IEEE Transactions on Antennas and Propagation [volume 59 number 4]

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Citation preview

APRIL 2011

VOLUME 59

NUMBER 4

IETPAK

(ISSN 0018-926X)

PAPERS

Antennas Reconfigurable Axial-Mode Helix Antennas Using Shape Memory Alloys ..... ......... ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... .... S. Jalali Mazlouman, A. Mahanfar, C. Menon, and R. G. Vaughan Electrical Separation and Fundamental Resonance of Differentially-Driven Microstrip Antennas ...... ....... Y. P. Zhang A Modal Approach to Tuning and Bandwidth Enhancement of an Electrically Small Antenna . ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... . J. J. Adams and J. T. Bernhard Multi-Beam Multi-Layer Leaky-Wave SIW Pillbox Antenna for Millimeter-Wave Applications ....... ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ .. M. Ettorre, R. Sauleau, and L. Le Coq A Modal-Based Iterative Circuit Model for the Analysis of CRLH Leaky-Wave Antennas Comprising Periodically Loaded PPW ... ......... ......... ........ ......... ......... ........ ......... .. J. S. Gomez-Diaz, A. Álvarez-Melcon, and T. Bertuch Evolved-Profile Dielectric Rod Antennas ......... . ....... .. S. M. Hanham, T. S. Bird, A. D. Hellicar, and R. A. Minasian A Compact UWB Antenna for On-Body Applications .. ......... ....... N. Chahat, M. Zhadobov, R. Sauleau, and K. Ito Design of a Corner-Reflector Reactively Controlled Antenna for Maximum Directivity and Multiple Beam Forming at 2.4 GHz ........ ......... ........ ......... ....... T. D. Dimousios, S. A. Mitilineos, S. C. Panagiotou, and C. N. Capsalis Arrays Energy Patterns of UWB Antenna Arrays With Scan Capability ......... ........ ... C.-H. Liao, P. Hsu, and D.-C. Chang Method of Moments Analysis of Slotted Substrate Integrated Waveguide Arrays ....... . .... E. Arnieri and G. Amendola Substrate-Integrated Cavity-Backed Patch Arrays: A Low-Cost Approach for Bandwidth Enhancement ....... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... . M. H. Awida, S. H. Suleiman, and A. E. Fathy 3D Power Synthesis with Reduction of Near-Field and Dynamic Range Ratio for Conformal Antenna Arrays ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... ..... M. Comisso and R. Vescovo Adaptive Wideband Beamforming With Frequency Invariance Constraints ..... ....... Y. Zhao, W. Liu, and R. J. Langley A 4-Element Balanced Retrodirective Array for Direct Conversion Transmitter ........ L. Chiu, Q. Xue, and C. H. Chan Dual Grid Array Antennas in a Thin-Profile Package for Flip-Chip Interconnection to Highly Integrated 60-GHz Radios .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ....... Y. P. Zhang, M. Sun, D. Liu, and Y. L. Lu Evaluation of a New Wideband Slot Array for MIMO Performance Enhancement in Indoor WLANs . ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ .. J. R. Costa, E. B. Lima, C. R. Medeiros, and C. A. Fernandes Parameter Estimation of Damped Power-Law Phase Signals via a Recursive and Alternately Projected Matrix Pencil Method ......... ......... ........ ......... ......... ........ ......... ......... ........ .. K. Chahine, V. Baltazart, and Y. Wang Lenses for Circular Polarization Using Planar Arrays of Rotated Passive Elements .... R. H. Phillion and M. Okoniewski

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(Contents Continued on p. 1069)

(Contents Continued from Front Cover) Electromagnetics and Propagation Automated Analytic Continuation Method for the Analysis of Dispersive Materials .... ......... . K. Inan and R. E. Diaz 3D-Aggregate Quantitative Imaging: Experimental Results and Polarization Effects .... ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ C. Eyraud, J.-M. Geffrin, and A. Litman Characterization of Metamaterials Using a Strip Line Fixture ... ...... .... ... L. Yousefi, M. S. Boybay, and O. M. Ramahi LF Ground-Wave Propagation Over Irregular Terrain ... ......... ......... ........ ......... L. Zhou, X. Xi, J. Liu, and N. Yu Ultrawideband Multi-Static Scattering Analysis of Human Movement Within Buildings for the Purpose of Stand-Off Detection and Localization .... ......... ......... ........ ......... ......... ........ ......... ...... M. Thiel and K. Sarabandi Analytical Propagation Modeling of BAN Channels Based on the Creeping-Wave Theory ...... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ T. Alves, B. Poussot, and J.-M. Laheurte Numerical Techniques Fast Optimization of Electromagnetic Design Problems Using the Covariance Matrix Adaptation Evolutionary Strategy .. .. ........ ......... ......... ........ ......... ......... ........ ......... ........ M. D. Gregory, Z. Bayraktar, and D. H. Werner Self-Adaptive Differential Evolution Applied to Real-Valued Antenna and Microwave Design Problems ...... ......... .. .. ........ ......... ......... ........ ......... ..... S. K. Goudos, K. Siakavara, T. Samaras, E. E. Vafiadis, and J. N. Sahalos Improving the Accuracy of the Second-Kind Fredholm Integral Equations by Using the Buffa-Christiansen Functions .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... .... S. Yan, J.-M. Jin, and Z. Nie A Low-Dispersion Realization of Precise Integration Time-Domain Method Using a Fourth-Order Accurate Finite Difference Scheme ..... ........ ......... ......... ....... .. ......... ......... ........ ........ Z.-M. Bai, X.-K. Ma, and G. Sun High-order Div- and Quasi Curl-Conforming Basis Functions for Calderón Multiplicative Preconditioning of the EFIE .. .. ........ ......... ......... ........ ......... ......... ........ ......... F. Valdés, F. P. Andriulli, K. Cools, and E. Michielssen On the FDTD Formulations for Modeling Wideband Lorentzian Media . ....... .. .... Z. Lin, Y. Fang, J. Hu, and C. Zhang An Analytical Expression for 3-D Dyadic FDTD-Compatible Green’s Function in Infinite Free Space via z-Transform and Partial Difference Operators ....... ......... ........ ......... ......... ........ ......... ......... ........ ........ S.-K. Jeng Pulsed Beams Expansion Algorithms for Time-Dependent Point-Source Radiation. A Basic Algorithm and a Standard-Pulsed-Beams Algorithm .... ......... ........ ......... ......... ........ ......... ......... . Y. Gluk and E. Heyman

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COMMUNICATIONS

Compact UWB Antenna With Multiple Band-Notches for WiMAX and WLAN ........ ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ...... M.-C. Tang, S. Xiao, T. Deng, D. Wang, J. Guan, B. Wang, and G.-D. Ge A Wideband Stacked Offset Microstrip Antenna With Improved Gain and Low Cross Polarization .... ......... ......... .. .. ........ ......... ......... ....... V. P. Sarin, M. S. Nishamol, D. Tony, C. K. Aanandan, P. Mohanan, and K. Vasudevan Dual-Band Circularly-Polarized CPW-Fed Slot Antenna With a Small Frequency Ratio and Wide Bandwidths ........ .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... .. C.-H. Chen and E. K. N. Yung Rectangular Dielectric Resonator Antennas With Enhanced Gain ........ ........ . ......... .... A. Petosa and S. Thirakoune Design of a Microstrip Monopole Slot Antenna With Unidirectional Radiation Characteristics . ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... ........ C.-J. Wang and T.-L. Sun Compact and Tunable Slot-Loop Antenna ........ ........ ........ .. ......... ........ ... P.-L. Chi, R. Waterhouse, and T. Itoh Design of SIW Cavity-Backed Circular-Polarized Antennas Using Two Different Feeding Transitions ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... . D.-Y. Kim, J. W. Lee, T. K. Lee, and C. S. Cho UWB Dielectric Resonator Antenna Having Consistent Omnidirectional Pattern and Low Cross-Polarization Characteristics . ......... ........ ......... ......... ........ ......... ......... ........ ......... ....... K. S. Ryu and A. A. Kishk Analog Direction of Arrival Estimation Using an Electronically-Scanned CRLH Leaky-Wave Antenna ........ ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ..... S. Abielmona, H. V. Nguyen, and C. Caloz On the Mathematical Link Between the MUSIC Algorithm and Interferometric Imaging ...... G. Hislop and C. Craeye On the Nature of Oscillations in Discretizations of the Extended Integral Equation ..... ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ....... G. Fikioris, N. L. Tsitsas, and I. Psarros Time Domain UTD-PO Solution for the Multiple Diffraction of Spherical Waves for UWB Signals ... ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... ........ ......... ...... P. Liu, J. Tan, and Y. Long ...... ........ .... J. S. Gardner Asymptotic Expansion of the Associated Legendre Function Over the Interval Experimental Verification of Link Loss Analysis for Ultrawideband Systems ... ......... ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ . K. Rambabu, A. E.-C. Tan, K. K.-M. Chan, and M. Y.-W. Chia The Design of an Ultrawideband T-Pulse With a Linear Phase Fitting the FCC Mask ... ......... ........ ......... ......... .. .. ........ ......... ......... ........ ......... ......... ........ ......... ......... .... Z. Mei, T. K. Sarkar, and M. Salazar-Palma

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Digital Object Identifier 10.1109/TAP.2011.2137012

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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 59, NO. 4, APRIL 2011

Reconfigurable Axial-Mode Helix Antennas Using Shape Memory Alloys Shahrzad Jalali Mazlouman, Member, IEEE, Alireza Mahanfar, Member, IEEE, Carlo Menon, Member, IEEE, and Rodney G. Vaughan, Fellow, IEEE

Abstract—Reconfigurable structures based on smart materials offer a potential solution to realize adaptive antennas for emerging communication devices. In this paper, a reconfigurable axial mode helix antenna is studied. A shape memory alloy spring actuator is used to adjust the height of a helix antenna. With the total length of the helix wire fixed, the pitch spacing and pitch angle are varied as the height is varied. This in turn can alter the antenna pattern in order to adjust to altered operating conditions. In order to undertake the design, the Kraus equations for the axial mode helix are compared with simulation results, and their range of applicability is clarified. It is shown that based on these equations, antenna gain variation is possible by varying the height of the antenna, while keeping its conductor length fixed. We then show that a pattern can be reconfigured using a two-helix structure. Finally, a proof-of-concept helix antenna is implemented using a shape memory alloy actuator. Measurement results confirm that the pattern can reconfigure while maintaining a reasonable impedance match. Index Terms—Conical helix, helix antenna, reconfigurable antenna, shape memory alloy, smart antenna, steerable beam.

I. INTRODUCTION

H

ELIX antennas have widely been used since the 1950s [1]–[3], including conical, and other shapes. Helix antennas have different modes of operation [3]. The helix is operating in the axial mode when the circumference is about one freespace wavelength, although miniaturization is possible e.g., [4]–[6]. Axial-mode helix antennas are surface wave antennas and so have medium gain, and wideband characteristics [1], [2], [6]–[8]. In particular, helices are for circular polarization (CP) [9], and this was Kraus’ motivation for his pioneering developments. The helix antenna continues to appear in new designs and research papers. Several variations focus on optimizing the length, pitch angle or radius of the helix antenna for a certain application. For example, in [10], the pitch angle of an axial-mode

Manuscript received April 05, 2010; revised August 18, 2010; accepted September 20, 2010. Date of publication January 31, 2011; date of current version April 06, 2011. This work was supported by the Natural Science and Engineering Research Council of Canada (NSERC). S. Jalali Mazlouman, C. Menon, and R. G. Vaughan are with the School of Engineering Science, Simon Fraser University, Burnaby, BC V5A 1S6, Canada (e-mail: [email protected]; [email protected]). A. Mahanfar is with the Mobile Device Strategy and Commercialization (MDSC) Division, Microsoft Corporation, Redmond, WA 98052 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.2011.2109686

helix antenna is varied in a non-linear manner from a relatively small angle at the feed end to a large angle at the open end of the antenna, to optimally match the surface wave velocity to that of the free space, and to provide multiple peak gains. In [11], exponential pitch spacing is proposed in order to increase the CP bandwidth of the axial mode. A spring-tunable normal mode helix whip antenna is built in [12] for vehicular mounting. In addition, a tri-band normal mode helix antenna to cover the EGSM/GPS/PCS bands is designed in [13] using a dual pitch helix. In [14], a dielectric tuning element is used to fine-tune the normal-mode. In [15], a variable pitch design is given for a variable scanning-mode helix. There are several other examples, too numerous to list here, but these are representative of the activity and interest. However, little attention has been paid to adaptively controlling the axial mode helix structure. Emerging wireless communication devices call for antennas that can dynamically adjust antenna characteristics such as the far-field radiation pattern, centre frequency or main lobe direction (squint), in response to changing operating conditions. For example, reconfigurable antennas can dynamically alter their pattern in order to improve reception or transmission (c.f., switched diversity) by improving gain to the wanted or suppressing interference, or a combination of these. The adjustment of antenna characteristics can be realized through electrical, mechanical or other means [16]. Conventional methods include solid-state switches such as PiN diodes, e.g., [17], [18], and RF-MEMS switches, e.g., [19]. Usually, the focus of these methods is to alter the electric length of the antenna by activating and deactivating critically located switches, which in turn vary the operating (impedance match) frequency. Alternatively, the pattern can be varied by switching parasitic elements, which has the great advantage of not locating the switch in the signal path. However, solid state switches can suffer from non-linearity and low isolation [20]. In addition, their cost and reliability can be prohibitive, mostly in the case of MEMS switches. Usually, only certain discrete configurations can be attained using these methods. Continuous operation is possible, in principle, from parasitic element continuous reactance loading (e.g., using a varactor, although these also have practical problems such as small tuning range and affecting the input impedance [16]). Mechanical reconfiguration of antennas is slow but is also claimed to deliver the most dramatic antenna parameter variations [21]. In mechanical reconfiguration, actuators can be smart materials such as piezoelectric actuators [21], electro-active polymers (EAPs) [22] and shape memory alloys (SMAs) [23]. Unlike switches, electromechanical approaches do not

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JALALI MAZLOUMAN et al.: RECONFIGURABLE AXIAL-MODE HELIX ANTENNAS USING SHAPE MEMORY ALLOYS

introduce non-linearity in the RF path. One feature is that these can provide continuous change and therefore a smooth transition between antenna parameters (e.g., the pattern). In this paper, a reconfigurable axial-mode helix antenna is implemented using an axially located SMA spring as an actuator. It is shown that by applying a direct current to the SMA spring, the height of the helix antenna and therefore its pitch spacing is varied. The helix makes an excellent reconfigurable antenna because, unlike other antenna types, the spring-like helix structure is deformable without imposing too much stress to the conductors. Although the variation of the axial mode helix height does not tilt the beam, it is demonstrated how multiple reconfigurable axial mode helices can be used to steer the main lobe. The rest of the paper is as follows: the concept of reconfigurable axial-mode helix antennas is discussed in Section II. It is shown that axial-mode helix antenna pattern parameters such as gain and half-power beamwidth (HPBW) can be varied by varying the pitch spacing (height) of the antenna and keeping the conductor length fixed. These variations are studied for a regular and conical helix antenna, using Finite Integration Technique (FIT) numerical methods (CST) and empirical Kraus equations. We clarify the range of configurations for the axial mode helix that is covered by the Kraus design equations. In addition, we show that a two-element helix antenna configuration can be used to attain significant pattern reconfigurability. In Section III, proof-of-concept experimental results are presented for an implemented reconfigurable helix actuated by an SMA spring-based on the idea proposed in Section II. Finally, Section IV concludes the paper. II. THE RECONFIGURABLE AXIAL-MODE HELIX ANTENNA A. The Regular Reconfigurable Helix Antenna The helix antenna is operating in the axial mode when the circumference is about one free space wavelength, viz., [1]. Let be the radius of the helix, the wavelength in free space, the spacing between turns (pitch spacing), the number of turns, the total height of the helix and the pitch angle, we have [1]

(1) The height refers to the length of the helix antenna, following Kraus, and some authors use “length” to describe this. We follow Kraus’ terminology and use “length” for the length of the wire making up the helix. Empirical equations were developed by Kraus [1] for the axial-mode helix which link the pitch spacing and the directivity of the antenna (2) is the spacing between turns in free space wavewhere lengths, so the height of the helix is wavelengths. In addition, the HPBW is related to the helix pitch spacing as [1] (3)

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Equations (2) and (3) are restricted to pitch angles of [1, pp. 281–284], although Kraus’ original results [2] indicate “optimal contours,” based on axial ratio (AR), impedance, and beam pattern, can be extended outside ([2]; [1, p. 308]). these pitch angles to about King and Wong [24] also built many helix antennas and developed further empirical equations for gain and HPBW over pitch angles 11.5 to 14.5 . In this section, numerical experiments are used to investigate pitch angles well outside of Kraus’ 12 –14 limits. This is motivated by the potential of reconfiguring the helix by varying in the pitch angle (height), with a constant length helical wire. Strictly speaking the radius of the helix changes, but this is small because we are dealing with small pitch angles. FIT simulation results (CST) are shown in Fig. 1(a), (b) for , a thin-wire (0.7 mm diameter) copper helix with , and varying from 35 to 195 mm (about ). A plastic hollow cylindrical base ( , ), inside the helix, has a radius of 7.5 mm. This is to provide mechanical stability, and to reduce its bending as the height is varied. The effect of the actuator (SMA spring) is also modeled by including another helical wire , wire thickness of , of radius and Nitinol conductivity of inside the plastic tube with the same height as the helical antenna. More information on the mechanical structure and square copper the SMA is given in Section III. A ground plane is used. Fig. 1 depicts, for different heights, vari; and (b) the 4.35 GHz ) pattern for ations of: (a) plane. the In Fig. 1(a), a reasonable match is observed for heights 70 mm or more over a wide bandwidth around 4.35 GHz. This matching is attained by tuning the feed angle (initial angle of wire at ground plane) as explained in [25] in all the numerical simulations (i.e., all configurations), as well as the experiments. As expected from the empirical equations in [1], the axial mode pattern of the helix antenna generally becomes more directive as the antenna pitch spacing (height) increases, i.e., the maximum gain/directivity increases and the HPBW decreases. It is evident from the figures that the axial mode pattern can be changed by varying the pitch spacing while maintaining the same operating frequency. To further inspect variations of the helix antenna parameters outside of the specified pitch spacing range, the maximum gain and HPBW are determined numerically and results are presented in Fig. 2. Also, the empirical curves [1], [2] for these parameters are presented for comparison. The directivity curve follows the mean form of the numerical experiments. The oscillating form of the experiments is expected from surface radiation principles (which were developed mainly after Kraus’ analysis). Consequently, the directivity formula is not very accurate for a given structure (with a given surface wave velocity) but is an excellent rule of thumb for the mean gain (over different pitch angles or surface wave velocities) of a copper helix. The HPBW curve holds for larger pitch angles, but not for smaller pitch angles. Both results confirm that different pitch spacings (or height for fixed wire length) of an axial mode helix will give different gains. A reconfigurable axial mode helix antenna, with

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Fig. 1. CST Simulation results for variations of a reconfigurable helix antenna for various heights: (a) the S ; (b) the 4.35 GHz, pattern cut, ' = 0 .

Fig. 2. Simulated (CST) and empirical formula computations of maximum gain and HPBW of the axial-mode helix antenna versus. height (pitch angle). Kraus’ formulas are only for a pitch angle of 12 to 14 , but it is clear that they apply over a much larger range.

pitch angles between 2 to 16 looks promising. For example, Figs. 1 and 2 show a gain variation between 7 to 13 dB, when the height is varied between 40 to 110 mm. An associated HPBW of between 60 to 45 can be attained if the height is varied between 55 and 80 mm. B. The Conical Reconfigurable Helix Antenna The conical axial-mode helix antenna can be used as a reconfigurable helix antenna in a similar manner as the regular helix antenna. The conical helix offers axial radiation over a much wider bandwidth [26]. The conical spiral helix is more mechanically stable when its height is varied. In principle, no plastic support is required. of a conical helix with a maxSimulation results for the imum radius of 18 mm and a radius ratio of 0.55, with 6 turns and its height varied between 50 to 90 mm confirm reasonable matching for a wideband around 3 GHz. Similar to the cylindrical helix, the patterns of the conical helix at 3 GHz (not shown) demonstrate that by varying the height of the helix from 40 to 95 mm, the gain ranges from 10 to 11.5 dB and the HPBW from 65 to 50 . The radiation mechanism, where the main axial radiation is from the region of a one wavelength circumference, tends to compensate for the height (and pitch) change.

C. The Axial-Mode Dual-Helix Antenna The application of the variable helix is not limited to gain variations. Various configurations of commonly fed multiple helices with variable heights provide more significant pattern reconfigurability. The number of helices, their height, as well as their relative location can be configured to attain desired pattern configurations. For example, the beam can be squinted, as in Fig. 4(a), discussed below. One such configuration consists of two helix antennas, one right handed with height , and the other left handed with height , located at a distance of apart, as shown in Fig. 3. The and are otherwise helices are of the same radius identical. Note that since the two antenna elements are not identical, the array equations do not apply. Each helix antenna height can be independently controlled, similar to the structures discussed in the previous subsections. The two helix antennas are fed axially from a fixed corporate feed via a common stripline with the same line length to each helix from the feed line split. No attempt was made to balance the powers in each helix element, although the single port match remained reasonable (see Fig. 6 below, at a frequency of 4.5 GHz). The separation of the helices and the location of the feed point are configured to provide a changing radiation pattern with reasonable impedance matching around the design centre frequency, respectively.

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Fig. 3. The axial-mode counter-wound two-helix antenna structure.

Fig. 4 depicts the variations of the 4.5 GHz pattern by controlling the heights of the two helix antennas as described above, apart: (a) for located at a distance of ; (b) for , ; and (c) , . The power pattern (both polarizations) is no longer a beam in the axial direction, in general. The pattern can be reconfigured, spanning different polarizations and gain directions. Some detail is depicted in the 4.5 GHz pattern cut at , of Fig. 5. The gain is not array-enhanced, as expected since the elements are not identical. The polarized patterns are not displayed here. The change of gain direction is evident parand (40,50). ticularly for the configurations The directional gain is low, but the patterns from these configurations are essentially orthogonal, and therefore useful in diversity/MIMO applications. Diversity correlation coefficients (inner products) [27] for the patterns indicated in Fig. 5 are reported in Table I. As evident from this table, low correlation coefficients show the near-orthogonality of the patterns for the larger height differences. of the two helix anFig. 6. depicts the variations of the tenna. As seen in these figures, while different patterns can be attained by varying the heights of the two helices, reasonable below about 10 dB) is maintained impedance matching ( for all the configurations shown, at 4.5 GHz. The matching bandwidth could probably be improved by further adjustment of the feed. III. EXPERIMENTAL RESULTS A. Shape Memory Alloys (SMAs) SMAs are materials that can restore their original configuration by heating after they are plastically deformed at low temperature. Previous applications of the SMA actuators for antennas include contour optimization of large space reflector antennas [28] and deployable space antennas and structures [29]. One of the most common shape memory alloys is Nitinol, an alloy of Nickel and Titanium. The temperature variation can be realized by passing a DC current through it. The SMA used in this work is the BMX-150 Biometal spring by Toki [30]. These actuators can be elongated at room temperature, typically by an external force, and contracted by applying the DC current. Quick cooling and appropriate design of the actuator can provide subsecond return time to the original shape [30]. Variations of a load-free BMX-150 length by passing DC currents through it, in a test at room temperature of 20 is shown

Fig. 4. Variations of the radiation pattern of an axial-mode counterwound twohelix antenna array with a constant distance of d and: (a) for h h , (b) for h ,h , and (c) h , h , around the 4.5 GHz frequency.

= 40 mm = 40 mm

= 40 mm

= 25 mm = 100 mm

= = 100 mm

in Fig. 7. As can be seen in this figure, up to 40 mm of length variation (42%) can be attained by applying up to 180 mA DC current. The disadvantages of SMA materials include their potential sensitivity to ambient temperature, their low energy efficiency ( 5%), and their non-linear characteristics such as hysteresis properties [31]. Hysteresis problems can cause difficulties in the length control of the SMAs, but can be resolved by use of feedback control systems [31]. SMAs should be isolated from the ambient temperature if dramatic changes are expected due

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=0

Fig. 5. Variations of the 4.5 GHz power pattern at ' counterwound, axial-mode helices with variable heights.

Fig. 7. Variations of the total length of a load-free BMX-150 SMA spring against the DC current passed through it. Up to 40 mm of length variation (42%) can be attained using a maximum DC current of 180 mA. plane for the two

TABLE I CORRELATION COEFFICIENTS FOR PATTERNS OF FIG. 5

Fig. 8. The reconfigurable axial-mode helix antenna: (a) full model, (b) building blocks as an exploded diagram, (c) the prototype at original height, when no current is applied, and (d) the compressed helix prototype when DC current is passed through the SMA. The SMA spring is connected from the top of the helix to below the groundplane by a thin wire which is between the helix and the SMA spring.

S

d=

Fig. 6. Variations of the of an axial-mode two-helix structure with and various heights of the two helix independently-tunable-height antennas, fairly good impedance matching around 4.5 GHz.

25 mm

to their sensitivity to the environment conditions. The effect of minor changes in the ambient temperature is however minor. This can accounted for using a closed loop control circuitry in an adaptive antenna design. For example, the maximum temperature of the SMA used in this implementation is given 50 . In this paper, SMA spring actuators are used to vary the height of the helix. The reconfigurable helix antenna system using this spring is explained in the following subsection. B. Hardware Prototype As a proof-of-concept, a reconfigurable axial-mode helix antenna was implemented as shown in Fig. 8, comprising an

11-turn helix made of copper wire, a radius of 9.9 mm and wire thickness 0.7 mm. This antenna is wound loosely around a cylindrical hollow plastic base (shown in Fig. 8(a), (b) and the prototype shown in Fig. 8(c), (d)) and fed from the center of a square copper ground plane. Two SMA spring actuators in series (BMX-150 Biometal Springs by Toki) are passed through the plastic tube with one end connected to the open end of the helix and the other end passed through a small hole and fixed under the ground plane, where it is connected to the DC circuitry. Note that the SMA is located inside the plastic cylinder and is only visible in Fig. 8(b), which depicts an exploded diagram of the structure. When sufficient DC current is applied to the SMA springs, they shrink, thereby decreasing the height and the spacing beof the helix antenna. In this way, the height tween turns of the antenna can be changed by changing the current passing

JALALI MAZLOUMAN et al.: RECONFIGURABLE AXIAL-MODE HELIX ANTENNAS USING SHAPE MEMORY ALLOYS

Fig. 9. Simulation and measurement results for the reconfigurable helix S for some selected sweep points.

through the SMAs. When no current is applied, an external reverse force spring is required to expand the SMA springs and return the helix antenna to its original height. The helix antenna itself acts as this external force spring to return the antenna to its original state. Note that the SMA and the antenna are electrically isolated. The SMA spring actuators showed little hysteresis effect. They require a maximum DC current of 200 mA for full actuation, equivalent to shrinking (height variation) by 90 mm. Note that the variation is continuous and therefore smooth transitions between various antenna pattern configurations are feasible. The and patterns for different heights of the helix antenna are presented in the following section. C. Measurement Results The reconfigurable helix system is measured using a 5071 measurements and a Satimo StarLab aneAgilent VNA for choic chamber for pattern measurements. Measurements were done under normal room temperature. measurement results for some heights of Fig. 9 depicts the helix antenna (60, 70, 80 mm) as well as the simulation results for the 80 mm, for comparison. It can be seen that good impedance matching is attained around 4–4.5 GHz for all sweep points (not all sweep points are shown, for brevity), as also expected from the simulation results. The match is partly a result of manually adjusting the angle of the wire feed. The simulations below included the plastic base and the SMA spring. As expected, the prototype axial-mode helix is a fairly wideband structure and maintains its wide frequency bandwidth while mechanically reconfigured. Reasonable agreement is observed around the matching frequencies, i.e., 4–4.5 GHz between simulation and measurement results. The discrepancies are partly due to the uneven pitch spacing in the physical helix structure (c.f., Fig. 8(c) and (d)). A mechanical structure using a more elastic wire for the antenna (or an actual spring with sufficiently low spring coefficient to be tunable using an SMA actuator), could improve the uniformity of pitch spacing. Note that as stated before, the plastic is a thin cylinder for supporting the helix conductors. It is not an essential part of the

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antenna for radiation or actuation. It was chosen from readily available material. It could be replaced by Styrofoam or any other type of low loss material. The effect of the plastic is noticeable on the resonance frequency and is included in the simulation. The SMA actuator does not have a significant impact on the antenna radiation pattern. This is because it has a much smaller radius than the helix and is centrally located. The presence of a small center conductor in an axial mode helix is known to not affect the patterns much. Many axial mode helices are in fact constructed this way. The radius of axial-mode helix is relatively large and the distance to the shape memory alloy (SMA) in the axis of the helix is substantial, given the radius of SMA is sub-millimeter. Therefore the contribution of the SMA in overall radiation pattern is not significant. These effects are modeled in numerical simulations and confirmed by measurements. Fig. 10(a)–(c) depict radiation pattern measurement results at 4.35 GHz for selected heights from 40 to 150 mm. These measurement results report the realized gain and therefore include the effect of the plastic support and the SMA spring. As expected from simulation results, the pattern can be adjusted by varying the helix height. Fig. 10(d) compares the maximum gain (empirical directivity), simulation (gain), and the Fig. 10(a) measurement (realized gain) results for various heights. (Impedance match is maintained over the band). The gain of the helix can be adjusted from about 7.4 to 12.4 dB. However, this trend is not monotonic, as expected from surface wave radiation principles. The speed of the axial mode surface wave along the helical structure and the separation distance of the feed and open end (the locations of radiation) are both changed. Comparing Figs. 10 and 2, it can be seen that simulation pattern results match the measurements reasonably well. axial Fig. 11 depicts the experimental results for the ratio of the helix antenna versus variations of its height. It can be seen that the circular polarization is well maintained within the HPBW. The axial direction has a worst case axial ratio of 2, or a cross polar ratio of about 18 dB. The dual helix structure has not been implemented physically. However, the simulated patterns can be expected to be reliable and we have demonstrated the SMA actuation of a single element. The dual helix structure is a subject of future development. IV. CONCLUSION AND SUMMARY In this paper, a reconfigurable helix antenna is implemented using shape memory alloy spring actuators. The height and therefore pitch spacing of the helix is governed by the length of an SMA spring along the axis of the helix. Applying a direct current to the SMA spring causes it to shrink which decreases the pitch spacing and height of the helix antenna, while the radius is essentially constant owing to the small pitch angles. Observations from both simulations and physical measurements confirm the Kraus empirical relations for the axial mode helix. The directivity equation is a good fit apart from the oscillations expected from the surface wave radiation. The axial mode is dominant over a very wide range of pitch angles, and we use this for a reconfigurable axial mode antenna. Experimental results demonstrate a reconfigurable axial-mode helix antenna that can maintain a reasonable impedance match and axial ratio

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Fig. 10. Experimental results for variations of the implemented reconfigurable helix antenna parameters for some selected heights: the 4.35 GHz radiation pattern: (a) ' , (b) ' , (c)  , and (d) the maximum gains at ' (measurements are from (a), for example), compared with simulation results. The gain from the simulations is compared to the realized gain of the measurements, (however, the impedance match is reasonable as seen in Fig. 9) and to the empirical directivity.

=0

= 90

= 90

Fig. 11. Experimental results of '

=0

=0

AR of the prototype vs. height variations within HPBW. The AR increases drastically beyond the depicted range.

over a wide range of height variations. A dual helix structure allows beam squint through mutual reconfiguration. REFERENCES [1] J. D. Kraus, Antennas, 2nd ed. New York: McGraw-Hill, 1988. [2] J. D. Kraus, “Helix beam antennas for wideband applications,” Proc. IRE, vol. 37, no. 3, pp. 263–272, Mar. 1949. [3] R. Mittra, “Wave propagation on helices,” IEEE Trans. Antennas Propag., vol. 11, no. 5, pp. 585–586, Sep. 1963. [4] G. G. Rassweiler, “Helical and log conical antennas loaded with an isotropic material,” Univ. Michigan Radiation Lab., Report No. 7848-3-Q, Nov. 1966.

[5] D. T. Warren, “Full Core Loaded Sheath Helix,” M.Sc. thesis, Syracuse Univ., NY, Jan. 1969. [6] A. H. Safavi-Naeini and O. Ramahi, “Miniaturizing the axial mode helical antenna,” in Proc. IEEE Conf. Communications and Electronics, ICCE, Jun. 2008, pp. 374–379. [7] H. Nakano, Y. Samada, and J. Yamauchi, “Axial mode helical antennas,” IEEE Trans. Antennas Propag., vol. 34, no. 9, Sep. 1986. [8] R. M. Barts and W. L. Stutzman, “A reduced size helical antenna,” in IEEE Antennas and Propagation Soc. Int. Symp. Digest, Jul. 1997, vol. 3, pp. 1588–1591. [9] R. G. Vaughan and J. B. Andersen, “Polarization properties of the axial mode helix antenna,” IEEE Trans. Antennas Propag., vol. 33, no. 1, pp. 10–20, Jan. 1985. [10] W. D. Killen, “Variable pitch angle, axial mode helix antenna,” U.S. patent 5 892 480, Apr. 6, 1999.

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[11] C. Chen, E. Yung, B. Hu, and S. Xie, “Axial mode helix antenna with exponential spacing,” Microw. Opt. Technol. Lett., vol. 49, no. 7, pp. 1525–1530, 2007. [12] T. J. Wilson, “Spring tunable helix whip antenna,” U.S. patent 4 163 981, Aug. 7, 1979. [13] Y. Zhang, “Design of tri-band (EGSM/GPS/PCS) antenna with parasitic element for mobile-phone application,” Microw. Opt. Technol. Lett., vol. 48, no. 7, pp. 1347–1350, 2006. [14] A. Chufarovsky and A. D. Arns, “Helix antenna element,” U.S. patent 6 111 554, Aug. 29, 2000. [15] R. G. Vaughan, N. L. Scott, and C. A. Jenness, “Steerable beam helix antenna,” U.S. Patent 5 612 707, Mar. 1997. [16] J. T. Bernhard, Reconfigurable Antennas. London, U.K.: Morgan & Claypool, 2007. [17] J. Sarrazin, Y. Mahé, S. Avrillon, and S. Toutain, “Pattern reconfigurable cubic antenna,” IEEE Trans. Antennas Propag., vol. 57, no. 2, Feb. 2009. [18] S. Chen, J. Row, and K. Wong, “Reconfigurable square-ring patch antenna with pattern diversity,” IEEE Trans. Antennas Propag., vol. 55, no. 2, Feb. 2007. [19] J. Kiriazi, H. Ghali, H. Radaie, and H. Haddara, “Reconfigurable dualband dipole antenna on silicon using series MEMS switches,” in Proc. IEEE/URSI Int. Symp. Antennas Propag., 2003, pp. 403–406. [20] N. P. Cummings, “Active antenna bandwidth control using reconfigurable antenna elements,” Ph.D. dissertation, Virginia Polytechnic Institute & State Univ., Blacksburg, 2003. [21] J. T. Bernhard, E. Kiely, and G. Washington, “A smart mechanically-actuated two-layer electromagnetically coupled microstrip antenna with variable frequency, bandwidth, and antenna gain,” IEEE Trans. Antennas Propag., vol. 49, pp. 597–601, Apr. 2001. [22] A. Mahanfar, C. Menon, and R. G. Vaughan, “Smart antennas using electro-active polymers for deformable parasitic elements,” IET Electron. Lett., 2008. [23] X. Huang, G. J. Ackland, and K. M. Rabe, “Crystal structures and shape memory behaviour of NiTi,” Nature Mater., no. 2, pp. 307–311, 2003. [24] H. King and J. Wong, “Characteristics of 1 to 8 wavelength uniform helical antennas,” IEEE Trans. Antennas Propag., vol. 28, no. 3, pp. 291–296, Mar. 1980. [25] J. D. Kraus, “A 50-Ohm input impedance for helical beam antennas,” IEEE Trans. Antennas Propag., vol. 25, no. 6, p. 913, Nov. 1977. [26] J. S. Chatterjee, “Radiation field of a conical helix,” J. Appl. Phys., vol. 24, 1953. [27] R. G. Vaughan and J. B. Anderson, “Antenna diversity in mobile communication,” IEEE Trans. Veh. Technol., vol. VT-36, pp. 149–1987. [28] G. Song, B. Kelly, and B. N. Agrawal, “Active position control of a shape memory alloy wire actuated composite beam,” J. Smart Mater. Struct., vol. 9, no. 5, 2000. [29] S. H. Mahdavi and P. J. Bentley, “Evolving noise tolerant antenna configurations, using shape memory alloys,” presented at the 2nd Int. Conf. on Comp. Intel., Robotics and Auton. Syst. (CIRAS 2003), Dec. 2003. [30] BMX Biometal Springs Datasheet. Toki, Japan [Online]. Available: http://www.toki.co.jp/BioMetal/ [31] J. Jayender, R. V. Patel, N. Nikumb, and M. Ostojic, “Modeling and control of shape memory alloy actuators,” IEEE Trans. Control Syst. Tech., vol. 16, no. 2, Mar. 2008.

Shahrzad Jalali Mazlouman (M’09) received the B.Sc. and M.Sc. degrees in electrical engineering (electronics) from Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran, in 2001 and 2003, respectively, and the Ph.D. degree in electrical and computer engineering from University of British Columbia (UBC), Vancouver, BC, Canada, in 2008. She has worked on several mixed-signal, RF, and antenna system designs. In 2007, she worked as a mixed-signal intern at PMC-Sierra, Burnaby, BC, Canada. Since 2009, she has been a Postdoctoral Fellow at the School of Engineering Science, Simon Fraser University (SFU), Burnaby, BC, Canada, where she is working on reconfigurable RF and antenna systems for wireless communication devices, using smart material actuators.

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Alireza (Nima) Mahanfar (S’99–M’05) received the B.S. (honors) and M.S. degrees from Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran, in 1997 and 1999, respectively, and the Ph.D. degree from XLIM (formerly IRCOM), Limoges, France, in 2005, all in electrical engineering. From 2006 to 2007, he was with Simon Fraser University, Burnaby, Canada, as a Research Associate. From 1998 to 2005, he was with a number of organizations including Electromagnetics Lab (Tehran Polytechnic), Tehran, and Wireless 2000, Sierra Wireless and Nokia Mobile Phones, all in BC, Canada. Since December 2009, he has been with Mobile Device Strategy and Commercialization (MDSC) Division, Microsoft Corporation, Redmond, WA, where he is involved in the research and development of antennas for portable devices. His research interests are design of antennas and radio-frequency circuits. Dr. Mahanfar is the recipient of an NSERC Postdoctoral Fellowship (2005), and URSI Young Scientist Award (2007). Carlo Menon (M’04) received the Laurea degree in mechanical engineering from the University of Padua, Italy, in 2001 and the Ph.D. degree in space sciences and technologies from the Centre of Studies and Activities for Space—“G. Colombo,” Italy, in 2005. He was a Visiting Scholar at Carnegie Mellon University, in 2004, and a Research Fellow at the European Space Agency, The Netherlands, in 2005 and 2006. Since 2007, he has been an Assistant Professor at Simon Fraser University (SFU), Burnaby, Canada, where he leads the MENRVA Research Group (http:// menrva.ensc.sfu.ca). He is an Associate Member to both the School of Biomedical Physiology and Kinesiology and the Institute of Micromachine and Microfabrication Research at SFU. His research team is focusing on mechatronics, smart materials and structures, robotics, and bioinspired systems with applications especially in the biomedical and space sectors. Dr. Menon is an AIAA, ASME, BIONIS, and IAF member. He received the International IAF Luigi G. Napolitano Award, Spain, in 2006, and the International BIONIS Award on Biomimetics, U.K., in 2007. He is currently a Reviewer for about 20 international journals and is on the editorial board of the Journal of Bionic Engineering. Rodney G. Vaughan (F’07) received the Bachelor and Masters degrees from the University of Canterbury, New Zealand, in 1975 and 1976 respectively, and the Ph.D. degree from Aalborg University, Denmark, in 1985, all in electrical engineering. He worked with the New Zealand Post Office (now Telecom NZ Ltd) and the NZ Department of Scientific and Industrial Research, and Industrial Research Limited (IRL). Here he undertook a wide variety of practical mechanical and electrical projects including network analysis and traffic forecasting, and developed microprocessor and DSP technology for equipment ranging from abattoir hardware to communications networks. He was an URSI Young Scientist in 1982 for Fields and Waves, and in 1983 for Electromagnetic Theory. He developed research programs and personnel working in communications technology for IRL, revolving around signal processing, multipath communications theory (electromagnetic, line and acoustic media), diversity design, signal theory, and DSP. Industrial projects included the design and development of specialist antennas for personal, cellular, and satellite communications, large-N MIMO communications systems design; and also capacity theory and spatial field theory. In 2003, he became Professor of Electrical Engineering and Sierra Wireless Chair in Communications, at the School of Engineering Science, Simon Fraser University, Burnaby, BC, Canada. His current research for mobile communications involves propagation theory, communications signal processing and theory and design of antennas. Recent projects include compact mammalian bio-implantable antennas; multielement antenna design and evaluation; circularly polarized antennas, multifaceted structures for large arrays; microelectronic antenna structures, MIMO capacity realization; and blind-, precoding- and interference mitigation-techniques for OFDM. Dr. Vaughan has guest-edited for several special issues including the IEEE ANTENNAS AND PROPAGATION TRANSACTIONS Special Issue on Wireless Communications. He is a Fellow of the BC Advanced System Institute, an URSI Correspondent, and continues as the New Zealand URSI Commission B (Fields and Waves) representative.

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Electrical Separation and Fundamental Resonance of Differentially-Driven Microstrip Antennas Y. P. Zhang, Fellow, IEEE

Abstract—This paper studies the electrical separation and fundamental resonance of differentially-driven microstrip antennas with dual-probe feeds on both electrically thick substrate of high permittivity and electrically thin substrate of low permittivity. The electrical separation is defined as the ratio of the distance of the dual-probe feeds to the free-space wavelength . It is found that the occurrence of resonance of the fundamental mode is related with the electrical separation of the dual-probe feeds. When the electrical separation is satisfied, the resonance occurs. Otherwise, the resonance does not occur. It is shown that the empirical factor is smaller for the electrically thicker substrate of higher permittivity than that for electrically thinner substrate of low permittivity and is smaller for the circular patch than that for the rectangular patch. To validate the relationship of the occurrence of fundamental resonance with the electrical separation, several differentially-driven microstrip antennas were fabricated on the electrically thin substrate of the low permittivity and measured. It is observed that the simulated and measured results are in acceptable agreement for these differentially-driven microstrip antennas. Thus, the electrical separation condition derived in this paper should be very useful in guiding the design of differentially-driven microstrip antennas. Index Terms—Input impedance, microstrip antenna, resonance.

