing simulations similar to those described in [23] and [24] may provide additional insight and be a useful next step. In all cases and both band groups, we found that the reduction in path gain associated with human presence was well-approximated by a quadratic expression in distance of the form (2) where , and are constants and is a zero-mean Gaussian random variable with a standard deviation of that accounts for location variability. In each case, we determined , and by applying regression analysis to the constants the measured data. We estimated by subtracting the quadratic and fitting regression line from the measured values of the results to a Gaussian distribution. The values of the parameters in each case are presented in Table I. In the C-to-H

CHIU AND MICHELSON: EFFECT OF HUMAN PRESENCE ON UWB RADIOWAVE PROPAGATION

Fig. 3. Reduction in path gain with respect to distance for band group 1 for (a) ceiling-to-headrest, (b) ceiling-to-armrest, and (c) headrest-to-armrest configurations.

configuration, the maximum decrease in mean path gain due to human presence is relatively low (no more than a few dB), as one might expect given that the C-to-H paths are relatively unobstructed by human presence. In the C-to-A and H-to-A configurations, the maximum decrease in mean path gain is much greater (up to 10 dB), as one might expect given that the C-to-A and C-to-H paths are much more obstructed by passengers.

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Fig. 4. The normalized power delay profiles for band group 2 for the ceiling-toarmrest path type that were observed at row 13 for occupancy levels of: (a) empty, (b) partially full, and (c) completely full.

IV. EFFECT OF HUMAN PRESENCE ON TIME DISPERSION IN THE AIRCRAFT ENVIRONMENT Our first step in characterizing time dispersion within the cabin was to convert the CFRs that we measured into CIRs. Following [27], we truncated the CFRs into band groups and zero-padded them to restore the original length and thus preand are the upper and serve the temporal resolution. If

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TABLE I LARGE-SCALE PATH GAIN PARAMETERS FOR THE AIRCRAFT PASSENGER CABIN ENVIRONMENT

Note: C

= Ceiling, H = Headrest, A = Armrest

TABLE II LARGE-SCALE DELAY SPREAD PARAMETERS FOR THE AIRCRAFT PASSENGER CABIN ENVIRONMENT

lower frequency boundaries of band group , respectively, then the complex CFR for band group is given by if otherwise.

(3)

Following the approach described in [31], we applied a Kaiser to the CFR in order to suppress disperwindow with sion of energy between delay bins. We then applied an inverse Fourier transform (IFT) directly to the complex baseband of the CFR to yield a CIR. We expressed the result in the form of a power delay profile (PDP) (4) are the amplitudes (expressed in units of power) of where MPCs at different delays . Measured PDPs typical of the C-to-A configuration under empty, partially filled and completely filled conditions in the

aircraft are given in Fig. 4. It is immediately apparent that the passenger cabin is rich with scatterers leading to a high density of MPCs in the PDPs. For LOS channels, we define the start of the PDP as the first MPC that arrives within 10 dB of, and 10 ns, before the peak MPC. For NLOS channels, we define the start of the PDP as the first MPC that arrives within 10 dB of, and 50 ns, before the peak MPC. We remove the propagation delay by setting the start time of the first arriving MPC to zero. These criteria are based upon those adopted by IEEE 802.15.4a and used in [32]. Using regression techniques, we estimated the decay time constants, , i.e., the reciprocal of the slope of the scattered components in the PDPs, for various path types, degrees of occupancy and band groups. The values are given in Table II. As the density of occupancy increased from empty to half full, the decay rate of the scattered components in the PDP almost doubled. Further increases in the density of occupancy had little effect, however.

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A. Delay Spread The normalized first-order moment of a PDP gives the mean excess delay (5) while the square root of the second central moment of a PDP gives the rms delay spread (6) where (7) Before we estimated the rms delay spread, we removed all MPCs with amplitudes that are more than 25 dB below the peak scattered component. In Fig. 5, we show how rms delay spread depends upon the transmitter-receiver separation distance for the three different path types (C-to-H, C-to-A, H-to-A) in band group 2. We model the distance dependence as (8) where is the mean rms delay spread at , is the is a zero-mean Gaussian random distance exponent, and that accounts for locavariable with a standard deviation of tion variability. The values of these parameters for various path types, degrees of occupancy and both band groups are given in Table II. In all cases where the aircraft was empty, the rms delay spread increased rapidly with distance while increasing the density of occupancy to half-full generally caused to decrease by a factor of nearly four. Increasing the density of occupancy caused little further reduction in . The decrease in is likely the result of energy in the scattered components being blocked or attenuated as the number of passengers aboard the aircraft increase. The rms delay spread generally decreases with increasing center frequency, which is likely a consequence of the corresponding increase in attenuation and diffraction losses with frequency. Although increasing from band group 1 to 2 for the C-to-H path type causes the rms delay spread to drop by 15–20%, doing so for the C-to-A and H-to-A path types results in little if any reduction. The mean excess delay and rms delay spread that we observed for the C-to-H and C-to-A cases for band group 2 as a function of threshold levels of 5, 10, 15 and 20 dB below the strongest MPC are summarized in Table III and Table IV, respectively. When assessing the performance of practical systems, it may be more realistic to apply a dynamic noise threshold that accounts for the diminishing signal-to-noise ratio and the tendency of weaker multipath components to drop below the noise floor at greater ranges. B. Number of Significant Paths We define a significant path as a resolvable MPC that exceeds a given threshold of 5, 10, 15 and 20 dB below the strongest

Fig. 5. RMS delay spread with respect to distance for band group 2 for (a) ceiling-to-headrest, (b) ceiling-to-armrest, and (c) headrest-to-armrest configurations.

MPC. In Table III and Table IV, respectively, we have summarized, as a function of the threshold level, the number of significant paths that we observed for the C-to-H and C-to-A cases and band group 2 and the percentage of energy that each set captures.

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TABLE III CEILING-TO-HEADREST CONFIGURATION—MEAN EXCESS DELAY, RMS DELAY SPREAD, NUMBER OF SIGNIFICANT PATHS AND ENERGY CAPTURED FOR DIFFERENT THRESHOLD LEVELS

Note: C

= Ceiling, H = Headrest, A = Armrest

TABLE IV CEILING-TO-ARMREST CONFIGURATION—MEAN EXCESS DELAY, RMS DELAY SPREAD, NUMBER OF SIGNIFICANT PATHS AND ENERGY CAPTURED FOR DIFFERENT THRESHOLD LEVELS

We found that the PDPs associated with band group 2 have between 10 and 30% fewer significant paths at a given threshold than those associated with band group 1. Moreover, we found that the PDPs measured in a full aircraft have between 40 and

45% fewer significant paths at a given threshold than those measured in an empty aircraft. These results are consistent with our observation that the duration of the PDP shrinks with increased occupancy and increased carrier frequency.

CHIU AND MICHELSON: EFFECT OF HUMAN PRESENCE ON UWB RADIOWAVE PROPAGATION

V. CONCLUSION Because the passenger cabin has a confined volume and may be densely occupied, human presence affects radiowave propagation within a midsized airliner more than in conventional indoor environments such as homes, offices and industrial sites. In order to assess the effect of human presence in such environments, we collected channel frequency response data over the range 3.1 to 6.1 GHz within the passenger cabin of a Boeing 737-200 aircraft. Despite the essentially square layout and short extent of the widebody scenario considered in [26] compared to the long and narrow extent of the case considered here, the results obtained within the two environments showed remarkable consistency. In particular, increasing occupancy tended to increase path loss by a few dB and lower delay spread by a few tens of nanoseconds. Otherwise, the much different geometry of the two scenarios precludes meaningful detailed comparison. Our investigation of path gain over point-to-multipoint links within the narrowbody cabin with the transmitting antenna in the front of the cabin reveals that: (1) the decrease in path gain that occurs as occupancy increases reaches a maximum near the mid-point of the cabin, decreases thereafter, and is well-approximated by a quadratic function, (2) the maximum decrease in path gain becomes more acute as: (a) the transmitting antenna drops from the ceiling to the headrest level and (b) as the receiving antenna drops from the headrest to armrest, (3) in the ceiling-to-headrest configuration, the maximum decrease in the mean path gain due to human presence is only a few dB; in the ceiling-to-armrest or headrest-to-armrest cases, the maximum decrease in the mean path gain is up to 10 dB. Although our measurement data are insufficient to reveal the physical cause of the distance-dependent behaviour, numerical simulations similar to those described in [23] and [24] may provide additional insight and might be a useful next step. Our investigation of time dispersion within the narrowbody cabin reveals that: (1) the channel impulse response always presents a dense single cluster regardless of the level of occupancy, (2) the rms delay spread generally increases with distance when the aircraft is empty but is essentially uniform when the aircraft is partially or fully occupied, (3) both the rms delay spread and the number of significant paths reduces by up to half as the level of occupancy increases from empty to half occupied, and (4) increasing the level of occupancy from half to full has little additional effect. In summary, our results: (1) suggest that human presence substantially affects radiowave propagation within an aircraft passenger cabin and should be considered when characterizing the performance of in-cabin wireless systems and (2) will be helpful to those wishing to validate the results of software simulations of in-cabin wireless propagation. Further measurements in different aircraft will be required to assess how seatbacks that incorporate in-flight entertainment units contribute to excess shadowing on ceiling-to-armrest and headrest-to-armrest links. ACKNOWLEDGMENT The authors would like to thank Associate Dean J. Baryluk, Chief Instructor/Hangar Supervisor G. Johnson and ATC Chair

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L. Kurk of the BCIT Aerospace Technology Campus at Vancouver International Airport for providing them with access to their Boeing 737-200 aircraft and their outstanding cooperation during the course of this study. They would also like to thank W. Muneer, W. Liu, R. White, F. Limbo and J. Wang for their assistance during the measurement sessions and the many volunteers who served as part of the propagation environment within the aircraft cabin. REFERENCES [1] M. Ghaddar, L. Talbi, and T. A. Denidni, “Human body modeling for prediction of effect of people on indoor propagation channel,” Electron. Lett., vol. 40, no. 25, pp. 1592–1594, Dec. 9, 2004. [2] M. Ghaddar, L. Talbi, T. A. Denidni, and A. Sebak, “A conducting cylinder for modeling human body presence in indoor propagation channel,” IEEE Trans. Antennas Propag., vol. 55, no. 11, pp. 3099–3103, Nov. 2007. [3] R. Ganesh and K. Pahlavan, “Effects of traffic and local movements on multipath characteristics of an indoor radio channel,” Electron. Lett., vol. 26, no. 12, pp. 810–812, Jun. 7, 1990. [4] K. I. Ziri-Castro, W. G. Scanlon, and N. E. Evans, “Prediction of variation in MIMO channel capacity for the populated indoor environment using a radar cross-section-based pedestrian model,” IEEE Trans. Wireless Commun., vol. 4, no. 3, pp. 1186–1194, May 2005. [5] K. I. Ziri-Castro, N. E. Evans, and W. G. Scanlon, “Propagation modeling and measurements in a populated indoor environment at 5.2 GHz,” in Proc. AusWireless, Mar. 13–16, 2006, pp. 1–8. [6] S. L. Cotton and W. G. Scanlon, “Characterization and modeling of the indoor radio channel at 868 MHz for a mobile bodyworn wireless personal area network,” IEEE Antennas Wireless Propag. Lett., vol. 6, pp. 51–55, Dec. 2007. [7] T. B. Welch et al., “The effects of the human body on UWB signal propagation in an indoor environment,” IEEE J. Sel. Areas Commun., vol. 20, no. 9, pp. 1778–1782, Dec. 2002. [8] A. Kara, “Human body shadowing variability in short range indoor radio links at 3–11 GHz,” Int. J. Elec., vol. 96, no. 2, pp. 205–211, Feb. 2009. [9] J. Karedal, A. J. Johansson, F. Tufvesson, and A. F. Molisch, “Shadowing effects in MIMO channels for personal area networks,” in Proc. IEEE VTC 2006 Fall, Sep. 25–28, 2006, pp. 1–5. [10] A. Fort, J. Ryckaert, C. Desset, P. De Donecker, P. Wambacq, and L. Van Biesen, “Ultra-wideband channel model for communication around the human body,” IEEE J. Sel. Areas Commun., vol. 24, no. 4, pp. 927–933, Apr. 2006. [11] S. L. Cotton and W. G. Scanlon, “A statistical analysis of indoor multipath fading for a narrowband wireless body area network,” in Proc. IEEE PIMRC’06, Sep. 2006, pp. 1–5. [12] A. Alomainy, Y. Hao, A. Owaldally, C. G. Parini, Y. I. Nechayev, C. C. Coonstantinou, and P. S. Hall, “Statistical analysis and performance evaluation for on-body radio propagation with microstrip patch antennas,” IEEE Trans. Antennas Propag., vol. 55, no. 1, pp. 245–248, Jan. 2007. [13] N. R. Diaz and M. Holzbock, “Aircraft cabin propagation for multimedia communications,” in Proc. EMPS, Sep. 25–26, 2002, pp. 281–288. [14] A. Jahn et al., “Evolution of aeronautical communications for personal and multimedia services,” IEEE Commun. Mag., vol. 41, no. 7, pp. 36–43, Jul. 2003. [15] N. R. Diaz and J. E. J. Esquitino, “Wideband channel characterization for wireless communications inside a short haul aircraft,” in Proc. IEEE VTC-Spring, May 17–19, 2004, pp. 223–228. [16] R. Bhagavatula, R. W. Heath, and S. Vishwanath, “Optimizing MIMO antenna placement and array configuration for multimedia delivery in aircraft,” in Proc. IEEE VTC Spring, Apr. 22–25, 2007, pp. 425–429. [17] A. Kaouris, M. Zaras, M. Revithi, N. Moraitis, and P. Constantinou, “Propagation measurements inside a B737 aircraft for in-cabin wireless networks,” in Proc. IEEE VTC Spring, May 11–14, 2008, pp. 2932–2936. [18] A. F. Molisch, J. R. Foerster, and M. Pendergrass, “Channel models for ultrawideband personal area networks,” IEEE Wireless Commun., vol. 10, no. 6, pp. 14–21, Dec. 2003. [19] A. F. Molisch et al., “A comprehensive standardized model for ultrawideband propagation channels,” IEEE Trans. Antennas Propag., vol. 54, no. 11, pp. 3151–3165, Nov. 2006.

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[20] J. Chuang, N. Xin, H. Huang, S. Chiu, and D. G. Michelson, “UWB radiowave propagation within the passenger cabin of a Boeing 737-200 Aircraft,” in Proc. IEEE VTC 2007 Spring, Apr. 22–25, 2007, pp. 496–500. [21] J. Jemai et al., “UWB channel modeling within an aircraft cabin,” in Proc. IEEE ICUWB, Sep. 10–12, 2008, pp. 5–8. [22] G. A. Breit, H. Hachem, J. Forrester, P. Guckian, K. P. Kirchoff, and B. J. Donham, “RF propagation characteristics of in-cabin CDMA mobile phone networks,” in Proc. Digital Avionics Syst. Conf., Oct. 30–Nov. 3 2005, pp. 9.C.5-1–9.C.5-12. [23] M. Youssef and L. Vahala, “Effects of passengers and internal components on electromagnetic propagation prediction inside Boeing aircrafts,” in IEEE AP-S Int. Symp. Dig., Jul. 9–14, 2006, pp. 2161–2164. [24] T. Hikage, T. Nojima, M. Omiya, and K. Yamamoto, “Numerical analysis of electromagnetic field distributions in a typical aircraft,” in Proc. EMC Europe, Sep. 8–12, 2008, pp. 1–4. [25] M. P. Robinson, J. Clegg, and A. C. Marvin, “Radio frequency electromagnetic fields in large conducting enclosures: Effects of apertures and human bodies on propagation and field-statistics,” IEEE Trans. Electromagn. Compat., vol. 48, no. 2, pp. 304–310, May 2006. [26] M. Jacob et al., “Influence of passengers on the UWB propagation channel within a large wide-bodied aircraft,” in Proc. EuCAP, Mar. 23–27, 2009, pp. 882–886. [27] ECMA International, High rate—ultrawideband (UWB) background [Online]. Available: www.ecma-international.org/activities/communicaitons/tg20_UWB_Background.pdf [28] A. F. Molisch, “Ultrawideband propagation channels: Theory, measurement, and modeling,” IEEE Trans. Veh. Technol., vol. 54, no. 5, pp. 1528–1545, Sep. 2005. [29] J. Wang, A. S. Mohan, and T. A. Aubrey, “Angles-of-arrival of multipath signals in indoor environments,” in Proc. IEEE VTC, Apr. 28–May 1 1996, pp. 155–159. [30] W. Q. Malik, D. J. Edwards, and C. J. Stevens, “Frequency dependence of fading statistics for ultrawideband systems,” IEEE Trans. Wireless Commun., vol. 6, no. 3, pp. 800–804, Mar. 2007. [31] S. S. Ghassemzadeh, R. Jana, C. W. Rice, W. Turin, and V. Tarokh, “Measurement and modeling of an ultra-wide bandwidth indoor channel,” IEEE Trans. Wireless Commun., vol. 52, no. 10, pp. 1786–1796, Oct. 2004. [32] C. C. Chong and S. K. Yong, “A generic statistical-based UWB channel model for high-rise apartments,” IEEE Trans. Antennas Propag., vol. 53, no. 8, pp. 2389–2399, Aug. 2005. [33] V. Erceg et al., “A model for the multipath delay profile of fixed wireless channels,” IEEE J. Sel. Areas Commun., vol. 17, no. 3, pp. 399–410, Mar. 1999.

Simon Chiu was born in Hong Kong, China in 1984. He received the B.A.Sc. and M.A.Sc. degrees in electrical engineering from the University of British Columbia, Vancouver, BC, Canada in 2006 and 2009, respectively. His main research interests focus on UWB propagation in passenger aircraft cabins and outdoor industrial environments as well as the effects of human presence.

David G. Michelson (S’80–M’89–SM’99) received the B.A.Sc., M.A.Sc., and Ph.D. degrees in electrical engineering from the University of British Columbia (UBC), Vancouver, BC, Canada. From 1996 to 2001, he served as a member of a joint team from AT&T Wireless Services, Redmond, WA, and AT&T Labs-Research, Red Bank, NJ, where he was concerned with the development of propagation and channel models for next-generation and fixed wireless systems. The results of this work formed the basis for the propagation and channel models later adopted by the IEEE 802.16 Working Group on Broadband Fixed Wireless Access Standards. From 2001 to 2002, he helped to oversee the deployment of one of the world’s largest campus wireless local area networks at UBC while also serving as an Adjunct Professor with the Department of Electrical and Computer Engineering. Since 2003, he has led the Radio Science Laboratory, Department of Electrical and Computer Engineering, UBC, where his current research interests include propagation and channel modeling for fixed wireless, ultra wideband, and satellite communications. Prof. Michelson is a registered professional engineer. He serves as the Chair of the IEEE Vehicular Technology Society Technical Committee on Propagation and Channel Modeling and as an Associate Editor for Mobile Channels for IEEE Vehicular Technology Magazine. In 2002, he served as a Guest Editor for a pair of Special Issues of the IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS concerning propagation and channel modeling. From 2001 to 2007, he served as an Associate Editor for the IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY. From 1999 to 2007, he was the Chair of the IEEE Vancouver Section’s Joint Communications Chapter. Under his leadership, the chapter received Outstanding Achievement Awards from the IEEE Communications Society in 2002 and 2005 and the Chapter of the Year Award from IEEE Vehicular Technology Society in 2006. He received the E. F. Glass Award from IEEE Canada in 2009 and currently serves as Chair of IEEE Vancouver Section.

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BAN-BAN Interference Rejection With Multiple Antennas at the Receiver Imdad Khan, Yuriy I. Nechayev, Member, IEEE, Khalida Ghanem, Member, IEEE, and Peter S. Hall, Fellow, IEEE

Abstract—Rejection of interference from a nearby body area network (BAN) becomes significant in BAN applications, where a number of BANs are operating in the near vicinity of each other. Two interference rejection combining (IRC) techniques, optimum combining (OC) and Weiner-Hopf (WH) solution, are applied to the on-body wireless communication channels, and a more robust interference rejection technique, applicable to on-body channels, is proposed and compared to the conventional IRC algorithms. Three on-body channels are investigated by real time measurements of the desired and interference signals in an indoor environment with the antennas mounted on two human subjects walking around each other. The output signal to interference plus noise ratio improvement achieved with each algorithm is presented for the three channels and the dependence of the interference rejection gain on the average desired signal to interference ratio has been shown along with the covariance matrices for each channel. OC gives around 3–5 dB interference rejection gain for belt-wrist and belt-head channels. The WH solution gives the worst performance for all the channels. The proposed algorithm gives a flexible handle on the interference rejection gain and gives better rejection than the two conventional algorithms with proper interval selection. Index Terms—Body area networks (BAN), correlation, interference rejection, multiple antennas, optimum combining.

I. INTRODUCTION

W

ITH the ever increasing use of body-worn devices in the personal area networks (PAN) and body-area networks (BAN), on-body channels are prone to interference from neighboring BANs. Rejection of the interference from a nearby BAN becomes more significant when the BANs are operating very close to each other and the level of the desired and interference signals are almost the same. The use of multiple antennas at the receiver side, i.e., receiver diversity is very effective to combat fading. Receiver diversity can be exploited to enhance the desired signal and reject the unwanted interfering signal in variety of ways [1]–[7]. If an appropriate combining technique is used at the receiver side equipped with multiple antennas,

Manuscript received March 17, 2009; revised July 29, 2009. First published December 28, 2009; current version published March 03, 2010. The work of K. Ghanem was supported by the Research Fund on Nature and Technologies, Canada. I. Khan is with University of Birmingham, Birmingham B15 2TT, U.K. and also with COMSATS Institute of Information Technology, Abbottabad, Pakistan (e-mail: [email protected]; [email protected]). K. Ghanem is with University of Birmingham, Birmingham B15 2TT, U.K. and also with the University of Southampton, Southampton SO17 1BJ, U.K. (e-mail: [email protected]). Y. I. Nechayev and P. S. Hall are with University of Birmingham, Birmingham B15 2TT, U.K. (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2009.2039312

a sufficient level of interference rejection can be achieved. A combining method, commonly known as interference rejection combining (IRC), is applied such that the interference signal is suppressed at the receiver and the output signal to interference plus noise ratio (SINR) is increased. Interference rejection can be employed by using adaptive arrays at the receiver side [7]–[9]. An IRC technique, known as optimum combining, is described in [7]. Optimum combining needs to calculate complex optimum weight vector for the receiving antenna array signals. The weight vectors are updated using various algorithms. For rejection of the interference, smart antenna arrays generate nulls in the direction of the interferer [4], [7]. To steer the null and/or maximum beam direction, the direction of arrival of the interference and desired signal is required. Receivers with interference rejection capabilities can become very complex and a lot of signal processing is required to be done if smart antenna techniques are applied. This complexity can add up to the cost and size of the receivers. It is desirable that devices mounted on the body should be smaller and less expensive. Thus, low complexity IRC algorithms are needed for BAN applications. Most of the work on diversity and multiple antennas emphasize on the improvement in the desired signal strength and increase the diversity gain [10]–[12]. No significant work has been done so far on the BAN-BAN interference rejection for the on-body channels to the best of the authors’ knowledge. This paper presents some experimental results and significance of the interference rejection for the on-body wireless communication channels. The optimum combining technique [5]–[7] and the Weiner-Hopf optimum solution [7]–[9] are applied to the interference limited BAN systems, and their ability to suppress the interference coming from a neighboring BAN, and thus improve the output SINR, is discussed. Due to the high correlation and power imbalance between the received signals at the two diversity branch antennas for some of the on-body channels in a 2-branch diversity receiver [10]–[12], the conventional IRC algorithms may not provide significant amount of interference rejection. To calculate the weight vector, the Weiner-Hopf solution relies on the covariance between the branch signals [8], [9]. High covariance of the branch signals for some of the on-body channels thus degrades the performance of such algorithms. In this paper, a simple interference rejection technique is proposed, referred to as Interference Cancellation with Interrupted Transmission (ICIT). This technique, which relies on switching the desired signal off at regular intervals, is applicable to the on-body channels and may not be feasible for the mobile communication channels, where one base station transmits to many receivers. The performance of the conventional optimum combining techniques is compared with that of the proposed algorithm. The Interference Rejection Gain (IRG), defined here

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versity receiver, the received signal vector, vector, , can be defined as

, and the weight

(3) (4) where output,

represents the transpose of the vector. The array , is then (5)

Similarly, the desired signal vector, , interference signal , are given as vector, , and RMS noise vector, Fig. 1 Simplified diversity combiner.

as the improvement in the output SINR, over the highest SINR among the branches at 1% probability, is used as the metric for comparison. Extensive measurement campaign was carried out by mounting the desired signal transmitting antennas and the receiving arrays on a human body and the interference signal transmitting antenna on another person’s body. Three on-body channels, which show importance in the current applications, were investigated namely, the belt-chest, belt-head, and beltwrist channels. The paper is organized as follows. The system model along with a brief overview of IRC techniques is described in Section II. The description of the antennas used in this study, and the measurement setup are presented in Section III. The results and analysis are given in Section IV. Section V summarizes the results with some conclusions. II. SYSTEM MODEL AND IRC TECHNIQUES Consider a narrowband -branch diversity combiner, as shown in Fig. 1, with receiving antennas mounted on a human body. The desired transmitting antenna is also mounted on the same body to form an on-body channel. It is also assumed that the receiving antennas also receive an interference signal coming from another BAN in the close proximity. The received at the antenna of the diversity combiner is thus signal,

(6) represents the mean value. Various IRC schemes where have been proposed in the literature [1]–[7]. A brief description of the optimum combining and the Weiner-Hopf (WH) solution is given below. The details of the proposed ICIT scheme are also given. A. Optimum Combining (OC) In the model presented above, the weight vector, optimum combining is generated as [5]–[7]

, for the

(7) is the desired channel transfer gain vector and variance matrix of interference plus noise the error covariance matrix [5]–[7], i.e.

is the co, called

(8) represents the complex conjugate transpose and where is the expected value operator. can be estimated by using a known transmitted training sequence and the channel response knowledge [5], [6].

(1) B. Weiner-Hopf (WH) Solution is the received desired signal, is the received where interference signal, and is the additive white Gaussian branch antenna. If the signal transmitted from noise, at the , then the desired transmitting antenna is (2) branch. where is the channel transfer function of the The received branch signals are combined in an optimum way to suppress the interference signal and improve the output SINR. The aim of an optimum-combining scheme is to calculate the optimum weight, , for each branch [7]. For a two branch di-

The weight matrix, , can be calculated by using the WeinerHopf solution as [8], [9] (9) where and

is the covariance matrix of the received signals, , and is defined as [8], [9] (10)

is the correlation matrix of the received signal, , and the . The reference signal can be achieved by reference signal

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various methods given in [8, Ch. 7]. It must be correlated to the desired signal and uncorrelated to the interference signal. For processing the measured data in this work, the sum of the desired signals at the two receiving antennas was taken as the reference signal, i.e.

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ruption, which results in throughput degradation. The larger the averaging window, the more is the throughput degradation and vice versa. Considering the model presented above, if at any time instant, , the average estimated interference signal at the branch is (13)

(11) then the weight vector,

, at that instant, , is calculated as

(12) (14) represent the complex conjugate operation. where In both of the algorithms defined above, an optimum weight vector is calculated for a given channel and its value is updated if the channel changes significantly. Thus, the covariance and correlation matrices are computed over a large portion of the channel data for which the channel is stationary. The larger the portion of channel data, the larger will be the computation time taken to calculate the covariance matrix and its inverse. This conventional solution may not give the optimum solution for the on-body channels due to the fast variation and non-stationary nature of the channels. For this reason, both the algorithms are applied at each instant of the measured data, i.e., an optimum weight vector is calculated for each sample of the data for the on-body channels under investigation. To calculate the covariance and correlation matrices in this situation, a local sliding window, constituting the previous samples, was selected. The previous local samples were selected because in real systems, the future values cannot be used. A training sequence is needed for the estimation of the channel during the first local window, whereas, the rest of the values can be estimated from the received data. C. Interference Cancellation With Interrupted Transmission (ICIT) Scheme A simpler approach to interference rejection, which is applicable to BAN, is proposed for the two-branch diversity receiver system. It works on the principle that at a certain instance, if the amplitude and phase of the interference signal is known, the two received branch signals can be combined with such a weight vector, which can make the interference signal at one branch equal and out of phase to the interference signal at the other branch. A mechanism is thus required to measure the phase and amplitude of the interference signals at the two receiving antennas, to reasonable accuracy, at a certain time instant. To do this, the desired signal transmitter is turned off temporarily at regular intervals, using a pre-defined algorithm. Thus, at a particular instance, , when the desired signal transmitter is turned off, only the interference signal is received. The phase and amplitude of the interference signal is measured and stored. The estimated value of the interference signal is then used as an estimate to calculate the weight vector, , for the forthcoming time interval until the desired signal is interrupted and turned off again. Such a signal, however, is subject to fast fading and thus should be averaged over time to achieve an accurate estimate. This leads to a larger amount of desired signal inter-

where (15) A quick look at (14) reveals that the weight of the first branch is 1, i.e., the signal goes unchanged. The second branch has the weight associated with it, which makes the interference at this branch equal in magnitude and 180 out of phase to the interferis then used for ence signal at branch 1. The same value of the rest of the interval until the next interruption. The phase difference and amplitude difference calculated with the estimated interference value changes slowly with time and may be independent of the modulation except for very high data rates, where the variation in the phase or amplitude with time is much faster than the time delay between the two copies received at the two antennas. Hence, it can be valid for a certain period depending upon the angle of arrival and speed of motion of the two human subjects. The performance of the scheme depends upon the accuracy of the estimate and hence on the interruption period (time interval after which the desired transmitted signal is to be switched off). If the interruption period is too long, the estimate of the interference may be outdated and the performance will be degraded. As an upper limit, with estimation of interference signal at every instant, complete cancellation of interference can be achieved and the output SINR becomes equal to the output signal to noise ratio (SNR). Keeping in view (5), the input and the output SINR for each case can be calculated as [2] (16) (17)

III. MEASUREMENT PROCEDURE Measurements were performed at 2.45 GHz. The receiving antenna was an array of two microstip-fed planar inverted-F antennas (PIFA) on 0.8 mm thick FR4 substrate, as shown in Fig. 2. The ground plane size was the same as the substrate size, which was 45 mm 40 mm. The thickness of the radiating plate of PIFA was 1 mm and the distance between the short-circuit pin and the feeding pin was 3 mm. Other dimensions of the antenna are shown in Fig. 2. The mutual coupling between the two PIFA elements was 12.5 dB. Due to the mutual coupling, the radiation efficiency of the antennas was slightly degraded. Very little

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Fig. 2. Top and side view of the PIFA array.

Fig. 4. Placement of the antennas on the body The Rx antenna array was placed at the three positions separately for the three on-body channels. Tx antennas remained at the waist position.

Fig. 3. Measured radiation pattern as a function of with the second element terminated by 50 ohms.

' of each PIFA element

detuning was observed when the antennas were mounted on the body, but the return loss for all antenna ports at the desired frequency was still above 10 dB. The measured radiation patterns of each of the two PIFA elements, when the other element is terminated by 50 ohms, are shown in Fig. 3 for the xy-plane, which is the plane of interest for the on-body channels. The transmitting antenna was a similar single element PIFA antenna. For each on-body channel, the antenna transmitting the desired signal was placed at the waist (belt) position on the front left side of the body, about 100 mm away from the body centre line. The receiving array was placed, alternately, at the right side of the head, right side of the chest, and the right wrist positions, as shown in Fig. 4, thus forming three on-body channels named belt-head, belt-chest, and belt-wrist. The antenna transmitting the interference signal was mounted at the same belt position on another person. All the antennas were oriented such that vector , shown in Fig. 2, was pointing downwards for the transmitting antennas and upwards for the receiving array for all the three channels measured, assuming the subject standing straight. The distance between the body and the antennas mounted on the body was kept to about 7–10 mm including the clothing. The coaxial cables used during the measurement were firmly strapped to the body to minimize the effect of moving cables over the duration of the channel measurement.

Measurements were performed in an indoor environment, which was a 7.5 m 9 m sized laboratory containing equipment, tables, chairs, and computers thus providing a rich multipath propagation environment. The two transmitting antennas, i.e., the desired and the interferer, were connected through an RF switch to a signal generator operating at 2.45 GHz. The switching time of the RF switch was 40 , which was much less than the coherence time of the channel [11]. The two receiving antennas were connected to the two ports of a Vector Network Analyzer (VNA) calibrated in tuned dual channel receiver mode with a single frequency sweep at 2.45 GHz. The calibration was done by connecting the signal generator to each port of the VNA through the cables used, to normalize the total power delivered to the transmitting antenna port to be 0 dBm. The signal generator and the VNA were synchronized by using the 10 MHz reference output signal from the signal generator. The noise floor for the measurement was at 90 dBm. A total of 1600 points were collected for one sweep of 12 s duration. Each collected sample contained amplitude and phase of the . Thus, each receiving antenna was collecting channel gain, 1600 samples with alternate samples from each transmitter with a sampling time of 15 ms, giving 800 samples each for the desired and the interference signals. A total of 6 such sweeps were carried out giving 4800 samples each for the desired and interference signals. The three interference rejection algorithms were applied in post processing to these data sets. During the measurement, the two subjects were walking around each other in the room in a random manner. The distance between them was randomly changed within the range of about 0.1 m to 4 m. IV. RESULTS As explained in Section II, the weight vector was calculated at each instant and a local sliding window was selected to calculate the covariance and correlation matrices in case of optimum combining and Weiner-Hopf solution. Various window

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Fig. 5. IRG versus interval period with various averaging window sizes for belt-head channel. Fig. 6. Data rate degradation versus interval period for various averaging window sizes.

sizes were tried for both the algorithms and the window size giving the best IRG was used. These were 600 ms for belt-wrist and belt-head channels and 225 ms for the belt-chest channel. In the ICIT scheme, various averaging window sizes were used for the estimation of the interference signal, and the performance of the algorithm was compared. In addition, the interval period was varied to see its effect on the SINR improvement. The IRG as a function of interval period is plotted for belt-head channel in Fig. 5 for various averaging window sizes. For a single instant (no averaging), the desired signal transmitter is kept shut(switching time of the switch used in the meadown for 40 surement). The other two channels showed similar trend. It can be noted from the figure that increasing the averaging window size does not provide any significant improvement in the IRG over the single instant estimate of the interference signal. On the other hand, averaging degrades the throughput of the system, as the desired signal is turned off for a larger amount of time. An estimate of the throughput degradation can be achieved by the ratio of the averaging window size (time for which the desired signal is kept off) to the total interval period (time until the next interruption). This degradation of the data rate with different averaging window sizes is shown in Fig. 6 as a function of interval period. Due to the fact that time averaging of the estimated interference signal does not provide significant improvement in the IRG but degrades the throughput of the system, an averaging window size of one instant is used for the rest of the results, as this gives negligible data rate reduction. The IRG versus interval period is shown in Fig. 7 without averaging, i.e., interference signal estimated at a single instant for the three on-body channels. It was noticed that for the beltchest channel, the interruption interval period can be as high as about 60 ms to achieve some reasonable IRG values (around 2 dB or more). For belt-wrist and belt-head channels, the period can be increased to about 250 ms and 600 ms, respectively. Increasing the period size from the values specified for each channel, the IRG value decreases significantly and no effective improvement is observed in the output SINR. The IRG reaches a certain minimum level (on average) upon increasing the interval length. If the IRG is sacrificed for less interruption of the desired

Fig. 7. IRG versus interval period for ICIT with interference estimated at a single instant.

signal, this minimum level can be used. However, for belt-chest channel, IRG goes to negative values close to about 80 ms interval period, which signifies that the performance is worse than the system without interference rejection. The shorter duration for the belt-chest channel reveals that the interference estimate is outdated very quickly. This is because the interference signal transmitter is shadowed much strongly and more frequently due to the body shadow area by the virtue of the receiver position compared to the other channels. For consistency, an interval period of 60 ms was used for the ICIT scheme for all of the three channels. The cumulative distribution functions (CDFs) of the branch and output SINR were plotted for each of the three IRC algorithms and the three on-body channels, as shown in Figs. 8–10. The IRG was calculated from the CDFs at 1% probability and the results are presented in Table I. It can be seen from Table I and Figs. 8–10 that WH solution does not give better interference rejection when compared to the other two schemes. The poor performance of the WH solution may be due to the fact that it relies on correlation between the two received branch signals, which is high in the on-body

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TABLE I RESULTS FOR THE THREE CHANNELS

Fig. 8. SINR CDFs for belt-head channel.

the belt-chest channel. The same is true for the optimum combining, which depends upon the covariance of the noise plus interference signals (R) at the two branches. The covariance maof the branch signals and the error covariance matrices trices (R), calculated for the whole data set in each case, are shown below for the three channels. As is slightly lower than , the optimum combining gives better performance than the WH solution Belt-Head

Fig. 9. SINR CDFs for belt-wrist channel.

Belt-Wrist

Belt-Chest

Fig. 10. SINR CDFs for belt-chest channel.

communication scenario [11], [12]. The belt-chest channel has the highest correlation between the two branch signals due to the line of sight (LOS) link [11], [12]. The presence of LOS leads to a low degree of scattering and hence the branch signals are highly correlated. For the belt-chest channel, the covariance of the received signals, i.e. , is high compared to the covariance of the other two channels and the weight vector will have low values, resulting in low IRG for this channel. The other two channels show somewhat better performance compared to

The comparison of the ICIT scheme with the two conventional algorithms reveal that ICIT can perform better interference rejection if a proper interruption interval period is selected. In contrast to the other two schemes, there is a handle on the performance of ICIT combiner, which is the interruption interval period. If a high level of interference rejection is desirable, the interval period can be made shorter and hence the rejection gain can be increased significantly. The drawback of ICIT is the frequent interruption of the desired signal, which can degrade the

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average SIR value for the measured case, as shown in Table I. Due to the short range and line-of-sight communication between the desired signal transmitter and the receiver, the level of interference is much lower than the desired signal level for most of the time. Hence, interference rejection may not be required for this channel. The other two channels showed reasonable improvement with the WH solution and the optimum combining schemes. The belt-head channel, which is the most important on-body channel for the handset-headset communication, shows much better improvement. V. CONCLUSION

Fig. 11. IRG versus average SIR for belt-head channel with interval length for ICIT = 60 ms.

overall throughput of the system. In fact, in packet radio systems, the gap between two transmitted packets can be used to estimate the interference and the throughput degradation can be avoided. However, the disadvantage will be increased power consumption at the receiver end, as the sleep time for the receiver will be decreased and utilized in estimating the interference signal. On the other hand, it provides a simple and robust solution for interference rejection compared to the complex and expensive smart antenna techniques. The mean desired and interference signal powers were also calculated and presented in Table I. The table also contains the , which was average desired signal to interference ratio calculated as given below in (18) (18) The level of interference was varied to see its effect on the performance of the IRC algorithms. In other words, the average SIR value was decreased gradually from a higher value to in a lower value and the IRG was plotted as a function of Fig. 11 for belt-head channel, as an example. The same behavior was observed for the other two channels. It is clear from the figure that IRG shows a downward trend with increasing values of average SIR (low interference signal level compared to the close to desired signal level) and is fairly constant with 0 dB. This means that the IRC algorithms work well to suppress the interference signals which are comparable with the desired signal level. For interference levels that are much lower than the desired signal levels, none of the algorithms gives reasonable rejection gains. It is important to note that the WH solution values higher than 5 dB. gives negative IRG values at For the belt-chest channel, the situation is worse with negative values. This suggests that WH IRG values even for low solution is not recommended for the on-body applications. Out of the three on-body channels, the belt-chest channel shows worst performance for all the three algorithms. This is due to two reasons. Firstly, due to the relatively high covariance between the branches and secondly, due to a relatively high

In a situation, where two BANs are operating in close proximity of each other, BAN-BAN interference can significantly disturb the performance of the systems, and interference rejection becomes an important aspect of the receiver design. Interference rejection schemes can be deployed using two-branch diversity at the receivers. The WH solution and optimum combining IRC schemes do not provide significant interference rejection for the on-body communication case, but these algorithms are still useful in keeping the interference level low and not allowing it to reach or exceed the desired signal level. The new proposed IRC algorithm, i.e. ICIT, provides a flexible way to handle the BAN-BAN interference. The performance of the ICIT scheme is greatly depended upon the interval period of interruption. The belt-chest channel gives the lowest values of IRG due to the presence of strong LOS links and very high SIR values. The belt-head and belt-wrist channels show considerable improvement in output SINR with the three IRC techniques. The interference rejection gain depends upon the average SIR value at the receiving antennas. IRC works better when the SIR values are close to 0 dB, and the IRG values are almost constant for a certain range of SIR. The IRG decreases rapidly with increasing SIR outside that range. REFERENCES [1] C. C. Ling and Z. Chunning, “Low-complexity antenna diversity receivers for mobile wireless applications,” Wireless Personal Commun., vol. 14, pp. 65–81, 2000. [2] C. Braun, M. Nilsson, and R. D. Murch, “Measurement of the interference rejection capability of smart antennas on mobile telephones,” presented at the IEEE Veh. Technol. Conf., 1999. [3] J. Karlsson and J. Heinegard, “Interference rejection combining for GSM,” presented at the 5th IEEE Int. Conf. on Universal Personal Communications, 1996. [4] M. L. McCloud, L. Scharf, and M. K. Varanasi, “Beamforming, diversity, and interference rejection for multiuser communication over fading channels with a receiver antenna array,” IEEE Trans. Commun., vol. 51, no. 1, pp. 116–124, Jan. 2003. [5] E. Tiirola and J. Ylitalo, “Performance of smart antenna receivers in WCDMA uplink with spatially coloured interference,” presented at the IST Mobile Communications Summit, Barcelona, Spain, Sep. 9–12, 2001. [6] D. Bladsjo, A. Furuskar, S. Javerbring, and E. Larsson, “Interference cancellation using antenna diversity for EDGE-enhanced data rates in GSM and TDMA/136,” presented at the 50th IEEE Vehicular Technol. Conf., Fall, 1999. [7] J. H. Winters, “Optimum combining in digital mobile radio with co-channel interference,” IEEE J. Sel. Areas Commun., vol. SAC-2, no. 4, pp. 528–539, Jul. 1984. [8] R. T. Compton, Adaptive Antennas, Concepts and Performance. Englewood Cliffs, NJ: Prentice-Hall, 1988.