I. INTRODUCTION ICROSTRIP antennas have many unique and attractive properties—low in profile, light in weight, compact and conformable in structure, and easy to fabricate and to be integrated with solid-state devices [1], [2]. Therefore, they have been widely used in radio systems for various applications. Radio systems have been traditionally designed for single-ended signal operation, so have been microstrip antennas. Recently, radio systems have begun to be designed for differential signal operation [3]. This is because the differential signal operation is more suitable for high-level integration or single-chip solution of radio systems. Radio systems that adopt differential signal operation require differential antennas to get rid of bulky off-chip and lossy on-chip balun to improve the receiver noise performance and transmitter power efficiency [4]. There have been a few papers about differential microstrip antennas [5]–[10]. Of which [5]–[8] focus on integration with

M

Manuscript received January 29, 2010; revised August 13, 2010; accepted November 23, 2010. Date of publication January 31, 2011; date of current version April 06, 2011. The authors is with the Integrated Systems Research Lab, School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798, Singapore (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.2011.2109680

solid-state circuits and only [9] extends the cavity model to analyze differentially-driven microstrip antennas. The extended cavity model provides physical insight into the differential signal operation of microstrip antennas but it fails to yield accurate prediction of the performance, especially when an electrically thick substrate of high permittivity is used in attempting to achieve miniaturization and larger bandwidth. For these situations, a rigorous full-wave method has to be used [10]. In this paper we study differentially-driven microstrip antennas using the HFSS simulator from Ansoft. The HFSS is a rigorous full-wave electromagnetic simulator based on the finite element method. We analyze the relationship of the occurrence of resonance of the fundamental mode with the electrical separation of the differentially-driven microstrip antennas in Section II. We describe the experiment and discuss the measured results to validate the analysis in Section III. Finally, we summarize the conclusions in Section IV. II. ANALYSIS OF ELECTRICAL SEPARATION AND FUNDAMENTAL RESONANCE A microstrip antenna and a coordinate system are illustrated in Fig. 1(a). The microstrip patch located on the surface of a grounded dielectric substrate with thickness , dielectric relative permittivity , and dielectric loss tangent is differentially and with dual-probe feeds. In driven at points this figure, denotes a shape for the microstrip patch and means its area. Typical rectangular and circular microstrip patch shapes as shown in Fig. 1(b) and (c) are considered in this paper. of the differentially-driven miThe input impedance crostrip antenna is given by [9] (1) where the parameters are defined at the driving points and and they can be easily calculated with the HFSS simulator. If the differentially-driven microstrip antenna is indeed symmetric, (1) simplifies to (2) which reveals that there is a cancellation mechanism, which may is required improve the impedance bandwidth. The value of in the design of matching network between the differentiallydriven microstrip antenna and the differential active circuitry in calculated from is given by a radio system. The

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an electrically thick substrate of high permittivity has a smaller patch size and a larger bandwidth; while a microstrip antenna on an electrically thin substrate of low permittivity achieves good radiation efficiency and reasonable bandwidth. In the following we will analyze the relationship of the fundamental resonance with the electrical separation of the dual-probe feeds of differentially-drive rectangular and circular microstrip antennas on both electrically thick substrate of high permittivity and electrically thin substrate of low permittivity, respectively. A. Electrically Thick Substrate of High Permittivity

Fig. 1. Differentially-driven microstrip antenna: (a) Arbitrary patch in threedimensional view, (b) rectangular patch, and (c) circular patch.

where is 100 impedance

. When the imaginary part of the input

(4) happens, the resonance occurs. The fundamental resonance is of importance because the microstrip antenna is usually designed to operate near the resonant frequency of the fundamental mode. It is found that the fundamental resonance is related with the electrical separation of the dual-probe feeds. The electrical separation is defined as the ratio of the distance between the dual-probe feeds to the free-space wavelength , which is simof a subilar to what we define the electrical thickness and strate. The substrate is electrically thick when . A microstrip antenna on is electrically thin when

Consider a square RT/duriod 6010 substrate of side length (about one at 2.24 GHz), thickness , dielectric constant and dielectric loss tangent . A rectangular microstrip patch that has the length in the Y direction 19 mm and the width in the X direction 30 mm is on the middle of the substrate. The diameter of the probes is 1.0 mm. Simulations show that the rectangular is matched microstrip antenna driven at to a 50- single-ended signal source at 2.26 GHz. Measurements indicate that the rectangular microstrip antenna driven at is matched to a 50- single-ended signal source at 2.24 GHz [11], [12]. Hence, the simulator can satisfactorily predict the occurrence of resonance of the fundamental mode of this rectangular microstrip antenna. We locate at the mirror point of the second driving point along the line within the patch for the best differential signal operation. It is found that the rectangular microstrip and is antenna driven at also matched to a 100- differential signal source at 2.26 GHz. The simulated resonance of the fundamental mode of the differentially-driven rectangular microstrip antenna occurs indicating at 2.23 GHz. The electrical thickness is that the differentially-driven rectangular microstrip antenna is indeed on an electrically thick substrate. Table I shows the simas a function of electrical separation. ulated input impedance occurs Note that the resonance of the fundamental mode . The larger the for the electrical separation electrical separation, the lower the resonant frequency is and the higher the resonant resistance is. For example, the resonant frequency is 2.22 GHz and the resonant resistance is 420 for , while the resonant frethe electrical separation quency decreases to 2.2 GHz and the resonant resistance in. It creases to 935 for the electrical separation is also seen that the resonance of the fundamental mode does not occur for the electrical separation . The input resistance is quite small and the input impedance is inat 2.25 GHz for the ductive. For example, . electrical separation Then consider a circular microstrip patch with the radius 9.92 mm on the middle of the substrate that has the same electrical properties as we used for the rectangular microstrip patch but slightly smaller side length 111 mm (about one at 2.71 GHz). The diameter of the probes is also 1.0 mm. It is found that the is circular microstrip antenna driven at matched to a 50- single-ended signal source. The simulated occurs at 2.58 GHz. resonance of the fundamental mode

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TABLE I RESONANCE AND ELECTRICAL SEPARATION OF THE DIFFERENTIALLY-DRIVEN RECTANGULAR MICROSTRIP ANTENNA ON THE ELECTRICALLY THICK SUBSTRATE OF HIGH PERMITTIVITY

TABLE II RESONANCE AND ELECTRICAL SEPARATION OF THE DIFFERENTIALLY-DRIVEN CIRCULAR MICROSTRIP ANTENNA ON THE ELECTRICALLY THICK SUBSTRATE OF HIGH PERMITTIVITY

The measured resonance of the fundamental mode occurs at 2.71 GHz [11]. The simulator can predict the occurrence of resonance of the fundamental mode of the circular microstrip antenna; however, the predicted resonant frequency has 4.8% error from the measured result. We locate the second at the mirror point of along the driving point within the patch. It is found that the cirline cular microstrip antenna driven at and is matched to a 100- differential signal source. of The simulated resonance of the fundamental mode the differentially-driven circular microstrip antenna occurs at indicating 2.575 GHz. The electrical thickness is that the differentially-driven circular microstrip antenna is indeed on an electrically thick substrate. Table II shows the simuas a function of electrical separation. lated input impedance occurs Note that the resonance of the fundamental mode . The larger the for the electrical separation electrical separation, the lower the resonant frequency is and the higher the resonant resistance is. For example, the resonant frequency is 2.55 GHz and the resonant resistance is 545 for , while the resonant frethe electrical separation quency decrease to 2.49 GHz and the resonant resistance in. It creases to 1200 for the electrical separation is also seen that the resonance of the fundamental mode does not occur for the electrical separation . The input resistance is small and the input impedance is inductive. at 2.575 GHz for the electrical For example, . separation

TABLE III RESONANCE AND ELECTRICAL SEPARATION OF THE DIFFERENTIALLY-DRIVEN RECTANGULAR MICROSTRIP ANTENNA ON THE ELECTRICALLY THIN SUBSTRATE OF LOW PERMITTIVITY

B. Electrically Thin Substrate of Low Permittivity Consider a square RT/duriod 5880 substrate of side length (about one at 3.8 GHz), thickness , dielectric constant and dielectric loss . A rectangular microstrip patch that has tangent the length in the Y direction 25 mm and the width in the X direction 40 mm is on the middle of the substrate. The diameter of the probes is 1.0 mm. It is found that the rectangular microstrip is matched to a 50antenna driven at single-ended signal source. The simulated resonance of the funoccurs at 3.83 GHz. The measured resodamental mode occurs at 3.92 GHz [12]. nance of the fundamental mode The simulator can acceptably predict the occurrence of resonance of the fundamental mode of the rectangular microstrip anat the mirror tenna. We locate the second driving point along the line within the patch point of for the best differential signal operation. It is found that the rectand angular microstrip antenna driven at is matched to a 100- differential signal source. The simulated resonance of the fundamental mode of the differentially-driven rectangular microstrip antenna still occurs at 3.83 GHz, the same as that of the single-ended rectangular microstrip antenna. The electrical thickness is indicating that the differentially-driven rectangular microstrip antenna is indeed on an electrically thin substrate. Table III shows the simulated input impedance as a function of electrical separation. Note that the resonance occurs for the electrical sepaof the fundamental mode ration . The larger the electrical separation, the lower the resonant frequency is and the higher the resonant resistance is. For example, the resonant frequency is 3.8 GHz and the resonant resistance is 280 for the electrical separation , while the resonant frequency decreases to 3.78 for the GHz and the resonant resistance increases to 488 . It is also seen that the electrical separation does not occur for resonance of the fundamental mode the electrical separation . The input resistance is small and the input impedance is inductive. For example, at 3.83 GHz for the electrical separation . Then consider a square RT/duriod 5870 substrate of side (about one at 3.8 GHz), thickness length

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TABLE IV RESONANCE AND ELECTRICAL SEPARATION OF THE DIFFERENTIALLY-DRIVEN CIRCULAR MICROSTRIP ANTENNA ON THE ELECTRICALLY THIN SUBSTRATE OF LOW PERMITTIVITY

, dielectric constant and dielectric loss tangent . A circular microstrip patch with the radius 14.85 mm is on the middle of the substrate. The diameter of the probes is also 1.0 mm. It is found that the circular microstrip antenna driven at is matched to a 50- single-ended signal source. The simulated occurs at 3.72 GHz. resonance of the fundamental mode also The measured resonance of the fundamental mode occurs at 3.72 GHz [12]. Hence, the simulator can accurately predict the occurrence of resonance of the fundamental mode of the circular microstrip antenna. We locate the second driving at the mirror point of along the line point within the patch. It is found that the circular and microstrip antenna driven at is matched to a 100- differential signal source. of The simulated resonance of the fundamental mode the differentially-driven circular microstrip antenna still occurs at 3.72 GHz, the same as that of the single-ended circular miindicrostrip antenna. The electrical thickness is cating that the differentially-driven circular microstrip antenna is indeed on an electrically thick substrate. Table IV shows the as a function of electrical separasimulated input impedance oction show that the resonance of the fundamental mode . The larger curs for the electrical separation the electrical separation, the lower the resonant frequency is and the higher the resonant resistance is. For example, the resonant frequency is 3.69 GHz and the resonant resistance is 550 for the electrical separation , while the resonant frequency increases to 3.675 GHz and the resonant resistance de. It creases to 800 for the electrical separation is also seen that the resonance of the fundamental mode does not occur for the electrical separation . The input resistance is quite small and the input impedance is inducat 3.775 GHz for the tive. For example, . electrical separation C. On Electrical Separation and Fundamental Resonance Having observed the dependence of occurrence of resonance of the fundamental mode on electrical separation of dual-probe feeds for differentially-driven microstrip antennas, we now explain it as follows. For a dual-probe-feed microstrip antenna, it is known that the dual-probe feeds introduce not only the self

Fig. 2. Magnitude of electric field at 3.75 GHz on the patch of the circular microstrip antenna with dimensions given in this Section II-B.

and mutual inductances but also the capacitance between them. The mutual inductance and the capacitance depend on electrical (or physical) separation, the larger an electrical separation is, the smaller the mutual inductance and the capacitance are. It is found that regardless of electrical separation, the dual-probe feeds contribute inductively rather than capacitively to the input over the frequency range. This is because the impedance capacitance between the dual-probe feeds, although relatively larger for a smaller electrical separation, is still too small; while the inductance of the dual-probe feeds, which is the sum of the positive self and negative mutual inductances, is still quite large even for the smaller electrical separation. Hence, one can conclude that it is not the inductance of the dual-probe feeds that prevents the occurrence of resonance of the fundamental mode for the differentially-driven microstrip antenna when the electrical separation condition is satisfied. Rather, it is the fundamental mode that cannot be well excited when the dual-probe feeds are brought physically closer to a certain extent or the electrical separation condition is satisfied. Under such circumstances, the dual-probe feeds are located in the weak field region of the fundamental mode, thus making it impossible to strongly excite the fundamental mode but higher order modes. Since the resonant wavelength of higher order modes is shorter for the given microstrip patch dimension, the resistance generally decreases with mode order, thus resulting in a smaller resistance and inductive impedance [2]. Fig. 2 shows the magnitude of electric field on the patch of the circular microstrip antenna driven differentially with the dual-probe feeds and with a single-probe feed, respectively. The weak field region for the single-probe feed, as predicted from the cavity model, is an approximately-elliptical zone near a diameter of the patch. The minor axis of a zone is always along the line drawn from the feed point to the patch center, while the major axis of the zone is, of course, along the patch diameter. The weak field region for the dual-probe feeds is also an approximately-elliptical zone but its position may not be always near a diameter of the patch. It depends on the location

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of the dual-probe feeds. For instance, the dual-probe feed location in Fig. 2(a) forces the zone in a vertical position; while the dual-probe feed location in Fig. 2(c) makes the zone in a horizontal position. The zone is a consequence of the vector addition of the field components excited, respectively, by the dual-probe feeds. Hence, the zone may deviate from a diameter of the patch if the dual-probe feeds are located asymmetrically to the diameter. The distribution of the weak-field region of the fundamental mode of the rectangular microstrip antenna driven differentially with the dual-probe feeds is more complicated than that driven with a single-probe feed. For the single-probe feed, the weakfield region of the excited fundamental mode is always in the vicinity of the patch middle line parallel to the X axis. This line is defined here as the H line. While for the dual-probe feeds and due to their differential nature, the weak-field region of the excited fundamental mode may not be always in the vicinity of the H line. It is in the vicinity of the H line only if the dual-probe feeds are placed along the same line or respectively two different lines parallel to the Y axis and symmetrically to the H line as well. For such a case, the rectangular microstrip antenna is an efficient differential radiator and the electrical separation is an important and useful design parameter. The differential signal from a practical circuit such as a power amplifier exhibits the amplitude difference and phase imbalance. If such an imperfect differential signal is used to drive the rectangular microstrip antenna, the dual-probe feeds should be placed along the same line or respectively two different lines parallel to the Y axis and asymmetrically to the H line. If one feed is above and the other is below the H line, the distribution of the weak-field region of the excited fundamental mode will still be in between the dual-probe feeds but deviate from the vicinity of the H line, the fundamental mode still can be excited and the rectangular microstrip antenna still radiate if the electrical separation condition is still satisfied. However, the fundamental mode is not excited as strong as the previous case and the rectangular microstrip antenna is not an efficient differential radiator any longer. If the dual-probe feeds are both either above or below the H line, the distribution of the weak-field region of the fundamental mode will move as a function of time and not necessarily be in between the dual-probe feeds. The fundamental mode cannot be well excited and the microstrip patch antenna radiates quite poorly. Furthermore, for impedance matching the single-probe feed is often located at a point away from the patch middle line, which is perpendicular to the X axis and is defined here as the V line. This suggests that the rectangular microstrip antenna be driven differentially with the dual-probe feeds away from the V line but symmetrically to it. For this case, although the dual-probe feeds are located in the strong-field region of the fundamental mode, the fundamental mode cannot be excited but destroyed due to a vertical weak-field region created by the differential dual-probe feeds. Since no fundamental mode is excited, neither does the resonance occur. The input resistance is near to zero and the input impedance is inductive. The electrical separation becomes meaningless. The weak field region should be more appropriately termed as the non-resonant or inductive region. It is found that the non-resonant region is smaller for the same microstrip antenna

Fig. 3. Slot effect on Magnitude of electric field at 3.75 GHz on the patches of both circular and rectangular microstrip antennas with dimensions given in this Section II-B and the same color scale as in Fig. 2.

Fig. 4. Input impedance as a function of frequency with and without the slot: (a) circular and (b) rectangular microstrip antennas with dimensions given in this Section II-B.

driven differentially with dual-probe feeds than that driven with a single-probe feed. This implies that the fundamental mode cannot be well excited by the single-probe feed but still can be well excited differentially by the dual-probe feeds with one located at the same place of the single-probe feed and the other at the mirror image place if the electrical separation condition is satisfied. Now that the electrical separation condition is related with the weak field region, the perturbation of the weak field region will affect the degree of electrical separation and may cause the occurrence of resonance of the (quasi) fundamental mode even if the electrical separation condition is not satisfied. Fig. 3

ZHANG: ELECTRICAL SEPARATION AND FUNDAMENTAL RESONANCE OF DIFFERENTIALLY-DRIVEN MICROSTRIP ANTENNAS

Fig. 5. Measured and calculated input impedance as a function of frequency for the rectangular differentially-driven microstrip antennas: (a) = : and (b) = : .

= 0 11

= 0 08

shows that the weak field regions on the patches of both circular and rectangular microstrip antennas have been perturbed by the small rectangular slots [13]. Fig. 4 compares their input as a function of frequency with and without the impedance slots for fixed electrical separations. It is evident from these figures that the slots do make the resonances of the (quasi) fundamental modes occur, but at lower frequencies, as expected. III. EXPERIMENTAL VALIDATION AND DISCUSSION Both rectangular and circular differentially-driven microstrip antennas were constructed on the RT/duriod 5880 substrate for experimental validation. To have with a sufficient physical separation so as to make it possible to feed the antennas with 3.5-mm SMA connectors, we designed differentially-driven microstrip antennas to operate at 2.4 GHz . For rectangular microstrip antennas, patches and are on the middle of of the substrates; while for circular microstrip antennas, patches are on the middle of the substrates. The of diameter of the feed probes is 1.0 mm. An Agilent network analyzer E5062A was used to measure their -parameters in an anechoic chamber. The measured -parameters can be [9]. converted to the differential input impedance as a function Fig. 5 shows the measured input impedance of frequency for the rectangular differentially-driven microstrip

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Fig. 6. Measured and calculated input impedance as a function of frequency : and for the circular differentially-driven microstrip antennas: (a) = (b) = : .

= 0 08

= 0 07

antennas with electrical separations and 0.11, respectively. It is clear from the figure that the fundamental mode does not resonate if the electrical separation condition is not satisfied but does resonate if the electrical separation condition is as a satisfied. Fig. 6 shows the measured input impedance function of frequency for the circular differentially-driven miand crostrip antennas with electrical separations 0.08, respectively. Again, when the electrical separation condition is satisfied, the fundamental resonance occurs. Otherwise, it does not occur. are also shown in Figs. 5 The simulated input impedance and 6. There are small frequency (the resonant frequency or the frequency at which the resistance reaches the peak) differences less than 1% between the simulated and measured input impedances. They are mainly caused by the patch size tolerance [14]. One slightly reduces the circular patch diameter and increases the rectangular patch length, the frequency differences diminish. There are large magnitude differences between the simulated and measured input impedances. The simulated magnitudes are generally smaller than the measured ones, which can be attributed to the enlarged electrical separation due to warpage. The different stresses between the metal ground plane and the dielectric substrate yield the noticeable warpage in the fabricated microstrip antenna. The warpage was not included in simulations. However, the warpage effect was modeled by a slight increment in the electrical separation in our simulations.

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As expected, the enlarged electrical separation makes the magnitude differences reduce significantly. IV. CONCLUSION The electrical separation, an important parameter in the design of differentially-driven microstrip antennas, has been defined in this paper. It has been demonstrated that the occurrence of the fundamental resonance of a differentially-driven microstrip antenna is related with the electrical separation. When the electrical separation condition is satisfied, the resonance occurs. Otherwise, the resonance does not occur. The electrical separation condition is an empirical factor . It has been found for the first time that is smaller for electrically thicker substrate of higher permittivity than that for electrically thinner substrate of low permittivity and is smaller for circular patch than that for rectangular patch. More importantly, it has also been found that the electrical separation condition is related with the weak field region under the patch. A simple technique of cutting a slot in the patch to perturb the weak field region so as to alter the degree of electrical separation has been simulated. It has been shown that the slot in the patch can cause the occurrence of the fundamental resonance even if the electrical separation condition is not satisfied. The weak field region is more appropriately termed as non-resonant or inductive region. It has been found that the non-resonant region is smaller for the same microstrip antenna driven differentially with dual-probe feeds than that driven with a single-probe feed. ACKNOWLEDGMENT The author would like to thank Mr. B. Zhang for his assistance in this work. REFERENCES [1] Y. T. Lo, D. Solomon, and W. F. Richards, “Theory and experiment on microstrip antennas,” IEEE Trans. Antennas Propag., vol. 27, no. 2, pp. 137–145, 1979. [2] W. F. Richards, Y. T. Lo, and D. D. Harrison, “An improved theory for microstrip antennas and applications,” IEEE Trans. Antennas Propag., vol. 29, no. 1, pp. 38–46, 1981. [3] A. R. Behzad, M. S. Zhong, S. B. Anand, L. Li, K. A. Carter, M. S. Kappes, T. H. Lin, T. Nguyen, D. Yuan, S. Wu, Y. C. Wong, V. Gong, and A. Rofougaran, “A 5-GHz direct-conversion CMOS transceiver utilizing automatic frequency control for the IEEE 802.11 a wireless LAN standard,” IEEE J. Solid-State Circuits, vol. 38, no. 12, pp. 2209–2220, 2003. [4] Y. P. Zhang, J. J. Wang, Q. Li, and X. J. Li, “Antenna and transmit/receive switch for single-chip radio transceivers of differential architecture,” IEEE Trans. Circuits Syst. I, vol. 55, no. 11, pp. 3564–3570, 2008. [5] W. R. Deal, V. Radisic, Y. X. Qian, and T. Itoh, “Integrated-antenna push-pull power amplifiers,” IEEE Trans. Microwave Theory Tech., vol. 47, no. 8, pp. 1418–1425, 1999.

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[6] W. Wong and Y. P. Zhang, “0.18-  CMOS push-pull power amplifier with antenna in IC package,” IEEE Microwave Wireless Comp. Lett., vol. 14, no. 1, pp. 13–15, 2004. [7] P. Abele, E. Ojefors, K. B. Schad, E. Sonmez, A. Trasser, J. Konle, and H. Schumacher, “Wafer level integration of a 24 GHz differential SiGe-MMIC oscillator with a patch antenna using BCB as a dielectric layer,” in Proc. 11th GAAS Symp., Munich, 2003, pp. 419–422. [8] T. Brauner, R. Vogt, and W. Bachtold, “A differential active patch antenna element for array applications,” IEEE Microwave Wireless Comp. Lett., vol. 13, no. 4, pp. 161–163, 2003. [9] Y. P. Zhang and J. J. Wang, “Theory and analysis of differentiallydriven microstrip antennas,” IEEE Trans. Antennas Propag., vol. 54, no. 4, pp. 1092–1099, 2006. [10] Y. P. Zhang, “Design and experiment on differentially-driven microstrip antennas,” IEEE Trans. Antennas Propag., vol. 55, no. 10, pp. 2701–2708, 2007. [11] D. H. Schuabert, D. M. Pozar, and A. Adrian, “Effect of microstrip antenna substrate thickness and permittivity: Comparison of theories with experiment,” IEEE Trans. Antennas Propag., vol. 37, no. 6, pp. 677–682, 1989. [12] Y. B. Gan, C. P. Chua, and L. W. Li, “An enhanced cavity model for microstrip antennas,” Microw. Opt. Technol. Lett., vol. 40, no. 6, pp. 520–523, 2004. [13] K. F. Tong, K. M. Luk, K. F. Lee, and R. Q. Lee, “A broad-band u-slot rectangular patch antenna on a microwave substrate,” IEEE Trans. Antennas Propag., vol. 48, no. 6, pp. 954–960, 2000. [14] D. M. Pozar, “Input impedance and mutual coupling of rectangular microstrip antennas,” IEEE Trans. Antennas Propag., vol. 30, no. 6, pp. 1191–1196, 1982. Y. P. Zhang (M’03–SM’07–F’10) received the B.E. and M.E. degrees from Taiyuan Polytechnic Institute and Shanxi Mining Institute of Taiyuan University of Technology, Shanxi, China, in 1982 and 1987, respectively, and the Ph.D. degree from the Chinese University of Hong Kong, Hong Kong, in 1995, all in electronic engineering. From 1982 to 1984, he was with the Shanxi Electronic Industry Bureau; from 1990 to 1992, the University of Liverpool, Liverpool, U. K.; and from 1996 to 1997, City University of Hong Kong. From 1987 to 1990, he taught at the Shanxi Mining Institute and from 1997 to 1998, the University of Hong Kong. He was promoted to a Full Professor at Taiyuan University of Technology in 1996. He is now an Associate Professor and the Deputy Supervisor of Integrated Circuits and Systems Laboratories with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore. He has broad interests in radio science and technology and published widely across seven IEEE societies. Dr. Zhang received the Sino-British Technical Collaboration Award in 1990 for his contribution to the advancement of subsurface radio science and technology. He received the Best Paper Award from the Second International Symposium on Communication Systems, Networks and Digital Signal Processing, July 18–20, 2000, Bournemouth, U.K., and the Best Paper Prize from the Third IEEE International Workshop on Antenna Technology, March 21–23, 2007, Cambridge, U.K. He has organized/chaired dozens of technical sessions of international symposia. He was awarded a William Mong Visiting Fellowship from the University of Hong Kong in 2005. He has delivered scores of invited papers/keynote addresses at international scientific conferences. He was a Guest Editor of the International Journal of RF and Microwave Computer-Aided Engineering and an Associate Editor of the International Journal of Microwave Science and Technology. He serves as an Editor of ETRI Journal, an Associate Editor of the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, and an Associate Editor of the International Journal of Electromagnetic Waves and Applications. He also serves on the Editorial Boards of a large number of Journals including the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES and IEEE MICROWAVE AND WIRELESS COMPONENTS LETTERS.

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A Modal Approach to Tuning and Bandwidth Enhancement of an Electrically Small Antenna Jacob J. Adams, Graduate Student Member, IEEE, and Jennifer T. Bernhard, Fellow, IEEE

Abstract—We describe the physical phenomena that contribute to the behavior of an electrically small 10 antenna using characteristic mode theory. The application of characteristic modes to antenna tuning and bandwidth enhancement serves as demonstration of the broad utility of the modal technique. A modal analysis of the 10 antenna’s impedance match yields several interesting observations as to the nature of resonances and antiresonances, which has implications for the impedance matching of small antennas in general. Furthermore, to overcome the bandwidth limitations inherent in small antennas, we determine that multiple resonances must be combined and use a conductance ratio as a figure of merit for design. We then investigate the 10 antenna’s potential for multiresonant operation by examining different candidate modes. Using the appropriate characteristic modes to form multiple resonances, we show how the bandwidth of the 10 antenna can be designed to be nearly double that expected from the physical limit for a single resonance.

TM

TM

TM

TM

Index Terms—Bandwidth, characteristic modes, electrically small antenna, multimode, multiresonant, Q, spherical antenna.

I. INTRODUCTION

E

LECTRICALLY small antennas have been long studied in the antenna community because of the obvious advantages of miniaturization and the challenges of small antenna design. An early seminal work by Chu [1] showed that the size of an electrically small antenna constrains its minimum radiation . Radiation is critical for a singly resonant quality factor antenna, because the antenna’s bandwidth is inversely proportional to [2], making a low antenna desirable. The lower limit determined by the electrical size of a sphere of radius which circumscribes the antenna is [2], [3]

(1) where is the wavenumber and is the antenna efficiency. This bound is commonly called the “Chu limit” and we will use this name. A singly resonant antenna with a that approaches Chu’s limit has been sought for decades, but it remains an elusive

Manuscript received March 06, 2010; revised August 13, 2010; accepted November 19, 2010. Date of publication March 03, 2011; date of current version April 06, 2011. This work is supported under a National Science Foundation Graduate Research Fellowship. The authors are with the Electromagnetics Laboratory, Department of Electrical and Computer Engineering, University of Illinois, Urbana, IL 61801 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.2011.2109683

target. However, several recent developments have led to improving small antenna designs. First, Foltz and McLean showed that an antenna occupying a spherical volume achieves the minimum , rather than a dipole or planar structure [4]. Also, because Chu’s analysis did not include energy stored inside the radius sphere, Thal reconsidered the problem and showed that a higher limit exists on the minimum for spherical antennas [5]. In addition to defining a new limit about 1.5 times larger than that predicted by Chu’s analysis, Thal also found that the spherical mode has the lowest [5]. Following the conclusions of Foltz, McLean, and Thal, the present authors developed a hemispherical, low antenna that excites the mode [6], [7]. Our fundamental design approach resulted in a versatile design that is easily matched regardless of electrical size, operating frequency, or system impedance. To further elucidate the fundamental phenomena at work in this electrically small antenna, we now analyze the antenna using characteristic mode theory (CMT) [8]–[10]. Although discussions of modes abound in the literature, CMT describes modes that are rigorously defined and provide physical insight. The characteristic mode approach is valuable because it reveals information (namely, the individual modes and their eigenvalues) that is otherwise lost by standard electromagnetic simulation techniques [11]. CMT has been occasionally applied to a range of antenna problems [12]–[14] including packaging [15], [16] and MIMO design [17]–[19] but has not been widely used. We first review characteristic mode theory in Section II, and in Section III we discuss the characteristic modes of the structure. The analysis of this particular antenna is just one example of how CMT can be applied, and the conclusions we draw from this analysis are widely applicable for electrically small antenna design. In Section IV, we examine the distinct origins of resonances and antiresonances based on characteristic mode theory. The fundamental differences between these types of resonances have some interesting implications for small antennas. Finally, we consider a method of extending small antenna bandwidth beyond what Chu’s limit suggests by combining multiple resonances. After discussing a general approach to multiresonant small antenna design in Section V, we apply characteristic mode theory to select modes for multiresonant operation in Sections VI and VII. Using the modes discussed in Section VII, antenna with bandwidth sigwe design a multiresonant nificantly greater than the Chu limit. II. CHARACTERISTIC MODE THEORY The Theory of Characteristic Modes originated as a result of work by Garbacz and Turpin [8] and was refined soon after by Harrington and Mautz [9], [10]. The theory proposes a set of

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characteristic current modes that form an orthonormal basis for the current on a metal structure such as an antenna. A recent review of CMT can be found in [20]. mode there is an associated real eigenvalue For the and eigenvector which are the solution to the generalized eigenvalue problem

(2) where and are the real and imaginary parts of the Moment Method impedance matrix, respectively. This eigenvalue problem is solved at every frequency of interest, yielding a large set of eigenmodes at each frequency. The eigenvectors represent the cophasal modal current distribution, and the eigenvalues determine how capacitive or inductive the mode is and how well it radiates. The eigenvalues and eigenvectors both vary with frequency, but the eigenvectors tend to vary slowly, retaining the same general modal pattern. To solve the eigenvalue problem, we first used FEKO [21] to model the antenna and generate the impedance matrix. We then solve the eigenvalue problem and organize solutions in Matlab, exporting the results back to FEKO where current distributions and fields can be plotted. The total current on the antenna can be represented as a weighted sum of the eigencurrents. Similarly, the input admittance can be expressed as a summation of the admittances of each mode. For the case of a gap voltage source on a wire wire segment, the input admittance can be located at the calculated as [22]

Fig. 1. A 4-arm TM monopole. The wire pitch, p, is measured on the outer radius of the arms. The helical coils have radius r and their centerlines follow a circle of radius b from the center of the structure. The width of the trace is w and the probe feed makes contact from underneath the ground plane at the center of the feed trace.

Fig. 2. Characteristic modes of TM antenna. Unfilled arrows represent currents on the planar feed lines, solid arrows represent currents in the helical arms and solid circles represent current nulls. (a) Mode 1; (b) Mode 2.

(3) For small structures, there are two types of characteristic modes: resonant and non-resonant modes. Resonant modes are capacitive at low frequencies, resonate, and become inductive beyond their resonance. Non-resonant modes (inductive modes) begin as inductive modes and never resonate, only ever contributing inductive susceptance to the antenna. III. CHARACTERISTIC MODES OF THE

ANTENNA

The

antenna fundamentally operates by exciting the spherical mode, which Thal showed to have the lowest possible [5]. The mode can be excited by a surface current distribution of the form

(4) The antenna structure consists of helical wires coiled along the constant- lines of a sphere which support the current distribution of (4) [7]. The sphere is bisected by a ground plane . A conducting trace suspended above the ground along plane contacts all of the arms at their base. The trace is fed in the center by a probe from behind the ground plane. Fig. 1 shows the structure using four arms.

The mode can be excited at different electrical sizes by changing the wire pitch. We have previously demonstrated the and [7], and simulated results antenna at have shown even smaller values are feasible. In this paper, . we focus on the behavior of the antenna when antenna Since we are interested in the behavior of the in the electrically small region, the number of significant characteristic modes is small when excited at the feed location indicated. As we will show, the superposition of just two modes results in a tunable, low antiresonance and allows a multiresonant, wideband response if properly combined. Fig. 2 illustrates the currents of the first and second characteristic modes. Mode 1 looks very similar to the current distribution spherical mode that the antenna was designed to of the excite, further verifying our initial design methodology. Mode 2 has a larger current along the feed trace than in the arms and a null near the base of each arm. Both characteristic modes are normal resonant modes as discussed in Section II (i.e., they are not special inductive modes). The modal eigenvalues are often presented in terms of the [20]. Modes are characteristic angle, , inductive when capacitive when , and resonant when . The characteristic angles of the two modes over frequency are shown in Fig. 3.

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Fig. 5. An approximate circuit model for antenna input impedance from characteristic mode theory.

Fig. 3. Characteristic angle of TM of the full structure noted.

antenna modes with resonant frequencies

Fig. 4. Normalized elevation and azimuth patterns of the first two characteristic modes of TM antenna on a substrate with dielectric constant  = 2:94 at 1050 MHz. Only the  component is shown because cross-polarization is insignificant. (a) Elevation; (b) Azimuth.

Mode 1 has the smallest eigenvalue magnitude over the operating band and provides most of the antenna’s radiation. The Mode 1 resonance falls at the same frequency as the resonance of the entire structure. As with all resonant characteristic modes, the mode is capacitive before its resonance and inductive after its resonance. The Mode 2 resonance is significantly higher in frequency than either the first resonance or antiresonance of the entire antenna. Thus it presents a capacitive admittance in the operating band of the antenna. Fig. 4 shows the polarized radiation pattern of each mode at 1050 MHz. Cross-polarization is insignificant for both modes. IV. MODAL INTERACTIONS AND ANTENNA INPUT IMPEDANCE As discussed in Section II, the input admittance of an antenna can be expressed as a sum of the admittances of its characteristic modes from (3). Individual characteristic modes behave as series RLC circuits in that they have declining capacitive susceptance before their single resonance and increasing inductive susceptance after their resonance (except in the case of special inductive modes which are always inductive). When multiple modes are excited in the same structure, it is similar to connecting these series RLC resonators in parallel since the admittances add directly from (3). This circuit model, shown in

Fig. 5, suggests that resonances and antiresoof the entire structure are caused by funnances damentally different phenomena. Antenna resonances are generally caused by the resonance of a single dominant characteristic mode while antiresonances must be the result of the interaction of two or more characteristic modes, some that are capacitive and some that are inductive (either non-resonant or at a frequency above their resonance). Because resonances are the result of a single characteristic mode, resonant properties depend almost entirely on the modal properties. Resonant conductance is typically very high for electrically small resonant modes due to large currents at the feed. This is the cause of the small resonant resistance that is often observed in electrically small antennas [23], [24]. Since the resonant conductance is set by the feed-point current squared from (3), the ability to reduce conductance at the electrically small antenna’s resonance through geometric manipulation is limited. Thus the modal theory suggests that a different approach to impedance matching is warranted. Consider the antenna model in Fig. 5 with just two modes (i.e., higher order modes appears as open circuits). The radiaand cannot be significantly changed to tion resistances adjust resonant properties, but the antiresonant resistance is a function of many variables, including each mode’s radiation resistance, , and resonant frequency. If these other parameters can be adjusted, the antiresonance can be matched to a much wider range of system impedances. A. Tuning the

Antenna

As a result of its two significant characteristic modes discussed in Section III, the helical antenna exhibits both a resonance and an antiresonance in the electrically small region [7]. According to Fig. 3, the antenna’s resonance occurs at nearly the same frequency as the Mode 1 resonance, and only Mode 1 contributes significantly to the admittance at that frequency. However, consider the antenna’s antiresonance, which falls between the Mode 1 resonance and the Mode 2 resonance. We can postulate that the antiresonance occurs when the capacitive susceptance of Mode 2 cancels, then exceeds, the inductive susceptance of Mode 1. To further investigate the source of the antenna’s antiresonance, the modal admittances of the antenna are calculated using (3). The conductance and susceptance of the two significant modes are shown in Fig. 6 for several different values of , the substrate dielectric constant. From the figure, it is clear Mode 2

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changes in the feed lines and substrate. On the other hand, Mode 1’s resonance is moved primarily by changing the wire pitch since the majority of the current is in the arms. For an impedance match to 50 , the conductance at resonance or antiresonance should be 20 mS. In all cases, it is clear that the conductances at the modal resonances are orders of magnitude above this value and are difficult to change. When the air substrate is used, the conductance at the antiresonance is 1 mS. As the modes are moved closer together, the antenna antiresonance occurs at a frequency where the con, the antiresonant conducductance is higher, and when tance is 20 mS. Thus, by modifying the spacing between the modes, the antenna’s antiresonant resistance can be controlled with small changes in the operating frequency as observed in [7]. V. MULTIPLE RESONANT BEHAVIOR OF SMALL ANTENNAS

Fig. 6. Admittance of the significant characteristic modes for substrates with varying dielectric constant. The total antenna admittance is shown by the black ; (b)  : ; (c)  . line. (a) 

=1

= 2 94

=6

resonates at a significantly lower frequency when a higher dielectric constant material is used, while Mode 1’s resonant frequency changes little. Since Mode 2 has a larger current along the feed trace than in the arms, its resonant frequency is more strongly influenced by

Chu’s limit suggests that the antenna’s cannot be further decreased by much, but this does not necessarily mean its bandwidth cannot be enhanced. One of the implicit assumpis that the antenna’s tions required to relate bandwidth to impedance response can be modeled as a single series or parallel RLC circuit over the frequency range of interest. However, two or more resonances created near enough to each other cannot be modeled by a single RLC circuit [25]. Consider the modal circuit model in Fig. 5 where at least two resonators must exist if an antiresonance exists. While Chu’s limit still applies to the of each mode, the inverse relationship between and bandwidth no longer holds [26]. Then, it is possible to overcome the implicit bandwidth limitation imposed by Chu, although not the limit. The Goubau antenna is likely the most famous multimode design [27], [28]. Goubau achieved a very large impedance bandwidth, but the antenna was not electrically small ( was slightly larger than 1). Few multiresonant electrically small antennas have been reported. Recently Stuart and Tran de[29]. With a signed a multiresonant antenna with matching network, the antenna achieved 2:1 VSWR bandwidth approximately equal to that expected for a singly resonant antenna at the Chu limit. Stuart and Best also reported a wideband and 1.7 times the impedance bandwidth antenna with predicted by Chu’s limit [30]. For singularly resonant small antennas, is the salient factor in evaluating an antenna’s bandwidth potential. However, when combining multiple modes, several additional factors contribute to the attainable bandwidth, including the frequency spacing of the modes and their conductance maxima and minima. The of an individual mode no longer has critical importance, although it will affect how far apart in frequency modes can be before VSWR specifications are violated. To reduce the complications that these additional factors bring, we wish to develop a simple figure of merit to evaluate a multimode design. Consider matching an antenna to an arbitrary system . In order for the VSWR to remain below a deimpedance sired value at both a resonance and an antiresonance, the ratio of the resonant conductance to the antiresonant conductance must be less than or equal to the square of the desired VSWR. This

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observation is easily derived from the equations for VSWR. At then the VSWR can be written a resonant point

(5) and at We then assume that at a resonance, since we desire that the anan antiresonance tenna’s impedance encircle the center of the Smith chart for minimum VSWR over the entire bandwidth. If we assume at both resonances, we find that and should be chosen as that . Of course, this approach neglects the behavior of the susceptance and of the modes and is not a guarantee of multiresonant matchability but rather a minimum requirement. While approximate, this resonance-antiresonance conductance ratio (CR) is useful because it allows us to quickly evaluate an antenna’s multiresonant potential even when the conductance values are not centered around the desired system conductance. If the conductance has a swing greater than the square of the maximum VSWR between resonances and antiresonances, it is clear that it cannot have a VSWR below the desired value at both the resonant and antiresonant frequencies. In the next section, we use anthe CR to evaluate the multiresonant potential of the tenna. VI. LOW ORDER CHARACTERISTIC MODES FOR MULTIRESONANT BEHAVIOR Next we take this approach based on the conductance ratio antenna to make it multiresonant. We and apply it to the will focus on designing a structure with so that the antenna is electrically small. The most obvious way to create an additional mode in the antenna is to offset the pitches of half of the helices so that two low order, -like modes appear and resonate at different frequencies. The initial design consists of two arms and two arms with pitch , where the arms with with pitch the same pitch are placed opposite each other. As anticipated, this approach leads to two characteristic modes with slightly offset resonant frequencies. In the low frequency mode (Mode 1a), the current is highest is the arms with the smaller pitch and in the high frequency mode (Mode 1b), the current exists in the arms with the larger pitch. Fig. 7 contains a schematic showing these modes. However, while both modes are generally well excited by the feed location, an anomalous drop in the conductance of Mode 1a occurs with this configuration. A typical example of this response can be seen in Fig. 8. When the conductance drops to nearly zero in the band of interest, the CR becomes very large and multiresonant behavior is not possible. According to (3), such a drop in the conductance can either come from a large increase in the modal eigenvalue at that frequency or a sharp drop in the feed point current of the mode. In this case, the mode is near resonance and the eigenvalue is actually decreasing. Thus, there must be a drop in the feed point current. Fig. 7(a) shows the currents of Mode 1a. There are two notable current nulls on

Fig. 7. Characteristic modes of TM antenna with offset pitches. The arms have one of two pitches, p or p , as shown, where p < p . Filled arrows represent currents in the helical wires and unfilled arrows represent currents on the feed lines. Circles represent current nulls. (a) Mode 1a; (b) Mode 1b.

Fig. 8. Modal input conductance of a two mode antenna with offset pitches. The anomalous falloff of Mode 1a conductance is circled.

the feed traces associated with the inactive arm. As frequency increases, the nulls are observed to move along the trace toward the center until they reach the feed point and cause the conductance null. The current nulls approach the feed symmetrically along the inactive arms. Observing the characteristic modes, it seems that the symmetry of the structure supports the current null anomaly. To eliminate the nulls, the symmetry of the structure is broken by placing the arms with the same pitch adjacent to each other, rather than across from each other. This asymmetric configuration eliminates the current null and allows distinct Modes 1a and 1b to be excited as seen in Fig. 9. However, while the CR is lower than symmetric configuration, it is still much too high for multiresonant operation (Fig. 9(a)). While the conductance peaks are fundamental to the mode and difficult to change, the antiresonance can often be adjusted to some extent as discussed in Section IV. If the modes were closer together, their conductance curves might cross at a higher value and result in a higher antiresonance conductance minimum. and are moved closer together. To test this, the pitches Fig. 9(b) shows the conductance response under this scenario. However, the effect is not entirely as intended; the modes move closer together but Mode 1a begins to deteriorate as it moves close to 1b. As the pitches move even closer, Mode 1a continues -like to deteriorate, eventually collapsing into the single mode that appears when the pitches are the same. In order to operate the antenna at multiple resonances, a different set of

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Fig. 10. Modal input conductance of two mode, 8 arm antenna on 0.050 Duroid 6010 with b : ,r , ,p : ,w : . and wire

= 21 mm = 2 0 mm diameter = 0 6 mm

Fig. 9. Modal input conductance of two mode asymmetric, offset pitch antenna : ,p : and (b) p : ,p : . with (a) p , Both designs are on a 0.050 Duroid 6002 substrate and have b r : ,w , and wire diameter of 0.2 mm.

= 3 0 mm = 9 5 mm = 2 5 mm = 2 mm

= 3 5 mm

= 4 5 mm = 21 mm

modes that have either a lower resonant conductance or a higher antiresonant conductance must be found. VII. MULTIRESONANT BEHAVIOR USING HIGHER ORDER CHARACTERISTIC MODES Combining the two first order modes to yield a multiresonant structure seems impossible because of the very high CR that was found. However, higher order modes can also be excited in the structure in the electrically small region. As shown in Section IV, when the pitches are equal, a higher order mode (Mode 2) is excited. In Section IV-A, we used this mode to tune the impedance of a single antiresonance. Now we will attempt to use it to reduce the CR so that several resonances can be combined. Looking back to Fig. 6, we see that the CR decreases sigis changed. nificantly as the substrate dielectric constant Mode 2’s resonance decreases greatly when increases while the Mode 1 resonance hardly moves. As this happens, the minimum conductance increases and the CR decreases. As these modes get closer together, they do not degenerate into one as was observed with the low order modes. Thus, the combination of Mode 1 and Mode 2 appear to be better suited for achieving a small CR than the two low order modes. To further reduce the CR, variations of the original design were simulated. Geometric parameters such as the pitch of the arms, feed trace width, substrate dielectric constant, and number of helical arms were varied and their effects cataloged. These

= 6 mm

= 2 5 mm

data are beyond the scope of this paper and will not be exhaustively presented here, but several key points are highlighted. First, an increasing number of arms tends to decrease the CR. More arms allows the currents to better approximate the mode, resulting in a that is closer to the lower bound and in turn leading to a lower conductance ratio. However, the coupling between the more closely spaced arms requires a smaller wire pitch to achieve the same operating frequency. Next, when the feed arms are very wide and the substrate dielectric constant is high, an interesting effect is observed. The currents of Mode 2 differ somewhat from those shown in Fig. 2. Because the electrical length of the feed trace is now quite large, the null in the Mode 2 distribution, which was previously in the arms, is shifted to the feed trace. As a result, the Mode 2 current peak is now located near the base of the arms, which have a -like current distribution similar to Mode 1. Under this configuration, both modes have a low and both are effective radiating modes. However, since Mode 2 resonates at a frequency approximately 50% higher than Mode 1, it has a significantly lower conductance. This makes it a much better candidate for multimode combination with the antiresonance between Modes 1 and 2. Instead of combining the low-order mode (Mode 1) with the following antiresonance, we mode (Mode 2) with its precombine the higher-order ceding antiresonance and find that a much lower CR can be achieved. Furthermore, these two modes have very similar radiation patterns, resembling that of a monopole, so the total pattern remains consistent across the band and cross-polarization is very low. A. Multiresonant Antenna With Matching Network To demonstrate this concept, an 8 arm antenna on 0.050-inch thick Duroid 6010 was designed with the following parameters, , , , , and . Simulations of the lossless antenna wire were performed in FEKO. Given that the antenna structure has not changed significantly from the antenna in [7], we expect the efficiency to be 80–90% when fabricated. The input conductance of the two modes is shown in Fig. 10. -like mode was resonant around 850 MHz. In Fig. 6, the Now, the higher order mode is resonant at 1050 MHz, and we expect that this a larger electrical size, the conductance of the

ADAMS AND BERNHARD: A MODAL APPROACH TO TUNING AND BANDWIDTH ENHANCEMENT OF AN ELECTRICALLY SMALL ANTENNA

Fig. 11. Simulated VSWR of the multimode antenna with shunt capacitor connected to a 21 system impedance.

resonance will be lower. This is precisely what is observed: the resonant conductance of Mode 2 in Fig. 10 is an order of magnitude lower than Mode 1 in Fig. 6. Meanwhile the antiresonant conductance is approximately the same as it was in Fig. 6(c). Therefore, the CR is reduced by an order of magnitude to approximately 12; however, there is a large inductive susceptance. Agilent ADS was used to simulate connecting the antenna to a 21 system impedance and using a 43 pF shunt capacitor to resonate it. The matching network discussed here is only theoretical, but could be integrated into the feed network. We are also developing a practical matching technique that requires no external components to match directly to 50 . Fig. 11 shows a VSWR plot of the antenna using the theoretical matching network. Centered at approximately 1050 MHz, the antenna has 120 MHz of 2:1 VSWR bandwidth, which is about 11.4%. The antenna’s electrical size, from the bottom of the substrate to the outer radius of the furthest helix, . Using Chu’s limit to estimate the bandwidth of a is singly resonant antenna at this electrical size, we find that this antenna’s fractional bandwidth is about 31% larger than that achievable with a single resonance in the ideal case. Compared to the more practical limit derived by Thal [5], the antenna has 96% greater bandwidth than an ideal singly resonant antenna. It is important to emphasize that this does not violate Chu’s fundamental principle, which is a constraint only on the of an antenna. If we consider the non-radiating mode to be an additional “matching circuit” for the radiating mode and apply Fano’s matching limitations, the bandwidth of the mode can be at improved by at most a factor of 2.31 [31]. The ideal is approximately 11 [5], so the optimal 2:1 VSWR bandwidth for such an antenna is 15%, close to the value we have attained. On the other hand, if we consider an ideal mode, the attainable bandwidth matching network for a is over 60% at this electrical size [32], suggesting significant room for improvement through further research. VIII. CONCLUSION Our study of the characteristic modes of the electrically small antenna provides insight into the individual current modes that comprise the total current on the antenna.

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Observations of how the modes interact lead us to match the antenna’s antiresonance rather than its resonance. The modal study explains how we can use a non-radiating mode to match a large range of resistances without affecting the radiation properties. This straightforward modal impedance model will be beneficial to developing simple but rigorous models of new structures. To achieve bandwidth larger than the Chu limit, we concluded that we need to break the inverse -bandwidth relationship by exciting modes at closely spaced frequencies. We determined that a reasonable design approach is to keep the resonance-antiresonance conductance ratio to a minimum. Applying characteristic mode theory, we identified modes which are candidates for multiresonant operation and eliminate those which -like are not. The first design attempted with a low order mode was found to exhibit an interesting modal null, but was ultimately not useful for multiresonant operation. By using a -like characteristic mode to achieve a small higher-order CR, an electrically small multiresonant antenna was designed with half-power bandwidth greater than Chu’s limit by 31% and than Thal’s limit by 96% for an ideal singly resonant antenna. The usefulness of this multiresonant approach to bandwidth is enhancement of electrically small antennas around clear as bandwidth has been increased by a significant amount. may However, very electrically small structures require new approaches because the fundamentally large conductances associated with the resonances result in very large conductance ratios that are difficult to correct. Wideband operation may still be possible in terms of half power bandwidth , but the CR needed to achieve wideband operation in terms of bandwidth may be difficult. In future work, we ultimately intend to investigate the theoretical limits of multiresonant operation in electrically small anstructures. The possibility of further increasing the tenna’s bandwidth by combining additional modes will also be studied. REFERENCES [1] L. J. Chu, “Physical limitations of omni-directional antennas,” J. Appl. Phys., vol. 19, no. 12, pp. 1163–1175, Dec. 1948. [2] A. D. Yaghjian and S. R. Best, “Impedance, bandwidth, and of antennas,” IEEE Trans. Antennas Propag., vol. 53, no. 4, pp. 1298–1324, Apr. 2005. [3] J. S. McLean, “A re-examination of the fundamental limits on the radiation of electrically small antennas,” IEEE Trans. Antennas Propag., vol. 44, no. 5, pp. 672–676, May 1996. [4] H. D. Foltz and J. S. McLean, “Limits on the radiation of electrically small antennas restricted to oblong bounding regions,” in Proc. IEEE Antennas and Propagation Int. Symp., 1999, vol. 4, pp. 2702–2705. [5] H. L. Thal, “New radiation limits for spherical wire antennas,” IEEE Trans. Antennas Propag., vol. 54, no. 10, pp. 2757–2763, Oct. 2006. [6] J. J. Adams and J. T. Bernhard, “A low electrically small spherical antenna,” in Proc. IEEE Antennas and Propagation Int. Symp., 2008. [7] J. J. Adams, “Tuning method for a new electrically small antenna with low ,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 303–306, 2009. [8] R. J. Garbacz and R. H. Turpin, “A generalized expansion for radiated and scattered fields,” IEEE Trans. Antennas Propag., vol. 19, no. 3, pp. 348–358, May 1971. [9] R. F. Harrington and J. R. Mautz, “Theory of characteristic modes for conducting bodies,” IEEE Trans. Antennas Propag., vol. 19, no. 5, pp. 622–628, Sep. 1971.

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[10] R. F. Harrington, “Computation of characteristic modes for conducting bodies,” IEEE Trans. Antennas Propag., vol. 19, no. 5, pp. 629–639, Sep. 1971. [11] P. Hazdra and P. Hamouz, “On the modal superposition lying under the MoM matrix equations,” Radioengineering, vol. 17, no. 3, pp. 42–46, Sept. 2008. [12] R. J. Garbacz and D. M. Pozar, “Antenna shape synthesis using characteristic modes,” IEEE Trans. Antennas Propag., vol. 30, no. 3, pp. 340–350, May 1982. [13] B. D. Raines, K. A. Obeidat, and R. G. Rojas, “Characteristic modebased design and analysis of an electrically small planar spiral antenna with omnidirectional pattern,” presented at the Antennas and Propagation Int. Symp., 2008. [14] K. A. Obeidat, B. D. Raines, and R. G. Rojas, “Design and analysis of a helical spherical antenna using the theory of characteristic modes,” presented at the Proc. Antennas and Propagation Int. Symp., 2008. [15] E. H. Newman, “Small antenna location synthesis using characteristic modes,” IEEE Trans. Antennas Propag., vol. 27, no. 4, pp. 530–531, Jul. 1979. [16] C. T. Famdie, W. L. Schroeder, and K. Solbach, “Optimal antenna location on mobile phones chassis based on the numerical analysis of characteristic modes,” in Proc. 37th European Microwave Conf., Munich, Germany, Oct. 2007, pp. 987–990. [17] N. Belmar-Moliner, A. Valero-Nogueira, M. Cabedo-Fabres, and E. Antonino-Daviu, “Simple design for cost-effective diversity antennas,” Microw. Opt. Technol. Lett., vol. 49, no. 4, pp. 994–996, Apr. 2007. [18] J. Ethier, E. Lanoue, and D. McNamara, “MIMO handheld antenna design approach using characteristic mode concepts,” Micow. Opt. Technol. Lett., vol. 50, no. 7, pp. 1724–1727, Jul. 2008. [19] J. Ethier and D. A. McNamara, “The use of generalized characteristic modes in the design of MIMO antennas,” IEEE Trans. Magn., vol. 45, no. 3, pp. 1124–1127, Mar. 2009. [20] M. Cabedo-Fabres, E. Antonino-Daviu, A. Valero-Nogueira, and M. F. Bataller, “The theory of characteristic modes revisited: A contribution to the design of antennas for modern applications,” IEEE Antennas Propagat. Mag., vol. 49, no. 5, pp. 52–68, October 2007. [21] “EM Software and Systems,” FEKO Suite 5.5. Stellenbosch, South Africa, 2009. [22] A. O. Yee and R. J. Garbacz, “Self- and mutual-admittances of wire antennas in terms of characteristic modes,” IEEE Trans. Antennas Propag., vol. 21, no. 6, pp. 868–871, November 1973. [23] H. A. Wheeler, “Fundamental limitations of small antennas,” Proc. IRE, vol. 35, no. 12, pp. 1479–1484, Dec. 1947. [24] S. R. Best, “A discussion on the properties of electrically small selfresonant wire antennas,” IEEE Antennas Propag. Mag., vol. 46, no. 6, pp. 9–22, Dec. 2004. [25] P. E. Mayes and W. Gee, “Using multiple resonant radiators for increasing the bandwidth of electrically small antennas,” in Proc. Antenna Applications Symp., 2000, pp. 246–269. [26] H. R. Stuart, S. R. Best, and A. D. Yaghjian, “Limitations in relating quality factor to bandwidth in a double resonance small antenna,” IEEE Antennas Wireless Propag. Lett., vol. 6, pp. 460–463, 2007. [27] G. Goubau, “Multi-element monopole antennas,” in Proc. ECOM-ARO Workshop on Electrically Small Antennas, 1976, pp. 63–67. [28] C. B. Ravipati and S. R. Best, “The Goubau multi element monopole antenna—Revisited,” in Proc. Antennas and Propagation Int. Symp., June 2007, pp. 233–236. [29] H. R. Stuart and C. Tran, “Small spherical antennas using arrays of electromagnetically coupled planar elements,” IEEE Antennas Wireless Propag. Lett., vol. 6, pp. 7–10, 2007. [30] H. R. Stuart and S. R. Best, “A small wideband multimode antenna,” presented at the Proc. IEEE Antennas and Propagation Int. Symp., 2008. [31] R. C. Hansen, “Correct impedance-matching limitations,” IEEE Antennas Propag. Mag., vol. 51, no. 3, pp. 122–124, Jun. 2009. [32] A. Hujanen, J. Holmberg, and J. C.-E. Sten, “Bandwidth limitations of impedance matched ideal dipoles,” IEEE Trans. Antennas Propag., vol. 53, no. 10, pp. 3236–3239, Oct. 2005.

Jacob J. Adams (S’05–GSM’10) was born in Plain City, OH. He received the B.S. and M.S. degrees in electrical engineering from the Ohio State University, in 2005 and 2007, respectively, and the Ph.D. degree from the University of Illinois at Urbana-Champaign, in 2010. He is currently a Postdoctoral Research Associate at the University of Illinois at Urbana-Champaign. His research interests include electrically small antennas, modal analysis of microwave structures, novel materials for electromagnetic devices, and bioelectromagnetics. Dr. Adams is a member of Tau Beta Pi and Eta Kappa Nu. He is the recipient of a Graduate Research Fellowship from the National Science Foundation, the Dean’s Distinguished University Fellowship from the Ohio State University, and the Mavis Memorial Fellowship from the University of Illinois.

Jennifer T. Bernhard (S’89–M’95–SM’01–F’10) was born on May 1, 1966, in New Hartford, NY. She received the B.S.E.E. degree from Cornell University, New York, in 1988, and the M.S. and Ph.D. degrees in electrical engineering from Duke University, Durham, NC, in 1990 and 1994, respectively, with support from a National Science Foundation Graduate Fellowship. While at Cornell, she was a McMullen Dean’s Scholar and participated in the Engineering Co-op Program, working at IBM Federal Systems Division in Owego, New York. During the 1994–95 academic year she held the position of Postdoctoral Research Associate with the Departments of Radiation Oncology and Electrical Engineering at Duke University, where she developed RF and microwave circuitry for simultaneous hyperthermia (treatment of cancer with microwaves) and MRI (magnetic resonance imaging) thermometry. At Duke, she was also an organizing member of the Women in Science and Engineering (WISE) Project, a graduate student-run organization designed to improve the climate for graduate women in engineering and the sciences. From 1995-1999, she was an Assistant Professor in the Department of Electrical and Computer Engineering, University of New Hampshire, where she held the Class of 1944 Professorship. Since 1999, she has been with the Electromagnetics Laboratory, Department of Electrical and Computer Engineering, University of Illinois at Urbana-Champaign, where she is now a Professor. Her industrial experience includes work as a Research Engineer with Avnet Development Labs and, more recently, as a private Consultant for members of the wireless communication and sensors community. Her research interests include reconfigurable and wideband microwave antennas and circuits, wireless sensors and sensor networks, high speed wireless data communication, electromagnetic compatibility, and electromagnetics for industrial, agricultural, and medical applications, and has four patents on technology in these areas. Prof. Bernhard is a member of URSI Commissions B and D, Tau Beta Pi, Eta Kappa Nu, Sigma Xi, and ASEE, and is a Fellow of the IEEE. She was an NASA-ASEE Summer Faculty Fellow at the NASA Glenn Research Center in Cleveland, OH, in 1999 and 2000. She received the NSF CAREER Award in 2000. She is also an Illinois College of Engineering Willett Faculty Scholar and a Research Professor in Illinois’ Coordinated Science Laboratory, and the Information Trust Institute. She and her students received the 2004 H. A. Wheeler Applications Prize Paper Award from the IEEE Antenna and Propagation Society for their paper published in the March 2003 issue of the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION. She served as an Associate Editor for the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION from 2001–2007 and served as an Associate Editor for the IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS from 2001–2005. She is also a member of the editorial board of Smart Structures and Systems. She served as an elected member of the IEEE Antennas and Propagation Society’s Administrative Committee from 2004–2006, and was President of the IEEE Antennas and Propagation Society in 2008.

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Multi-Beam Multi-Layer Leaky-Wave SIW Pillbox Antenna for Millimeter-Wave Applications Mauro Ettorre, Member, IEEE, Ronan Sauleau, Senior Member, IEEE, and Laurent Le Coq

Abstract—This work proposes a novel multi-beam leaky-wave pillbox antenna. The antenna system is based on three main parts: feeding part (integrated horns), quasi-optical system and radiating part. The radiating and input parts are placed in two different stacked substrates connected by an optimized quasi-optical system. In contrast to conventional pillbox antennas, the quasi-optical system is made by a pin-made integrated parabola and several coupling slots whose sizes and positions are used to efficiently transfer the energy coming from the input part to the radiating part. The latter consists of a printed leaky-wave antenna, namely an array of slots etched on the uppermost metal layer. Seven pin-made integrated horns are placed in the focal plane of the integrated parabola to radiate seven beams in the far field. Each part of the antenna structure can be optimized independently, thus facilitating and speeding up the complete antenna design. The antenna concept has been validated by measurements (around 24 GHz) showing a scanning capability over 30 in azimuth and more than 20 in elevation thanks to the frequency scanning behavior of the leaky-wave radiating part. The proposed antenna is well suited to low-cost printed circuit board fabrication process, and its low profile and compactness make it a very promising solution for applications in the millimeter-wave range. Index Terms—Integrated reflector antennas, leaky-wave antennas (LWA), millimeter-wave antennas, multi-beam antennas, pillbox antennas, planar antennas.

I. INTRODUCTION

I

N recent years, the growing demand of high-performance, low cost, compact scanning antennas for telecommunication and surveillance applications has boosted the development of planar electronic scanning antennas. The operation of a generic electronically-scanned multi-beam antenna can be radiators are fed by represented schematically as in Fig. 1: input ports by means of a beam forming network (BFN) that should provide the required phase and amplitude to each radiator in order to obtain the desired far field pattern and pointing direction. Besides, to steer the antenna main beam, the BFN should be able to control the phase gradient provided

Manuscript received June 11, 2010; revised August 26, 2010; accepted September 09, 2010. Date of publication January 31, 2011; date of current version April 06, 2011. This work was carried out in the framework of the “Radar ACC” project from the French competitiveness cluster “IdforCAR” and in the framework of the FP-7 coordination action ARTIC. The authors are with the Groupe Antennes & Hyperfréquences, Institut d’Electronique et de Télécommunications de Rennes (IETR), UMR CNRS 6164, Université de Rennes 1, 35042 Rennes Cedex, France (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.2011.2109695

M

Fig. 1. Block diagram for electronic scanning antennas. In the case of beam input ports corresponds a direcswitching scanning antennas to each of the tion in the far field whereas, for phased array solutions to each of the radiators is associated a transmitter/receiver module.

N

to each radiator. In case of phased arrays [1] a dedicated transmitter/receiver (T/R) module is used for each radiator for a continuous 2D scanning of the antenna main beam at the expense of losses and increasing cost. Among the other solutions, those based on the substrate integrated waveguide (SIW) technology use particular BFNs such as Blass [2], Nolen [3] and Butler matrices [4], [5] or integrated quasi-optical focusing systems as Luneburg [6], R-KR [7] and Rotman lenses [8], [9]. In all these cases the main beam is steered by switching from one input port to another one. In other words, to each of the input ports corresponds a pointing direction or beam in the far field (Fig. 1). However, the size, losses and need of cross-overs (Butler matrix) and delay lines, are the major drawbacks for all these solutions for compact, efficient, scanning antennas. Based on the same idea of integrated focusing systems as Luneburg and Rotman lenses, a single off-set parabolic reflector [10] and dual-offset Gregorian system [11] have been synthesized in (SIW) technology. In these two systems [10], [11], the energy launched by an integrated feed is confined between the two metal plates of the antenna substrate and is collimated to the radiating part, laying in the same substrate (single-layer configuration), by using either an offset parabolic reflector [10] or a Gregorian system [11] to avoid the back scattering and blockage from the source. As shown in [12] and [13], the back reflection can also be avoided using a double-layer structure or “pillbox configuration” where the feed and radiating part are located in two different substrates connected by a 180 parallel plate bend of parabolic profile. In such a way both the size and modularity of the antenna are improved. However, the classical 180 parallel plate bend of parabolic profile turns out to be narrow band and efficient only for a narrow range of illumination angles [14]. In addition, it introduces phase distortion for the transmitted mode,

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Fig. 2. Multi-beam multi-layer leaky-wave pill-box antenna. (a) 3D view. (b) 2D view in xz -plane. The integrated parabola and the input horns are made by vertical metallic vias.

high-coupling and mismatching for the feeds in the focal plane of the integrated parabolic reflector, precluding the possibility of using such transition for multi-beam systems where more than one feed is present in the focal plane of the parabolic bend. In the present work, a new optimized multi-beam multi-layer SIW leaky-wave pillbox antenna is introduced (Fig. 2). The three main building blocks of the structure are the following: the input part, the quasi-optical system, and the radiating system. The vertical walls present in the structure are implemented by means of metallic pins. The complete system suits very well for low-cost printed circuit board (PCB) technology. As a major difference with classical pillbox antennas [12]–[14], the 180 parallel plate bend coupler (quasi-optical system) has been optimized in order to improve the quality of the transmitted wave front in terms of bandwidth and efficiency. The optimization has been achieved by replacing the classical unique slot bordering the parabolic surface (Fig. 3(a)) by several optimized small slots (Fig. 3(b)). Here, the input part is composed by seven pin-made integrated horns located in the focal plane of the parabolic reflector. To each feed in the focal plane corresponds a radiated beam in the far field as in common beam switching 3D imaging systems and multi-beam reflector antennas. A leaky-wave solution (LWA) based on an array of slots etched on the uppermost metal layer (M.3) has been chosen as a radiating part for its high radiation efficiency [15]. The paper is organized as follows. The proposed multi-slot quasi-optical system is presented in Section II and compared with the classical one. The input part and the transition used for measurements are described in Section III. Section IV introduces the leaky-wave radiating part. Section V presents the

Fig. 3. Top view of the quasi-optical transition. (r; ') are the usual cylindrical coordinates, and F is the focal length of the parabolic surface at which the input part is placed. (a) Classical pillbox transition. 1 is the width of the slot. (b) Multi-slot transition. r , 1 l and w refer to the ith slot and are respectively its position along r , offset from the parabolic profile, length and width.  is the distance between two adjacent slots.

experimental results obtained at clusions are drawn in Section VI.

. Finally, con-

II. INTEGRATED QUASI-OPTICAL SYSTEM The proposed quasi-optical system is shown in Fig. 2 and Fig. 3(b). In the present case, it consists of a parabolic reflecting surface made by vertical metallic pins connecting the metal layer M.1 and M.3 and several coupling slots, located in the intermediate metal layer M.2 between the two substrates Sub.1 and Sub.2. The role of the quasi-optical system is to efficiently transfer the quasi-transverse electromagnetic (quasi-TEM) mode coming from the input part in the first substrate (Sub.1) to second substrate of the antenna (Sub.2) and at the same time transform its phase wave front from cylindrical to plane. A. Design Procedure The design of the multi-slot transition is based essentially on three parameters: the length , the width and the poof the th slot [16]. By resorting to the geometrical sition optics (GO) tracing of the rays associated to the dominant parallel plate waveguide (PPW) quasi-TEM mode coming from the feed, the th slot is placed along a path described by (1) where flector,

is the focal length of the 2D integrated parabolic reand are the usual cylindrical coordinates, and

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is the distance of the slot center from the parabolic profile. The symmetry of the structure along the parabola axis is preserved by the slot position and size. The starting parameters for the design are those corresponding to a normal incidence of the quasi-TEM mode coming from the input part, that is when the rays of the incoming energy impinge normally to the transition and parabolic reflector. In this case the parabolic reflector and multi-slot transition can be approximated respectively by a plane vertical wall and several coupling slots along a line parallel to the wall. Such a structure is equivalent to a classical coupler between ) and distance from the PPWs [17], [18]. The slot size ( , are then chosen to improve the coupling reflecting wall among the two PPWs and reduce the back-scattering to the input PPW. Finally, the real illumination of the transition is taken into account, and several full-wave simulations [19] are used to tune and distance of the various slots to improve the length the matching of each source in the focal plane of the parabolic reflector and reduce the coupling among them.

B. Comparison With a Classical Pillbox Transition In order to perform a comparison with the classical pillbox transition, and demonstrate the relevance of the proposed design, the structure shown in Fig. 4 has been considered. In contrast to Fig. 2(a), Sub.2 is closed on a perfectly matched layer condition (PML boundary condition in [19]), and the radiating part removed. The thickness and permittivity of the two sub, and strates Sub.1 and Sub.2 are , respectively. Seven H-plane integrated horns have been placed one close to the other in the focal place of the parabolic reflector (Fig. 3). The horns present a radiating aperture (where is the wavelength in the dielectric at ) of 1.5 and provide a 10 dB edge illumination (Section III). The focal length and diameter of the parabolic reflector equal and , respectively. The parabolic reflector has been oversized to avoid side effects from the finiteness of the structure. For the single slot case (Fig. 3(a)) the width of the slot , whereas for the multi-slot case (Fig. 3(b)) is , a total number of 48 slots have been used with , and a ratio between the first (on the right of the , of 0.62 parabola axis) and last slot length, , and position, and 1.03 respectively ( , ). The reflection coefficient at the seven input ports is given in Fig. 5 for both configurations. As clearly shown, the multi-slot transition introduces an improvement of more than 10 dB in matching level and mutual coupling thanks to a reduction of back scattering from the parabolic reflector. The wide bandwidth of the transition can also be appreciated. However, the scattering parameters do not provide any detail about the phase front of the mode traveling in Sub.2. The phase front quality is important since the quasi-TEM mode traveling in the second substrate feeds the radiating part of the antenna structure, and then phase aberations or distortions of the feeding mode would deteriorate the far field pattern of the radiating part. The phase along the line (Fig. 4) has then been

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Fig. 4. 3D view of the structure used to compare the proposed and classical pillbox transitions. For the classical pillbox case a single slot bordering the parabolic reflector is considered as in Fig. 3(a). The PML condition in the upper substrate Sub.2 avoids back scattering to the quasi-optical transition.

derived by using the near field capabilities of the HFSS simulong, lator [19] for the two cases analyzed. The line is 16.4 corresponding to a 10 dB tapering of the field coming from quasi-optical system, and it is placed at 4.9 from the vertex of the parabola. Ideally, the wave traveling in Sub.2 should present a plane wave front and propagates with the medium propagation . Besides, the directions of propagation constant of such mode are fixed by the quasi-optical system as for 3D cases. In particular to each feed position in the focal plane corresponds a direction of propagation in the second substrate of the TEM-mode given approximately by [1] (2) where is the distance of the considered feed from the parabola axis. It is worth noting that, in the present case, we assumed a beam deviation factor (BDF) associated to the quasi-optical system of 1 [1]. The phase along the line for an ideal TEMmode is then given by (3) where is the position along the line. The phase fronts achieved for the classical and proposed transitions are reported in Fig. 6, when feed #4, #3, #2 and #1 are activated separately. For comparison purposes, the phase of an ideal TEM-mode given by (3) is also reported. The maximum phase deviation with respect for a clasto the phase of an ideal TEM-mode is about for the multi-slot one. sical pillbox transition and only It is evident the improvement in terms of phase front quality and distortion of the transmitted wave achieved by the multi-slot transition. III. INPUT PART: INTEGRATED HORNS The input part is located in the focal plane of the optical system in the first substrate and consists of one or more elementary sources. Its main role is to launch and shape a TEM-mode, polarized along , in the first substrate of the antenna. Ideally, the launched TEM mode should present a cylindrical wave front and be tapered in such a way to control the field along

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Fig. 5. Scattering parameters for the input horns in the focal plane of the integrated parabola. (a) Reflection coefficient. (b)–(d) Coupling among the input horns. The dashed and continuous line correspond to the classical and proposed transitions, respectively.

Fig. 6. Phase characteristic for the classical (dashed line) and proposed pillbox transitions (continuous line). (a)–(d) Phase along the line l in Fig. 4 for horns #4, #3, #2, #1, respectively. The ideal phase variation Eqn. (3) is given in dotted line as a reference.

Fig. 8. Normalized amplitude at f of the cylindrical wave launched by horn #4 in Fig. 7. Fig. 7. Input part: seven integrated H-plane sectoral horns fed by a current probe transition. Seven Mini-SMP connectors are soldered on the Mini-SMP pads for measurements purposes. The metallic pins connects the metal layer M.1 and M.2 of Fig. 2.

the parabolic profile to reduce spill over and improve the radiated pattern quality. In this work, the input part consists of seven integrated H-plane sectoral horns (Fig. 7) made by metallic pins connecting the metal layers M.1 and M.2 (Fig. 2). The horn aperin Fig. 7, is chosen in order to get a 10 dB ture size, 1.5 taper along the surface of the bend. The calculated pattern in the first substrate of the antenna for the input horn #4 is shown in is achieved. Finally, Fig. 8. A 10 dB beamwidth of a coplanar waveguide (CPW) to current probe transition is used to feed each horn [20], and seven Mini-SMP connectors are soldered on the Mini-SMP pads in Fig. 7 to the coplanar waveguide lines for measurements purposes.

IV. RADIATING PART: LEAKY-WAVE RADIATION The radiating part of the antenna lies in the uppermost substrate and is fed by the quasi-TEM mode coming from the quasioptical system. In the present case it is made by an array of slots etched on metal layer M.3 (Fig. 9). The first three slots of the radiating part present different sizes in order to reduce the reflection of the guided mode by the radiating part. Such reflection would deteriorate the antenna performances and increase the total losses [11]. The geometrical parameters of the radiating part are provided in Table. I. The basic radiation mechanism of the radiating part is based on an efficient leaky-wave phenomenon as the one used in [11]. In absence of the slots the quasi-TEM mode coming from the quasi-optical system would propagate along the surface with a real propagation constant essentially equal to the one of the medium. This slow mode

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Fig. 9. Geometry of the radiating slots. The array of slots radiates thanks to the 1 indexed leaky-wave mode of the corresponding Floquet mode expansion.

0

TABLE I GEOMETRICAL DIMENSIONS (in mm) OF THE RADIATING PART

would not radiate in upper hemisphere, but once reached the ra, diating part, thanks to the slot region and its periodicity it is transformed in a fast leaky-wave mode: the dominant 1 indexed leaky-wave mode of the Floquet mode expansion for periodic structures. The knowledge of the propagation constant of this mode can be used to derive the pointing angle of the radiating part [11]. In the entire operating bandwidth, the periodicity of the slots along produces a backward radiation in the -plane, in particular at the operating frequency a squinted beam around 40 is expected for feed #4 in the focal plane of the quasi-optical system. V. PILL-BOX ANTENNA PROTOTYPE: EXPERIMENTAL RESULTS The various parts presented above have been finally used to build and test an antenna prototype. Two laminates Rogers Duroid 5880 with a thickness of 0.508 mm and permittivity have been adopted for the substrates Sub.1 and Sub.2 in Fig. 2. The final antenna prototype is shown in Fig. 10 where a top and bottom view are provided. The final thickness of the antenna is lower than 1.2 mm, resulting in a really compact and low profile antenna. Besides, the final structure is totally shielded by vias connecting the metal layer M.1 and M.3 (Fig. 2). During the entire test campaign, the Mini-SMP connectors of each input port have been connected to the vector network analyzer through Mini-SMP to K adaptors. The complete scattering parameters ( -matrix) for the antenna have been measured. The reflection coefficient and mutual coupling among the various ports are shown in Fig. 11. A reflection coefficient lower than 10 dB is achieved in the band 23.5–25.75 GHz, corresponding to a 9% relative bandwidth. In this band and further over, the mutual coupling between the various ports is lower than 20 owing dB. The only exception is the parameter to lateral back scattering from the side walls shielding the antenna structure. Indeed, the integrated parabola has not been adequately oversized because of size limitations in the fabrication process. The radiation patterns for the various input ports have been measured in the far field range of IETR. 3D measurements have been made in order to get a complete overview of the radiation patterns corresponding to each feed. For feed #4 (source

Fig. 10. Antenna prototype. (a) Top view. (b) Bottom view with the seven Mini-SMP connectors used to feed the integrated horns soldered on the antenna. A legend for the various feed is also provided.

lying on the parabola axis), the E- and H-plane patterns have also been measured by using a linear scanning along the corresponding planes. For the considered input horn, the E-plane corresponds to -plane of Fig. 2 whereas the H-plane at a certain frequency is a plane orthogonal to -plane and passing through the direction of maximum radiation (see inset of Fig. 17). A comparison between the measured and simulated radiais provided tion patterns in E- and H-planes at in Figs. 12 and 13, respectively. A very good agreement between expected and measured results is obtained. The measured E and H-plane patterns for horn #4 in the complete bandwidth 23.8–25.4 GHz are shown in Figs. 14 and 15, respectively. The typical frequency scanning behavior of the leaky-wave radiating part can be noticed in the E-plane radiation patterns. In the band 23.5–25.75 GHz a 23 variation of the pointing angle is found, as expected. The Side Lobe Level (SLL) in both E- and H-planes is lower than 20 dB over the entire band. The 3 dB beamwidths of the radiation patterns in both principal planes are given in Fig. 16. In both planes the measured beamwidth is roughly constant, with an average value of 15 and 6 in E- and H-planes, respectively. The small difference between the measured and simulated beamwidth in E-plane is due to losses and thickness of the metal layers not taken into account in the full wave analysis. It is worth noting that the beamwidth on these two planes are independently controlled by the quasi-optical system (H-plane) and the leaky-wave phenomenon (E-plane). Fig. 17 shows the measured 3 dB contour beams for feed #1–#5 in the focal plane of the structure in the frequency range 23.6–25 GHz. For symmetry reasons, the results for feed #6 and #7 would correspond to those of #2 and #1, respectively, as it is the case for feed #3 and #5. An elevation over azimuth representation has been used for the measured results. In such a

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Fig. 11. Scattering parameters (S -matrix) for the antenna prototype measured in the band 22.5–26.5 GHz. (a) Reflection coefficient. (b)–(d) Mutual coupling between the various ports.

Fig. 13. Measured and simulated H-plane radiation patterns for feed #4 at f = = 24:2 GHz.  is the elevation angle starting from the direction of maximum

Fig. 12. Measured and simulated E-plane radiation patterns for feed #4 at f .  is the usual elevation angle starting from the normal to the plane : of the antenna.

24 2 GHz

representation, the complete radiation sphere is sampled along cut-planes passing through the -axis in Fig. 2, as sketched in the inset of Fig. 17. The scanning in azimuth at a fixed frequency is made along a conical profile associated to the leaky-wave radiating part. It is worth noting that the dispersion properties of this leaky wave radiation change for oblique incidence of the quasi-TEM mode coming from the quasi-optical system [21]. This could explain why some differences among the various beams in the elevation plane especially for large scan angles in azimuth. For clearness, the variation of pointing direction with frequency is shown in Fig. 18 for all the beams. Note that the and pointing direction of each beam is constant in azimuth with the frequency due to the only varies in elevation

radiation in the H-plane.

leaky-wave radiation behavior. Besides, in Table II the measured gain, SLL and cross-pol level for feed #2-#5 are also provided in the frequency range 23.6–25.8 GHz. The results for feed #1 have not been provided since degraded for the reasons already mentioned. It is clear from all the previous results, that the proposed antenna can steer its main beam in an angular sector of about 20 in elevation by frequency scanning and over in azimuth by beam switching with a measured gain variation in the range 18.51–23.78 dB and high radiation performances in terms of SLL and cross-pol. It is worth noting that the scanning performances of the proposed antenna depends on the ratio between the focal distance and its diameter . In the present case a of the parabola moderate value for this ratio (about 0.6) has been chosen, and

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TABLE II RADIATION PATTERN CHARACTERISTICS FOR FEED #2–#5: GAIN, SLL AND CROSS-POLARIZATION LEVEL

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Fig. 14. Measured E-plane radiation patterns for feed #4 in the frequency range 23.8–25.4 GHz.  is the usual elevation angle starting from the normal to the plane of the antenna.

Fig. 17. 3 dB measured contour beams in the band 23.6–25 GHz for feed #1-#5 in the focal plane of the integrated reflector. An elevation over azimuth representation have been used as shown in the inset of the figure.

Fig. 15. Measured H-plane radiation patterns for feed #4 in the frequency range 23.8–25.4 GHz.  is the elevation angle starting from the direction of maximum radiation in the H-plane.

Fig. 18. Variation of the pointing angle for feed #1–#5 in the focal plane of the integrated reflector in the band 23.6–26 GHz.

system a 3 dB crossing among the various beams in the azimuth plane (beam switching) can be achieved; this has not been the goal of the present design where the main antenna characteristics were under investigation. VI. CONCLUSION

Fig. 16. Measured

03 dB beamwidth in E and H-planes for feed #4.

then higher values would directly improve the scanning performances of the complete system [1]. Besides, by choosing ratio and illumination of the quasi-optical properly on the

A novel multi-beam multi-layer leaky-wave pillbox antenna has been presented. The antenna structure is based on three independent building blocks: input part, quasi-optical system and radiating part. For the quasi-optical system a novel transition made by several slots has been introduced to overcome the limitations in terms of bandwidth, coupling and phase front of classical single slot transitions used in pillbox configurations. A prototype composed by 7 input horns in the focal plane of the quasi-optical transition has been tested. A bandwidth of 9% has been achieved for all the input horns together with high radiation performances in terms of side lobe level and cross-polarization.

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The measured prototype could cover a large angular sector by steering its main beam in about 20 in elevation by frequency in azimuth by beam switching with a scanning and over measured gain variation in the range 18.51–23.78 dB. Thanks to the wide scanning angular range capability, compactness, and compatibility with low-cost fabrication process as PCB technology, the proposed antenna is a very attractive solution for many millimeter-wave applications, as automotive radars. ACKNOWLEDGMENT The authors would like to thank F. Boutet for his support during the measurement campaign. REFERENCES [1] J. L. Volakis, Antenna Engineering Handbook, 4th ed. New York: McGraw Hill, 2007. [2] P. Chen, W. Hong, Z. Kuai, and K. Wu, “A double layer substrate integrated waveguide Blass matrix for beamforming applications,” IEEE Microw. Wireless Comp. Lett., vol. 19, no. 6, pp. 374–376, June 200. [3] T. Djerafi and N. J. G. Fonseca, “Planar Ku-band 4 4 Nolen matrix in SIW technology,” IEEE Trans. Microwave Theory Tech., vol. 58, no. 2, pp. 259–266, Feb. 2010. [4] P. Chen, Z. Kuai, J. Xu, H. Wang, J. Chen, H. Tang, J. Zhou, and K. Wu, “A multibeam antenna based on substrate integrated waveguide technology for MIMO wireless communications,” IEEE Trans. Antennas Propag., vol. 57, no. 6, pp. 1813–1821, Jun. 2009. [5] C.-H. Tseng, C.-J. Chen, and T.-H. Chu, “A low-cost 60-GHz switched-beam patch antenna array with Butler matrix network,” IEEE Antennas Wireless Propag. Lett., vol. 7, pp. 432–435, Dec. 2008. [6] Y.-J. Park and W. Wiesbeck, “Offset cylindrical reflector antenna fed by a parallel-plate Luneburg lens for automotive radar applications in millimeter-wave,” IEEE Trans. Antennas Propag., vol. 51, no. 9, pp. 2481–2483, Sep. 2003. [7] Y. J. Cheng, W. Hong, and K. Wu, “Design of a substrate integrated waveguide modified R-KR lens for millimetre-wave application,” IET Microw., Antennas Propag., vol. 4, no. 4, pp. 484–491, Apr. 2010. [8] Y. J. Yu Cheng, W. Hong, K. Wu, Z. Q. Kuai, C. Yu, and J. X. Chen, “Substrate integrated waveguide (SIW) Rotman lens and its Ka-band multibeam array antenna applications,” IEEE Trans. Antennas Propag., vol. 56, no. 8, pp. 2504–2513, Aug. 2008. [9] H.-H. Fuchs and D. Nübler, “Design of Rotman lens for beamsteering of 94 GHz antenna array,” Electron. Lett., vol. 35, no. 11, pp. 854–855, May 1999. [10] Y. J. Yu Cheng, W. Hong, and K. Wu, “Millimeter-wave substrate integrated waveguide multibeam antenna based on the parabolic reflector principle,” IEEE Trans. Antennas Propag., vol. 56, no. 9, pp. 3055–3050, Sep. 2008. [11] M. Ettorre, A. Neto, G. Gerini, and S. Maci, “Leaky-wave slot array antenna fed by a dual reflector system,” IEEE Trans. Antennas Propag., vol. 56, no. 10, pp. 3143–3149, Oct. 2008. [12] W. Rotman, “Wide-angle scanning with microwave double-layer pillboxes,” IRE Trans. Antennas Propag., vol. 6, no. 1, pp. 96–105, Jan. 1958. [13] T. Teshirogi, Y. Kawahara, A. Yamamoto, Y. Sekine, N. Baba, and M. Kobayashi, “Dielectric slab based leaky-wave antennas for millimeterwave applications,” in Proc. IEEE AP-S Int. Symp., Jul. 2001, vol. 1, pp. 346–349. [14] V. Mazzola and J. E. Becker, “Coupler-type bend for pillbox antennas,” IEEE Trans. Microwave Theory Tech., vol. 15, no. 8, pp. 462–468, Aug. 1967. [15] J. Hirokawa and M. Ando, “Efficiency of 76-GHz post-wall waveguide-fed parallel-plate slot arrays,” IEEE Trans. Antennas Propag., vol. 48, no. 11, pp. 1742–1745, Nov. 2000. [16] M. Ettorre and R. Sauleau, “Antenne Multicouche à Plans Parallèles de Type Pillbox et Système d’Antennes Correspondants,” French patent, FR0952158, Apr. 2009. [17] D. M. Pozar, Microwave Engineering, 2nd ed. New York: Wiley, 1998, ch. 6. [18] N. Marcuvitz, Waveguide Handbook. New York: McGraw-Hill, 1951, ch. 6–7. [19] Ansoft HFSS version 12.0 1984–2010, Ansoft Corporation.

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[20] D. Deslandes and K. Wu, “Analysis and design of current probe transition from grounded coplanar to substrate integrated rectangular waveguides,” IEEE Trans. Microwave Theory Tech., vol. 53, no. 8, pp. 2487–2494, Aug. 2005. [21] P. Baccarelli, P. Burghignoli, F. Frezza, A. Galli, P. Lampariello, G. Lovat, and S. Paulotto, “Modal properties of surface and leaky waves propagating at arbitrary angles along a metal strip grating on a grounded slab,” IEEE Trans. Antennas Propag., vol. 53, no. 1, pp. 36–46, Jan. 2005. Mauro Ettorre (M’08) was born in Tricarico, Matera, Italy, in 1979. He received the Laurea degree (summa cum laude) in telecommunication engineering and the Ph.D. degree in electromagnetics from the University of Siena, Italy, in 2004 and 2008 respectively. During his Master’s degree studies he spent five months at the Technical University of Denmark (DTU), Lyngby, Denmark. Part of his Ph.D. was developed at the Defence, Security and Safety Institute of the Netherlands Organization for Applied Scientific Research (TNO), The Hague, The Netherlands, where afterwards he worked as an Antenna Researcher. From 2008 to 2010, he was a Postdoctoral Fellow at the Institut d’Electronique et de Télécommunications de Rennes (IETR), Université de Rennes 1, France. Since October 2010, he has been appointed CNRS Researcher at IETR. His research interests include the analysis and design of leaky-wave antennas, periodic structures and compact planar antennas. Dr. Ettorre received the Young Antenna Engineer Prize during the 30th ESA Antenna Workshop 2008 in Noordwijk, The Netherlands.

Ronan Sauleau (SM’10) graduated in electrical engineering and radio communications from the Institut National des Sciences Appliquées, Rennes, France, in 1995 and received the Agrégation degree from the Ecole Normale Supérieure de Cachan, France, in 1996, and the Doctoral degree in signal processing and telecommunications and the “Habilitation à Diriger des Recherche” degree from the University of Rennes 1, France, in 1999 and 2005, respectively. He was an Assistant Professor and Associate Professor at the University of Rennes 1, between September 2000 and November 2005, and December 2005 and October 2009. He has been a full Professor in the same University since November 2009. His current research fields are numerical modelling (mainly FDTD), millimeter-wave printed and reconfigurable (MEMS) antennas, lens-based focusing devices, periodic and non-periodic structures (electromagnetic bandgap materials, metamaterials, reflectarrays, and transmitarrays) and biological effects of millimeter waves. He has received six patents and is the author or coauthor of 75 journal papers and more than 200 contributions to national and international conferences and workshops. Dr. Sauleau received the 2004 ISAP Conference Young Researcher Scientist Fellowship (Japan) and the first Young Researcher Prize in Brittany, France, in 2001 for his research work on gain-enhanced Fabry-Perot antennas. In September 2007, he was elevated to Junior member of the “Institut Universitaire de France.” He was awarded the Bronze medal by CNRS in 2008.

Laurent Le Coq received the electronic engineering and radiocommunications degree and the french DEA degree (M.Sc.) in electronics in 1995 and the Ph.D. degree in 1999, all from the National Institute of Applied Science (INSA), Rennes, France. In 1999, he joined the Institute of Electronics and Telecommunications of Rennes (IETR), University of Rennes 1, as a Research Lab Engineer, where he is responsible for measurement technical facilities up to 110 GHz. His activities in antenna measurements and development of related procedures involved him in more than twenty research contracts of national or European interest. He is the author or coauthor of 20 journal papers and 30 papers in conference proceedings.

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A Modal-Based Iterative Circuit Model for the Analysis of CRLH Leaky-Wave Antennas Comprising Periodically Loaded PPW Juan Sebastian Gomez-Diaz, Student Member, IEEE, Alejandro Álvarez-Melcon, Senior Member, IEEE, and Thomas Bertuch, Member, IEEE

Abstract—A novel modal-based iterative circuit model is described for the calculation of the complex propagation constant of mushroom-like parallel-plate composite right/left handed leaky-wave antennas (PPW CRLH LWAs). The conventional lossless CRLH unit cell circuit is modified in order to consider the electromagnetic coupling to free space through a slot. For this purpose, a slot equivalent radiative structure, based on phased-array theory, is analyzed using a mode matching approach combined with Floquet’s theorem. A direct correspondence between lumped elements and this radiative structure is found, leading to a frequency-dependent unit cell circuit model. A quickly converging iterative algorithm is then employed to determine the final element values of the unit cell. The proposed method is accurate, and it takes into account the structure physical dimensions. It also allows to obtain a balanced CRLH unit cell design without requiring any full-wave simulation, is several orders of magnitude faster than full-wave simulations, and provides a deep insight into the physics of the antenna radiation mechanism. Index Terms—Composite right/left-handed (CRLH) metamaterials, leaky-wave antennas (LWAs), modal analysis (MA), parallelplate waveguides (PPW).

I. INTRODUCTION

M

ETAMATERIAL (MTM) leaky-wave antennas (LWA) are designed to operate in their fundamental guided (e.g., [1] and [2]), while conventional periodic mode [3]. LWAs mostly use their first space harmonic Several types of MTM LWAs have been presented in literature. All antenna types are based on the same underlying design principle, which is the periodic loading of a host transmission line (TL) in such a way that the resulting TL becomes a so-called composite right/left-handed (CRLH) TL [1] or negative refractive index (NRI) TL [2], respectively. In connection with frequency scanning LWAs, the term “CRLH” seems more

Manuscript received March 17, 2010; revised August 05, 2010; accepted September 11, 2010. Date of publication January 28, 2011; date of current version April 06, 2011. This work was supported in part by the Spanish Ministry of Education and Science under Grant FPU-AP2006-015 and in part by Project TEC2007-67630-C03-02. J. S. Gomez-Diaz and A. Álvarez-Melcon are with the Technical University of Cartagena, Cartagena, E-30202, Spain (e-mail: [email protected]). T. Bertuch is with the Fraunhofer Institute for High Frequency Physics and Radar Techniques (Fraunhofer FHR), 53343 Wachtberg, Germany (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.2011.2109357

adequate for describing the behavior of the loaded TL and thus it will be used synonymously for the term “MTM” throughout this article. The main distinctive feature of all MTM LWAs found in literature is the type of host TL used. Typical host TLs are microstrip (MS) lines [4], [5], coplanar waveguides (CPW) [6], or coplanar striplines (CPS) [7]. Also the metallo-dielectric surfaces of the mushroom-type [8] have been employed to generate LW radiation [9]. The type of host TL determines the polarization of the radiated field. LWAs based on MS lines or CPWs generate transverse magnetic (TM) polarization whereas LWAs comprising CPS lines radiate transverse electric (TE) fields. The mushroom surfaces can produce both TM [1], [10] and TE [9] polarization depending on the kind of excitation. In order to analyze these types of antennas, circuit models are usually employed [1], [2]. These models are able to accurately represent the antenna dispersive behavior [i.e., the prop] but they have difficulties to characterize agation constant ]. Therethe amount of radiated power [i.e., leaky factor fore, the radiation characteristic of the antenna cannot completely be determined with these methods. This is an important limitation of the existing techniques, since then the attenuation factor cannot be controlled in the design of antennas for practical applications. In addition, a considerable number of time-consuming full-wave simulations are usually required for the design of CRLH LWAs. This makes the CRLH LWA design procedure a tedious task. Moreover, full-wave analysis does not provide any deep insight into the physics of the radiation phenomena, which is extremely important to understand and to speed-up the design process. In this paper, a novel circuit model is employed for the analysis of mushroom-like CRLH LWAs comprising periodically loaded parallel-plate waveguides [11]. The elements of the circuit model are determined by an iterative algorithm combined with modal analysis. Specifically, the attenuation factor of the antenna is rigorously computed for the first time using an equivalent radiating structure, which is based on phased-array theory. The analysis of this structure leads to the accurate definition of a frequency-dependent circuit model, which relates the radiation characteristics with the physical dimensions of the antenna. An iterative algorithm is proposed to determine the values of the equivalent circuit. The main advantage of the method is that it models and describes in a simple way the complex CRLH LWA radiation phenomena using equivalent dispersive circuits. Furthermore, the proposed approach also serves to compute the

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Fig. 1. Topology of a CRLH LWA comprising a periodically loaded PPW (top) and equivalent circuit model (bottom) representing a unit cell of the periodic one-dimensional CRLH TL. The loading is obtained by wires and slots. The slots also provide the coupling to free space, which is rigorously modeled by the dispersive lumped elements C (! ) and R (! ).

physical dimensions of a balanced CRLH unit cell design. Note that the proposed technique is accurate and very efficient, requiring just minutes to analyze a complete LWA. The paper is organized as follows. Section II describes the CRLH LWA comprising the loaded parallel-plate waveguides (PPW) under analysis. Section III presents the circuit model employed for the antenna characterization, and shows how the antenna complex propagation constant can be derived from the equivalent circuit. In Section IV, a mode-matching analysis approach combined with Floquet’s theorem is employed to analyze a slot-equivalent radiating structure, leading to a frequency-dependent lumped circuit model for the CRLH LWA unit cell. The dispersive values of this model are obtained using a quickly converging iterative algorithm, presented in Section V. In Section VI, the proposed method is employed to design and analyze a single CRLH unit cell first, and then a ten unit cell LWA. Full-wave simulations are included to completely validate the proposed technique. Finally, conclusions are given in Section VII. II. CRLH LWA COMPRISING LOADED PPW The top of Fig. 1 shows the topology of the proposed CRLH LWA including two laterally attached PPW feeding sections. At the bottom an equivalent circuit for the unit cell of the LWA’s CRLH TL section is given. It will be discussed in the next section.