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[9] X. N. Tran, T. Taniguchi, and Y. Karasawa, “Subband adaptive array for multirate multicode DS-CDMA,” presented at the IEEE Tropical Conf. on Wireless Commun. Technol., Honolulu, HI, Oct. 15–17, 2003. [10] A. A. Serra, A. Guraliuc, P. Nepa, G. Manara, and I. Khan, “Diversity gain measurements for body-centric communication systems,” Int. J. Microw. Opt. Technol., vol. 3, no. 3, pp. 283–289, Jul. 2008. [11] I. Khan and P. S. Hall, “Multiple antenna reception at 5.8 and 10 GHz for body-centric wireless communication channels,” IEEE Trans. Antennas Propag., vol. 57, no. 1, pp. 248–255, Jan. 2009. [12] I. Khan, P. S. Hall, A. A. Serra, A. R. Guraliuc, and P. Nepa, “Diversity performance analysis for on-body communication channels at 2.45 GHz,” IEEE Trans. Antennas Propag., vol. 57, pp. 956–963.

Imdad Khan received the B.Sc. and M.S. degrees in electrical engineering in from NWFP University of Engineering and Technology, Peshawar, Pakistan, in 2000 and 2003, respectively, and the Ph.D. degree from the University of Birmingham, Birmingham, U.K, in 2009. He was with NWFP University of Engineering and Technology, from 2000 until 2001. He has been with COMSATS Institute of Information Technology, Abbottabad, Pakistan, since 2001. His major field of research is diversity and MIMO for body-centric wireless communication channels.

Yuriy I. Nechayev (M’00) received the Diploma of Specialist in Physics (honors) from the Kharkiv State University, Ukraine, in 1996 and the Ph.D. degree in electronic and electrical engineering from the University of Birmingham, Birmingham, U.K., in 2004. Since 2003, he has been with the University of Birmingham as a Research Associate, and later, a Research Fellow, working on the problems of on-body propagation channel. He has coauthored a book chapter, an IEEE magazine article, and a number of technical papers on radio propagation in urban environments and around human body. His research interests include radio wave propagation modeling and measurements, propagation in random media, and electromagnetics.

Khalida Ghanem (S’03–M’09) received the B.Sc. (Eng.) degree in electronics from the Ecole Nationale Polytechnique (ENP), Algiers, Algeria, and the M.Sc. and Ph.D. degrees in telecommunications from INRS-EMT, University of Quebec, Montreal, Canada, in 2004 and 2007, respectively. From 1999 to 2000, she was with the Department of Mathematics, Electrical and Computer Engineering, University of Quebec at Rimouski (UQAR), Rimouski, Canada, where she worked as a Research assistant on smart antenna systems. She is now with University of Birmingham as a Research Postdoctorate at the school of Electronic, Electrical and Computer Engineering. Her research interests lie in the area of wireless and mobile communications and antenna and propagation. Dr. Ghanem is a recipient of various awards namely the Quebec government FQRNT Postdoctoral Fellowship in 2007. She has been the Chair of the Montreal Chapter IEEE Women In Engineering society, a member of the Technical Program and organizing committees for various conferences.

Peter S. Hall received the Ph.D. degree in antenna measurements from Sheffield University, Sheffield, U.K. He spent three years with Marconi Space and Defence Systems, Stanmore, working largely on a European Communications satellite project. He then joined The Royal Military College of Science as a Senior Research Scientist, progressing to Reader in Electromagnetics. He joined The University of Birmingham, Birmingham, U.K., in 1994. Currently, he is a Professor of communications engineering, Leader of the Antennas and Applied Electromagnetics Laboratory, and Head of the Devices and Systems Research Centre in the Department of Electronic, Electrical and Computer Engineering at the University of Birmingham, Birmingham, U.K. He has researched extensively in the areas of microwave antennas and associated components and antenna measurements. He has published five books, over 250 learned papers and taken various patents. Prof. Hall is a Fellow of the Institution of Engineering Technology (IET, formerly Institution of Electrical Engineers (IEE)) and the IEEE and a past IEEE Distinguished Lecturer. His publications have earned six IEE premium awards, including the 1990 IEE Rayleigh Book Award for the Handbook of Microstrip Antennas. He is a past Chairman of the IEE Antennas and Propagation Professional Group and past coordinator for Premium Awards for IEE Proceedings on Microwave, Antennas and Propagation and is currently a member of the Executive Group of the IEE Professional Network in Antennas and Propagation. He is a member of the IEEE AP-S Fellow Evaluation Committee. He chaired the organizing committee of the 1997 IEE International Conference on Antennas and Propagation and has been associated with the organization of many other international conferences. He was Honorary Editor of IEE Proceedings Part H from 1991 to 1995 and currently on the editorial board of Microwave and Optical Tech Letters. He is a member of the Executive Board of the EC Antenna Network of Excellence.

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Characterization of UWB Channel Impulse Responses Within the Passenger Cabin of a Boeing 737-200 Aircraft Simon Chiu, James Chuang, and David G. Michelson, Senior Member, IEEE

Abstract—With its confined volume, cylindrical structure and high density of seating, the passenger cabin of a typical midsize airliner is significantly different from the residential, office, outdoor and industrial environments previously considered by IEEE 802.15.4a. We have characterized the shape of the ultrawideband (UWB) channel impulse response (CIR) and the fading statistics experienced by individual multipath components (MPCs) within that environment based upon 3300 complex frequency responses that we measured over the range 3.1–10.6 GHz at various locations aboard a Boeing 737-200 aircraft. We found that: (1) the shape of the CIR generally follows IEEE 802.15.4a’s dense single-cluster model, but with negligible rise time if the link is line-of-sight, (2) both the mean and variance of the exponential decay constant tend to increase with transmitter-receiver separation and also as the receiving antenna drops from the headrest to the footrest of the passenger seats, and (3) small-scale fading of individual MPCs at each measurement location within the aircraft tends to follow a Nakagami distribution with a lognormally-distributed -parameter that has a mean value of 0.2 dB and a standard deviation of 1.1 dB. We have modified IEEE 802.15.4a’s CIR simulator to generate responses similar to those seen in the cabin. Index Terms—Aircraft, channel impulse response, channel model, fading channels, multipath channels, ultrawideband (UWB) propagation.

I. INTRODUCTION HE channel modeling committees of the IEEE 802.15.3a and 802.15.4a task groups devoted considerable effort to developing ultrawideband (UWB) wireless channel models applicable to systems that operate between 3.1 and 10.6 GHz under both line-of-sight (LOS) and non-line-of-sight (NLOS) conditions in residential, office, outdoor, industrial and body-centric environments at ranges up to 15 m. The standard channel models and channel impulse response (CIR) simulator that they developed allow fair comparison between alternative UWB systems over a range of representative channel conditions and deployment scenarios [1]–[3]. So that developers can effectively predict and compare the performance of UWB wireless communication systems in an

T

Manuscript received January 06, 2009; revised July 03, 2009. First published December 04, 2009; current version published March 03, 2010. This work was supported in part by grants from Nokia Products, Bell Canada (through its Bell University Labs program), and Western Economic Diversification Canada. The authors are with the Radio Science Laboratory, Department of Electrical and Computer Engineering, University of British Columbia, Vancouver, BC V6T 1Z4, Canada (e-mail: [email protected]; [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2009.2037707

environment of interest, both the shape and structure of the CIR, and the small-scale fading statistics experienced by individual multipath components (MPCs) within the CIR, must be accurately modeled. The results affect many important design issues, including selection of the number and placement of the fingers in rake receivers used to implement temporal diversity in spread spectrum systems and the selection of the guard-time and the design of cyclic prefixes used to mitigate multipath fading in OFDM systems. Because unclustered CIR models tend to overestimate link capacity if the MPCs are indeed clustered, it is useful to determine the extent to which clustering occurs [4]. The shape of the CIR also affects the performance of UWB ranging and positioning algorithms because it determines how well the algorithm will be able to detect the first arriving MPC. In practice, the CIR is often expressed in the form of a power delay profile (PDP) that excludes the phase information associated with each MPC. UWB wireless systems hold great promise for: (1) enabling high data rate in-flight entertainment (IFE) and network access within the passenger cabin of an aircraft and (2) facilitating operations and maintenance through deployment of low power UWB-based sensor networks [5]. Early concerns that UWB-based systems would interfere with aircraft systems have largely been allayed by recent NASA studies [6], [7]. However, with its confined volume, cylindrical structure and high density of seating, an aircraft passenger cabin is fundamentally different from previously modeled UWB propagation environments. Although several research groups have made considerable progress in characterizing aircraft passenger cabins in support of deployment of conventional wireless technologies [8]–[14], and a few groups, including us, have reported results regarding large-scale aspects of UWB propagation in aircraft passenger cabins [15], [16], little has been reported concerning the detailed structure of UWB CIRs and the fading and correlation properties of their MPCs in such environments. Here, we characterize the shape and structure of the UWB CIR, and the fading statistics and correlation properties of individual MPCs within the passenger cabin of a typical mid-sized airliner with the intent of developing a UWB CIR simulation model useful in analysis and design. Our results are based upon over 3300 complex channel frequency responses (CFRs) that we measured over the range 3.1–10.6 GHz aboard a Boeing 737-200 aircraft with an omnidirectional transmitting antenna mounted near the cabin ceiling and an omnidirectional receiving antenna mounted at selected locations throughout the cabin. We refer to this as a point-to-multipoint (p-to-mp) configuration. So that we could assess the spatial statistics of the UWB CIR, i.e., the spatial

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average and the spatial correlation, we collected the CIRs across a 300-mm-by-300-mm spatial sampling grid with 50-mm spacing. The remainder of this paper is organized as follows. In Section II, we describe the configuration and calibration of our VNA-based channel sounder, our procedure for collecting channel frequency response (CFR) data in the aircraft and our measurement database. In Section III, we present our proposed model for the shape and structure of the PDPs that we observed within the aircraft passenger cabin. In Section IV, we report upon the fading statistics experienced by MPCs and the fading correlation between MPCs that are either: (1) in adjacent delay bins with the antenna at the same point on the sampling grid or (2) in the same delay bin but with the antenna at an adjacent point on the sampling grid. In Section V, we describe how we modified the standard channel impulse response simulation code developed by IEEE 802.15.4a to generate CIRs representative of those observed in the aircraft passenger cabin environment and verified that its output is consistent with our measurement results. Finally, in Section VI, we summarize our key findings and contributions.

TABLE I LINK BUDGET FOR THE UWB CHANNEL SOUNDER

Calculated using a path loss exponent of 2.2

II. MEASUREMENT APPROACH A. UWB Channel Sounder Configuration and Calibration Our UWB channel sounder consists of an Agilent E8362B vector network analyzer (VNA), 4-m FLL-400 SuperFlex and 15-m LMR-400 UltraFlex coaxial cables, a pair of Electrometrics 6865 UWB omnidirectional biconical antennas, a 0.5-m-by-0.5-m two-dimensional antenna positioner based upon Velmex BiSlide positioning slides, the tripods and fixtures that we used to mount the antennas at various locations throughout the cabin, and a laptop-based instrument controller equipped with a GPIB interface. During data collection, a MATLAB script running on the laptop controlled both the VNA and the two-dimensional positioner, and logged the received data. We configured the VNA to sweep from 3.1 to 10.6 GHz in 6401 steps with an IF bandwidth of 3 kHz. The resulting displayed average noise level (DANL) is 107.2 dBm. In order to meet RF emission limits imposed upon us by the Research Ethics Boards at the University of British Columbia and the British Columbia Institute of Technology for the human presence study to be conducted as a follow-on to the present work, we set the transmit power to 5 dBm. The frequency sampling interval of 1.1716 MHz corresponds to a maximum unambiguous excess delay of 853 ns or a maximum observable distance of 256 m. The frequency span of 7.5 GHz corresponds to a maximum temporal resolution of 133 ps or a maximum spatial resolution of 40 mm. In Table I, we give the principal elements of the system link budget at 3.1, 6.85 and 10.6 GHz, i.e., the bottom, mid-point and top of the UWB frequency band for a transmitter-receiver separation distance of 15 m. The average antenna gain refers to the average over all angles and directions. The path loss exponent of 2.2 used in the Table is the worst case that we observed both here and in our previous work [15]. We used through-line calibration to remove the frequency distortion introduced by the VNA and the coaxial cables that connect the VNA to the transmitting and receiving antennas. We applied a Kaiser window with to the CFRs in order to suppress dispersion of energy into adjacent delay bins. After applying moderate flexion and torsion to the RF cables, we applied a Fourier

Fig. 1. Temporal resolution and dynamic range of the channel sounder after , and through-line calibration and application of a Kaiser window with after the RF cables have undergone moderate flexion and torsion.

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transform to the resulting complex frequency response in order to reveal the resolution and dynamic range of the instrument under typical conditions. The result is shown in Fig. 1. The transmitting and receiving antennas are vertically polarized, omnidirectional and identical. The measured channel response includes elements of both the actual response of the propagation channel and the response of the transmitting and receiving antennas. This result is often referred to as the response of the radio channel. In order to perfectly de-embed the propagation channel response from the radio channel response, one would need to measure the frequency-dependent double-directional channel response that accounts for the angle-of-departure (AoD) and angle-of-arrival (AoA) of each ray and the frequency-dependent three-dimensional radiation pattern of each antenna [2]. Implementing the required measurement setup within the confines of the aircraft passenger cabin would be problematic, however. The antenna calibration problem is simplified considerably if we can assume that the environment is rich with scatterers so that the physical MPCs arrive from all possible directions and each resolvable MPC includes many physical MPCs. Because the directivity of any antenna averaged over all directions would be unity for all frequencies, the measured CIR would be independent of the radiation patterns of the transmitting and receiving antennas. Moreover, the CIR would take on a charac-

CHIU et al.: CHARACTERIZATION OF UWB CIRs WITHIN THE PASSENGER CABIN OF A BOEING 737-200 AIRCRAFT

teristic form in which every delay bin would contain MPCs and every MPC would exhibit Rayleigh fading. In such cases, after appropriate account has been taken for the frequency-dependent return loss of the antennas, the measured channel response would be equivalent to the actual channel response. As we shall show, the density of the MPCs in the CIRs and the Rayleigh fading distribution displayed by each resolvable MPC that we measured in the aircraft passenger cabin suggests that many of these conditions are at least partly met. Because the receiving antenna pattern is essentially uniform in the horizontal plane, the effective antenna pattern given by the convolution of the free space antenna pattern and the AoA distribution in that plane is also uniform regardless of the actual AoA distribution. Thus, this condition is automatically met. However, the receiving antenna pattern in the vertical plane is decidedly non-uniform so the effective antenna pattern will be uniform only if the actual AoA distribution is uniform. Previous work in conventional indoor environments has shown that the AoA distribution in the vertical plane broadens considerably as the size of the enclosed space becomes smaller [17]. While this suggests that the AoA distribution in the vertical plane within the aircraft is likely to be broad, it is not likely to be uniform. Other previous work using the same biconical antennas found remarkable differences in the spatial correlation between 2 and 12 GHz, which were also related to differences in the antenna patterns, particularly in the vertical plane [18]. In such work, when the frequency was increased, the spatial correlation was increased as well (for the same wavelength), which indicated higher directivity on radio channels, and lower delay spread. Thus, although we believe that our measured CIR provides a reasonable indication of the actual CIR, our results strictly apply to the radio channel and slightly different results may be obtained if other transmitting and receiving antennas with different radiation patterns in the vertical plane are used. B. Data Collection We collected our CFR measurements within the passenger cabin of a Boeing 737-200 aircraft. The cabin, which can seat over 100 passengers, is 3.54 m in width, 2.2 m in height and 21 m in length of which 18 m actually includes passenger seating. Plan and cross-sectional views of the passenger cabin are shown in Fig. 2(a) and (b), respectively. Other modern mid-sized airliners, such as the CRJ series from Bombardier, the A320 family from AirBus Industries and the ARJ21 family from ACAC, have similar cross-sections. Only the lengths of the passenger cabins, which range from 12 to 43 m, are appreciably different. Here, we have considered a p-to-mp wireless system configuration in which the transmitting antenna is mounted along the centerline of the cabin ceiling in the manner of an access point and the receiving antenna is placed at the headrest, armrest and footrest levels of the passenger seats throughout the aircraft, as suggested by Fig. 2(a) and (b). The different receiving antenna mounting positions not only represent typical use cases such as using a cell phone (at headrest level), a laptop (at armrest level) or devices that might be contained in passengers’ carry-on baggage (at footrest level) but also represent both LOS (at the headrest and aisle armrest) and NLOS (at the outboard armrest and footrest) channels.

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In Section IV, we present MPC fading statistics within a local area that we estimated using methods similar to those described in [3]. Although standard practice would be to move the receiving antenna across the spatial sampling grid, this is difficult to do when the antenna is mounted close to the passenger seats. Because moving the transmitting antenna instead was shown to yield good results in [4], we did so here, too. With the receiving antenna mounted at headrest, armrest and footrest levels, we collected 49 spatial samples by mounting the transmitting antenna at ceiling level at row 2 and moving it across a 7-by-7 grid with a spacing of 50 mm, as shown in Fig. 2(a). By setting the spacing equal to half of the wavelength of the lowest frequency, we sought to ensure that the spatial samples are independent. This, however, does not allow unambiguous resolution of the direction of a given ray, which requires the spacing be equal to half of a wavelength at the highest frequency [2]. Previous work suggests that: (1) approximately nine samples are sufficient to average out the small-scale fading and permit the true shape of the PDP to be recovered [19], and (2) approximately 50 samples are sufficient to determine the underlying fading distribution [2]. Here, we have elected to use 49 spatial samples per measurement location because it permits use of a symmetrical 7-by-7 sampling grid. C. Consistency Checks Before we collected production data, we conducted a series of development runs in order to: (1) verify that the channel is static and show that we could exploit the bilateral and translational symmetry inherent in the cabin layout to dramatically reduce the number of measurements needed to characterize propagation within the aircraft, and (2) verify that the shape and structure of the CIRs are consistent within a local area and that any differences between the CIRs that we observed over that local area are mostly due to multipath fading of individual MPCs. We did so by comparing: (1) the shapes of the average power delay profiles (APDPs) based upon CIRs measured at nine points on a 100-mm-by-100-mm grid with the receiving antenna mounted , at rows 4, 7, 11, 15 and 19, and (2) the mean excess delay, and RMS delay spread, , based upon CIRs measured at 49 points on a 300-mm-by-300-mm grid with the receiving antenna mounted at rows 4, 11 and 19. The mean excess delay and RMS delay spread were calculated using a threshold of 25 dB below the peak scattered component. Although it is difficult to set an absolute criterion for consistency, support for the conjecture is given by: (1) visual inspection of the APDPs and the plot of RMS delay spread vs. distance in Fig. 3 and (2) observation that the standard deviations of the mean excess delay and RMS delay spread over all measurement locations are, on average, less than 1.5 and 1 ns, respectively. In Section III-A, we describe the details of the processing steps that we followed when estimating APDPs from measured CIRs. D. Measurement Database Our measurement database includes both development and production data. During our development runs, we collected two sets of data. In the first set, we considered three transmitter locations and over 50 receiver locations. For selected paths, we collected multiple sweeps in succession and verified that: (1) the channel is static and (2) our channel sounder yielded consistent results. In the second set, we used a single transmitter

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Fig. 2. Locations of the transmitting antenna ( ) and receiving antennas ( = headrest and armrest,  = footrest) within a Boeing 737-200 aircraft in (a) plan and (b) cross-section view.

receiving antenna placed at headrest, armrest and footrest levels. The two sets combine to yield over 700 CFRs. During our production runs, we placed the receiving antenna at headrest, armrest and footrest levels throughout the port side of the cabin. When we mounted the receiving antenna at the headrest and armrest, we collected the CFRs at 24 different locations and when we mounted the antenna at the footrest level, we collected CFRs at five locations. These measurement locations are shown in Fig. 2. In both cases, we used a 49-point spatial sampling grid, yielding 2597 CFRs. In total, our development and production runs yielded over 3300 CFRs. III. SHAPE AND STRUCTURE OF THE POWER DELAY PROFILE Fig. 3. RMS delay spread as a function of distance when the receiving antenna is mounted on the headrest.

location and we measured the channel response at five locations across either 9-point or 49-point spatial sampling grids with the

A. Initial Processing of the Channel Impulse Response Whether measured in the time or frequency domain, a measured channel response has a finite bandwidth that is determined by either the instrument or the measurement process. The result is equivalent to convolving the true CIR with a sinc function

CHIU et al.: CHARACTERIZATION OF UWB CIRs WITHIN THE PASSENGER CABIN OF A BOEING 737-200 AIRCRAFT

whose duration is inversely proportional to the bandwidth of the measurement. Before processing a measured CIR, one must first remove the effects of the finite bandwidth either by windowing or deconvolution. Here, we applied a Kaiser window with to the CFRs in order to suppress dispersion of energy into adjacent delay bins. We converted the CFRs into complex baseband CIRs by applying an inverse Fourier transform (IFT). We normalized the CIRs so that they contained unit energy. Because we know the precise separation between the transmitter and receiver, it was a simple matter to determine the propagation delay, , and then set the start time of the first arriving MPC to zero. (When the precise separation is not known, previous workers have defined the start of a LOS CIR as the first MPC that arrives within 10 dB of, and 10 ns before, the peak MPC. They further defined the start of a NLOS CIR as the first MPC that arrives within 10 dB of, and 50 ns before, the peak MPC. Such an approach is not required here.) After we removed the initial delays, we aligned the first arriving MPCs in each PDP and averaged the MPCs directly in the time domain to yield the small-scale APDP [20], [21]. Unless otherwise indicated, we removed all MPCs with amplitudes that are more than 25 dB below the peak MPC before we extracted any model parameters. These criteria are based upon those cited in [4] and employed by the IEEE 802.15.4a channel modeling committee. As others have noted, the fine delay resolution of a UWB PDP may cause a physical MPC that arrives at a certain delay when observed at a certain grid point to fall in a different delay bin when observed at another grid point [2], [21]. Although the process of averaging will smear the PDP, the result will affect dense single cluster PDPs (in which a resolvable MPC consists of several physical MPCs) differently than sparse multi-cluster PDPs (in which a resolvable MPC may correspond to a single physical MPC and many delay bins are empty). Following the method described in [20], we reduced our delay resolution by a factor of 10, i.e., from 133 ps to 1.33 ns in order to reduce the smearing effect. We observed that the APDPs with reduced time resolution present the same shape and structure as the original APDPs. B. IEEE 802.15 CIR Models Our next task was to identify the channel impulse response model that offers the best description of time dispersion within the aircraft passenger cabin. We began by considering the two standard UWB channel models that were adopted by the IEEE 802.15.3a and 4a task groups [1], [3]. The sparse multi-cluster model is based upon the SV model given by (1) , and Here, the MPCs are modeled as Dirac delta functions, and are the amplitude and phase of the th MPC in the th cluster, is the total number of clusters in the CIR and is the total number of MPCs within the th cluster. and represent the arrival time of the th cluster and the th MPC in the th cluster, respectively. Because path loss is frequency dependent, the MPCs are distorted as described in [2], [3]. IEEE 802.15.4a used a modified form of the SV model that accounts

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for such distortion to describe the UWB CIRs in six of the eight scenarios they considered. The shape of the corresponding PDP is described by the product of two exponential functions, (2) are the inter-cluster and intra-cluster decay where and constants, respectively. The dense single-cluster model is used to describe dense scattering environments, e.g., the office and industrial environments under NLOS conditions. In these environments, one can no longer discern clustering within the CIR and the envelope of the PDP can be described as (3) where denotes the attenuation of the first component, determines how fast the PDP rises to its local maximum and represents the decay at later times [3]. If the scattering environment is sufficiently dense, e.g., an industrial NLOS environment, then every time resolution bin contains an MPC. Accordingly, the PDP can be modeled as a tapped delay line with a fixed arrival time, , that is given by the inverse of the signal bandwidth. Where scatterers are less dense but the single cluster response still applies, e.g., an office NLOS environment, then the convention is to model the arrival rate of the MPCs by a Poisson distribution [3]. C. Modeling the Shape of the Power Delay Profile In Fig. 4(a), we present a typical APDP of a LOS channel based upon measurement data collected when the transmitting antenna was mounted near the cabin ceiling and the receiving antenna was mounted on the headrest of a passenger seat. When the receiving antenna is mounted on the armrest of an aisle seat, the resulting channel is also LOS and the CIR resembles that of the headrest. As in the case of industrial LOS channels, the MPCs form a continuous exponential decay with no distinct clusters. In many cases, we observed a few strong spikes or impulses early in the APDP, as described below. In Fig. 4(b) and (c), we present typical APDPs observed over NLOS channels where the receiving antenna was mounted on an outboard armrest or footrest, respectively. Similar to industrial NLOS channels, both cases display a gentle rise before reaching the local maximum described by the dense single cluster model. We also observe that the footrest case exhibits a slower rise time than the armrest case. This is likely because the initial MPCs in the footrest case encounter more and/or denser obstacles and thus are more severely attenuated than the initial MPCs in the armrest case. Based upon our measurement results, we propose the following model for the PDP of LOS channels in aircraft passenger cabins, i.e., where the receiving antenna is mounted on a headrest or aisle armrest. First, we model the shape of the scatter components of the APDP as a simple exponential decay (4)

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Fig. 5. Linear ratio of the total energy in the delayed impulses to the energy in the initial impulse. The initial impulse corresponds to line-of-sight propagation while the delayed impulses likely result from specular reflection in the cabin.

is the power in the LOS component and the denomwhere inator is the expected power at the beginning of the exponential decay described using (4). On LOS channels, we often observe random impulses within the first 30 ns of the initial response. We suspect that they are due to specular reflection from the cabin bulkhead or floor and note that similar impulses have been observed in industrial environments [21]. The ratios of the energy in the initial (LOS) and delayed impulses in the APDPs that we observed when the receiving antenna is mounted on the headrest are shown in Fig. 5. The delayed impulses contain only a very small fraction of the energy in the CIR and, on average, carry only 15% of the energy in the LOS component. The development of a statistical model that captures their occurrence, amplitude distribution and arrival rate would require much more data than we have available. Accordingly, we leave further efforts to model them for future study. Although the IEEE 802.15.4a channel modeling committee did not account for the distance dependence of the CIR model parameters, we have done so here. In UWB scenarios, increases in RMS delay spread with distance are generally associated with a decrease in the SV model’s cluster decay constant, , or the single cluster model’s exponential decay constant, . Using methods similar to those employed in [22] and [23], we model the variation in the exponential decay constant and the excess amplitude with distance for LOS channels by

Fig. 4. The spatially averaged PDP observed when the receiving antenna is mounted at row 19 on (a) the headrest, (b) the outboard armrest and (c) the footrest.

where is the exponential decay constant. Next, we model the excess amplitude of the LOS MPC above the exponential decay curve at the propagation delay, . In linear units, we define the excess amplitude as (5)

(6) and (7) where and are the intercepts, and are and are zero-mean Gaussian random varithe slopes, and , respectively, and ables with standard deviations is the distance between the transmitting and receiving antennas. The regression lines given by (6) and (7) are shown in

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TABLE III POWER DELAY PROFILE MODEL PARAMETERS—NLOS CASES (OUTBOARD ARMREST AND FOOTREST)

case does not pass; it is skewed towards higher values and exhibits positive kurtosis. Our measurement data is insufficient to explain this single instance. Further study of this scenario, perhaps using ray-tracing methods, may be warranted. A summary of the LOS channel model parameters that we extracted is given in Table II. For NLOS channels, i.e., the receiving antenna mounted upon an outboard armrest or a footrest, we modeled the envelope of the PDP using (3). We describe the distance dependence of the parameters by (8) (9) and (10) Fig. 6. Shape parameters of the power delay profile as a function of distance for headrest channels: (a) the exponential decay constant, , and (b) the excess amplitude of the LOS path, .

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TABLE II POWER DELAY PROFILE MODEL PARAMETERS—LOS CASES (HEADREST AND AISLE ARMREST)

In (8), (9) and (10), , and are the intercepts and , and are the slopes. , and are zero-mean Gaussian and , rerandom variables with standard deviations, , spectively, and is the distance between the transmitting and receiving antennas. A summary of the NLOS channel model parameters that we extracted is given in Table III. IV. SMALL-SCALE FADING AND INTERDEPENDENCE OF MPCS A. Small-Scale Fading

Fig. 6(a) and (b), respectively. The regression lines for the exponential decay constant for the aisle and non-aisle cases are essentially identical so we have treated the two cases as a single case in Fig. 6(a). The regression lines for the excess amplitude of the LOS component for aisle and non-aisle cases are quite different so we have presented them separately in Fig. 6(b). and are generally well described by zero-mean Both normal distributions in ns and dB, respectively, and pass the Anderson-Darling goodness-of-fit test at the 5% significance level in most cases and at the 1% level in all cases except one. The in the aisle-headrest bell-shaped distribution presented by

We determined the distribution that best describes the smallscale fading of individual MPCs by processing the CIRs that we sampled at 49 points within a 300-mm 300-mm grid, extracting the amplitudes of the taps over all delays, computing the corresponding CDFs, and comparing them to standard distributions. In the past, others have found that the small scale fading distributions observed in residential environments are well approximated by a lognormal distribution [23] while others have found that a Nakagami distribution fits well [3]. However, our results show that the small-scale fading distribution of individual MPCs in the aircraft environment is well-approximated by a Rayleigh distribution. This is a reasonable outcome given that the aircraft passenger cabin is a dense scattering environment and it is likely that each delay bin or MPC consists of several rays. Moreover, others have reported that small-scale fading follows Rayleigh statistics in other dense scattering environments such as industrial plants [21].

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We refined our understanding of the distribution of smallscale fading by fitting it to the more general Nakagami distribution that has been used to model small-scale fading in other UWB environments. The Nakagami distribution is given by (11) where is the Nakagami -factor (or the shape paramis the Gamma function, and is eter of the distribution), the mean-squared value of the amplitude (or the spread parameter of the distribution). For each delay bin, we estimated the -factor of the Nakagami distribution by applying the inverse normalized variance estimator [24] to the 49 spatial samples. The estimate of the -factor is given by (12) where (13) and where is the number of spatial sampling points and is the complex amplitude of the th path. A scatter plot of the -factor estimates for the first 200 ns of delay bins for the receiving antenna mounted on the headrest at row 19 is shown in Fig. 7(a). Although a few MPCs at the beginning of the PDP (typically when the delay is less than 30 ns) exhibit large -factors, the vast majority of the 1501 MPCs shown in Fig. 7(a) exhibit -factors of approximately 1 and, as noted previously, their fading distributions are therefore well approximated by Rayleigh statistics. The fading distributions of MPCs observed at the armrest and footrest are also well approximated by Rayleigh statistics. Other researchers have found that the -parameter follows a lognormal distribution given by

m

Fig. 7. Estimates of the -factors (in dB) that describe the MPC fading distribution when the receiving antenna was mounted on the headrest of row 19: (a) as a function of delay and (b) expressed as a CDF and compared to the best fit normal distribution. TABLE IV SMALL-SCALE FADING PARAMETERS

(14) and are the mean and variance of the -factors where and are by convention given in decibels [19], [21]. The initial impulses in the PDP for the LOS channel are characterized by a , which is typically much larger than deterministic -factor, at other delays. In [3], it was found that both and may depend on the delay of the MPC within the CIR. As shown in Fig. 7(a), we did not find any evidence of such dependence. We tends to decrease with increasing distance, also observe that and are effectively independent of distance. Acwhile and simply by taking cordingly, we have characterized the average over all distances in each case and we model by (15) is the intercept and is the slope, is a where zero-mean Gaussian random variable with standard deviation , and is the distance between the transmitting and receiving antennas. The small-scale fading parameters that we extracted are summarized in Table IV. The CDF of the estimated

-factors is compared to the CDF of the best-fit lognormal distribution for the case of the receiving antenna mounted on the headrest at row 19 in Fig. 7(b). The 12 strongest taps (out of 1501 taps in total) deviate greatly from the lognormal distribution. They correspond to a few strong impulses that arrived near the leading edge of the response and we consider them to be outliers. B. Interdependence of MPCs The fading correlation between MPCs that are either: (1) in adjacent delay bins with the antenna at the same point on the sampling grid, which we shall refer to as temporal correlation, or (2) in the same delay bin but with the antenna at other points on

CHIU et al.: CHARACTERIZATION OF UWB CIRs WITHIN THE PASSENGER CABIN OF A BOEING 737-200 AIRCRAFT

the sampling grid, which we shall refer to as spatial correlation, is of interest for several reasons. First, if the MPCs in adjacent delay bins cannot be modeled as independent random variables, then the complexity of the channel model will increase dramatically. Second, we want to verify that the fading observed at a given delay at each point in the spatial sampling grid is reasonably independent from that observed at other points in the grid so that we have confidence that we have a sufficient number of independent samples to estimate the fading statistics. Third, some have recently proposed that WiMedia UWB systems be equipped with antenna arrays so that the direction-of-arrival of incoming signals can be estimated and adaptive array techniques can be used to reduce the susceptibility of the system to interfering signals. UWB-MIMO systems have also been proposed. Knowledge of the spatial correlation properties of the channel is required in order to determine the required antenna element spacing [25]. While the spatial correlation results presented here provide a useful first indication, our grid spacing of 5 cm does not allow unambiguous resolution of angular components at higher UWB frequencies. Thus, practical design of adaptive array antennas to be used at higher UWB frequencies will require that our measurements be supplemented by new data with finer spatial resolution. The temporal correlation is given by (16) where denotes expectation, and are the amplith MPC respectively, as observed tudes of the th and and in the CIRs measured at all 49 points in the grid, and are the corresponding mean values across all 49 points [4]. For all receiving antenna positions and locations considered, the mean value of the temporal correlation for the different delay taps is 0.13 with no value exceeding 0.56. Because the correlation between MPCs in adjacent delay bins is low, we can reasonably treat the path amplitudes at each delay as uncorrelated independent random variables. The spatial correlation between the MPCs at a given delay is given by

(17)

and are the amplitudes of the th MPC in the where PDPs that are observed at the th pair of points that are sepaand are the mean rated by a distance . The parameters amplitudes seen at all pairs of observation points that satisfy the above criteria [26]. In Fig. 8, we show the spatial correlation coefficient as a function of separation distance averaged over all delay bins. When the separation distance is greater than or equal to 50 mm, both the mean and standard deviation of the spatial correlation coefficient are always less than 0.1 and 0.3, respectively. Thus, we can reasonably assume that the path amplitudes observed at any two grid points at the same delay are uncorrelated and that our grid spacing of 50 mm was sufficient

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Fig. 8. Spatial correlation averaged over delay as a function of distance between spatial sampling points when the receiving antenna is mounted on the headrest of row 19.

for obtaining independent samples for fading statistics estimation. Although this result implies that UWB-MIMO arrays can be realized within aircraft passenger cabins with antenna spacings as small as 50 mm, further measurements will be required to determine if an even smaller spacing is practical. V. A SIMULATION MODEL FOR UWB CIRS IN AN AIRCRAFT PASSENGER CABIN With their final report, the IEEE 802.15.4a channel modeling committee released a MATLAB-based simulation code that uses their models to generate CIRs typical of those encountered in residential, office, outdoor and industrial environments. We have modified their channel simulation code so that it can be used to generate UWB CIRs typical of p-to-mp scenarios with the transmitting antenna located at the cabin ceiling and the receiving antenna located at the headrest, armrest and footrest level in the aircraft passenger cabin environment. In our version of the channel simulation code, scenarios AC 1 through AC 4 refer to transmission from the cabin ceiling to the headrest, aisle armrest, outboard armrest and footrest, respectively. The four main parts of the IEEE 802.15.4a code are concerned with: (1) assignment of the channel model parameters, (2) generation of CIRs using random processes that simulate: (a) the arrivals of the clusters and rays and (b) the path amplitudes based upon the shape of the PDP and the small-scale fading distribution, (3) prediction of the frequency dependent path loss, and (4) conversion of the result from continuous time to discrete time. The original simulator is based upon statistics that have been averaged over distance. Here, we use our new models to account for distance explicitly. The headrest and aisle armrest scenarios correspond to LOS channels and the APDPs are modeled by a single exponential decay as described by (4)–(7). The outboard armrest and footrest scenarios correspond to NLOS channels and are modeled using (3) and (8)–(10). Finally, we have modeled the small-scale fading of individual MPCs using (14) and (15). To verify that the modified channel simulator produces reasonable results, we generated CIRs using parameters for a given distance and then compared the results with the measured CIRs

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ulated CIRs over all ranges between 2 and 13 m also compare well. The authors will supply a copy of the modified version of the channel simulator code upon request. VI. CONCLUSION

Fig. 9. Comparison of the measured and regenerated APDP for receiving antennas mounted at the (a) headrest and (b) footrest.

Based upon channel response data collected within the passenger cabin of a typical mid-size airliner in p-to-mp configurations, we have proposed a pair of statistical models that describe the UWB channel impulse responses observed over LOS and NLOS channels, respectively. The models describe the shape of the power delay profile, characterize the fading experienced by individual multipath components and give the spatial and delay dependence of the correlation between fading on adjacent MPCs. We have observed the following trends: (1) For LOS channels, e.g., cabin ceiling to headrest or aisle armrest, the shape of the PDP generally follows IEEE 802.15.4a’s dense singlecluster model, but with negligible rise time and, on many occasions, one or more impulses or spikes within 30 ns of the leading edge of the response. (2) For NLOS channels, e.g., cabin ceiling to outboard armrest or footrest, the shape of the PDP follows IEEE 802.15.4a’s dense single-cluster model and the rise time is up to 10 ns. (3) The mean and variance of the exponential decay constant (hence the RMS delay spread) tends to increase with path length and as the receiving antenna drops from the headrest to the footrest. (4) Small-scale fading of MPCs tends to follow a Nakagami distribution with a lognormally-distributed -parameter that is close to 0 dB (which corresponds to Rayleigh fading) with a small variance, as has been found in other rich scattering environments. In most cases, our results take the form of the parameters of the corresponding models recommended by the IEEE 802.15.4a channel modeling committee and can be used directly in simulations of UWB propagation in an aircraft interior. Moreover, we have modified the standard channel impulse response simulation code developed by IEEE 802.15.4a so that it can generate CIRs representative of those observed in the aircraft passenger cabin environment. Accordingly, our results will assist: (1) those who are planning UWB deployments and field trials in aircraft and (2) those who need to simulate UWB systems in aircraft using realistic channels. ACKNOWLEDGMENT

Fig. 10. Distributions of simulated and measured RMS delay spreads for different receiving antenna mounting positions. For clarity, the distributions for the aisle armrest, outboard armrest and footrest cases are offset by 10, 20 and 30 ns, respectively.

The authors thank Associate Dean J. Baryluk and Chief Instructor/Hangar Supervisor G. Johnson of the BCIT Aerospace Technology Campus at Vancouver International Airport for providing them with access to their Boeing 737-200 aircraft and for their outstanding cooperation during the course of this study. The authors would also like to thank W. Muneer, W. Liu, R. White, C. Woodworth and C. Yeung for their considerable assistance during the measurement sessions. REFERENCES

observed at the same distance. As shown in Fig. 9, the measured and simulated APDPs for both the headrest and outboard armrest scenarios compare well. As shown in Fig. 10, the CDFs of the RMS delay spreads associated with measured and sim-

[1] A. F. Molisch, J. R. Foerster, and M. Pendergrass, “Channel models for ultrawideband personal area networks,” IEEE Wireless Commun., vol. 10, no. 6, pp. 14–21, Dec. 2003. [2] A. F. Molisch, “Ultrawideband propagation channels: Theory, measurement, and modeling,” IEEE Trans. Veh. Technol., vol. 54, no. 5, pp. 1528–1545, Sep. 2005.