The structure is assumed to be finite in the - and -directions and infinitely periodical in the -direction. Effective wave -plane. The LWA consists of a propagation occurs in the planar substrate layer of thickness with homogeneous and and isotropic material characterized by relative permittivity relative permeability . Its back side is completely metalized. The front side is also metalized except for a finite number of parallel and equidistant slots. The first and last slots, used for matching to the input/output ports, have a width of , whereas all other slots have the width . The metal strips oriented along the -direction between adjacent slots are connected to the back side metalization by a rectangular grid of metalized via holes . The via holes are placed symmetrically with diameter and in the center between adjacent slots with a spacing of along the - and -directions, respectively. The vias and the slots constitute the loading of the PPW and when operated in the proper frequency band the loaded section behaves as a CRLH TL. Note that this CRLH line is attached to two conventional right-handed PPW at its ends, which constitute the antenna feeding and matching load. Without any slots present, the rectangular grid of via holes creates a so-called artificial dielectric (AD) or wire medium (WM) which exhibits strongly dispersive properties, similar to the ones observed in hollow . In cylindrical waveguides, including a cut-off frequency the past, open slabs of AD material have been used extensively to create forward scanning LWAs [12]–[15]. The AD medium’s dispersive properties are fundamental for the operation of the

GOMEZ-DIAZ et al.: MODAL-BASED ITERATIVE CIRCUIT MODEL FOR THE ANALYSIS OF CRLH LWAs

proposed CRLH LWA. If the distance between the metal planes is sufficiently small that only the fundamental parallel plate mode (PPM) can propagate in the unloaded PPW, the WM acts like a high pass filter. The WM’s cut-off frequency can be computed resorting to derivations given in [16]. Below cut-off, there is no propagation in the WM and above cut-off the WM supports RH propagation with an effective wave number along the -direction that is always smaller than the free space wave number . Introducing the slots in the upper metalization has two effects. On the one hand, a coupling between the region above and the region inside the PPW is established, and on the other hand, a CRLH TL is created which may support left-handed (LH) propagation below the WM’s cut-off frequency. Coupling of the regions facilitates leaky-wave (LW) radiation as long as the magnitude of the effective wave number’s real part of the CRLH TL is smaller than the free space wave number . At frequenthe propagating mode in the loaded PPW will cies above be RH and thus, forward LW radiation will be observed. Dethe loaded PPW may pending on the geometry, below support LH propagation which will result in backward LW radiation. If the CRLH TL exhibits the so-called “balanced” behavior [1], a smooth transition from left-handed to right-handed frequency regions is possible, as frequency varies. However, even in the case that the structure is completely balanced, the antenna presents a reduction in the radiation efficiency at broadside. This is because the PPW loading only provides a series resistor in the unit cell equivalent circuit, representing radiation losses, whereas a shunt resistor is also required to efficiently radiate at broadside [17], [18]. III. EQUIVALENT CIRCUIT MODEL The equivalent circuit model related to a single unit cell (with and a width ) of the CRLH LWA is shown in a length Fig. 1 (bottom). The layout of the equivalent circuit assumes a symmetric composition of the unit cell along the direction of , have wave propagation. Two right-handed TLs, of length been employed to model the PPW behavior (i.e., the host TL). These TLs are described by their characteristic impedance and propagation constant . It is very important to distinguish . The former is related to the host TL (unbetween and loaded PPW) and it is typically real, as long as material losses are neglected. The latter is related to the total CRLH unit cell and it is complex, because it also includes the radiations losses of the structure. The LH behavior is achieved by a via-hole and by two half-slots (see Fig. 1). The loading is modeled in the equivand two symmetrialent circuit with a shunt inductance cally placed dispersive circuit elements, composed of the parand a capacitor . allel connection of a resistor The parallel connection of the two elements is convenient to represent the radiation mechanism through the slot. In the limiting case of a narrow slot, the capacitor becomes very large, and tends to reduce the radiation by short-circuiting the resistance. This correctly models the radiation reduction that occurs in the real structure for very narrow slots. Note that this dispersive circuit rigorously takes into account the effects of the slot, including the physical parameters of the structure, coupling

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to free-space, reactive fields, coupling to other slots, radiation losses, and the capacitive behavior required to balance the unit cell. An equivalent radiating structure and the modal analysis employed to derive the values of the equivalent circuit elements, including dispersion will be explained in the following section. In order to compute the complex propagation constant of the CRLH unit cell, we represent the equivalent circuit in terms of transmission matrices. This helps to obtain the value of the for the given geometry, and to determine shunt inductance the complex propagation constant of the unit cell. In the next discussion it is assumed that the physical dimensions of the CRLH unit cell are known. In the following sections, we will explain how to accurately obtain these physical dimensions, without the need to use full-wave simulations. The first step required for the analysis is to obtain the characteristic impedance and the propagation constant of the unloaded and width . PPW, related to a single unit cell of length These values may be obtained as (1) (2) and are the permittivity and the permeability of where vacuum, respectively, and is the angular frequency. Then, the may be extransmission matrix of the host TL of length pressed as

(3) Next, the PPW loaded by a grid of via-holes is considered. This creates an artificial dielectric with strong dispersive properties. Similar to hollow waveguides, where the metallic side walls introduce the same effect, the AD acts like a high pass on of the AD the fundamental PPM. The cut-off frequency can be found by solving (e.g., numerically) the following dispersion equation (see [16]): (4)

where is the intrinsic wave number of the substrate material. Note that the effective wavelength in the AD becomes infinite at the cut-off frequency, which means that the propagation constant tends to zero. Therefore, this frequency corresponds to the transition frequency of a CRLH TL. The transmission matrix of the shunt inductance is given by (5) In order to determine the value of the inductance, we will analyze the CRLH unit cell at the transition frequency. At this frequency, the phase shift at the ports of the unit cell becomes zero, and the model of Fig. 1 (bottom) reduces to the circuit of Fig. 2,

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Fig. 2. Equivalent circuit model of a unit cell related to a PPW loaded by a periodic grid of wires. The circuit model of Fig. 1 reduces to this model at the CRLH TL transition frequency [1], assuming that the cell is balanced.

as is demonstrated in [1]. The boundary conditions applied to and the voltages may be formuthe currents lated as

Fig. 3. Cross section of one-dimensional periodic array of infinitely long slots radiating into free space, employed to rigorously model the CRLH LWA radiation mechanism. Periodic boundary conditions in free space are imposed at the limits of the unit cell. Each slot is attached to a PPW T-junction with two PPW ports. Port 1 serves as excitation of the array element.

(6) and

is determined by solving (7)

where is the unitary matrix. The transmission matrix which models the slot in the upper , which will metalization of the host TL is represented by be derived in the next section. The behavior of half of a slot (required to maintain the unit cell symmetry) is obtained as the matrix, and it is denoted by . square root of the The transmission matrix associated to the total CRLH unit cell is then obtained by a simply multiplication of the transmission matrices related to the unit cell elements, as follows:

(8) Note that the diagonal elements of are identical, due to the equivalent circuit symmetry. related to the total The complex propagation constant unit cell may be then determined by solving (9) which yields the complex value of (10) Finally, note that the complex propagation constant can also be obtained using alternatives approaches (such as the one described in [19]), once the different transmission matrixes employed in the analysis are known. IV. EQUIVALENT RADIATING STRUCTURE In this section, a rigorous computation of the transmission , which characterizes the unit cell slot behavior, matrix

is presented. For this purpose, an equivalent phased-array antenna model is employed. It consists of a one-dimensional periodic array of infinitely long slots in a metal plane. At this point, we assume an infinite number of elements (slots) along the -direction. This assumption is not critical for the analysis of leaky-wave antennas [3], since they are usually several wavelengths long. Using phased-array theory, we assume that all array elements are fed with a progressive phase shift. Each slot is individually attached to a T-junction formed with the PPWs, as shown in Fig. 3. In the figure, the horizontal PPWs which feed the slots are not interconnected, in order to clearly identify the different unit-cells. The total length of the whole feeding PPW , which must be greater than . Note that the influence is of this TL will be removed at the end, in order to characterize an isolated slot in an external array environment. Due to this, it is sufficient to consider a single unit cell (array element) with imposed periodic boundary conditions in free space along the -direction (see Fig. 3). Moreover, the imposed phase shift at a given frequency is determined by the effective wave number of . the CRLH TL unit cell as Then, the single array element is studied using a multi modematching (MM) approach [20] combined with Floquet’s theorem. The reason to use a multimode analysis is that not only propagative modes, but also evanescent modes must be rigorously taken into account. This is especially important to model the coupling from the PPW to the slot, from where the energy is radiated. In order to perform the analysis, the equivalent radiating structure of Fig. 3 is split into a PPW E-plane T-junction (see Fig. 4) and into a slot fed by a vertical PPW (see Fig. 5). Then, the general scattering matrix (GSM) [21] associated to and ) is each individual structure ( obtained by using mode-matching techniques [20]. Next, both , related GSMs are combined into a single matrix to the total radiating array element. Note that the radiation . At mechanism of the structure is embedded into this point, this matrix is further simplified, considering only the fundamental PPW mode. This approximation is accurate,

GOMEZ-DIAZ et al.: MODAL-BASED ITERATIVE CIRCUIT MODEL FOR THE ANALYSIS OF CRLH LWAs

Fig. 4. Cross section of an E-plane T-junction of parallel-plate waveguides.

because although evanescent modes couple to the slot and have strong influence on the radiation, they are strongly attenuated as they propagate down the ports. In this way we obtain a matrix , which contains the scattering parameters related to the total radiating array element. However, we are interested only in modeling the slot. Consequently, we de-embed the reference (as shown in Fig. 3), planes of the ports to the plane . Finally, we perform a resulting into the scattering matrix matrix to the transmission simple transformation from the [21]. matrix It can be expected that the magnetic field inside the slots and above the slots in Fig. 1 will be primarily polarized parallel to them. Hence, it will be sufficient, in the following modal analysis to consider TM waves. The reference directions of these waves change as a function of the PPW orientation (from to -direction according to Fig. 4). The steps to perform the analysis described above are detailed in the next subsections, including a validation of the approach using full-wave simulations.

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Fig. 5. Cross-section of an open-ended parallel-plate waveguide radiating in an array environment. Periodic boundary conditions, related to the complex propagation constant of the complete CRLH LWA unit cell, are imposed in the free-space region.

in the case of ports 1 and 2 and to in the case and are the mode of port 3, as shown in Fig. 4), and wavenumbers. B. Modal Analysis of Open-Ended PPW Array The study of an array of dielectric-filled waveguides radiating into free space has already been performed in the past [20], [25]. The structure is shown in Fig. 5, including periodic boundary conditions for the free-space radiation. Its simple geometry allows to perform a modal analysis, resorting to the procedures described in [20] and combined with Floquet’s theorem. The field within the vertical PPW in Fig. 5 is expanded in . This means that the electric mode functions TM to vector potential is zero and the magnetic vector potential is (12) where is the unit vector in the -direction and scalar wave potential, which is given by

is the

A. Modal Analysis of a PPW E-Plane T-Junction The E-plane T-junction in a parallel-plate waveguide has extensively been studied in the past [22]–[24]. A general crosssection of this junction is depicted in Fig. 4. In order to build the GSM associated to it, we individually excite each port of (where the junction with an incident TM mode denoted by is the incident port number and is the mode number). Then, we obtain the complex mode ampli, where is the observation port and tudes ( is the observation mode) using the analytic series solution method proposed in [23]. Note that the modal coefficients are referred to the T-junction borders (dashed line in Fig. 4), and that these coefficients directly correspond to the generalized scattering parameters. Then, the exact length of the T-junction ports are taken into account by moving the reference plane of each modal coefficient, using (11) is a complex mode amplitude related to the origin where and are the lengths of the waveof the observation port guide sections related to ports and (which corresponds to

(13) Here,

is the intrinsic wave number of the filling material, , and the -directed wave number as determined by the separability condition is (14) In (13), we excite the waveguide with the -th mode, which has . This mode propagates along the waveguide an amplitude until it reaches the aperture. There, some energy is reflected back towards the waveguide. This field is expanded into an infinite , as series of modes with complex amplitudes indicated in (13). Furthermore, some energy is coupled to free space. In this region, the field is expanded in mode functions which implies again that the electric vector TM to potential is zero. Note that the propagation direction has been

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axis. In this situation the magnetic

(15) with (16) and (17) where is the intrinsic wavenumber of free space and is the effective wave number related to the CRLH TL unit cell as determined by (10). It is important to remark that Floquet’s theorem has been employed in (16), because of periodicity, leading instead of the usual conto a discrete set of complex modes tinuous spectrum obtained in the single slot case [20]. Also, note that the imposition of the continuity of the electromagnetic field components across the boundary will assure that the slot radiation mechanism entirely depends on the complex propagation . This allows us to estabconstant of the CRLH TL unit cell lish a fundamental relationship between the CRLH TL unit cell and the modal analysis performed of the equivalent radiating structure, which are closely interrelated. For the analysis of the T-junction in Fig. 4 the propagation direction is the axis for ports 1 and 2, and the modal expansion modes with respect to the used is based on the traditional propagation direction. For port 3, the propagation direction is changed along the axis, but the modal expansion used in this for a correct matching of region is still based on modes the fields. Therefore, in the parallel plate region of Fig. 5 these modes are hybrid with respect to the new propagation direction along- . With the help of [20], the components of the electromagnetic fields within the PPW and in the free space region are calculated from (12) and (15). Applying boundary conditions for the , and utilizing and -components of the fields in the plane the orthogonality properties of the harmonic functions involved in the formulation for the scalar wave potentials, expressions and are derived as a function for the modal amplitudes . Finally, the generalized scattering matrix of the excitation for the structure of Fig. 5 is obtained after performing a modal analysis for each incident mode. C. Analysis of the Total Equivalent Structure The equivalent radiating structure shown in Fig. 3 can now easily be modeled using the generalized scattering matrices and , which are connected as indicated in Fig. 6(a). This connection can be further simplified, leading to a single matrix [see Fig. 6(b)]. , a matrix formulation is develIn order to derive this oped. Specifically, boundary conditions are applied at the matrices’ interconnection. The outgoing waves from the third port of the T-junction are considered the input waves to the aperture waveguide, meanwhile the reflected waves from the slot,

Fig. 6. Representation of the equivalent radiating structure of Fig. 3 using generalized scattering matrices (GSM). (a) Using the GSM related to the T-junction (see Fig. 4) combined with the GSM related to the aperture (see Fig. 5). (b) Using a single equivalent GSM.

due to the air discontinuity, are treated as the exciting waves at the T-junction’s third port. Then, the input and output waves are expressed as a function of all components of of and . Therefore, the whole behavior of these two matrices is embedded into the matrix shown in Fig. 6(b). matrix is further simplified. SpecifIn addition, the ically, all higher order modes are neglected and only the fundamental PPW mode is considered. Note that the higher order modes have rigorously been taken into account to model the coupling from the T-junction to the slot, and to model the aperture radiation. However, since they are evanescent, they are strongly attenuated while propagating down the ports, and their contributions can be neglected. Therefore, the equivalent radiating structure may now be represented by a simple (2 2) scattering matrix relating the fundamental modes at the two . ports At this point, it is important to remember that the goal is to model the effect of the slot in the PPW (including its radiation characteristics in a periodic environment) in order to be included into the CRLH unit cell model of Fig. 1 (bottom). Examining that model, one realizes that the effect of the host parallel-plate waveguide has already been considered. Therefore, we need to shift the reference plane of the last scattering matrix to the position of the slot, which may easily be done as (18) where are the port numbers and and are the propagation constant of the fundamental mode and physical length related to the feeding parallel-plate waveguide, respectively. This matrix is then transformed into the transmission matrix [26], which may be expressed as (19) In this last transformation, the PPW characteristic impedance [see (1)] has been employed as a reference impedance [26]. This normalizes the resulting transmission matrix with respect to the unit cell width . models the entire radiation It is important to note that mechanism of the equivalent structure (see Fig. 3), including radiation losses, coupling to free space, reactive fields, coupling to other slots, and the slot influence within the PPW. Also, note that this matrix relates the electrical behavior of the slot with the physical dimensions of the structure. Finally, the transmission

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Fig. 7. Comparison of the scattering parameters (S and S ) of the equivalent radiating structure (see Fig. 3) computed by HFSS and by the proposed modal analysis (MA), as a function of both, frequency and phase shift between unit cell elements. The parameters of the unit cell are ` : ,g : ,t : , : mm. and s

= 23 54 = 0 5 = 3 65

= 0 05

matrix , employed in (8), is derived as the square root of the matrix, exploiting the concatenation property of two transmission matrices [26]. reveals that it has the simple form A numerical study of of two parallel connected impedances (20) This direct correspondence with lumped elements is expected, since the slot radiation losses can be modeled by the resistor, whereas the capacitor takes into account the slot capacitive behavior within the host parallel-plate waveguide as well as the field coupling to free space (reactive fields). In the next sections, it will be demonstrated that the approximation (20) is accurate, introducing very small errors. From this matrix, the radiation and series capacitor are determined, for losses as a particular angular frequency (21) (22)

This correspondence with lumped elements also allows us to derive the complete equivalent dispersive circuit model related to the PPW CRLH LWA unit cell shown in Fig. 1 (bottom). Finally, note that the radiation losses are only modeled by a resistor in the series branch, and that there is no radiation contribution from the shunt branch. As it is explained in [17], radiation at broadside is only achieved when the radiation losses are distributed over both the series and the shunt branches of the CRLH unit cell. Otherwise, the attenuation constant tends to zero at the transition frequency. Therefore, it is expected that the type of CRLH LWAs proposed here suffers from an important drop in efficiency when radiating at broadside. However, they are still able to radiate at backward and forward directions, . Note that although using the fundamental harmonic there is a drop in the broadside radiation efficiency, the CRLH TL is still balanced. This means that the propagation constant does not exhibit a bandgap around the transition frequency. D. Validation Against Full-Wave Simulations In this section, we present a complete validation of the modal analysis of the equivalent radiating structure (see Fig. 3). For this purpose, let us consider this structure with dimensions

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, , , and mm. Note , because the that we can chose any value for the length influence of the auxiliary feeding PPW ports is removed in the analysis. For intermediate calculations, we usually set . For a complete validation of the technique, the and related to this structure are scattering parameters computed for all possible phase shifts, using the proposed modal approach (18), and are then compared with the commercial software Ansoft HFSS (see Fig. 7). As can be observed in the figure, an excellent agreement is obtained in all cases. Also, note that the modal technique needs about 35 min to perform the proposed analysis, while full-wave simulations spend more than one day to obtain the same results. V. ITERATIVELY REFINED APPROACH FOR COMPLEX PROPAGATION CONSTANT DETERMINATION In the previous sections we have explained how to compute as a the CRLH unit cell complex propagation constant , and how to compute function of the transmission matrix this matrix as a function of the physical dimensions of the structure and of the CRLH unit cell complex propagation constant . Therefore, one can easily realize that these variables are closely interdependent. In order to determine the equivalent circuit elements of the CRLH LWA unit cell [see Fig. 1 (bottom)], from previously known physical dimensions, an iterative algorithm is proposed. The description of the algorithm flow-chart, shown in Fig. 8, is as follows: initially, the nondispersive elements of the circuit model and the CRLH transition frequency are obtained using the procedures described in Section III. After that, an initial value of at all zero is assumed for the complex propagation constant is then derived emfrequencies. The transmission matrix ploying the proposed modal analysis, taking into account and the physical dimensions of the structure. Once this matrix is computed based on the has been obtained, the value of . This procedure is repeated until convercurrent value of gence is reached. This iterative algorithm leads to an accurate , and to a final commodel of the slot, through the matrix . In the last step, the frequency plex propagation constant and are extracted from the transdependent values of mission matrix . In this way, all circuit parameters related to the unit cell are determined. It is important to remark that this iterative algorithm is quickly convergent. Numerical simulations (see next section) demonstrate that 20–30 iterations are enough to achieve a relative error between two consecutive steps over the whole less than frequency range. VI. ANALYSIS OF 1D CRLH LWAS In this section, we carefully study a CRLH LWA comprising a periodically loaded parallel-plate waveguide. Specifically, we will design a balanced CRLH TL with a transition frequency set to 3.0 GHz. For this purpose, we choose a host waveguide filled , loaded by by a material with relatively permittivity mm. The unit cell’s total via-holes with diameter length is set to mm.

Fig. 8. Flow chart of the proposed iterative algorithm that determines the element values of the unit cell equivalent circuit [see Fig. 1 (bottom)] and the CRLH TL complex propagation constant.

The analysis steps are as follows: first, in Section VI-A we will derive the waveguide’s and slot’s physical dimensions required to obtain a balanced design. Second, in Section VI-B we rigorously analyze a single unit cell, obtaining its associated complex propagation constant (including radiation losses), and we will validate the result using HFSS. Furthermore, it is numerically demonstrated that the approximation employed to extract and is accurate. Fithe frequency-dependent elements nally, in Section VI-C a complete CRLH LWA composed of ten unit cells is satisfactorily analyzed using the proposed method and the results are validated using full-wave simulations. A. Balancing the CRLH Unit Cell In the case of a balanced unit cell, its associated phase constant must be equal to zero at the transition frequency (i.e., ). This allows to obtain a CRLH unit cell with a smooth transition from the left-handed to the right-handed frequency region, avoiding the stopband which appears in the unbalanced case [1]. In order to determine the slot and waveguide physical dimensions, we apply the iterative algorithm developed in Section V. First, we set some default physical dimensions. In this case, we mm and a metal thickness of choose a slot width of mm, which approximates an infinitesimally thin metal (see Fig. 3). The value of is chosen to make the fabrication process easier. Then, the idea is to obtain the complex propa, for a range of wavegation constant at the frequency guide heights . From this analysis we select the value of

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Fig. 10. Dispersive behavior of the CRLH LWA under analysis, computed with the proposed iterative algorithm after i = 1 and i = 30 (convergence reached) iterations. (a) Brillouin diagram, validated using HFSS. (b) Attenuation (radiation) losses versus frequency.

Fig. 9. Determination of the physical dimensions of the unit cell required for a balanced CRLH design, i.e., Re(k ) = 0. (a) Evolution of the propagation constant as a function of the waveguide height (t), for a fixed value of the slot width (g = 0:5 mm.). (b) Evolution of the propagation constant as a function of the slot width (g ), for a fixed value of the waveguide height (t = 3:65 mm.)

which makes zero the real part of the complex propagation constant. Fig. 9(a) presents this analysis, which yields a final wavemm. This provides a balanced unit cell guide height of design. In order to indeed show that the unit cell is balanced, the procedure is repeated again, but fixing now the waveguide mm) and varying the slot height to the new value ( width. The analysis result is shown in Fig. 9(b), which demonmm is indeed the slot width which balances strates that the CRLH unit cell for the given waveguide height ( mm). This completes the CRLH unit cell balancing method. It is important to remark that the proposed procedure is able to accurately balance the CRLH LWA unit cell, without requiring any full-wave simulations of the complete unit cell. In fact, with the technique proposed the modal analysis is only applied to the slot problem, and not to the complete unit cell structure. Usually, a considerable number of extremely time-consuming full-wave simulations are required to obtain a balanced-design. This is completely avoided using the proposed method, which is able to determine the physical dimensions of a balanced structure in less than 4 min. In addition, note that the iterative algorithm is quickly convergent, requiring just eight iterations to obtain a between two consecutive steps. relative error of less than B. Analysis of a Single CRLH Unit Cell The complex propagation constant of the CRLH unit cell is then obtained for the desired frequency region applying the iterative algorithm. A maximum of 30 iterations are required to obbetween two tain convergence (for a relative error below consecutive step for all frequencies). The result of the analysis and is shown in Fig. 10, for the case of iterations (convergence reached). As it can be observed, a balanced dispersive behavior, with a transition frequency of 3.0 GHz, is clearly

obtained. This is further confirmed using simulation data for the dispersion curve, which has been obtained using HFSS. In Fig. 10(a) it can be observed that the dispersion curve and of the of the TM mode and the RH parts of the dispersion curve coincide very well with the real part of for . However, there is a discrepancy in the frequency range between 1.95 and 2.5 GHz. This is because the equivalent circuit only reproduces the propagation phenomenon of waves traveling inside the loaded PPW, while the full-wave eigenmode analysis of HFSS also considers waves which propagate in free-space above the CRLH TL. This leads to a bandgap due to the coupling between opposite waves propagating above and below the slotted surface (in the case that both types of waves are excited, which occurs in the eigenmode analysis). Note that the equivalent circuit only considers the excitation of the waves traveling inside the CRLH TL, which is correct for predicting the behavior of the proposed LWA. In Fig. 10(b) the radiation losses of the antenna are presented. A significant decrease of the antenna efficiency at the broadside direction (i.e., at the transition frequency of the antenna) can be observed. As explained in Section IV-C, this is expected for this type of unit cell configuration. In addition, the computed radiation losses accurately complete the study of the antenna radiation behavior as a function of frequency and of the physical dimensions of the structure. Usually, circuit models [1], [2] are only able to predict the phase constant, and use curve fitting to obtain a frequency-independent resistor value which models the losses. Furthermore, note that usual commercial full-wave software have also difficulties to obtain this parameter in infinitely periodic configurations, because they usually assume a purely real phase shift between the unit cell limits. and In order to complete the analysis, the resistor , normalized with respect to the unit cell length capacitor , are shown as a function of frequency in Fig. 11. It is interesting to note that the bandpass frequency region of the TL (approximately from 2 to 6 GHz) is clearly visible in this figure. In particular, within this frequency range the capacitor exhibits smooth variations, while the value of the radiation resistance

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Fig. 12. Maximum absolute error of the T matrix elements of the equivalent radiating structure (see Fig. 3) with respect to the ideal T matrix related to the equivalent circuit (where T = T = 1 and T = 0), as a function of frequency.

Fig. 11. Frequency dependent behavior of the dispersive lumped components shown in Fig. 1 (bottom), calculated for the CRLH unit cell described in Section VI-B. (a) Series capacitor C (! ). (b) Series resistor R (! ).

experiences a slow decrease. Also, around the lower and upper cut-off frequencies of the structure the capacitance shows an abrupt increase. This is related to the larger stored energy of the structure close to the bandpass edges, very well known in filter theory [27]. In addition, note that although the radiation losses decrease at the broadside transition [see Fig. 10(b), at 3 GHz] this does not correspond to a decrease of the dispersive lumped resistor . This is due to the complex relationship between value these two quantities, as explained in Section III. Furthermore, note that the approximation employed to obtain the dispersive lumped parameters is very accurate. This is demonstrated in two different ways. First, the complex propagation constant ob[i.e., only with the circuit elements of Fig. 1 tained using (bottom)] directly superimpose the full-wave results presented in Fig. 10. Second, the maximum absolute error of the terms , and as compared with the is very small, as can be observed in same elements of Fig. 12. This confirms that the proposed dispersive equivalent circuit is indeed accurate. C. Analysis of a Ten Unit Cells CRLH LWA of a complete CRLH LWA Finally, a single strip of width identical unit cells with mm consisting of is analyzed combining the single unit cell results obtained in the previous subsection with an ABCD matrix approach [1]. In order to correctly match the antenna, the width of the first and must accurately be derived. The goal is to obtain last slots a width which behaves as a half-slot in the infinite array environment. In this way, the first and last unit-cells of the antenna rigorously follow the equivalent circuit model of Fig. 1, and they see the PPW as a kind of continuation of the periodic structure. This leads to a smooth transition from the start/end of the CRLH

structure and the unperturbed PPW within the whole frequency range. Following this strategy, the first and last slots present a . The approximate value capacitive behavior close to mm. Furthermore, found using this procedure is note that is responsible for the connection of the CRLH TL to the feeding TL, but it does not influence the propagation and radiation characteristics of the line. Reference results for the antenna under analysis have been obtained by full-wave time domain simulation using CST Microwave Studio (MWST). The full-wave model was made up of a strip with perfectly magnetically conducting (PMC) boundary conditions applied to the lateral walls of the simulation volume, in order to represent a laterally periodic structure of infinite extension. The magnitude of the computed scattering parameters and and the radiation efficiency (only lossless materials were considered) are plotted in Fig. 13. As it can be observed in the figure, a very good agreement between the proposed method and the full-wave simulation is obtained. Furthermore, Fig. 14 presents the scanning capabilities of the antenna, as a function of the operating frequency. Specifically, a scanning of the main lobe from the radiation angle degrees up to degrees is shown. As expected, a decrease in the radiation efficiency is found at the . Note that the directivity is higher broadside direction in the RH region than in the LH frequency region . This is related to the fact that the radiation losses are higher in the LH region [as shown in Fig. 10(b)]. Therefore, the input power is radiated in a few unit-cells, leading to a reduced effective length of the antenna (and, therefore, a lower directivity). On the other hand, radiation losses are lower at the RH region, and the power is radiated along the whole structure, leading to a larger effective length of the antenna (and, consequently, to a higher directivity). The above full-wave validations demonstrate that the proposed iterative method is able to efficiently and rigorously analyze PPW CRLH LWAs, taking into account the real physical dimensions of the structure. Furthermore, the proposed method is able to perform the analysis in just six minutes, instead of eight hours required by the full-wave simulations. This allows the application of the proposed modal-based iterative method in

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composite right/left-handed leaky-wave antennas (PPW CRLH LWAs). The conventional lossless CRLH unit cell configuration has been modified, including an equivalent circuit which takes into account the structure’s coupling to free-space. This coupling has been modeled employing a unit cell equivalent radiating structure, which is rigorously analyzed using a multimode approach combined with Floquet’s theorem. The resulting transmission matrix has accurately been represented by lumped elements, leading to a frequency-dependent unit cell model. Then, a quickly converging iterative algorithm has been employed to determine the final element values of the unit cell. The proposed technique was found to be accurate, and it can take into account the structure physical dimensions. The technique also allows to obtain a balanced CRLH unit cell design, it is much faster than full-wave simulations, and it provides a deep insight into the physics of the antenna’s radiation mechanism. ACKNOWLEDGMENT The authors would like to thank their colleagues C. Löcker and Dr. T. Vaupel for many fruitful discussions. REFERENCES

Fig. 13. Comparison of scattering parameters and radiation efficiency computed by CST Microwave Studio (CST MWST) and by the proposed iterative circuit method (EQC) of a single strip CRLH LWA consisting of ten identical reference unit cells.

Fig. 14. Radiation pattern of the proposed CRLH LWA at different operating frequencies, showing the space scanning capabilities of the antenna.

the analysis of practical antennas, or even to include this technique into a CAD tool for the analysis, design, and optimization of mushroom based CRLH LWAs. VII. CONCLUSION This contribution has presented a novel modal-based iterative circuit model for the calculation of the complex propagation constant related to mushroom-like parallel-plate waveguide

[1] C. Caloz and T. Itoh, Electromagnetic Metamaterials: Transmission Line Theory and Microwave Applications. Hoboken, NJ: Wiley-Intersci., 2006. [2] , G. V. Eleftheriades and K. G. Balmain, Eds., Negative-Refraction Metamaterials: Fundamental Principles and Applications. Hoboken, NJ: Wiley/IEEE, 2005. [3] A. A. Oliner and D. R. Jackson, “Leaky-wave antennas,” in Antenna Engineering Handbook, J. L. Volakis, Ed., 4th ed. New York: McGraw-Hill, 2007. [4] L. Liu, C. Caloz, and T. Itoh, “Dominant mode leaky-wave antenna with backfire-to-endfire scanning capability,” Electron. Lett., vol. 38, no. 23, pp. 1414–1416, Nov. 2002. [5] S. Lim, C. Caloz, and T. Itoh, “A reflecto-directive system using a composite right/left-handed (CRLH) leaky-wave antenna and heterodyne mixing,” IEEE Microw. Wireless Compon. Lett., vol. 14, no. 4, pp. 183–185, Apr. 2004. [6] A. Grbic and G. V. Eleftheriades, “Leaky CPW-based slot antenna arrays for millimeter-wave applications,” IEEE Trans. Antennas Propag., vol. 50, no. 11, pp. 1494–1504, Nov. 2002. [7] M. A. Antoniades and G. V. Eleftheriades, “A CPS leaky-wave antenna with reduced beam squinting using NRI-TL metamaterials,” IEEE Trans. Antennas Propag., vol. 56, no. 3, pp. 708–721, Mar. 2008. [8] D. Sievenpiper, L. Zhang, R. F. J. Broas, N. G. Alexópolous, 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. [9] D. Sievenpiper, J. Schaffner, J. J. Lee, and S. Livingston, “A steerable leaky-wave antenna using a tunable impedance ground plane,” IEEE Antennas Wireless Propag. Lett., vol. 1, pp. 179–182, 2002. [10] C. A. Allen, C. Caloz, and T. Itoh, “Leaky-waves in a metamaterialbased two-dimensional structure for a conical beam antenna application,” in Proc. IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2004, vol. 1, pp. 6–11. [11] T. Bertuch, “A TM leaky-wave antenna comprising a textured surface,” presented at the Int. Conf. Electromag. Adv. Appl. (ICEAA), Sep. 2007. [12] I. J. Bahl and K. C. Gupta, “A leaky-wave antenna using an artificial dielectric medium,” IEEE Trans. Antennas Propag., vol. 22, no. 1, pp. 119–122, Jan. 1974. [13] I. J. Bahl and K. C. Gupta, “Frequency scanning by leaky-wave antennas using artificial dielectrics,” IEEE Trans. Antennas Propag., vol. 23, no. 4, pp. 584–589, Jul. 1975. [14] I. J. Bahl and K. C. Gupta, “Radiation from a dielectric-artificial dielectric slab,” IEEE Trans. Antennas Propag., vol. 24, no. 1, pp. 73–76, Jan. 1976. [15] I. J. Bahl and P. Bhartia, “Leaky-wave antennas using artificial dielectrics at millimeter wave frequencies,” IEEE Trans. Antennas Propag., vol. 18, pp. 23–26, June 1980.

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[16] R. J. King, D. V. Thiel, and K. S. Park, “The synthesis of surface reactance using an artificial dielectric,” IEEE Trans. Antennas Propag., vol. 31, no. 3, pp. 471–476, May 1983. [17] S. Paulotto, P. Baccarelli, F. Frezza, and D. R. Jackson, “Full-wave modal dispersion analysis and broadside optimization for a class of microstrip CRLH leaky-wave antennas,” IEEE Trans. Microw. Theory Tech., vol. 56, no. 12, pp. 2826–2837, Dec. 2008. [18] C. S. Paulotto, P. Baccarelli, F. Frezza, and D. R. Jackson, “A novel technique for open-stopband suppression in 1-D periodic printed leakywave antennas,” IEEE Trans. Antennas Propag., vol. 57, no. 7, pp. 1894–1906, Jul. 2009. [19] S. Marini, A. Coves, V. Boria, and B. Gimeno, “Efficient modal analysis of periodic structures loaded with arbitrarily shaped waveguides,” IEEE Trans. Microw. Theory Tech., vol. 58, no. 3, pp. 529–536, Mar. 2010. [20] R. F. Harrington, Time-Harmonic Electromagnetic Fields. New York: McGraw-Hill, 1961. [21] D. Pozar, Microwave Engineering, 3rd ed. Hoboken, NJ: Wiley, 2005. [22] F. Arndt, I. Ahrents, U. Papziner, U. Wiechmann, and R. Wilkeit, “Optimized E-plane t-junction series power dividers,” IEEE Trans. Microw. Theory Tech., vol. 35, no. 11, pp. 1052–1059, Nov. 1987. [23] K. H. Park, H. J. Eom, and Y. Yamaguchi, “An analytic series solution for E-plane t-junction in parallel-plate waveguide,” IEEE Trans. Microw. Theory Tech., vol. 42, no. 2, pp. 356–358, Feb. 1994. [24] Y. H. Cho, “New iterative equations for an E-plane t-junction in a parallel-plate waveguide using green’s functions,” Microw. Opt. Technol. Lett., vol. 37, no. 7, pp. 447–449, Jun. 2003. [25] N. Marcuvitz, Waveguide Handbook. Boston, MA: MIT Press, 1964, MIT Radiation Lab. Ser.. [26] D. M. Pozar, “Analysis and design of cavity coupled microstrip couplers and transitions,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 3, pp. 1034–1044, Mar. 2003. [27] C. Ernst and V. Postoyalko, “Prediction of peak internal fields in directcoupled-cavity filters,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 1, pp. 64–73, Jan. 2003. Juan Sebastian Gomez-Diaz (S’07) was born in Ontur (Albacete), Spain, in 1983. He received the Telecommunications Engineer degree (with honors) from the Technical University of Cartagena (UPCT), Spain, in 2006, where he is currently working toward the Ph.D. degree. In 2007, he joined the Telecommunication and Electromagnetic group (GEAT), UPCT, as a Research Assistant. From November 2007 to October 2008, he was at Poly-Grames, École Polytechnique de Montréal, as a visiting Ph.D. student, where he was involved in the impulse-regime analysis of linear and nonlinear metamaterial-based devices and antennas. From September 2009 to March 2010, he was a visiting Ph.D. student with the Fraunhofer Institute for High Frequency Physics and Radar Techniques (FHR), working on the modal analysis of leaky-wave antennas. His current scientific interests also include IE and numerical methods and their application to the analysis and design of microwave circuits and antennas.

Alejandro Álvarez-Melcón (M’99–SM’07) was born in Madrid, Spain, in 1965. He received the Telecommunications Engineer degree from the Technical University of Madrid (UPM), Madrid, in 1991, and the Ph.D. degree in electrical engineering from the Swiss Federal Institute of Technology, Lausanne, in 1998. In 1988, he joined the Signal, Systems and Radiocommunications Department, UPM, as a Research Student, where he was involved in the design, testing, and measurement of broad-band spiral antennas for electromagnetic measurements support (EMS) equipment. From 1991 to 1993, he was with the Radio Frequency Systems Division, European Space Agency (ESA/ESTEC), Noordwijk, The Netherlands, where he was involved in the development of analytical and numerical tools for the study of waveguide discontinuities, planar transmission lines, and microwave filters. From 1993 to 1995, he was with the Space Division, Industry Alcatel Espacio, Madrid, and was also with the ESA, where he collaborated in several ESA/European Space Research and Technology Centre (ESTEC) contracts. From 1995 to 1999, he was with the Swiss Federal Institute of Technology, École Polytechnique Fédérale de Lausanne (EPFL), where he was involved with the field of microstrip antennas and printed circuits for space applications. In 2000, he joined the Technical University of Cartagena, Spain, where he is currently developing his teaching and research activities. Dr. Álvarez Melcón was the recipient of the Journée Internationales de Nice Sur les Antennes (JINA) Best Paper Award for the best contribution to the JINA’98 International Symposium on Antennas, and the Colegio Oficial de Ingenieros de Telecomunicación (COIT/AEIT) Award to the best Ph.D. thesis in basic information and communication technologies.

Thomas Bertuch (M’97) received the Diplom-Ingenieur and Ph.D. (Doktor der Ingenieurwissenschaften) degree from RWTH Aachen University, Aachen, Germany, in 1996 and 2003, respectively. His Ph.D. was developed at the Research Institute for High Frequency Physics and Radar Techniques of the Research Establishment for Applied Natural Science e.V. (FGAN-FHR), Wachtberg, Germany. Throughout 2004, he was a Senior Antenna Scientist with the Defence, Security and Safety Institute of The Netherlands Organization for Applied Scientific Research (TNO), The Hague, The Netherlands. Since 2005, he has been working with the FGAN-FHR which, since 2009, is the Fraunhofer Institute for High Frequency Physics and Radar Techniques (Fraunhofer FHR) where he currently holds the position of Team Leader Antennas and Front-End Technology. From 2000 to 2009, he has been a Lecturer of RF engineering at the Bonn-Rhine-Sieg University of Applied Sciences, Sankt Augustin, Germany. His main research activities concern antenna and microwave circuit design, engineered electromagnetic materials (metamaterials), radar system design, and electromagnetic modeling.

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Evolved-Profile Dielectric Rod Antennas Stephen M. Hanham, Member, IEEE, Trevor S. Bird, Fellow, IEEE, Andrew D. Hellicar, Member, IEEE, and Robert A. Minasian, Fellow, IEEE

Abstract—A systematic approach is presented for the design of profiled dielectric rod antennas that satisfy specified radiation pattern objectives. The approach uses a body of revolution method of moments technique to rapidly analyze arbitrarily profiled dielectric rods while a genetic algorithm is used to achieve the design objectives. As examples we present dielectric rod designs optimized for maximum gain and low sidelobes. These designs are compared with a conventional linear profile design. Measured results are presented and these are shown to agree well with the calculated radiation patterns. We show that improved gain and sidelobe performance is achieved using a non-linear rod profile compared to a standard linear profile. The generality of the approach is demonstrated with a shaped beam antenna design that has a cosecant-squared pattern. Index Terms—Antenna theory, dielectric rod antennas, endfire antennas, genetic algorithm.

I. INTRODUCTION

T

HE relationship between the radiation pattern of a dielectric rod antenna and its profile is not well established [1], [2]. There exists in the literature a number of design guidelines for achieving specific radiation pattern objectives such as maximum gain and minimum sidelobes [1]; however, in general the optimal profile for a given radiation pattern can be difficult to determine. Dielectric rod antennas are commonly designed using the discontinuity radiation concept (DRC) [1]. In this approach the antenna is treated as a linear array of effective sources which are due to radiation from discontinuities experienced by the surface-wave travelling along the antenna and the feed radiation due to power not converted into the surface-wave by the exciting feed. The total radiated field is the sum of the fields due to these sources. For a dielectric rod designed to produce maximum gain on boresight it is common practice to gently taper the rod profile toward its end. This tapering achieves several outcomes. Firstly,

Manuscript received May 04, 2010; revised August 27, 2010; accepted September 27, 2010. Date of publication January 31, 2011; date of current version April 06, 2011. S. M. Hanham was with the School of Electrical and Information Engineering, University of Sydney, Sydney NSW 2006, Australia and the CSIRO ICT Centre, Epping NSW 1710, Australia. He is now with the Department of Physics, Blackett Laboratory, Imperial College London, London SW7 2AZ, U.K. (e-mail: [email protected]). T. S. Bird and A. D. Hellicar are with the CSIRO ICT Centre, Epping NSW 1710, Australia. R. A. Minasian is with the School of Electrical and Information Engineering, University of Sydney, Sydney NSW 2006, Australia. 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.2011.2109689

a larger starting diameter corresponds to a slower surface-wave which is more tightly bound to the rod and permits a higher excitation efficiency of the surface-wave. Secondly, a smaller terminating diameter causes the surface-wave to become weakly mode field bound to the rod and consequently allows the to expand further into the surrounding free-space region. This increases the size of the effective aperture at the rod termination producing an increase in gain. The tapering also reduces the difference between the surface-wave propagation constant and the free-space propagation constant at the rod termination, minimizing reflection of the surface-wave at the end of the rod. The taper profile must also ensure that appropriate phasing is achieved between the feed radiation and radiation occurring along the rod profile, and, in particular, at the rod termination, to maximize gain. Commonly adopted rod profile tapers are linear, exponential and curvilinear [1]–[3]. Ando et al. [2] demonstrated that high order curvilinear tapers could be used to minimize the conversion of the surface-wave into radiation modes along the length of the rod and hence maximize the amount of the power remaining in the surface-wave at the rod termination. The taper also increased the radial extent of the equiphase region at the rod termination due to the surface-wave, improving the gain. Kishk [4] demonstrated that shaping the end of very short, waveguide-fed, dielectric rods also lead to higher gains. A notable example of a non-linear profiled surface-wave antenna is the disc-on-rod cigar antenna of Simon [5] where the non-linear profile employed modulates the surface-wave causing the surface-wave to be more tightly bound than on a comparable single-mode surface for the same propagation constant [6], [7]. This allowed greater directivity by ensuring the correct phasing between the radiating elements. A periodically modulated surface can also be used to excite a fast wave, which radiates at a specific angle away from boresight, such as in [8]. Compared with these previous studies, we investigate directly the optimal profile for short dielectric rods that satisfy chosen radiation pattern objectives. As examples, we consider rods optimized separately for maximum gain and low sidelobes as well as with a shaped beam. Initial work on this approach was described in [9]. Dielectric rod antennas are often used as feed elements in a focal plane array (FPA) due to their potential for dense packing and integration. Applications of dielectric rod antenna FPAs include millimeter-wave and terahertz imaging [10]. A dielectric rod antenna which produces a radiation pattern with low sidelobes is useful in a FPA to reduce the interaction between array elements and the effect of stray radiation entering the sidelobes. A dielectric rod with maximum gain for a given rod length is attractive for applications such as for millimeter-wave communications where an antenna with moderate gain is required.

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Optimized dielectric rod antennas with shaped beams that peak off-boresight could be used for satellite [11], [12] and wireless network applications [13]–[15] where a constant flux of illumination over a footprint is desired. In this paper a ring-slot source is used as the feed. This type of source has recently been shown to achieve very high excitation efficiencies [16] of over 90%. We use this feed here so the influence of the feed on the radiation pattern is minimized; however, the profile optimization approach presented is also valid for other feeds. A wider bandwidth feed might be adopted, such as a waveguide taper [1], for applications where broadband operation is required. In Section II we outline a method of moments (MoM) technique that assists in the rapid analysis of dielectric rod antennas that are excited by ring slot sources. The efficiency of this method allows a genetic algorithm to be employed for the optimization of the rod profile. The implementation of the genetic algorithm is described in Section III. Three dielectric rods with different optimized profiles have been designed and fabricated, and these are described in Section IV. The measured results are presented in Section V and their implications are discussed in Section VI. Also given in Section VI are the results for a dielectric rod antenna whose profile has been optimized to achieve a shaped beam which has a cosecant-squared pattern.

Fig. 1. Body of revolution for analysis of a profiled dielectric rod antenna.

then be represented as a sum of along the generating curve

triangular basis functions

(2)

II. NUMERICAL ANALYSIS METHOD

(3)

A body of revolution method of moments (BoR-MoM) code was developed to rapidly compute the radiation pattern for an arbitrarily profiled dielectric rod excited by a ring slot feed. We employed the Poggio-Miller-Chang-Harrington-Wu (PMCHW) MoM formulation as described by Mautz and Harrington [17], [18] and Kishk [19]. The full equations and details of this technique are given in these references and herein we only summarize its modifications for the present problem. The dielectric rod is represented as a homogeneous dielectric as body of revolution around the axis with permittivity illustrated in Fig. 1. The field radiated by the dielectric rod is formulated in terms of equivalent electric and magnetic currents over the body surface . The dielectric rod is excited by a ring slot of radius in a ground plane which covers the base of the rod. This ring slot is represented in the MoM model by an equivalent magnetic current distribution given by

(1) This current distribution has previously been shown to excite mode on a dielectric rod with high efthe fundamental ficiency [16]. The symmetry of this current distribution allows the Fourier series representing the electric and magnetic current variation with about the axis of revolution that is used in the BoR formulation to be reduced to the first two exponential . That is, only modes with a single azimuthal period terms will be excited by the source. The symmetry between the solutions for these two terms necessitates only the calculation of term. These simplifications greatly reduce the computhe tational complexity of the method. The equivalent currents can

where

The superscript or denotes the tangential or phi component, , , and are the unknown current corespectively. efficients to be determined. The Galerkin method is employed for solving for the unknown coefficients. The resulting matrix equation is solved as described in [17], [18] with the source terms calculated using (1). From these coefficients the equivalent currents are obtained from (2) and (3). The fields radiated by the dielectric rod can then be calculated from the computed currents by means of the free-space Greens function. A Gaussian quadrature with 24 points was used to evaluate the integrals while approximately 100 triangular basis functions evaluated at 4 points (as in [18]) were used to represent the profile. Evaluating a profile at a single frequency took 3.3 seconds on a computer with a 32-bit 2.2 GHz processor and 2 GB RAM, whereas CST Microwave Studio took several hours on a computer with a 64-bit 3 GHz processor and 8 GB RAM, depending upon the meshing density. III. ROD PROFILE OPTIMIZATION A cubic spline with knot points uniformly distributed along the -axis was used to represent the dielectric rod profile. This ensured a smooth and continuous rod profile whilst minimizing the number of parameters to optimize. Fig. 2 illustrates

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algorithm can be found in [23]. The GA used a population size of 20 chromosomes with each representing a candidate rod profile. The chromosomes consisted of a vector of the radius component of the knot points defining the spline profile. A uniform stochastic selection rule was used to select the chromosomes which would become the parents for the next generation of solutions. The cross-over operation was used to produce 80% of the next generation solutions with the mutation operation used for the remainder except for the two best or elite solutions which were transferred to the next generation unmodified. The mutation operation modified each knot point radius in a selected chromosome with a probability of 0.01 by replacing it with a randomly chosen value within the range of allowed values.

Fig. 2. Spline representation of the rod profile.

the spline representation of the rod. Each knot point was con, and for radii smaller than strained to vary between 0 and the radius would clip at . This was a practical radius beyond which it was considered that the rod could not be fab, it ricated. By allowing the knot points to be smaller than allowed spline profiles where the radius would rapidly decrease to the minimum radius and then flatten out at the minimum . This shape was found to be useful for maximum radius gain designs. The starting radius of the rod was fixed to keep the excitation efficiency of the surface-wave constant so that the feed radiation did not vary and mask the effect of rod profile variations. It should be noted that the spline representation does not allow rod profiles with step discontinuities in the radius and hence this class of profiles is not considered. A genetic algorithm (GA) was used to optimize the rod profile by minimizing a fitness function , which measures how close the rod radiation pattern is to a desired radiation pattern objective [20]–[22]. Two different fitness functions were employed. be the radiation pattern in dBi for an arbitrarily Letting profiled dielectric rod, where is the elevation angle and the azimuthal angle, the definition of G for maximum gain on boresight is

(4) and for a more general upper- and lower-bound constrained radiation pattern

(5)

where is a function which returns the maximum of and . and are the upper- and lower-bound radiation pattern envelope constraints, respectively, over the range and , respectively. The fitness function was evaluof ated at a number of discrete frequencies over a given bandwidth and its value averaged. A standard implementation of the genetic algorithm written in Matlab was used for the optimization. A full description of this

IV. EVOLVED ROD DESIGNS As a demonstration of our approach we designed two dielectric rod antennas with profiles optimized using the fitness functions described in Section III, and a third linear-profiled rod for comparison. These three rods shall be referred to as: • linear profiled (LIN); • minimum sidelobe level (SL); • maximum boresight gain (MG). In this work, we consider short, high permittivity dielectric rods which are applicable to millimeter-wave and terahertz applications where fabrication of long rods may be difficult [24]; however, the approach is equally applicable to longer rods with with permitlower permittivities. A rod length of tivity was chosen, where is the free-space wavelength. A high permittivity is advantageous because it increases the radiation of the ring slot into the dielectric giving a good front-to-back ratio. The starting diameter of the rod and magnetic ring current was chosen to be and , respectively, to radius maximize the excitation efficiency according to [16] and operate at the resonant frequency of the ring slot. The minimum allowable radius was set to for the LIN, SL and MG rods. Originally, this minimum radius but it was subsequently discovered was set to that the optimized MG rod profile included a section of rod that was difficult to fabricate due to its narrowness. The maximum ), rod radius was set to be the starting radius (i.e. which prevented the rod from becoming thicker than at its start. It was found that using 20 knot points for the spline profile gave a good balance between performance and optimization time. The MG design was optimized to maximize the gain on boresight over a bandwidth of 4% using the fitness function given in (4). The impedance bandwidth of a ring-slot is typically 10%; however, it was found that the rods optimized for maximum gain exhibited a narrower pattern bandwidth and hence a value of 4% was used. The narrower pattern bandwidth is a consequence of the oscillatory profile. Wider bandwidths can be achieved by evaluating the fitness function over a wider range of frequencies; however, this additional bandwidth is accompanied by reduced radiation performance. The SL design was optimized to minimize the level of the sidelobes of the antenna by applying an upper-bound envelope over a 10% bandwidth using (5). constraint for

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Fig. 3. Dielectric rod profiles with crosses representing spline knot points. From top to bottom: linear profile (LIN); low sidelobe (SL); maximum gain (MG); another maximum gain design.

The dielectric rods optimized for maximum gain and minimum sidelobes were characterized by very different rod profiles. It should be possible to simultaneously optimize the dielectric rod antennas for both maximum gain and minimum sidelobe characteristics; however, here the aim was to give examples with fewer constraints than would be required in the combined design and minimize the possibility of compromise. The LIN rod was designed with a linear taper which maximized the boresight gain for comparison purposes. A direct search method was employed for this optimization rather than a genetic algorithm since the starting diameter was fixed and thus it resulted in a simple, single variable optimization problem. The GA optimization was run for several hundred generations until the algorithm stalled and no further improvement occurred. Running the algorithm numerous times showed that many different types of rod profiles were possible that satisfied the fitness function. Ultimately, the designs chosen were those that gave the best performance and were easiest to manufacture. The three chosen dielectric rod profiles are shown in Fig. 3. An additional rod profile optimized for maximum gain is also shown for comparison purposes in the final profile in Fig. 3. It can be seen that there is a strong similarity between the two maximum of length and both antennas gain antennas over the first feature a flare at the end of the rod. This rod was optimized with a smaller minimum diameter than the MG rod. When the minimum radius constraint was set to zero the optimizer produced rod profiles with sections that featured a zero radius. V. FABRICATION The LIN, SL and MG dielectric rods with profiles shown in Fig. 3 were designed and fabricated for operation at 10 GHz.

Fig. 4. (a) Fabricated antennas with feed before attachment. (b) MG dielectric rod with feed attached.

The rods were milled from Emerson & Cuming Eccostock HiK , loss tangent 0.002) using a CNC rods (measured lathe. The source ring slot and its input were etched on a 0.010 inch thick, Rogers RT/duroid 6010 high frequency laminate . Photographs of the fabricated dielectric rods and the feed are shown in Fig. 4. The ring slot feed consisted of a coplanar waveguide (CPW) network, which included a quarter wavelength section for matching the ring slot impedance to a section of CPW with a 50 characteristic impedance. The ring was tuned slightly for each rod slot radius of using CST Microwave Studio (MWS) after optimization with the BoR-MoM technique to ensure the ring slot was operating at its resonant frequency. The slot width was chosen to be 0.3 mm. The feed circuit board was kept as small as possible to minimize its influence on the radiation pattern of the antenna. The feed circuit boards were bonded to the back of the dielectric rods. VI. RESULTS The radiation patterns of the three dielectric rod antennas described in the previous two sections were calculated using the BoR-MoM approach outlined in Section II and with MWS. The BoR-MoM approach uses a magnetic ring current to model the ring slot feed while MWS uses a more accurate model of the fabricated ring slot. The comparison of results with MWS and

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TABLE I SUMMARY OF ANTENNA PERFORMANCE WITH DIFFERENT PROFILED RODS

the simple ring-slot model indicate the significance of the practical feed on the overall design. The radiation patterns of the three rods were measured at 10 GHz in the CSIRO far-field range and these are compared with the simulated results in Figs. 7 to 9. The computed and measured gain and first sidelobe level in the principal planes are compared for all antennas in Table I. Plots of the electric field from MWS when the LIN, SL and MG antennas are radiating are shown in Figs. 5 and 6 to illustrate the different radiation behavior. This will be discussed in the following section. The field plots were generated assuming an input power of 1 W and are plotted on the same scale for ease of comparison. The measured impedance bandwidth of the LIN, SL and MG rods was 6%, 5% and 6%, respectively. This is less than the typical bandwidth of a ring-slot due to the CPW feed and matching section and the effect of reflections upon the ring-slot impedance which are not considered in the BoR-MoM code. To further demonstrate the generality of the optimization approach presented here, the profile of a dielectric rod was designed to produce a shaped beam. As an example, a target radiation pattern was chosen with a cosecant-squared variation with elevation angle which peaks between 80 and 90 degrees. This type of pattern may be suitable for wireless networking [13]–[15] or for satellite applications [11], [12] where more power is required at elevation angles away from boresight. The fitness function was defined to be the integral of the absolute difference between the dielectric rod radiation pattern and the plus an upper-bound term for target pattern for to restrict the rear radiation. The optimized profile is shown in Fig. 10 and the simulated radiation pattern in Fig. 11. VII. DISCUSSION The measured gain of the three antennas is less than the simulated results, partially due to losses not included in the simulation such as dielectric loss and feed loss. Further analysis revealed the dielectric loss to be approximately 0.2 dB and loss in the feed circuit board to be 0.4 dB. The connector loss is estimated to be 0.1 dB. Loss due to impedance mismatch with the ring slot was calculated to be approximately 0.1 dB for the MG and SL rods and 0.3 dB for the LIN rod. This gives a total calculated loss of 0.8 dB for the MG and SL rods and 1 dB for the LIN rod. This compares well with a 0.9 dB difference between measured and calculated boresight gain for the MG rod. The reduction in boresight gain between measured and simulated was higher in the case of the LIN and SL rods due to the ring slot

Fig. 5. Instantaneous magnitude of the radiating electric field for the (a) linear (LIN), (b) low sidelobes (SL), and (c) maximum gain (MG) antenna.

operating slightly off resonant frequency, which modifies the ring slot radiation characteristics. The gain calculated using the BoR-MoM technique agrees quite well with the more accurate feed model results that are calculated with MWS using lossless materials (see Table I).

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Fig. 6. Amplitude of the radiating electric field for the maximum gain antenna (MG). Arrows indicate the direction of radiation of the surface-wave.

There is reasonable agreement between the shape of the measured and simulated radiation patterns for the LIN and SL rods. The agreement for the more complex profiled MG rod is better and it is thought that this is due to the resonant frequency being closer to 10 GHz. The introduction of the ring slot feed disrupts the rotational symmetry of the structure and slightly modifies the radiation pattern from that predicted by the BoR-MoM method. The effect of the feed can be seen in the MWS and measured radiation patterns where there is an observable difference with the BoR-MoM results for theta angles greater than 60 . The gain and sidelobe levels presented in this paper were achieved using a high excitation efficiency source with a high permittivity rod which helped reduce the back radiation. However, the same gain and sidelobe level could be achieved using a rod with a much lower permittivity in combination with a high efficiency feed. One of the potential applications of the profiled dielectric rod antennas is as a feed and this generally requires a stable phase centre and good cross-polarization performance. Simulations revealed that the phase centre was relatively stable and varied across the 10% bandwidth of the LIN and by less than SL rods and 4% bandwidth of the MG rod. It was also found that the rotational symmetry of the rods meant that the cross-polarization was not significantly influenced by the rod profiling and was mainly determined by the interaction between the back radiation and feed. The individual results for the dielectric rods are now discussed below. A. Linear Profile (LIN) The LIN antenna has a simulated boresight gain of 10.3 dBi which is significantly lower than the 12.4 dBi and 13.2 dBi simulated gains of the SL and MG rods, respectively. The measured maximum sidelobe level was approximately 9.6 dB in the E-plane and 10.3 dB in the H-plane, below boresight. A plot of the electric field magnitude in the H-plane, given in Fig. 5(a), shows that the surface-wave power is concentrated

Fig. 7. Radiation patterns in the (a) E-plane and (b) H-plane of the linear tapered (LIN) rod antenna at 10 GHz. Theory (dashed) and experiment (solid).

inside the rod until after its midpoint, where the wavefront begins to emerge from the rod and expands into the surrounding free-space region. Only about half of the length of the rod has been utilized for expanding the surface-wave wavefront outside the rod, which is required for high gain. Also of note is the strong curvature of the surface-wave wavefront when it radiates from the end of the rod. Ando et al. [2] showed that a straighter equiphase region of the surface-wave at the rod termination was important for achieving higher gains. It can be observed from the present work that a rod profile that decreases more rapidly at its start will provide more time for the surface-wave to expand and the wavefront to straighten leading to higher gains. The transverse expansion of the surface-wave into the surrounding free-space corresponds to a reduction in the surface-wave propagation constant and increase in the surface-wave wavelength. B. Minimum Sidelobes Level Profile (SL) A number of different rod profiles were optimized which exhibited low sidelobes. Several designs had oscillatory profiles

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where the oscillation amplitudes were too small to radiate significant power along the length of the rod. However, the rod profile which so far has given the best sidelobe performance over a 10% bandwidth has an optimized profile that is close to a third-order power curvilinear. This profile has the approximate form

(6) are the radius of the rod at the feed end and where and termination end, respectively, and is the rod length. Additionally, the rod has a slight flare at its termination. Ando et al. [2] showed that third-order and higher curvilinear profiles minimize radiation along the length of the dielectric rod, maximizing the amount of energy reaching the end. This leads to an inherently broadband design because the pattern is almost entirely determined by the radiation from the rod termination rather than the interference between multiple radiation points along the rod length. An examination of the electric field in Fig. 5(b) shows that the rod profile causes a gentle expansion of the surface-wave along the entire length of the rod with no observable radiation from the rod except at its termination. The ring slot produces a small amount of feed radiation which radiates at the back side of the rod and diffracts around the rod base, interfering with the field in the forward direction; however, this is a minor effect. The measured radiation pattern, shown in Fig. 8, demonstrates that the antenna has a 1st sidelobe level of better than 21 dB in the E- and H-planes. This is greater than the 17 dB cited for a minimum sidelobe design by Zucker [1] for a longer long dielectric rod antenna. This is partially a result of the low feed radiation due to the high launching efficiency of the ring slot. A simulated sidelobe level of greater than 17 dB below boresight is maintained over a 7% bandwidth; however, this level begins to degrade for lower frequencies. The gain improvement of 2.8 dB over the linear profiled rod is due to the better utilization of the rod length for expanding the surface-wave field. The measured radiation pattern varies slightly from the simulated results due to the interaction of the ring slot radiation with the ring slot feed circuit board. C. Maximum Gain Profile (MG) A large variety of rod profiles were evolved that satisfied the maximum gain criteria, indicating that within the search space of allowed rod profiles there are a number of possible solutions that are local minima of the fitness function. It should be noted that a genetic algorithm is not guaranteed to converge to a global minimum. The rod profiles optimized for maximum gain generally had the characteristic of an oscillatory profile that commenced with a thick rod that gradually thinned with damped oscillations before flaring towards the rod termination. For the MG rod, it can be seen that the spacing between the peaks in diameter increase towards the end of the rod. These oscillations serve several purposes. The first purpose is to guide the majority of power in the fundamental surface-wave from the feed to the end of the rod where it can radiate giving a maximum gain on boresight. Secondly, the overall narrowing and increased peak spacing of the

Fig. 8. Radiation patterns in the (a) E-plane and (b) H-plane of the minimum sidelobes (SL) rod antenna at 10 GHz. Theory (dashed) and experiment (solid).

rod helps expand the equiphase region of the surface-wave in the transverse direction, increasing the size of the effective aperture at the rod termination and consequently the gain. It can be seen in the plot of the electric field in Fig. 5(c) that the shape of the equiphase region of the radiating surface-wave appears to be straighter than for the LIN and SL rods which also helps improve gain as demonstrated in [2]. The straightening of the surface-wave wavefront occurs primarily over the first third of the rod. Another effect of the rod diameter peaks is to cause the surface-wave to radiate a small amount of power at an angle away from boresight, shown in Fig. 6; however, this effect is only slight. Interestingly, a large proportion of the antennas optimized for maximum gain did not have a terminal taper to help match ) as recthe surface-wave to free-space (i.e. allow ommended in [1], and instead flared at the termination. Simulations revealed that the flare acts like a lens, helping to extend and further straighten the equiphase region of the radiating

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Fig. 10. Dielectric rod profile for the shaped beam antenna with crosses representing spline knot points.

Fig. 11. Simulated radiation patterns in the E- and H-plane for the shaped-beam antenna.

Fig. 9. Radiation patterns in the (a) E-plane and (b) H-plane of the maximum gain (MG) rod antenna at 10 GHz. Theory (dashed) and experiment (solid).

surface-wave. As noted above, it also causes part of the surface-wave to break off and radiate at an angle off boresight. A study by Zucker [1] showed that the gain of a surface-wave end-fire antenna could be expressed by

(7) where for a maximum gain design for a length and for a Hansen-Woodyard design. Zucker further that a 30% increase in noted that for lengths shorter than gain was possible. This equation can be used as a comparison for the results presented in this paper. The simulated gain of the MG antenna was 13.2 dBi, which is more than 2.9 dB higher than the linearly profiled rod and slightly higher than that predicted by for a maximum gain design. It should be noted (7) with that this high gain was achieved without employing a large, directive feed. Further investigation found that rods optimized for to achieved a gain maximum gain having lengths from corresponding to an value in (7) of between 10 and 11.

The maximum gain antennas exhibited a smaller pattern bandwidth than those optimized for low sidelobes and the MG dielectric rod had a 1 dB gain bandwidth of 7% despite being optimized over a bandwidth of 4%. The MG antenna gain is sensitive to variations in the rod dielectric permittivity. For example, a 6% variation in the permittivity will reduce the gain by approximately 1 dB. The measured sidelobe levels for the MG rod are 13.7 dB and 12.6 dB, below the boresight level in the E- and H-plane, respectively. This is lower than the 10 dB and 11 dB levels cited as typical of maximum gain designs [1]. D. Shaped Beam Antenna It can be seen from Fig. 11 that the shaped beam antenna is able to achieve the basic shape of the target pattern; however, it does not track the target pattern exactly. It also exceeds the upper-bound defined for the rear radiation for part of the angular range. The profile of the shaped beam antenna was defined by 15 knot points and was chosen to be in length rather than . The constraint that the rod radius be less than or equal to the starting rod radius was relaxed to allow thicker rods, i.e., . These two changes were found to lead to improved performance for rod antennas with peak gain at angles other than

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boresight. Further improvement may be possible using different rod lengths and increasing the number of knot points to increase the degrees of freedom in the design; however, this requires further investigation.

VIII. CONCLUSION A systematic approach for the design of dielectric rod antennas to achieve specific radiation pattern objectives was presented. This approach used a spline representation for the dielectric rod profile and a genetic algorithm to optimize the rod’s profile. A BoR-MoM technique was described to rapidly analyze an arbitrarily profiled dielectric rod antenna excited by a ring slot. As a demonstration of the technique the profile of several dielectric rod antennas was optimized for maximum gain and minimum sidelobes characteristics, as well as for a shaped beam. The rod profile optimized for minimum sidelobes was found to be close to a third-order power curvilinear profile which decreased rapidly at the start of the rod before flattening out with a slight flare at the rod termination. This profile minimized radilength rod produced ation along the rod’s length and for a a measured boresight gain of 10.9 dBi and a first sidelobe level of better than 21 dB below boresight. The maximum gain rod had an oscillatory profile with a flare at the rod termination which produced a termination field with a relatively straight equiphase region. The simulated gain was 13.2 dBi which is higher than that predicted by Zucker for a length rod antenna; however in a practical realization, due to system losses, a boresight gain of 12.3 dBi was measured with a first sidelobe level of better than 12 dB below boresight. These results indicate that profiled dielectric rods can lead to improved performance over the standard linear profiled dielectric rod antenna. The profiling allows the radiation along the length of the rod and at its termination to be controlled. Furthermore, it allows the surface-wave field at the rod termination to be shaped to optimize specified radiation characteristics. It was also demonstrated that using this profile optimization approach it is possible to design dielectric rod antennas with shaped beams having maximum gain at angles other than boresight. Future work is concentrating on designing profiles where the rod length is variable and the maximum radius constraint is significantly larger to allow the consideration of lens-like structures.

ACKNOWLEDGMENT The authors would like to thank J. Mautz for providing his original plane-wave MoM Fortran code, J. Kot, C. Granet and S. Hay for their advice regarding the profile optimization, C. Holmesby for fabricating the antennas and K. Smart for assisting with the antenna measurements.

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REFERENCES [1] F. J. Zucker and W. F. Croswell, “Surface-wave antennas,” in Antenna Engineering Handbook, J. L. Volakis, Ed., 4th ed. New York: McGraw-Hill, 2007, ch. 10. [2] T. Ando, I. Ohba, S. Numata, J. Yamauchi, and H. Nakano, “Linearly and curvilinearly tapered cylindrical-dielectric-rod antennas,” IEEE Trans. Antennas Propag., vol. 53, no. 9, pp. 2827–2833, Sep. 2005. [3] J. Richter and L.-P. Schmidt, “Dielectric rod antennas as optimized feed elements for focal plane arrays,” in Proc. IEEE AP-S Int. Symp., Jul. 3–8, 2005, vol. 3A, pp. 667–670. [4] A. Kishk and L. Shafai, “Radiation characteristics of the short dielectric rod antenna: A numerical solution,” IEEE Trans. Antennas Propag., vol. 35, no. 2, pp. 139–146, Feb. 1987. [5] J. C. Simon and V. Biggi, “Un nouveau type d’aerien et son application a la transmission de television a grande distance,” L’Onde Elect., no. 332, Nov. 1954. [6] J. T. Bolljhan, “Synthesis of modulated corrugated surface-wave structures,” IRE Trans. Antennas Propag., vol. 9, no. 3, pp. 236–241, May 1961. [7] A. S. Thomas and F. J. Zucker, “Radiation from modulated surface wave structures—I,” Proc. IRE Int. Conv. Record, vol. 5, pp. 153–160, Mar. 1957. [8] H. Kubo, K. Yamaguchi, and I. Awai, “Radiation characteristics of HE -type wave from dielectric rod antenna with periodic surface,” IEEE Trans. Antennas Propag., vol. 44, no. 8, pp. 1063–1066, Aug. 1996. [9] S. M. Hanham, T. S. Bird, A. D. Hellicar, and R. A. Minasian, “Optimized dielectric rod antennas for terahertz applications,” presented at the IRMMW-THz, Busan, South Korea, Sep. 21–25, 2009, R4B06. 0530. [10] S. Hanham, T. S. Bird, and B. Johnston, “A ring slot excited dielectric rod antenna for terahertz imaging,” in Proc. IEEE AP-S Int. Symp., Honolulu, HI, Jul. 10–15, 2007, pp. 5539–5542. [11] H. E. King, J. L. Wong, and C. J. Zamites, “Shaped-beam antennas for satellites,” IEEE Trans. Antennas Propag., vol. 14, no. 5, pp. 641–643, 1966. [12] A. W. Love, “Two hybrid mode, earth coverage horn for GPS,” in Proc. IEEE AP-S Int. Symp., Jun. 1985, pp. 575–578. [13] J. S. Kot, N. Nikolic, R. A. Sainati, and T. S. Bird, “Aspects of antenna designs for indoor wireless millimetre-wave systems,” J. Electr. Electron. Eng. Aust., vol. 15, no. 2, pp. 145–150, Jun. 1995. [14] C. A. Fernandes and L. M. Anunciada, “Constant flux illumination of square cells for millimeter-wave wireless communications,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 11, pp. 2137–2141, Nov. 2001. [15] J. C. Brégains, G. Franceschetti, A. G. Roederer, and F. Ares, “New toroidal beam antennas for WLAN communications,” IEEE Trans. Antennas Propag., vol. 55, no. 2, pp. 389–398, Feb. 2007. [16] S. M. Hanham and T. S. Bird, “High efficiency excitation of dielectric rods using a magnetic ring current,” IEEE Trans. Antennas Propag., vol. 56, no. 6, pp. 1805–1808, Jun. 2008. [17] J. R. Mautz and R. F. Harrington, “H-Field, E-Field and Combined Field Solutions for Bodies of Revolution,” Syracuse University, 1977, Tech. Rep.. [18] J. R. Mautz and R. F. Harrington, “Electromagnetic Scattering from a Homogeneous Body of Revolution,” Syracuse University, 1977, Tech. Rep.. [19] A. Kishk and L. Shafai, “Different formulations for numerical solution of single or multibodies of revolution with mixed boundary conditions,” IEEE Trans. Antennas Propag., vol. 34, no. 5, pp. 666–673, May 1986. [20] T. S. Bird and G. T. Poulton, “Envelope-constrained pattern synthesis for satellite antenna,” presented at the IREE 20th Int. Convention, Melbourne, Australia, 1985. [21] C. Granet and T. S. Bird, “Optimization of corrugated horn radiation patterns via a spline-profile,” in Proc. ANTEM, Montreal, Canada, Jul. 27–29, 2002, pp. 307–310. [22] T. S. Bird and A. W. Love, “Horn antennas,” in Antenna Engineering Handbook, J. L. Volakis, Ed., 4th ed. New York: McGraw-Hill, 2007, ch. 14. [23] D. Goldberg, Genetic Algorithms in Search, Optimization and Machine Learning. Boston, MA: Addison-Wesley Longman, 1989. [24] S. M. Hanham, T. S. Bird, B. F. Johnston, A. D. Hellicar, and R. A. Minasian, “A 600 GHz dielectric rod antenna,” presented at the EuCAP, Berlin, Germany, Mar. 23–27, 2009.

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Stephen M. Hanham (S’03–M’10) received the B.E. and B.Sc. degrees from the University of Western Australia, Perth, in 2004 and the Ph.D. degree from the University of Sydney, Sydney, Australia, in 2010. From 2008 to 2010, he was a Research Project Officer and later a Systems Engineer at the Commonwealth Scientific and Industrial Research Organisation (CSIRO), Sydney, Australia. Currently, he is employed at Imperial College London, London, U.K., as a Research Associate in the area of near-field terahertz imaging and spectroscopy. His research interests include electromagnetics, antennas, terahertz technologies and microwave photonics.

Trevor S. Bird (S’71–M’76–SM’85–F’97) received the B. App. Sc., M. App. Sc., and Ph.D. degrees from the University of Melbourne, Melbourne, Australia, in 1971, 1973, and 1977, respectively. From 1976 to 1978, he was a Postdoctoral Research Fellow at Queen Mary College, University of London, London, U.K., followed by five years as a Lecturer in the Department of Electrical Engineering, James Cook University of North Queensland. During 1982 and 1983, he was a Consultant at Plessey Radar, U.K., and in December 1983, he joined CSIRO, Sydney, Australia. He has held several positions with CSIRO and is currently a CSIRO Fellow and Chief Scientist in the CSIRO ICT Centre. He is also an Adjunct Professor at Macquarie University, Sydney. He has published widely in the areas of electromagnetics and antennas, particularly related to waveguides, horns, reflectors, wireless and satellite communication applications, and holds 12 patents. Dr. Bird is a Fellow of the Australian Academy of Technological and Engineering Sciences, the IEEE, the Institution of Electrical Technology, U.K., and an Honorary Fellow of the Institution of Engineers, Australia. In 1988, 1992, 1995 and 1996 he received the John Madsen Medal of the Institution of Engineers, Australia, for the best paper published annually in the Journal of Electrical and Electronic Engineering, Australia, and in 2001 he was co-recipient of the H. A. Wheeler Applications Prize Paper Award of the IEEE Antennas and Propagation Society. He was awarded a CSIRO Medal in 1990 for the development of an Optus-B satellite spot beam antenna and again in 1998 for the multibeam antenna feed system for the Parkes radio telescope. He received an IEEE Third Millennium Medal in 2000 for outstanding contributions to the IEEE New South Wales Section. Engineering projects that he played a major role in were given awards by the Society of Satellite Professionals International (New York) in 2004, the Engineers Australia in 2001, and the Communications Research Laboratory, Japan, in 2000. In 2003, he was awarded a Centenary Medal for service to Australian society in telecommunications and also named Professional Engineer of the Year by the Sydney Division of Engineers Australia. His biography is listed in Who’s Who in Australia. He was a Distinguished Lecturer for the IEEE Antennas and Propagation Society from 1997 to 1999, Chair of the New South Wales joint AP/MTT Chapter from 1995 to 1998, and again in 2003, Chairman of the 2000 Asia Pacific Microwave Conference, Member of the New South Wales Section Committee from 1995–2005 and was Vice-Chair and Chair of the Section in 1999–2000 and 2001–2002 respectively, Associate Editor of the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION from 2001 to 2004, a member of the Administrative Committee of the IEEE Antennas and Propagation Society from 2003–2005, a member of the College of Experts of the Australian Research Council (ARC) from 2006–2007 and Editor-in-Chief of the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION from 2004 to 2010. He has been a member of the technical committees of numerous conferences including JINA, ICAP, AP2000, IRMMW-THz and the URSI Electromagnetic Theory Symposium. Currently, he is member of the Editorial Boards of the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES and the Journal of Infrared, Millimeter and Terahertz Waves.

Andrew D. Hellicar (M’08) received the Ph.D. degree in electronic and computer systems engineering from Monash University, Victoria, Australia, in 2007. In 2003, he joined the Commonwealth Scientific and Industrial Research Organisation (CSIRO), Marsfield, NSW, Australia, as a Research Scientist, and in 2006, took on the role of leading a project in the field of millimeter-wave and terahertz imaging. His interests in the area of computational electromagnetism have been supplemented to include researching and building systems that detect millimeter-wave and terahertz frequencies.

Robert A. Minasian (S’78–M’80–SM’00–F’03) received the B.E. degree from the University of Melbourne, Melbourne, Australia, the M.Sc. degree from the University of London, University College, London, U.K., and the Ph.D. degree from the University of Melbourne, Melbourne, Australia. He is currently a Chair Professor with the School of Electrical and Information Engineering, University of Sydney, Australia. In addition, he is the Director of the Fibre-optics and Photonics Laboratory, and has also served as the Head of the School of Electrical and Information Engineering, University of Sydney. His research encompasses optical signal processing and telecommunications, and currently centers on photonic signal processing, microwave photonics, terahertz/gigahertz photonics in communication and radar systems, and optically controlled phased array antennas. He has contributed 260 technical publications in these areas. He also holds several patents and has done consulting work with industry. He is an Associate Editor of Optical Fiber Technology. Prof. Minasian is a Fellow of the IEEE, the Optical Society of America, and the Institute of Engineers, Australia. He was the recipient of the ATERB Medal for Outstanding Investigator in Telecommunications, awarded by the Australian Telecommunications and Electronics Research Board. He has served on the Australian Research Council as a member of the College of Experts. He is a member of the Technical Committee on Microwave Photonics of the IEEE Microwave Theory and Techniques Society (IEEE MTT-S), and has served and is on the program committees for several international conferences including the IEEE International Meeting on Microwave Photonics, (MWP2003-MWP2010), the Asia Pacific Microwave Conference, the IEEE International Microwave Symposium, the Asia-Pacific Microwave Photonics Conference (APMP2006-APMP2010), and the IEEE LEOS Annual Meeting.

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A Compact UWB Antenna for On-Body Applications Nacer Chahat, Student Member, IEEE, Maxim Zhadobov, Member, IEEE, Ronan Sauleau, Senior Member, IEEE, and Koichi Ito, Fellow, IEEE

Abstract—A new compact planar ultrawideband (UWB) antenna designed for on-body communications is presented. The antenna is characterized in free space, on a homogeneous phantom modeling a human arm, and on a realistic high-resolution whole-body voxel model. In all configurations it demonstrates very satisfactory features for on-body propagation. The results are presented in terms of return loss, radiation pattern, efficiency, and E -field distribution. The antenna shows very good performance within the 3–11.2 GHz range, and therefore it might be used successfully for the 3.1–10.6 GHz IR-UWB systems. The simulation results for the return loss and radiation patterns are in good agreement with measurements. Finally, a time-domain analysis over the whole-body voxel model is performed for impulse radio applications, and transmission scenarios with several antennas placed on the body are analyzed and compared. Index Terms—Body-area network (BAN), body-centric wireless communications, compact antenna, printed antenna, ultrawideband (UWB) antenna.

I. INTRODUCTION

B

ODY-AREA NETWORKS (BAN) are wireless communication systems that enable communications between wearable and/or implanted into the human body electronic devices. Such systems are of great interest for various applications including sport, multimedia, health care, and military applications [1], [2]. Ultrawideband (UWB) antennas have been identified as a highly promising solution for BAN. One of the major advantages of the UWB systems at 3.1–10.6 GHz is their high data-rate-transmission capabilities (typically 100 Mbps) with low power spectral densities ( 41.3 dBm/MHz) [3], ensuring thereby low interference with other narrow-band wireless devices. Designing an antenna for UWB body-centric communications is a challenging task, as the antenna needs to fulfill several fundamental requirements, such as: 1) optimized characteristics in frequency-and time-domains; 2) small size and low profile; 3) good on-body propagation. Indeed, both frequency-and time-domain responses should be considered and characterized [4]–[7], and the interaction with the human body, i.e., the changes in the antenna performance Manuscript received April 12, 2010; revised August 04, 2010; accepted September 24, 2010. Date of publication January 28, 2011; date of current version April 06, 2011. This work was supported in part by the “Agence Nationale de la Recherche” (ANR), France by Grants ANR-09-VERS-003 (METAVEST project) and ANR-09-RPDOC-003-01 (Bio-CEM project), “Région Bretagne” (Dose_ULB project), and in part by the “Centre National de la Recherche Scientifique” (CNRS), France. N. Chahat, M. Zhadobov, and R. Sauleau are with the Institute of Electronics and Telecommunications of Rennes (IETR), UMR CNRS 6164, University of Rennes 1, 35042 Rennes, France (e-mail: [email protected]). K. Ito is with the Graduate School of Engineering, Chiba University, Chiba 263-8522, Japan (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2011.2109361

due to the presence of the body and power losses in the tissues, should be carefully taken into account [8]. Besides, the miniaturization of UWB antennas is particularly important for wearable applications. Significant research efforts have been undertaken to reduce the size of the radiating structures, and some interesting miniaturization techniques have been proposed [9]–[11]. Finally, for on-body applications, the antenna needs to exhibit suitable on-body propagation features. However, in most of the proposed body-centric UWB communication scenarios, omnidirectional planar antennas are placed parallel to the human body, and, as a result, the efficiency of these antennas is significantly reduced [12], and the on-body propagation is not optimal. Indeed, this configuration is more suitable for off-body communication, i.e., for communication between an antenna mounted on the body and a remote device or base station, as demonstrated in several studies introducing textile antennas with high potential for off-body communications [13]–[15]. It was shown that the antenna -field polarization needs to be normal to the body surface in order to improve the on-body propagation [16]. In particular, it was demonstrated that a quarter-wavelength monopole antenna is appropriate for on-body communication for the following reasons: 1) it has an omnidirectional pattern with maximum radiation along the body surface; 2) -field is normal to the body surface [16]. Furthermore, a comparison between two different UWB antennas has been performed showing that the planar inverted cone antenna (PICA) with an omnidirectional monopole-like pattern demonstrates very good performances for on-body communications [17]. Nevertheless, the quarter-wavelength monopole antenna and the PICA have a relatively large ground plane and heights. To overcome this problem we introduce here a reduced-size UWB antenna suitable for on-body communications since the -field is polarized perpendicularly to the body surface. This paper is organized as follows. The antenna design and a two-thirds muscle equivalent phantom used for numerical and experimental characterizations are introduced in Section II. The main characteristics of the proposed antenna, namely its reflection coefficient, radiation patterns, and efficiency are then given in Section III. Experimental results, obtained using the phantom, are also compared to calculations. In addition, the -field distribution around a homogeneous arm model and whole-body voxel phantom is also studied numerically to investigate the on-body propagation. Finally the time-domain capabilities of the antenna are analyzed in Section IV, transmission scenarios between several antennas placed on the body are numerically studied, and the effect of modulation schemes on the on-body system performance is discussed. All calculations are carried out using the finite integration technique implemented in CST Microwave Studio.

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Fig. 2. Reflection coefficients S of the proposed antenna simulated in free space for different slot shapes. Three values for the total slot length are considered: 6.9 mm, 11.9 mm, and 16.4 mm.

Fig. 1. (a) Geometry of the proposed antenna. (b) Antenna on the two-thirds muscle-equivalent phantom.

Fig. 3. Reflection coefficient S different ground plane heights. . 111111g .

10 mm

= 15 mm

of the proposed antenna in free space for . g . g

g

= 1 mm

= 5 mm

=

II. EXPERIMENTAL MODELS A. Antenna Design An antenna was designed and manufactured for UWB on-body applications. It consists of a compact microstrip-fed printed monopole [Fig. 1(a), ] printed on a 1.6-mm-thick AR350 substrate . In this study, the antenna performance is evaluated and optimized in the following configurations: 1) antenna located on a homogeneous phantom with two-thirds muscle-equivalent dielectric properties; 2) antenna mounted on a high-resolution nonhomogeneous human body model (Sections III-C, III-E, and IV). To our best knowledge, the smallest UWB antennas have a significant height (around 17 mm) [18]–[21] and thus might be not suitable for a perpendicular configuration ( -field normal to the body). Here, the ground plane size and radiator height have been reduced by 8 mm and 7 mm, respectively, leading to a total antenna height of only 10 mm. This choice enables us to minimize distortions in the antenna performance compared to the original design [22]. This size reduction results in a significant decrease of the current path, thus in an increase of the lower band limit. To decrease the lower frequency, one solution consists in extending the total slot length (white dotted line in Fig. 1(a)). The impact of this slot length is illustrated in Fig. 2 where we compare the reflection coefficients for three slot length values: 6.9 mm, 11.9 mm, and 16.4 mm. Selecting a long slot enables to slightly enlarge the 10 dB return loss bandwidth and cover the full UWB frequency range (solid line in Fig. 2).

TABLE I CROSS POLARIZATION LEVELS, PEAK GAINS, AND EFFICIENCY FOR THE PROPOSED ANTENNA WITH GROUND PLANE HEIGHT OF 1 mm/10 mm

As a wearable antenna, the antenna ground plane is voluntarily small to minimize the antenna height. The impact of the ground plane size on the reflection coefficient is illustrated in Fig. 3. Although slightly degraded around 4 and remains below 10 dB between these two 10 GHz, the frequencies for larger ground plane dimensions. Furthermore, the antenna size reduction (and particularly, the small ground plane), affects the cross-polarization component and peak gain (Table I). However, along the body surface (i.e., and ), the cross-polarization ratio remains below 10 dB. B. Phantom The human arm is modeled as a two-thirds muscle-equivalent phantom, and the antenna is located 1 mm above [Fig. 1(b) and Fig. 4]. This phantom has a parallelepipedic shape . A water-based semisolid phantom

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Fig. 4. Antenna prototype mounted on the two-thirds muscle-equivalent phantom.

[23] was chosen as a tissue-equivalent model for UWB measurements. The complex permittivity and conductivity of the phantom are adjusted using the polyethylene powder and sodium chloride, respectively. Agar is used to maintain the shape of the phantom, sodium azide is a preservative, and TX-151 improves the phantom stickiness [23].

Fig. 5. Measured and computed characteristics of the phantom.    Measurement. Calculation.

Target.

III. FREQUENCY-DOMAIN ANALYSIS In this section, the dielectric properties of the phantom are provided in the 3–11 GHz range. Then computational and measured results are presented for the antenna return loss and radiation pattern. Finally, -field distributions are computed for the arm and human body models. A. Numerical Model and Dielectric Properties of the Phantom in the UWB Band

Fig. 6. Simulated reflection coefficients of the optimized antenna (the total slot Antenna in free space. Antenna mounted on length equals 16.4 mm). the phantom.

In order to model accurately the antenna in presence of the phantom, the dielectric properties of the phantom should be determined carefully in the 3.1–10.6 GHz range. Two-thirds of the muscle permittivity was used as a target value for the phantom. The muscle dielectric properties are well characterized up to 20 GHz [24]. For the numerical modeling, the complex dielectric permittivity of the phantom is expressed as a Debye’s dispersion equation [25] (1) where is the angular frequency, is the static permittivity, is the optical permittivity, and is the relaxation time. The best fit of this theoretical model to the target values was obtained for , , and . Theoretical permittivity and conductivity model is in a very good agreement with target values over the considered frequency range (3–11 GHz), confirming thereby that the choice of the Debye model is appropriate (Fig. 5). The phantom has been built as explained in Section II-B and characterized using the dielectric probe kit 85070E (Agilent Tech., CA). The measured complex permittivity is in satisfactory agreement with the numerical results (Fig. 5). B. Reflection Coefficient and Impact of the Feed Connector of the proposed UWB anThe reflection coefficients tenna are represented in Fig. 6 assuming the antenna is either in free space or on the arm. Here, the numerical models do not take into account the feed connector. These results show is very slightly affected by the presence of the that the

Fig. 7. Reflection coefficient of the antenna with feed connector mounted on Simulation results.  Measurements. the phantom.

phantom and remains below 10 dB within the 3–11.3 GHz range. The fabricated prototype is fed by a tiny coaxial probe connector (BL58-3123-00, Orient Microwave Corp., Japan). Its reflection coefficient measured for the antenna mounted on the phantom is represented in Fig. 7. The agreement with simulations (accounting this time for the feed connector) is very satisfactory. The 10 dB return loss bandwidth almost covers the full 3.1–10.6 GHz UWB frequency range. C. Radiation Patterns and The antenna radiation patterns in planes have been computed at three frequency points (4, 7, and 10 GHz) for three configurations: antenna alone, antenna on the phantom, and antenna on the whole-body model (Duke model from Virtual Family [26]). Duke represents a 34 year-old 174 cm-tall adult weighting 70 kg. Compared to the free-space scenario, the antenna mounted on the phantom shows

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Fig. 8. Measured and simulated radiation patterns at 4, 7, and 10 GHz. (a)  voxel phantom.

= 90

. (b) '

=0

.

Free space.

On homogeneous phantom.

On

Measurement.

TABLE II PEAK AND AVERAGE GAINS OF THE ANTENNA IN FREE SPACE AND ON THE PHANTOM

which corresponds to a practical situation where the antenna is mounted on clothes. However, the antenna efficiency can be improved using a larger ground plane size as shown in Table I. E. Computed E-Field Distributions

better front-to-back ratio. This is essentially related to the absorption in and reflections from the phantom. In addition, comand parison with measurements at 4 and 10 GHz in the planes using the two-thirds muscle-equivalent phantom confirms the very satisfactory matching between experimental and numerical results. The average and peak values of the antenna gain are given in Table II. These data confirm the gain plane due to the presence of the phantom, as reduction in already mentioned in other studies (e.g. [27], [28]). However, plane, the gain is reduced at 4 GHz where absorptions in are higher, and at 7 and 10 GHz, the gain increases due to the reflection caused by the arm. D. Radiation Efficiency Radiation efficiencies of 19.1%, 38.2%, and 28.4% are estimated on the homogeneous phantom, at 4, 7, and 10 GHz, respectively. As expected, the radiation efficiency is quite low since the gap between the antenna and the body is set to 1 mm,

To estimate possible range of application scenarios for the proposed antenna, it is important to consider electromagnetic field distribution in, on, and around the body. Fig. 9 shows the computed electric field distributions in the plane in cross-section plane of the numerical arm model [ Fig. 1(b)] at 4, 7, and 10 GHz. It is important to highlight that, component is the dominant one, in this configuration, the which means that the major contribution to the overall electric field comes from the component perpendicular to the upper side of the phantom. Based on these results, several observations can be made: • The electromagnetic field propagates along the phantom surface and, as expected, attenuates nonlinearly; • Within the considered frequency range, higher frequencies correspond to the more localized energy absorption. The field propagation around the body is analyzed in Fig. 10 using Duke model (the spatial resolution is ). The antenna is mounted on the left arm where most part of the electric field is confined. Fig. 10 shows that the electromagnetic field propagates in both directions along the left arm and that shadowing due to the human body plays a key role since the electric field is much stronger at the source side compared to the opposite side (the attenuation is at least 70 dB). Therefore, communications between antennas situated on opposite sides of the body might be difficult in this configuration. However, wireless radio links between hand/head or hand/foot are absolutely

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Fig. 9. Electric field distributions (root-sum-square) in x

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0 y cross section of a homogeneous model.

Fig. 10. Electric field distribution around and within the Duke model at 4, 7, and 10 GHz.

achievable, and the time-domain behavior for such communication scenarios is studied in Section IV. IV. PROPAGATION AROUND THE BODY A. Time-Domain Analysis A UWB system can be either a traditional pulse-based system transmitting each pulse that occupies the entire UWB bandwidth, or a carrier system such as, for instance, multiband orthogonal frequency-division multiplexing (MB-OFDM) which has been adopted by WiMedia. Time-domain analyses

are of great importance to evaluate the capabilities of UWB antennas for impulse radio systems (IR-UWB). A UWB antenna excited by nanosecond pulses behaves as a pulse-shaping filter. Therefore, a suitable antenna for UWB communications has to demonstrate minimum distortion of the pulse in time-domain to reduce the complexity of the detection mechanisms at the receiver terminal. The quality of the received pulse is determined by (2)

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Fig. 11. (a) Normalized excitation signal. (b) Its spectral density.

Fig. 13. (a) Signals monitored by virtual probes. (b) Their spectra.

Fig. 12. Locations of the probes around the antenna.

where the source pulse and received pulse are normalized by their respective energies. The source pulse signal chosen here [Fig. 11(a)] is a fifthorder derivative Gaussian pulse satisfying the FCC power mask [Fig. 11(b)]

(3) where and denote the amplitude and spread of the Gaussian pulse, respectively. In the numerical simulations, the output signal is monitored in three directions using three probes located 50 cm apart from the antenna (Fig. 12). The pulse received at each is excellent: probe is represented in Fig. 13(a). Its fidelity 98.8%, 98.3%, and 95.2% for probes A, B, and C, respectively. A similar study has been conducted for on-body propagation. To this end, three transmission scenarios between four antennas are considered: the transmitting antenna is mounted on the left

wrist (TX1), the three receiving antennas are placed on the left arm (RX1), left ear (RX2), and left leg (RX3) (Fig. 14). The distortion of the received pulses (Fig. 15) depends on the antenna position, and their fidelity equals 75%, 46%, and 81%, for RX1, RX2, and RX3, respectively. The most pronounced distortion is observed for the antenna on the head, suggesting that a direct communication in this scenario might be delicate. From these results, it is clear that pulses experience strong distortions in on-body scenarios. The signal fidelity can be very low (e.g., 46% for RX2), and therefore an appropriate modulation scheme needs to be implemented. For instance binary phase-shift keying (BPSK) is not appropriate in this case because of strong distortions of the transmitted signal. However, in IR-UWB systems, pulse-position modulation (PPM) and on-off keying (OOK) are excellent candidates for on-body communications. These two modulation schemes can be implemented with noncoherent receivers using energy detection mechanisms instead of correlation for coherent systems. With a noncoherent receiver, the pulse shape is secondary and the low fidelity can be overcome. We focus our attention here on IR-UWB system using the entire UWB-band. However, these modulations schemes can also be used as multiband schemes [29], [30].

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Fig. 16. Path loss of the proposed antenna mounted on the homogeg . neous phantom with different ground plane heights. g .

   = 10 mm

222

= 1 mm

TABLE III MEAN PATH GAIN AND PATH GAIN VARIABILITY (RANGE) FOR SOME SPECIFIC ON BODY PATH LOSS

Fig. 14. Transmission scenario with four antennas mounted on the body.

efficiency (Table I) and as consequences the path loss becomes higher. The propagation path loss for this on-body scenario is presented in Table III. The mean path gain and variability within the 3.1–10.6 GHz UWB band are given for all receiving antennas. The wrist-to-arm link, being the shortest, has the lower loss, with an average loss of 53.9 dB, and has a peak-to-peak variation of approximately 32.9 dB within the whole UWB frequency band. The wrist-to-head link has the higher average loss, (this is mainly due to the longer link length), and its path loss variability is estimated around 34.4 dB. In the wrist-to-calf link, an average loss of 60.5 dB is evaluated, and the variability is much lower compared to the other links.

Fig. 15. Wave forms of the received pulses.