CHIU et al.: CHARACTERIZATION OF UWB CIRs WITHIN THE PASSENGER CABIN OF A BOEING 737-200 AIRCRAFT

[3] A. F. Molisch et al., “A comprehensive standardized model for ultrawideband propagation channels,” IEEE Trans. Antennas Propag., vol. 54, no. 11, pp. 3151–3165, Nov. 2006. [4] C. C. Chong and S. K. Yong, “A generic statistical-based UWB channel model for high-rise apartments,” IEEE Trans. Antennas Propag., vol. 53, no. 8, pp. 2389–2399, Aug. 2005. [5] A. Jahn et al., “Evolution of aeronautical communications for personal and multimedia services,” IEEE Commun. Mag., vol. 41, no. 7, pp. 36–43, Jul. 2003. [6] J. J. Ely, W. L. Martin, G. L. Fuller, T. W. Shaver, J. Zimmerman, and W. E. Larsen, “UWB EMI to aircraft radios: Field evaluation on operational commercial transport airplanes,” in Proc. Digital Avionics Syst. Conf., Oct. 24–28, 2004, pp. 9.D.4-1–9.D.4-11. [7] J. J. Ely, W. L. Martin, T. W. Shaver, G. L. Fuller, J. Zimmerman, and W. E. Larsen, UWB EMI to Aircraft Radios: Field Evaluation on Operational Commercial Transport Airplanes NASA TP-2005-213606, vol. 1, Jan. 2005. [8] N. R. Diaz and M. Holzbock, “Aircraft cabin propagation for multimedia communications,” in Proc. EMPS, Sep. 25–26, 2002, pp. 281–288. [9] N. R. Diaz and J. E. J. Esquitino, “Wideband channel characterization for wireless communications inside a short haul aircraft,” in Proc. IEEE VTC 2004—Spring, May 17–19, 2004, pp. 223–228. [10] S. Fisahn, M. Camp, N. R. Diaz, R. Kebel, and H. Garbe, “General analysis of leaky section cables for multi-band aircraft cabin communications with different measurement techniques,” in Proc. Ultra-Wideband Short-Pulse Electromagnetics, Jul. 12–16, 2004, pp. 509–516. [11] G. Hankins, L. Vahala, and J. H. Beggs, “Propagation prediction inside a B767 in the 2.4 GHz and 5 GHz radio bands,” in IEEE AP-S Int. Symp. Dig., Jul. 3–8, 2005, pp. 791–794. [12] C. P. Niebla, “Topology and capacity planning for wireless heterogeneous networks in aircraft cabins,” in Proc. IEEE PIMRC, Sep. 11–14, 2005, pp. 2088–2092. [13] G. A. Breit, H. Hachem, J. Forrester, P. Guckian, K. P. Kirchoff, and B. J. Donham, “RF propagation characteristics of in-cabin CDMA mobile phone networks,” in Proc. Digital Avionics Syst. Conf., Oct. 30–Nov. 3, 2005, pp. 9.C.5-1–9.C.5-12. [14] R. Bhagavatula, R. W. Heath, and S. Vishwanath, “Optimizing MIMO antenna placement and array configuration for multimedia delivery in aircraft,” in Proc. IEEE VTC 2007—Spring, Apr. 22–25, 2007, pp. 425–429. [15] J. Chuang, N. Xin, H. Huang, S. Chiu, and D. G. Michelson, “UWB radiowave propagation within the passenger cabin of a Boeing 737-200 aircraft,” in Proc. IEEE VTC—Spring, Apr. 22–25, 2007, pp. 496–500. [16] J. Jemai et al., “UWB channel modeling within an aircraft cabin,” in Proc. IEEE ICUWB, Sep. 10–12, 2008, pp. 5–8. [17] J. Wang, A. S. Mohan, and T. A. Aubrey, “Angles-of-arrival of multipath signals in indoor environments,” in Proc. IEEE VTC, Apr. 28–May 1, 1996, pp. 155–159. [18] D. Porrat and Y. Serfaty, “Sub-band analysis of NLOS indoor channel responses,” in Proc. IEEE PIMRC, Sep. 15–18, 2008, pp. 1–5. [19] C. W. Kim, X. Sun, L. C. Chiam, B. Kannan, F. P. S. Chin, and H. K. Garg, “Characterization of ultra-wideband channels for outdoor office environment,” in Proc. IEEE WCNC, Mar. 13–17, 2005, pp. 950–955. [20] D. Cassioli, M. Z. Win, and A. F. Molisch, “The ultra-wide bandwidth indoor channel: From statistical model to simulations,” IEEE J. Sel. Areas Commun., vol. 20, no. 6, pp. 1247–1257, Aug. 2002. [21] J. Karedal, S. Wyne, P. Almers, F. Tufvesson, and A. F. Molisch, “A measurement-based statistical model for industrial ultra-wideband channels,” IEEE Trans. Wireless Commun., vol. 6, no. 8, pp. 3028–3037, Aug. 2007. [22] J. A. Dabin, A. M. Maimovich, and H. Grebel, “A statistical ultra-wideband indoor channel model and the effects of antenna directivity on path loss and multipath propagation,” IEEE J. Sel. Areas Commun., vol. 24, no. 4, pp. 752–758, Apr. 2006. [23] S. S. Ghassemzadeh, L. J. Greenstein, T. Sveinsson, A. Kavcic, and V. Tarokh, “UWB delay profile models for residential and commercial indoor environments,” IEEE Trans. Veh. Technol., vol. 54, no. 4, pp. 1235–1244, Jul. 2005.

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[24] A. Abdi and M. Kaveh, “Performance comparison of three different estimators for the Nakagami parameter using Monte Carlo simulation,” IEEE Commun. Lett., vol. 4, no. 4, pp. 119–121, Apr. 2000. [25] A. K. Marath, A. R. Leyman, and H. K. Garg, “DOA estimation of multipath clusters in WiMedia UWB systems,” in Proc. IEEE SAM’08, Jul. 21–23, 2008, pp. 108–112. [26] K. Makaratat and S. Stavrou, “Spatial correlation technique for UWB antenna arrays,” Electron. Lett., vol. 42, no. 12, pp. 675–676, Jun. 8, 2006.

m

Simon Chiu was born in Hong Kong, China, in 1984. He received the B.A.Sc. and M.A.Sc. degrees in electrical engineering from the University of British Columbia, Vancouver, BC, Canada, in 2006 and 2009, respectively. His main research interests focus on UWB propagation in passenger aircraft cabins and outdoor industrial environments, including the effects of human presence.

James Chuang was born in Tainan, Taiwan, in 1982. He received the B.A.Sc. and M.A.Sc. degrees in electrical engineering from the University of British Columbia, Vancouver, BC, Canada, in 2004 and 2007, respectively. His main research interests focus on UWB propagation in passenger aircraft cabins and computer-assisted identification of clusters in UWB channel impulse responses.

David G. Michelson (S’80-M’89-SM’99) received the B.A.Sc., M.A.Sc., and Ph.D. degrees in electrical engineering from the University of British Columbia (UBC), Vancouver, BC, Canada. From 1996 to 2001, he served as a member of a joint team from AT&T Wireless Services, Redmond, WA, and AT&T Labs-Research, Red Bank, NJ, where he was concerned with the development of propagation and channel models for next-generation and fixed wireless systems. The results of this work formed the basis for the propagation and channel models later adopted by the IEEE 802.16 Working Group on Broadband Fixed Wireless Access Standards. From 2001 to 2002, he helped to oversee the deployment of one of the world’s largest campus wireless local area networks at UBC while also serving as an Adjunct Professor with the Department of Electrical and Computer Engineering. Since 2003, he has led the Radio Science Laboratory, Department of Electrical and Computer Engineering, UBC, where his current research interests include propagation and channel modeling for fixed wireless, ultrawideband, and satellite communications. Prof. Michelson is a Registered Professional Engineer. He serves as the Chair of the IEEE Vehicular Technology Society Technical Committee on Propagation and Channel Modeling and as an Associate Editor for mobile channels for the IEEE Vehicular Technology Magazine. In 2002, he served as a Guest Editor for a pair of Special Issues of the IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS concerning propagation and channel modeling. From 2001 to 2007, he served as an Associate Editor for the IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY. From 1999 to 2007, he was the Chair of the IEEE Vancouver Section’s Joint Communications Chapter. Under his leadership, the chapter received Outstanding Achievement Awards from the IEEE Communications Society in 2002 and 2005 and the Chapter of the Year Award from IEEE Vehicular Technology Society in 2006. He received the E. F. Glass Award from IEEE Canada in 2009 and currently serves as Chair of IEEE Vancouver Section.

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Landmobile Radiowave Multipaths’ DOADistribution: Assessing Geometric Models by the Open Literature’s Empirical Datasets Kainam Thomas Wong, Senior Member, IEEE, Yue Ivan Wu, and Minaz Abdulla

Abstract—“Geometric modeling” idealizes the spatial geometric relationships among the transmitter, the scatterers, and the receiver in a wireless propagation channel—to produce closed-form formulas of various channel-fading metrics (e.g., the distribution of the azimuth angle-of-arrival of the arriving multipaths). Scattered in the open literature are numerous such “geometric models,” each advancing its own closed-form formula of a fading metric, each based on a different idealization of the spatial geometry of the scatterers. Lacking in the open literature is a comprehensive and critical comparison among all such single-cluster geometric-model-based formulas of the arriving multipaths’ azimuth direction-of-arrival distribution. This paper fills this literature gap. The comparison here uses all empirical data legibly available in the open literature for landmobile wireless radiowave propagation. No one geometric model is best by all criteria and for all environments. However, a safe choice is the model with a Gaussian density of scatterers centered at the transmitter. Despite this model’s simplicity of having only one degree of freedom, it is always either the best fitting model or offers an LSE within one third of an order-of-magnitude as the best fitting model for all empirical dataset of all environments. Index Terms—Communication channels, direction-of-arrival (DOA), dispersive channels, fading channels, geometric modeling, multipath channels, scatter channels.

fading,” “local fading,” or “microscopic fading”—because the multipaths’ vector-summation would vary greatly in magnitude even if the receiver is displaced by a small distance at fractions of a wavelength. “Small-scale fading” is also called “fast fading,” because a moving receiver would experience the small-scale fading’s spatial variability as a fast temporal variability. “Small-scale fading” contrasts against “large-scale fading” (a.k.a. “slow fading”), which is caused by propagation-distance-related path-loss. “Small-scale fading” also contrasts against “shadowing,” which is caused by sizeable obstacles blocking the receiver from the transmitter. It is important to model the wireless channel’s DOA distribution at the receiver, for the development and analysis of smart-antennas spatial-diversity schemes, such as space-division frequency re-use, beamforming, emitter localization, etc. This DOA distribution may be obtained by “normalizing” the arriving multipaths’ power distribution over all directions-of-arrival, by magnitude-scaling the multipaths’ arrival-power distribution so that the power distribution sums to one over the entire range of the direction-of-arrival. B. “Geometric Models” Versus Other Modeling Approaches of Microscopic Channel Fading

I. INTRODUCTION A. Distribution of the Azimuth Direction-of-Arrival of the Arriving Multipaths N wireless communications, a transmitted signal reaches a receiver via multiple propagation paths, undergoing various sequences of reflection, diffraction, and scattering. Each such “multipath” carries its own propagation history, resulting in its particular amplitude, propagation delay, direction-of-arrival, polarization, and Doppler shift. At the receiving antenna, these multipaths are phasor-summed, constructively or destructively, to produce that antenna’s measured data. Hence, the receiver “sees” the transmitter in space not as a geometrically point-like source, but as spatio-temporally spread over a range of time-of-arrival (TOA) and direction-of-arrival (DOA). The above propagation phenomenon is labeled “small-scale

I

Manuscript received December 03, 2008; revised October 17, 2009. First published December 04, 2009; current version published March 03, 2010. K. T. Wong and Y. I. Wu are with the Department of Electronic and Information Engineering, Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong (e-mail: [email protected]). M. Abdulla is at Toronto, Ontario, Canada (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2009.2037698

There exist various strategies to mathematically model the propagation channel. The most direct and the most site-specific approach is empirical measurement at the particular site/terrain/building of interest. Another approach, more labor saving but still site-specific, is to approximate the particular site under investigation as an electromagnetic-physics-based ray-tracing computer-model. These site-specific/terrain-specific/building-specific approaches are faithful to the particular site’s idiosyncratic electromagnetic and spatio-temporal complexities. Each such simulation produces a quantitatively accurate model, but each simulation applies to only that one particular propagation setting under investigation (e.g., a particular city’s particular cross-sectional street corner under a particular weather). With many simulations over many scenarios, the ray-tracing approach can be generalized to a wider class of environments (e.g., the class of “bad urban” settings of high-rises in all downtowns). In contrast, a “geometric model” can encapsulate the essence of a wide class of diverse propagation settings. “Geometric modeling” idealizes the wireless electromagnetic propagation environment via a geometric abstraction of the spatial relationships among the transmitter, the scatterers, and the base-station. (For example, scatterers could be idealized as distributed evenly on only a small disc centered around the mobile [9], [11], [12], [20], [38].) Geometric models attempt to

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TABLE I PROPAGATION & MEASUREMENT ENVIRONMENT FOR EMPIRICAL DATASETS WITH A Uni-MODAL HISTOGRAM

TABLE II PROPAGATION & MEASUREMENT ENVIRONMENT FOR EMPIRICAL DATASETS WITH A Non-UNI-MODAL HISTOGRAM

embed measurable fading metrics (e.g., the DOA distribution) integrally into the propagation channel’s idealized geometry, such that only a very few geometric parameters (e.g., the single model-parameter of the ratio between the aforementioned disc’s radius and the transmitter-receiver distance ) would affect these various fading metrics in an inter-connected manner to conceptually reveal the channel’s underlying geometric dynamics. This modeling’s generic abstract geometry involves no site-specific or terrain-specific or building-specific information, such as those used in empirical measurements or in any one ray-shooting/ray-tracing computer-simulation. Much literature on “geometric models” involves little or no mathematically rigorous derivation of the received signal’s measurable fading statistics, due to the inherent mathematical difficulties of such a rigorous derivation. Instead, a limited series of Monte Carlo simulations would approximate the numerical values of the channel-fading metrics. Such simulations can be performed only at relatively limited number of pre-set numerical values, which are geometrically independent of the model parameters. Hence, this would produce no closed-form mathematical relationship among the fading metrics, in terms of the geometric-model’s independent parameters. Such simulations thereby limit the insight obtainable from such a geometric model. This survey will focus only on those “geometric models” for which rigorous analytical derivation have closed-form expressions of the uplink azimuth direction-of-arrival distribution, explicitly in terms of the geometric parameters.

C. The Purpose of This Work Geometric models of propagation-channels have been used in [30], [46], [48], [57], [58] (among others) to analytically predict the performance of communications systems (and not merely by computer-simulations). However, numerous “geometric models” have emerged in the past decade, each based on a different spatial distribution of the scatterers. Each would thus offer a competing closed-form distribution-formula for the azimuth-DOA of the multipaths arriving at the receiver. Many authors proposed their geometric models without verification by empirical data, though a few were validated by a few empirical datasets pre-selected by the authors themselves. It remains unclear which “geometric model” is how best under what field scenarios and why. This literature gap is perhaps due to the labor-intensive nature of such an investigation. This present work aims to be an impartial third party, to thoroughly compare and contrast the accuracy of these competing geometric models’ derived azimuth direction-of-arrival distribution in landmobile radiowave communications against the open literature’s empirically measured data. More specifically, for every such empirical dataset available in the open literature (and listed in Tables I and II), it is used herein to calibrate every known “geometric model” (listed in Table III) for which a closed-form explicit formula has been analytically derived for the azimuth direction-of-arrival. Such two-dimensional modeling admittedly ignores the elevation, but often justifiably so, especially in a macro-cell situation where the transmitter-receiver separation

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TABLE III TWO-DIMENSIONAL “GEOMETRICAL MODELS” FOR OUTDOOR RADIOWAVE CELLULAR COMMUNICATION’S UPLINK AZIMUTH DIRECTION-OF-ARRIVAL DISTRIBUTION

D

( denotes the spatial separation between the base-station receiver and the mobile transmitter. The azimuth angle linking the mobile to the base-station.)

would greatly exceeds the heights of the transmitter or the receiver. Conclusions are then drawn as to which, how, and why specific geometric models best fit what field situations. Admittedly, partial listings of these “geometric models” can be found in [10], [17], [49]; however, those partial listings offer no

 is defined with respect to the axis

comparative assessment of various “geometric models” against empirical data. This present work will complete this missing link. The rest of this manuscript is organized as follows: Section II will survey various competing “geometric models”. Section III will characterize the empirical data-sets to be used to calibrate

WONG et al.: LANDMOBILE RADIOWAVE MULTIPATHS’ DOA-DISTRIBUTION: ASSESSING GEOMETRIC MODELS

Fig. 1. Azimuth DOA distributions for various “Rx outside” geometric models.

the geometric models. Section IV will define the least-squares errors (LSE) metric to measure how well any geometric model fits any empirical data-set, as well as fine points in the calibration algorithm. That section will also present calibration leastsquares errors. Section V-A will discuss, for unimodal datasets, which “geometric models” best fits what types of field-scenarios and why, whereas Section V-B will do the same for bimodal or multimodal datasets. Section VI will conclude this work. II. THE CANDIDATE “GEOMETRIC MODELS” FOR THE ARRIVING MULTIPATHS’ AZIMUTH-DOA DISTRIBUTION Numerous two-dimensional “geometric models” [2], [9], [11], [12], [14], [20], [38], [41], [47], [55] have been proposed for the radiowave outdoor landmobile cellular communication uplink’s azimuth direction-of-arrival distribution. “Geometric models” typically model a multipath as the bouncing of the transmitted signal off one scatterer. A multipath’s azimuth direction-of-arrival is thus determined by the spatial location of the scatterer off which the multipath is reflected before reaching the receiver. Hence, one pivotal character of any geometric model is how the model characterizes the scatterers’ spatial distribution in relation to the transmitter and the receiver. Various geometric models differently idealize the scatterers’ spatial distribution in relation to the transmitter and the receiver. Table III comparatively summarizes these two-dimensional geometric models’ contrasting scatterer spatial distributions and corresponding azimuth direction-of-arrival distributions. Figs. 1 and 2 graphically contrast these direction-of-arrival distributions at comparable model parameter values. All above-mentioned geometric models make these common assumptions: a) all transmitting and receiving antennas are omnidirectional; b) polarizational effects may be ignored;

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Fig. 2. Azimuth DOA distributions for various “Rx inside” geometric models.

c) each propagation path, from the mobile to the base-station, reflects off exactly one scatterer; d) each scatterer acts (independently of other scatterers) as an omnidirectional lossless re-transmitter; e) negligible complex-phase effects in the receivingantenna’s vector-summation of its arriving multipaths. That is, all arriving multipaths arriving at each receivingantenna are assumed to be temporally in-phase among themselves. All above models (except [14]) also ignore “propagation loss,” i.e., the power loss experienced as a signal travels outwards from the transmitter, due to the signal wavefront’s expanding area. These models’ different scatterer-distributions may be classified according to several perspectives: A) whether the scatterers surround only the transmitter, or surround also the receiver; B) the shape of spatial density of the scatterers around the transmitter; C) unimodal versus bimodal versus multimodal spatial densities for the scatterers; The following subsections will analyze these categories one by one. A. Geometric-Model Classification by Whether the Receiver Lies Within/Outside the Scatterers’ Spatial Region For an elevated base-station receiver (Rx) in a macro-cell, most significant scatterers concentrate locally around the street-level transmitter (Tx) but away from the elevated receiver. Hence, a “geometric model” could idealize its scatterers’ spatial support region as enclosing(and centering around) the mobile transmitter, but as excluding the base-station receiver itself. This is a “local scattering model” and is exemplified by the following models: 1) a uniform density within a circular-disc support region of radius , which is less than the transmitter-receiver separation [9], [11], [12], [20], [38];

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Fig. 3. (a) The support region of the “uniform hollow-disc (Rx outside)” model. (b) The support region of the “uniform pie-cut (Rx inside)” model.

2) a uniform density within a hollow circular-disc support re[47]. Please refer to Fig. 3(a); gion of outer radius 3) an inverted-parabolic density within a circular-disc support [55]; region of radius 4) a conical density within a circular-disc support region of radius [9]; 5) a uniform density within an elliptical-disc support region centered at the transmitter but excluding the receiver [11]. On the other hand, for a micro-cell with a relatively low basestation height, significant scatterers may locate near the basestation. This is modeled with the scattering region enclosing both the base-station receiver and the mobile transmitter. The multipaths’ DOAs could impinge from any direction 360 . The following models fall under this class: (6) a uniform density within a circular-disc support region of radius [9], [12], [50]; (7) a uniform density within a support region of a pieshaped cut of a circular-disc of radius [50] (for a directional transmitter with a azimuth beam-width). Please refer to Fig. 3(b); (8) a conical density within a circular-disc support region of radius [9]; (9) a uniform density within an elliptical-disc support region focused at the transmitter and the receiver [20]. (10) a Gaussian density centered at the transmitter [41], [45], [52],1; (11) a Rayleigh density centered at the transmitter [14].2 B. Geometric-Model Classification by the Spatial Concentration of the Scatterers Around the Transmitter The six “geometric models” in rows #1–4 and 8–9 of Table III have uniform densities; however, the remaining five models

p

p

1The Gaussian spatial distribution is also investigated in [19], but its deerfc( D cos = 2 ). This rived formula is (A=2 2 )e formula disagrees with that derived in [41] for the same model and appears incorrect to the present authors. Hence, [19] will be ignored thereafter. Any subsequent reference to a Gaussian scatterer model would mean [41]. 2The Rayleigh scatterer distribution of [14] assumes that (R=D ) 1, at which the DOA distribution would approach that of the Gaussian scatterer model in [41]. For (R=D ) 1, the DOA distribution f ( ) could become negative, unless (and unstated in [14] that) the azimuth angle,  , is restricted to ( (=2); =2). This restriction turns out to be moot in this present work, as all empirical data-sets here satisfy the restriction.

0



0



have unimodal densities peaking at the transmitter. Among the non-uniform densities, the “conical circular” model [9] has the most concentrated scatterers around the transmitter followed by the “inverted-parabolic circular (Rx outside)” model [55], then the “Rayleigh circular (Rx outside)” model [14], and lastly the “Gaussian” model [41] (which has an infinite spatial support region for the scatterers). The greater concentration of scatterers can be intuitively justified as follows: Recall that all aforementioned “geometric models” idealize every scatterer as an omnidirectional lossless transmitter, thereby overlooking any power loss due to scattering. A unimodal concentration is an indirect way to account for this neglected scattering loss. The bounce off a distant scatterer in the model may correspond to only the last bounce in an actual sequence of consecutive physical reflections farther and farther away from mobile. Each such reflection incurs power loss. Hence, the farther from the transmitter is a scatterer, the weaker its reflected path would be in actuality. Rather than accounting for such scattering-loss explicitly in the mathematical derivation, it is mathematically simpler to assume a denser distribution of “last-bounce” scatterers closer to the transmitter. Far-off scatterers (like mountains, high-rises) could increase the angular spread and may be accounted for in the “geometric model” by a larger scattering area. A larger “normalized” radius leads to less concentration of scatterers around the receiver. For , the various “circular-disc (Rx outside)” models [9], [11], [12], [20], [38], [55] or “uniform hollow-disc (Rx outside)” model [47] can have multipaths arriving from only . The circular-disc models’ azimuth-DOA distribution’s unimodal peak would have a width equal to radians in the azimuth direction-of-arrival. As decreases, becomes narrower and “taller,” such that as . Similar trends hold for the “Gaussian” model’s [41], the “Rayleigh circular (Rx outside)” model’s [14], and the “uniform elliptical (Rx outside)” model’s [11]. C. Geometric-Model Classification by the Modality of the Scatterers’ Spatial Density: Unimodal, Bimodal, or Multi-modal All aforementioned “geometric models” produce unimodal probability densities for the azimuth direction-of-arrival, except for the “uniform pie-cut (Rx inside)” model (row # 3 in Table III) and the“uniform hollow-disc (Rx outside)” model (row # 4 in Table III). The “uniform hollow-disc (Rx outside)” model [47] has a bimodal DOA-density. It generalizes the “uniform circular (Rx outside)” model of [9], [11], [12], [20], [38]. Fig. 3(a) shows the “uniform hollow-disc (Rx outside)” model’s allowable locations for the scatterers. When the “uniform hollow-disc (Rx outside)” model has , it becomes the “uniform circular (Rx outside)” model. As increases for the “uniform hollow-disc (Rx outside)” model, the azimuth direction-of-arrival distribution’s two peaks become narrower and “taller,” as well as getting further apart from each other. The “uniform pie-cut (Rx inside)” model has a trimodal DOA-density.

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TABLE IV LEAST-SQUARE ERRORS (LSE) WHEN EACH “GEOMETRICAL MODEL” OF TABLE III IS CALIBRATED BY EACH EMPIRICAL DATA-SET OF TABLES I AND II

III. EMPIRICAL DATA FROM THE OPEN LITERATURE Spread through the open literature are empirical data for the uplink azimuth direction-of-arrival’s distribution in radiowave wireless landmobile communications. The present authors have done an exhaustive search for such empirical data, which are listed in Tables I and II. Surprisingly, only about a dozen readable data-sets can be located. To assure consistence in extracting numerical data from data graphs, the present authors use the software GetData instead of human visual reading. See http:// www.getdata.com/ Excluded from Table I and Table II are many illegible graphical data from the open literature, often presented in poor-quality three-dimensional plots or contour maps, from which no numerical data can be reliably extracted. Examples of such numerically illegible empirical datasets include: Laurila [37, Figs. 7, 11, 13, 17 and 19]; DeJong [22, Figs. 8 and 9]; DeJong [23, Figs. 5, 9 and 10 ]; of DeJong [24, Fig. 4]; Kuchar [27, Figs. 5, 8, 9, 13, 14, 16, and 21]; Martin [16, Fig. 9 ]; Steinbauer [33, Figs. 15–18]; Thoma [29, Figs. 7 and 8]; Zhao [39, Fig. 11]; Zhu [32, Figs. 6–9]; Zhu [35, Fig. 6]; Toeltsch [36, Fig. 1]; Blanz [4, Fig. 4]; Kalliola [8, Figs. 4 and 6]; and Larsson [26, Fig. 1]. Tables I and II describe each numerically legible empirical data-set’s physical environment and setting, the channel-sounding signal’s frequency, heights of the transmitting antenna and the receiving antenna—where such information is given in the corresponding reference. However, not all references give all of the above information. Tables I and II’s data-sets will provide the basis on which to compare what geometric model(s) can best describe what types of empirical propagation environment. The open literature appears to offer no such systematic and comparative validation of

various competing geometric models. This literature gap is filled by this work. Tables I and II’s data-sets may be classified by the measurement’s field environment and by the measured data’s histogram shape. A. Empirical Data-Set Classification by “Rural” Versus “Suburban”Versus “Urban” The measurement’s field environments may be roughly divided into the categories of “rural,” “suburban,” or “urban” as follows. • (R) The “rural” environment consists of flat or hilly terrains with large open spaces. It is mainly nature, possibly with forests or very few buildings. • (S) The “suburban” environment consists of small buildings of 3 to 5 stories, with much less open space than does the rural environment. An example is a suburban residential neighborhood in North America. • (U) The “urban” environment consists of highrises with narrow streets and no open space. An example is a downtown metropolis. These categories are admittedly fuzzy but nonetheless often used in the literature. The “suburban” versus “urban” classification partly depends on the researcher’s location. Many European “urban” environments may well be considered as “suburban” in Northeast Asia. Moreover, as subsequent sections will show, a equally critical consideration is the height of the transmitting antenna or receiving antenna relative to the surrounding buildings’ height. Nonetheless, Tables I and II’s rural/suburban/urban classification mostly honors each paper’s

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Fig. 5. Curve-fitting various geometric model to the empirical data in Pedersen [13, Fig. 1]. Fig. 4. Curve-fitting various geometric model to the empirical data in Matthews [1, Fig. 7].

own self-characterization.3 The following datasets have no self-classification: Matthews [1, Figs. 7 and 8] and Kloch [34, Fig. 6]. B. Empirical Data-Set Classification by Histogram’s Modality Another classification criterion is by the measured data’s histogram shape. Table I lists all unimodal datasets, whereas Table II lists all bimodal and higher-modal datasets. This division will aid comparison with the “geometric models,” most of which are unimodal but one is bimodal and another is trimodal. Among Table II’s five non-unimodal empirical data sets: four are “urban,” only one is “suburban,” and none is “rural.” This is intuitively reasonable, because multiple clusters of scatterers are more likely in densely built-up environments. IV. THE GOODNESS-OF-FIT METRIC AND THE CALIBRATION RESULTS For each empirical dataset available in Table I and Table II, this paper will use that dataset to calibrate each “geometric model” in Table III. Conclusions will then be drawn in the next section as to what, how, when and why specific geometric models best fit what field situations. 3The dataset from [34] is reclassified from “suburban” to “urban,” because its receiving antenna was on the street level and was surrounded by two story buildings. The dataset from [25, Fig. 3] is reclassified here as “urban,” despite its self-classification as “suburban.” This reclassification is because both the transmitter and the receiver were placed atop buildings, thereby allowing LOS propagation.

Fig. 6. Curve-fitting various geometric model to the empirical data in Kuchar [25, Fig. 3].

WONG et al.: LANDMOBILE RADIOWAVE MULTIPATHS’ DOA-DISTRIBUTION: ASSESSING GEOMETRIC MODELS

Fig. 7. Curve-fitting various geometric model to the empirical data in Takada [40, Fig. 4].

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Fig. 8. Curve-fitting various geometric model to the empirical data in Fleury [18, Fig. 16].

The goodness-of-fit of any calibrated geometric model to the calibrating empirical data-set is the least-squares error (LSE) between the two. The first calibration-step is to normalized each empirical dataset to give unity area under the data-set, to match the unity area under each geometric-model’s DOA density-distribution. The least-squares error (LSE) is defined as (1) where denotes the normalized represents the geometric model’s azempirical dataset, refers to the imuth direction-of-arrival density distribution, dataset’s number of data points, and is a nuisance-parameter to align the data-set’s transmitter-receiver line-of-sight DOA. Many empirical datasets do not state this transmitter-receiver line-of-sight DOA. The calibration here will search to identify the LSE. Note also that through all values of may be unevenly spaced along the coordinate. When a reference paper graphically presents its will be evenly empirical data as curves, spaced because a uniform grid is used with the GetData softmay be non-uniformly ware. However, spaced when the reference presents its data as discrete icons. does not conMoreover, tribute to the LSE. For most empirical data sets, is not near or . Hence, it is unlikely that were zero for or for . Rather, the empirical dataset has zero for

Fig. 9. Curve-fitting various geometric model to the empirical data in Mogensen [6, Fig. 3].

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Fig. 10. Curve-fitting various geometric model to the empirical data in Pedersen [28, Fig. 4] Aarhus.

Fig. 11. Curve-fitting various geometric model to the empirical data in Pedersen [28, Fig. 4 ] Stockholm.

been truncated on both ends of the histogram. Consequently, . the LSE should be computed only for Table IV lists the LSE for each of Table III’s geometric model, calibrated by each empirical data-set of Tables I and II. The geometric models, that “well fit” each empirical dataset of Tables I and II, are listed in the second-to-last column thereof. This includes any geometric model with a calibration-LSE within 110% of the best-fitting geometrical model’s. Figs. 4 to 14 each plot one empirical data-set of Tables I and II, along with the DOA-distributions of the geometric models calibrated to that empirical data-set. V. INSIGHTS FROM CALIBRATION A. Insights From the Unimodal Empirical Datasets For the uni-model datasets, the well-fitting models are “Rayleigh circular (Rx outside),” “Gaussian,” “uniform elliptical (Rx outside),” and (in only one case) “uniform elliptical (Rx inside).” In both the “Rayleigh circular (Rx outside)” and the “Gaussian” models, the scatterers become denser as they are closer to the transmitter. Indeed, for whichever empirical dataset well-fit by either the “Gaussian” model or the “Rayleigh circular (Rx outside)” model, the other model is also well-fitting for that data-set. In such well-fitting cases, the calibrated model , 0.15, for both of these parameters geometric models. (Please refer to Table III for all symbol-definitions in this section.) Moreover, such a range of values for implies that the receiver is far the “Gaussian” model’s

Fig. 12. Curve-fitting various geometric model to the empirical data in Matthews [1, Fig. 8].

from most scatterers, even though the “Gaussian” model has a nominally infinite spatial support region for the scatterers.

WONG et al.: LANDMOBILE RADIOWAVE MULTIPATHS’ DOA-DISTRIBUTION: ASSESSING GEOMETRIC MODELS

Fig. 13. Curve-fitting various geometric model to the empirical data in Pedersen [13, Fig. 5].

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The well-fitting “uniform elliptical (Rx outside)” and the “uniform elliptical (Rx inside)” models have the model-param, i.e., the receiver is just marginally eter inside or marginally outside the ellipse. Moreover, it is on the ellipse’s longer axis that the receiver lies, showing that the “depth” is more important than the “breadth” (i.e., the azimuth-spread) of the scatterers’ spatial distribution between the transmitter and the receiver. Table V lists the azimuth-spreads of the arriving multipaths for the several empirical datasets that are well-fit by the “uniform elliptical (Rx outside)” geoin all those metric model. As the model-parameter of cases, the azimuth-spread approximately equals . Table V shows that the arriving multipaths’ azimuth-spread increases as the propagation environment setting moves from “rural” to “suburban” to “urban,” fitting the intuitive expectation that the more clustered environment will result in multipaths arriving from a wider azimuth-spread. Note that the “uniform elliptical (Rx outside)” model is the only unimodal geometric model with two degrees of freedom. The “Rx inside” models are not well-fitting, except for one “urban” case. This conforms to the intuitive expectations that the more urban is the propagation environment setting, the transmitter needs to be modeled as located more among the scatterers. Which of the four above-mentioned well-fitting models is best for performance-analysis of a communication system? Recall from Table III that both the “Gaussian” model and the “Rayleigh circular (Rx outside)” model have open-form expressions for the arriving multipaths’ DOA-distribution; however, Gaussianity may ease further mathematical analysis. As these two geometric models are comparable in their calibration-LSE, the “Gaussian” model may be preferred over the “Rayleigh circular (Rx outside)” model. If a closed-form DOA-distribution is required, the choice will be the “uniform elliptical (Rx outside)” geometric model. B. Insights From the Bimodal & Higher-Model Empirical Datasets For the five bimodal and trimodal empirical datasets in Table II, the best-fitting model is either the “uniform pie-cut (Rx inside)” model or the “uniform hollow-disc (Rx outside)” model.4 Both models have two degrees of freedom. These two models are in fact the only two geometric models with more than one peak in the DOA-distribution: the “uniform pie-cut (Rx inside)” model is trimodal, whereas the “uniform hollow-disc (Rx outside)” model is bimodal. For the two tetramodal empirical data-sets, they are both best-fit by the “uniform pie-cut (Rx inside)” geometric model, which alone (among all geometric models) offers three peaks. Considering the three empirical datasets best-fit by the “uniform pie-cut (Rx inside)” geometric model. a. Two empirical datasets are “urban,” while one is “suburban.” This dove-tails with the intuitive expectation that a more clustered propagation-environment would more likely produce a non-unimodal DOA-distribution.

Fig. 14. Curve-fitting various geometric model to the empirical data in Pedersen [28, Fig. 14].

4The “uniform circular (Rx inside)” model comes in second for the one dataset from Kloch [34, Fig. 6]. There, the receiver at the street level surrounded by two-storey buildings.

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TABLE V COMPARING THE ARRIVING MULTIPATHS’ AZIMUTH-SPREADS FOR THE EMPIRICAL DATA-SETS WELL-FIT BY THE “UNIFORM ELLIPTICAL (RX OUTSIDE)” GEOMETRIC MODEL

Fig. 15. Curve-fitting various geometric model to the empirical data in Kloch [34, Fig. 6].

b. All calibrated “uniform pie-cut” models have a beamwidth under 45 . c. All calibrated “uniform pie-cut” models have the model, i.e., the receiver is at parameter or very close to the pie-cut rim. This suggests that the scatterers at the receiver’s backside are of only marginal importance. The “Gaussian” model, though best fitting for none of the five non-unimodal datasets in Table II, is at worst only roughly double the lowest LSE. The “Gaussian” model can thus offer modeling simplicity for an LSE still within about one third of an order-of-magnitude of the best fitting model. VI. CONCLUSION For the uni-modal datasets, the well-fitting geometric models are mainly “Rayleigh circular (Rx outside),” “Gaussian,” and “uniform elliptical (Rx outside).” The “Gaussian” model may be preferred over the “Rayleigh circular (Rx outside)” model, because Gaussianity may ease further mathematical analysis of a communication system’s performance. If a closed-form DOAdistribution is required, the choice will be the “uniform elliptical (Rx outside)” geometric model. The non-uni-modal empirical datasets are best-fit by the “uniform pie-cut (Rx inside)” geometric model or the “uniform hollow-disc (Rx outside)” geometric model, which have three and two peaks, respectively. Though no one geometric model is best by all criteria and for all environments, a safe choice is the “Gaussian” model, with a Gaussian density of scatterers centered at the transmitter. Despite this model’s simplicity with only one degree of freedom, it is always either the best fitting model or offers an LSE within one third of an order-of-magnitude as the best fitting model—The only other model that offers such robust fitting is the “Rayleigh” model with two degrees of freedom. ACKNOWLEDGMENT

Fig. 16. Curve-fitting various geometric model to the empirical data in Eggers [43, Fig. 6].

This paper categorically expands (and substantially corrects) M. Abdulla’s M.A.Sc. thesis, submitted in April 2005 to the Department of Electrical and Computer Engineering, University of Waterloo, Waterloo, ON, Canada.

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[46] W. T. Ng and V. K. Dubey, “Effect of employing directional antennas on mobile OFDM system with time-varying channel,” IEEE Commun. Lett., vol. 7, no. 4, pp. 165–167, Apr. 2003. [47] A. Y. Olenko, K. T. Wong, and E. H.-O. Ng, “Analytically derived TOA-DOA statistics of uplink/downlink wireless multipaths arisen from scatterers on an hollow-disc around the mobile,” IEEE Antennas Wireless Propag. Lett., vol. 2, no. 22, pp. 345–348, 2003. [48] A. Giorgetti, M. Chiani, M. Shafi, and P. J. Smith, “Level crossing rates and MIMO capacity fades: Impacts of spatial/temporal channel correlation,” in Proc. Int. Conf. on Commun., 2003, vol. 5, pp. 3046–3050. [49] S. Mahmoud, Z. M. Hussain, and P. O’Shea, “Space-time geometricalbased channel models: A comparative study,” presented at the Australian Telecommunications, Networks and Applications Conf., 2003. [50] L. Jiang and S. Y. Tan, “Simple geometrical-based AOA model for mobile communication systems,” Electron. Lett., vol. 40, no. 19, pp. 1203–1205, Sep. 2004. [51] N. Blaunstein and E. Tsalolihin, “Signal distribution in the azimuth, elevation, and time-delay domains in urban radio communication links,” IEEE Antennas Propag. Mag., vol. 46, no. 5, pp. 171–178, Oct. 2004. [52] D. D. N. Bevan, V. T. Ermolayev, A. G. Flaksman, and I. M. Averin, “Gaussian channel model for mobile multipath environment,” EURASIP J. Appl. Signal Processing, vol. 2004, no. 9, pp. 1321–1329, 2004. [53] L. Jiang and S. Y. Tan, “Geometrical-based propagation model for mobile communication systems,” in Proc. Int. Conf. on Microw. Millimeter Wave Technol., 2004, pp. 834–837. [54] L. Jiang and S. Y. Tan, “Geometrically-based channel model for mobile-communication systems,” Microw. Optical Technol. Lett., vol. 45, no. 6, pp. 522–527, Jun. 2005. [55] A. Y. Olenko, K. T. Wong, and M. Abdulla, “Analytically derived TOA-AOA distributions of uplink/downlink wireless-cellular multipaths arisen from scatterers with an inverted-parabolic spatial distribution around the mobile,” IEEE Signal Processing Lett., vol. 9, no. 7, pp. 516–519, Jul. 2005. [56] M. T. Simsim, N. M. Khan, R. Ramer, and P. B. Rapajic, “Time of arrival statistics in cellular environments,” in Proc. IEEE Veh. Technol. Conf., Spring, 2006, vol. 6, pp. 2666–2670. [57] F. Bohagen, P. Orten, and G. E. Oien, “Design of optimal high-rank line-of-sight MIMO channels,” IEEE Trans. Wireless Commun., vol. 6, no. 4, pp. 1420–1425, Apr. 2007. [58] I. Sarris and A. R. Nix, “Design and performance assessment of highcapacity MIMO architectures in the presence of a line-of-sight component,” IEEE Trans. Veh. Technol., vol. 56, no. 4, pp. 2194–2202, Jul. 2007. [59] N. M. Khan, M. T. Simsim, and P. B. Rapajic, “A generalized model for the spatial characteristics of the cellular mobile channel,” IEEE Trans. Veh. Technol., vol. 57, no. 1, pp. 22–37, Jan. 2008.