B. On-Body Propagation As a non-coherent receiver is preferred, the path loss needs to be investigated. The path loss of the proposed antenna mounted on the homogeneous phantom with two different ground plane heights is presented in Fig. 16. For each distance, each point represents the path loss result at one frequency point. The antenna with a 10-mm-height ground plane demonstrates a better on-body propagation performance. Using a larger ground plane improves the path loss by 7 dB whereas the cross-polarization component remains fairly the same between these two antenna models. However a larger ground plane improves the antenna

V. CONCLUSION A compact planar UWB monopole antenna has been designed for on-body communications. The antenna shows good impedance matching and satisfactory on-body propagation features. It was shown that, in spite of the small distance between the antenna and the body, the latter does not affect significantly the antenna input matching. The reflection coefficient and radiation patterns were successfully measured using a two-thirds muscle homogeneous phantom. The electric field distributions around a realistic whole-body model have been computed at different frequencies, and suitable on-body propagation features have been highlighted. Compared to the parallel configuration, the perpendicular one demonstrates higher gain, better on-body propagation, and the antenna impedance performance is less affected. The time-domain behavior of the proposed antenna has been fully investigated on a realistic body model. Significant pulse distortions have been observed and thus a noncoherent modulation is advised for on-body communication. Hence, for

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IR-UWB systems, PPM and OOK are excellent solutions for on-body scenarios. Finally, excellent path gain results have been demonstrated for specific transmission scenarios with several antennas placed on the body. ACKNOWLEDGMENT The authors would like to thank all members of Prof. Ito and Prof. Takahashi laboratory (Chiba University, Chiba, Japan) and especially Prof. K. Saito, Basari, N. Haga, and R. Watanabe for their kind assistance in measurements. REFERENCES [1] P. S. Hall and Y. Hao, Antennas and Propagation For Body Centric Communications Systems. Norwood, MA: Artech House, 2006, 10: 1-58053-493-7. [2] P. S. Hall and Y. Hao, “Antennas and propagation for body centric communications,” in Proc. First Eur. Conf. Antennas Propag., Nice, France, Nov. 6–10, 2006. [3] Federal Communication Commission, First Rep. Order Feb. 14, 2002. [4] Z. N. Chen, Antennas for Portable Devices. Hoboken, NJ: Wiley, 2007, 10: 0470030739. [5] X. N. Low, Z. N. Chen, and T. S. P. See, “A UWB dipole antenna with enhanced impedance and gain performance,” IEEE Trans. Antennas Propag., vol. 57, no. 10, pp. 2959–2966, Oct. 2009. [6] A. Alomainy, A. Sani, A. Rahman, J. G. Santas, and Y. Hao, “Transient characteristics of wearable antennas and radio propagation channels for ultrawideband body-centric wireless communications,” IEEE Trans. Antennas Propag., vol. 57, no. 4, pp. 875–884, Apr. 2009. [7] S. Chamaani and S. A. Mirtaheri, “Planar UWB monopole antenna optimization to enhance time-domain characteristics using PSO,” in Proc. Int. Symp. Antenna Propag. (ISAP 08), 2008, pp. 553–556. [8] W. T. Chen and H. R. Chuang, “Numerical computation of human interaction with arbitrarily oriented superquadric loop antennas in personal communications,” IEEE Trans. Antennas Propag., vol. 46, no. 6, pp. 821–828, Jun. 1998. [9] K. Bahadori and Y. Rahmat-Samii, “A miniaturized elliptic-card UWB antenna with band rejection for wireless communications,” IEEE Trans. Antennas Propag., vol. 55, no. 11, pp. 3326–3332, Nov. 2007. [10] A. M. Abbosh, “Miniaturization of planar ultrawideband antenna via corrugation,” IEEE Antennas Wireless Propag. Lett., vol. 7, pp. 685–688, 2008. [11] A. M. Abbosh, “Miniaturized microstrip-fed tapered-slot antenna with ultrawideband performance,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 690–692, 2009. [12] M. Klemm and G. Troester, “EM energy absorption in the human body tissues due to UWB antennas,” Progress in Electromagn. Res., vol. 62, pp. 261–280, 2006. [13] C. Hertleer, H. Rogier, L. Vallozzi, and L. Van Langenhove, “A textile antenna for off-body communication integrated into protective clothing for firefighters,” IEEE Trans. Antennas Propag., vol. 57, no. 4, pp. 919–925, Apr. 2009. [14] C. Hertleer, H. Rogier, and L. Van Langenhove, “A textile antenna for protective clothing,” in Proc. IET Seminar on Antennas and Propag. Body-Centric Wireless Commun., Apr. 2007, pp. 44–46. [15] C. Hertleer, A. Tronquo, H. Rogier, L. Vallozzi, and L. Van Langenhove, “Aperture-coupled patch antenna for integration into wearable textile systems,” IEEE Antennas Wireless Propag. Lett., vol. 6, pp. 392–395, 2007. [16] P. S. Hall, Y. Hao, Y. I. Nechayev, A. Alomainy, C. C. Constantinou, C. Parini, M. R. Kamarudin, T. Z. Salim, D. T. M. Hee, R. Dubrovke, A. S. Owadally, W. Song, A. Serra, P. Nepa, M. Gallo, and M. Bozzetti, “Antennas and propagation for on-body communication systems,” IEEE Antennas Propag. Mag., vol. 49, no. 3, pp. 41–58, Jun. 2007. [17] A. Alomainy, Y. Hao, C. G. Parini, and P. S. Hall, “Comparison between two different antennas for UWB on-body propagation measurements,” IEEE Antennas Wireless Propag. Lett., vol. 4, pp. 31–34, 2005. [18] M. Sun and Y. P. Zhang, “A chip antenna in LTCC for UWB radios,” IEEE Trans. Antennas Propag., vol. 56, no. 4, pp. 1177–1180, Apr. 2008. [19] Q. Ye, Z. N. Chen, and T. S. P. See, “Miniaturization of small printed UWB antenna for WPAN applications,” in IEEE Int. Workshop on Antenna Technol. (iWAT), Mar. 2–4, 2009, pp. 1–4.

[20] L. Guo, S. Wang, Y. Gao, X. Chen, and C. Parini, “Miniaturisation of printed disc UWB monopoles,” in Proc. Int. Workshop on Antenna Technol.: Small Antennas and Novel Metamater. (iWAT), Mar. 4–6, 2008, pp. 95–98. [21] M. Sun and Y. P. Zhang, “Miniaturization of planar monopole antennas for ultrawide-band applications,” in Proc. Int. Workshop on Antenna Technol.: Small and Smart Antennas Metamater. Appl. (IWAT), Mar. 21–23, 2007, pp. 197–200. [22] Z. N. Chen, T. S. P. See, and X. M. Qing, “Small printed ultrawideband antenna with reduced ground plane effect,” IEEE Trans. Antennas Propag., vol. 55, no. 2, pp. 383–388, Feb. 2007. [23] Y. Okano, K. Ito, I. Ida, and M. Takahashi, “The SAR evaluation method by a combination of thermographic experiments and biological tissue-equivalent phantoms,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 11, pt. 2, pp. 2094–2103, Nov. 2000. [24] S. Gabriel, R. W. Lau, and C. Gabriel, “The dielectric properties of biological tissues: II. Measurements in the frequency range 10 Hz to 20 GHz,” Phys. Med. Biol., vol. 41, pp. 2251–2269, 1996. [25] O. G. Martinsen, S. Grimmes, and H. P. Schwan, “Interface phenomena and dielectric properties of biological tissue,” in Encyclopedia of Surface and Collied Science. New York: Marcel Dekker, 2002. [26] IT’IS Foundation, The Virtual Family [Online]. Available: http://www. itis.ethz.ch/index/index_humanmodels.html [27] A. Cai, T. S. P. See, and Z. N. Chen, “Study of human head effects on UWB antenna,” in Proc. IEEE Int. Workshop on Antenna Technol.: Small Antennas and Novel Metamater. (IWAT), Mar. 7–9, 2005, pp. 310–313. [28] Z. N. Chen, A. Cai, T. S. P. See, and M. Y. W. Chia, “Small planar UWB antennas in proximity of the human head,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 4, pp. 1846–1857, Jun. 2006. [29] S. Woods and S. Aiello, Essentials of UWB. Cambridge, U.K.: Cambridge Univ. Press, 2008, 10: 0521877830. [30] S. Hernandez and R. Kohno, “Ultra low power UWB transceiver design for body area networks,” in Proc. Biomed. Commun. Technol. Int. Symp., Nov. 24–27, 2009, pp. 1–4. Nacer Chahat (S’09) was born in Angers, France, in 1986. He graduated in electrical engineering and radio communications from the Ecole Supérieur d’ingénieur de Rennes (ESIR) and received the Master’s degree in telecommunication and electronics in 2009. Since 2009, he has been working toward the Ph.D. degree at the Institute of Electronics and Telecommunications of Rennes (IETR), University of Rennes 1, Rennes, France. His current research fields are electrically small antennas, millimeter-wave antennas, and the evaluation of the interaction between the electromagnetic field and human body. In 2009, he accomplished a six-month Master’s training period as a special research student at the Graduate School of Engineering, Chiba University, Chiba, Japan.

Maxim Zhadobov (S’05–M’07) was born in Gorky, Russia, in 1980. He received the M.S. degree in radiophysics from Nizhni Novgorod State University, Nizhni Novgorod, Russia, in 2003, and the Ph.D. degree in bioelectromagnetics from the Institute of Electronics and Telecommunications of Rennes (IETR), University of Rennes 1, Rennes, France, in 2006. He accomplished Postdoctoral training with the Center for Biomedical Physics, Temple University, Philadelphia, PA, in 2008, and then rejoined IETR as an Associate Scientist with the Centre National de la Recherche Scientifique (CNRS). He has authored or coauthored more than 50 scientific contributions. His main scientific interests are in the field of biocompatibility of electromagnetic radiations, including interactions of microwaves, millimeter waves and pulsed radiations at the cellular and subcellular levels, health risks and environmental safety of emerging wireless communication systems, biocompatibility of wireless noninvasive biomedical techniques, therapeutic applications of nonionizing radiations, bioelectromagnetic optimization of body-centric wireless systems, experimental, and numerical electromagnetic dosimetry. Dr. Zhadobov was the recipient of the 2005 Best Poster Presentation Award from the International School of Bioelectromagnetics, the 2006 Best Scientific Paper Award from the Bioelectromegnetics Society, and Brittany’s Young Scientist Award in 2010.

CHAHAT et al.: COMPACT UWB ANTENNA FOR ON-BODY APPLICATIONS

Ronan Sauleau (M’04–SM’06) graduated in electrical engineering and radio communications from the Institut National des Sciences Appliquées, Rennes, France, in 1995. He received the Agrégation degree from the Ecole Normale Supérieure de Cachan, France, in 1996, and the Doctoral degree in signal processing and telecommunications and the “Habilitation à Diriger des Recherche” degree from the University of Rennes 1, France, in 1999 and 2005, respectively. He was an Assistant Professor and Associate Professor with the University of Rennes 1, between September 2000 and November 2005, and between December 2005 and October 2009. He has been a full Professor with the same University since November 2009. His current research fields are numerical modeling (mainly FDTD), millimeter-wave printed and reconfigurable (MEMS) antennas, lens-based focusing devices, periodic and nonperiodic structures (electromagnetic bandgap materials, metamaterials, reflectarrays, and transmitarrays), and biological effects of millimeter waves. He has received four patents and is the author or coauthor of 70 journal papers Dr. Sauleau has more than 180 contributions to national and international conferences and workshops. He received the 2004 ISAP Conference Young Researcher Scientist Fellowship (Japan) and the first Young Researcher Prize in Brittany, France, in 2001 for his research work on gain-enhanced Fabry-Perot antennas. In September 2007, he was elevated to Junior member of the “Institut Universitaire de France.” He was awarded the Bronze medal by CNRS in 2008.

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Koichi Ito (M’81–SM’02–F’05) received the B.S. and M.S. degrees from Chiba University, Chiba, Japan, in 1974 and 1976, respectively, and the D.E. degree from the Tokyo Institute of Technology, Tokyo, Japan, in 1985, all in electrical engineering. From 1976 to 1979, he was a Research Associate at the Tokyo Institute of Technology. From 1979 to 1989, he was a Research Associate with Chiba University. From 1989 to 1997, he was an Associate Professor with the Department of Electrical and Electronics Engineering, Chiba University, and is currently a Professor with the Department of Medical System Engineering, Chiba University. From 2005 to 2009, he was Deputy Vice-President for Research, Chiba University. From 2008 to 2009, he was Vice-Dean of the Graduate School of Engineering, Chiba University. Since April 2009, he has been Director of Research Center for Frontier Medical Engineering, Chiba University. In 1989, 1994, and 1998, he visited the University of Rennes I, France, as an Invited Professor. His main research interests include analysis and design of printed antennas and small antennas for mobile communications, research on evaluation of the interaction between electromagnetic fields and the human body by use of numerical and experimental phantoms, microwave antennas for medical applications such as cancer treatment, and antennas for body-centric wireless communications. Dr. Ito is a Fellow of the Institute of Electronics, Information and Communication Engineers (IEICE) of Japan, a member of the American Association for the Advancement of Science, the Institute of Image Information and Television Engineers of Japan (ITE) and the Japanese Society for Thermal Medicine. He served as Chair of the Technical Group on Radio and Optical Transmissions, ITE from 1997 to 2001, Chair of the Technical Committee on Human Phantoms for Electromagnetics, IEICE from 1998 to 2006, Chair of the IEEE AP-S Japan Chapter from 2001 to 2002, TPC Co-Chair of the 2006 IEEE International Workshop on Antenna Technology (iWAT2006), Vice-Chair of the 2007 International Symposium on Antennas and Propagation (ISAP2007) in Japan, General Chair of iWAT2008, Co-Chair of ISAP2008, and an AdCom member for the IEEE AP-S from 2007 to 2009. He currently serves as an Associate Editor for the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, a Distinguished Lecturer for the IEEE AP-S, and Chair of the Technical Committee on Antennas and Propagation, IEICE. He has been appointed as General Chair of ISAP2012 to be held in Nagoya, Japan in 2012.

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Design of a Corner-Reflector Reactively Controlled Antenna for Maximum Directivity and Multiple Beam Forming at 2.4 GHz Themistoklis D. Dimousios, Stelios A. Mitilineos, Member, IEEE, Stylianos C. Panagiotou, and Christos N. Capsalis

Abstract—Electronically steerable passive array radiator (ESPAR) antennas constitute a promising research field and are expected to play important role in future wireless communications. In this paper, a new approach in ESPAR antenna design for base station applications is proposed. A corner-plate reflector is combined with active and passive (reactively loaded) elements in order to implement a corner-reflector ESPAR (CR-ESPAR) configuration. It is shown that when combined with corner reflectors in order to sectorize the coverage area, an ESPAR antenna offers multiple radiation patterns with higher directivity and resolution. A case study of a CR-ESPAR suitable for 2.4 GHz ISM applications is demonstrated, where the performance of the structure is optimized with respect to resonance frequency, input impedance, and multiple switched-beam patterns configuration. The optimization of the array is performed using a Genetic Algorithm (GA) tool as a method of choice, achieving a maximum gain equal to 14 dBi for a 30 3 dB-beamwidth and a gain of 11 dBi for a 45 3 dB-beamwidth, while the VSWR is kept below 1.7 in all cases. Due to its limited physical size, the proposed CR-ESPAR can be used as a portable antenna for deployment in WiFi, WLAN and other applications. Index Terms—Beam steering, corner reflector-electronically steerable passive array radiator CR-ESPAR, radiation pattern oscillation, reflector, transmit diversity.

I. INTRODUCTION HE rapid development of wireless communications has imposed new demands and expectations to antenna systems designers. In a modern wireless system, an antenna must be of high directivity in order to cover the desirable area and at the same time reject noise and interference. On the other hand, adaptive antenna arrays are considered to become key component in future wireless systems due to their proven beneficial impact [1]. However, fully adaptive arrays come with increased development and maintenance costs due to the complex calibration and signal processing tasks that need to be undertaken. Moreover, a problem of space availability emerges, since the typical element separation distance in such antennas is usually of the order of half wavelength.

T

Manuscript received October 14, 2009; revised August 10, 2010; accepted October 25, 2010. Date of publication January 28, 2011; date of current version April 06, 2011. T. D. Dimousios, S. C. Panagiotou, C. N. Capsalis are with the National Technical University of Athens, 15773 Zografou, Athens, Greece (e-mail: [email protected]). S. A. Mitilineos is with the Technological Education Institute of Piraeus, 12244 Aigaleo, Athens, Greece (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.2011.2109348

The concept of reactively controlled directive arrays was early introduced by Harrington [2], while it has been also reported for multipath situations in [3] and built and demonstrated using monopoles on a finite ground plane in [4]. Electronically steerable passive array radiator (ESPAR) antennas have been recently revisited as an efficient approach of significant interest for adaptive arrays development [5], [6]. ESPAR antennas are considered as an attractive alternative to fully adaptive arrays due to their compact size, low cost and restricted requirements on signal processing capabilities. Beamforming in ESPAR antennas is performed by controlling the reactance loaded to a number of passive elements, while there is only one RF output [7], [8]. All elements are in short distances to one another, and the consequent strong coupling among them is used in order to control the current flowing into the passive elements. By electrically controlling the loading reactance of the passive elements, thus changing their electrical size, directional beams can be formed and steered throughout the azimuth plane [7], [9]–[11]. ESPAR antenna applications include adaptive filtering and beamforming, using steepest gradient, stochastic and MSE techniques [12]–[14], while space-time adaptive filtering using ESPAR antennas has been also proposed [15]. Furthermore, direction-of-arrival (DoA) estimation may be implemented using ESPAR antennas [16], [17], yielding applications such as base-station tracking, satellite direction control and wireless vehicle homing systems [18], [19]. In the literature, the design of ESPAR antennas has been performed using various analysis and optimization techniques. Modal expansion analysis of ESPAR is presented by Wang and Shen [20], while the method of moments (MoM) and the finite elements method (FEM) have been introduced for ESPAR design by Lu et al. [21]. Schlub et al. developed a method based on genetic algorithms (GAs) and FEM in order to design and implement an ESPAR antenna using reactance capacitors loaded to the SMA connectors of array elements [11]. In the same work, the concept of a ground skirt was also introduced in order to improve horizontal directivity—this concept was further investigated in [22]. Shibata and Furuhi fabricated a dual-band ESPAR antenna for WLAN applications [23], while Kato and Kuwahara proposed an ESPAR design using sets of capacitive loads instead of varactors which are optimized using a Direct Search method [24]. Finally, Lu et al. reduced the size of an ESPAR antenna by embedding it into a dielectric rod, with the help of FEM and a GA [25]. On the other hand, corner reflectors have been proposed in the literature in order to achieve higher directivity and narrower

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DIMOUSIOS et al.: DESIGN OF A CR REACTIVELY CONTROLLED ANTENNA FOR MAXIMUM DIRECTIVITY

Fig. 1. The structure of the proposed CR-ESPAR antenna.

beamwidth [6], [26]–[31]. Herein, the configuration of a cornerreflector ESPAR (CR-ESPAR) antenna is proposed and a case study suitable for 2.4 GHz ISM applications is demonstrated. In the case where a corner plate surrounds the active and passive elements the resolution of an ESPAR’s available beams will increase, thus offering more granulated scanning of the coverage area. Moreover, the proposed design inherently eliminates electromagnetic coupling among similar CR-ESPARs deployed in close proximity but aiming to different angles, since the corner reflector is connected to the ground. Therefore, the proposed design may be used in base station applications, where a number of identical CR-ESPARs may be deployed covering selected sectors of the desired area. Moreover, due to the compact size of the final design, the proposed CR-ESPAR may be also used in portable applications. Thus, the proposed configuration of CR-ESPAR may provide a cost-effective means of developing antenna array systems for base stations or portable devices. II. CR-ESPAR CONFIGURATION The proposed CR-ESPAR consists of 7 dipole elements, namely six passive and one active, as well as a corner reflector which surrounds all elements, as illustrated in Fig. 1. The corner of the reflector is selected to be equal to 90 as a value of choice, in order to cover a quadrant of the horizontal plane. The active element is placed on the bisector of the corner from its acme. Each plate’s angle at a certain distance passive element is reactively loaded by, , . The and positioned inside the area of the quadrant at passive elements are formed in symmetrical pairs around the bisector of the reflector. However, their electrical length may differ since each passive element is able to be loaded with a different reactive load value. In a practical application, dipole elements may be supported by a platform of electrically transparent material, an approach that has been used in the literature in order to develop dipole arrays [32]. The transparent material will also be useful as a means of mechanically sustaining the corner reflector and the array feeding cables. The structure of the CR-ESPAR is generated using the SuperNEC electromagnetic simulation software package, while the currents flowing into the elements as well as the radiation

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pattern of the array are calculated using the method of moments (MoM). SuperNEC is a hybrid MoM-UTD (uniform theory of diffraction) antenna and electromagnetic simulation program. The MoM primitives available in the code are wire segments, whereas the UTD primitives supported are dielectrically coated plates and elliptical cylinders [33], [34]. The characteristics of the reflector and the array elements were optimized using a GA. GAs constitute a stochastic optimization class of algorithms that employ the mechanisms of natural selection and genetic engineering (crossover, mutation) in order to explore non-linear and discontinuous solution spaces [35]. GAs are suitable for multi-dimensional and multi-variable problems, such as the design and synthesis of antennas, where a large number of parameters need to be optimized in order to meet a set of performance criteria (e.g., gain, front-to-back ratio, input impedance, etc.) [36]. A GA was employed herein as an optimization method of choice, due to their successful employment in similar problems recently [11], [25]. Optimization parameters included the height and width of the corner plate, the position, length and thickness of the active and passive elements, and the reactive load values of the passive elements. The SuperNEC code results were then evaluated via the Finite Elements Method using HFSS. The case study of the CR-ESPAR configuration presented herein is suitable for base station as well as portable applications at the 2.4 GHz ISM band, either as a stand-alone array or in a complex configuration of four identical CR-ESPARs covering the entire horizontal plane. The CR-ESPAR antenna may be used as a switched-beam receiver, where the signal quality of each beam is measured and the beam that provides the strongest signal is selected. Similarly, the proposed array may be used as a switched-beam transmitter in the case where the transmitter acquires feedback regarding the link quality by the receiver. Moreover, significant diversity gain can be achieved at the mobile receiver by oscillating the transmitting antenna’s radiation patterns [37]–[41]. The oscillation of the antenna pattern, achieves both directionality and transmit diversity, while at the same time creates a time varying channel with a controllable coherence time. With the proposed antenna at the base station (BS) transmitting CI/MC-CDMA signals and a single omnidirectional antenna at the mobile receiver, the controllable coherence time is used by the mobile to exploit time diversity and enhance performance. The multi-fold-diversity created via beam pattern scanning leads to increased directionality gain and corresponding network capacity. Furthermore, the antenna’s benefits against similar types of antennas are worth mentioning and make the proposed antenna an ideal solution when considering antennas for base station WiFi applications for both fixed and portable scenarios. From what can be seen from the results section, the CR-ESPAR provides significant directional gain, greater than the other simple ESPAR antenna structures proposed in the literature. Additionally, CR-ESPAR provides an oscillating antenna pattern using only one feeding element, making its construction easy to handle and signal processing cost extremely low, in contrast to most commonly used MIMO systems where time varying delays and phase variation is used in multiple feeding points. Furthermore, the proposed antenna offers more points of oscillation compared to a switched parasitic array (SPA) with a

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TABLE I INPUT PARAMETERS AND RESULTS OF THE GA FOR THE CR-ESPAR AT 2.4 GHZ

similar structure and the same amount of elements, making the CR-ESPAR much more directional. Finally, it’s the usage of the varactors that makes the CR-ESPAR antenna capable of forming virtually any desired radiation pattern, as pointed out in the results section of this paper. III. DESIGN AND OPTIMIZATION PROCEDURE A. Optimization Parameters and Goals The goal of the optimization procedure was to produce seven radiation patterns with maximum gain and minimum reflection coefficient at 2.4 GHz (75 Ohm input impedance for a 75 Ohm feeding transmission line). Five switching radiation patterns of 30 3 dB-beamwidth each, with the main beam oriented towards and 30 respectively, will cover the quadrant of the corner plate with high resolution. Two radiation patterns of 45 3 dB-beamwidth each, with the main beam oriented and 22.5 respectively will cover the quadrant of towards, the corner plate with lower resolution, and one radiation pattern of 90 3 dB-beamwidth with the main beam oriented towards 0 will cover the entire quadrant in the case where no switched beams are desired. The last three radiation patterns indicate the multiple radiation patterns forming ability of the proposed antenna. Due to

the structure’s symmetry toward the corner plate’s bisector, only and the radiation patterns oriented towards the radiation pattern oriented towards 0 needed to be optimized. The 30 , 22.5 and 15 degree radiation patterns may be achieved by mutually swapping the electrical condition of the symmetrical dipoles (mutual load swapping). As already mentioned, optimization parameters included and width of the corner plate, the posithe height —and length of the active and passive elements, tion— and the reactive load values of the passive elements. However, the lengths of all passive elements are identical and denoted , while the length of the active element is denoted by by . Also, the positions of the paired passive elements are symmetrical with respect to the x-axis (bisector of the corner reflector), and thus for each pair only two parameters needed to be optimized regarding their placement instead of four and (two symmetrical passive elements are placed at ). Regarding the position of the active element, only its x-axis coordinate was optimized. However, each passive element is independently loaded; therefore all six reactance values of the passive elements participated in the optimization procedure. The optimization parameters are depicted in Fig. 1 and also tabulated in Table I further below.

DIMOUSIOS et al.: DESIGN OF A CR REACTIVELY CONTROLLED ANTENNA FOR MAXIMUM DIRECTIVITY

Fig. 2. Schematic of a passive element’s matching network.

Fig. 3. Simulation results for CR-ESPAR passive elements’ reactive loads.

B. Optimization Range of Reactive Loads In order to determine the range of the optimization parameters regarding the reactive loads, the matching network of a passive element was simulated using ADS by Agilent. The configuration schematic of the matching network is illustrated in Fig. 2, where the vertical open stub is used in order to emulate an inductor and achieve positive values of the reactive loads range. The variable capacitor C2 is used in order to achieve variable reactive load, while the capacitor C3 is used in order to push the optimization range to negative values and balance the effect of the vertical stub. The left-sided termination is connected to the passive element, while the right-end microstrip is grounded. In order to demonstrate a generic approach, the well-known and popular FR4 substrate was used for simulation, with a thickness . of 1.6 mm and A very large number of commercially available voltage-controlled variable capacitors were evaluated, and finally the JDV2S71E diode by Toshiba was selected. The JDV2S71E diode is an SMD silicon epitaxial diode, with a variable capacitance controlled by in-line DC voltage. Its capacitance values range from 0.64 pF (maximum low capacitance) to 6 pF (minimum high capacitance) for a control voltage ranging from 25 V to 1 V respectively. The simulation results for the schematic of Fig. 2 when using the JDV2S71E diode are depicted in Fig. 3. The achieved passive element’s reactive load Ohms to Ohms. These values were ranges from used by the GA in order to generate realistic values for the optimization of the CR-ESPAR’s matching networks. C. Objective Function of the GA The objective function is the driving force behind the GA. It is called from the GA to determine the fitness of each solution string generated during the search. As already pointed out, the main goal was to offer radiation patterns with certain

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characteristics (maximum gain at specific directions, specific 3-dB beamwidth, and relative sidelobe levels as low as possible), while attaining input impedance close to 75 Ohms at 2.4 GHz. Due to the structure’s symmetry with respect to the corner plate’s bisector, only the radiation patterns oriented toand the radiation pattern oriented wards towards 0 needed to be optimized. The 30 , 22.5 and 15 degree radiation patterns may be achieved by mutually swapping the electrical condition of the symmetrical dipoles. Since various radiation patterns pointing at various directions and of different 3 dB beamwidth are required, the optimization of the proposed antennas was selected to be performed in more than one stages. First, a full GA with all optimization parameters participating was carried out, in order to deliver the and of radiation pattern with the main lobe pointing at 30 3 dB beamwidth (first factor of the function below). As regards the remaining azimuth radiation pattern pointing at and the two radiation patterns oriented towards 0 , the structure extracted as a result of the first GA underwent four more GA optimizations (one for each remaining factor of function below), but this time the only optimization the parameters were the loads of the parasitic elements. The spatial parameters of the structure remained the ones that stemmed from the optimization procedure concerning the first radiation with a 30 3 dB beamwidth) since pattern (pointing at the antenna maintains its physical attributes at all five radiation patterns. The objective function that satisfied the aforementioned demands was the following. — One set of 360 points was used to form the desired direpresents the desired rectivity pattern. Each point normalized directivity pattern value at angular position . is formed with angular step of 1 and is the normalized directivity pattern calculated by the softis the requested 3 dB beamwidth each time ware. and a side lobe ratio of 9 dB. The first relative error term is given by (1) where

(2) — Input impedance matching is also required. So, a relevant error term was taken into consideration (3) where and represent the real and imaginary part of the input impedance respectively and a characteristic impedance of 75 was taken into account.

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— The cumulative error was given by a weighted sum of the terms described above as in

TABLE II RESULTS OF THE GA FOR THE CR-ESPAR AT 2.4 GHZ CONCERNING THE RADIATION PATTERNS

(4) where are the weights of respectively. — The objective function was defined as (5) The simulation frequency was set to be 2.4 GHz, the frebe the wavelength at quency used broadly at WLANs. Let 2.4 GHz. In Table I, the range of variation of each component parameter taking part into the optimization procedure is tabulated. Since the spatial parameters of the antenna under examination are expressed in terms of , apart from the coordinates of each pair of dipoles, the electrical dimensions of the antenna remain constant and the derived data is suitable for application in other frequencies as well. The wire radius of each element is . The total population comprised of 250 generations with 60 chromosomes per generation. The selection method was population decimation, while adjacent fitness pairing was the mating type. The crossover point was chosen randomly and each chromosome was divided at a gene level, while the mutation probability was set equal to 0.15 [41]–[43]. It should be noted herein that due to its stochastic nature the GA may in general deliver different results for different runs. Thereupon, a number of 50 runs of the GA were executed; however, the optimized parameters were not differing more than 0.1% among GA runs, indicating that the final results of the GA presented herein are stable. Table I presents the range of variation of each component parameter taking part into the optimization procedure, where all physical dimensions are expressed in terms of . IV. NUMERICAL RESULTS Due to the demand of specific radiation patterns, several GA runs took place for different values of the weight factors incorporated in the objective function (see (4)). In the case of the radiation patterns of 30 3 dB beamwidth pointing at and 0 best results were obtained for three different sets of and , or and weight factors: either , or and . Regarding the radiation pattern of 45 3 dB beamwidth pointing at 22.5 , best results and . Finally, for the radiawere obtained for tion pattern of 90 3 dB beamwidth pointing at 0 , best results were obtained for and (see Table II). The final results of the GAs for each parameter are shown in Table I, the output structure designed by both SNEC and HFSS is depicted in Fig. 4 and all five radiation patterns, produced by simulating the output structure using SNEC and HFSS, are presented in Fig. 5. It is noted that, in practical cases, the resolution of the impedance loads included in Table I may not be easily achieved. Therefore, a sweep analysis was performed, where the impedance loads were changing randomly following a uniform distribution centered at their nominal value and within %. After 100 simulation runs, it was concluded a range of

Fig. 4. CR-ESPAR antenna designed by SNEC and HFSS.

that the change in the gain and other structure characteristic (VSWR, scanning angle, beamwidth) did not change more than 5.63% around their nominal values. It is also noted that the GA exhausts the limits of the corner reflector dimensions. As expected, larger reflector dimensions result to higher gain; nevertheless, there is a threshold over which this trend fades out. In our case, it was selected that the reflector dimensions remain at 21.25 21.25 cm , as a trade-off between large gain, low VSWR and limited antenna size and weight. Table II summarizes the radiation specifications of the proposed design. It is shown that all resulted radiation patterns exhibit a 3 dB beamwidth pretty close to the desired one, satisfactory side and back lobe suppression, they point to the demanded direction and, at all cases, the VSWR factor is less than 1.75, making the request of input impedance matching at 2.4 GHz granted. Furthermore, the maximum gain of each radiation pattern is quite large (10–14 dBi with a 3 dB-beamwidth of 30 ), compared to the gain of other ESPAR antennas in the literature operating at the same frequency (ranging from 5 dBi to 8.1 dBi) [8], [19], [22]. As an example, consider the gain of the pattern towards 0 , with a 3 dB-beamwidth equal to 84 and a gain equal

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Fig. 5. SNEC and HFSS results for the radiation patterns and the achieved oscillation.

to 8.6 dBi. The gain of the array proposed in [8] has a lower gain with a lower beamwidth (8.08 dBi and 60 respectively). The increased gain of the proposed CR-ESPAR is partly due to the presence of the corner reflector which adds to the physical size of the antenna; namely, the reflector’s physical dimensions are equal to 21.5 21.5 21.5 cm , while the physical dimensions of the array in [8] are equal to cm . Similarly, the dielectric-embedded ESPAR (DE-ESPAR) reported in [22], has even lower gain (5.1 dBi), but is delivered in a smaller form factor (physical dimensions equal to cm ).

Fig. 5(f) depicts the oscillation achieved using the radiation patters of 30 3 dB beamwidth pointing at and 30 , one of the key features of the proposed antenna, which results to a multi-fold-diversity created via beam pattern scanning and leads to increased directionality gain and corresponding network capacity. Using only one feeding point, the CR-ESPAR is of significantly lower development and operating cost compared to an SPA similar antenna structure proposed and at 0 in [40]. The radiation patterns pointing at of 45 and 90 3 dB beamwidth accordingly, presented in Fig. 5(a)–(e), indicate that the usage of varactors makes the

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proposed antenna virtually capable of forming any desired radiation pattern which gives the CR-ESPAR a significant advantage against other, fixed rotation pattern, antenna proposals. Finally, the physical dimensions of the antenna, presented in Table I, demonstrate a quite compact antenna thus satisfying the original demand of a limited sized antenna structure. V. CONCLUSION Smart antenna systems design has known significant progress during the last years. The aim of designing multi-purpose, directional, electrically steerable antennas in order to cover specific locations, avoid interferences, has met new horizons with the ESPAR antennas. The CR-ESPAR presented in this paper meets all the above challenges. Achieving increased directionality, input impedance matching and more than great maximum gain, it can be used ideally as a transmitter in WiFi/WLAN systems operating at 2.4 GHz. Its selective directionality gives the chance to focus on specific coverage area (transmitter) or chose from a variety of hot spots the one with the best signal strength (receivers) and its oscillating beam capability achieves both directionality and transmit diversity exploited by the mobile users to enhance performance. Finally, its compact dimensions make it ideal for both fixed and mobile reception and transmission since there are no weight or size limitations. REFERENCES [1] A. Paulraj, R. Nabar, and D. Gore, Introduction to Space-Time Wireless Communications. Cambridge, U.K.: Cambridge Univ. Press, 2003. [2] R. F. Harrington, “Reactively controlled directive arrays,” IEEE Trans. Antennas Propag., vol. AP-26, no. 3, May 1978. [3] R. G. Vaughan, “Switched parasitic elements for antenna diversity,” IEEE Trans. Antennas Propag., vol. 47, no. 2, pp. 399–405, Feb. 1999. [4] N. L. Scott, M. O. Leonard-Taylor, and R. G. Vaughan, “Diversity gain from a single-port adaptive antenna using switched parasitic elements illustrated with a wire and monopole prototype,” IEEE Trans. Antennas Propag., vol. 47, no. 6, pp. 1066–1070, Jun. 1999. [5] D. V. Thiel, “Impedance variations in controlled reactance parasitic antennas,” in Proc. IEEE Antennas and Propagation Society Int. Symp., Washington, DC, Jul. 3–8, 2005, vol. 3A, pp. 671–674. [6] D. V. Thiel and S. Smith, Switched Parasitic Antennas for Cellular Communications. Boston, MA: Artech House, 2001. [7] K. Gyoda and T. Ohira, “Design of electronically steerable passive array radiator (ESPAR) antennas,” in Proc. IEEE Antennas and Propagation Society Int. Symp., Jul. 2000, vol. 2, pp. 922–925. [8] T. Ohira and K. Gyoda, “Electronically steerable passive array radiator antennas for low-cost analog adaptive beamforming,” in Proc. IEEE Int. Conf. on Phased Array Systems and Technology, May 2000, pp. 101–104. [9] T. Ohira and K. Iigusa, “Electronically steerable parasitic array radiator antenna,” Electron. Commun. Jpn., vol. 87, no. 10, pt. 2, pp. 25–45, 2004. [10] C. Sun, N. C. Karmakar, and T. Ohira, “Experimental studies of radiation pattern of electronically steerable passive array radiator smart antenna,” in Proc. IEEE Antennas and Propagation Society Int. Symp., Jun. 2003, vol. 3, pp. 884–887. [11] R. Schlub, J. Lu, and T. Ohira, “Seven-element ground skirt monopole ESPAR antenna design from a genetic algorithm and the finite element method,” IEEE Trans. Antennas Propag., vol. 51, no. 11, pp. 3033–3039, Nov. 2003. [12] J. Cheng, Y. Kamiya, and T. Ohira, “Adaptive beamforming of ESPAR antenna using sequential perturbation,” in IEEE MTT-S Int. Microwave Symp. Digest, May 2001, vol. 1, pp. 133–136. [13] B. Shishkov and T. Ohira, “Adaptive beamforming of ESPAR antenna based on stochastic approximation theory,” in Proc. Asia-Pacific Microwave Conf., Dec. 2001, vol. 2, pp. 597–600. [14] C. Sun, A. Hirata, T. Ohira, and N. C. Karmakar, “Fast beamforming of electronically steerable passive array radiator antennas: Theory and experiment,” IEEE Trans. Antennas Propag., vol. 52, no. 7, pp. 1819–1831, Jul. 2004.

[15] K. Yang and T. Ohira, “Realization of space-time adaptive filtering by employing electronically steerable passive array radiator antennas,” IEEE Trans. Antennas Propag., vol. 51, no. 7, pp. 1476–1485, Jul. 2003. [16] C. Plapous, J. Cheng, E. Taillefer, A. Hirata, and T. Ohira, “Reactance domain MUSIC algorithm for electronically steerable parasitic array radiator,” IEEE Trans. Antennas Propag., vol. 52, no. 12, pp. 3257–3264, Dec. 2004. [17] E. Taillefer, A. Hirata, and T. Ohira, “Direction-of-Arrival estimation using radiation power pattern with an ESPAR antenna,” IEEE Trans. Antennas Propag., vol. 53, no. 2, pp. 678–684, Feb. 2005. [18] S. L. Preston, D. V. Thiel, T. A. Smith, S. G. O’Keefe, and J. W. Lu, “Base-station tracking in mobile communications using a switched parasitic antenna array,” IEEE Trans. Antennas Propag., vol. 46, no. 6, pp. 841–844, Jun. 1998. [19] J. Cheng and T. Ohira, “ESPAR antenna signal processing for DOA estimation,” in Advances in Direction of Arrival Estimation, S. Chandran, Ed. Norwood, MA: Artech House, 2006, pp. 395–418. [20] X. Wang and Z. Shen, “Modal expansion analysis of electrically steerable passive array radiators (ESPAR),” in Proc. Antennas and Propagation Society Int. Symp., Jul. 2005, vol. 4B, pp. 27–30. [21] J. Lu, D. Ireland, and R. Schlub, “Development of ESPAR antenna array using numerical modeling techniques,” in Proc. 3rd Int. Conf. on Computational Electromagnetics and its Applications, Nov. 2004, pp. 182–185. [22] H. Kawakami and T. Ohira, “Electrically steerable passive array radiator (ESPAR) antennas,” IEEE Antennas Propag. Mag., vol. 47, no. 2, pp. 43–49, Apr. 2005. [23] O. Shibata and T. Furuhi, “Dual-band ESPAR antenna for wireless LAN applications,” in Proc. Antennas and Propagation Society Int. Society Symp., Jul. 2005, vol. 2B, pp. 605–608. [24] H. Kato and Y. Kuwahara, “Novel ESPAR antenna,” in Proc. Antennas and Propagation Society Int. Symp., Jul. 2005, vol. 4B, pp. 23–26. [25] J. Lu, D. Ireland, and R. Schlub, “Dielectric embedded ESPAR (DEESPAR) antenna array for wireless communications,” IEEE Trans. Antennas Propag., vol. 53, no. 8, pp. 2437–2443, Aug. 2005. [26] N. P. Yeliseyeva, “Optimizing gain of finite size corner-reflector antenna,” in Proc. 10th Int. Crimean Microwave and Telecommunication Technology Conf., 2000, pp. 302–303. [27] L. Desclos, M. Madihian, and J. M. Floc’h, “A 1.8–6 GHz corner reflector based on shaped monopole excitation,” Microw. Opt. Technol. Lett., vol. 21, no. 3, pp. 196–199, Apr. 1999. [28] V. P. Joseph and K. T. Mathew, “A novel corner reflector antenna,” Microw. Opt. Technol. Lett., vol. 30, no. 6, pp. 403–404, Aug. 2001. [29] N. Okamoto, “Electronic lobe switching by 90 deg corner reflector antenna with ferrite cylinders,” IEEE Trans. Antennas Propag., vol. AP-23, pp. 527–531, Jul. 1975. [30] Y. Lu, X. Cai, and Z. Gao, “Optimal design of special corner reflector antennas by the real-coded genetic algorithm,” in Proc. Asia-Pacific Microwave Conf., 2000, pp. 1457–1460. [31] A. Nesic, Z. Micic, S. Jovanovic, and I. Radnovic, “Millimetre wave printed antenna array with high side lobe suppression,” in Proc. IEEE Antennas and Propagation Society Int. Symp., Jul. 9–14, 2006, pp. 3051–3054. [32] S. C. Panagiotou, S. A. Mitilineos, T. D. Dimousios, and C. N. Capsalis, “A broadband, vertically polarized, circular switched parasitic array for indoor portable DVB-T applications at the IV UHF band,” IEEE Trans. Broadcast., vol. 53, no. 2, pp. 547–552, Jun. 2007. [33] T. D. Dimousios, C. I. Tsitouri, S. C. Panagiotou, and C. N. Capsalis, “Design and optimization of a multipurpose tri-band electronically steerable passive array radiator (ESPAR) antenna with steerablebeam-pattern for maximum directionality at the frequencies of 1.8, 1.9 and 2.4 GHz with the aid of genetic algorithms,” in Proc. Antennas and Propagation Conf., LAPC 2008, Loughborough, Mar. 17–18, 2008, pp. 257–260. [34] SuperNec v. 2.4 MOM Technical Reference Manual. [35] D. E. Goldberg, Genetic Algorithms in Search, Optimization, and Machine Learning. New York: Addison-Wesley, 1989. [36] B. Orchard, “Optimizing Algorithms for Antenna Design,” M.Sc. thesis, Univ. Witwatersrand, Johannesburg, South Africa, 2002. [37] S. A. Zekavat and C. R. Nassar, “Smart antenna arrays with oscillating beam patterns: Characterization of transmit diversity in semi-elliptic coverage,” IEEE Trans. Commun., vol. 50, no. 10, pp. 1549–1556, Oct. 2002, appears in:. [38] S. A. Zekavat and C. R. Nassar, “Fading channel characterization for oscillating-beam-pattern smart antennas using geometric-based stochastic channel modelling with circular coverage area,” in Proc. IEEE 54th Vehicular Technology Conf. Fall, 2001, vol. 3, pp. 1452–1456.

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[39] S. A. Zekavat, C. R. Nassar, and S. Shattil, “Oscillating-beam smart antenna arrays and multicarrier systems: Achieving transmit diversity, frequency diversity, and directionality,” IEEE Trans. Veh. Technol., vol. 51, no. 5, pp. 1030–1039, Sep. 2002. [40] S. A. Zekavat and C. R. Nassar, “Achieving high-capacity wireless by merging multicarrier CDMA systems and oscillating-beam smart antenna arrays,” IEEE Trans. Veh. Technol., vol. 52, no. 4, pp. 772–778, Jul. 2003. [41] S. A. Zekavat and C. R. Nassar, “Transmit diversity via oscillatingbeam-pattern adaptive antennas: An evaluation using geometric-based stochastic circular-scenario channel modelling,” IEEE Trans. Wireless Commun., vol. 3, no. 4, pp. 1134–1141, Jul. 2004. [42] Y. R. Samii and E. Michielssen, Electromagnetic Optimization by Genetic Algorithms. New York: Wiley, 1999. [43] T. D. Dimousios, S. C. Panagiotou, and C. N. Capsalis, “Design and optimization of a switched parasitic corner plated antenna for maximum directionality and diversity gain at the WiFi frequency of 2.4 GHz with the aid of genetic algorithms,” in Proc. Int. Conf. on Electromagnetics in Advanced Applications, ICEAA 2007, Sep. 17–21, 2007, pp. 731–734.

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Stelios A. Mitilineos (M’07) was born in Athens, Greece, in 1977. He received the Diploma in electrical and computer engineering and the Ph.D. degree from the National Technical University of Athens (NTUA), Greece, in October 2001 and June 2007, respectively, with a fellowship from the Institute of Informatics and Telecommunications, National Center Scientific Research “Demokritos” (IIT-NCSR). Previously, he was a Postdoctoral Researcher with the IIT-NCSR, where he was also involved in European projects. Currently, he is a Lecturer with the Technological Education Institute of Piraeus, Athens, Greece. He has participated in numerous EU and national research projects and has published more than 30 papers in scientific journals and international conferences with more than 80 citations. His main research interests are in the areas of antennas and propagation, smart antennas and microwave components, mobile communications and mobile channel simulation, and position location.

Stylianos C. Panagiotou was born in Athens, Greece, in 1980. He received the Diploma in electrical and computer engineering from the National Technical University of Athens (NTUA), Greece, in 2003, where he is currently working toward the Ph.D. degree. His main research interests are in the fields of multipath propagation, smart antennas, wireless MIMO systems and antenna design.

Themistoklis D. Dimousios was born in Athens, Greece, in 1980. He received the Diploma in electrical and computer engineering from the National Technical University of Athens (NTUA), Greece, in 2004, where he is currently working toward the Ph.D. degree. His main research interests are in the fields of mobile communications, smart antennas and antenna design.

Christos N. Capsalis was born in Nafplion, Greece, in 1956. He received the Diploma in electrical and mechanical engineering from the National Technical University of Athens (NTUA), in 1979, the B.S. degree in economics from the University of Athens, in 1983, and the Ph.D. degree in electrical engineering from NTUA, in 1985. He is currently a Professor in the Department of Electrical and Computer Engineering, NTUA. His current scientific activity concerns satellite and mobile communications, antenna theory and design, and electromagnetic compatibility.

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Energy Patterns of UWB Antenna Arrays With Scan Capability Chao-Hsiang Liao, Student Member, IEEE, Powen Hsu, Fellow, IEEE, and Dau-Chyrh Chang, Fellow, IEEE

Abstract—A beam scanning technique is developed with a time delay for ultrawideband (UWB) arrays. In addition, we give a formulation for the time domain array factor for UWB antenna arrays. An UWB comb taper slot antenna array with large element spacing and impulse excitation is studied and four-element UWB linear array systems with element spacings of 2.5, 8.5, and 18 cm in the broadside direction and 20 degree scanning are implemented. The voltage response, spectrum response, energy pattern and energy gain for from simulation and measurement are in good agreement. The periodic grating lobes of the UWB antenna array do not occur in the energy pattern, even if the element spacing and scan angle are large. Simulation and measurement results for the energy pattern and voltage response in various directions are studied to validate the theory of an UWB antenna array with scan capability. The measured peak energy gain of arrays in the broadside direction and 20 degree beam scanning with element spacings of 8.5 and 18 cm are 10.79/10.78 dBi and 9.53/9.06 dBi, respectively. Index Terms—Antenna array, beam scanning, energy pattern, impulse excitation, ultrawideband (UWB) array.