Kainam Thomas Wong (SM’01) received the B.S.E. degree in chemical engineering from the University of California at Los Angeles, in 1985, the B.S.E.E. degree from the University of Colorado, Boulder, in 1987, the M.S.E.E. degree from the Michigan State University, East Lansing, in 1990, and the Ph.D. in degree in electrical engineering from Purdue University, West Lafayette, IN, in 1996. He was a manufacturing Engineer at the General Motors Technical Center, Warren, MI, from 1990 to 1991, and a Senior Professional Staff Member at the Johns Hopkins University Applied Physics Laboratory, Laurel, MD, from 1996 to 1998. From 1998 and 2006, he was on the faculties of Nanyang Technological University, Singapore, the Chinese University of Hong Kong, and the University of Waterloo, Waterloo, ON, Canada. Since 2006, he has been with the Hong Kong Polytechnic University as an Associate Professor. His research interest includes signal processing for communications as well as sensor-array signal processing. Prof. Wong received the Premier’s Research Excellence Award from the Canadian province of Ontario in 2003. He has been an Associate Editor of Circuits, Systems, and Signal Processing, the IEEE SIGNAL PROCESSING LETTERS, the IEEE TRANSACTIONS ON SIGNAL PROCESSING, and the IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY.

Yue Ivan Wu received the B.Eng. degree in communication engineering and the M.Eng. degree in communication and information systems from the University of Electronic Science and Technology of China, Chengdu, in 2004 and 2006, respectively. He is currently working toward the Ph.D. degree at the Hong Kong Polytechnic University. His research interest is in wireless propagation channel modeling and space-time signal processing.

Minaz Abdulla received the B.A.Sc. degree in computer engineering and the M.A.Sc. degree in electrical and computer engineering from the University of Waterloo, Waterloo, ON, Canada, in 2002 and 2005, respectively. He worked as a Navigation Core Engineer at Destinator Technologies from 2004 to 2007. He is currently a freelance Engineer. His technical interests are software development for wireless mobile applications, routing algorithms, and optimization problems.

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Communications The Beam Pattern of Reflector Antennas With Buckled Panels A. Greve, D. Morris, J. Peñalver, C. Thum, and M. Bremer

Abstract—On high precision reflector telescopes the transient thermal panel buckling can have an effective rms-value comparable to the errors in the adjustment of the reflector panels. Under this condition, high signal-to-noise radio holography of high spatial resolution can reveal the characteristic signature of panel buckling in the beam pattern and can map the surface deformation of the buckling, while lower signal-to-noise Moon limb scans may see the buckling only under favorable conditions. Detailed diffraction calculations, and some observations, indicate (1) that the panel buckling produces diffraction rings and/or diffraction spokes, (2) that panel buckling in azimuthal direction may have a smaller degrading effect than panel buckling in radial direction because for azimuthal buckling the energy is spread more uniformly over a large solid angle, and (3) that the coverage of the reflector aperture with buckled panels determines the multiplicity of the diffraction rings and/or diffraction spokes. Index Terms—Beam pattern, reflector surface deformations, thermal panel buckling.

I. INTRODUCTION The reflector surface of radio telescopes often consists of panels, either of triangular, rectangular, or hexagonal shape. The panels are attached to the backup structure at 3, 4, or 5 support points, by screws or actuators. The panels follow the deformation of the backup structure (BUS), caused by gravity, temperature, and wind. However, the panels themselves, or the panel support frames, may deform under these influences, in particular under temperature effects. This leads to thermal panel buckling, a pattern of repetitive surface deformations, and the characteristic degradation of the beam pattern from grating-like deformations. Since the panels, or panel frames, are relatively small compared to the reflector diameter, the influence of panel buckling appears at large distances from the beam center, while the amount of beam degradation depends on the geometry and the amplitude of the buckling. The effect of thermal panel buckling, often of transient nature, has been observed on several high precision radio telescopes, especially those with a low error beam from random surface deformations (for references see [1]). Thermal panel buckling can occur on centimeter and millimeter wavelength telescopes as an effect of differential thermal expansion when using different kinds of materials for the BUS and the panels/panel frames, and/or as an effect of thermal gradients through the panel/panel frames [2], [3]. The buckling depends furthermore on the panel/panel frame fixation, i.e., a stiff connection or a hinged connection with the BUS. Panel buckling across the reflector aperture can be mapped in holography measurements [4] or seen as an integrated effect of beam pattern degradation, for instance derived from differentiated total power Manuscript received January 29, 2009; revised August 19, 2009. First published December 28, 2009; current version published March 03, 2010. C. Thum and M. Bremer are with the Institut de Radioastronomie Millimétrique (IRAM), 38406 St. Martin d’Heres, France. J. Peñalver is with IRAM, Granada 18012, Spain A. Greve and D. Morris (both retired) were with the Institut de Radioastronomie Millimétrique (IRAM), 38406 St. Martin d’Heres, France Digital Object Identifier 10.1109/TAP.2009.2039299

scans across the limb of the Moon [1], [5]. As model calculations have shown [1], [6], the on-axis (main beam) effect of buckled panels can be determined when using the effective rms-value 3  =3, with  the buckling amplitude, in the Ruze relation [7]. A reflector has also random surface errors rd , of certain correlation length L and rms-value  , onto which the buckling deformations b superpose so that the total deformation is  = b + rd . The buckling effect disappears in the Gaussian-type error beam whenever  3 <  ; the buckling is noticeable on the error beam whenever    3 . The reflector surface therefore must have a high precision in order to see the buckling effect in high signal-to-noise holography measurements, or even in Moon limb scans. Thermal panel buckling on the IRAM 30-m radio telescope [8], [9] was noticed as transient shoulders and peaks in the beam patterns derived from differentiated Moon limb scans, as shown in Fig. 1. These measurements revealed that the shoulders/peaks scale with (frequency)01 , as to be expected from a grating-like effect, while the fine structure of the degradation remained hidden in the relatively low signal-to-noise of the measurements. From the night time and day time reflector surface error map derived from 39 GHz holography measurements [4] the beam pattern was calculated for the highest frequencies regularly used on the IRAM 30-m telescope; the corresponding beam patterns at 240 GHz, averaged in azimuth direction of the polar diagram, are shown in Fig. 2(a). The calculated beam patterns show double or single diffraction rings superposed on the extended error beam from random surface errors. From continued surface adjustments and the gradual decrease of the random errors from 0:07 mm (rms) to 0:05 mm (rms) [4], the errorbeam [5] is now sufficiently low so that under favorable conditions (i.e., observations at high elevation and stable atmosphere, strong radiative cooling of the reflector towards the cold sky and hence strong buckling) the double structure of the 1st order diffraction ring is sometimes also visible in differentiated high frequency Moon limb scans, as shown in Fig. 2(b). Although Fig. 2 are snapshots of the thermal life of the reflector surface, the basic diffraction phenomenon of single and double peaked rings (and spokes) and of an extended and more or less enhanced background is evident. II. BUCKLING OF RECTANGULAR PANELS Dependent on the construction of a panel (single or multi-layered panel, different materials like aluminum or CFRP), its shape, and its fixation on the backup structure, and dependent on the construction of panel frames [10] and the backup structure, the buckling 1 b () may occur in radial direction () only, producing diffraction rings; the buckling 2 b (') may occur in azimuthal direction (') only, producing diffraction spokes; or may occur in both directions 3 b (; '), producing a combination of diffraction rings and spokes. Appropriate mathematical functions b describing the radial, azimuthal and combined thermal buckling are given in [1]. Evidently, the influence of the buckling on the diffraction pattern depends on the buckling amplitude . On the IRAM 30-m telescope the rectangular panels follow the buckling of the panel frames, which extend approximately 0.6 m into the temperature controlled backup structure [11]. As finite element calculations have shown, the buckling of the panel frames is due to the measured axial temperature gradient (front to back) in their structure [12]. As evident from Fig. 3, the temperature gradient of the frames, and

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Fig. 3. IRAM 30-m telescope. Temperature gradient from front to back through a panel frame. A reversal of the temperature gradient, and by this the direction of the buckling, occurs within 1 to 2 hours during morning and evening time.



Fig. 1. IRAM 30-m telescope. Thermal buckling of reflector panels observed as shoulders or peaks (indicated by the dashed lines) on the beam patterns derived from differentiated Moon limb scans [5]. The position of the shoulders/peaks scales with (frequency) (indicated in GHz). For clarity the scans are displaced in vertical direction.

Fig. 2. IRAM 30-m telescope. (a) Beam pattern (averaged in azimuth direction) at 240 GHz derived from a day time holography surface error map (D) and a night time holography surface error map (N; shifted by 5 dB for clarity). Note the double peak diffraction rings of 1st, 2nd and 3rd order at 120; 240, and 380 arcsec from the beam center, and the lower and single peak diffraction rings during night time. The gray line is the underlying error beam pattern. (b) Differentiated Moon limb scan at 230 GHz showing the 1st order double peak diffraction ring.

0







by this the panel buckling, reverses direction within the relatively short time of 1 to 2 hours, at sunrise and sunset. The panel frames also buckle under gravity though with a significantly smaller amplitude [10]. III. BEAM PATTERNS The radial buckling produces on the IRAM 30-m telescope double peak and single peak diffraction rings (Fig. 2) which are reproduced when using the function 1 b () [1]. Double peak diffraction rings, with different intensities of the peaks, are predicted for certain circular

optical gratings, though not for all [13]–[15]. However, radial panel buckling is not similar to an optical transmission grating consisting of concentric circular slits, but rather is similar to a grating with periodic radial concentric phase shifts. For illustrative calculations we choose the IRAM 30-m telescope with respect to panel size, panel arrangement (7 zones, 24 sectors), wavelength of observation ( = 1:3 mm, 230 GHz), and buckling of amplitude  = 0:1 mm (and effective rms-value 3 = =3  0:03 mm), as for instance derived from Fig. 2. A simplified random surface error distribution rd of 1/4 panel dimension, correlation length (L) and  = 0:05 mm (rms) is superposed on the buckling deformations b (for the more complicated actual random error distribution see [5]). A 015 dB edge taper is applied. In the Fourier transform calculations we used 66000 points in the aperture, or 280 points per panel frame of 2 m 2 2 m area. Fig. 4(a) shows the repetitive reflector surface deformations corresponding to the panel arrangement of the IRAM 30-m telescope and radial, azimuthal and radial and azimuthal buckling deformations. It is assumed that all panels buckle in the same way. The calculated 230 GHz beam patterns, without underlying random surface errors ( = 0), are shown in Fig. 4(b). The radial buckling [Fig. 4(1)] produces diffraction rings, the azimuthal buckling [Fig. 4(2)] produces diffraction spokes; the combined case [Fig. 4(3)] produces rings and spokes. A good picture of the degradation is obtained when averaging the beam patterns in azimuthal direction of the polar diagram as shown in Fig. 4(c) for the case of no additional random surface errors ( = 0), and in Fig. 4(d) including additional random surface errors as specified above ( = 0:05 mm), producing the error beam of Fig. 2(a). The radial buckling produces double peak diffraction rings of 1st, 2nd, . . . j-th order superposed on the diffraction pattern, without or with error beam. The position j (in the focal plane) of the diffraction ring of j-th order is obtained from the grating equation [16]

h i;

sinj = j= L

j = 1; 2; 3

111

(1)

The frequency (wavelength) scaling of the first order diffraction ring 1 on the IRAM 30-m telescope is seen in Fig. 1. From the position of the shoulders/peaks follows hLi  2 m which is the average dimension of the panel frames projected into the aperture plane and weighted by the taper function. At 230 GHz, (1) gives 1  130 arcsec, 2  250 arcsec, and 3  380 arcsec, as seen in Fig. 2. Relation (1), however, does not explain the double peak structure of the diffraction rings; this follows only from the theory of circular gratings [13]–[15]. In case of an underlying error beam, as illustrated in Fig. 4(1,d), the diffraction rings are less pronounced, and more diffcult to detect. A different situation occurs for azimuthal buckling, and the combination of radial and

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Fig. 5. Holography map (night time) showing that the reflector was only partially covered with buckled panels. Fig. 4. Beam pattern calculation of radial (1), azimuthal (2), and radial and : mm. (a) Deformation of azimuthal (3) panel buckling of amplitude  the reflector surface; (b) corresponding diffraction pattern at 230 GHz, without random surface errors (gray scale between dB and dB); (c) azimuth averaged 230 GHz beam pattern with panel buckling: black line, without panel buckling: gray line; (d) azimuth averaged 230 GHz beam pattern for panel buckling and random surface errors ( : mm): black line. The box (1,d) shows also the perfect beam pattern (lower curve) and the beam pattern of Fig. 2(a): gray line. The gray line in box 2(d) and box 3(d) is the error beam without additional buckling effect.

= 01 030 065

= 0 05

azimuthal buckling. As Fig. 4(2) illustrates, for azimuthal buckling regular radial spokes extend outwards approximately from the axial distance of the 1st diffraction ring; for azimuthal and radial buckling radial spokes and rings appear, with the rings being more pronounced. IV. SINGLE PEAK DIFFRACTION RINGS Fig. 2(a) shows that single peak diffraction rings can occur. Inspection of the night time holography map shown in Fig. 5, on which the beam pattern calculation of Fig. 2(a, N) is based, reveals that the reflector surface was only partially covered with buckled panels. From this we infer that the appearance of a double or single peak diffraction ring may be due to the extent of the presence of buckled panels on the reflector surface. Since the literature explains only the case of uniform coverage, in order to explore the effect of incomplete coverage we selected a heuristic example and calculated the diffraction pattern once of a reflector uniformly covered with panels with radial buckling, and once of a reflector of which only 1/2 of the surface is covered with buckled panels. The result of the calculation is shown in Fig. 6. The uniformly covered reflector shows double peak diffraction rings [Fig. 6(a)], the asymmetric buckling distribution shows single peak diffraction rings [Fig. 6(b)]. A similar situation is found for azimuthal buckling where the double spokes become blurred in case the reflector is only partially covered with buckled panels. These heuristic examples suggest that the feature of a double or single peak diffraction ring (spoke) depends on the buckled panel distribution on the reflector surface.

Fig. 6. A possible explanation for the occurrence of single and double peak diffraction rings in terms of a different distribution of buckled panels over the reflector surface. The case shown is for radial buckling. Upper panel: reflector surface with uniform distribution of buckled panels (a), and asymmetric distribution of buckled panels (b). Lower panel: corresponding calculated beam pattern at 230 GHz. The inserted box shows an artificial radio source, the arrow is the scan direction of the observation (see Section V).

V. CONSEQUENCES FOR OBSERVATIONS As explained in [1], panel buckling of amplitude  introduces a main beam degradation which can be calculated from the Ruze relation [7] when using the effective rms-value 3  =3. However, the effect of the beam deformation from random surface errors and additional panel buckling may become important in the mapping of extended and structured astronomical sources, for instance the mapping of galaxies like M 82 [17], [19] and M 51 [18], [20]. The astronomical question in

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diffraction spokes. Dependent on the coverage of the reflector surface with buckled panels the diffraction rings may be double peaked or more or less single peaked; dependent on the coverage, the diffraction pattern of panels with azimuthal buckling may have clean or blurred spokes. On the IRAM 30-m telescope the thermal panel buckling seems to be more pronounced in the radial direction. The on-axis beam degradation can be calculated from the Ruze relation and is, for equal amplitude, the same for the three cases of buckling. However, when using the IRAM 30-m telescope as example it is shown that in scanning observations the 1st order diffraction ring from radial buckling may have a more pronounced influence, leading to spurious features at the 1%-level at arcminute distances from the source. These are transient features.

Fig. 7. Model calculations of a central scan through a source of 90 arcsec extension and 30 arcsec Gaussian width (see Fig. 6) with beam patterns of different degree of degradation. S: edge of the source; 1 : scan with the beam of a perfect telescope (11 acrsec beam width, FWHP), the Nyquist spacing is indicated by the dots; 2 : scan with a telescope with random surface errors; 3 : scan with a telescope with random surface errors and azimuthal panel buckling; 4 : scan with a telescope with random surface errors and radial panel buckling.

these publications is, for instance, the existence of a weak halo around a strong central source (M 82), or the contrast of 230 GHz CO emission in and between spiral arms which are separated by a few beam widths (M 51). A model calculation illustrates the situation for the IRAM 30-m telescope at 230 GHz where the beam width is 11 arcsec (FWHP). It is important to remember in these calculations that panel buckling is a transient effect, although Fig. 3 indicates that the time without a temperature gradient through the panel frames, and hence the time without buckling, is short. We calculated scans through the center of a 90 arcsec wide source [y-direction] of Gaussian profile of 30 arcsec width (FWHP) [x-direction]. The source is scanned (convolution of source and beam) in x-direction (Fig. 6) with the beams shown in Fig. 4(d). The result of the calculation is summarized in Fig. 7 which shows one edge of the source (S) and the intensities (100% at the center position of the source) measured with the perfect telescope with no surface errors and no panel buckling (1); measured with a telescope with error beam from random surface errors (2) ( = 0:05 mm); measured with a telescope with error beam from random surface errors and azimuthal panel buckling (3) (2 = 0:1 mm); and measured with a telescope with error beam from random surface errors and radial panel buckling (4) (1 = 0:1 mm). The influence of the 1st order diffraction ring from radial panel buckling is the dominating effect. At the distance of the 1st order diffraction ring at 1  130 arcsec from the center of the source appears a spurious feature at the 1%-level. The influence of azimuthal panel buckling, or azimuthal and radial panel buckling, is similar to an enhancement of the error beam. VI. CONCLUSION Thermal panel buckling, which mostly is of transient nature, becomes a noticeable degradation of the beam pattern when the effective rms-value =3 (with  the amplitude of the buckling) is comparable to the precision of the reflector surface setting. This situation may occur primarily on high precision mm-wavelength radio telescopes where the thermal panel deformations are of the same order as 1/15 to 1/10 of the wavelength of observation. A rectangular panel, or panel frame, can buckle in radial or azimuthal direction, or in both directions. Radial buckling produces diffraction rings; azimuthal buckling produces

REFERENCES [1] A. Greve and D. Morris, “Repetitive radio reflector surface deformations,” IEEE Trans. Antennas Propag., vol. AP-53, pp. 2123–2126, 2005. [2] W. N. Christiansen and J. A. Högbom, Radiotelescopes, 2nd ed. Cambridge: University Press, 1985. [3] S. Von Hoerner, “Accuracy and thermal deformations of ESSCO panels,” 25-m Telescope Memo 81 NRAO, USA, 1977. [4] D. Morris, M. Bremer, G. Butin, M. Carter, A. Greve, J. W. Lamb, B. Lazareff, J. Lopez-Perez, F. Mattiocco, J. Penalver, and C. Thum, “Surface adjustment of the IRAM 30-m radio telescope,” IET Microw., Antennas Propag., vol. 3, no. 1, pp. 99–108, 2009. [5] A. Greve, C. Kramer, and W. Wild, “The beam pattern of the IRAM 30-m telecope,” Astron. Astrophys. Suppl., vol. 133, pp. 271–284, 1998. [6] A. Greve, “Strehl number degradation of large-scale systematic surface deformations,” Appl. Opt., vol. 19, pp. 2948–2951, 1980. [7] J. Ruze, “Antenna tolerance theory-A review,” Proc. IEEE, vol. 54, pp. 633–640, 1966. [8] J. W. M. Baars, B. G. Hooghoudt, P. G. Mezger, and M. J. De Jonge, “The IRAM 30-m millimeter radio telescope on Pico Veleta, Spain,” Astron. Astrophys., vol. 175, pp. 319–326, 1987. [9] J. W. M. Baars, A. Greve, H. Hein, D. Morris, J. Penalver, and C. Thum, “Design parameters and measured performance of the IRAM 30-m millimeter radio telescope,” Proc. IEEE, vol. 82, pp. 687–696, 1994. [10] H. Eschenauer, H. Gatzlaff, and H. W. Kiedrowski, “Entwicklung und Optimierung hochgenauer Paneeltragstrukturen,” Techn. Mitt. Krupp Forschg.-Ber. pp. 43–57, 1980, Band 38. [11] A. Greve, M. Bremer, J. Peñalver, P. Raffin, and D. Morris, “Improvement of the IRAM 30-m telescope from temperature measurements and finite element calculations,” IEEE Trans. Antennas Propag., vol. AP-53, pp. 851–860, 2005. [12] D. Plathner, “Studies on 30-m Panel Frames Under Thermal Gradient,” IRAM, France, IRAM Working Report, No. 244, 1997. [13] D. Tichenor and R. N. Bracewell, “Fraunhofer diffraction of concentric annular slits,” J. Opt. Soc. Am., vol. 63, pp. 1620–1622, 1973. [14] I. Amidror, “Fourier spectrum of radially periodic images,” J. Opt. Soc. Am. A, vol. 14, pp. 816–826, 1997. [15] I. Amidror, “Fourier spectra of radially periodic images with a non-symmetric radial period,” J. Opt. A: Pure Appl. Opt., vol. 1, pp. 621–625, 1999. [16] M. Born and E. Wolf, Principles of Optics. London, U.K.: Pergamon Press, 1980. [17] N. Neininger, M. Guélin, U. Klein, S. García-Burillo, and R. Wielebinski, “13 CO at the center of M 82,” Astron. Astrophys., vol. 339, pp. 737–744, 1998. [18] S. García-Burillo, M. Guélin, and J. Cernicharo, “CO in M51—Molecular spiral structure,” Astron. Astrophys., vol. 274, pp. 123–147, 1993. [19] F. Walter, A. Weiss, and N. Scoville N., “Molecular gas in M 82: Resolving the outflow and streamers,” Astrophys. J., vol. 580, pp. L 21–L 25, 2002. [20] K. F. Schuster, C. Kramer, M. Hitschfeld, S. Garcia-Burillo, and B. Mookerjea, “12 CO 2-1 map of M 51 with HERA (I),” Astron. Astrophys., vol. 461, pp. 143–151, 2007.

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The Periodic Half-Width Microstrip Leaky-Wave Antenna With a Backward to Forward Scanning Capability Yuanxin Li, Quan Xue, Edward Kai-Ning Yung, and Yunliang Long

Abstract—The periodic half-width microstrip leaky wave antenna (MLWA) with a backward to forward scanning capability is presented in this paper. The proposed antenna consists of a series of half-width MLWA. The radiating periods place on the different sides of the transmission line. The radiating period works in cutoff region of the first higher order mode, the periodic construction radiates the slow wave out along the edge. The experimental results show the main lobe scans electronically and continuously from 149 to 28 in H-plane ( - plane) toward end fire ( direction) when the operating frequency increases from 4.2–.9 GHz.

+

Index Terms—Backward to forward, half-width microstrip leaky wave antenna (MLWA), periodic.

I. INTRODUCTION The microstrip leaky wave antenna (MLWA) has been investigated for a long time since it was developed by W. Menzel in 1979 [1]. Compared with the conventional beam-scanning antenna, MLWA has the advantages of simple structure, low profile, and easy to match. The frequency-scanning pencil beam in H-plane makes it suitable to be the choice of automobile radar system or the other consumer products [2], [3]. Traditionally, the uniform MLWA operates in the first higher order mode of a microstrip line with a complex propagation constant: kz = z 0 jz [4]. z is the attenuation constant and z is the phase constant. While the attenuation constant z is proportional to the beamwidth of main lobe; and the phase constant z is related to  , that is

=



01 z

2 0 sin

k0

(1)

where k0 is the free-space wave number. The main lobe elevates from the horizontal plane as specified by the angle of scanning  . As a result, the angle of scanning could be modified by a minor change in the frequency of operation. On the other hand, the reduced size half-width MLWA with the beam-scanning capability is achieved compared with the conventional whole-width MLWA. In 2007, a half-width MLWA design was reported by G.M. Zelinski, which theoretically showed that the vertical wall used in the half-width MLWA [5]. In the previous works, a quasi microstrip leaky-wave antenna with a 2-dimensional beam-scanning capability was presented [6]. Either the conventional uniform whole-wide antenna or the half-width antenna works in the first higher order mode with the positive phase constants. The main lobe of the uniform leaky wave antenna just scans from broadside to the end fire direction with the varying of the operating frequency in the H-plane when z  z < k0 . A simple microstrip leaky wave antenna design, with a pencil beam scanning capability in either side of the broadside, may be a great candidate in application in future. Manuscript received February 14, 2009; revised April 25, 2009. First published December 28, 2009; current version published March 03, 2010. This work was supported in part by the Natural Science Foundation of China under Grant 60901028 and in part by the NSFC-Guangdong under Grant U0635003. Y. Li and Y. Long are with the Department of Electronics and Communication Engineering, Sun Yat-Sen University, Guangzhou, China (e-mail: [email protected]). Q. Xue and E. K.-N. Yung are with the State Key Laboratory of Millimeter Waves, City University of Hong Kong, Kowloon, Hong Kong, China. Digital Object Identifier 10.1109/TAP.2009.2039304

963

There are several researches focus on the antenna design with a backward-to-forward beam scanning capability. The conventional periodic leaky wave antenna consists of the rectangular waveguide or an array of metal strips over a ground plane, and it has the advantage of the main lobe scanning in either backward or forward direction due to either the positive or negative phase constant [7]. But the constructions of the conventional periodic leaky wave antennas are so complex, and the profile is high. Meanwhile, the right/left handed (CRLH) metamaterials has been applied as the leaky wave antenna in the last few years [8], [9]. In 2002, a leaky-wave antenna with a backfire-to-endfire scanning capability was introduced [10]. In this antenna design, the phase constant increased from negative to positive values with the increase of the frequency, so the main lobe scanned from backward to forward. On the other hand, the CRLH microstrip structure was demonstrated as a fix-frequency beam-scanning leaky wave antenna [11]. The beam scanning from backward to forward at a fix-frequency was obtained by modulating the capacitances of the structure by adjusting the bias voltage applied to the varactors. Though the main lobe scanning over all quadrants of the hemispherical space is achieved, the scanning angle region is small. The measured beam scanning angle regions of above antenna designs are not more than 50 . [12] reports a novel 2D CRLH transmission line leaky wave antenna array capable of phase and frequency scanning a pencil beam in full-space. The beam scanning angle region has been improved to about 60 . A periodic half-width MLWA with a backward to forward scanning capability is presented. The proposed antenna as shown in Fig. 1 consists of a series of half-width MLWA. The radiating period works in slow-wave region of the first higher order mode, the periodic construction radiates the slow wave out along the edge. The main lobe could scan in either backward direction or forward direction with a simple structure. The far-field H-plane (y -z plane) radiation patterns of this antenna have been measured. The measured and the calculated phase constant have been compared. The reflection coefficient of the antenna has been shown too. II. THEORY FOR PERIODIC HALF-WIDTH MLWA A periodic half-width MLWA with a backward to forward scanning capability is shown in Fig. 1. This periodic microstrip antenna consists of a series of half-width MLWA, and the half-width MLWA are placed alternatively on the different sides of a transmission line. As mentioned above, the periodic construction makes the phase constant increase from negative to positive values with the increase of the frequency. The complex propagation constant kzn = zn 0 jz of the wave in a conventional periodic leaky wave antenna is given as follows [7]:

kzn

= kz + 2n ; d

n = 61; 62; 631 1 1

(2)

where kz is the wave number of the fundamental Floquet wave in the uniform structure, d is the period, n is the order of the space harmonics. The periodic antenna working in either the leaky region (fast-wave region, j zn j < k0 ) or the bound region (slow-wave region, j zn j > k0 ) determines the character of the single space harmonic. In the fast-wave region 0k0 < zn < k0 , the direction of the main lobe is given by

=



01 zn :

2 0 sin

k0

(3)

As shown in Fig. 1, a single half-width MLWA period is seen as the uniform structure in this periodic microstrip antenna design. The half-width MLWA consists of the half of the conventional whole width MLWA with a shorting circuit edge. The half-width MLWA achieves

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)

k0 . According to (2) and (9), the normalized phase constant zn =k0 of the periodic half-width MLWA with a backward and forward beam scanning capability is given by zn k0

= k0 + 2dkn0 = k0 0 d0 Re !2 " 0 k2 0 = 0d k0 z

z

r

x

(10)

=2

= 1

where k0 =0 , 0 is the free-space wavelength, and n 0 . The attenuation constant z is calculated by the basic (9) of the half-width microstrip leaky wave antenna, due to the attenuation constant is the same for all the order of the space harmonics. When 0 < zn =k0 < , a part of energy leaks from the periodic construction into the free space, the angle of the main lobe is related to the normalized phase constant, and (3) could be rewritten as follows:

1

1

=

Fig. 1. (a). The 3D view of the proposed periodic half-width MLWA with backward to forward scanning capability. (b) The layout of this periodic half-width ,W ,s : ,T : , MLWA. (L l l , r ,h : ," : ). ,a

+ = 225 mm = 21 mm = 2 2 mm = 11 6 mm = 21 mm = 3 mm 2 = 1 mm = 0 8 mm = 2 65



01 2 0 sin Re

!2 "r 0 kx2 k0

0 d0

:

(11)

With the increase of the operating frequency, the phase constant zn varies from negative to positive values and then the main lobe of the periodic half-width MLWA scans from the backward direction to the forward direction. The surface wave propagates away from the feed, but the power leaks in the backward direction with 0k0 < zn < .

0

III. DESIGN the main lobe frequency-scanning with the reduced size and the radiation characters are similar to the conventional whole-width MLWA. The complex propagation constant could be calculated by a simple method mentioned in [6], and the experimental results match well with the calculated normalized propagation constant. So the equivalent extension T of the half-width MLWA is given by

1

1

:3 h+0l:262 1T = 0:412h "e"e0+00:258 +1 l h+0:813 2 2 2 1 = 2a ln r1 0 4a2r + 0:601 a"2r 0 "r + 1 "r 0 1 5 h 0(1=2) "e = 2 + 2 1+ T

(4) (5) (6)

where is the expression of the parallel-plate circuit model of a series of shorting pins, a is the space between each of shorting pins, r is the radius of shorting pins. h is the thickness of substrate, l is the length of half-width MLWA, T is the width of half-width MLWA. The methods to obtain the complex propagation constant express as follows:

" 1T = 120h0 + j k0120  k 0 !y exp(jk 2T ) = 0 k + !y y!

x

r

x

!

x

!

(7) (8)

so the complex propagation constant of an uniform half-width MLWA is

kz

=

!2 "r 0 kx2 :

(9)

The uniform segments work in the first higher order mode, and they are in the cutoff mode (slow-wave region, z > k0 ) in the frequency region of operation. The periodic construction makes the slow wave radiate out along the antenna. The propagation constant kzn could be obtained from (2) when the space harmonics , and the phase constant zn of the periodic construction is in the fast-wave region j zn j
8 dB from 21.5 to 25.9 GHz with a measured peak gain of 9.8 dB at 24 GHz. HFSS reports a directivity of 10.3 dB with a gain of 9.9 dB at 24 GHz, and this corresponds to a radiation efficiency > 90%, which is collaborated by our experiment (within measurement error). III. CONCLUSION This communication presented a millimeter-wave CPS-fed Yagi-Uda antenna with a folded dipole feed, relatively wideband operation (21–25 GHz), medium gain (8–10 dB) and very low cross-polarization levels (< 022 dB). The antenna can be scaled to 60–94 GHz for automotive radars and high data-rate communication systems, and is ideally suited for differential RFIC connections.

A Finite Edge GTD Analysis of the H-Plane Horn Radiation Pattern Maifuz Ali and Subrata Sanyal

Abstract—The earlier geometrical theory of diffraction (GTD) approach to the H-plane horn radiation problem is reconsidered with spherical source excitation. Corner diffraction terms are included to provide a GTD model for the finiteness of the horn edges. A new heuristic corner slope diffraction (CSD) coefficient for a finite edge in a conducting plane is presented. The H-plane horn pattern, obtained with the addition of the corner diffraction and the new CSD terms to the earlier infinite edge GTD approach, is found to be in better agreement with measured results compared to earlier GTD formulations. Index Terms—Corner diffraction, corner slope diffraction, geometrical theory of diffraction (GTD), horn antenna, H-plane radiation pattern.

I. INTRODUCTION The horn antenna invented by J. C. Bose [1] in the 1890s is a good canonical problem as it has a broad range of low response in the backward (LRB) directions [2]. Keller’s GTD and parallel plate waveguide mode approximation was used by Yu et al. to obtain the H-plane pattern in [3]. The GTD prediction was in good agreement with measurements in the forward direction. In the back lobe direction it was shown that the E-edge contributions were significant for observation angle  at and around 180 . However, there remained wide differences in LRB region. For  > 105 , better results in the average was obtained by taking into account the finite edge effect of the E-edges using current

REFERENCES [1] L. Zhu and K. Wu, “Model-Based characterization of CPS-fed printed dipole for innovative design of uniplanar integrated antenna,” IEEE Microw. Guided Wave Lett., vol. 9, pp. 342–344, Sep. 1999. [2] N. Kaneda, W. R. Deal, Y. Qian, R. Waterhouse, and T. Itoh, “A broadband planar quasi-Yagi antenna,” IEEE Trans. Antennas Propag., vol. 50, no. 8, pp. 1158–1160, Aug. 2002.

Manuscript received October 28, 2008; revised August 23, 2009. First published December 04, 2009; current version published March 03, 2010. The authors are with the Department of Electronics and Electrical Communication Engineering, Indian Institute of Technology, Kharagpur-721 302, India (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TAP.2009.2037762

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Fig. 1. Geometry of a pyramidal horn. (a) Diverging wavefront at the aperture. (b) Side view (E-plane). (c) Front view. (d) Top view (H-plane).

Fig. 2. Conical diffraction for top E-edge.

elements. The dip at 180 was conjectured to be caused by the detector mount used. These works were later followed by those of Mentzer et al. who used the extended equivalent current concept [4]. The finite-difference time-domain (FDTD) technique was applied by Taflove and Umashankar [2] and later Tirkas and Balanis [5] to the horn radiation problem and results obtained for small aperture horns. In this paper the formulation given in [3] for H-plane horn radiation is reconsidered with spherical source excitation and in conjunction with corner diffraction to provide a GTD model of the uniform theory of diffraction (UTD) type [6] which takes into account the finiteness of the horn edges. A new corner slope diffraction (CSD) coefficient is presented. Addition of the CSD terms is found to provide further refinement to the GTD model. II. ANALYSIS

OF

H-PLANE HORN RADIATION PROBLEM GTD METHODS

BY

Fig. 3. Geometry for E-edge diffracted corner incident ray.

Geometry of a pyramidal horn along with it’s top (H-plane), side (E-plane) and front view is shown in Fig. 1. The TE10 mode at the waveguide horn junction gives rise to a tube of rays and the diverging wavefront shown emerging from the aperture in Fig. 1(a). The axial ray, which is surrounded by the tube of rays, coincides with the Z-axis. The horn E and H-planes intersect the principal directions of the wavefront surface. The principal radii of curvature of the diverging phase front are (E ; H ) at the aperture. Along the E and H-planes the phase variations of the diverging wavefront are coincident with excitations of an anisotropic spherical source at S . In the H-plane rays from S are assumed to have electric vector parallel to the H-edges (the edges perpendicular to principal H-plane) and geometrical optics (GO) ray-optical expansion given by

Ei

=

E0 cos

  A

cos H tan 

0jkR) :

exp(

R

(1)

The incident field at the center Q1 of the H-edge is zero while the slope of the incident field is non zero. This gives rise to the edge slope d (P ) at the observation point P obtained using ([7], diffracted field EQ d from the point Q4 can be (77)). The edge slope diffracted field EQ calculated in the same way. The rays incident at Q, points off the midpoint on the top and bottom E-edges (edges perpendicular to the principal E-plane) emanate from S (see Figs. 1 and 2), give rise to conical diffraction (see Fig. 2) and the d resulting diffracted field contributions to the H-plane pattern ETop E d and EBottomE are computed using [6, Eq. (60)].

Fig. 4. Geometry for corner diffraction from q at the end of the bottom wall q q S.

The edge diffracted ray from the off-center top E-edge point Q2 (see Fig. 3) is incident on corner q4 . The corner q4 is first considered to be at one end of the bottom wall q3 q4 S (see Fig. 4). The Y -component d (P ) of the corner diffracted field is obtained using the heuristic EQ q

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Fig. 6. Geometry for corner diffraction.

Fig. 5. Geometry for corner diffraction from q at the end of the side wall q q S.

diffraction coefficient [8, Eq. (20)]. Next, the corner q4 is considered to be at the bottom end of the side wall q1 q4 S (see Fig. 5). However, the Y -component of this corner diffracted field is found to be zero. Similarly the contributions of the corners q1 ; q2 ; q3 are obtained. Adding the above diffracted fields and using symmetry the H-plane radiation pattern is i

E (P ) = E (P )jfor jj Fig. 7. Geometry for side wall corner slope diffraction.

d

+ EQ (P )

for 0

 180

for 0

 0 and RHCP for z < 0, as can be deduced from the magnetic current distributions in Fig. 3. The measured and simulated antenna gains in the +z direction within the 3-dB AR band of 2075–3415 MHz (see Fig. 7) are found within 2.6–4.2 dBic, which are large enough for short-range wireless communications.

Fig. 7. Measured and simulated antenna gains of Antenna 3 in the

+z direction.

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IV. CONCLUSION A CPW-fed square slot antenna has been proposed for CP operations. The lightening-shaped feedline protruded from the CPW and the pair of inverted-L grounded strips both contribute to generation of CP radiation. The vertical and horizontal tuning stubs have effectively widened the VSWR  2 impedance band apart from slightly broadening the 3-dB AR band. The designed antenna was measured to show a 3-dB ARBW of up to 48.8%, along with an even larger impedance bandwidth of 51.4% and an antenna gain of around 3.4 dBic in the +z direction. ACKNOWLEDGMENT The authors would like to thank the reviewers for their careful review and helpful comments. The authors are also grateful to the National Center for High-Performance Computing for computer time and facilities.

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Optimization and Modeling of Sparse Conformal Retrodirective Array Jim S. Sun, Darren S. Goshi, and Tatsuo Itoh

Abstract—Conformal scanning arrays are important for scanning ranges beyond 90 degrees and the retrodirective array (RDA) is known as a promising candidate for this application. We discuss the optimization of a previously realized conformal sparse RDA by means of the genetic algorithm, and the RDA numerical model that is used in this approach. In particular, we improve the numerical model of the RDA by identifying practical factors that previous model failed to account for. The impact of each practical factor on the RDA monostatic pattern is visualized, and the improved model provides much better prediction of the monostatic pattern. Index Terms—Conformal array, genetic algorithm (GA), retrodirective array, thinned array.

REFERENCES

I. INTRODUCTION

[1] J. S. Row, “Design of square-ring microstrip antenna for circular polarisation,” Electron Lett., vol. 40, pp. 93–95, Jan. 2004. [2] Y. F. Lin, H. M. Chen, and S. C. Lin, “A new coupling mechanism for circularly polarized annular-ring patch antenna,” IEEE Trans. Antennas Propag., vol. 56, pp. 11–16, Jan. 2008. [3] K. L. Wong, C. C. Huang, and W. S. Chen, “Printed ring slot antenna for circular polarization,” IEEE Trans. Antennas Propag., vol. 50, pp. 75–77, Jan. 2002. [4] J. S. Row, “The design of a squarer-ring slot antenna for circular polarization,” IEEE Trans. Antennas Propag., vol. 53, pp. 1967–1972, Jun. 2005. [5] J. Y. Sze, C. I. G. Hsu, M. H. Ho, Y. H. Ou, and M. T. Wu, “Design of circularly polarized annular-ring slot antennas fed by a double-bent microstripline,” IEEE Trans. Antennas Propag., vol. 55, pp. 3134–3139, Nov. 2007. [6] X. M. Qing and Y. W. M. Chia, “Broadband circularly polarised slot loop antenna fed by three-stub hybrid coupler,” Electron Lett., vol. 35, pp. 1210–1211, Jul. 1999. [7] J. Y. Sze, K. L. Wong, and C. C. Huang, “Coplanar waveguide-fed square slot antenna for broadband circularly polarized radiation,” IEEE Trans. Antennas Propag., vol. 51, pp. 2141–2144, Aug. 2003. [8] R. P. Xu, X. D. Huang, and C. H. Cheng, “Broadband circularly polarized wide-slot antenna,” Microwave Opt. Technol. Lett., vol. 49, pp. 1005–1007, May 2007. [9] I. C. Deng, J. B. Chen, Q. X. Ke, J. R. Chang, W. F. Chang, and Y. T. King, “A circular CPW-fed slot antenna for broadband circularly polarized radiation,” Microwave Opt. Technol. Lett., vol. 49, pp. 2728–2733, Nov. 2007. [10] Y. B. Chen, X. F. Liu, Y. C. Jiao, and F. S. Zhang, “CPW-fed broadband circularly polarised square slot antenna,” Electron Lett., vol. 42, pp. 1074–1075, Sep. 2006. [11] L. Y. Tseng and T. Y. Han, “Microstrip-fed circular slot antenna for circular polarization,” Microwave Opt. Technol. Lett., vol. 50, pp. 1056–1058, Apr. 2008. [12] T. Y. Han, Y. Y. Chu, L. Y. Tseng, and J. S. Row, “Unidirectional circularly-polarized slot antennas with broadband operation,” IEEE Trans. Antennas Propag., vol. 56, pp. 1777–1780, Jun. 2008. [13] J. Y. Sze and C. C. Chang, “Circularly polarized square slot antenna with a pair of inverted-L grounded strips,” IEEE Antennas Wireless Propag. Lett., vol. 7, pp. 149–151, 2008. [14] T. N. Chang and C. P. Wu, “Microstripline-fed circularly-polarized aperture antenna,” in IEEE AP-S Int. Symp. Dig., 1998, pp. 1372–1375.

Conformal antenna and array has been a hot research topic in antenna area for many years due to its capability to be mounted on non-planar surfaces, which represents a practical situation. In [1], [2], excitation synthesis for uniform circular and cylindrical phased array by optimization algorithms have been demonstrated, and it was pointed out that since now the element patterns for a conformal array are facing in different directions, this effect must be included in the pattern calculation for each antenna element. In [3], the effect of curving the antenna elements to fit to the surface and the interaction between antenna element and the conformal surface on which it is mounted are studied. Element pattern for each antenna is affected by both factors and need to be accounted for. These literature and its references constitute quite a complete discussion on conformal antenna and arrays. However, the conformal surfaces discussed are still quite regular, and the number of elements is considered large since they are all uniform arrays. Furthermore, the excitation synthesis requires heavy DSP calculation, and the complex non-uniform excitation needed may complicate the whole system, which is not desirable if the platform on which the array is mounted is power and payload limited, such as unmanned aerial vehicles (UAVs) [4]. Hence, for such applications, a conformal array capable of beam scanning with thinned non-uniform configuration and simple excitation is needed. In recent years, more attention has been paid to the retrodirective array (RDA) as an automated real-time phased array. A RDA, when being illuminated by an interrogating signal (RF signal) from an arbitrary direction, it can automatically form a beam and re-transmit the signal back to the signal source without prior knowledge of the position of the interrogator or sophisticated DSP algorithms. Furthermore, since in the heterodyne technique that is usually used to realize RDA the phase conjugation condition needed for retrodirectivity is achieved independently in each element, it promises the realization of retrodirective array on general curved surfaces [5], [6]. As discussed previously, for UAV application, a thinned RDA is desirable for power saving purpose, since each antenna element needs a Manuscript received July 31, 2008; revised June 29, 2009. First published December 28, 2009; current version published March 03, 2010. The authors are with the Electrical Engineering Department, University of California, Los Angeles, Los Angeles, CA 90095 USA (e-mail: [email protected]; [email protected]; [email protected]). Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2009.2039291

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Fig. 1. A retrodirective array using heterodyne technique [5].

As shown in [4], because of the curvature of the surface, scanning at different angles by this conformal RDA are all separate radiation problems. However, GA is still capable of finding a solution with optimized scanning patterns in the prescribed scanning range, in this case, 0120  120 . In this development, this is achieved by sampling the patterns at five discrete scanning angles for evaluation in the optimization. These angles are shown as arrows in Fig. 2. As long as angles sampled are close enough, the characteristic of the patterns in between should not have significant degradation. In the GA, the whole aperture area is broken down into grids of 0:05o , so that the elements can be placed almost anywhere on the aperture, and the array configuration is encoded based on which grid the antenna elements occupy. In each array configuration, the element spacing was constrained to be in the range of 0:5o  1:5o to avoid excessive mutual coupling and large element distance resulting in inefficient aperture usage. A population about 80 individuals was created for GA mating. The new generation of the population is created by tuning the array configuration from the old population. First, some of the elements in an array configuration are chosen randomly, and then their positions are randomly tuned in the range which complies with all the positioning restrictions. This tuning is done to every individual in the population, and then the GA eliminates the inferior individuals and keeps the good ones. This process is repeated until the best individual remains the same for a certain number of iterations. This means the evolution in this stage reaches its peak, and the number of the elements tuned is reduced for further fine tuning. The algorithm terminates when the number of elements need to be tuned becomes zero. At this point, the best configuration survives all the fine tuning, and should be a relatively superior solution P attern

Fig. 2. The shape and the physical size of the aperture [4].

separate set of receiving/transmitting circuit. Genetic algorithm (GA) is an iterative optimization algorithm that imitates the evolutionary process in the nature [7], and has been applied extensively to array thinning problems [8], [9]. In [4], an in-house code was developed to adopt GA to tackle with this conformal RDA thinning problem. The uniqueness of this RDA thinning problem is also illustrated in [4]. In this communication, we extended the study in [4]. In particular, the GA realization of the conformal RDA is revisited in more detail first. Since GA will evaluate a lot of possible array configurations, fullwave simulation for each configuration is impossible. Thus, following the detail realization of the GA, we discuss the RDA model used in the GA. The model is improved from [4] by including more practical factors, and is used to discuss the causes of the discrepancy between the predicted and measured monostatic RDA patterns.

In this section, the GA for conformal RDA synthesis is introduced in detail. The conformal aperture on which this RDA prototype will be mounted is the same as in [4], which is shown here again in Fig. 2 for convenience. The hollowed circles represent the antenna elements, and the physical size of the aperture resembles the real size of the nose of a UAV. Notice that the aperture area is limited, and the array is quite small and includes only 16 elements. While the aperture can accommodate 24 elements with 0:5o spacing when fully occupied, this case represents a thinned conformal array for power saving purpose.

M n=m

( )

( ) exp (j (phn () 0 phn (o )) :

(1)

fn o fn 

For each array configuration, the patterns at five prescribed scanning angles are calculated and the sidelobe levels (SLLs) evaluated according to (1), where o is the angle of incoming wave, fn () is the nth element pattern taking its orientation into account, fn (o ) is the amplitude of excitation for the nth element reflecting the fact that it is excited by the received signal, and phn (o ) is the phase of the received RF signal. The minus sign before the term phn (o ) is due to the phase conjugation operation, where the phase of the excitation is the inverse of the phase of the received signal. Since not all the elements are excited, the summation was performed from mth to M th elements only [4]. The cost function of the GA that is used to separate the good array configurations from the bad ones is then formed by (2) using the five SLLs calculated. In the equation, max(SLLs) is the maximum SLL in dB of all the five patterns, and SLLn where n = 1  4 is the SLLs excluding the maximum one C

II. GA FOR CONFORMAL RDA

=

= 20 2 max(SLLs) +

4 n=1

SLLn

0 4 2 max(

)

SLLs

:

(2)

In (2), the primary factor that determines the value of the cost function is the maximum SLL of the five patterns due to the heavy weighting, but additional reduction of cost function is made from each pattern if it has SLL lower than the maximum SLL. When the objective of the GA is minimizing the cost function, the maximum SLL of the five patterns would be minimized, and the SLLs for other four patterns is also further minimized from the maximum SLL, guaranteeing good scanning patterns for the whole scanning range. Due to the complexity of the structure, the optimization took about 9 hours to converge on a laptop with 1.8 GHz CPU and 1.5 GB RAM.

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Fig. 4. Illustration of the near-field calculation.

Fig. 3. Comparison between monostatic pattern predicted by simple cosine model and by measurement.

The resulted optimized eight element positions of the right half of the array are d = [0:35; 1:45; 2:05; 2:60; 3:15; 3:65; 4:30; 5:75]o along the arc of the aperture, starting from the axis of symmetry, and the left half of the array would be the mirror image of the right half of the array. This optimized conformal sparse RDA configuration is shown also in Fig. 2. As reported in [4], the resulting RDA achieved SLL in the measured bistatic scanning pattern consistent with those predicted by GA in the whole prescribed scanning range. However, further observation reveals large discrepancy in the measured and predicted monostatic pattern. This issue will be addressed by the RDA model in the next section. III. RDA MODELING Since performing full-wave simulation for each possible array configuration that is evaluated by GA would be too time-consuming to be practical, a simpler numerical model of the conformal RDA is used instead to calculate the radiation. In this section, the detail of this RDA radiation calculation will be discussed. Furthermore, the monostatic RDA pattern that is presented in [4] will be used to check the validity of the RDA model used. This model is also improved in this section by including more non-ideal factors in the actual measurement. Fig. 3 is the comparison between the measured and predicted monostatic patterns by GA in [4]. We can see three main deficiencies between these two patterns: the peak location shifted, the broadside radiation is a lot stronger than predicted, and the measured pattern has lower power level on the left when ideally it is predicted to be symmetric. The main reason for these discrepancies is attributed to the oversimplified element pattern fn () used in [4], which is just a simple cosine pattern. In the content that follows, non-ideal factors both in radiation point of view and circuit point of view will be gradually included into the simple cosine model. Furthermore, as can be seen later, in this case each deficiency in the two patterns can be attributed to one non-ideality that is going to be incorporated into the model.

Fig. 5. Near-field model element pattern in the far end position.

merely a constant phase shift due to its displacement from the array center. A little more involved calculation for each element pattern as shown in Fig. 4 and (3) is needed. In Fig. 4, rm is the distance from the patch to the observation points, ra is the position vector of the patch with respect to the array center, and R is the distance from array center to the observation point. When doing measurement, the horns stays on a circle with center coincide with the array center, and radius R which is 1.5 m for receiving horn and 2 m for interrogating horn. As can be observed from (3), because R is not far enough relative to ra , the amplitude and phase of the radiation are both functions of rm instead of a constant, and the cosine pattern must be evaluated in the direction of rm instead of the direction of R

R 0jk(r e rm

A. Near-Field Effect From Fig. 2 we can find that the largest possible array aperture has dimension  4:75o . Thus, to be in the far-field, a distance of about 2.7 m is required at 5 GHz. However, in [4] the transmitting and receiving horns are placed at distance of 2 m and 1.5 m, respectively, to the array center. This invalidates the simple cosine model in the way that, now the pattern of each element cannot be modified from cosine pattern by

pattern =

16

i=1

0R) 3 (cos(^rm ))

[P Ri (o )]3 [P Ti ()] :

(3) (4)

This effect is most manifested for the element in the far end. In Fig. 5, we compare the element pattern where the element is at the array center

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TABLE I MISMATCH SCALING FACTOR

Fig. 6. Monostatic pattern by measurement and by near-field model prediction.

Fig. 7. General circuit model of an antenna.

and at the far end position of the straight part of the array. It is a transmitting pattern which corresponds to R = 1:5 m. It is obvious that for the amplitude pattern, the maximum shifted from the expected position of 90 and is larger than unity. This is because at some angles, rm can be smaller than R, so the radiation is stronger. Also, the phase is not a constant anymore. This pattern correction procedure is repeated for each element, both receiving and transmitting patterns. Then the RDA pattern is calculated by (4), where PRi (o ) is the complex receiving pattern of the ith element from the interrogating direction o in the near-field model, and PTi () is the complex radiation pattern of the ith element. Since these complex patterns already embed the phase information, the path delay has been included in this calculation. The conjugate operation on the PRi (o ) accounts for the phase conjugate mixing process in the RDA. The monostatic pattern calculated by this near-field model is shown in Fig. 6. We can see that now this near-field model predicts the position of maximum power and the monostatic pattern at large scanning angles correctly. B. Antenna Mismatch and Frequency Drift Although the patch antenna performance is simulated to be acceptable at 5 GHz, when fabricated, the resonance frequency, and hence input impedance, may differ from one to another. Fig. 7 is the general circuit model for an antenna. Zo is the port impedance 50 Ohm, and R + jX is the input impedance of the antenna. Since the antenna is mismatched, the radiated power is smaller than the incident power due to reflection, and is calculated as in (5) where Er is the radiated field amplitude and Ea is the incident E-field amplitude at the port. This scaling is calculated for both receiving (5 GHz) and transmitting (5.01 GHz), and the product of the two would be the total scaling of the re-radiated signal. This is illustrated in (6), where the

Fig. 8. Monostatic pattern of the near-field-mismatch model compared to measurement.

near-field model (4) is modified to include this mismatch effect. WR is the receiving scaling factor from (5) evaluated at 5 GHz, while WT is the transmitting scaling factor evaluated at 5.01 GHz

p

2 Zo R Er Ea = jR + jX + Zo j exp j

pattern =

16

i=1

1 R + jX + Zo

[WR 3 PRi (o )]3 [WT 3 PTi ()] :

(5) (6)

Table I summarize the amplitude of these scaling factors for the patch elements used in the measurement. We can see that the left side contains more poorly matched elements, and thus a lower radiation power on the left side is expected. Fig. 8, which is the comparison between the measured monostatic pattern and the one predicted by this near-fieldmismatch model, shows that this is indeed the case. Although the power level on the left side of the array is still predicted to be a little higher than measurement, the prediction should improve after including the conversion loss of the mixers, which has a similar trend as the scaling factors in Table I. It is noticed that in (5), the mismatch scaling factor also has a phase shifting effect on the receiving and transmitting signal. This phase shift will be canceled by the phase conjugating process if receiving and transmitting is at the same frequency. In this RDA case, we have 5 GHz as receiving frequency and 5.01 GHz as transmitting frequency, resulting in different input impedance in the two situations. Hence, the phase will not be canceled perfectly and a non-uniform residual phase shift will be present in the excitations for the elements. However, from

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Fig. 9. Array simulation setup in HFSS for the purpose of element pattern extraction. Fig. 11. Comparison of monostatic patterns between the near-field model, the scattering model, and the measurement.

ment in Fig. 11. We can see that after including this scattering effect, the power level in the broadside scanning is brought up. Although the match of the two patterns is not quite close yet at the broadside, but we can safely conclude that this deficiency at the broadside in Fig. 3 can be attributed to the scattering effect rendering the patch patterns. IV. CONCLUSION

Fig. 10. HFSS extracted element pattern compared to near-field model.

the measured input impedance of the patches, these residual phases have maximum only about 3 degree and have little effect to both monostatic and bistatic patterns. C. Scattering Effect Up until now the model is still built on the assumption that the element pattern is cosine or modified from cosine pattern. However, this pattern will be affected by the elements nearby, by mutual coupling or scattering. To include this effect in the model, an HFSS simulation is built to estimate and extract the element pattern for each patch. This simulation setup is shown in Fig. 9, and is a simplified model of the array setup and measurement environment. Because of the size of the problem, only half of the array setup is built in the simulation. However, by applying PEC and PMC symmetry boundary condition, we can simulate even and odd excitation for any symmetric pair of the elements, thus extracting the element pattern for each patch. The extracted transmitting amplitude pattern (R = 1:5 m) of the same element as in Fig. 5 is plotted in Fig. 10, normalized to unity and compared to the near-field model. We can see that the transmitting pattern is rendered severely by the scattering of other elements in the array. In this RDA, since the element spacing is restricted to be larger than 0:5o , the mutual coupling between two elements is simulated to be less than 018 dB and its effect on the active input impedance of the elements can be safely neglected. The scattering model is built by substituting the receiving and transmitting pattern in (6) by these newly extracted patterns for each element, and the resulting monostatic pattern is compared with measure-

In this communication, the synthesis of a sparse RDA on a conformal aperture resembling the nose of a UAV is discussed. A GA was developed to sparsely synthesize the aperture with low sidelobe level in the scanning patterns. The resulting array demonstrates scanning patterns with reasonable sidelobes within the whole scanning range of 0120  120 . It should be noted that, unlike the usual phased array, low sidelobes are automatically obtained in all patterns without deliberate control of the excitation in this RDA. Thus, it represents a simple solution to the real-time beamforming on a power and payload limited platform. Furthermore, the numerical model that was used for the GA implementation is introduced and discussed in detail. This numerical model is further improved by taking more realistic factors into account. The impact of each factor on the RDA operation is illustrated, and formula to modify the RDA model is developed and presented.

REFERENCES [1] K. L. Virga and D. Beauvarlet, “The effect of the element factor on low sidelobe circular arc array performance,” in Proc. IEEE AP-S, Jul. 2000, vol. 3, pp. 1206–1209. [2] J. A. Ferreira and F. Ares, “Pattern synthesis of conformal arrays by the simulated annealing technique,” Electron. Lett., vol. 13, no. 14, pp. 1187–1189, 1997. [3] R. J. Allard, D. H. Werner, and P. L. Werner, “Radiation pattern synthesis for arrays of conformal antennas mounted on arbitrarily-shaped three-dimensional platforms using genetic algorithms,” IEEE Trans. Antennas Propag., vol. 51, no. 5, pp. 1054–1062, May 2003. [4] J. S. Sun, D. S. Goshi, and T. Itoh, “A sparse conformal retrodirective array for UAV application,” in Proc. IEEE MTT-S Microw. Symp., 2008, pp. 795–798. [5] R. Y. Miyamoto and T. Itoh, “Retrodirective arrays for wireless communication,” IEEE Microw. Mag., pp. 71–79, March 2002. [6] C. W. Pobanz and T. Itoh, “A conformal retrodirective array for radar applications using a heterodyne phased scattering element,” in IEEE MTT-S Int. Microw. Symp. Digest, May 1995, vol. 2, pp. 905–908. [7] J. M. Johnson and Y. Rahmat-Samii, “Genetic algorithms in engineering electromagnetics,” IEEE Antennas Propag. Mag., vol. 39, no. 4, pp. 7–21, Aug. 1997. [8] R. L. Haupt, “Thinned arrays using genetic algorithms,” IEEE Trans. Antennas Propag., vol. 42, no. 7, pp. 993–999, Jul. 1994. [9] R. L. Haupt, “Optimized element spacing for low sidelobe concentric ring arrays,” IEEE Trans. Antennas Propag., vol. 56, no. 1, pp. 266–268, 2008.

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Retrodirective Array Performance in the Presence of Near Field Obstructions Vincent F. Fusco and Neil Buchanan

retrodirective array can maintain self-focussing action into its near field, [13]. In this communication we aim to provide by experimental means further insight into the topic of object blockage in the near field of a retrodirective array. II. RETRODIRECTIVE ARRAY HARDWARE CONFIGURATION

Abstract—We investigate the situation where there are obstructing elements present in the near field of a retrodirective array. We describe three scattering cases, (1) by an array of straight wires, (2) by low loss medium density fibre board partially obscuring the array, and (3) by concrete blocks, totally and then partially obscuring the array. For all scenarios retrodirective action was shown to be able to provide various degrees of automatic compensation for loss in gain relative to that which would have occurred for a conventional (non-retrodirective) array in the presence of the same scattering screens. Gain improvements of up to 10 dB were observed when the retrodirective array was used. In addition we show how the induced variation of received and re-transmited amplitudes across the array, caused by the scattering screens, is the principle mechanism causing deterioration of the retrodirective arrays monostatic response. Index Terms—Retrodirective antennas, retrodirectivity, scattering.

I. INTRODUCTION By means of phase conjugation retrodirective antennas operate by automatically retransmitting a signal back in the direction of an incident pilot tone. Applications for this technology include tracking of high altitude platforms, communication between unstablized platforms such as UAVs [1], solar power satellite (SPS) systems, [2]. Previously reported retrodirective antenna measurements [3]–[7] have been carried out over unobstructed free space propagation channels between antennas which are located in each others far field. Whilst these measurements prove the self steering ability of this antenna array class they do not show how these arrays might operate in conditions such as those which may occur when obstructions are present near to the radiating aperture of the retrodirective array. This class of operation is important when if such technology is to be used as part of through wall detection, or communication system, [8]. To date a very limited amount of work on the topic of how scatterers affect the performance of retrodirective arrays has been conducted, [9]–[12]. For example in [12] a 4 element software based retrodirective array was measured for its ability to transmit two channels of multipath data in a non line of sight environment. The retrodirective array used required to have static phase and amplitude settings applied for each specific environment, whereas the solution presented here is able to dynamically and automatically compensate for propagation path changes in real time. In [9]–[11] obstructions were not deliberately introduced into the near field of the retrodirective array, although it is known that an unobstructed Manuscript received December 17, 2008; revised June 24, 2009. First published December 28, 2009; current version published March 03, 2010. This work was supported by the UK Engineering and Physical Science Research Councils Grants EP/E01707X/1, EPD045835/1, the Northern Ireland Department of Education and Learning Strengthening all Island Grant: Mobile Wireless Futures and is subject to United Kingdom Patent Application No 0811635.2, 0811635.2, P104036.GB.01, Retrodirective Antenna Systems. The authors are with the The Institute of Electronics, Communications and Information Technology (ECIT), Queen’s University Belfast, Northern Ireland Science Park, Belfast BT3 9DT, Northern Ireland (e-mail: [email protected]; [email protected]; [email protected]). Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2009.2039293

A ten element 1D (azimuth scanning) retrodirective array constructed according to the principles elaborated in [14] was mounted in the anechoic chamber as shown in Fig. 1(a). Here two separate sets of microstrip patch antennas at 0:48 spacing are used for receive @ 2400.5 MHz and for retransmit @ 2409.5 MHz, both sets of elements are vertically polarized. The individual antenna elements are linearly polarized microstrip patch antennas with S11 = S22 = 025 [email protected] GHz, Gain = 5 dBi, HPBW = 110 . These are separated in the H plane by 0:9, yielding >20 dB isolation@ 2405 MHz. The monostatic far field response of the unobstructed 10 element retrodirective array exhibits 110 3 dB beamwidth, Fig. 1(b), with over 640 azimuth scan angle each of the arrays bistatic patterns having a 3 dB beamwidth of approximately 12 . The receive and re-transmit gain of the array are respectively 15 dBi. III. NEAR FIELD SCATTERING EFFECTS The aim of the experiments reported in this section is to determine how the retrodirective array described above behaves in an environment where objects are positioned in its near field. A. Effect of Wire Scattering Screen Positioned in Near Field of Retrodirective Array In Fig. 2 a 10 2 3 array of vertically orientated (i.e., aligned with the patch polarization) wire scatterers of length tapering from 35 mm to 80 mm left to right and with fixed spacing 0:48 were positioned on expanded foam ("r  1, tan   0) and placed in front of the retrodirective array at a distance of 15 cm from it. The scattering elements were made progressively shorter across the surface in order to provide different amounts of phase reflection across the aperture. Measurements were carried out at various distances between the wire scatterer and the array, two scenarios were investigated. First a basic transmit only antenna array whose main beam lies along broadside to the array and consisting of an identical aperture arrangement and EIRP, 22 dBm, as the retrodirective array, was set up and measured. Next the retrodirective array was substituted for the basic array and the arrangement measured in terms of its bistatic and monostatic behavior. It was found that the retrodirective array had some ability to compensate for the scattering induced by the wire screen positioned in front of it. Fig. 3 shows the radiation patterns obtained. Examination of the results in Fig. 3 shows that the monostatic response of the retrodirective array is largely maintained despite the presence of the screen, albeit with additional ripple and on average about 2 dB gain reduction due to scattering loss is observed over the entire monostatic 3 dB half power azimthual scanning range. At only 15 cm separation retrodirective action is being preserved with deviation between free space and scatterer position monostatic response occurring out to += 0 60 . The basic array on the other hand is yielding about 4 dB gain loss along its fixed boresight direction. The presence of the scatterering screen increases the sidelobe response of both the basic and the retrodirective bistatic array patterns. The ripple in the monostatic responses in Fig. 3(a) suggests that the scattering screen was subjecting the retrodirective array to variations in

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Fig. 1. Ten element retrodirective array. (a) Antenna array in anechoic chamber; (b) monostatic and bistatic patterns for ten element retrodirective array.

Fig. 2. Wire scattering screen positioned in front of Retrodirective Array (RDA). (a) Wire scatterer in front of RDA; (b) wire scattering array.

Fig. 3. Radiation patterns showing effect that wire scattering screen positioned 15 cm from the retrodirective array has on its far field behavior. (a) Retrodirective array; (b) basic array.

amplitude of the signal it was receiving. To investigate this effect further, the amplitudes and phases of the signals re-transmitted by individual retrodirective elements was measured in the retrodirective array far field with and without the scattering screen in position. This was carried out by switching ON each individual element (with all other elements switched OFF) and noting the boresight amplitude and phase, the results are shown in Fig. 4. Fig. 4(a) shows that with no scattering screen positioned in front of the array, each individual element re-trans-

mits almost identical signal strength. This indicates that the signal received by each element in the retrodirective array is, as expected, approximately equal. However when the scattering screen is placed 15 cm in front of the array, Fig. 4(a), large amplitude variation are present resulting in a 020 dB dip at element 6 relative to outer elements of the array. This amplitude variation gives rise, after automatic phase compensation, to the 4 dB variation in measured monostatic response observed in the presence of the wire scattering screen. In order to re-

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C. Effect of Concrete Screen Positioned in Near Field of Retrodirective Array

Fig. 4. Normalized amplitude distributions across retrodirective array elements in the presence of wire screen.

Fig. 5. Retrodirective array partially obscured by MDF panel.

duce this difference automatic gain control of signal strength across all re-transmitted elements would be required in order to equalise the energy being re-transmitted from all of the elements in the retrodirective array. This feature was not included in the 10 element array used in this communication and this aspect will be the subject of further study elsewhere.

By placing concrete blocks in front of the retrodirective array, the effect of a lossy dielectric on the behavior of the retrodirective array was established. The arrangement is shown inlaid on the graphs of Fig. 7. The concrete blocks were placed 15 cm from the array, and each had dimensions of 20 2 10 2 8 cm, the aggregate filled concrete blocks typically had a dielectric constant of "r = 12 and tan  = 5 [16]. Radiation patterns were recorded for seven and for four concrete blocks positioned in front of the array. The radiation pattern results of Fig. 7 show that in both cases the retrodirective array improves boresight amplitude response when compared to those obtained for the basic array, 10.74 dB gain improvement for the four block case. The case involving seven concrete blocks with no spacing between them showed a less marked improvement of 1.2 dB. We also note that in both cases the bistatic response sidelobes are much lower than those obtained from the basic array operating into the concrete screen. These results indicate that the retrodirective antenna can offer advantage to a normal array even when operated through a solid concrete block wall. In order to further investigate why such a large retrodirective gain increase over the basic array is being obtained for the four block case we measured the amplitude and phase variation across the array in the presence of both types of obstruction. The results as shown in Fig. 8 indicate amplitude distribution variation due to the lossy nature of the blocks, but more importantly for the four block case phase varies in such a way that the phase difference between pairs of adjacent elements approaches 180 . This explains why in the case of a basic array boresight amplitude is considerably reduced due to phase cancellation. The retrodirective array automatically compensates for this phase variation thereby yielding considerable improvement in boresight gain. However, as explained above, since the retrodirective array described in this communication is equipped to compensate only for phase, the amplitude variation across the array prevents any further improvement in performance being obtained. This has also been shown mathematically in [9] where the performance of a retrodirective array in a reflective environment was considered. IV. CONCLUSION

B. Effect of Medium Density Fibreboard (MDF) Screen Positioned in Near Field of Retrodirective Array The purpose of this experiment was to see to what extent that the retrodirective array could compensate in an environment where there was principally only phase distortion present. This was carried out by placing a 30 cm 2 15 cm sheet of 18 mm thick medium density fibreboard (MDF) ("r  3:2, tan   0:01) [15] such that it covered only one half of the array, Fig. 5. The monostatic results in Fig. 6(a) suggest that the MDF sheet is nearly lossless since the far and near field monostatic patterns, (boresight bistatic only pattern shown here for clarity), are almost identical. The gain of the retrodirective array is reduced by only 0.7 dB in the presence of the MDF sheet illustrating the ability of the retrodirective antenna to self compensate while operating through a low loss dielectric sheet positioned in its near field. The results in Fig. 6(b) show the effect that the addition of the MDF sheet has on the conventional (i.e., non-retrodirective) array. Here, due to the partial obscuration of the array the beam squints 3 to the left resulting in a boresight amplitude reduction of 02.2 dB at boresight.

In this communication we have specifically addressed the situation where there are scattering elements present in the near field of a retrodirective array. We have characterized by experimental means three cases: (1) a wire scattering screen, (2) a low loss (MDF) dielectric screen partially obscuring the array and (3) a lossy dielectric screen (concrete blocks) totally and partially obscuring the array. In all scenarios the retrodirective action was shown to be able to provide compensation for the presence of these obstructions by delivering gain increase over a range of azimuthal angular positions. It was shown that the retrodirective antennas ability to self compensate for phase distortions allowed it to recover nearly 11 dB gain with respect to a basic antenna when operating through a gapped concrete block arrangement. Further the variation of received and then the un-equalized re-transmited energy across the array was shown to be the principle mechanism causing deterioration of the retrodirective arrays monostatic response in the presence of lossy material placed in front of the retrodirective array. The work in this communication has for the first time shown that there is practical advantage to operationally deploy retrodirective arrays in difficult propagation environments such as through wall imaging in a building environment.

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Fig. 6. Ability of retrodirective antenna to compensate for MDF screen. (a) Retrodirective compensation for MDF screen; (b) effect of MDF screen on basic array.

Fig. 7. Effect of brick scatterers on retrodirective and array responses. (a) 7 blocks; (b) 4 blocks.

Fig. 8. Amplitude and phase distribution across the retrodirective array in the presence of concrete block obstructions. (a) Amplitude; (b) phase.

REFERENCES [1] DoD, Unmanned Aircraft Systems, Roadmap 2005–2030 Department of Defense, Washington, DC, Aug. 2005. [2] H. Matsumoto, “Research on solar power satellites and microwave power transmission in Japan,” IEEE Microw., vol. 3, no. 4, pp. 36–45, Dec. 2002. [3] C. Y. Pon, “Retrodirective array using the heterodyne technique,” IEEE Trans. Antennas Propag., vol. AP-12, pp. 176–180, 1964.

[4] L. D. DiDomenico and G. M. Rebeiz, “Digital communications using self-phased arrays,” in IEEE MTT-S Int. Microw. Symp. Digest, Jun. 2000, vol. 3, pp. 1705–1708. [5] J.-Y. Park and T. Itoh, “A 60-GHz 4th subharmonic phase-conjugated retrodirective array,” in Proc. 34th Eur. Microw. Conf., Oct. 2004, vol. 3, pp. 1277–1280. [6] L. Chiu, Q. Xue, and C. Chan, “A planar circular phase conjugated array with full scanning range,” in Proc. IEEE MTT-S Int. Microwave Symp., Atlanta, GA, Jun. 2008, pp. 599–602.

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[7] B. Y. Toh, V. F. Fusco, and N. Buchanan, “Assessment of performance limitations of PON retrodirective arrays,” IEEE Trans. Antennas Propag., vol. 50, no. 10, pp. 1425–1432, Oct. 2002. [8] K. M. Yemelyanov, J. A. McVay, N. Engheta, and A. Hoorfar, “Polarization-contrast sensing in complex environment for through-the-wall microwave imaging applications,” in Proc. IEEE Antennas and Propagation Int. Symp., Jun. 9–15, 2007, pp. 1473–1476. [9] V. F. Fusco, R. Roy, and S. L. Karode, “Reflector effects on the performance of retrodirective antenna array,” IEEE Trans. Antennas Propag., vol. 48, no. 6, pp. 946–953, Jun. 2000. [10] S. Karode and V. F. Fusco, “Use of an active retrodirective antenna array as a multipath sensor,” IEEE Microw. Guided Lett., vol. 7, no. 12, pp. 399–401, Dec. 1997. [11] J. Tuovinen, G. S. Shiroma, W. E. Forsyth, and W. A. Shiroma, “Multipath communications using a phase-conjugate array,” in 2003 IEEE Microw. Theory and Techniques Symp. Digest, pp. 1681–1684. [12] B. E. Henty and D. D. Stancil, “Multipath-enabled super-resolution for RF and microwave communication using phase-conjugate arrays,” Phys. Rev. Lett., vol. 93, p. 243904, 2004. [13] S. L. Karode and V. F. Fusco, “Near field focusing properties of an integrated retrodirective antenna,” in Proc. IEE National Conf. on Antennas and Propag., Mar. 31–Apr. 1 1999, pp. 45–48. [14] V. F. Fusco and N. B. Buchanan, “Dual mode retrodirective/phased array,” IET Electron. Lett., vol. 45, no. 3, pp. 139–141, Jan. 2009. [15] National Physical Laboratory, “Kaye and Laby Tables of Physical and Chemical Constants,” sec. 2.2.5, Dielectric Properties of Materials, 16th ed., 1995. [16] D. K. Misra and K. Fenske, “Dielectric materials at microwave frequencies,” Appl. Microw. Wireless, vol. 12, no. 12, pp. 92–1000, Oct. 2000.

Experimental Two-Element Time-Modulated Direction Finding Array Alan Tennant

Abstract—A two-element time-modulated array system is configured to operate as a direction finding antenna. The signal from each element of the array is time switched to provide a phase modulated output in which the depth of modulation is dependent on the angle of arrival of the received signal. Details of an experimental system designed to operate at 2400 MHz are presented and the results are compared to theoretical predictions.

Fig. 1. Two element time-switched array and definition of switching waveforms.

simple two-element array and compare measured data with theoretical predictions. Time-modulated arrays were first proposed as a means of producing low sidelobe antenna patterns by using simple on-off switching of the array elements [3], [4]. The technique allows conventional array amplitude weighting patterns to be synthesized in a time-average sense by switching the array elements on for a period that corresponds to the relative amplitude weight of the array element. An inherent problem normally associated with time-modulated arrays is that they generate unwanted harmonics, or sidebands, at multiples of the switching frequency and most research into time-modulated arrays has concentrated on minimizing or controlling these harmonics [4]–[9]. More recently work has been carried out to investigate the use of phase centre motion to control sidelobe levels [10] and the application of time-modulated arrays in Doppler radar [11]. In previous publications we have demonstrated how the harmonics of two and four element time-switched arrays can be configured to produce a simple direction finding system with active null scanning capabilities [2], [3]. In this contribution we describe a prototype two element time-switched array system and present measured data to illustrate the performance of the array. II. SYSTEM DESCRIPTION

Index Terms—Antenna, array, direction finding.

I. INTRODUCTION Conventional radio direction finding (RDF) systems often use an array of two or more antennas and use either phase-comparison or amplitude-comparison of the received signals to determine direction of arrival information [1]. In both of these techniques directional information is derived by processing array data at the receive signal frequency. In recent publications we proposed an alternative approach to direction finding using the concept of a time-switched array [2], [3]. The time-switched array system uses simple signal processing techniques to provide a directional main beam and pattern nulls at harmonic frequencies. In this contribution we present experimental results of a Manuscript received January 30, 2009; revised August 03, 2009. First published January 15, 2010; current version published March 03, 2010. The author is with the Department of Electronic and Electrical Engineering, University of Sheffield, Sheffield S1 3JD, U.K. (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2009.2039301

The time-switched array system is shown schematically in Fig. 1 and consists of two antenna elements separated by a distance d. Each element is assumed to exhibit an Omnidirectional radiation pattern in the plane of the diagram. The output from each element is connected to the array output via a switch in the feed line; the element switching waveforms and switching period are also defined in Fig. 1. A complete analysis of the system may be found in [2] but for the purpose of this contribution we will assume that the elements are alternately switched on and off with an equal mark-space ratio square wave switching function such that 1 = 2 = T =2 If we assume that the spacing between the elements is a half wavelength, the output from the array may be described [2] by

( ) = 9 cos !t; where 1; nT  t < nT + T2 = ejsin ; nT + T  t < (n + 1)T

A ; t

2

:

Hence the output of the array is phase modulated and the depth of modulation is dependent on the angle of arrival of the incident signal.

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Fig. 2. Angular pattern of the two element array at the fundamental (solid line) and the first harmonic (dashed line).

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Fig. 4. Measured spectrum of the transmit signal from the uncalibrated timemodulated array at broadside.

Fig. 3. Details of the measurement set-up.

For illumination broadside to the array, the array output is not modulated. However, as the illumination angle increases the depth of modulation increases until at endfire illumination ( = 90 ) the array output exhibits 180 phase shift modulation. A plot of the array receive pattern at the illumination frequency, ! , is given in Fig. 2 and shows a directional response with a peak on broadside and a null response at 690 . The directional pattern occurs because as the depth of phase modulation increases energy is redistributed from the carrier into sidebands, and at endfire there is zero energy at the carrier frequency. If we now examine the output of the time-switched array at the first harmonic (Fig. 2) we observe that the angular response has a deep null at broadside (corresponding to the zero phase modulation condition) with maximum gain at 690 . Hence the array exhibits the desired characteristics of a “mono-pulse” tracking system. An experimental time-switched array was designed and constructed to operate at 2400 MHz using quarter-wave monopoles as the two array elements. The monopoles were constructed from 1.5 mm copper wire mounted on a 200 mm diameter ground plane using chassis mounting 3.5 mm connectors. Because the array was only required to operate with equal mark-space ratio square-wave switching, a SPDT switch was constructed based on a design provided by Philips Semiconductors using BAP51-02 surface mount PIN diodes [10]. The switch was constructed using microstrip on standard FR4 circuit board. The switch was mounted on the underside of the array ground plane and connections were made to the monopoles using connected to the monopoles using short lengths of co-axial cable. However as the outputs of the SPDT switch were not phase matched, a co-axial line stretcher was inserted into one of the monopole connections to allow phase compensation. For ease of measurement the array was tested when operating in transmit mode. The measurements were made in an anechoic antenna test chamber using an HP 8350B as the RF source and an Agilent E4407B spectrum analyzer as a receiver. The SPDT switch was modulated with a square wave derived from a standard laboratory

Fig. 5. Measured spectrum of the transmit signal from the phase compensated time-modulated array at broadside.

Fig. 6. Measured angular transmit pattern of the array at the fundamental and first harmonic frequency (black line) compared to theoretical predictions (gray line).

signal generator at a frequency of 20 kHz. Fig. 3 shows the received spectrum measured broadside to the array prior to phase compensation. The spectrum contains a component at the fundamental frequency (2400 MHz) and also harmonic components at 20 kHz intervals due to the fact that the outputs from the RF are not phase matched. It is also evident that although the array was modulated with an equal mark-space ratio switching waveform, even harmonics are present in the spectrum. However the even harmonics are at a low level (more that 25 dB below the unmodulated fundamental) and may be attributed to a combination of many factors including a non-ideal switching waveform and small differences in the amplitudes of the signals radiating from the two

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monopoles. With the array still orientated at broadside the line stretcher was adjusted until the harmonics of the received spectrum were minimized and this result is shown in Fig. 4. With the array calibrated, measurements of the received power levels were recorded at the fundamental and the first harmonic frequency over an angular range of +=0108 deg at 6 deg intervals. The measured results are plotted in Fig. 5 along with theoretical predictions obtained from a simple model based on array factor.

A New Procedure for Assessing the Sensitivity of Antennas Using the Unscented Transform Leonardo R. A. X. deMenezes, Antônio J. M. Soares, Franklin C. Silva, Marco A. B. Terada, and Davi Correia

Details of an experimental two element time-switched array system have been presented. The signals to each element of the array are time switched to provide a phase modulated output signal in which the depth of modulation is dependent on the angle of arrival of the received signal. The angular response of the array at the fundamental frequency of the received signal exhibits a directional response at broadside, while the response at the first harmonic of the switching frequency exhibits a deep null. Measured results from the experimental system confirm the basic operation of the system.

Abstract—In this communication, we introduce a new procedure to analyze the sensitivity of antennas. The procedure is based on the method of unscented transform (UT) developed by Julier and Uhlman in 1997. Although the UT is used in control engineering, the method has only recently been applied to electromagnetic (EM) problems. This work describes the application of the UT both for single and multiple random variables cases. The UT method is applied to problems using examples involving different types of antennas. In each example, the UT method is combined with different numerical methods to perform the sensitivity analysis of wire or reflector antennas. In the first example, the results were validated with results from the Monte Carlo technique. The second example investigated the convergence of the UT procedure, and the final example compared UT results with measured data. The main conclusion is that UT is a feasible alternative to the popular Monte Carlo technique, with the advantage of being far less computationally expensive.

REFERENCES

Index Terms—Antennas, electromagnetic simulation, Monte Carlo methods, sensitivity, unscented transform.