I. INTRODUCTION

T

HE application of ultrawideband (UWB) arrays leads to narrow beam widths for sparse arrays with large element spacing; this suits many potential applications in both wireless communications and radar systems [1]–[3]. The UWB array waveform is an effective way of improving the resolution for microwave imaging. This makes an UWB array with impulse waveforms suitable for radar direction finding and imaging applications. To provide high-resolution radar imaging, the antenna beam width should be narrow. Since the 3 dB beam width of the array antenna is inversely proportional to the size of the aperture, the larger aperture of the UWB array produces a narrower beam width. Unfortunately, a larger aperture of a in an UWB array with larger element spacing generates periodic grating lobes in the power pattern; however, the grating lobes do not occur in the energy pattern, even if the element spacing Manuscript received October 04, 2009; revised August 04, 2010; accepted September 03, 2010. Date of publication January 28, 2011; date of current version April 06, 2011. This work was supported by the National Science Council, Taiwan, R.O.C., under Contracts NSC 97-2221-E-161-003 and NSC 97-2221-E-002-061-MY3. C. H. Liao is with the Graduate Institute of Communication Engineering, National Taiwan University, Taipei 10617, Taiwan, R.O.C. P. Hsu is with the Department of Electrical Engineering and the Graduate Institute of Communication Engineering, National Taiwan University, Taipei 10617, Taiwan, R.O.C. D.-C. Chang is with the Oriental Institute of Technology, Taipei, Taiwan, R.O.C. (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.2011.2109352

of the array is larger than one wavelength. Frequency domain array theory, by itself, is not sufficient to assess the influence of the UWB antenna array on system performance; however, the unusual antenna pattern properties may be demonstrated analytically by deriving the energy pattern. It is well known that antenna parameters are function of the frequency. For narrowband applications it is possible to analyze these parameters within the narrow band. For wider bandwidths, antenna parameters change greatly from frequency to frequency. The parameters as functions of frequency are not sufficient for the characterization of the communication system. If using the proposed solution, it is easier to describe the energy pattern, instead of the power pattern, for UWB antennas and arrays. Impulse radiating arrays have recently been proposed for several applications [4] in radar, communications, and remote sensing, exploiting their advantages of reduced sidelobes and high resolution. Impulse radiating arrays are driven by pulsed waveforms, and their scanning performance is controlled by inter-element time delays. Preliminary physical investigations have been reported in [5], [6], where the coupling between radiating elements is usually neglected, and emphasis is placed on the conditions for sidelobe absence, permitting the design of sparse arrays. The frequency bandwidth of a conventional phased array antenna is ultimately limited by the array element (antenna elements, amplifiers, and beam-forming network) bandwidth [7]–[9]. However, a more severe limitation is often caused by the use of phase shifters to scan the beam [10]. True time delay (TTD) technology potentially eliminates this bandwidth restriction; however, standard time-delay technology consists of switched transmission line sections and the weight, loss, and cost increase rapidly with the phase-tuning resolution. A delay line concept was previously presented in a proof-of-principle demonstration, where a linear delay line was used as a low-loss, beam scanning controlled and broadband TTD line [11]. Traditional radar systems have a narrow band and high power; the proposed radar system affords an ultra wide band and low power. This paper presents an UWB four-element beam scanning array, controlled by a linear delay line. The system consists of various wideband components, including a microstrip UWB power divider, UWB comb tapered slot antenna array, linear delay line, and broadband transitions. The combination of these wideband elements provides significant flexibility for a broadband system design. For the current system, delay line elements support use from 3.1 to 10.6 GHz; however, the same system components and implementation can be used at frequencies up to 18 GHz, using a delay line with lower loss, to produce an extremely wideband array system.

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where

(3)

Fig. 1. UWB time-delay array for beam scanning in 

.

II. THEORY OF UWB ANTENNA ARRAY IN TIME DOMAIN The transient properties of UWB arrays have been discussed in the literature [12], [13]. The antenna array design was based on modified array theory; the beam maximum scan angle for a given antenna spacing between adjacent antenna elements, as shown in Fig. 1, is given in (1)

(1) is the relative excitation time delay between adjacent where antenna elements, is the excitation time delay for the th element, and is the speed of light. Typically, the transmit signal is a Gaussian pulse. UWB arrays require true time delay components to steer the maximum beam direction. To achieve the maximum signal in the scan direction, the array outputs must be in phase. For a linear array with uniform spacing between the antennas, the relative excita. If the tion time delay is given in (1) for a scanned angle at output signals are not in phase, the summed signal will not have a maximum in the desired beam direction. Compared with traditional phased array antennas, the UWB time domain antenna array is realized through different time delays among antenna elements—a phase shifter is unnecessary. The transient transmission characteristic of an UWB array is calculated as the convolution of the antenna element transient response with the time-domain array factor. In UWB antenna arrays, each antenna element radiates impulses. The superposition of these signals in free space is determined by both the impulse waveform impulse and the relative time delay. For a uniformly spaced one-dimensional linear array with elements excited by identical impulses, if the mutual coupling is can be obnegligible, the convolution time waveform tained by (2)

(2)

is the radiated waveform for an antenna element in where the direction, is the array factor in the time domain, is the convolution operator, is the number of antenna elements, is the amplitude excitation, is the excitation time delay for the th element, is the relative space time delay between neighboring elements if the wave is incident in the direction, is the light speed, and is the antenna spacing between adjacent antenna elements. To achieve maximum signal in the desired direction, the array ; when the output outputs must be in phase , the summed signals signals are not in phase will spread in the time domain and the voltage will not increase in the desired direction. III. ANTENNA ENERGY PATTERN Ultra wideband (UWB) antenna and array are widely used in UWB wireless communication, radar and warfare in recent years. Power patterns in the frequency domain are usually used to describe the antenna performance in narrowband communication systems; however, they are not sufficient for ultra wideband (UWB) antennas in wideband communication systems. The antenna power pattern in the frequency domain is the spatial variation of the radiation intensity along a constant radius. The UWB antenna has a very wide band and many frequencies, thus, its patterns represent a very complex problem. But if the UWB antenna pattern is defined as the energy pattern in the time domain, the problem is solved easily. The energy pattern of the antenna is the total response of the power patterns in the frequency domain. This provides a simple representation of antenna behavior as that of a large number of power patterns. It is easier to describe the energy pattern, instead of the power pattern, for UWB antennas and arrays. in the direcThe energy pattern (energy intensity) in the time domain was defined in [14] and is given tion by (4) (4) is the intrinsic impedance of free space (377 ohm) and is the radiated electric field intensity. The energy pattern is the energy per unit solid angle. Although the electric field intensity depends on the range , the energy pattern will not depend on the range in the far field range. The unit of the power pattern (power intensity) is in watt per unit solid angle, while the unit of the energy pattern is in joule per unit solid angle. Typically, UWB time waveform envelopes decay rapidly; to accelerate the computation time, the energy pattern will be sim. The energy patulated or measured and time gated over direction can be modified as in (5) tern in the where

(5)

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Fig. 2. Prototype of the microstrip UWB power divider.

IV. THE UWB ARRAY There are two parts in this section covering the UWB array: with and without beam scanning. The four-element UWB linear array system consists of various wideband components, including a four-way microstrip UWB power divider, four UWB comb tapered slot antennas, four identical coaxial cable lines (ULA-316) for arrays without beam scan, and four different coaxial linear delay lines (ULA-316) for arrays with beam scanning. The impedance of coaxial cable line is 50 ohms and the relative permittivity inside the coaxial cable is 2.1. The transmission loss of coaxial cable is 2.5 dB at 10.6 GHz for a 60 cm length. The relative length difference between adjacent coaxial cables is . The relationship among expressed as the relative time delay , element spacing , and beam scan angle is given in (1). Three types of UWB antenna arrays, with element spacings of 2.5, 8.5, and 18 cm, are studied. In this paper, the impulse time-domain antenna measurement system (ITDAMS) [15] is used to measure both the energy pattern in the time domain and the power pattern in the frequency domain. The key components of ITDAMS are the commercially available trigger generator and receiver. The measurement impulse which from trigger generator is a Gaussian pulse with pulse width 30 ps and amplitude 20 V. The receiver is a wideband digital sampling oscilloscope. The output impedances for the trigger generator and digital sampling oscilloscope are 50 ohm.

Fig. 3. Measured phase response and voltage response of the power divider. (a) Phase response; (b) impulse voltage response.

A. Array Without Beam Scanning The performance of phase linearity and time response of power divider are important for the feeding network of the UWB array. A five-step Chebyshev transformer is designed to match the four wideband output impedances. A 4:1 power divider was fabricated using microstrip lines on a FR4 substrate, with a relative permittivity of 4.4, thickness of 1.6 mm, and loss , tangent of 0.0254. The power divider is as shown in Fig. 2. The measured transmission phase response in the frequency domain and impulse voltage response in the time domain are shown in Fig. 3(a) and (b), respectively. The transmission phase responses of the four outputs are very similar. The transmission time delay and fidelity of voltage of the four outputs are almost identical. This 4:1 power divider will be used for the UWB array feeding network.

Fig. 4. UWB comb taper slot antenna.

An UWB comb taper slot antenna, as shown in Fig. 4, was chosen as the antenna element. The antenna is designed on a FR4 substrate with relative permittivity of 4.4, thickness of 0.8 mm, and loss tangent of 0.0254, and the size of the antenna . The antenna can achieve ultra wideband is

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Fig. 5. H-plane UWB antenna array.

Fig. 6. Comparison of simulation and measurement for single-element and four-element arrays with element spacings of 2.5, 8.5, and 18 cm. (a) Simulation (X-Z plane); (b) measurement (X-Z plane).

performance owing to its elegant transition from the microstrip line. The microstrip transition at the input is circularly tapered to parallel strips for the antenna feed. The corrugations

Fig. 7. Simulated voltage of the array with the designed BFN in the 20 degree direction, (a) and (b) for wave direction in 20 degree, (c) (d) for wave direction in 20 degree. (a) Voltage output at the four elements before the BFN; (b) voltage output after the BFN; (c) voltage output at the four elements with before the BFN; (d) summation of the voltage after the BFN at a wave direction at 20 degree.

0



0

along the sides reduce the antenna width, improve the voltage standing wave ratio (VSWR) over a wide frequency range, and

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Fig. 8. Simulated field intensity (a) and measured voltage (b) in various directions for an array with 8.5 cm element spacing. (a) Simulation; (b) measurement.

suppress the sidelobe levels [17]–[19]. The UWB antenna array is composed of four antenna elements aligned in the H-plane, a multistage, with a four-way microstrip UWB power divider, and four identical coaxial cable lines, as shown in Fig. 5. Several simulation and experimental examples will be given to illustrate the points discussed above. Fig. 6 shows the results of simulation and measurement of the H-plane energy patterns of single-element and four-element arrays with element spacings of 2.5, 8.5, and 18 cm. Minor differences between the simulation and measurement, may be caused by measurement error and defects in the hardware implementation. The larger the element spacing of the array is, the higher the peak side lobe level of the energy pattern will be. This is because the four voltage outputs cannot be a coherent summation if away the boresight direction. There is only one peak side lobe level for the energy pattern of an UWB array with large element spacing, unlike the periodic grating lobes will appear in the power pattern of an array with large element spacing. Far away, in the direction of the peak side lobe, the array energy pattern decays rapidly, due to the taper energy pattern of the antenna element. A detailed explanation is given in the discussion of the UWB array with beam scanning. The beam width of the main lobe narrowed by increasing the array aperture size.

Fig. 9. Simulated field intensity and measured voltage in various directions for an array with 18 cm element spacing. (a) Simulation; (b) measurement.

B. Array With Beam Scanning An array with an element spacing of 8.5 cm for 20 degree beam scanning is described here. To simulate the radiation patof the tern of the UWB array, the transient response antenna element is used in (2). For beam scan at 20 degree with array element spacing of 8.5 cm, the relative excitation is 0.097 time delay at the beam-forming network (BFN) ns. The simulation result is the electric field intensity (V/m) in the far-field region. The simulated excitation waveform is a Gaussian pulse at the port of antenna and the simulated far field distance is 20 m. Fig. 7 shows the details of a wave incident at 20 degrees and the designed BFN with beam scanning is 0.097 at 20 degree. The relative space time delay ns for a wave incident at 20 degrees. Fig. 7(a) shows the simulated field intensity of the four antenna elements before the , i.e. ), the BFN. After the BFN ( simulated field intensity will be time coherent and the summed voltage is shown in Fig. 7(b). The summation voltage will increase from 1.7 V/m to 6.8 V/m for a beam direction at 20 degree. If the wave direction is at 20 degree, i.e., the relative is 0.097 ns, then the total of relative time space time delay is 0.194 ns. The simulated field intensity of delay the four antenna elements before the BFN is shown in Fig. 7(c).

LIAO et al.: ENERGY PATTERNS OF UWB ANTENNA ARRAYS WITH SCAN CAPABILITY

Fig. 10. Simulated and measured spectrum response for element spacing of 8.5 cm. (a) Simulation; (b) measurement.

The simulated field intensity of the four antenna elements after the BFN is shown in Fig. 7(d). The array output with a wave direction at 20 degree is not in phase, and the summed signal will not have a maximum, as shown in Fig. 7(d). The simulation result is the electric field intensity (V/m) in the far-field region, the far field is simulated with a perfect source excitation delivering a Gaussian pulse at antenna input port. The measurement is performed with the same Gaussian pulse voltage source exciting a double ridge horn as the transmission antenna and the array is used as the receiving antenna. The measurement result is the voltage amplitude (V) at the display of digital sampling oscilloscope. In this paper, it is only rough comparison between the measured voltage and the simulated field intensity as shown in Figs. 8 and 9. The spectrum response of Figs. 8 and 9 are also compared in Figs. 10 and 11. The simulated field intensity and measured voltage in the , , , , time domain in various directions ( ) in the H-plane, for an array with element spacings of 8.5 and 18 cm at a designed BFN for 20 degree beam scanning,

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Fig. 11. Simulated and measured spectrum response for element spacing of 18 cm. (a) Simulation; (b) measurement.

are shown in Figs. 8 and 9. The total relative time delay for an array with 18 cm element spacing is larger than that for one with spacing of 8.5 cm. For a beam direction in 20 degree with array element spacings of 8.5/18 cm, the relative space time is 0.097/ 0.205 ns, the relative excitation time delay is 0.097/0.205 ns, and the summation of relative delay is 0 ns. There is a maximum output in the time delay 20 degree beam direction. For a beam direction of 20 degree, the relative space time delay and relative excitation time are 0.097/0.205 ns for element spacings of 8.5/18 delay is cm, and the summation of relative time delay 0.194/0.41 ns. The array output is not in phase and the summed signal will not have a maximum. The simulation and measurement results are in agreement for element spacings of 8.5 and 18 cm. The spectrum response of Figs. 8 and 9 are shown in Figs. 10 and 11. Except for the beam direction in 20 degree, the spectrum response will have periodic response. There are more periodic spectrum responses with narrower bandwidth if the beam direction is far from the designed beam direction—this is more

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TABLE I SIMULATED AND MEASURED ENERGY GAIN OF UWB ARRAY FOR DIFFERENT ELEMENT SPACINGS WITH BEAM SCANNING AT 0 AND 20

0/20 degrees is 11.08/10.45 dBi and 10.79/9.53 dBi. The simulated and measured energy gain with an element spacing of 18 cm in 0/20 degrees is 11.10/10.24 and 10.78/9.06 dBi, respectively. These results show that the energy gain of the array will not increase with a larger aperture for higher directivity. This may be because most of the energy is distributed in the side lobe. The array scanning loss in the frequency domain will also used in energy patterns in the time domain. Fig. 12. Simulation and measurement of the energy pattern with 20 degree beam scanning for an element spacing of 8.5 cm. (X-Z plane).

Fig. 13. Simulation and measurement of the energy pattern with 20 degree beam scanning for an element spacing of 18 cm. (X-Z plane).

obvious with larger element spacing. The array will have maximum instantaneous bandwidth if the BFN is designed in the desired beam direction. The maximum instantaneous bandwidth will not depend on the element spacing. The simulated and measured normalized energy patterns of Figs. 8 and 9 are shown in Figs. 12 and 13. The larger the array aperture is, the higher the directivity of the UWB array will be. The beam width of an array with larger element spacing is narrower than for smaller element spacing. The peak side lobe level is about 5 dB (angular region between 10 and 40 degrees) 40 degrees, the below the main beam level. Far from the energy pattern decays very fast; this is caused by the antenna element energy pattern, as shown in Fig. 6. The simulated and measured energy gain with a gated time window of 6 ns for beam scanning of 0 and 20 degrees is listed in Table I. The simulated and measured energy gain with an element spacing of 8.5 cm in

V. CONCLUSION UWB antenna arrays were analyzed on the basis of their transient response, which provides all relevant information about their transient radiation and reception characteristics. The farfield voltage consists of the time-domain superposition of the pulse radiated by each of the individual elements. The transient response is given by the convolution of the transient response for the single element with the time-domain array factor. The theory of the UWB time-domain antenna array is derived for computation of the time-domain array factor and beam scanning direction. A uniformly spaced linear UWB antenna array with various element spacings for beam scanning was investigated. An UWB comb taper slot antenna was selected as the array element and beam scanning capability with a linear delay line concept was demonstrated. The antenna energy pattern was presented for the UWB antennae from the time-domain field. Simulated and measured energy patterns of several UWB arrays, with and without beam scanning, were compared. The energy pattern does not have periodical grating lobes even if the element spacing of the UWB array is large. To achieve the maximum signal in a desired beam direction, the outputs of the array BFN must be in phase. When the output signals of the array BFN are not in phase, the signal output will not have a maximum in the desired beam direction. REFERENCES [1] C. T. Rodenbeck, S.-G. Kim, W.-H. Tu, M. R. Coutant, S. Hong, M. Li, and K. Chang, “Ultra-wideband low-cost phased-array radars,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 12, pp. 3697–3703, Dec. 2005. [2] L. Yang, N. Ito, C. W. Domier, N. C. Luhmann, Jr., and A. Mase, “18–40-GHz Beam-shaping/steering phased antenna array system using fermi antenna,” IEEE Trans. Microw. Theory Tech., vol. 56, no. 4, pp. 3697–3703, Apr. 2008. [3] S.-G. Kim and K. Chang, “Ultra wideband 8 to 40 GHz beam scanning phased array using antipodal exponentially-tapered slot antennas,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2004, vol. 3, pp. 1757–1760. [4] M. G. M. Hussain, “Principles of space-time array processing for ultrawide-band impulse radar and radio communications,” IEEE Trans. Veh. Technol., vol. 51, pp. 393–403, May 2002. [5] A. Shlivinski and E. Heyman, “A unified kinematic theory of transient arrays,” in Proc. Ultra-Wideband, Short-Pulse Electromagn., New York, 2002, vol. 5.

LIAO et al.: ENERGY PATTERNS OF UWB ANTENNA ARRAYS WITH SCAN CAPABILITY

[6] L. D. DiDomenico, “A comparison of time versus frequency domain antenna pattems,” IEEE Trans. Antennas Propag., vol. 50, no. 11, pp. 1560–1566, 2002. [7] C. T. Rodenbeck, S.-G. Kim, W.-H. Tu, M. R. Coutant, S. Hong, M. Li, and K. Chang, “Ultra-wideband low-cost phased-array radars,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 12, pp. 3697–3703, Dec. 2005. [8] C.-C. Chang et al., “True time phased antenna array systems based on nonlinear delay line technology,” in Proc. Asia-Pacific Microw. Conf., Dec. 2001, pp. 795–800. [9] K. S. Yngvesson et al., “The tapered slot antenna—A new integrated element for millimeter-wave application,” IEEE Trans. Microw. Theory Tech., vol. 37, no. 2, pp. 365–374, Feb. 1989. [10] R. J. Mailloux, Phased Array Antenna Handbook. Norwood, MA: Artech House, 1994. [11] W.-M. Zhang, R. P. Hsia, C. Liang, G. Song, C. W. Domier, and N. C. Luhmann, Jr., “Novel low-loss delay line for broadband phased antenna array applications,” IEEE Microwave Guided Wave Lett., vol. 6, pp. 395–397, Nov. 1996. [12] R. W. Ziolkowsky, “Properties of electromagnetic beams generated by ultra-width bandwidth pulse-driven arrays,” IEEE Trans. Antennas Propag., vol. 40, no. 8, pp. 888–905, Aug. 1992. [13] S. Werner, C. Strum, and W. Wiesbeck, “Impulse response of linear UWB antenna arrays and the application to beam steering,” in Proc. Proc. Int. Ultrawideband Conf., Sep. 2005, pp. 275–280. [14] J. S. McLean, H. Foltz, and R. Sutton, “Pattern descriptors for UWB antennas,” IEEE Trans. Antennas Propag., vol. 53, pp. 553–559, Jan. 2005. [15] D. C. Chang, C. H. Liao, and C. H. Wu, “CATR without both reflector edge treatment and RF anechoic chamber,” IEEE Antennas Propag. Mag., vol. 46, no. 4, pp. 27–37, Aug. 2004. [16] D. M. Pozar, Microwave Engineering. New York: Addison-Wesley, 1990, pp. 313–318. [17] S. Sugawara, Y. Maita, K. Adachi, K. Mori, and K. Mizuno, “A mm-wave taper slot antenna with improved radiation pattern,” in IEEE MTT-S Int. Microw. Symp. Dig., 1997, pp. 959–962. [18] H. Sato, K. Sawaya, Y. Wagatsuma, and K. Mizuno, “Broadband FDTD analysis of fermi antenna with narrow width substrate,” in IEEE AP-S Int. Symp., Dig., 2003, vol. 1, pp. 261–264. [19] D. C. Chang, B. H. Zeng, and L. C. Liu, “Modified antipodal Fermi antenna with piecewise-linear approximation and shaped-comb corrugation for ranging applications,” IET Microw. Antennas Propag., vol. 4, no. 3, pp. 399–407, 2010.

Chao-Hsiang Liao (S’09) was born in Taipei, Taiwan. He received the B.S. and M.S. degrees in electrical engineering from Da Yeh University, Changhua, Taiwan, in 2002 and 2005, respectively. He is currently working toward the Ph.D. degree at National Taiwan University, Taipei. From 2002 to 2005, he was with the Wireless Communication Research Center (WCRC), Da Yeh University, where he has been engaged in the analysis and design of reflector antennas, and also the development of the Da Yeh University compact antenna test range (CATR). He has done considerable work on the optimum design of reflector antennas for DBS. His main research interests are design of ultrawideband (UWB) antennas, performance analysis of UWB arrays with beam scan capability, optimum design of reflector antennas for direct broadcast satellite, high resolution microwave imaging for radar applications. Mr. Liao received the Best Student Paper Award from the 2008 International Symposium on Antennas and Propagation (ISAP 2008), the third prize Best Paper Award from the 4th International Conference on Electromagnetic NearField Characterization and Imaging (ICONIC 2009), and the Best Paper Award from the 2010 International Conference on Applications of Electromagnetism and Student Innovation Competition Awards (AEM2C 2010).

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Powen Hsu (M’86–SM’98–F’10) was born in Taipei, Taiwan. He received the B.S. degree in physics from the National Tsing-Hua University, Hsinchu, Taiwan, in 1972, the M.S. degree in physics from the University of Maryland, College Park, in 1976, and the M.S. and Ph.D. degrees in electrical engineering from the University of Southern California, Los Angeles, in 1978 and 1982, respectively. From 1982 to 1984, he was with ITT Gilfillan, Van Nuys, CA, where he was engaged in research and development pertaining to radar antenna systems. In 1984, he joined the faculty of the National Taiwan University, Taipei, Taiwan, where he is currently a Professor with the Electrical Engineering Department. From 1992 to 1995, he was the Department Chairperson there. In August 1997, he established the ninth college, College of Electrical Engineering and Computer Science, at the National Taiwan University, and served as the first Dean of the College until 2003. His current research interests include the design and analysis of slot antennas, microstrip antennas, and microwave and millimeterwave integrated circuits. Prof. Hsu is a Fellow of IEEE and a Distinguished Professor of National Taiwan University.

Dau-Chyrh Chang (F’04) was born in Taiwan. He obtained the B.S. and M.S. degrees from Chung-Cheng Institute of Technology, in 1970 and 1973, respectively, and the Ph.D. degree in electrical engineering from the University of Southern California, in 1981. He spent 17 years (1981–1998) in antenna R&D at CSIST (Chung Shan Institute of Science and Technology). For 12 of these years (1986–1998), he served as the Director of the Antenna Section. During his employment at CSIST, he developed the reflector antennas, phased array antennas, slot array antennas, and communication antennas. He also established the new and innovative antenna test ranges (near field range in 1986, and compact range in 1987) in Taiwan. In 1998, he left his post as the Director of the Antenna Section to become the Dean of the Engineering School at Da-Yeh University. He had been invited to be the Dean of College of Electrical and Communication Engineering, Oriental Institute of Technology, in 2006, where he is also the Chair Professor. His achievements include (i) Developments of hybrid near field and compact ranges for various kinds of antenna testing; (ii) applications of impulse time domain system; (iii) developments of various kinds of UWB antenna and smart antennas for communication system. Prof. Chang established the IEEE AP-S Taipei Chapter and was the first Chair in 2001, Chair of the IEEE MTT-S Taipei Chapter (2000–2002), and President of Chinese Microwave Association (2000–2002). He has been the General Chair of CSTRWC2001, CSTRWC2008, ISAP2008, ICONIC2009, and AEM2C2010. He is a Fellow of IEEE, IET, and EMA.

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Method of Moments Analysis of Slotted Substrate Integrated Waveguide Arrays Emilio Arnieri, Member, IEEE, and Giandomenico Amendola, Member, IEEE

Abstract—A method of moments analysis is described for substrate integrated waveguide slot arrays. The array structure is modeled as a parallel plate waveguide on which via holes and slots are created. The field into the waveguide is expressed by the sum of the parallel plate’s contribution and the field scattered by the metallic through holes, with both contributions expressed by means of an expansion in terms of vectorial cylindrical eigenfunctions. The slots are modeled as unknown equivalent magnetic current distributions which are found by solving an integral equation derived from the continuity of the field on the slot surface. Antenna arrays already presented in the literature will be analyzed, and the results compared with HFSS simulations. It will be shown that the proposed method is efficient and gives results in excellent agreement with the most common simulation tools. Index Terms—Dyadic Green’s functions, method of moments, slot arrays, substrate integrated circuits (SICs), substrate integrated waveguide (SIW).

I. INTRODUCTION

S

UBSTRATE-integrated waveguides (SIW), also called post-walled or laminated waveguides, have been introduced recently to provide a means to create waveguide based, high frequency circuits at low cost [1]. Many passive and active SIW devices have been presented in the recent literature [2], [3]. SIW-based antennas have also been investigated, and arrays of slots have been shown to be the most successful designs [4], [5]. Besides reduced costs, SIW technology allows for the easy integration of active devices, enables sophisticated packaging technology, and permits the integration of complex beam-forming networks and antennas on the same board [6], [7]. In the past, slot array antennas have been extensively investigated by engineers and a considerable number of studies and implementations, dealing with slot arrays of different sizes and characteristics, have been published. The analysis of large arrays of conventional waveguides by means of the method of moments can be made very efficient, in particular thanks to the availability of effective representations of the waveguide’s Green’s function [8]. With SIW structures, on the other hand, due to the presence of the vias fences which form the lateral walls of the waveguides, a finite difference or finite element type of solution needs to be employed, which can be both time-consuming and demanding of memory. An approximate Manuscript received March 08, 2010; revised August 04, 2010; accepted August 19, 2010. Date of publication January 28, 2011; date of current version April 06, 2011. The authors are with the Dipartimento di Elettronica, Informatica e Sistemistica, Universita’ della Calabria, 87936 Rende (CS), Italy (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2011.2109356

Fig. 1. SIW structure with slots.

method can be found by considering metallic waveguides that are equivalent to the substrate-integrated ones and resorting to the techniques adopted for conventional waveguides [9]. This approach is computationally efficient but it does not ensure the correctness of the final design results, and it does not permit the simulation of the beam-forming network. Recently in [10] the authors presented a rigorous and efficient full-wave analysis of nonradiating SIW devices based on the dyadic Green’s function technique [11], [12]. In this present paper the same approach will be used to efficiently solve the problem of large SIW slot-array including the beam forming network. In [13], the authors presented preliminary results which were limited to the analysis of arrays of small size without beam forming network. In what follows, the field radiated by a magnetic-current source inside the waveguide is computed by considering the Green’s function of the parallel plate, expanded in terms of vector eigenfunctions, and including the field scattered by the metallic posts. The slots are represented by unknown equivalent magnetic currents which radiate inside and outside the SIW structure and that are determined using the method of moments. To validate the proposed method SIW arrays having various sizes of inclined slots were analyzed. A comparison with published results and finite element simulations is presented, which shows how the method is both accurate and efficient. II. SIW DYADIC GREEN’S FUNCTION As done previously in [10], the integrated circuit is modeled as a cavity delimited by metallic plates on top and bottom and fences of via holes, and excited through coaxial or waveguide ports (Fig. 1). The magnetic field into the post walled structure is expressed as

0018-926X/$26.00 © 2011 IEEE

(1)

ARNIERI AND AMENDOLA: METHOD OF MOMENTS ANALYSIS OF SLOTTED SIW ARRAYS

where is the source current distribution defined in the and is the dyadic Green’s function volume of the parallel-plate waveguide which has the following form:

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A. Integral Equation Using the field-equivalence principle, in the equation the slots are replaced by prefect conductor having an equivalent magflowing on the inner surface and on the netic current outer surface. The slots considered are narrow slots, so only the transverse electric field component – across the slots – needs to be taken into account, resulting in magnetic currents that are directed along the slots axis. In the equation the continuity of the tangential magnetic field across the surface of the slots requires that for each slot one have

(2) with

,

, and

for for (3) represents the field scattered by the The term through holes when the incident field is taken to be the field in the parradiated by a magnetic current source allel-plate waveguide without cylinders. is expressed as the series of outgoing cylindrical waves centered on the metallic spans cylinders, see (4) at the bottom of the page. In (4), the number of verticals modes, spans the number of radial modes, spans the number of cylinders, and is the position of the center of the cylinder . The scattering coefficients, (TM mode) and (TE mode), are evaluated by taking the total tangential electric field on the cylinders surface to be null. This results in a system of equations to be inverted, with the scattering coefficients as the unknowns [10]. It can be seen that the coefficients depend on the order of the modes along and that for each a scattering problem has to be solved [10]. If TM modes and TE modes are used, then matrix inversions need to be performed. III. ANALYSIS OF RADIATING SLOTS The analysis of the radiating slots is carried out using the moments method. The slots are modelled as equivalent magnetic currents expressed as a sum of subsectional sinusoidal basis functions having unknown coefficients. The continuity of the tangential component of the magnetic field is imposed on the surface of the slots, and an integral equation is thus obtained. The unknown coefficients are computed with the application of the Galerkin Moment Method. In the present case the thickness of the conducting plates is not taken into account, and the slots are considered to be infinitely thin.

(5) where is the surface of the th aperture, is the slot-normal is the incident magnetic field in the parallel plate versor, waveguide with metallic post when all of the slots are short-cirand , respectively, are the internal magnetic cuited; and the external magnetic field field due to the slot currents due to the slot currents , with spanning the slots. The inner and outer magnetic fields can be expressed as follows:

(6) is the dyadic Green’s function of the parallel where plate waveguide when all the slots are short-circuited, given in is the source-current distribution defined in the (2), volume and is the field scattered by the metallic posts to the current , given in (4). B. Formulation of Moment Method The equivalent magnetic currents for a slot (Fig. 2) are

and elsewhere (7)

(4)

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probe, with inner radius and outer radius , is modeled as a ring of magnetic current given by (10)

Fig. 2. Shape of the basis functions.

where is the center of the th basis function on the slot , is the number of slots, are the unknown expansion coeffiused to reprecients, spans the number of basis functions sent the current on the slot. Substituting (6) and (7) in (5) and applying Galerkin’s method, the integral equations can be converted into a system linear equations with unknowns of

where is the voltage between the inner and outer conductors, is the position of the center of the inner conductor, and is a point inside the annular region where the magnetic current is flowing. By substituting (10) into the final equation in (6), the following expression is obtained [14], see (11) at the bottom of is the number of cylinders. the page, where C. Treatment of External Region In the integral relevant to the exterior region, the Green’s function has the following form: (12)

(8) with

where spans over the basis functions, while and . number of slots, The quantities in (8) are given by

span the

. in (9) can be computed numerically using Gauss (two separate slots), while the singular quadrature for , can be evaluated by using a case, which will occur for suitable change of variables, yielding the following expression [8], [14]:

(9)

where . When the integrals in (9) are calculated and the equation system solved, the unknown equivalent magnetic currents are found. is the magnetic field radiIn the expressions in (9), ated by a port source into the structure where slots have been metalized. In [10], coaxial and waveguide ports were used, but the present paper considers only the coaxial case. The coaxial

(13)

D. Treatment of Internal Region From the expressions in (9) it is seen that the interior region’s contribution can be decomposed in two parts: the contribution

(11)

ARNIERI AND AMENDOLA: METHOD OF MOMENTS ANALYSIS OF SLOTTED SIW ARRAYS

Fig. 3. Accuracy of Y

(int) versus number of vertical modes.

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Fig. 4. Accuracy of Y

(int) versus number of vertical modes.

of the parallel plates and the contribution coming from the field scattered by the conducting cylinders. The parallel plates’ contribution can be evaluated numerically (two separate slots). using Gauss quadrature for can be solved using The singular case that will occur for the following relation:

(14) is easily integrated by adding and subThe singularity tracting the small argument expression of the Hankel function and using integration by parts. The scattered field contribution is numerically evaluated using (4).

Fig. 5. Simulated S

of 1

2 10 array as in [16].

V. VERIFICATION OF RESULTS IV. CONVERGENCE ANALYSIS As shown in the expressions in (9), the elements of the admittance matrix are the sums of two terms: the inner term is relevant to the parallel plate region and the external one is pertinent to the outer region. The contribution of the inner term depends on the number of modes used in the expansion of the Green’s function and, hence, an analysis of the convergence of the number of modes along is needed. Fig. 3 shows the percentage error for a slot self-admittance, ), as when only one expansion function is considered ( a function of the number of vertical modes for a substrate with and . It is seen that thickness spanning between even for thick substrates a limited number of vertical modes is sufficient. Fig. 4 shows the percentage error for mutual admittance between two slots with only one expansion function ), as a function of the number of vertical modes when ( the sources are placed at a decreasing distance normalized with respect to the guide wavelength. In this case, the substrate thick. In this case as well, convergence is ness is set to achieved using just a small number of modes.

The theory presented in the preceding sections was implemented in a MATLAB code. Different structures were analyzed in order to confirm the validity of the proposed model. The results were compared with published results and with finite-element simulations carried out using commercially available software (Ansoft HFSS) [15]. A. Input Impedance and Radiation Patterns Firstly, the SIW-based array antenna proposed by Zhang and Lu [16] was considered. The array, printed on a substrate with and thickness , has slant radiation slots and reflection-canceling vias and an operating frequency of 24 GHz. In Fig. 5 the reflection coefficient of the array antenna is shown, and compared with results obtained from HFSS. Fig. 6 shows the total field patterns in the elevation plane and the azimuth plane at 24 GHz. A 10-element array was used to design dual-polarized arrays with 2 10, 8 10 elements. In and Figs. 7 and 8 are shown computed S parameters and patterns, taken at as in [5], for the 2 10 array, compared with results obtained from HFSS. Figs. 10 and 11. show

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Fig. 9. Top view of the 8

Fig. 6. Simulated gain pattern of 1

2 10 array as in [16].

Fig. 10. Simulated S

Fig. 7. Simulated S

and S

2 10 slot array antenna.

of dual-polarized 2

and S

of 8

2 10 array.

2 10 array.

Fig. 11. Simulated gain pattern of 8

2 10 array with feeding network.

from the pattern given by HFSS for the 2 10 case is compared to the pattern computed using the present method. In this last case the power divider is not used and each waveguide is fed through a coaxial port. As can be seen, the absence of the power divider brings the cross-polar pattern back down to a level close to the ideal case. Fig. 8. Simulated gain pattern of dual-polarized 2

2 10 array.

the same quantities for the 8 10 case for which a by four power divider, shown in Fig. 9, has also been included in the simulation. In this case HFSS was unable to simulate the structure with the computer resources available. A high cross-polar level is observed, which is due to the presence of the power divider. This fact is clearly shown in Fig. 12, where the array factor computed

B. CPU Time and Storage Requirements As has been explained in the previous sections, the method described in the present paper requires two matrices to be filled and inverted: one is necessary in order to evaluate the field scattered by the metallic vias holes and the other one to compute the unknown currents with which the slots have been modeled. The size of the two matrices, and consequently the CPU processing time and memory requirements, depend on the number

ARNIERI AND AMENDOLA: METHOD OF MOMENTS ANALYSIS OF SLOTTED SIW ARRAYS

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TABLE I CPU TIME (INTEL QUAD-CORE XEON @2.33 GHZ. 16 GB RAM)

TABLE II MEMORY REQUIREMENTS

Fig. 12. Simulated gain pattern of 8

2 10 array without feeding network. TABLE III CPU TIME (INTEL QUAD-CORE XEON @2.33 GHZ. 16 GB RAM)

of cylinders , on the number of slots and on the subof modes along strate thickness, which dictates the number that will be used. The number of matrix inversions required to evaluate the scattering from the ensemble of metallic posts is , where is the number of ports. In the arrays analyzed in this paper, modes along higher than the first one are cut off, as normally happens in SIW structures. Under these conditions the field scattered by any single metallic post is a combination of rapidly decaying evanescent waves. In this case, only near interactions between posts (for distances less than one wavelength) need to be taken into account. As a consequence, the matrices to be inverted in order to find the field scattered by the metallic through holes have terms, corresponding to interactions between cylinders placed at a relative distance larger than a wavelength, that can be set to zero. These matrices turn out to be very sparse, and the coefficients of the expansion (4) can be found with an iterative solver. Taking these considerations into account, the following schema was adopted in the simulations: 1. The matrix that corresponds to the first (propagating) mode ) is a full matrix and it is inverted using a direct ( solver. The inverse of the matrix is stored, and the coefficients in (4) are computed by multiplying the inverse by the right-hand sides. 2. Coefficients in (4) corresponding to modes in cutoff ( ) are computed with an iterative solver which is run times. 3. The matrix corresponding to system (8) is of a smaller size and is inverted using a direct solver. Tables I and II report the CPU processing time and storage used in simulating the arrays. For the sake of completeness, the 4 10 case has been included in the table as well. In all the cases presented, three basis functions were used on each slot. As can be observed, a considerable savings in time and storage capacity is achieved with respect to HFSS, making it possible to simulate larger structure with limited computing resources. Further improvements in performance may be achieved by exploiting the multiprocessing capabilities of the multicore CPUs currently available on the market. In the present case, no attempt has been made to create truly parallel routines, but the MATLAB program was reorganized into independent serial

segments of code that could be executed as parallel tasks. In principle these tasks could run on multiprocessor machines, but even executing the code on a single multicore CPU yields considerable performance improvements. In Table III are shown the CPU times for the two largest structures considered in Table I, simulated by dispatching the tasks to four different cores. As can be seen, CPU times decrease to 50% in both cases when all four cores are used. VI. CONCLUSION In this paper, a semianalytical method for analyzing substrate-integrated waveguides with radiating slots is presented. It is shown that the results compare favorably with those obtained with the most commonly used software. The proposed method is shown to be resource-efficient, and can be applied to the analysis of structures of large extension. REFERENCES [1] D. Deslandes and K. Wu, “Integrated microstrip and rectangular waveguide in planar form,” IEEE Microw. Wireless Compon. Lett., vol. 11, pp. 68–70, Feb. 2001. [2] M. Bozzi, L. Perregrini, K. Wu, and P. Arcioni, “Current and future research trends in substrate integrated waveguide technology,” Radio Eng., vol. 18, no. 2, pp. 201–209, Jun. 2009. [3] C. Zhong, J. Xu, Z. Yu, and Y. Zhu, “Ka-band substrate integrated waveguide Gunn oscillator,” IEEE Microw. Wireless Compon. Lett., vol. 18, pp. 461–463, Jul. 2008. [4] W.-K. 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. AP-46, pp. 625–630, May 1998. [5] S. Park, Y. Okajima, J. Hirokawa, and M. Ando, “A slotted post-wall waveguide array with interdigital structure for 45 linear and dual polarization,” IEEE Trans. Antennas Propag., vol. 53, pp. 2865–2871, Sep. 2005. [6] T. Djerafi, N. J. G. Fonseca, and K. Wu, “Planar Ku-band 4 4 nolen matrix in SIW technology,” IEEE Trans. Microw. Theory Techn., vol. 58, pp. 259–266, Feb. 2010. [7] P. Chen, W. Hong, Z. Kuai, J. Xu, H. Wang, J. Chen, H. Tang, J. Zhou, and K. Wu, “A multibeam antenna based on substrate integrated waveguide technology for MIMO wireless communications,” IEEE Trans. Antennas Propag., vol. 57, pp. 1813–1821, Jun. 2009.

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[8] G.-X. Fan and J.-M. Jin, “Scattering from a large planar slotted waveguide array antenna,” Electromagn., vol. 19, no. 1, pp. 109–130, 1999. [9] L. Yan, W. Hong, G. Hua, J. Chen, K. Wu, and T. J. Cui, “Simulation and experiment on SIW slot array antennas,” IEEE Microw. Wireless Compon. Lett., vol. 14, pp. 446–448, Sep. 2004. [10] E. Arnieri and G. Amendola, “Analysis of substrate integrated waveguide structures based on the parallel-plate waveguide Green’s function,” IEEE Trans. Microw. Theory Tech., vol. 56, pp. 1615–1623, 2008. [11] H. Chen, Q. Li, L. Tsang, C.-C. Huang, and V. Jandhyala, “Analysis of a large number of vias and differential signaling in multilayered structures,” IEEE Trans. Microw. Theory Techn., vol. 51, pp. 818–829, Mar. 2003. [12] L. Tsang et al., Scattering of Electromagnetic Waves: Numerical Simulations. New York: Wiley InterSci., 2001. [13] E. Arnieri, G. Amendola, and L. Boccia, “Analysis of integrated waveguide slot array antennas,” presented at the Eur. Microw. Conf. (EuMC), Sep. 29-Oct. 1 2009. [14] E. Arnieri, “Full wave analysis of substrate integrated circuits,” Ph.D. dissertation, Univ. Mediterranea Reggio Calabria, Calabria, Italy, 2007. [15] HFSS. Pittsburgh, PA, Ansoft Corp., vol. 10. [16] Q. Zhang and Y. Lu, “45 linearly polarized substrate integrated waveguide-fed slot array antennas,” presented at the ICMMT, 2008. [17] M. Zhang and Z. Wu, “The application of MOM and EECS on EM scattering from slot antennas,” Progr. Electromagn. Res. (PIER), vol. 21, pp. 307–318, 1999.

Emilio Arnieri (S’05–M’07) was born in Cosenza, Italy, in 1977. He received the degree (with honors) in information technology engineering from the University of Calabria, Rende, Italy, in 2003 and the Ph.D. degree in electronic engineering from the University “Mediterranea” of Reggio Calabria, in 2007. Currently, he is a Research Engineer with the Dipartimento di Elettronica, Informatica e Sistemistica, University of Calabria. His main research activities concern the development of microstrip antennas and millimeter-wave components, and in particular, the development of numerical methods for the electromagnetic modeling of microwave and millimeter-wave circuits (substrate integrated circuits, slotted substrate integrated waveguide arrays, and substrate integrated waveguide resonators).

Giandomenico Amendola (M’96) received the degree in electrical engineering from the University of Calabria, Rende, Italy. From 1988 to 1992, he was a Research Fellow with the Proton Synchrotron Division, European Center for Nuclear Research (CERN), Geneva, Switzerland. He is currently an Associate Professor with the Dipartimento di Elettronica, Informatica e Sistemistica, University of Calabria. His main research interests are in the areas of antennas and of microwave- and millimeter-wave circuits.

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Substrate-Integrated Cavity-Backed Patch Arrays: A Low-Cost Approach for Bandwidth Enhancement Mohamed H. Awida, Student Member, IEEE, Shady H. Suleiman, Senior Member, IEEE, and Aly E. Fathy, Fellow, IEEE

Abstract—The substrate-integrated waveguide (SIW) technology is utilized as an alternative low-cost approach in fabricating cavity-backed patch antennas. The proposed antenna arrays combine the attractive features of the conventional metalized cavity-backed patch arrays like surface wave suppression, high radiation efficiency, and enhanced bandwidth, yet provide a low manufacturing cost. A previously developed design of a 2 2 SIW cavity-backed microstrip patch sub-array is extended here and used as a basic building block to attain larger arrays of 2 4, 4 4, and 8 8 elements. The fabricated arrays have been measured and demonstrate good agreement with their simulated performance. The design and performance of these arrays are compared to other conventional bandwidth enhancement techniques, which prove SIW as a viable alternative. Index Terms—Cavity-backed, microstrip array, substrate integrated waveguide.

I. INTRODUCTION

M

ICROSTRIP patch antennas have drawn the attention of antenna engineers since the 1970s due to their attractive features of being low profile, compact size, light weight, and amenable to low-cost PCB (Printed Circuit Board) fabrication processes. However, patch elements are basically resonating at a single frequency, typically have less than 2% bandwidth [1]–[4], which is a major drawback in their utilization. Multiple techniques have been proposed to address the aforementioned bandwidth limitation. The simplest one is to use a thick substrate that potentially could enhance the bandwidth to 20% upon using 0.2 -thick substrate as discussed in [5]. However that would degrade the antenna efficiency due to undesired surface wave excitation [3]. Therefore, it is preferred to use either patches that are built with suspended substrates [6], [7], or ones that are backed by cavities [8], [9], or even a combination of both [10] to achieve a wider operating bandwidth while minimizing surface wave excitation. But, the fabrication of the cavity-backed patches is not common, as it would require the integration of metal cavities in the back—demanding two fabrication processes. The first process is a conventional PCB process that is used to print the microstrip patch layer and the second is a CNC (Computed Numerically Controlled) machining or metal Manuscript received February 01, 2010; revised August 16, 2010; accepted September 28, 2010. Date of publication January 31, 2011; date of current version April 06, 2011. M. H. Awida and A. E. Fathy are with the Department of Electrical Engineering, University of Tennessee at Knoxville, Knoxville, TN 37996 USA (e-mail: [email protected]). S. H. Suleiman is with the Winegard Company, Burlington, IA 52601 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.2011.2109681

casting process to fabricate the metalized cavities. The previously prescribed fabrication scenario increases the total fabrication cost and complicates the structure assemblage. Nevertheless, cavity-backed patches exhibit superior performance due to their significant surface wave suppression, reduced coupling, better matching, and wider scan performance in an infinite array environment [8], [9]. Recently, SIW was suggested as an alternative technology to facilitate the low-cost implementation of waveguide-like components using the standard PCB fabrication processes [11]. In principle, the waveguide metal walls could be emulated using via-holes that are properly spaced at approximately one twentieth of a guided wavelength, which is similar to the previously suggested laminated [12] and post-wall waveguides [13]. Several waveguide-like components were successfully attained [14] and the technology has been also used for implementing lowprofile slotted array antennas [13], [15]–[17]. Along these lines, we have recently developed an SIW 2 2 cavity-backed microstrip patch antenna. In our implementation, the cavities were emulated using a circular array of via holes and the patches were fed using microstrip line feed networks. Over 70% in efficiency and 9% in fractional bandwidth have been demonstrated [18]. In this paper, the 2 2 sub-array design described in [18] is utilized as a basic building block for implementing larger arrays of 2 4, 4 4, and 8 8 elements. Performance is thoroughly compared to other conventional techniques for bandwidth enhancement including thick microstrip substrates [5], suspended substrates [6], [7], metalized cavity-backed patches [8], [9], and cavity-backed suspended patches [10]. The paper is organized as follows: in Section II, the SIW cavity-backed antenna configuration is described. The design approach is covered in Section III with corresponding design charts. In Section IV, the performance of the proposed SIW cavity-backed antennas are evaluated against alternative common bandwidth enhancement techniques. The measurements carried out for the various realized prototypes are explored in Section V. Conclusions of our investigation are described in Section VI. II. ANTENNA CONFIGURATION Fig. 1 illustrates the proposed SIW cavity-backed antenna for an 8 8 array with its constituting layers spaced apart. The proposed array basically consists of a stack of two substrates: the top one, a “microstrip substrate” where an array of patch elements are printed, and the bottom one, a “cavity substrate” where many via holes spaced along circular openings are drilled and then through-plated, constituting the SIW circular cavities

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Fig. 1. 3D model of the SIW cavity-backed patch 8 stituting layers spaced apart.

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2 8 array showing the con-

Fig. 3. Basic one-to-four probe-fed microstrip feed network. (a) 3D model. (b) Simulated loss and efficiency performance vs. the normalized substrate thickness (simulated at Ku-band for " : , and  : ).