III. CONCLUSION

[1] M. J. Wilson and B. S. Halo, The ARRL Handbook for the Radio Amateur, 65th ed. Washington, DC: American Radio Relay League, 1988. [2] A. Tennant and B. Chambers, “A two-element time-modulated array with direction finding properties,” IEEE Antennas Wireless Propag. Lett., vol. 6, pp. 64–65, 2007. [3] A. Tennant and B. Chambers, “Direction finding using a four-element time-switched array,” presented at the Loughborough Antennas and Propagation Conf., Loughborough, U.K., Mar. 17–18, 2008. [4] H. E. Shanks and R. W. Bickmore, “Four-dimensional electromagnetic radiators,” Canad. J. Phys., vol. 37, p. 263, 1959. [5] W. H. Kummer, A. T. Villeneuve, T. S. Fong, and F. G. Terrio, “Ultra-low sidelobes from time-modulated arrays,” IEEE Trans Antennas Propag., vol. 11, pp. 633–639, 1963. [6] S. Yang, Y. B. Gan, and A. Qing, “Sideband suppression in time-modulated linear arrays by the differential evolution algorithm,” IEEE Antennas Wireless Propag. Lett., vol. 1, pp. 173–175, 2002. [7] S. Yang, Y. B. Gan, and P. K. Tan, “A new technique for power-pattern synthesis in time-modulated linear arrays,” IEEE Antennas Wireless Propag. Lett., vol. 2, pp. 285–287, 2003. [8] J. Fondevila, J. C. Bregains, F. Ares, and E. Moreno, “Optimising uniformly excited linear arrays through time modulation,” IEEE Antennas Wireless Propag. Lett., vol. 3, pp. 298–300, 2004. [9] A. Tennant and B. Chambers, “Control of the harmonic radiation patterns of time-modulated antenna arrays,” in Proc. IEEE APS, San Diego, CA, Jul. 5–12, 2008, pp. 1–4. [10] S. Yang, Y. B. Gan, and P. K. Tan, “Linear antenna arrays with bidirectional phase center motion,” IEEE Trans. Antennas Propag., vol. 53, no. 5, pp. 1829–1835, May 2005. [11] G. Li, S. Yang, Y. Liu, and Z. Nie, “The application of time modulated arrays in pulse Doppler radar,” presented at the IEEE AP-S Int. Symp., San Diego, CA, Jun. 5–12, 2008. [12] Phillips Semiconductors, “2.45 GHz T/R, RF Switch for Bluetooth Applications Using Pin-Diodes,” Application Note: AN10173-01.

I. INTRODUCTION The design of complex antenna systems usually relies on the use of accurate electromagnetic simulation tools. An accurate simulation tool provides results that are in conformity with the exact ones (in the IEEE Dictionary of Electrical and Electronic Terms accuracy is defined as “The quality of freedom from mistake or error, that is, of conformity to truth or to a rule”). Among of the most used EM analysis techniques are the finite difference time domain (FDTD), physical optics (PO) and method of moments (MoM). All these methods use several deterministic assumptions in relation to the input parameters. One assumption is that the geometry of the simulated antenna is known with complete accuracy. In reality, the assembly process adds uncertainty to any real antenna. Naturally, the evaluation of such effects within the EM simulator could improve both the design and the prototyping stages. Modeling the uncertainty with random variables is the usual procedure and these variables are themselves modeled according to a specific probability distribution. The simplest form to integrate the randomness into an electromagnetic simulation is using the Monte-Carlo method [2], which can be very computationally demanding. In some cases, it is necessary to perform several hundred thousand simulations. This makes the Monte-Carlo method not suitable to be used in conjunction with EM simulators. Another way to represent randomness is the UT method, developed by Julier and Uhlman in 1997 [1]. The idea is to approximate the nonlinear mapping of a continuous random variable by the mapping of a set of deterministically selected points (sigma points). All statistical moments are available by a weighted average of the mapped values at the sigma points. Since any kind of numerical processes may be viewed as Manuscript received January 20, 2009; revised July 03, 2009. First published December 04, 2009; current version published March 03, 2010. The authors are with the Antenna Group, Department of Electrical Engineering, University of Brasilia-Brazil, Brasilia 70919-970, Brazil (e-mail: [email protected]; [email protected]; [email protected]; [email protected]; [email protected]; [email protected]). Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2009.2037838

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TABLE I WEIGHTS, SIGMA POINTS AND CORRESPONDING PROBABILITY DENSITY FUNCTIONS

a mapping, it is straightforward to combine the UT with any EM simulation tool. This work presents the combination of the UT technique with EM simulators as a direct approach to model uncertainty. The technique does not require modifications to analysis software, because it is used as a pre and post-processing tool. As a result, it can be used to quantify the effect of manufacturing tolerances, propagation of electromagnetic waves in complex media, and random configurations of geometrical parameters. II. THE THEORY

UNSCENTED TRANSFORM—ONE RANDOM VARIABLE

OF THE

One possible interpretation of the UT is an approximation of the conu) of a random variable u^ by a tinuous probability density function w(^ discrete distribution wi , existing only at discrete points Si . The approximation is performed so that the continuous (integral representation) and discrete (summatory representation) distributions have the same moments

E fu^

k

g=

u^ w(^u)du^ =

wS :

k

i

k i

Fig. 1. Cumulative probability distribution for discrete and continuous cases.

(1)

i

This formulation is analogous to a Gaussian quadrature integration scheme [3] of a function uk , where Si are selected integration points and wi are weights. The advantage of viewing the UT as an integration scheme is two-fold. First, the weights wi and sigma points Si are calculated from (1). Second, there is a number of integration schemes that can be used to accurately represent the moments. The quadrature scheme simplifies the calculation of the weights and sigma points. In u) is the probability distribution of the random this case the function w(^ variable. The nonlinear mapping is represented as

w G(S ) = G(^u)w(^u)du: ^ i

i

The Gaussian quadrature schemes are some of the best approximations to the integration in the one random variable case. Table I shows the probability density functions and the corresponding interpolation polynomials. The higher the order of the polynomial, the better is the approximation of the higher order moments. The sigma points and weights are computed with Rodrigues formula [4]

d (w(x)Q(x) ) p (x) = w(1x) dx w(x) xp 0(xx) dx: w = 1 = n

n

i

(2)

i

Equation (2) shows that the UT expression for the expected value is an approximation of the integral of the nonlinear mapping function G(^u) weighted by a function w(^u). The nonlinear mapping represents ^. The process may the process that will affect the random variable u be a analytical equation or a numerical computation. Since (2) also u) with a weight function represents an integration of the function G(^ w(^u), the calculation of the integral may be performed by an appropriate quadrature scheme. In this scheme the roots of the quadrature interpolation polynomial are related to the weights and sigma points of u). the UT. Naturally, the polynomial is dependent on the function w(^

n

n

n

dp dx

x

x

(3)

i

Where pn (x) is the polynomial and Q(x) is a different weight function that is defined according to the probability density function w(x). The weights wi calculated with (3) are positive definite with a total sum of 1. The sigma points are always part of the original distribution. Both constitute a discrete set that satisfies all the requirements for a probability distribution [5]. Table I shows the weights and sigma points for the different probability density functions. Since the discrete probability function is an approximation of the continuous one, as the number of sigma points grows, the discrete function tends to the continuous one. Fig. 1 shows the cumulative distribution function for the discrete and continuous normal distributions.

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TABLE II NUMBER OF SIGMA POINTS VERSUS NUMBER OF RANDOM VARIABLES (RVS). ONLY THE POLYNOMIAL OF SECOND ORDER IS CONSIDERED

Fig. 1 also shows that, in between sigma points, the value of the discrete and continuous distributions coincides. The larger the number of sigma points, the closer the value of the midpoints (between sigma points) is to the continuous cumulative distribution function. The same is true in the uniform and exponential cases. By a similar argument, after a nonlinear mapping the resulting discrete probability distribution will be closer to the continuous one as the number of sigma points grows. The UT can also be formulated using the Taylor expansion of the nonlinear mapping [5]. The truncation of the Taylor polynomial determines the number and value of the sigma points Si as well as the discrete probabilities wi of the UT. Once the sigma points are known, it is straightforward to apply them to the nonlinear mapping and calculate the statistical moments. The higher order central moments are given by

E

G(^u) 0 G

k

wi G(Si ) 0 G k :

=

(4)

i

The UT may be used to calculate any moments of the mapping as well as the resulting distribution. III. THE THEORY OF THE UNSCENTED TRANSFORM—MULTIPLE RANDOM VARIABLES If there are several independent random variables, the resulting multidimensional polynomials are the multiplication of one-dimensional ones. Therefore, the sigma points and weights are given by

IV. APPLYING THE UT TO ANTENNA PROBLEMS The UT may be combined as a pre and post processing tool with any numerical method used for antenna or EM propagation analysis. This section will present a few examples of the combination of the UT with different numerical techniques in different problems. A. Applying the UT With the Method of Moments The first example shows the effect of a hypothetical uncertainty on the length of a wire dipole antenna in free space. The uncertainty is modeled as a Gaussian random variable. The combination of the UT with the method of moments [6] can be used to calculate the expected input impedance, the standard deviation and the probability density function of the impedance. In the example, the antenna has a length of 0:5 and a diameter of 0:005. The standard deviation of the relative length varies from 0:01 to 0:05. The results are validated with 40,000 Monte Carlo simulations for each standard deviation point. In a second order UT scheme, a total of three simulations by stanL= dard deviation points are necessary. The first simulation uses S 1p p 0:5 0 3 , the second S 2 L = 0:5, and the last S 3 L = 0:5 + 3 . Therefore, for the standard deviation of 0.01 the lengths are 0:4827, 0:5, and 0:5173. Once the results are available, the expected value and variance are calculated using

f= i



2

1

2

1

6

3

6

wi f (Si L) = f (S1 L)+ f (S2 L)+ f (S3 L)

=

wi f (Si L) 0 f

2

i

pn (u; v) = pn (u)pn (v) wij = wi wj =

1 dp du

2

u

=u 1

dp dv

v

=v

=

pn (u) 0 ui ) w(u)du pn (v) w(v)dv: (v 0 vj )

(u

1 6

f (S1 L) 0f

2

+

2 3

f (S2 L) 0 f

One disadvantage of this approach is that the number of necessary sigma points grows rapidly. If 3 sigma points per random variable are used, in the case of 3 random variables there will be a total of 27 sigma points. This is shown in Table II in contrast to the approach presented in [5]. It is interesting to notice that in the two-random variables case, the number, weights, and sigma points are the same in the two approaches. If the random variables are not independent, then the covariance matrix approach shown in [5] can be used.

+

1 6

f (S3 L) 0 f 2 : (6)

The simulation of the three lengths results in the impedances: and 104:713 + 51:756j . The expected value of the impedance calculated using (6) is 97:121 + 40:994j . The standard deviation is the square root of the variance calculated with (6), its value is 7.356. As a validation procedure, the same problem was also calculated with 40,000 Monte Carlo repetitions. The results are presented in Table III. It is important to point the computational savings that the UT provides in comparison to the Monte Carlo method: each computation took about a tenth of a second to run. While the UT needed only three computations (0.3 seconds), Monte Carlo used 40,000 repetitions (about 1 hour and 5 minutes). The UT can also be used estimate the probability density function (PDF). The definite integral of this function shows the probability of the impedance falling within a given interval (defined by the integration 90:395 + 30:326j , 97:040 + 40:971j ,

(5)

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TABLE III COMPARISON BETWEEN EXPECTED VALUE OF THE INPUT IMPEDANCE OF THE DIPOLE—

Fig. 2. Probability density function (PDF) for the wire dipole case—standard deviation of 0.01. The real part corresponds to the resistance of the dipole and the imaginary part to the reactance.

limits). Obviously, the higher the polynomial order, the better the approximation. In this example, the second order UT scheme was used to calculate the PDF for the impedance (resistance and reactance). Fig. 2 shows the calculated PDF with the reactance having high probability of being in the 20 to 60 interval and the resistance in the 80 to 120

interval. B. Reflector Antennas and Physical Optics The next example evaluates the effect of uncertainty in the position of the feed with respect to the focal point of a parabolic reflector system. In this case, the electric behavior of the antenna is sensitive to displacements of the feed, due to the defocusing of the system [7], [8]. The problem is an attempt to investigate the level of accuracy needed when installing a feed in practice. As before, the uncertainty is modeled as a Gaussian random variable. The UT is herein combined with physical optics [7], [8] to determine the gain of the reflector antenna at boresight, the expected value, the standard deviation and the probability density function. The code WebPRAC [9] is employed in the simulations. The reflector configuration used in the simulations is listed in Table IV. The configuration results in a gain at boresight of 52.36 dBi, as computed by WebPRAC. The standard deviation of the displacement of the feed with relation to the focal point was set to 1.39 cm. This value corresponds to a variation of 1% of the 240-cm focal length. Nevertheless, it is important to note that the focal length is kept constant in the simulations, and the feed is displaced from the focal point along the feed axis. This cause the reflector system to be defocused and the pattern may scan in the

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MC (Monte Carlo) 2 UT

Fig. 3. Cumulative probability function (CDF) for the parabolic reflector case—the probability of a gain smaller than 48 dBi is smaller than 10%.

plane of offset. Furthermore, the gain deteriorates as the defocusing increases. The gain values considered in our numerical example are always taken at boresight, in the plane normal to the offset plane. This particular choice seemed to suit well the purpose of illustrating the application of UT to reflector antennas. Finally, it is worth mentioning that the code WebPRAC is accurate only for small feed displacements around the focal point. In this example, the displacements are limited to about 6 cm along the feed axis. In order to investigate the convergence of the UT, simulations use N sigma points (N = 3, 5, 7 and 9). The results are presented in Table V. For this specific example, displacing the feed by 1% of the focal length leads to a deterioration of the gain at boresight from 52.36 dBi (gain with the feed at the focal point) to 50.88 dBi (expected value computed with N = 9) with a linear standard deviation of 48, 643. 7051. The cumulative probability function for this example is shown in Fig. 3. C. Wire Antennas and FDTD The last example analyzes the effect of uncertainty in a Yagi antenna with finite ground plane. The antenna was built and tested in the open range measurement facility at Universidade de Brasília [10]. The EM simulation uses the FDTD method with Gaussian modeled uncertainty of 2.31 mm on the length of the elements (95% confidence interval). Instead of relying on Monte Carlo or higher order sigma points approximation, the validation of this example uses measured data with the appropriate confidence intervals. This shows that the UT can be used to

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TABLE IV REFLECTOR ANTENNA CONFIGURATION

TABLE V RESULTS FOR A STANDARD DEVIATION OF 1.39 FOR THE DISPLACEMENT OF THE FEED

TABLE VI DIMENSIONS OF SIMULATED YAGI ANTENNA

Fig. 4. Yagi antenna with finite ground plane.

provide adequate bounds of simulated antenna performance regarding actual measurements. The simulation uses a domain of (504 2 264 2 176) cells with slightly distinct timesteps due to the discretization of the problem for each sigma point. The antenna is shown in Fig. 4. There were three simulations with the sigma points shown in Table VI. Each one ran for 16384 timesteps. Both VSWR and E-field radiation pattern are calculated. The results are compared to the measurements to define the bounds and confidence intervals of VSWR and E-field radiation pattern using

Fig. 5. VSWR uncertainty bounds (without confidence intervals) of the Yagi antenna response compared to measurements.

the UT. The results are presented in Fig. 5 and Fig. 6. Few measurements are outside the predicted bounds. That is expected since the uncertainty in the measurement itself was not taken into account.

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60-GHz Wideband Substrate-Integrated-Waveguide Slot Array Using Closely Spaced Elements for Planar Multisector Antenna Masataka Ohira, Amane Miura, and Masazumi Ueba

Fig. 6. E-field radiation pattern uncertainty bounds (with confidence intervals) of the Yagi antenna response compared to measurements.

V. CONCLUSION The UT technique was applied to EM simulations. To show that the technique does not depend on the simulation method, three different problems were presented, each one with a different EM simulation method. The UT combination technique was validated using both numerical simulations and measurements. The technique may be used to assess uncertainty in the design of antennas, due to its significant reduction in computational requirements in contrast to the Monte Carlo method.

REFERENCES [1] S. J. Julier and J. K. Uhlmann, “Unscented filtering and nonlinear estimation,” Proc. IEEE, vol. 92, no. 3, pp. 401–422, Mar. 2004. [2] A. Papoulis, Probability, Random Variables, and Stochastic Processes, 3rd ed. New York: McGraw-Hill, 1991. [3] E. W. Weisstein, 2007 Jul., “Gaussian quadrature,” From MathWorld—A Wolfram Web Resource [Online]. Available: http://mathworld.wolfram.com/GaussianQuadrature.html [4] E. W. Weisstein, 2007 July, “Rodrigues representation,” From MathWorld—A Wolfram Web Resource [Online]. Available: http://mathworld.wolfram.com/RodriguesRepresentation.html [5] L. de Menezes, A. Ajayi, C. Christopoulos, P. Sewell, and G. A. Borges, “Efficient computation of stochastic electromagnetic problems using unscented transforms,” IET Sci. Meas. Technol., accepted for publication. [6] L. de Menezes, A. Ajayi, C. Christopoulos, P. Sewell, and G. A. Borges, “Efficient extraction of statistical moments in electromagnetic problems solved with the method of moments,” in Proc. Int. Microw. and Optoelectron. Conf. Salvador SBMO/IEEE MTT-S, 2007, vol. 1, pp. 757–760. [7] W. L. Stutzman and G. A. Thiele, Antenna Theory and Design, 2nd ed. New York: Wiley, 1998. [8] M. Terada, “Reflector antennas,” in Wiley Encyclopedia of RF and Microwave Engineering, K. Chang, Ed. New York: Wiley, 2005, vol. 5, pp. 4450–4474. [9] R. Rabelo, M. Terada, and W. Stutzman, “Analysis of reflector antennas through the world wide web,” IEEE Antennas Propag. Mag., vol. 49, pp. 113–116, Apr. 2007. [10] P. R. de Moura, F. C. Silva, and A. J. M. Soares, “Analysis and experimental results of wire antennas on a finite conductor ground plane,” presented at the Int. Microw. and Optoelectron. Conf. SBMO/IEEE MTT-S, Brazil, 2005.

Abstract—A new wideband waveguide slot array is proposed for covering a wide elevation-angle range with a planar sector antenna of a multigigabit wireless LAN in the 60-GHz band. The requirement of millimeter-wave sector antennas is to realize both wide bandwidth of 11.2% and wide elevation-angle coverage with a required gain of over 10 dBi. To satisfy both the required gain and bandwidth, the important structural features of the proposed antenna are that multiple slot elements are successively arranged at an interval of the length less than one quarter guide wavelengths and that the slot length increases from the shorted waveguide end to the feed end. The operational principle of the slot array is based on multisection quarter-wavelength reflection cancellation, so the slot array can achieve the required bandwidth. Furthermore, this antenna produces a unidirectional radiation pattern by using the radiators and the reflectors, resulting in the required gain over a wide elevation-angle range. In the design, the structural parameters of the slot array are selected to meet the requirements for 60-GHz-band sector antenna. The developed substrate-integrated-waveguide (SIW) slot array achieves both the 10 dB return-loss bandwidth of 15% and the specified gain in 88–96% of the one-sector coverage range of 59–66 GHz. The effectiveness of the proposed slot array for a sector antenna is verified by the good agreement between the simulated and measured results. Index Terms—Millimeter-wave band, sector antenna, slot array antenna, wideband, wireless local area networks.

I. INTRODUCTION The unused millimeter wave bands allotted for wireless local area networks (WLANs) [1] are promising for achieving a speed of over 1 Gbit/s. In Japan, the millimeter-wave band between 59–66 GHz has been allocated for use by WLAN systems, and we have investigated and developed a WLAN system in this band [2], [3]. There are some difficulties in realizing millimeter-wave communications, such as a large propagation loss. Therefore, the gain of millimeter-wave antennas has to be high to maintain a predetermined transmission quality. The antenna is also required to have a broad radiation pattern so that communications can be carried out at arbitrary locations in a room. However, a broader beamwidth leads to a lower gain. To solve this problem, a switchable multisector antenna [4]–[10] is one of the promising antenna configurations. The sector antenna covers the entire azimuth direction by switching several high-gain antennas. Furthermore, the high-gain antenna in each sector covers a wide elevation-angle range by using a tilt beam in the elevation plane. The wide elevation-angle coverage with required gain of over 10 dBi has been realized by three-dimensional antenna elements [5], [6]. However, the three-dimensional antennas, including the monopole Yagi-Uda array Manuscript received April 30, 2009; revised July 09, 2009. First published December 31, 2009; current version published March 03, 2010. This work is part of the “Research and development of ultra-high-speed gigabit-rate wireless LAN systems” funded by the National Institute of Information and Communication Technology (NICT). M. Ohira and M. Ueba are with ATR Wave Engineering Laboratories, Keihanna Science City, Kyoto 619-0288, Japan (e-mail: [email protected]; [email protected]; [email protected]). A. Miura was with ATR Wave Engineering Laboratories, Keihanna Science City, Kyoto 619-0288, Japan. He is now with the National Institute of Information and Communications Technology, Koganei-shi, Tokyo 184-8795, Japan (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2009.2039329

0018-926X/$26.00 © 2010 IEEE

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TABLE I SPECIFICATIONS OF ANTENNA

Fig. 1. Wireless LAN environment using eight-sector antennas.

[7], are not suitable for a user terminal, since they do not have a planar structure. To realize a low profile antenna, planar-type sector antennas have been investigated so far [8]–[10]. However, conventional array antennas designed for sector antennas have very narrow bandwidth, although the frequency of 11.2% bandwidth is required in the 60-GHz band. This is because the antenna elements used for the array antennas have essentially a narrow bandwidth of less than 5%. For obtaining wideband characteristics, a densely packed slot array fed by a microstrip line [11] has been proposed, but the front-to-back ratio is very low, which means that the antenna cannot realize required gain of over 10 dBi. To our knowledge, no antenna has been developed with both a wideband of 11.2% and a wide elevation-angle coverage with required gain of over 10 dBi for millimeter-wave sector antennas. This communication proposes a new wideband waveguide slot array covering a wide elevation-angle range with a required gain for a planar sector antenna in the 60-GHz band. In the proposed antenna, multiple slot elements with different lengths are successively arranged at an interval of less than one quarter guide wavelength to realize both a required gain and bandwidth for the WLAN. This slot configuration is inspired by the log-periodic array having a wide bandwidth [12], [13], but the fundamental principle of the proposed antenna differs from that of the log-periodic array. The wide bandwidth is realized by introducing a multisection configuration of a reflection-cancelling slot pair [14]. Consequently, the proposed antenna can realize over 15% bandwidth. The required gain over a wide elevation-angle range is obtained by the radiators and the reflectors, which produce a unidirectional pattern with the endfire radiation. An elevation-angle coverage of 66 with a maximum gain of over 14.5 dBi can be achieved with the proposed antenna. We design the slot array antenna in the 60-GHz band, and fabricate it using the substrate-integrated waveguide (SIW) technology [15]–[19]. The reflection and radiation characteristics are evaluated by experiment to show the effectiveness of the proposed slot array.

Fig. 2. Proposed structure for wideband slot array antenna for 60-GHz-band sector antenna.

To achieve the maximum gain of 13 dBi at max = 66 , this WLAN system introduces the eight-sector antenna, as shown in the inset of Fig. 1. Each antenna of AP and UTs has eight sectors. The required gain per one sector is represented by G(; )  Gspec (; ) in the angle range given in Table I. More specifically, the required gains at max = 66 and min = 0 are 13 dBi and 9 dBi, respectively. The antenna in each sector is selected by a single-pole eight-throw (SP8T) switch. A millimeter-wave SP8T switch is also being developed, but the details are out of this communication’s scope. To connect the eightsector antenna to the SP8T switch, the antenna in each sector is fed by the V -band coaxial line. The target of the relative bandwidth in this communication is over 11.2% to realize the return loss of 0(f )  0spec in the full band of 59–66 GHz.

II. SPECIFICATIONS OF MULTISECTOR ANTENNA Fig. 1 illustrates a WLAN communication environment of access point (AP) and user terminals (UTs) in our proposed system [2], [3]. The AP is installed on the wall so that it can cover 180 degrees in the horizontal plane. In our system, the maximum communication distance r from the AP to the UT is assumed to be 5 m. The angle max is 66 when the AP is located at a height of h = 2 [m] from the position of the UT. Hence, the coverage range is 66 (0    66 ) in the vertical plane. In order to receive the signal at a certain constant power independent of the angle  , the gain pattern G(; ) of the AP and the UT must be proportional to sec  in the vertical plane so that the propagation loss can be compensated by the antenna gain. Accordingly, the required gain Gspec can be expressed by the equation in Table I. The maximum gain Gmax is determined from a link budget design. At the beginning stages of developing a millimeter-wave WLAN system [2], [3], the required maximum gain Gmax was 10 dBi. As a result of re-examination of the entire system, the maximum gain is set to Gmax = 13 [dBi] at  = 66 , as given in Table I.

III. WIDEBAND SIW SLOT ARRAY FOR PLANAR SECTOR ANTENNA A. Proposed Structure Fig. 2 shows the proposed wideband waveguide slot array for the planar eight-sector antenna. The antenna is constructed by successively arranging slot elements at an interval of length g in the direction of the x-axis to form a unidirectional pattern and by making an array in the direction of the y -axis to obtain the specified gain. The slot length li (i = 1; 2; 1 1 1 ; Nx ) increases from the shorted waveguide end to the feed end. The slot elements are fed by a waveguide feed circuit consisting of the SIW [15]–[19], resulting in a low-profile planar structure and a low fabrication cost. The number of slot elements per feed waveguide in the x-axis direction and that of the array in the y -axis direction are represented by Nx and Ny , respectively. Ny is the same as the number of the feed waveguides. The structural parameters of the slot elements are shown in Fig. 3. In Fig. 2, the feed waveguides are constructed by the SIW in the dielectric

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IV. DEPENDENCY OF BANDWIDTH AND GAIN ON SLOT PARAMETERS

Fig. 3. Structural parameters of proposed slot array.

substrate with relative permittivity "r and thickness t. In the design, the SIW is replaced by the dielectric-filled rectangular waveguide in order to reduce the computation burden. The width wg of the rectangular waveguide is equivalent to that of the SIW. The slot elements are symmetrical with respect to the center of the feed waveguide. The slot width is denoted by w . In the basic configuration, the slot length li is linearly increased as expressed by li

= lN + (Nx 0 i) 2 1

(1)

where 1 denotes the increment of the slot length. The width of the rectangular waveguide for the feed has to be large enough to feed the slot element with the longest length. To obtain a wide frequency bandwidth of return loss, we introduce the multistage configuration of the reflection-cancelling slot pair [14]. Similar to a multistage quarter-wavelength impedance transformer [20], the spacing between discontinuities, which are the slot elements, is about one quarter guide wavelength. Although the slot elements have the radiation unlike the impedance transformer, the reflections from the multiple slot elements can be cancelled out each other, thereby resulting in a wide bandwidth. B. Wide Elevation-Angle Coverage With Required Gain The wide elevation-angle coverage in the x 0 z plane is achieved by a unidirectional pattern with endfire radiation. A wide-beamwidth pattern such as a cardioid pattern is obtained by two isotropic point sources with identical amplitudes and a phase difference of 90 , and that are a quarter-wavelength apart. This radiation pattern can be realized by the slot pair [14] located at the shorted end in Fig. 3, since the two slots are radiators excited by the 90 phase difference. The beamwidth in the x 0 z plane is mainly determined by the array size, that is, the number of slot elements Nx and the slot spacing g . In the proposed structure, the array configuration of the radiators and the reflectors is employed. No directors are needed, since the two radiators placed at the waveguide end produce the endfire radiation to obtain the beam tilt. Namely, the proposed antenna has a small number of slot elements Nx . The gain enhancement at  = 66 and the radiation suppression in the backward direction of  < 0 are achieved by the reflectors, which are placed at the feed end. The optimum design of the slot parameters of the slot spacing g and the slot length li enables the gain of the unidirectional radiation pattern to be higher than that of the cosecant beam pattern in the range of min    max .

The dependency of the bandwidth and gain on the slot parameters is investigated by electromagnetic field simulation. Fig. 4 gives the bandwidth and the gains at min and max when the normalized slot spacing g=g0 , the normalized increment 1=g0 of the slot length, and the number of slots Nx are changed, where g0 denotes the guide wavelength at the center frequency. The bandwidth and the gains are calculated from the results obtained by the electromagnetic field simulation using Ansoft HFSS. The hatched region shown in the left-hand side figures indicates the structural parameters having the 15-dB return-loss bandwidth wider than 11.2%. To achieve low reflection, the return-loss bandwidth is evaluated at 15 dB greater than the specified value 0spec = 10 [dB]. The feed circuit composed of a four-way power divider and a coaxial-line to SIW transition is not included in the simulation. The bandwidth is calculated from the frequency characteristic of the one-dimensional waveguide slot array, that is Ny = 1. In the case of the antenna without the feed circuit, the bandwidth of the one-dimensional array is the same as that of the two-dimensional one. As the number of slot elements Nx increases, the optimum slot spacing g to obtain the wide bandwidth decreases. This is because the mutual coupling between the slot elements affects the bandwidth, and the reactive part of the mutual coupling makes the electrical length of the slot spacing shorter. Furthermore, the bandwidth becomes wider as the increment of the slot length increases. The results verify that the proposed slot configuration is effective for enhancing the bandwidth. In Fig. 4, the hatched regions of the middle and right-hand side figures show the parameters satisfying the directive gains of G(min = 0 )  9 [dB] and G(max = 66)  13 [dB], respectively. The directive gain shown in Fig. 4 does not consider the radiation efficiency and the mismatch loss. In the radiation characteristic, the number of arrays is fixed to Ny = 4 so that the maximum directive gain Gmax = 13[dB] can be obtained at max = 66 . As the increment 1 of the slot length increases, the directive gains at both 0 and 66 become higher, especially in the case of Nx = 3. The results indicate that the slot elements with increased lengths act as radiators and reflectors, as explained in the previous section. It can also be found from the two figures of the gains that the beam is tilted toward max = 66 . In either case of Nx = 3, 4, 5, the specification of G(min = 0 )  9 [dB] and G(max = 66 )  13 [dB] can be achieved by adjusting the appropriate slot parameters. Hence, the overlapped regions of the three figures in addition to the left-hand side figure are the slot parameters needed to achieve the wide bandwidth as well as the specified gain. V. DESIGN RESULTS AND EXPERIMENTAL EVALUATIONS A. Designed Slot Array Antenna The design results of the slot array antenna are shown in this section. A polytetrafluoroethylene (PTFE) substrate is used, since the dielectric loss is relatively low in the millimeter-wave band. The thickness t of the PTFE substrate is 1.2 mm, and the relative permittivity "r is 2.17 [22]. The number of the slot elements is chosen to be Nx = 4, since the overlapped region in Fig. 4(b) is wider than that in Fig. 4(a) and (c). Consequently, the reflection and radiation characteristics are not greatly affected by fabrication error, which is severe in the millimeter-wave band. The initial values of the optimization are determined from Fig. 4(b), and then the optimization of the structural parameters are performed. The initial values are slot length: l1 = 2:9, l2 = 2:5, l3 = 2:1, l4 = 1:7, slot width: w = 0:3, slot spacing: g = 0:8 in [mm]. They are indicated by the circle in Fig. 4(b). In the optimization, the slot width w and the slot spacing g are fixed. To adjust the operating frequency of the slot array, the optimization of the slot lengths is performed by evaluating the return loss and the gain every 1 GHz in 59–66

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=0 #

= 66

=4

#N = 3 68 [mm]

Fig. 4. Dependency of bandwidth and gains at  and  on slot parameters. The number of arrays is N . The slot length of element is l :  :  . The length from slot N to the shorted end is d :  :  [21], where the guide wavelength  : and the effective wavelength  : . The number of slot elements is (a) N , (b) 4, and (c) 5.

= 0 46

= 0 50

= 3 42 [mm]

GHz. During the optimization, the slot length l2 is closer to l1 , while l3 is closer to l4 by iterations. As a result, the following structural parameters are obtained: slot length: l1 = 2:9, l2 = 2:7, l3 = 1:9, l4 = 1:7 in [mm]. The structure of the designed slot array antenna is shown in Fig. 5, and the photographs of the fabricated antenna are also given in Fig. 6. The post diameter and the post spacing of the SIW are chosen to be 0.3 mm and 0.5 mm, respectively. The fabricated antenna in Fig. 6 is fed by the V -band coaxial line, and the coaxial-line connector is mounted at the back of the slot array. The transition from the coaxial line to the SIW is realized by a taper-stepped metallic post [22]. The uniform excitation of the feed circuit for the four-by-four slot array is confirmed by the numerical analysis. In the fabricated antenna, it was found from

= 0 48 = 0 52 =3

the measurement using a microscope that the etching errors of the slot spacing g , the slot length li and the slot width w are 65 m, 610 m, and 63 m, respectively. These errors are negligibly small, since they are less than about g0 =370 (= e0 =340). B. Return Loss and Radiation Patterns The simulated return loss of the designed one-dimensional slot array antenna, which is fed by a rectangular waveguide, is shown in Fig. 7. The frequency response is simulated by Ansoft HFSS. The 10-dB and 15-dB return-loss bandwidths are 19.3% and 17.7%, respectively. The obtained bandwidth is much wider than the specification of 11.2%. Fig. 8 shows the comparison of the return loss between the simulated

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Fig. 5. Top view of designed waveguide slot array for 60-GHz band sector antenna.

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Fig. 8. Reflection characteristics of designed two-dimensional slot array fed by coaxial line, comparing simulated and measured results.

Fig. 6. Fabricated slot array antenna. (a) Top and (b) bottom views.

Fig. 7. Simulated reflection characteristic of designed one-dimensional waveguide slot array fed by rectangular waveguide.

and the measured results in the case of the coaxial-line feed. The dielectric loss of tan  = 0:001 is taken into account in the simulation. The simulated result is in good agreement with the measured one. The designed slot array antenna is satisfied with the return loss of 0  10 [dB] at 59–66 GHz. The fabricated antenna also achieves a 10-dB return-loss bandwidth of 15%. Fig. 9 compares gain patterns at the center of sector (x 0 z plane at  = 0 ) of the designed slot array between the simulated and the measured results. In the simulation, the size of the ground plane is the same as that of the fabricated antenna. The measurement of the radiation patterns is performed by the cylindrical near-field antenna measurement method. The far-field patterns are obtained by a near-field to far-field transformation. Although the measured gains are a little lower than the simulated ones, the measured gain patterns agree well with the

Fig. 9. Comparison between simulated and measured results of gain pattern at center of sector ( = 0 ) of the designed slot array. (a) 59.0 GHz, (b) 62.5 GHz, and (c) 66.0 GHz.

simulated ones. The measured maximum gains are 14.5 dBi with 72% efficiency at 59.0 GHz, 14.9 dBi with 68% efficiency at 62.5 GHz, and 15.9 dBi with 76% efficiency at 66.0 GHz. The required gain pattern is also shown in Fig. 9. The designed slot array antenna satisfies the required gain in the cut plane at  = 0 . The effectiveness of the proposed slot array antenna is verified by the good agreement between the simulated and the measured results.

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antenna. An eight-sector antenna consisting of the slot array developed in this communication will be reported in the near future.

REFERENCES

Fig. 10. Difference between measured and required gains in one-sector coverage area. (a) 59.0 GHz (measured gain is higher than required one in 88% of coverage area), (b) 62.5 GHz (92%), and (c) 66.0 GHz (96%).

C. Experimental Evaluation of Gain in One Sector

The gain in the one-sector coverage range (0    66 , 011:25    11:25 ) of the developed slot array antenna is measured to evaluate the performance of the sector antenna. Fig. 10 plots the gain difference Gm (; ) 0 Gspec (; ) in decibels, where Gm represents the measured gain. The positive value of the gain difference means that the measured gain is higher than the required one. Although the gain does not reach the required gain at the edge of the one-sector coverage range, the antenna satisfies the required gain in 88%, 92%, and 96% of the coverage range at 59.0 GHz, 62.5 GHz, and 66.0 GHz, respectively. The coverage rate may be improved by sidelobe suppression in the radiation pattern and by lower insertion loss in the feed circuit. The experimental evaluation of the gain in one sector demonstrated that the proposed slot array antenna is useful for a wideband and high-gain millimeter-wave WLAN sector antenna. VI. CONCLUSION To overcome the difficulty of achieving both wideband and wide angle coverage with required gain in a millimeter-wave WLAN antenna, we have successfully developed a substrate-integrated-waveguide slot array for a 60-GHz-band planar multisector antenna. The slot elements of the proposed array are successively arranged at the interval of one quarter guide wavelength. Their different slot lengths make it possible to suppress the reflection in the wide bandwidth and also to enhance the gain in the one-sector coverage range. In the design, the structural parameters have been optimized so that the specified bandwidth and gain can be obtained. As a result, the fabricated slot array antenna, which includes a coaxial-line to the SIW transition and a four-way power divider, has achieved the 10-dB return-loss bandwidth of 15% as well as the maximum gain of over 14.5 dBi. Furthermore, a relatively high coverage rate of 88–96% in one sector was achieved at 59–66 GHz. The good agreement between the simulated and the measured return loss and radiation patterns validates the effectiveness of the developed slot array for a multisector

[1] Y. Takimoto, “Recent activities on millimeter wave indoor LAN system development in Japan,” in IEEE MTT-S Int. Microw. Symp. Dig., May 1995, vol. 2, pp. 405–408. [2] T. Ohira, “Prospective system design for gigabit rate wireless local area networks,” in Int. Joint Conf. of the 6th MINT Millimeter-Wave Int. Symp., Seoul, Feb. 2005, pp. 251–254. [3] M. Ueba, A. Miura, S. Kitazawa, S. Saito, and T. Ohira, “Feasibility study on millimetre wave multi-gigabit wireless LAN system,” in Proc. 37th Eur. Microw. Conf., Munich, Germany, Oct. 2007, pp. 688–691. [4] T. Seki and T. Hori, “Cylindrical multi-sector antenna with self-selecting switching circuit,” IEICE Trans. Commun., vol. E84-B, no. 9, pp. 2407–2412, Sep. 2001. [5] Y. Murakami, T. Kijima, H. Iwasaki, T. Ihara, T. Manabe, and K. Iigusa, “A switchable multisector antenna for indoor wireless LAN systems in the 60-GHz Band,” IEEE Trans. Microw. Theory Tech., vol. 46, no. 6, pp. 841–843, Jun. 1998. [6] T. Ohno and K. Ogawa, “A sector array using dielectric loaded antennas at 60 GHz,” in Proc. IEICE Int. Symp. on Antennas and Propag., Sendai, Japan, Aug. 2004, pp. 761–764. [7] T. Maruyama, K. Uehara, and K. Kagoshima, “Design and analysis of small multisector antenna for wireless LANs based on monopole Yagi-Uda elements,” Electron. Commun. Japan (Part I: Commun.), vol. 81, no. 12, pp. 80–90, Dec. 1998. [8] N. Honma, F. Kira, T. Maruyama, K. Cho, and H. Mizuno, “Compact six-sector antenna employing patch Yagi-Uda array with common director,” in IEEE Int. Antennas Propag. Symp. Dig., Jun. 2002, vol. 40, pp. 26–29. [9] M. Yamamoto, N. Kobayashi, T. Nojima, and K. Itoh, “A novel planartype sectored antenna consisting of a slot Yagi-Uda array with post-wall cavity,” in IEEE Int. Antennas Propag. Symp. Dig., Jun. 2003, vol. 41, pp. 569–572. [10] M. Yamamoto, K. Ishizaki, M. Muramoto, K. Sasaki, and K. Itoh, “Cavity-backed slot array antenna with backward excitation,” Electron. Commun. Japan (Part I: Commun.), vol. 85, no. 11, pp. 68–76, May 2002. [11] X. Liang and C. Y. W. Michael, “A microstrip-fed wide-band slot array with finite ground,” in IEEE Int. Antennas Propag. Symp. Dig., Jul. 2000, vol. 38, pp. 510–513. [12] D. E. Isbell, “Log periodic dipole arrays,” IRE Trans. Antennas Propag., vol. AP-8, pp. 260–267, May 1960. [13] W. E. McKinzie, III, J. J. Moncada, and T. L. Anderson, “A microstrip-fed log-periodic slot array,” in IEEE Int. Antennas Propag. Symp. Dig., Jun. 1994, vol. 32, pp. 1278–1281. [14] K. Sakakibara, J. Hirokawa, M. Ando, and N. Goto, “A linearly-polarized slotted waveguide array using reflection-cancelling slot pairs,” IEICE Trans. Commun., vol. E77-B, no. 4, pp. 511–518, Apr. 1994. [15] J. Hirokawa and M. Ando, “Single-layer feed waveguide consisting of posts for plane TEM wave excitation in parallel plates,” IEEE Trans. Antennas Propag., vol. 46, no. 5, pp. 625–630, May 1998. [16] D. Deslandes and K. Wu, “Integrated microstrip and rectangular waveguide in planar form,” IEEE Microw. Wireless Comp. Lett., vol. 11, no. 2, pp. 68–80, May 2001. [17] Y. Cassivi, L. Perregrini, P. Arcioni, M. Bressan, K. Wu, and G. Conciauro, “Dispersion characteristics of substrate integrated rectangular waveguide,” IEEE Microw. Wireless Comp. Lett., vol. 12, no. 9, pp. 333–335, Sep. 2002. [18] L. Yan, W. Hong, G. H. Jixin, K. Wu, and T. J. Cui, “Simulation and experiment on SIW slot array antennas,” IEEE Microw. Wireless Comp. Lett., vol. 14, no. 9, pp. 446–448, Sep. 2004. [19] Z. C. Hao, W. Hong, H. Li, H. Zhang, and K. Wu, “Multiway broadband substrate integrated waveguide (SIW) power divider,” in IEEE Int. Antennas Propag. Symp. Dig., Jul. 2005, vol. 1, pp. 639–642. [20] R. E. Collin, Foundations for Microwave Engineering. New York: McGraw-Hill, 1966. [21] T. Hirano, J. Hirokawa, and M. Ando, “Analysis of a waveguide matching crossed slot by the method of moments using numerical eigenmode basis functions,” in IEEE Int. Antennas Propag. Symp. Dig., Jun. 2001, pp. 258–261. [22] T. Kai, Y. Katou, J. Hirokawa, M. Ando, H. Nakano, and Y. Hirachi, “A coaxial line to post-wall transition for a cost-effective transformer between a RF-device and a planar slot-array antenna in 60-GHz band,” IEICE Trans. Commun., vol. E89-B, no. 5, pp. 1646–1652, Feb. 2006.

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 58, NO. 3, MARCH 2010

Efficient Modeling of Radiation and Scattering for a Large Array of Loops Panagiotis J. Papakanellos, Nikolaos L. Tsitsas, and Hristos T. Anastassiu

Abstract—A computationally efficient technique, based on the method of moments (MoM) formulation, is invoked in the characterization of radiation and scattering properties of an array of coaxial, circular, non-identical loops. A set of Pocklington-type integral equations for the loop currents is formulated and subsequently discretized by a standard procedure. Thanks to a suitable choice of the basis functions, the resulting matrix corresponding to the pertinent linear system is forced to consist of circulant blocks. This type of system is solvable by an innovative recursive algorithm, featuring several important advantages, such as lower memory and execution time consumption, over standard, purely numerical inversion. The overall procedure is simpler in implementation than already existing methods, based on Fourier analysis. The procedure invokes almost exclusively elementary functions, and is applicable to large arrays with respect to diameter or number of loops. Data for such configurations are presented for the first time in literature. Index Terms—Circulant matrices, integral equations, loop antenna arrays, recursive algorithms.

I. INTRODUCTION Circular loop antennas have been subject to intense research, thanks to their broad variety of applications, as well as to their construction simplicity, low weight and low cost. Like linear dipoles, loops can be arranged into efficient arrays with improved properties, such as increased directivity. Analysis of the current, input impedance and radiation pattern of such configurations has been presented in several papers and books, such as [1]–[12]. In [1], [2] the current distribution is assumed to be constant along the loops, i.e., independent of the azimuth angle, meaning that the results are sufficiently accurate only for very small diameters. In [3]–[12], current is computed via integral-equation modeling and subsequent Fourier analysis, which is an extension of the basic concept applied earlier to single loops [13]–[17]. Although this method is mathematically elegant and exact, it is not easily applicable to arbitrarily large loops. Arising difficulties become evident from the fact that early papers [13], [14] contained erroneous results or misinterpretations, finally corrected in [15], at the expense of highly elaborate mathematical expressions. Computation of the Fourier coefficients requires the use of complicated special functions, and involves demanding numerical integrations, which become time-consuming and potentially inaccurate for large orders, unless special treatment, such as prior regularization, is applied. Numerical difficulties and the necessity to use extremely high-order fast Fourier transforms (FFTs) prompted asymptotic evaluation of the coefficients in [17], where the inefficiency of brute-force numerical integration is emphasized. Results in [17] are Manuscript received August 07, 2009. First published December 31, 2009; current version published March 03, 2010. P. J. Papakanellos is with the Hellenic Air Force Academy, GR-1010 Dekelia, Greece (e-mail: [email protected]). N. L. Tsitsas is with the School of Applied Mathematical and Physical Sciences, National Technical University of Athens, GR-15773, Zografou, Athens, Greece (e-mail: [email protected]). H. T. Anastassiu is with the Defense Systems Department, Hellenic Aerospace Industry SA, GR-32009, Schimatari, Greece and also with the Hellenic Air Force Academy, GR-1010 Dekelia, Greece (e-mail: [email protected]) Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2009.2039337

999

exploited in a multiloop configuration in [18], focused on identical loop elements. This communication approaches the problem in an alternative way. Although it is still based on an integral-equation formulation, in conjunction with the method of moments (MoM), it does not attempt to construct a diagonal linear system, unlike the Fourier series methodology. However, it constructs a MoM matrix with circulant properties, which may be handled by efficient and robust numerical inversion algorithms. Thus, there is no need to resort to standard numerical solutions of linear systems, and hence instability problems due to ill-conditioning are less likely. The advantages of this method have been clearly highlighted in the case of a single loop [19], [20] and a double loop [21], and the analysis will herein be extended to multiple, coaxial, generally dissimilar loops. Similarly to [19]–[21], the pertinent integral equations for the currents are derived, and finally discretized via step-pulse (pulse triplet) basis functions with point matching. A delta-gap source or a magnetic frill current is used as an excitation, which may feed any one of the array elements, or even all of them. The consequences of using either of the two source models in the circular loop problem have been discussed in [22], whereas a related discussion for straight wires can be found in several papers including [23]. Also, plane wave excitation is discussed, to characterize the scattering properties of such an array. Although large loop diameters are normally of limited practical antenna use, scattering from very large loops is a realistic issue, indeed, since the frequency of incident waves on a given array is not necessarily controllable; this is why this subject has been investigated in a number of papers including [17] and [18]. The resulting linear system involves a block-square matrix, consisting of circulant sub-matrices. Such a matrix is invertible via the efficient recursive algorithm developed in [24], which is based on the feasibility of the exact, analytical inversion of a circulant matrix of arbitrary size. Therefore, the inherently discrete approximations of the current, the input impedance and the admittance can be given by straightforward, recursive mathematical expressions. One of the most important features of the procedure is that it invokes, almost exclusively, merely elementary functions, whose computation is trivial. It is therefore possible to extract numerically stable results, which are compared to reference solutions [6]–[9]. Novel data for very large arrays (in diameter or in number of elements) are also presented for the first time in literature. Finally, the computational efficiency of the method is compared to purely numerical matrix inversion. The algorithm facilitates efficient design of multiloop antenna arrays [7], [25], since it is significantly less time- and memory-consuming than standard, analytical or numerical schemes. II. MATHEMATICAL MODELING The geometry of the problem consists of 3, parallel, coaxial nonidentical loops, with radii b1 ; b2 ; . . . ; b3 and wire radii a1 ; a2 ; . . . ; a3 bl ; l = 1; . . . ; 3, respectively, as shown in Fig. 1. Assume that al i.e., the loop wires are supposed to fulfill the thin-wire approximation. Loops and are generally separated by vertical distances D (along the z axis). Using a standard formulation for the boundary conditions, based on the principles described in [19]–[21], and assuming a ej!t time dependence, the integral equation for the currents Il () (each one associated with the lth loop) can be compactly cast as follows:



(

) 1 '^ = j!

E i r; 

0018-926X/$26.00 © 2010 IEEE

1 0 j!"r

j 0 0 0 j 0 j cos( 0  )I ( )ds

e0jkjr 0r  r r0

4

l

@ e0jkjr 0r j dIl 0 0 ds r0 @  r r 0 d0

1

4

j0 j

( )

(1)

1000

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 58, NO. 3, MARCH 2010

Fig. 1. A multiloop antenna array of 3, parallel, coaxial non-identical loops, with radii b ; b ; . . . ; b and wire radii a ; a ; . . . ; a respectively.

where k is the wavenumber, ; " are the surrounding medium’s permeability and permittivity respectively, and E i is the incident electric field (due to the excitation). Also, Ll ; l = 1; . . . ; 3 denote the axial loop curves (circles), r (r; ); r 0 (r0 ; 0 ) are observation and current source ^ is points respectively, ds is the differential length along a curve, and  the unit vector along the azimuth direction. Obviously, for sufficiently thin wires, rl0  = bl ; l = 1; . . . ; 3. The unknown current on the lth loop, i.e., Il (0 ), will be calculated from (1), after suitable discretization and transformation to a linear system of equations, according to a standard point matching methodology and with the step-pulse basis functions used in [19]–[21]. The resulting linear system is written in compact form as

[Z ]fI g = fV g

(2)

where

fI g  [I1 1 ; I2 1 ; . . . ; I 1; I1 2 ; I2 2 ; . . . ; I 2 ; . . . ; I1 3 ; I2 3 ; . . . ; I 3 ] ;

;

N;

;

;

N;

;

;

T

N;

(3)

which is the column of unknown currents

fV g  [V1 1; V2 1; . . . ; V 1 ; V1 2; V2 2; . . . ; V 2 ; . . . ; V1 3 ; V2 3 ; . . . ; V 3 ] ;

;

N;

;

;

N;

;

;

N;

T

(4)

which is the column of excitation voltages at the matching points, related to the local incident electric field as follows:

Vm;

= b '^ 1 E (m 0 1) 2N

i

(5)

and finally [Z ] is the MoM impedance matrix, conveniently split into 32 blocks as follows: [Z ](11) [Z ](12) 1 1 1 [Z ](13) (21) (22) (23) [Z ] = [Z1]1 1 [Z1]1 1 11 11 11 [Z1]1 1 : (6) [Z ](31) [Z ](32) 1 1 1 [Z ](33) Superscripts ( ) specifying each block correspond to the and the

loop, and therefore the block simulates their mutual interaction. The

important property of all 32 blocks is their circulant nature, for reasons explained in [19]–[21]. Each block corresponds to the interaction between two loops in the array, and hence the entries in the block are essentially given by the expressions derived explicitly in [21]. It is remarkable that only elementary functions are invoked, the only exception being an elliptic integral [21]. However, even the latter can be readily calculated via standard algorithms found in easily accessible sources [26]. Therefore, unlike the classical Fourier formulation [3]–[16], which requires several elaborate special functions, this method is capable of computing the entries of the impedance matrix almost trivially. Moreover, the most significant property is its consistence of circulant blocks. An efficient inversion procedure of such a matrix is described in detail in [24], and is based on the analytic calculation of the eigenvalues of each circulant block [19]. In brief, if 3 is a power of 2, the algorithm originally transforms [Z ] into a matrix [Z D ] with diagonal blocks, via multiplication by a Fourier matrix. Next, it divides [Z D ] into 4, equally sized sub-matrices. Then, [Z D ]01 can be given formally in terms of the inverses thereof. To calculate the latter, each sub-matrix is further divided into 4 sub-sub-matrices. The procedure is performed recursively, until finally the inverse of merely diagonal matrices is required, which is a trivial task. If 3 is not a power of 2, then [Z ] is augmented via zero-padding, so as to reach an order which is equal to the smallest power of 2 exceeding 3. The above described recursive algorithm is then applied to the augmented matrix. One of the most important consequences of this inversion procedure is that, as rigorously shown in [24], the recursive inversion algorithm is computationally much less expensive (much faster) than the classical numerical inversion, based on LU decomposition. In terms of required memory, it can be shown that this algorithm is much more advantageous compared to LU. Indeed, the initial matrix [Z ] needs 32 2 N memory cells (locations) to store the 32 vectors of length N , determining each circulant block of order N . The aforementioned block diagonalization requires 32 2 N additional memory cells in order to transform each circulant block to a diagonal matrix by means of the FFT. Then, the algorithm implements a forward and backward recurrence procedure to invert [Z D ]. The largest number of memory cells is required in the first step of these procedures and is equal to 32 2 N . In the next steps of the recurrence procedure, the matrices involved are of lower order compared to those of the first step, and thus no additional memory cells are needed. The forward and backward recurrence yields [Z D ]01 , requiring 32 2 N memory cells, and finally, [Z ]01 is obtained via an inverse FFT, also requiring storage of 32 2 N complex numbers. Hence, the total number of memory cells necessary to the recursive inversion algorithm is 332 2 N , as opposed to 32 2 N 2 required by LU, which is N=3 times higher. For a typical example of 3 = 20 circular loops each modeled by N = 1000 basis functions, the LU inversion requires 4 2 108 memory cells, while the recursive inversion only 1:2 2 106 . Additional savings can be achieved by exploiting the matrix symmetry, i.e., the property [Z ]( ) = [Z ]( ) in (6), which allows the final memory requirements to drop down to 3=2 2 3 2 (3 + 1) 2 N . In view of all remarks above, very large loop arrays become tractable, due to two reasons: (a) the recursive inversion is extremely efficient, saving a lot of computational time and memory in comparison to standard numerical inversion algorithms, and (b) the entries in (6) are trivially computable, irrespective of the number of unknowns, unlike in the Fourier formulation. III. NUMERICAL RESULTS The results in Table I are for the Yagi arrays of circular loops examined in [7]. The arrays comprise a reflector with kb1 = 1:05, an exciter with kb2 = 1:1, and a number of directors with kbl = 0:9 for l = 3; 4; . . . ; 3. As in [7], the wire radius of all elements is selected

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TABLE I RESULTS FOR THE YAGI ARRAYS OF CIRCULAR LOOPS EXAMINED IN [7]

N = 100 kb = 1:1 2ln(2b =a) = 11 0:2

The computed input admittances and the corresponding execution times are for basis functions/points per , an exciter with , and a number of directors element. The arrays consist of a reflector with with . The wire radius of all elements is so that for . The distance , whereas the director spacing is set to . A delta-gap between the reflector and the exciter is taken to be is assumed to drive the exciting element of each array generator at

kb = 0:9

=0

l = 3; 4. . . ; 3

kb = 1:05 0:1

TABLE II RESULTS FOR UNIFORM ARRAYS

b= = 0:15

a= = 0:01

Results for uniform arrays of identical circular loops with and . The inter-element spacing basis . The computed input admittances and the corresponding execution times are given for is set to at is assumed to drive the exciting functions/points per element. A frill generator of radius element of each array.

0:2

so that 2 ln(2b2 =a) = 11 (or, equivalently, a= = 0:004496). A delta-gap generator is assumed to drive the exciter. The distance between the reflector and the exciter is taken to be 0:1, whereas the director spacing is set to 0:2. The results in Table I correspond to various numbers of directors, ranging from 2 to 10 (thus, 3 varies accordingly from 4 to 12). The computed input admittances in Table I correspond to N = 100. Apparently, these are in close agreement with the ones tabulated in [7], which are also contained in Table I for convenience. Execution times are also given for both a conventional LU solver and the recursive algorithm discussed above. The execution time corresponds to the total running time required for filling and inverting the associated matrices. All execution times given here were measured on the same platform (namely, a portable PC with a single-core 1.6 GHz Intel Pentium M processor and 504 MB of RAM). Further results for uniform arrays with more elements (up to 21) are presented in Table II. Reference results for that many loops are not known from the literature. The array consists of identical circular loops with b= = 0:15 and a= = 0:01. The inter-element spacing is set to 0:2. The computed input admittances and the corresponding execution times are given for N = 60 basis functions/points per element. A magnetic frill generator of radius af = = 0:023 at  = 0 is assumed to drive the central element of each array. Indicative results for the behavior of the computed currents as N grows are shown in Fig. 2, which depicts the real and imaginary part of the current distribution on the exciter of a Yagi array with 3 = 4 and the same dimensions as above. Due to symmetry, only half of the current distribution is shown. Obviously, the behavior of the current distribution is similar to that occurring when examining single loops

a = = 0:023  = 0

N = 60

Fig. 2. Computed current distributions, corresponding to three different values of (50, 100, 150), as functions of the azimuth angle along the circumference of the exciter of a four-element Yagi array with the characteristics of Table I. A delta-gap generator at is assumed to drive the exciting element of the array.

N



=0

[19]–[22]. Note that the imaginary part of the current for N = 150 exhibits some irregularity in the vicinity of the feeding point, which progressively deteriorates as N grows beyond maxf2bl =al g ( 156 in this example) in the sense that severe unnatural oscillations of increasing magnitude are finally encountered. Discussion on the origin of such oscillations is found in [22], [23]. Similarly with the single loop case, use of a magnetic frill model for the excitation, instead of

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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 58, NO. 3, MARCH 2010

REFERENCES

Fig. 3. Plot of the measured execution time as function of the total number of : and kb unknowns for a three-element array of identical loops with a= ranging from 0.6 to 15.7. The inter-element spacing is set to : . The array is = and  = [22]. The illuminated by an oblique plane wave with number of unknowns per element increases in proportion to the ratio b=a.

= 4

= 0 01 03 = 3 4

the delta gap, was found to eliminate these oscillations, at the expense of somewhat slower convergence. To further demonstrate the efficiency of the recursive algorithm, measured execution times are depicted in Fig. 3 for a three-element array of identical loops with a= = 0:01 and kb ranging from 0.6 to 15.7, illuminated by a plane wave, with a mathematical expression given in [22], adapted to a multiloop layout. The measured execution time is plotted as a function of the total number of unknowns 3N , with N determined from the ratio 4b=a. Obviously, the recursive algorithm is much less computationally demanding than the one based on LU decomposition, a fact that becomes crucial for the applicability of the method when focused on large arrays (that is, arrays of large numbers of elements) or arrays of large loops (namely, elements with large bl =al ). IV. SUMMARY AND CONCLUSION In this communication, mathematical characterization of an array of coaxial circular loop antennas has been carried out on the basis of a novel technique, exploiting the particular properties of the interaction matrix. Although early stages of the algorithm are closely related to a standard MoM with point matching, the linear system is finally solved via eigenvalue analysis. Applicability of the latter is due to the specific nature of the matrix involved, which is a priori forced to consist of circulant blocks. The advantages of the method can be summarized as follows. • It is simpler (from an algorithmic point of view) than standard, Fourier analysis, and also more efficient, since it does not require any special treatment, such as regularization, which may be necessary before the latter yields accurate results; • Modifications for various excitation models are trivial, unlike Fourier analysis, which requires extensive work. This is particularly important when examining receiver configurations (e.g., in a multiple input multiple output—MIMO environment) or composite feeding schemes; • It is more efficient than standard inversion algorithms, such as LU decomposition, in terms of required memory and speed; • It is capable of easily analyzing scattering arrays of very large size, which usually challenge standard methods severely.

[1] E. Ledinegg, W. Papusek, and W. Ninaus, “Low-frequency loop antenna arrays: Ground reaction and mutual interaction,” IEEE Trans. Antennas Propag., vol. AP-21, no. 1, pp. 1–8, Jan. 1973. [2] E. Ledinegg, W. Papusek, and W. Ninaus, “Low-frequency loop antenna arrays: Radiation field of systems of horizontally oriented loops,” IEEE Trans. Antennas Propag., vol. AP-22, pp. 464–467, May 1974. [3] K. Iizuka, R. W. P. King, and C. W. Harrison, Jr., “Self- and mutual admittances of two identical circular loop antennas in a conducting medium and in air,” IEEE Trans. Antennas Propag., vol. AP-14, no. 4, pp. 440–450, Jul. 1966. [4] S. Ito, N. Inagaki, and T. Sekiguchi, “An investigation of the array of circular-loop antennas,” IEEE Trans. Antennas Propag., vol. AP-19, no. 4, pp. 469–476, Jul. 1971. [5] A. S. Abul-Kassem and D. C. Chang, “On two parallel loop antennas,” IEEE Trans. Antennas Propag., vol. AP-28, no. 4, pp. 491–496, 1980. [6] A. Shoamanesh and L. Shafai, “Properties of coaxial Yagi loop arrays,” IEEE Trans. Antennas Propag., vol. AP-26, no. 4, pp. 547–550, Jul. 1978. [7] A. Shoamanesh and L. Shafai, “Design data for coaxial Yagi array of circular loops,” IEEE Trans. Antennas Propag., vol. AP-27, no. 5, pp. 711–713, Sept. 1979. [8] A. Shoamanesh and L. Shafai, “Multiply driven and loaded coaxial circular loop arrays,” IEEE Trans. Antennas Propag., vol. AP-28, no. 2, pp. 255–258, Mar. 1980. [9] A. Shoamanesh and L. Shafai, “Characteristics of Yagi arrays of two concentric loops with loaded elements,” IEEE Trans. Antennas Propag., vol. AP-28, no. 6, pp. 871–874, Jul. 1980. [10] T. Korekado, K. Okuno, and S. Kurazono, “Design method of Yagi-Uda two-stacked circular loop antenna arrays,” IEEE Trans. Antennas Propag., vol. AP-39, no. 8, pp. 1112–1118, Aug. 1991. [11] Y. Huang, A. Nehorai, and G. Friedman, “Mutual coupling of two collocated orthogonally oriented circular thin-wire loops,” IEEE Trans. Antennas Propag., vol. AP-51, no. 6, pp. 1306–1314, Jun. 2003. [12] S. Krishnan, L.-W. Li, and M.-S. Leong, “Entire domain MoM analysis of an array of arbitrarily oriented circular loop antennas: A general formulation,” IEEE Trans. Antennas Propag., vol. AP-53, no. 9, pp. 2961–2968, Sept. 2005. [13] E. Hallen, “Theoretical investigations into transmitting and receiving qualities of antennae,” Nova Acta Regiae Soc. Sci. Upsaliensis, vol. 4, pp. 1–44, 1938. [14] J. E. Storer, “Impedance of thin-wire loop antennas,” Trans. Am. Inst. Elec. Engrs., vol. 75, p. 606, 1956. [15] T. T. Wu, “Theory of thin circular loop antenna,” J. Math. Phys., vol. 3, pp. 1301–1304, 1962. [16] R. F. Harrington, Field Computation by Moment Methods. Malabar, FL: Krieger, 1982. [17] S. Li and R. W. Scharstein, “High frequency scattering by a conducting ring,” IEEE Trans. Antennas Propag., vol. 53, no. 6, pp. 1927–1938, Jun. 2005. [18] S. Li and R. W. Scharstein, “Edge effects in axial arrays of circular rings,” IEEE Trans. Antennas Propag., vol. 54, no. 2, pp. 544–553, Feb. 2006. [19] H. T. Anastassiu, “Fast, simple and accurate computation of the currents on an arbitrarily large circular loop antenna,” IEEE Trans. Antennas Propag., vol. AP-54, no. 3, pp. 860–866, Mar. 2006. [20] H. T. Anastassiu, “An efficient algorithm for the input susceptance of an arbitrarily large, circular loop antenna,” IET Electron. Lett., vol. 42, no. 16, pp. 897–898, 2006. [21] L. C. Tatalopoulos, A. I. Sotiropoulos, S. P. Skouris, and H. T. Anastassiu, “Efficient, numerically robust characterization of a large, double-loop antenna array,” IET Proc. Microw. Antennas Propag., vol. 3, no. 3, pp. 436–442, Apr. 2009. [22] G. Fikioris, P. J. Papakanellos, and H. T. Anastassiu, “On the use of nonsingular kernels in certain integral equations for thin-wire circular-loop antennas,” IEEE Trans. Antennas Propag., vol. 56, no. 1, pp. 151–157, Jan. 2008. [23] M. C. van Beurden and A. G. Tijhuis, “Analysis and regularization of the thin-wire integral equation with reduced kernel,” IEEE Trans. Antennas Propag., vol. 55, no. 1, pp. 120–129, Jan. 2007. [24] N. L. Tsitsas, E. G. Alivizatos, and G. H. Kalogeropoulos, “A recursive algorithm for the inversion of matrices with circulant blocks,” Appl. Math. Comput., vol. 188, pp. 877–894, 2007. [25] L. C. Shen and G. W. Raffoul, “Optimum design of Yagi array of loops,” IEEE Trans. Antennas Propag., vol. 22, no. 6, pp. 829–830, Nov. 1974. [26] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in FORTRAN, 2nd ed. Cambridge: Cambridge Univ. Press, 1992.

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Asymptotic Extraction Approach for Antennas in a Multilayered Spherical Media Salam K. Khamas

Abstract—An efficient algorithm is introduced to enhance the convergence of dyadic Green’s functions (DGF) in a layered spherical media where asymptotic expressions have been developed. The formulated expressions involve an infinite series of spherical eigenmodes that can be reduced to the simple homogenous media Green’s function using the addition theorem of spherical Hankel functions. Substantial improvements in the convergence speed have been attained by subtracting the asymptotic series representation from the original DGF. The subtracted components are then added to the solution using the homogenous media Green’s function format. Index Terms—Dyadic Green’s function, method of moments, spherical antennas.

I. INTRODUCTION Rigorous analysis of electromagnetic waves’ radiation and scattering in the presence of a layered dielectric sphere has been reported in a number of studies [1]–[4], where the required dyadic Green’s function has been expressed in the form of an infinite series of spherical eigenmodes. This series is convergent and hence can be truncated using a finite number of terms. However, the convergence speed depends on a number of factors such as sphere radius, permittivity and the distance between the source and field points, r0 and r, respectively. A large number of terms must be added in the summation when r0 and r are in the vicinity of each other and both are in the proximity of a dielectric interface. Once the source and the observation points are apart from an interface, convergence of the series can be achieved using a considerably reduced number of terms. Accelerating the infinite summation convergence can produce a computationally faster model. Furthermore, a common concern with adding a larger number of terms is the requirement to compute the spherical Hankel and Bessel functions, hn (kr) and jn (kr), of large orders. This is generally known to be accompanied by numerical over flows, or under flows; hence, it may result in a potentially unstable model. Several studies on speeding up the convergence are available in the literature using Watson [5] or Shanks [6], [7] transformations, where the former cannot be used when the source and field points are on the same radial line [7] and the latter needs additional numerical considerations. Further, the required DGF expansion coefficients poles need to be determined numerically for a multilayered structure when Watson transformation is used [8], which increases the complexity of the model. An alternative approach to reduce the required number of summation terms has been reported in [8], where a closed-form representation of the Green’s function has been attained using finite difference algorithm to model the layered sphere. This approach enhances the computation efficiency as the angular distance between source and field points increases. Other solutions have been reported in [9], [10] to accelerate convergence in the case of a radial monopole above, or connected, to a large PEC sphere. However, those solutions Manuscript received May 20, 2009; revised August 20, 2009. First published December 31, 2009; current version published March 03, 2010. The author is with the Communications Research Group, Department of Electronic and Electrical Engineering, University of Sheffield, Sheffield S1 3JD, U.K. (e-mail: [email protected]). Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2009.2039333

1003

cannot be adopted in the presence of conformal current sources or for layered spherical structures. A different approach to reduce the computation time has been proposed in [11], [12] through the development of novel closed-form expressions for the required numerical integrations in a method of moments (MoM) solution. A well-known technique that has been employed in the analysis of planar and cylindrical geometries is the asymptotic extraction approach [13]–[16], where the quasi-static images are extracted from the Sommerfeld-type integrals and then added back to the overall Green’s function using closed-form representations. In this article, the asymptotic extraction is adopted for spherical structures to expedite the infinite series convergence. This is based on developing asymptotic expressions for the DGF components as the summation index approaches infinity. These expressions have been incorporated into a new infinite series that can be expressed in a closed form by employing the spherical Hankel function addition theorem. A rapidly convergent model is then accomplished by subtracting the new series from the original summation. The subtracted series was subsequently added, albeit in a closed form, to the overall DGF expression. A method of moments model has been developed by adopting the introduced procedure, where it was found that the convergence speed is accelerated by several folds while accuracy is maintained. II. FORMULATION Fig. 1 illustrates a layered sphere that consists of four layers where each layer has a permittivity of "f and a permeability of f . The source and the field points could be located in any layer. For an antenna radiating in the vicinity of such a sphere, the DGF may be expressed as [1]–[3]

fs

Ge (r; r0 ) =

1  I+ 2 k

f

rr

0

e0jk R s (fs) f + Ges (r; r0 ) 4R

(1)

where the superscript fs refers to the layers of field and source points. The first term represents the DGF component owing to antenna radiating in an infinite homogenous media, while the second term is the scattering DGF that accounts for the presence of a layered sphere, given by [3] (fs) Ges (r; r0 ) 1 n jks 2n + 1 (n m)! 0 = 2 m 4 n (n + 1) (n + m)! n=0 m=0

0

0

2

(2) fs 0 fs 0(2) 11 Mmn (kf ) 12 AM Mmn (ks )+13 BM Mmn (ks ) (2) fs 0 fs 0(2) + 11 Nmn (kf ) 12 AN Nmn (ks )+13 BN Nmn (ks )

fs 0 fs 0(2) + 14 Mmn (kf ) 12 CM Mmn (ks )+13 DM Mmn (ks ) + 14 Nmn (kf )

fs 0 fs 0(2) 12 CN Nmn (ks )+13 DN Nmn (ks ) (2)

where Mmn and Nmn are the well-known spherical vector eigenfunctions of the transverse electric, T Emn , and transverse magnetic, T Mmn , modes, respectively, the superscript (2) refers to the second type spherical Hankel functions, 11 = 1 fL , 12 = 1 s1 , 13 = 1 sL , 14 = 1 f1 , uv is the Kronecker delta, and L is

0

0018-926X/$26.00 © 2010 IEEE

0

0

0

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both the source and field points are in the same layer, then, without loss of generality, it can be shown that

AiiM a = BMii a = CMii a = DMii  0 (4) a AiiN a = DNii a  0 (5) n (ki ai ) n (ki ai+1 ) BNii a = 0 wi j(2) (6) 0 wi+1 j(2) i 1 (7) i i01

n

n

i i02

BNLL a = CN11 a = 0 [3], wi = ("i 0 "i+1 )=("i + "i+1 ), wi+1 = 1 0 iL01 ("i+1 0 "i+2 )=("i+1 + "i+2 ); wi01 ; = ("i 0 "i01 )=("i + "i01 ), and wi02 = 1 0 i2 ("i01 0 "i02 )=("i01 + "i02 ). Local reflections at interfaces in the vicinity of the ith layer where

Fig. 1. Four layers dielectric sphere.

the number of spherical layers. Explicit expressions for the scattering fs fs fs DGF coefficients Afs M;N , BM;N , CM;N and DM;N are reported in [3]. The proposed model has been formulated by deriving asymptotic reflection and transmission coefficients to obtain the required scattering DGF coefficients, which are used to achieve the final asymptotic DGF, (fs) , where the subscript a denotes an asymptotic expression. Non es a magnetic materials have been considered, that is, f = o . However, the presented procedure can be extended to model magnetic materials with no difficulty.

G

A. Asymptotic Expansion Coefficients The development of asymptotic expressions for the equivalent reflections and transmission coefficients between dielectric spherical layers is an essential step toward the accomplishment of an asymptotic DGF. Detailed expressions of these coefficients are reported in [3, Eq. (18)]. The principal form asymptotic Bessel and Hankel functions formulas given in Appendix A have been employed through the substitution of (A8) and (A9) in the reflection and transmission coefficients to attain the following:

RPHf a  0 RFHf a  0

(3a) (3b)

"f RPV f a  0 ""f +1 0 f +1 + "f "f RFV f a  0 ""f +1 0 + "f f +1 TPHf a  jnj (k(kf +1aaf) ) n f f (2) h TFHf a  n(2)(kf +1 af ) hn (pkf af ) V TP f a  0 2" "f +"f +1 "f f +1 p 2 " " TFVf a  0 " f +f +1 " f +1

f

hn (kf af ) jn (kf af ) jn (kf af ) hn(2) (kf af ) (2)

The asymptotic scattering DGF coefficients

B. Asymptotic Dyadic Green’s Functions The asymptotic DGF components can be accomplished by substituting the coefficients given in (4)–(7) in the scattering DGF expression given by (2). For example, Grr ja can be derived as i Grr ja = jk 4

1

0

n=0

(2n + 1) n (n + 1) rr(r;0 kr ) Pn (cos ) 2

i

(8)

ii 0 ii 0 where  (r; r0 )= BN a hn (ki r ) hn (ki r )+ CN a jn (ki r ) jn (ki r ), Pn (cos ) is the Legendre polynomial of degree n, and cos = cos  cos 0 + sin  sin 0 cos( 0 0 ). The double summation of (2)

has been reduced to a single summation using the addition theorem of Legendre polynomial [4]. Employing (A5)–(A6) and assuming (n + 1)  n for larger n, Grr ja can be written in a more convenient form as

j @ 2 1 (2n + 1)(r; r0 )Pn (cos ): Grr ja = 4k i @r@r 0 n=0

(9)

(3c) (3d) (3e)

jn (kf +1 af ) jn (kf af ) hn(2) (kf +1 af ) : hn(2) (kf af )

have been considered because reflections from distant boundaries and multiple reflections decay rapidly as n increases; hence, they have no contribution to the DGF coefficients in (4)–(7). Furthermore, the TE modes coefficients asymptote to zero for larger n, while the corresponding TM modes coefficients depend on the wave reflections at the dielectric interfaces.

The other asymptotic DGF components can be derived in the same way as the component given in (9), and then incorporated into the following unified expression:

(3f)

Giies a = 40kj i rr0

(3g)

2

(3h)

fs Afs M;N , BM;N , a a

fs fs CM;N can be determined by substituting the coeffi, and DM;N a a

cients of (3) in the accordant expressions given in Appendix B. When

1 n=0

(2n + 1)Pn (cos )

n (ki ai ) n (ki ai+1 ) wi j(2) + wi+1 j(2) hn (ki ai ) hn (ki ai+1 ) (2) (2) 0 2 hn (ki r) hn ki r (2) (2) + wi01 hjnn ((kki iaai0i011)) + wi02 hjnn ((kki iaai0i022))

2 jn (ki r) jn ki r0 :

(10)

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1005

With the aid of (A1)–(A2), (10) may be expressed as

G

ii es

a

= 40kj rr

1 0

=0

i

2

wi

(2n + 1)P (cos ) n

n

(

ai

jn ki di

r

2 h(2)

ki r

n

+w

2

i0

) + w +1 a r+1 j (k d +1 )

+

0

ai02

i

i

wi01

n

ai01 r

(2) (ki di 2 )

hn

r

i i

(2) (ki di 1 )

hn

0

jn ki r

0

0

(11)

where d` = a2` =r . By invoking the addition theorem of the spherical Hankel function [4], (11) reduces to

G

ii es

1 rr = 4k 2

+1

i

0

a

i

=i 2

`

w`

a` e0jk

(12)

R`

r

0

R

where R` = r02 + d`2 0 2r0 d` cos . Therefore (1) can be expressed in a computationally efficient format as

G (r; r ) = I e4R + rr G + ii e

0jk

0

R

0

ii

(

Ges r; r0

)0 G

ii es

a

Fig. 2. Convergence of the input impedance of a spherical spiral using a PEC spherical core radius of 1 when the antenna is placed at the dielectric interface between a spherical substrate and free space, where i refers to the index of the antenna layer.

(13)

where G

1 = 4k 2 i

ss

and Ges

a

0jk

e

R

R

+

+1

i

=i 2

`

0

0jk

w`

a` e r

R`

R

(14)

is given by (11). III. RESULTS

To illustrate the significance of the presented procedure a moment method model has been developed for the analysis of spherically conformal antennas. The geometry of Fig. 1 has been modeled assuming the innermost layer to be a perfectly conducting, PEC, spherical core, the third layer is a dielectric substrate and the second layer is a spherical superstrate with the outermost layer represents free space. As an example, a conformal Archimedean spiral antenna printed on a grounded dielectric spherical substrate has been considered [17]. The spiral arm is defined by  = 0 + ', where 0 is the feeding segment length,  is the spiral constant and ' is the winding angle. The thin wire approximation has been adopted, piecewise sinusoidal current pulses have been employed and a delta gap voltage source has been used for excitation. The input impedance has been calculated in two cases: first, assuming the antenna is placed in the third layer that has a relative permittivity of "r3 = 2 and a thickness of 0.75 cm, and then when the spiral is located at the second layer. In both cases, the antenna has been positioned at the interface between the two layers. The permittivities of the first and second layers have been assumed to be "2 = "1 = "0 and the radius of the PEC spherical core chosen as 5 cm, that is, 1o at an operating frequency of 6 GHz. The spiral has been modeled using 0 = 0:163 cm,  = 0:0623 cm=rad, a maximum winding angle of 12.4 rad and a wire radius of 0.02 cm [17]. Fig. 2 shows the convergence of the input impedance when the infinite summation of (2) is truncated using asymptotic extraction approach compared to the case when the summation is implemented directly, that is, with no use of asymptotic extraction. It is evident from these results that the required number of terms to truncate the series has been reduced from over 100 to approximately 25 when the proposed

Fig. 3. Convergence of the input impedance of a spherical spiral positioned at dielectric interface using a PEC spherical core radius of 3 .

model is employed. As expected, the convergent input impedance is the same whether the antenna is positioned in the second or the third layer as long as it is located at the interface, with a slight difference in the imaginary part owing to the numerical computations of Hankel functions using different arguments. The required number of terms increases as the size of the sphere is increased, hence a structure with a larger sphere radius has been investigated. Fig. 3 presents the impedance convergence when a PEC spherical core of radius 3o is used, where, again, it can be seen that a convergent solution has been achieved using approximately 75 terms when asymptotic extraction is employed, compared with more than 300 terms when the summation is implemented directly. It should be mentioned that the impedance converges to 203 0 j 2 compared to 208 0 j 4

for an identical planar spiral when the antenna is located at the third layer. The convergence of the input impedance at the presence of a dielectric superstrate is then studied using "r3 = 2:2, "r2 = 4:2, a3 = 12 cm, a2 = 12:8 cm and a1 = 13:4 cm at a frequency of 5 GHz. The spiral parameters have been chosen as those reported in [18] for an equivalent spiral in an identical planar media, that is, 0 = 0:253 cm,

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Fig. 4. Convergence of the input impedance of a spherical spiral located at an interface of a dielectric substrate and a superstrate using a PEC spherical core radius of 2 .

= 0:097 cm=rad and a maximum winding angle of 12.4 rad. In this example the antenna has been assumed to be placed first in the second layer, i.e., the spherical superstrate, and then in the third layer, i.e., the spherical substrate, at the interface between those layers. The input impedance convergence is presented in Fig. 4 where a significant acceleration in the convergence is again accomplished when the asymptotic extraction is employed. The capability of the model to analyze structures with electrically thin spherical layers has been examined by modeling a probe-fed circular patch antenna that is printed on a 0.2 cm spherical substrate and covered by a superstrate with a similar thickness. The PEC spherical core radius and the substrate permittivity have been chosen as a3 = 10 cm and "r3 = 3, respectively. The patch antenna has been modeled using an arc radius of 1.88 cm, and it has been fed using a probe that is located at an arc distance of 0.94 cm from its center. The patch and the probe have been placed in the third layer with the patch antenna positioned at the dielectric boundary between the second and the third layers. Further, the structure has been analyzed using different superstrate permittivities: the first is "r2 = 2 and the second is "r2 = 3, which results in two resonance frequencies of 2.54 GHz and 2.49 GHz, respectively. Employing similar substrate and superstrate permittivities facilitates evaluation of the effectiveness of the model when the patch is close to, but not located at, a dielectric interface. Fig. 5 illustrates the convergence of the input impedance at the accordant resonance frequencies, where it can be observed that a substantially reduced number of expansion terms are sufficient to accomplish convergence when asymptotic extraction is employed. It can be seen from these results that the convergence of the direct summation improved noticeably when the antenna is shifted from the dielectric boundary. A two-element array has been analysed using the aforementioned microstrip antenna and sphere parameters, where the mutual coupling has been evaluated using "r2 = 2 at a frequency of 2.54 GHz, that is, the array is located at an interface between thin layers. The convergence has been investigated in two configurations: first using an arc distance of d = 2b between the centers of the patches, and then using a larger arc distance of 6b, where b is the patch radius. The mutual impedance has been obtained from the solution of the MoM block-Toeplitz impedance matrix, where 200 terms have been added for the self-term matrix entries to ensure convergence, while the added number of terms for the mutual coupling entries has been varied as shown in Fig. 6. Mutual coupling results show that

Fig. 5. Convergence of the input impedance of a conformal probe-fed patch antenna using " = 3 and different superstrate permittivities at the corresponding resonance frequencies.



Fig. 6. Convergence of the mutual impedance between two identical circular patches with a radius of b and an arc separation distance of d between their centers.

asymptotic extraction expedites convergence considerably for small as well as large angular separation distances. The numerical results illustrate the advantages of employing the introduced solution in the analysis of geometries that consist of electrically thick as well as thin spherical layers. Compared to other acceleration techniques, asymptotic extraction avoids the computation overheads that are associated with traditional approaches such as Watson and Shanks transformations [5]–[7]. The algorithm enhances the computation efficiency significantly for arbitrarily angular distances between the source and field points, which is different from a previously reported methodology that is suitable for larger separation distances [8]. Furthermore, in contrast to the solutions introduced in [9], [10], there is no restriction on the sphere size, number of layers, or antenna orientation. However, the accomplished improvement and the effectiveness of the model depend on the proximity of the antenna to a dielectric interface. This is owing to the rapid decline of the extracted quasi-static images’ contributions as the antenna is moved away from a dielectric boundary, where the direct summation of the infinite series converges using a considerably reduced number of spherical eigenmodes.

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IV. CONCLUSION An asymptotic extraction procedure has been established to accelerate the convergence of the infinite series of DGF in a multilayered spherical media. The superiority of the introduced model has been demonstrated in the MoM analysis of conformal antennas located at a spherical surface, where various configurations have been investigated. Early truncation of the series reduces the computation time considerably and eliminates the numerical limitations associated with a large-order Hankel and Bessel functions. In this study, attention is given to the problem of source and field points located in the same layer. The presented procedure can be followed to formulate asymptotic Green’s functions when the source and observation points are in different layers. The computations speed can be enhanced further by adopting the closed-form expressions reported in [11], [12] in conjunction with the proposed procedure in the analysis of a number of spherical antennas geometries. APPENDIX A

!1

When n , the spherical Bessel and Hankel functions can be approximated using the principal asymptotic expressions [19]

jn (kr)



hn(2) (kr)

j

ekr n+(1=2) 2n + 1 ekr 0n0(1=2)

1

2kr (2n + 1) 2

kr (2n + 1)

which lead to

jn (kf af +1 ) jn (kf af ) (2) hn (kf af ) hn(2) (kf af +1 )

2n + 1

af +1 n af af +1 n+1 af

 

Since n



1,

(

(A2)

(A4)

(

(

)

+ 1)

(

(

+ 1)

(A5) )

(A6)

)

(A7)

 0 nh(2) n kr : (

)

(A8)

 njn kr :

(A9)

(A7) can be expressed as

d (rjn (kr)) dr

(

)

APPENDIX B The following recurrence formulas can be employed to compute the scattering DGF coefficients [3] V;H fs f +1)s = AM;N + RFf C fs A(M;N (sf +1) (B1) V;H V;H TFf TFf M;N V;H V;H fs (f +1)s = BM;N + RFf Dfs + RFf  s BM;N (B2) M;N V;H V;H V;H f TFf TFf TFf fs V;H (f +1)s = R Afs + CM;N CM;N (B3) V;H M;N V;H

0

T T V;H fs (f +1)s = RP f B fs + DM;N DM;N M;N V;H TP f TPV;H f

+

fs

TPV;H f

:

BN11 a =

0

a 2n+1 a3 2n+1 w1 + w2 2 + w3 a1 a1 a3 2n+1 jn (k1 a1 ) +w1 w2 w3 (B5) a2 hn (k1 a1 )

where the first three terms correspond to local reflections at the dielectric interfaces and the last term accounts for multiple reflections, which is generally smaller than the other terms hence it can be neglected. Further, as ai+2 < ai for any spherical geometry the third term of (B5) . Therefore, subdeclines rapidly by a factor (ai+2 =ai )2n+1 as n stituting (A1)–(A2) into the first and second terms of (B5) gives

!1

0

j (k a ) BN11 a = w1 n 1 1 hn (k1 a1 )

0 w2 hjnn kk11aa22 : (

)

(

)

(B6)

Following a similar procedure, asymptotic representations of all the coefficients can be accomplished.

REFERENCES

(A3)

 nr jn kr  0 n r hn(2) kr  n jn kr

With the aid of (3) explicit asymptotic expressions can be derived for the aforementioned coefficients. For instance, in the four layers geometry shown in Fig. 1, it can be proved that

(A1)

and

djn (kr) dr dhn(2) (kr) dr d (rjn (kr)) dr d rhn(2) (kr) dr

1007

(B4)

[1] C. T. Tai, Dyadic Green’s Functions in Electromagnetics Theory. Scranton, PA: Intext Educational, 1971. [2] W. C. Chew, Waves and Fields in Inhomogeneous Media. New York: Van Nostrand, 1990. [3] L. W. Li, P. S. Kooi, M. S. Leong, and T. S. Yeo, “Electromagnetic dyadic Green’s function in spherically multilayered media,” IEEE Trans. Microw. Theory Tech., vol. 42, pp. 2302–2310, Dec. 1994. [4] R. F. Harrington, Time Harmonic Electromagnetic Fields. New York: Mc Graw-Hill, 1961. [5] G. M. Watson, “The diffraction of electric waves by the earth,” Proc. Royal Society, vol. 95, pp. 83–99, 1919. [6] D. Shanks, “Non-linear transformation of divergent and slowly convergent sequences,” J. Math. Phy., vol. 34, pp. 1–42, 1955. [7] F. M. Tesche, A. R. Neureuther, and R. E. Stovall, “The analysis of monopole antennas located on a spherical vehicle: Part 2, numerical and experimental results,” IEEE Trans. Electromagn. Compat., vol. 18, pp. 8–15, Feb. 1976. [8] V. I. Okhmatovski and A. C. Cangellaris, “Efficient calculation of the electromagnetic dyadic Green’s function in spherical layered media,” IEEE Trans. Antennas Propag, vol. 51, pp. 3209–3220, 2003. [9] B. D. Milovanovic, “Numerical analysis of radial thin-wire antenna in presence of conducting sphere,” Elec. Lett., vol. 16, no. 15, pp. 611–612, Jul. 1980. [10] L. W. Li, T. Fei, Q. Wu, and T. S. Yeo, “Convergence acceleration for calculating radiated fields by a vertical electric dipole in the presence of a large sphere,” in Proc. IEEE Antennas Propag. Soc. Int. Symp., Jul. 2005, vol. 2B, pp. 117–120. [11] K. W. Leung, “General solution of a monopole loaded by a dielectric hemisphere for efficient computation,” IEEE Trans. Antennas Propag., vol. 48, pp. 1267–1268, Aug. 2000. [12] K. W. Leung, “Analysis of the zonal and rectangular slots on a conducting spherical cavity’,” IEEE Trans. Antennas Propag., vol. 49, pp. 1739–1745, Dec. 2001. [13] D. R. Jackson and N. G. Alexopoulos, “An asymptotic extraction technique for evaluating Sommerfeld-type integrals,” IEEE Trans. Antennas Propag., vol. AP-34, pp. 1467–1470, Dec. 1986. [14] M. J. Tsai, F. D. Flaviis, O. Fordham, and N. G. Alexopoulos, “Modeling planar arbitrarily shaped microstrip elements in multilayered media,” IEEE Trans. Microw. Theory Tech., vol. 45, pp. 330–337, Mar. 1997. [15] C. Chen, W. E. McKinzie, III, and N. G. Alexopoulos, “Stripline-fed arbitrarily shaped printed-aperture antennas,” IEEE Trans. Antennas Propag., vol. 45, pp. 1186–1198, Jul. 1997. [16] J. Sun, C. F. Wang, L. W. Li, and M. S. Leong, “Further improvement for fast computation of mixed potential Green’s functions for cylindrically stratified media,” IEEE Trans. Antennas Propag., vol. 52, pp. 3026–3036, Nov. 2004.

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[17] S. K. Khamas, “Moment method analysis of an Archimedean spiral printed on a layered dielectric sphere,” IEEE Trans. Antennas Propag., vol. 56, pp. 345–352, Feb. 2008. [18] S. K. Khamas, P. L. Starke, and G. G. Cook, “Design of a printed spiral antenna with a dielectric superstrate using an efficient curved segments moment method with optimization using marginal distributions,” in Proc. Inst. Elect. Eng. Microw, Antennas Propag., Aug. 2004, vol. 151, pp. 315–320. [19] M. Abramowitz and I. A. Stegun, “Handbook of mathematical functions with formulas,” in Graphs, and Mathematical Tables. Washington, DC: Government Printing Office, 1964.

Miniature Internal Penta-Band Monopole Antenna for Mobile Phones Chia-Ling Liu, Yi-Fang Lin, Chia-Ming Liang, Shan-Cheng Pan, and Hua-Ming Chen

Abstract—A compact T-slit monopole antenna with slotted ground plane in the mobile phone for penta-band operation is proposed. In this configuration, the antenna comprises a T-slit monopole printed on the top ungrounded portion of an FR4 substrate of small size of 47 5 4 mm and a slotted ground plane etched on the back side of the substrate of size of 47 10 mm . In addition, an inverted-L copper strip is soldered to the end edge of the monopole for extending the electrical length of the antenna for GSM band; that is, the proposed antenna occupies a small volume of 47 10 5 mm inside the mobile phone and is suitable to operate as an internal antenna. By controlling the related parameters, the proposed antenna can resonates at different operating bands to cover GSM850/900 and DCS/PCS/UMTS operations independently. Index Terms—Mobile phone, penta-band, T-slit monopole antenna.

I. INTRODUCTION Recently with the rapid development of cellular communication, various types of antennas for mobile phones have been extensively presented and the trend of the mobile phones is getting smaller and slimmer because of the consumer’s needs and the multiplicity of functions. Conventional internal antennas for the mobile phones applications are generally in forms of monopole antennas because it can provides a wide impedance bandwidth [1]–[6]. These monopole antennas generally use two separate resonant paths of different lengths operated at their quarter-wavelength modes to cover the mobile phone’s operating bands. In this communication, we present a promising compact penta-band monopole antenna with an occupied volume of 10 2 47 2 5 mm3 in the mobile phone to operate GSM (824–894/890–960 MHz), Manuscript received March 11, 2009; revised August 30, 2009. First published December 28, 2009; current version published March 03, 2010. This work was supported by the National Science Council of Taiwan under Contract NSC 97-2221-E-151-010. C.-L. Liu, Y.-F. Lin, C.-M. Liang, and H.-M. Chen are with the Institute of Photonics and Communications, National Kaohsiung University of Applied Sciences, Kaohsiung 807, Taiwan (e-mail: [email protected]; hmchen@cc. kuas.edu.tw). S.-C. Pan is with the Department of Computer and Communication, Shu-Te University, Yen Chau, Kaohsiung 824, Taiwan. Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2009.2039309

Fig. 1. Configuration of the T-shaped slit monopole antenna with slotted ground plane for mobile phone application.

DCS (1710–1880 MHz), PCS (1850–1990 MHz), and UMTS (1920–2170 MHz) bands. The proposed antenna is easily printed on a thin substrate at low cost and fabricated by a bending metal plate shown a very low profile of 5 mm. The low profile of the antenna allows it very promising to be embedded inside the mobile phone as an internal antenna. In order to determine the performance of varying design parameters on bandwidth and resonance frequency, parametric study is carried out using simulation software HFSS and experimental results. Detailed design considerations of the proposed antenna are described in this article.

II. ANTENNA CONFIGURATION Fig. 1 shows the configuration of the T-slit monopole antenna with slotted ground plane for mobile phone application. A 0.8-mm thick FR4 substrate of relative permittivity 4.4 and of size 110 2 50 mm2 is used as the system circuit board with a ground plane of the same size. The dimensions of the system circuit board and ground plane considered here are practical for general mobile phones. In Fig. 1(a), the antenna comprises a T-slit monopole printed on the top ungrounded portion of an FR4 substrate of small size 47 2 5:4 mm2 and a slotted ground plane etched on the back side of the substrate of size 47 2 10 mm2 . In this study, an inverted-L copper strip is soldered to the end edge of the monopole for extending the electrical length of the antenna for GSM band. For the inverted-L copper strip, it comprises a horizontal section of size 37 2 2 mm2 and a vertical section of size 5 2 2 mm2 . In addition, a T-slit is etched on the monopole radiator to realize two major current paths and achieving an additional resonant mode. A 50-

microstrip feed line printed on the top side of the system circuit board has a length of 35 mm and a width of 1.5 mm. Fig. 1(b) shows the dimensions of the pattern of the ground plane on the back side of the substrate. The pattern comprises a narrow straight slot and a narrow slit. By varying the length of the slit, the various coupling energy between the feed line and the pattern in the ground plane can results in another excited resonant mode. Detailed dimensions of the antenna are given in Fig. 1.

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Fig. 2. (a) Geometry of the basic design monopole antenna (antenna 1). (b) Geometry of the T-slit monopole antenna (antenna 2). (c) The proposed antenna.

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Fig. 4. Simulated return loss for different antennas as shown in Fig. 3.

Fig. 5. Prototype of the proposed T-slit monopole antenna; (a) top view, (b) bottom view.

Fig. 3. Simulated current distributions at (a) 892 MHz (mode 1), (b) 2045 MHz (mode 2), (c) 1920 MHz (mode 3), (d) 1795 MHz (mode 4).

In this case, with the use of the proposed structure, it is easy to make it possible for generating desired operating bands. In order to check this operating mechanism, Fig. 2 shows the design process of the proposed antenna and the corresponding resonant modes are shown in Figs. 3 and 4. Fig. 2(a) is the basic design monopole antenna designated as antenna 1. In Fig. 3(a), the current flow from point A through points B, C, and D, then to point E, the total resonant length is about 84 mm which is about 0:250 at 892 MHz and the related resonant mode (mode 1) is shown in Fig. 4. The antenna 1 can also generate a quarter-wavelength mode (mode 2 as shown in Fig. 4) at 2045 MHz, which resonant length is about 39 mm [current flow from point C to point E shown as Fig. 3(b)]. It is also observed that the mode 2 is significantly affected when increasing the resonant length Lh from 35 to 39 mm with a step of 2 mm, which corresponding figure was not included in the text for space limitations. By embedding a T-slit in the basic monopole antenna (designated as antenna 2) leads to an additional excited mode (mode 3), are shown in Figs. 2(b) and 4, respectively. The current flow of mode 3 from points B, C to F in Fig. 3(c) has a length of about 40 mm, close to 0:250 at 1920 MHz. It can also be observed that when the length Lr is increased, as expected, the mode 3 is shifted to the lower frequency. However, the T-slit monopole antenna cannot cover DCS/PCS/UMTS operation for the antenna’s upper band. To achieve the resonant mode of the antenna for DCS1800 band, the energy coupling between the

ground plane and the feed line is used. In Fig. 2(c), a narrow straight slot and a narrow slit are etched in the ground plane for generating a quarter-wavelength mode (mode 4) as shown in [7]. In Fig. 3(d), the current flow length from point P through point Q, and to point H is about 38 mm, corresponding to about 0:230 at 1795 MHz. As expected, by adjusting the length of the slit Ls , the resonant frequency increases as Ls decreases. Finally, the dimensions of the proposed antenna can be optimized to get sufficient bandwidth to cover GSM and DCS/PCS/UMTS bands, which results in a penta-band operation for the proposed antenna with a very simple coupling mechanism. III. EXPERIMENTAL RESULTS AND DISCUSSIONS Based on the aforementioned optimized parameters, an antenna prototype was fabricated and measured as shown in Fig. 5. The optimal design parameters are given in Fig. 1. Fig. 6 shows the measured and simulated return loss for the fabricated antenna. Generally, a 6-dB return loss is acceptable for the mobile phone applications. In this study, the lower band is tuned at about 892 MHz and has a 3:1 VSWR (6-dB return loss) bandwidth of 125 MHz (840–965 MHz), which covers about GSM850/900 operation. For the upper band, a much wider bandwidth is obtained, which reaches 470 MHz (1705–2175 MHz) and satisfies the operating bandwidth of the DCS/PCS/UMTS band (1710–2170 MHz). Good agreement between the measured data and simulated results obtained by using High Frequency Structure Simulator (Ansoft HFSS) is seen. Fig. 7 shows the simulated return loss results for the length of ground plane varied from 70 to 110 mm. In mode 1, the impedance matching over the mode is degraded with a decreased ground plane length. In this study, the length is chosen as

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Fig. 6. Measured and simulated return loss of the proposed antenna.

Fig. 7. Simulated return loss for different ground plane length; other parameters are the same as studied in Fig. 6.

110 mm for good matching and is also suitable for the system ground plane in the mobile phone applications. The gain radiation patterns of the proposed antenna have been measured in an anechoic chamber at all the frequencies. Fig. 8(a) shows the measured radiation patterns at 892 MHz, center frequency of the GSM band. It is observed that the radiation patterns are similar to a conventional quarter-wavelength monopole antenna in the x-z and x-y planes, and near omnidirectional radiation in the x-y plane is obtained. Results at other frequencies of the GSM band are also measured, and very similar patterns as plotted here are obtained, indicating that stable radiation patterns are achieved. For the radiation patterns for frequencies over the upper band, Fig. 8(b)–(d) plot the measured gain radiation patterns at 1795, 1920, and 2045 MHz, which are center frequencies of the DCS, PCS, and UMTS bands, respectively. Again, similar radiation patterns are seen, that is, stable patterns are also obtained over the upper band. These radiation patterns show no special distinctions compared with those of the monopole antenna for DCS, PCS, or UMTS operation [1]. The measured and simulated antenna peak gain and radiation efficiency are presented in Fig. 9. The antenna peak gain and efficiency were measured in the anechoic 3D chamber. For the low band (GSM band) shown in Fig. 9(a), the measured peak gain is varied from about 2.0 to 2.9 dBi, and the measured radiation efficiency is larger than 60%. For the high band (DCS/PCS/UMTS bands) shown in Fig. 9(b), the measured peak gain is varied from about 01.7 to 2.8 dBi, and the measured radiation efficiency is varied from 30 to 80%. Good agreement between the measured data and simulated results are seen in the Fig. 9.

Fig. 8. Measured gain radiation pattern of the proposed antenna in x-z and x-y planes. (a) mode 1 (892 MHz), (b) mode 4 (1795 MHz), (c) mode 3 (1920 MHz), (d) mode 2 (2045 MHz).

It should be note that, in Fig. 9(b), there is a gain and efficiency drop at near 1920 MHz (mode 3). It is because of the opposite current flow (points B-C-F) in Fig. 3(c) that cancels the radiation power in the far-field. However, the measured peak gain in the drop band is about 01.7 dBi, which is acceptable for mobile phone applications. IV. CONCLUSION An internal monopole antenna with a volume of 47 2 10 2 5 mm3 for penta-band operation in the mobile phone has been proposed and fabricated. The excited multi modes of the proposed antenna are owing to the use of the promising resonant coupling mechanism with various

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Printed Single-Strip Monopole Using a Chip Inductor for Penta-Band WWAN Operation in the Mobile Phone Kin-Lu Wong and Shu-Chuan Chen

Abstract—A single-strip monopole capable of generating two wide operating bands at about 900 and 1900 MHz covering GSM850/900/1800/1900/ UMTS penta-band WWAN operation in the mobile phone is presented. The monopole has a simple structure of an inverted-L shape to be printed on the no-ground region of the system circuit board of the mobile phone. By simply embedding a chip inductor at the proper position in the strip monopole, the first two resonant modes of the monopole can have a frequency ratio of about 1 to 2 (instead of 1 to 3 for the traditional monopole) to respectively cover the desired wide 900 and 1900 MHz bands. In addition, the total strip length can be less than the required 0.25 wavelength (about 0.17 wavelength in this design) for the fundamental resonant mode excitation of the proposed monopole; this behavior is owing to the embedded chip inductor compensating for the increased capacitance seen at the feeding point with the decreasing strip’s resonant length. The SAR of the proposed monopole placed at the bottom position of the mobile phone is found to meet the SAR limit for practical applications. Index Terms—Handset antennas, internal mobile phone antennas, mobile antennas, multiband antennas, WWAN antennas.

I. INTRODUCTION

Fig. 9. Measured and simulated antenna gain and radiation efficiency of the proposed antenna. (a) The lower band for GSM operation. (b) The upper band for DCS/PCS/UMTS operation.

slots. The different coupling effects of the slotted structure make it controllable for the excitation of a quarter-wavelength resonant modes at about 1795, 1920, and 2045 MHz, respectively. The two excited resonant bands cover GSM and DCS/PCS/UMTS operations for the proposed antenna in this study.

REFERENCES [1] K. L. Wong, G. Y. Lee, and T. W. Chiou, “A low-profile planar monopole antenna for multiband operation of mobile handsets,” IEEE Trans. Antennas Propag., vol. 51, pp. 121–125, 2003. [2] Y. F. Lin, H. M. Chen, and K. L. Wong, “Parametric study of dual-band operation in a microstrip-fed uniplanar monopole antenna,” Proc. Inst. Elect. Eng. Microw. Antennas Propag., vol. 150, no. 6, pp. 411–414, 2003. [3] Z. N. Chen, M. Y. W. Chia, and M. J. Ammann, “Optimization and comparison of broadband monopoles,” Proc. Inst. Elect. Eng. Microw. Antennas Propag., vol. 150, no. 6, pp. 429–435, 2003. [4] C. H. Wu and K. L. Wong, “Printed compact S-shaped monopole antenna with a perpendicular feed for penta-band mobile phone application,” Microw. Opt. Technol. Lett., vol. 49, pp. 3172–3177, 2007. [5] K. L. Wong and P. Y. Lai, “Wideband integrated monopole slot antenna for WLAN/WiMAX operation in the mobile phone,” Microw. Opt. Technol. Lett., vol. 50, no. 8, pp. 2000–2005, 2008. [6] R. A. Bhatti and S. O. Park, “Octa-band internal monopole antenna for mobile phone applications,” Electron. Lett., vol. 44, no. 25, pp. 1447–1448, 2008. [7] S. I. Latif, L. Shafai, and S. K. Sharma, “Bandwidth enhancement and size reduction of microstrip slot antennas,” IEEE Trans. Antennas and Propag., vol. 53, pp. 994–1003, 2005.

It has been known that, by embedding a chip inductor, the monopole can have decreased resonant length for achieving its fundamental or lowest resonant mode excitation [1]–[3]. This behavior is owing to the additional inductance contributed by the embedded chip inductor to compensate for the increased capacitance resulting from the decreased resonant length of the antenna. Related works on the shortened dipole and monopole with this kind of inductive loading have also been available in the open literature. They include the shortened dipole with the inductive element placed in series with each radiating arm of the dipole [4] and the multi-branch monopole with one of its branches end-loaded with a dense meandered section as the inductive end section [5]. It has also been shown that the chip inductor should be embedded near the feeding point of the monopole [1] where the excited surface currents are strong for the fundamental resonant mode. In this case, the required resonant length is usually less than 0.2 wavelength for generating the fundamental resonant mode of the monopole. Also, it is found that the required resonant length of the second resonant mode of the monopole will be decreased as well such that the frequency ratio of the first two resonant modes maintains about 1 to 3, similar to that of the traditional monopole or dipole [6]–[8]. Hence, in order to obtain two wide bands at about 900 and 1900 MHz to cover GSM850/900/1800/1900/UMTS wireless wide area network (WWAN) operation, two separate radiating strips are usually required for the internal WWAN antenna for mobile phone applications [1], [9]. In this communication, we propose a printed single-strip monopole using a chip inductor for penta-band WWAN operation in the mobile phone. The embedded chip inductor can result in a decreased resonant length for the fundamental resonant mode excitation of the monopole. It can also generate an additional resonant mode such that the first two Manuscript received April 14, 2009; revised May 28, 2009. First published December 31, 2009; current version published March 03, 2010. The authors are with the Department of Electrical Engineering, National Sun Yat-sen University, Kaohsiung 80424, Taiwan (e-mail: [email protected]. edu.tw; [email protected]) Color versions of one or more of the figures in this communication are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2009.2039324

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Fig. 2. Measured and simulated (HFSS) return loss for the proposed antenna.

Fig. 1. Geometry of the proposed single-strip monopole antenna embedded with a chip inductor for penta-band WWAN operation in the mobile phone.

resonant modes of the monopole can be adjusted to have a frequency ratio of about 1 to 2 to respectively cover the desired wide 900 and 1900 MHz bands for GSM850/900 (824–894/880–960 MHz) and GSM1800/1900/UMTS (1710–1880/ 1850–1990/1920–2170 MHz) operation. This behavior of dual-band operation is similar to that of using an LC resonator applied to a PIFA [10] and that of using an inductive element loaded in each radiating arm of a dipole [4]. In this study, the use of a chip inductor applied to a single-strip monopole leads to a simple uniplanar structure with a small planar printed area 2 here) in the mobile phone. The printed area is much (about 250 smaller than that of the traditional two-strip monopoles for penta-band WWAN operation in the mobile phone that have been reported (about 2 2 in [1] and 420 in [9]). In addition, the uniplanar 350 structure of the antenna is very promising for thin-profile or slim mobile phone applications [11]–[17], which is becoming attractive for the mobile users. Also, when arranging the antenna to be at the bottom position of the mobile phone, the simulated SAR (specific absorption rate) [18]–[22] is found to meet the limit of 1.6 W/kg for 1-g head tissue and 2.0 W/kg for 10-g head tissue.

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II. PROPOSED ANTENNA Fig. 1 shows the geometry of the proposed antenna, which is mainly an inverted-L strip monopole embedded with a chip inductor of L 3 . The chip inductor of 0.5 2 0.5 2 1.0 separates the strip into two sections: a front section and an end section. The two sections 2 ) of the system cirare printed on the no-ground portion (15 2 60 2 cuit board, which uses a 0.8-mm thick FR4 substrate of 115 2 60 in this study. On the back side of the circuit board, a system ground 2 is printed. The dimensions of the system cirplane of 100 2 60 cuit board and ground plane are reasonable for practical mobile phones, especially for smartphones. The total length of the antenna from the feeding point A, through the chip inductor, to the open end has a length of about 57 mm only (about 0.17 wavelength at 900 MHz). Although the strip length is less than 0.25 wavelength at 900 MHz, a wideband resonant mode covering the GSM850/900 bands can be generated. This behavior is owing to the

mm mm

27 nH

mm

=

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additional inductance contributed by the chip inductor to compensate for the increased capacitance with decreased resonant strip length. On the other hand, at higher frequencies around 1900 MHz, the chip inductor provides a large inductive reactance such that the front section of length about 38 mm (about 0.24 wavelength at 1900 MHz) sees a virtual open circuit at the chip-inductor position. This condition leads to the excitation of a new quarter-wavelength mode at about 1900 MHz, which shows a wide bandwidth to cover GSM1800/1900/UMTS bands. Note that the wide bandwidths of the first two resonant modes of the antenna at about 900 and 1900 MHz are mainly resulted from the selection of widened widths (7.0 and 4.5 mm here) of the front and end sections. It has been shown that by increasing the width of the strip monopole, the antenna’s impedance matching can be improved, and hence the obtained bandwidth can be widened [23]. 2 Note that the antenna occupies a printed area of about 250 only. Owing to the small printed area, there is a large portion unused and available in the no-ground portion of the circuit board for accommodating the associated elements such as the speaker [24], [25], the lens of the digital camera [26], and so on. For testing the antenna in the experiment, a 50- microstrip feedline printed on the front side of the circuit board and connected through a via-hole to a 50- SMA connector on the back side of the circuit board is used. The obtained results are presented in Section III.

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III. RESULTS AND DISCUSSION Fig. 2 shows the measured and simulated return loss for the proposed antenna. Good agreement between the measured data and the simulated results obtained from Ansoft HFSS [27] is obtained. Two wideband resonant modes at about 900 and 1900 MHz are obtained. The first and second modes provide wide bandwidths of 155 MHz (810–965 MHz) and 515 MHz (1675–2190 MHz) to cover the GSM850/900 and GSM1800/1900/UMTS bands, respectively, for WWAN operation. Fig. 3 shows the simulated return loss for the proposed antenna, the case without the chip inductor (Ref 1), and the case with a chip inductor of 10 nH placed near the feeding point (Ref 2). Ref 1 can be consid2 ered as a simple monopole using a connecting strip of 1.0 2 1.0 to replace the chip inductor in the proposed antenna. It is seen that the lowest mode of Ref 1 occurs at about 1100 MHz, higher than those of the proposed antenna and Ref 2. In addition, the second mode of Ref 1 occurs at about 3300 MHz; that is, the first two modes have a frequency ratio of about 1 to 3. For Ref 2, with the chip inductor having than that in the proposed antenna, a smaller inductance of L the lowest mode can also occur at about 900 MHz, similar to the proposed antenna. Note that when the chip inductance has an inductance of 27 nH, the same as that in the proposed antenna, the lowest mode can be shifted to lower frequencies at about 700 MHz. Also, the second mode of Ref 2 occurs at about 2800 MHz, about three times that of the lowest mode. That is, both the first two modes of Ref 1 and Ref 2 have

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Fig. 5. Simulated (HFSS) return loss of the proposed antenna as a function of the end-portion length d. Fig. 3. Comparison of the simulated (HFSS) return loss for the proposed antenna, the case without the chip inductor (Ref 1), and the case with a chip inductor of 10 nH placed near the feeding point (Ref 2).

Fig. 4. Simulated (HFSS) return loss of the proposed antenna as a function of the inductance L of the chip inductor.

a frequency ratio of about 1 to 3, similar to the traditional monopole. Hence, an additional strip is required to generate a resonant mode at about 1900 MHz to cover the GSM1800/1900/UMTS bands. A longer strip is also required for Ref 1 to generate a resonant mode for operating in the 900 MHz band. Fig. 4 shows the simulated return loss as a function of the inductance L of the chip inductor. The increasing inductance causes the lowering of both the first two modes, and a frequency ratio of about 1 to 2 is maintained. It is noted that when a larger inductance is used, the obtained bandwidth of the lowest mode is decreased with the decreasing of its resonant frequency. Fig. 5 shows the simulated return loss as a function of the end-portion length d. In this case, when the length d varies, the total length of the antenna also varies. It is seen that, with the front-section length fixed as 23 mm, the second mode at about 1900 MHz is very slightly affected for various lengths of d from 10 to 20 mm. On the other hand, the lowest mode is shifted to lower frequencies when the length d increases. However, the impedance matching is also degraded when the lowest mode is shifted to lower frequencies. The results obtained in Figs. 4 and 5 indicate that the desired lower and upper modes of the antenna can be effectively controlled by adjusting the inductance L of the chip inductor and the length d of the end section. The radiation patterns of the proposed antenna are studied, and good agreement between the measured and simulated patterns is obtained. Dipole-like pattern is seen at 925 MHz for the 900 MHz band, while more directivity is obtained for the pattern at 1920 MHz

Fig. 6. Measured radiation efficiency and antenna gain for the proposed antenna. (a) GSM850/900 bands. (b) GSM1800/1900/UMTS bands.

for the 1900 MHz band. The obtained radiation patterns including the polarization information are similar to those of the two-strip monopole obtained in [1] and those of the conventional internal mobile phone antennas [28]. The measured radiation efficiency and antenna gain are shown in Fig. 6. The radiation efficiency varies from 55 to 88% for the GSM850/900 bands [see Fig. 7(a)] and from 65 to 96% for the GSM1800/1900/UMTS bands [see Fig. 7(b)]. The antenna gain is 0.4–1.8 dBi and 2.07–4.8 dBi for the GSM850/900 and GSM1800/1900/UMTS bands, respectively. The obtained radiation characteristics with efficiency better than 55% are sufficient (generally required to be larger than 50% over the bands) for practical mobile phone applications.

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REFERENCES

Fig. 7. SAR simulation model (SEMCAD) and the simulated SAR distributions on the phantom head for the proposed antenna at the bottom position of the mobile phone. The SAR distributions in the figure are normalized; 0 dB indicates the maximum SAR value.

The SAR results are analyzed with the aid of the SAR simulation model provided by SEMCAD [29]. The simulation model is shown in Fig. 7, and the simulated normalized SAR distributions on the phantom head for the proposed antenna at the bottom position of the mobile phone are shown in the figure. Note that it has been shown that this kind of internal mobile phone antenna (no ground plane below or behind the antenna’s radiating portion) is more suitable to be placed at the bottom position of the mobile phone to achieve decreased SAR values [1], [22]. In the simulation model, the system ground plane is spaced 5 mm from the phantom ear, and the mobile phone is oriented 60 to the vertical axis of the phantom head. The testing power is 24 dBm at 859 and 925 MHz, and 21 dBm at 1795, 1920 and 2045 MHz [1], [22]. From the results, the obtained SAR values for 1-g (10-g) head tissue are 1.25 (0.93), 1.08 (0.79), 0.68 (0.44), 0.60 (0.37) and 0.50 (0.31) W/kg at 859, 925, 1795, 1920, and 2045 MHz, respectively, which all meet the SAR limitation of 1.6 (2.0) W/kg [19]. Also note that there are two local SAR maxima observed at 1795, 1920 and 2045 MHz, which indicates that the antenna’s near-field radiation energy is more uniformly distributed; this behavior results in decreased SAR values. The obtained SAR results for the proposed antenna are also about the same as those of the two-strip monopole antenna obtained in [1]. IV. CONCLUSION A simple printed monopole using a chip inductor for achieving a frequency ratio of about 1 to 2 for its first two modes and moreover obtaining a small resonant length of about 0.17 wavelength for its first mode for WWAN operation in the mobile phone has been proposed. With a simple 2 structure, the antenna requires a small printed area of about 250 only. The antenna’s first two modes can be controlled to occur at about 900 and 1900 MHz to cover GSM850/900 and GSM1800/1900/UMTS bands. Good radiation characteristics over the five operating bands have been observed. The SAR of the antenna mounted at the bottom position of the mobile phone has been conducted. The obtained SAR values meet the requirements for practical applications.

mm

[1] T. W. Kang and K. L. Wong, “Chip-inductor-embedded small-size printed strip monopole for WWAN operation in the mobile phone,” Microw. Opt. Technol. Lett., vol. 51, pp. 966–971, Apr. 2009. [2] J. Thaysen and K. B. Jakobsen, “A size reduction technique for mobile phone PIFA antennas using lumped inductors,” Microw. J., vol. 48, pp. 114–126, Jul. 2005. [3] T. H. Chang and J. F. Kiang, “Meshed antenna reduction by embedding inductors,” in IEEE AP-S Int. Symp. and USNC/URSI Nat. Radio Sci. Meeting, Washington, DC, 2005, no. 78. [4] J. Carr, Antenna Toolkit, 2nd ed. Oxford, U.K.: Newnes, 2001. [5] V. Plicanic, B. K. Lau, A. Derneryd, and Z. Ying, “Actual diversity performance of a multiband diversity antenna with hand and head effects,” IEEE Trans. Antennas Propag., vol. 57, pp. 1547–1556, May 2009. [6] Y. W. Chi, K. L. Wong, and S. W. Su, “Broadband printed dipole antenna with a step-shaped feed gap for DTV signal reception,” IEEE Trans. Antennas Propag., vol. 55, pp. 3353–3356, Nov. 2007. [7] C. T. Lee, K. L. Wong, and Y. C. Lin, “Wideband monopole antenna for DTV/GSM operation in the mobile phone,” Microw. Opt. Technol. Lett., vol. 50, pp. 801–806, Mar. 2008. [8] W. L. Stutzman and G. A. Thiele, Antenna Theory and Design. New York: Wiley, 2003, ch. 5. [9] K. L. Wong and T. W. Kang, “GSM850/900/1800/1900/UMTS printed monopole antenna for mobile phone application,” Microw. Opt. Technol. Lett., vol. 50, pp. 3192–3198, Dec. 2008. [10] G. K. H. Lui and R. D. Murch, “Compact dual-frequency PIFA designs using LC resonators,” IEEE Trans. Antennas Propag., vol. 49, pp. 1016–1019, Jul. 2001. [11] K. L. Wong, Y. C. Lin, and T. C. Tseng, “Thin internal GSM/DCS patch antenna for a portable mobile terminal,” IEEE Trans. Antennas Propag., vol. 54, pp. 238–242, Jan. 2006. [12] K. L. Wong, Y. C. Lin, and B. Chen, “Internal patch antenna with a thin air-layer substrate for GSM/DCS operation in a PDA phone,” IEEE Trans. Antennas Propag., vol. 55, pp. 1165–1172, Apr. 2007. [13] Y. W. Chi and K. L. Wong, “Internal compact dual-band printed loop antenna for mobile phone application,” IEEE Trans. Antennas Propag., vol. 55, pp. 1457–1462, May 2007. [14] C. I. Lin and K. L. Wong, “Printed monopole slot antenna for internal multiband mobile phone antenna,” IEEE Trans. Antennas Propag., vol. 55, pp. 3690–3697, Dec. 2007. [15] R. A. Bhatti, Y. T. Im, J. H. Choi, T. D. Manh, and S. O. Park, “Ultrathin planar inverted-F antenna for multistandard handsets,” Microw. Opt. Technol. Lett., vol. 50, pp. 2894–2897, Nov. 2008. [16] R. A. Bhatti and S. O. Park, “Octa-band internal monopole antenna for mobile phone applications,” Electron. Lett., vol. 44, pp. 1447–1448, Dec. 2008. [17] Y. W. Chi and K. L. Wong, “Very-small-size printed loop antenna for GSM/DCS/PCS/UMTS operation in the mobile phone,” Microw. Opt. Technol. Lett., vol. 51, pp. 184–192, Jan. 2009. [18] J. C. Lin, “Specific absorption rates induced in head tissues by microwave radiation from cell phones,” Microwave, pp. 22–25, Mar. 2001. [19] Safety Levels With Respect to Human Exposure to Radio-Frequency Electromagnetic Field, 3 kHz to 300 GHz, ANSI/IEEE standard C95.1, Apr. 1999. [20] Y. W. Chi and K. L. Wong, “Compact multiband folded loop chip antenna for small-size mobile phone,” IEEE Trans. Antennas Propag., vol. 56, pp. 3797–3803, Dec. 2008. [21] M. R. Hsu and K. L. Wong, “Seven-band folded-loop chip antenna for WWAN/WLAN/WiMAX operation in the mobile phone,” Microw. Opt. Technol. Lett., vol. 51, pp. 543–549, Feb. 2009. [22] C. H. Chang and K. L. Wong, “Printed =8-PIFA for penta-band WWAN operation in the mobile phone,” IEEE Trans. Antennas Propag., vol. 57, pp. 1373–1381, May 2009. [23] Y. L. Kuo and K. L. Wong, “Printed double-T monopole antenna for 2.4/5.2 GHz dual-band WLAN operations,” IEEE Trans. Antennas Propag., vol. 51, pp. 2187–2192, Sep. 2003. [24] C. H. Wu and K. L. Wong, “Internal shorted planar monopole antenna embedded with a resonant spiral slot for penta-band mobile phone application,” Microwave Opt. Technol. Lett., vol. 50, pp. 529–536, Feb. 2008. [25] Y. W. Chi and K. L. Wong, “Half-wavelength loop strip fed by a printed monopole for penta-band mobile phone antenna,” Microw. Opt. Technol. Lett., vol. 50, pp. 2549–2554, Oct. 2008. [26] M. R. Hsu and K. L. Wong, “WWAN ceramic chip antenna for mobile phone application,” Microwave Opt. Technol. Lett., vol. 51, pp. 103–110, Jan. 2009. [27] Ansoft Corporation HFSS [Online]. Available: http://www.ansoft.com/ products/hf/hfss/ [28] K. L. Wong, Planar Antennas for Wireless Communications. New York: Wiley, 2003. [29] SEMCAD Schmid & Partner Engineering AG (SPEAG) [Online]. Available: http://www.semcad.com

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Comments and Replies Comments on “Fast Direct Solution of Method of Moments Linear System”

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A. Heldring, J. M. Rius, and J. M. Tamayo

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In [1], a new algorithm is presented for direct (non-iterative) frequency domain method of moments (MoM) calculations based on block-wise compression of the impedance matrix. The storage requirements and computational complexity of the algorithm are is the number of claimed to be 3=2 and 2 , respectively, if unknowns for a MoM surface discretization that is fixed with respect to the wavelength. A more rigorous analysis, explicitly using the asymptotic expression for the number of degrees of freedom (DoF) in the interaction between groups of scatterers, is presented here, showing that this should be 3=2 log and 2 log2 , respectively. The complexity analysis applies to problems that can be formulated as a surface integral equation (SIE) on surfaces in or between homogeneous media, such as a ship or an airplane. It does not apply to volume discretization MoM or multilayer problems. The results presented here are also relevant for block-wise compression methods using iterative solvers such as the Adaptive Cross Approximation algorithm presented in [2], since the compressed impedance matrix is the same. The number of DoF in the interaction between two separate sets of elementary scatterers is asymptotically proportional to [3], [4]

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Manuscript received May 08, 2009; revised August 04, 2009. First published December 31, 2009; current version published March 03, 2010. This work was supported by the Spanish “Comisión Interministerial de Ciencia y Tecnología (CICYT)” through the projects TEC2009-13897-C03-01 and TEC2007-66698C04-01/TCM and CONSOLIDER CSD2008-00068 and in part by the “Ministerio de Educación y Ciencia” through the FPU fellowship program. The authors are with the AntennaLab, Department of Signal Processing and Telecommunications, Universitat Politecnica de Catalunya, 08034 Barcelona, Spain (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2009.2039330

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