=22

tan = 0 0009

and thickness ). The circular opening has a radius constant , and these via holes are indented from its edge by a distance . equal to B. Microstrip Binary Feed Network Fig. 2. Proposed SIW cavity-backed patch single element (a) Top view, (b) side view.

backing the patches. The bottom layer of the microstrip substrate and the top layer of the cavity substrate have common circular openings underneath the patches. The 64 elements of the array are subdivided into sixteen 2 2 sub-arrays. An SMA of solder cup contact is employed to launch the microwave signal while a microstrip binary feed network is utilized to distribute the power to the patch elements. A. Array Element Fig. 2 depicts the top and side views of a single element of the array. Each element is a trimmed square patch of side length that is printed on a microstrip substrate (of dielectric constant and thickness ). Meanwhile, to emulate the cavity wall, eighteen via holes ) spaced along the common (each of radius circular opening are added in the cavity substrate (of dielectric

A simple microstrip binary feed network with quarter wave transformers is utilized to direct the signal from the central feed to the patches. The central feed is an integrated 50 coaxial probe feed topology similar to that proposed in [19] where many via holes were implemented to emulate the outer wall/shield of the coaxial feed probe. This structure provides a smooth transition and minimal unwanted feed radiation loss. For illustration, 3-D model of the one-to-four microstrip divider network employed in the 2 2 sub-array is shown in Fig. 3(a). III. DESIGN APPROACH The design of the SIW cavity-backed patch antenna involves the selection of both substrates—thickness and dielectric constant, and the determination of the patch and cavity dimensions. In this section, we provide brief design guidelines. A. Microstrip Substrate Properties and deThe microstrip substrate properties thickness termine the loss performance of the feed network. For instance, the insertion loss of the one-to-four divider shown in Fig. 3(a)

AWIDA et al.: SUBSTRATE-INTEGRATED CAVITY-BACKED PATCH ARRAYS: A LOW-COST APPROACH FOR BANDWIDTH ENHANCEMENT

Fig. 4. Design chart showing the fractional bandwidth and the normalized : a” patch side length a= vs. normalized cavity height adopting the “R design rule (assuming h :  ," : ,d :  ,L a= ).

= 0 016

=22 08

= 0 84 = 5

was calculated at Ku-band as a function of the substrate thickness, as shown in Fig. 3(b). In the previous calculation, the line impedances of the divider were constantly adjusted based on the utilized substrate thickness to keep a good input match. It is clear from Fig. 3(b) that increasing the microstrip substrate would lead to an increased insertion thickness beyond 0.02 loss for the divider. Even though the conductor loss would decrease with increasing the substrate thickness, but the additional losses of the surface waves excitation would still be more pronounced and would cause significant efficiency degradation of the divider performance. As an example, using a thin substrate would imply a 0.3 dB insertion loss for the divider. of 0.016 But, this feed insertion loss would increase to 1 dB if a four times thicker substrate is employed. Therefore, it is imperative to use relatively thin, low loss dielectric constant microstrip substrates to minimize the surface wave losses of the feed network; thus maximizing the antenna radiation efficiency. B. Cavity Substrate Selection is The selection of the cavity substrate—thickness and determined by the required fractional bandwidth of the antenna. The bandwidth is inversely proportional to the square root of the dielectric constant. Subsequently, using high dielectric constant substrates should also be avoided here as they tend to trap the energy which increases the quality factor of the patch, decreases the bandwidth, and lowers the antenna radiation efficiency. This effect has been reported before for conventional microstrip patches [3]. Fig. 4 shows the simulated fractional bandwidth of normalized to its center bandwidth ( frequency) of the basic 2 2 sub-array versus the cavity sub, , for a strate thickness assuming family of of 2.2, 3.2 and 4.5. It is clear that by increasing the cavity substrate thickness, the fractional bandwidth of the antenna can be enhanced. For instance, upon using a low diand height , electric constant substrate the fractional bandwidth has increased to approximately 13% compared to only 2% for the conventional microstrip antenna ). For even further bandwithout backing cavities (i.e., width enhancement, stacking patches at different layers could be

2 = a 5 = 0 066

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Fig. 5. Simulated reflection coefficient of the 2 2 array with different cavity size varied from 5.7 mm to 6.3 mm with a step of 0.3 mm (assuming h :  ," : ,d :  ,L = ,h :  ," : ).

0 016

=22 08

=22

=

used [20], [21], where over 18% fractional bandwidth has been achieved in [21]. Even though stacking more layers is compatible with SIW, it can render very thick overall antenna profile. C. Cavity and Patch Dimensions Selection The previous HFSS parametric study was continued to select the optimum cavity and patch dimensions. A 2 2 basic sub-array was numerically simulated assuming , , , and . In this parametric study, the patch side length was fixed at 7.5 mm, while the cavity radius was varied from 5.7 mm to 6.3 mm in 0.3 mm steps. The corresponding reflection coefficient performance is shown in Fig. 5 and indicates an adequate matched performance for a cavity with a radius of 6.3 mm-corresponding (where is the cavity radius and is the patch to side length). Hence, we adopted this ratio and developed a full design chart for the side length selection as well, as shown in Fig. 4. The design chart could be used as follow. • Select the cavity substrate thickness based on the required bandwidth; • Accordingly, select the patch resonant side length ; • The other design parameters can be calculated from and . For attaining larger arrays, the previously designed 2 2 subarray module is replicated along the and/or dimensions. Subsequently, 2 4, 4 4, and 8 8 SIW arrays were designed to operate at the Ku-band with a fractional bandwidth better than 9%. Their final design parameters are: microstrip and , cavity substrate substrate of and , patch side length , , array spacing of , and a cavity and radius of . IV. PERFORMANCE EVALUATION COMPARED TO CONVENTIONAL BANDWIDTH ENHANCEMENT TECHNIQUES A thorough investigation using HFSS [22] has been carried out to demonstrate the features of the proposed SIW cavitybacked structures relative to other structures commonly used to enhance the inherent patch limited bandwidth.

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TABLE I PERFORMANCE COMPARISON OF THE DIFFERENT SINGLE ELEMENTS AT 12.5 GHz

Fig. 6. Metalized cavity-backed suspended patch single element.

A. Single-Element Comparison Fig. 6 shows the general case of a suspended substrate patch backed by a metalized cavity. Obviously, this general case reduces to the special case of a microstrip structure, upon setting and , or to the suspended structure, upon set, or finally to the metalized cavity-backed structure ting . In our comparison, we kept the total height upon setting and the aperture size the same for the different investigated structures and equals to that of the SIW single el, aperture ). This ement ( comparison criterion is based on the fact that the antenna bandwidth is generally proportional to the volume of the microstrip antenna [2]. The simulated performance at 12.5 GHz of these various structures is shown in Table I indicating their fractional bandwidth, realized gain, and achieved aperture efficiency. We can infer the following from Table I. • As expected, the conventional microstrip patch built on a thick substrate has a degraded gain performance due to significant surface wave losses; • Upon using the suspended patch topology a substantial increase in bandwidth, gain, and efficiency is achieved;

• The highest gain (8.3 dBi) and aperture efficiency (86%) are achieved upon using the metalized-cavity, but with a slight decrease in the associated fractional bandwidth compared to the suspended case; • The largest fractional bandwidth of 8.5% is attained using the hybrid cavity-backed suspended patch, however with a realized gain of 7.7 dBi—lower than the metalized cavitybacked patch case; • Finally, the proposed SIW cavity-backed patch exhibits a gain of 8.0 dB corresponding to an aperture efficiency of 80%, slightly short of that of the metalized cavity-backed case but better than the other suspended or microstrip cases. The fractional bandwidth is 6.6% which is also, as expected, slightly lower than the metalized cavity-backed and suspended cases. In the aforementioned simulations, we have included the dielectric loss for the different single element cases, as we aswhich amounts to less sumed a loss tangent of than 0.1 dB, i.e., insignificant loss contribution. Clearly, the proposed SIW single element is far superior to the conventional microstrip patch structure as the implemented SIW cavity efficiently suppresses the surface waves, similar to the effect of band-gap structures [23]–[25]. Meanwhile, the SIW structure is simple to fabricate compared to the metalized cavitybacked case and is also easy to assemble compared to the suspended structures cases. B. Feed Network We have also compared the losses of the feed lines for the five topologies under investigation. The feed network would be either based on microstrip lines, for the cases of: microstrip, metalized cavity-backed, and SIW cavity-backed; or suspended lines for the cases of: suspended and cavity-backed suspended patches. Hence as a first step to develop general design rules we evaluated the insertion loss of both the microstrip and suspended lines as a function of their heights above the ground plane. Generally, it is preferable to use relatively narrow feed lines in order to decrease the mutual coupling between the microstrip feed network and the radiating elements, however increasing the height of these substrates would typically lead to unacceptable

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Fig. 8. Comparison between the SIW cavity-backed, metalized cavity-backed :  , and suspended cavity-backed 2 2 sub-arrays (assuming h " : ,d :  ,L a= ) as far as the fractional bandwidth and the normalized resonant side length vs. normalized cavity height (for the suspended cavity case, the height is the cavity height plus the air layer thickness).

= 22  08

Fig. 7. Feed loss comparison between microstrip and suspended lines. (a) Microstrip line. (b) Suspended line. (c) Simulated insertion loss and impedance h for the microstrip line case level vs. the normalized height h = (h h h for the suspended line case with h : ). and h

= +

=

= 0 38 mm

2 = 5

= 0 016

TABLE II SIMULATED REALIZED GAIN (dBi) OF THE DIFFERENT ARRAYS AT 12.5 GHz

wide lines if the impedance levels of the feed network lines were fixed. Alternatively, in our comparative study, we kept the lines’ widths constant and presented the corresponding lines’ impedstrip width, as indicated in Fig. 7, ances for the for both the microstrip and suspended structures cases. As expected, the suspended lines have better performance when compared to the microstrip lines due to their lower conductor and dielectric losses [26]. However, both the microstrip and suspended lines have significant losses upon using thicker substrates (i.e., ), as shown in Fig. 7(c), due to their associated surface wave losses. From that viewpoint, the cavity-backed patch structures proved to be advantageous over the thick substrate or suspended substrate topologies as the feed microstrip substrate can be kept relatively thin while the cavity height could be changed independently to achieve the required bandwidth. We also conclude that the feed losses of a thick microstrip are significantly worse than the other case and structures followed by the suspended case then the cavity-backed suspended patch case, as it has a smaller , then finally the SIW height above the ground . and metalized cavity-backed structures C. 2

2 Sub-Array

Fig. 8 depicts a comparison of the fractional bandwidth and normalized resonant patch side length between the 2 2 SIW ) and the corresponding metsub-array (assuming alized cavity-backed, and the suspended cavity-backed cases. Again, the SIW sub-array has a relatively narrow fractional bandwidth compared to the other two cases while the suspended cavity-backed patches case has the widest bandwidth which is expected from the single-element performance. Additionally, the SIW structure has a smaller patch size because of the miniaturization effect of the dielectric substrate loading [27].

Fig. 9. Calculated aperture efficiency versus frequency of the different array structures.

D. Larger Arrays Larger arrays of 4 4 and 8 8 sizes were numerically simulated taking into account their associated feed network losses. Table II compares the realized gain of the different five topologies for the different sized Ku-band arrays. Their corresponding aperture efficiency is plotted versus the array size in Fig. 9. Obviously, the SIW cavity-backed antennas have comparable performance to the cavity-backed suspended substrate arrays

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Fig. 11. Measured reflection coefficient of the different SIW cavity-backed patch arrays.

Fig. 10. Picture of the fabricated SIW cavity backed patch arrays. (a) 2 array. (b) 2 4 array. (c) 4 4 array. (d) 8 8 array.

2

2

2

22

while the cavity-backed patches with metallic cavities have the best performance, as shown in Fig. 9, delivering the highest aperture efficiency. SIW cavity-backed arrays come next with an approximately 8% reduction in efficiency for the same antenna height/thickness of 1.96 mm. Suspended cavity-backed arrays are close to the SIW arrays in efficiency with approximately 2% reduction. Clearly, the suspended arrays and the thick microstrip arrays are far inferior to the other structures with the aperture efficiency degradation is worse for large arrays (i.e., larger than 8 elements) as the lines of the feed network get longer. V. EXPERIMENTAL RESULTS Based on previously designed 2 2 sub-array [18], 2 4, 4 4, and 8 8 arrays have been fabricated, and are shown in Fig. 10. Rogers RT/duroid 5880 substrate with relative dielectric constant of 2.2, loss tangent of 0.0009 and thickness of 0.38 mm was utilized for the microstrip substrate while the same substrate, however with a thickness of 1.58 mm was used for the cavity substrate in all designs. Standard solder cup SMA connector was employed to launch the microwave signal to the feed network in each case. The two substrates were stacked up by soldering the cavity holes to the top microstrip substrate. Proper alignment and stacking of the two substrates are imperative to achieve the required performance.

Fig. 12. Normalized radiation pattern of the 2 (a) H-Plane. (b) E-Plane.

2 4 array measured at 12.5 GHz.

A. Reflection Coefficient Performance and Fractional Bandwidth The various fabricated arrays were tested using an Agilent E86386 network analyzer to inspect their reflection coefficient performance. Fig. 11 shows the measured reflection response of the different sized arrays. The various arrays exhibit slightly over 10% fractional bandwidth.

AWIDA et al.: SUBSTRATE-INTEGRATED CAVITY-BACKED PATCH ARRAYS: A LOW-COST APPROACH FOR BANDWIDTH ENHANCEMENT

Fig. 13. Normalized radiation pattern of the 4 (a) H-Plane. (b) E-Plane.

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2 4 array measured at 12.5 GHz. Fig. 14. Normalized radiation pattern of the 8 (a) H-Plane. (b) E-Plane.

2 8 array measured at 12.5 GHz.

B. Radiation Pattern The simulated and measured normalized far-field antenna gain patterns at 12.5 GHz are shown in Figs. 12–14 for the H-plane and E-plane cuts of the 2 4, 4 4, and 8 8 arrays, respectively. Simulated and measured co-polarization data are mostly in good agreement except of some side lobe discrepancies that could be related to a slight airgap stacking problem. For the 2 4 array, the measured side lobe level is , while the cross-pol is better than at about broadside, as shown in Fig. 12. The other arrays exhibit similar performances as shown in Figs. 13–14. C. Gain and Efficiency Performance The measured gain versus frequency is shown in Fig. 15 for the different arrays. It is clear that the relatively small arrays exhibit an almost flat gain over the 12.2 GHz to 12.7 GHz band, while the relatively large arrays have some gain ripples which have not been seen in our numerical simulation. We attribute this gain ripples to airgaps in the multilayer structure. However, that assembly problem could be minimized by using a special prepeg epoxy as recommended by Rogers Corp. instead of just soldering or bolting the stack together. Table III summarizes the measured characteristics of the different sized arrays. Only slight differences between the measured gain results and the predicted ones from the simulation

Fig. 15. Measured and simulated gain versus frequency of the different sized arrays.

(listed previously in Table II) can be seen. The efficiency has exceeded 70% for all cases except for the 8 8 case where only 50% has been achieved. This noticeable efficiency drop for the 8 8 array is related to the excessive losses associated with the relatively large feed network. However, waveguide feed networks could be implemented to retain the efficiency for such large arrays.

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TABLE III MEASURED PERFORMANCE SUMMARY OF THE DIFFERENT SIZED SIW CAVITY-BACKED ARRAYS AT 12.5 GHZ

VI. CONCLUSION SIW technology offers low-cost integrated cavity-backed patch antenna structures using the standard PCB fabrication process. The proposed SIW cavity-backed arrays consist of a stack of two substrates: the top substrate for the patches and microstrip feed network and the bottom one for the SIW cavities. The top microstrip substrate should be kept thin in order to minimize the surface waves and the associated feed network losses. Meanwhile, the bottom cavity substrate should be relatively thick for bandwidth enhancement. A design chart for the basic 2 2 sub-array has been presented and used with a modular design approach to realize larger arrays. The proposed SIW cavity-backed arrays outperform both the thick microstrip and suspended arrays in terms of gain and aperture efficiency. The SIW structure has a comparable performance to the cavity-backed suspended arrays and the conventional metalized cavity-backed arrays, but with a much lower fabrication cost. Various SIW array prototypes have been fabricated and experimentally tested. The fabricated structures, as predicted, have very good radiation characteristics, enhanced bandwidth, and high aperture efficiency. For further performance enhancement, waveguide feed networks could be utilized to substantially lower the feed loss and improve the efficiency of large arrays. Stacking more patches could also be employed to further widen the operating bandwidth. ACKNOWLEDGMENT The authors are grateful to the anonymous reviewers for their valuable feedback and comments that helped improve this manuscript. Many thanks go to Rogers Corporation for supplying the substrate boards. Special thanks go for G. Bull for his technical support and advice to better assemble the stack. Finally, Winegard Company’s help in offering the use of their near field setups in the antenna measurements is also greatly appreciated. REFERENCES [1] Y. T. Lo, D. Solomon, and W. F. Richards, “Theory and experiment on microstrip antennas,” IEEE Trans. Antennas Propag., vol. 27, pp. 137–145, 1979. [2] J. R. James and P. S. Hall, Handbook of Microstrip Antennas: IEE Electromagnetic Waves Series. London, U.K.: Inst. Elect. Eng., 1989. [3] D. M. Pozar, “Microstrip antennas,” Proc. IEEE, vol. 80, pp. 79–91, Jan. 1992. [4] C. A. Balanis, Antenna Theory: Analysis and Design, 3rd ed. Hoboken, NJ: Wiley, 2005. [5] E. Chang, S. A. Long, and W. F. Richards, “An experimental investigation of electrically thick rectangular microstrip antennas,” IEEE Trans. Antennas Propag., vol. 34, pp. 767–772, Jun. 1986.

[6] D. Busuioc, S. Safavi-Naeini, and M. Shahabadi, “High frequency integrated feed for front end circuitry and antenna arrays,” Int. J. RF Microw. Comput.-Aided Eng., vol. 19, pp. 380–388, May 2009. [7] M. Shahabadi, D. Busuioc, A. Borji, and S. Safavi-Naeini, “Low-cost, high-efficiency quasi-planar array of waveguide-fed circularly polarized microstrip antennas,” IEEE Trans. Antennas Propag., vol. 53, pp. 2036–2043, Jun. 2005. [8] N. C. Karmakar, “Investigations into a cavity-backed circular-patch antenna,” IEEE Trans. Antennas Propag., vol. 50, pp. 1706–1715, Dec. 2002. [9] F. Zavosh and J. T. Aberle, “Infinite phased-arrays of cavity-backed patches,” IEEE Trans. Antennas Propag., vol. 42, pp. 390–398, Mar. 1994. [10] S. Yang and A. E. Fathy, “Cavity-backed patch shared aperture antenna array approach for mobile DBS applications,” in Proc. Antennas Propag. Society Int. Symp., 2006, pp. 3959–3962. [11] D. Deslandes and K. Wu, “Single-substrate integration technique of planar circuits and waveguide filters,” IEEE Trans. Microwave Theory Tech., vol. 51, pp. 593–596, Feb. 2003. [12] H. Uchimura, T. Takenoshita, and M. Fujii, “Development of a “laminated waveguide”,” IEEE Trans. Microwave Theory Tech., vol. 46, pp. 2438–2443, Dec. 1998. [13] 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, pp. 625–630, May 1998. [14] M. Bozzi, L. Perregrini, K. Wu, and P. Arcioni, “Current and future research trends in substrate integrated waveguide technology,” Radioengineering, vol. 18, pp. 201–209, Jun. 2009. [15] G. Q. Luo, Z. F. Hu, L. X. Dong, and L. L. Sun, “Planar slot antenna backed by substrate integrated waveguide cavity,” IEEE Antennas Wireless Propag. Lett., vol. 7, pp. 236–239, 2008. [16] L. Yan, W. Hong, G. Hua, J. X. Chen, K. Wu, and T. J. Cui, “Simulation and experiment on SIW slot array antennas,” IEEE Microw. Wireless Compon. Lett., vol. 14, pp. 446–448, Sep. 2004. [17] S. Yang, S. H. Suleiman, and A. E. Fathy, “Low-profile multi-layer slotted substrate integrated waveguide (SIW) array antenna with folded feed network for mobile DBS applications,” in Proc. Antenna Application Symp., 2007, pp. 3137–3140. [18] M. H. Awida and A. E. Fathy, “Substrate-integrated waveguide Ku-band cavity-backed 2 2 microstrip patch array antenna,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 1054–1056, 2009. [19] H. Pawlak, M. S. Reuther, and A. F. Jacob, “High isolation substrate integrated coaxial feed for Ka-band antenna arrays,” in Proc. Eur. Microwave Conf., 2007, pp. 1507–1510. [20] F. Zavosh and J. T. Aberle, “Single and stacked circular microstrip patch antennas backed by a circular cavity,” IEEE Trans. Antennas Propag., vol. 43, pp. 746–750, Jul. 1995. [21] A. Panther, A. Petosa, M. G. Stubbs, and K. Kautio, “A wideband array of stacked patch antennas using embedded air cavities in LTCC,” IEEE Microw. Wireless Compon. Lett., vol. 15, pp. 916–918, 2005. [22] Ansoft HFSS 12.0 User Manual. [23] F. Caminita, S. Costanzo, G. D. Massa, G. Guarnieri, S. Maci, G. Mauriello, and I. Venneri, “Reduction of patch antenna coupling by using a compact EBG formed by shorted strips with interlocked branch-stubs,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 811–814, 2009. [24] M. Coulombe, S. F. Koodiani, and C. Caloz, “Compact elongated mushroom (EM)-EBG structure for enhancement of patch antenna array performances,” IEEE Trans. Antennas Propag., vol. 58, pp. 1076–1086, Apr. 2010.

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[25] 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, Oct. 2003. [26] J. M. Schellenberg, “CAD models for suspended and inverted microstrip,” IEEE Trans. Microwave Theory Tech., vol. 43, pp. 1247–1252, Jun. 1995. [27] J. L. Volakis and J. M. Jin, “A scheme to lower the resonant frequency of the microstrip patch antenna,” IEEE Microw. Guided Wave Lett., vol. 2, pp. 292–293, Jul. 1992.

systems for mobile digital broadcast satellite TV reception, antenna design for residential satellite applications, novel low cost antennas for medical telemetry applications, and antennas for VHF and UHF TV reception. From 1997 to 2002, he was a member of the Technical Staff and was Department Staff engineer at TRW Space and Defense, Redondo Beach, California. At TRW he was responsible for the design, analysis, and development of reflector and feed horn antennas for satellite applications. He has authored or coauthored over 20 technical papers and holds four patents. Dr. Suleiman is a member of Tau Beta Pi and the Golden Key International Honor Society.

Mohamed H. Awida (S’04) received the B.Sc. and M.Sc. degrees in electrical engineering from Ain Shams University, Cairo, Egypt, in 2002 and 2006, respectively. He is currently working towards the Ph.D. degree at the University of Tennessee at Knoxville. He has been a Research and Teaching Assistant at Ain Shams University, from 2002 to 2006. In summer 2005, he was a Visiting Scholar at Otto-Von-Guericke University, Magdeburg, Germany. From October 2008 to May 2009, he was a Research Assistant at the Spallation Neutron Source, Oak Ridge National Laboratory. His research interests include antenna arrays for mobile platforms, microwave passive planar structures, and microwave linear particle accelerators. Mr. Awida is a member of the Phi Kappa Phi Honor Society. He is listed in Who’s Who in the World 2006.

Aly E. Fathy (S’82–M’84–SM’92–F’04) received the B.S.E.E. degree, B.S. degree in pure and applied mathematics, and M.S.E.E. degree from Ain Shams University, Cairo, Egypt, in 1975, 1979, and 1980, respectively, and the Ph.D. degree from the Polytechnic Institute of New York, Brooklyn, in 1984. In February 1985, he joined the RCA Research Laboratory (currently the Sarnoff Corporation), Princeton, NJ, as a Member of the Technical Staff. In 2001, he became a Senior Member of the Technical Staff. While with the Sarnoff Corporation, he was engaged in the research and development of various enabling technologies such as high-Tc superconductors, low-temperature co-fired ceramic (LTCC), and reconfigurable holographic antennas. He was also an Adjunct Professor with the Cooper Union School of Engineering, New York. In August 2003, he joined the University of Tennessee, Knoxville, where he is currently a Professor and Head of the Antenna Labs. He has authored or coauthored numerous transactions and conference papers. He holds 11 U.S. patents. His current research interests include DBS Antennas, wireless reconfigurable antennas, see-through walls, UWB systems, and high-efficiency high-linearity combining of digital signals for base-station amplifiers. He has developed various microwave components/subsystems such as holographic reconfigurable antennas, radial combiners, direct broadcast antennas (DBSs), speed sensors, and low-temperature co-fired ceramic packages for mixed-signal applications. Dr. Fathy is a member of Sigma Xi and Eta Kappa Nu. He is an active member of the IEEE Microwave Theory and Techniques Society (IEEE MTT-S), International Microwave Symposium (IMS), Technical Program Committee (TPC), the IEEE Antenna and Propagation Symposium, and the IEEE Radio and Wireless Steering Committee. He was the General Chair of the 2008 IEEE Radio Wireless Conference. He was the recipient of five Sarnoff Outstanding Achievement Awards (1988, 1994, 1995, 1997, 1999).

Shady H. Suleiman (S’93–M’06) received the B.S. degree in engineering physics and the M.S. degree in electrical engineering from the Ohio State University, Columbus, in 1990 and 1992, respectively, and the Ph.D. degree in electrical engineering from the Georgia Institute of Technology, Atlanta, in 1997. Since 2002, he has been a Senior Antenna Engineer and Team Lead at Winegard Company, Burlington, IA. At Winegard Company, he established a state-of-the-art near-field antenna range test facility which is capable of testing antennas up to 40 GHz. His current research interests include low profile antenna

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3D Power Synthesis with Reduction of Near-Field and Dynamic Range Ratio for Conformal Antenna Arrays Massimiliano Comisso, Member, IEEE, and Roberto Vescovo, Member, IEEE

Abstract—A simple method of power synthesis for 3D radiation patterns of conformal antenna arrays is presented. The method enables to synthesize a desired far-field pattern simultaneously controlling the dynamic range ratio (DRR) of the array excitations and the electric field amplitude in a given region of interest close to the antenna. The power synthesis problem is reduced to a field synthesis one by introducing an auxiliary phase pattern, which is incorporated in a cost function together with the desired far-field pattern, the array pattern, the near electric field amplitude, and the array excitations. Such cost function is minimized by iteratively modifying the auxiliary phase pattern and the array excitations. Applications of the proposed method to conformal arrays of different geometries show that accurate results are obtained also in presence of stringent requirements and within acceptable CPU times, even when hundreds of elements are involved. Index Terms—Conformal antenna arrays, dynamic range ratio (DRR) reduction, near-field control, power synthesis.

I. INTRODUCTION

T

HE development of algorithms aimed to generate a far-field amplitude pattern having prescribed characteristics represents a fundamental concern for designers of antenna arrays. In many practical applications the array has to be mounted on surfaces of various shapes, such as, for example, those of terrestrial vehicles, aircrafts, missiles, satellites, and others. In these cases the array geometry must meet the surface shape, and the radiating structure is referred to as a conformal array. Several methods have been proposed in the literature for the synthesis of conformal arrays [1]–[10]. A power synthesis technique is proposed in [1], where the synthesized far-field pattern is seen as a solution of a fixed point problem. This algorithm is extended in [2] by the introduction of a weight function that allows to better match the far-field pattern requirements. The generalized method of successive projections is used in [3] to solve 3D power synthesis problems with constraints on the array excitations. A method of synthesis with phase only control, which uses the stationary phase pattern approach with a weight function, is presented in [4]. The method of simulated Manuscript received October 07, 2009; revised August 17, 2010; accepted September 13, 2010. Date of publication January 31, 2011; date of current version April 06, 2011. The authors are with the Dipartimento di Elettrotecnica, Elettronica ed Informatica, University of Trieste, 34127 Trieste, Italy (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.2011.2109674

annealing is used in [5] to minimize a cost function involving both the dynamic range ratio (DRR) and the desired side-lobe level (SLL) of the synthesized far-field pattern. The method of projections is also employed in [6] to develop a fast synthesis technique that allows to form nulls by phase-only control, exploiting the power azimuth spectrum of the undesired signals. In [7] the authors propose a fast phase-only control algorithm that enables to approximate a desired far-field amplitude pattern by minimizing a weighted cost function incorporating the design requirements. In [8] the author presents a synthesis method that allows to assign a prescribed value to each excitation amplitude, thus obtaining a desired DRR value. This method requires low CPU times in scenarios where the desired far-field amplitude pattern can be considered as zero in a large number of directions. The consistency between null constraints and DRR constraints is discussed in [9], where two iterative algorithms of far-field pattern synthesis with null and DRR constraints are presented. A fast method of 2D power synthesis, based on the minimization of a cost function, is proposed in [10], where an extension of the presented solution is also described to enable the DRR control. Many of the above cited synthesis methods have the capability of reducing the DRR of the array excitations, thus allowing to reduce the complexity of the feeding network. However, in some synthesis problems the requirement of reducing the electric field amplitude in a prescribed region located in the near-field zone should be added to the other requirements [11]–[15]. This requirement may be necessary not only to avoid undesired scattering effects produced by close conducting obstacles, but also to control the interference with electronic equipments in the proximity of the antenna, or for health safety reasons. In [11] the authors propose a 2D method for linear arrays that approximates a desired far-field pattern in the mean square sense, simultaneously minimizing the radiated power over the surface of an obstacle located in the near-field region. An algorithm for the 3D pattern synthesis of a phase-controlled reconfigurable conformal array is presented in [12], where an upper bound is imposed on the near-field in a prescribed region close to the antenna. In [13] the method of projections is used to find a far-field amplitude pattern belonging to a prescribed mask, simultaneously forming exact nulls in the near-field. A least mean square approach for linear and planar arrays is considered in [14], where nulls are imposed in the radiative near-field zone. A cost function is adopted in [15] to develop a fast method for the power synthesis of the 2D far-field pattern that simultaneously enables the reduction of the near-field in a prescribed region.

0018-926X/$26.00 © 2011 IEEE

COMISSO AND VESCOVO: 3D POWER SYNTHESIS WITH REDUCTION OF NEAR-FIELD AND DRR FOR CONFORMAL ANTENNA ARRAYS

In this paper we propose an iterative method of power synthesis for antenna arrays of arbitrary geometry. The method allows to generate a 3D far-field pattern having a desired shape, simultaneously reducing the DRR of the array excitations and the electric field amplitude in a given region close to the antenna. The method reduces the power synthesis problem to a problem of field synthesis by introducing an auxiliary pattern whose amplitude is given by the desired far-field pattern and whose phase distribution is unknown. The phase pattern and the array excitations are iteratively modified to minimize a cost function involving the far-field pattern, the electric near-field amplitude and the array excitations. The method is validated by synthesizing 3D patterns of conformal arrays involving some hundreds of elements, and proved to give accurate results with quite acceptable CPU times. The present study extends the approach adopted in [10] and [15] to perform the power pattern synthesis of large conformal arrays in a 3D scenario, simultaneously reducing the DRR and the electric field amplitude in a region close to the antenna. The paper is organized as follows. In Section II the problem is mathematically formulated. The method of synthesis is described in Section III, and numerical results are presented and discussed in Section IV. Section V concludes the work. II. FORMULATION OF THE PROBLEM Consider an antenna array of arbitrary geometry, consisting be the column vector of elements, and let of the complex excitations. With reference to a Cartesian system , the far-field pattern of the array, in the direction specified by the spherical coordinates and , is given by

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To this purpose, we minimize the following cost function:

(4) and specify the boundaries of the angular where , , is a real positive weight function, region of interest, and are proper positive weights, and is the space region, close to the antenna, where the electric field must be reduced. Note that, minimizing the first term in (4) allows the array amplitude pattern to approximate the desired amplitude pattern . The second term is inserted to reduce the electric field amplitude in the region of interest . Finally, minimizing the third term, which incorporates the squared differences between pairs of successive excitation amplitudes, allows to reduce the DRR. Hence, minimizing the cost function in (4) enables to approximate the desired far-field pattern, simultaneously reducing the DRR and the near-field in . A proper selection of the , and allows to reduce the corresponding weights terms in (4), according to the specific problem requirements. For leads to a stronger reduction of the example, an increase of near-field in the region . It is worth to note that this approach allows to better exploit the degrees of freedom of the array. In fact, minimizing the cost function in (4) is a problem of power synthesis, which does not involve the phase pattern, which usually is not of interest.

(1) III. THE METHOD OF SYNTHESIS where represents the far-field pattern of the th eleis the position of the th element, is the ment of the array, is wavelength, and the unit vector of the direction of observation, with , , and denoting the unit vectors of the coordinate axes , , and , respectively. The electric field generated by the array at the point can be expressed as

, we modify Introducing the generic phase pattern into the complex patthe desired amplitude pattern . Then, the problem of minimizing (4) betern comes equivalent to that of finding an excitation vector and a function that minimize the functional

(2) where is the electric field produced in by the excitation , having unity in the th vector position. , representing Consider now a real positive function the desired far-field amplitude pattern normalized with respect to unity. The problem that we want to solve consists in gen, whose amplitude approxerating an array pattern , simultaneously reducing the radiated electric imates field amplitude in a given region close to the antenna and the DRR of the array excitations, defined as (3)

(5) Substituting (1) and (2) into (5), after some manipulations we obtain:

(6)

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where the asterisk denotes the complex conjugate and (7a) (7b)

(7c)

quadrature approximation for the integral in (7c). Precisely, we into adjacent subinterdivide the elevation domain , and the azimuth vals of equal length domain into adjacent subintervals of equal length . Then, for each and , we evaluate the integrand function in the space , direction which is the center of the two-dimensional subdomain . Then, the integral in (7c) can be approximated as follows: (11)

(7d) for for

where

(7e)

For any phase function , the functional can be minimized with respect to the excitation vector by modifying the components one at a time, so as to minimize (6) iteratively. To this purpose, we rewrite (6) putting into evidence its dependence on the th component . After some manipuand , this lations, and noting that yields

, , and . Substituting (11) into (6) and putting into evidence on the generic phase sample , after the dependence of some algebra we obtain

where the samples , and

are seen as elements of a

(12) matrix

(8) where (9a) (9b) (9c) (13a) (13b)

(9d) and is the Kronecker delta ( if , otherwise). In order to minimize in (8) with respect to , we impose that the multiplicative coefficient of be minimum, and that be zero, thus obtaining the derivative of with respect to

Note that the term is independent of . The function in (12) can be minimized with respect to by imposing (14a) (14b) which yield

(10a) (10b) Equations (10a) and (10b) will be used to minimize the cost function in (6) iteratively with respect to the excitations , taken one at a time. with Now, let us discuss the process of minimization of respect to the function . To this purpose, we adopt a

(15) Hence, for any given excitation vector , the matrix in (12) is given by imizes

that min(16)

where with

and denoting an arbitrary integer.

are

matrices,

COMISSO AND VESCOVO: 3D POWER SYNTHESIS WITH REDUCTION OF NEAR-FIELD AND DRR FOR CONFORMAL ANTENNA ARRAYS

In conclusion, the method develops as follows. Given a , a starting phase matrix is starting point evaluated using (16), where is given by (13b). Subsequently, the first iteration is performed in two phases: during the first phase we modify the components of one at a time by (10), starting from the first one and continuing until the last one is . modified. This yields an excitation vector phase The second phase consists in modifying the . Simvalues by using (16), thus obtaining a phase matrix ilarly, during the second iteration we modify the excitations iteratively using the phase matrix , thus obtaining the vector , and then we calculate the matrix . Proceeding in this way, the algorithm generates the se. Therefore, at each quence of points iteration the algorithm calculates excitations and phase is minimized, values. Since at each step the function , it results: is nonincreasing, and is therethus the sequence fore convergent. The sequence is terminated at the step in which (17) where

and is a proper positive threshold. IV. RESULTS

In this section two examples of application of the proposed algorithm are presented. In both cases the excitation vector is selected as a starting point, samples of the phase pattern are taken in the elevation domain, samples are taken in the azimuth domain, and is the selected threshold in (17). Besides, the weight function in (4) is chosen in such a way as to take larger values in the angular regions where the desired far-field amplitude is lower. Precisely, we set (18) where

is a constant. In the presented examples we set . The mutual coupling between the array elements is taken into account by replacing each “isolated” element pattern by an equivalent element pattern obtained using the approach in described in [16]. The equivalent element patterns in (2) are calculated by using (1) and the electric fields the SuperNEC 2.7 electromagnetic simulator. In the following examples we employ arrays consisting of center-fed half-wavelength dipoles, and consider the component of the electric far-field as the co-polar pattern and the component of the electric far-field as the cross-polar pattern. The radiating structure adopted for the first example is a conelements arcentric ring array (CRA) consisting of plane on six circular rings centered at the ranged in the (Fig. 1). The array elorigin of the Cartesian system ements are center-fed half-wavelength dipoles parallel to the axis, resonating at 300 MHz and having a wire diameter equal has radius to 1 mm. The th ring and consists of equally spaced dipoles, where the first dipole of each ring is placed at . The desired far-

Fig. 1. CRA of

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N = 252 center-fed half-wavelength dipoles.

field pattern maximum at

is shown in Fig. 2(a) and (b): it has the , a maximum side-lobe level , and a null region , where a maximum pattern amplitude of 40 dB is required. Besides, a reduction of the radiated electric field amplitude is required in the region , which is close to the antenna and corresponds to a cube having edge length of . Fig. 2(a) shows the far-field pattern obtained in absence of both the near-field control and the in (4). Fig. 2(b) DRR reduction, that is, setting illustrates the far-field pattern obtained setting , in order to obtain a strong reduction of both the DRR and the near-field in the region . For reasons of clarity, in Fig. 3 not all the results concerning the near electric field amplitude are , where shown, but only those corresponding to the plane the lowest reduction is obtained (worst case), and those corre, where the highest reduction is obsponding to the plane tained (best case). Precisely, Fig. 3(a) and (b) shows the contour for , referred to the cases plots of and , respectively. Fig. 3(c) and (d) shows the contour plots of for , referred to the cases and , respectively. Table I and on the performance shows the effect of the weights of the presented algorithm. In order to obtain an accurate estimate of the near-field behavior, and in particular of the maximum near-field amplitude in , the region (19) which includes both the array and the region , is discretized , and each vector with a dense regular grid whose step is in (2) is calculated at all grid points to the aim function in (7b) with a summation. of approximating the integral As a consequence, the near-field minimization occurs only at the grid points. In order to verify the accuracy obtained with this approach not only in the grid points of , but also in other points, a statistical validation is performed by calculating the randomly generated points inside electric field in the region . This method of validation has been also used to choose the grid step, since it has revealed that grids having step or are not sufficiently dense to guarantee a satisof factory control of the near-field. Instead, the numerical results

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Fig. 2. 3D far-field pattern obtained using a CRA with

N = 252 elements: (a) w = w = 0. (b) w = w = 40000.

E (a;r)

Fig. 3. Contour plots of the near electric field amplitude j j obtained using a CRA with N w in the plane z  (worst case). (c) w w in the plane z case). (b) w (best case).

=

= 40000

=

=

have shown that, discretizing the region with a grid having a , the maximum amplitude of the near-field evalstep of uated in the random points is higher of no more than 2 dB compared to that evaluated in the grid points. Accordingly, each of

=0

= 252 elements: (a) w = w = 0 in the plane z =  (worst = 5 (best case), (d) w = w = 40000 in the plane z = 5

the values of the electric field amplitude in reported in Table I, is the maximum between the electric field amplitude in the grid points and the electric field amplitude in the randomly generated points. This maximum value is normalized with respect to the

COMISSO AND VESCOVO: 3D POWER SYNTHESIS WITH REDUCTION OF NEAR-FIELD AND DRR FOR CONFORMAL ANTENNA ARRAYS

TABLE I EFFECT OF THE WEIGHTS w AND w FOR THE FIRST EXAMPLE (CRA WITH N

maximum value obtained, with the approach above described, . For comparison purposes, in the region for Table I also shows the results obtained using the method presented in [13], here extended to the 3D case. In particular, with the latter method the near-field reduction in the region has of been obtained by imposing nulls in the 27 points , , , coordinates where , 2, 3. Besides, in the method in [13], the synthesized pattern is required to belong to a mask. Here, the upper bound of the mask is identical to the desired far-field , while the lower bound of the mask is obtained pattern and imposing a level of 200 by subtracting 0.5 dB to dB outside the main lobe. Figs. 2, 3, and Table I indicate that, using the proposed method, both the near-field and the DRR can be considerably and , mainreduced by a proper choice of the weights taining the SLL requirement and simultaneously satisfying, in all the examined cases, the upper bound of 40 dB in the desired far-field null region. Note, in particular, that in all the , a reduction of the near-field level cases where greater than 21 dB is obtained compared to the cases where . Besides, for a dramatic DRR reduction is and obtained. A comparison between the cases reveals that, in the latter case, the algorithm provides a 99.7% reduction of the DRR and simultaneously a reduction of 26.3 dB in the near-field. Furthermore, the CPU times listed in Table I reveal that no more than a few minutes are usually sufficient to calculate the array excitations. A comparison of the proposed algorithm with the method presented in [13], and extended to the 3D case, shows that similar performances are obtained in terms of SLL, near-field reduction, and CPU time, but the proposed algorithm is able to guarantee the fulfillment of all near and far-field requirements with a very low DRR. Besides, differently from the method in [13], where a direct control on the amount of the desired near-field reduction is not available, the presented technique allows to perform a tradeoff between far-field, near-field, and DRR requirements and . by properly selecting the weights The second example refers to an array composed by center-fed half-wavelength dipoles arranged around a perfectly and radius equal to conducting cylinder of height equal to (Fig. 4). The base of this cylinder lies in the plane , whose axis coincides with of the Cartesian system

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= 252 DIPOLES)

= 234

Fig. 4. Antenna array of N center-fed half-wavelength dipoles arranged around a perfectly conducting cylinder.

the cylinder axis. The array consists of three rings of 78 ele, 2, 3) has radius and lies ments. The th ring ( with respect to the at the height plane. These values for the radius and the heights are selected to between adjacent dipoles of the same obtain a distance of between the endings of dipoles bering, and a distance of longing to adjacent rings. The desired far-field pattern is illustrated in Fig. 5: it has a maximum at and . The space region where the electric field amplitude must be reduced is , thus consists of two separate regions. Fig. 5(a) and (b) shows the synand thesized far-field patterns obtained for , , respectively. Fig. 6(a) and (b) shows in the plane where the the contour plots of field reduction in is minimum (worst case) in the two cases and , , respectively. in the plane Fig. 6(c) and (d) shows the contour plots of , where the field reduction in is maximum (best case) in the two cases and , , reand can be inferred spectively. The effect of the weights from Table II, where the maximum electric field amplitude in is estimated using the same approach adopted in the previous

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Fig. 5. 3D far-field pattern obtained using ,w . (b) w

= 900

= 62500

N = 234 center-fed half-wavelength dipoles arranged around a perfectly conducting cylinder: (a) w = w = 0.

E (a;r) obtained using N = 234 center-fed half-wavelength dipoles arranged around a perfectly = 0:75 (worst case). (b) w = 900, w = 62500 in the plane z = 0:75 (worst case). (c) w = = 900 = 62500 in the plane z = 0 (best case).

Fig. 6. Contour plots of the near electric field amplitude j w in the plane z conducting cylinder: (a) w w in the plane z (best case). (d) w ,w

=0

=0

=

=0

j

example. In particular, the region in (19) has been used also in this second example, as it includes , and has been discretized . This second problem with a regular grid of step equal to has been solved also with the method in [13], extended to the of coor3D case, by imposing nulls in the 18 points , , , with dinates , 2, 3, and by generating the mask of the far-field

amplitude pattern with the same approach adopted in the first example. Nulling the radiated field in the above 18 points, which belong to the region , produces a strong field reduction in a neighborhood of such points, so a field reduction is expected in . Figs. 5, 6, and Table II show that, compared to the case , a DRR reduction in the order of 98.9% and a near-field

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TABLE II EFFECT OF THE WEIGHTS w AND w FOR THE SECOND EXAMPLE (N CENTER-FED HALF-WAVELENGTH DIPOLES ARRANGED AROUND A PERFECTLY CONDUCTING CYLINDER)

= 234

reduction of 19 dB was achieved by selecting and , at the cost of slight increase of the SLL with respect to the desired level. The figures reveal also that the algorithm allows to control the near electric field amplitude inside two separate regions simultaneously, thus allowing to satisfy stringent requirements on the near-field. Furthermore, the CPU times reported in Table II confirm that no more than a few minutes were sufficient to perform the synthesis, and, for more than half of the considered cases, the algorithm required less than one minute, although more than two hundred elements were used. A comparison of the proposed algorithm with the method in [13] extended to the 3D case, confirms that both techniques are able to provide a near field reduction, but the proposed algorithm allows to obtain a lower DRR. Besides, even if the method in [13] without near-field control provides a DRR equal to 17.8, which is lower than the DRR value obtained using the proposed algorithm for , the selection of or is able to reduce the DRR value to 7.9 or 3.2, respectively. It can also be noticed that, in the first example, the proposed algorithm achieves a near-field level of 46.1 dB for with an initial value of 19.8 dB for , while the method in [13] provides a level of 47.1 dB in presence of near-field control with an initial value of 28.1 dB in absence of near-field control. An analogous behavior can be observed in the second example. Hence, even if the two techniques move from different conditions in absence of near-field requirements, both tend to similar final values for the maximum electric field amplitude in the region . Moreover, for both examples, a further increase of for the proposed algorithm or an increase of the number of near-field nulls for the method in [13] do not provide significant near-field reductions. A. Influence of the Starting Point Table III shows the effect of the choice of the starting point on the solutions achieved by the presented algorithm, for the two considered examples. Together with the average value of the quantities of interest, the table reports the probability of matching the far-field pattern requirements, which are considered as satisfied if, with respect to the desired far-field pattern amplitude, the amplitude of the synthesized far-field pattern is at most 3 dB higher in the directions of the sidelobe or of the null regions, and at most 3 dB lower in the angular region of the main lobe. Three hundred pattern realizations have been performed for each of the four cases reported in the table, where, for each realization, the starting excitation vector has been ran-

TABLE III EFFECT OF THE STARTING POINT FOR THE PROPOSED ALGORITHM

domly generated according to a uniform distribution inside the interval [1, 5] for the amplitudes and inside the interval for the phases. As reported in Table III, in some cases the probability of matching the pattern requirements can be lower than one. However, a direct observation of the patterns synthesized in these cases reveals that the realizations not satisfying the pattern requirements do not provide unacceptable patterns, but simply patterns having a SLL 4 or 5 dB higher than the desired one. A comparison of the SLL value, of the maximum far-field amplitude in the null region (for the first example), and of the maximum electric field amplitude in the region , presented in Table III, with the corresponding values presented in Tables I and II, shows that similar performances are reached selecting different starting points. Similar performances are also obtained is in terms of DRR for the cases where a large value of selected, which confirms the ability of the proposed technique of reducing the DRR. Observe that, for the first example with , a larger difference can be noticed between the mean value of the DRR in Table III and that in Table I. However, for this specific case, a deeper analysis of the simulation results obtained using different starting points reveals that the root mean square deviation of the DRR is rather high, since it is approximately equal to 2088.4, and the minimum DRR is equal to 32.7. Thus, if desired, the starting point may be selected (by a certain number of random trials) to obtain a low DRR value . even for To better explain how the proposed algorithm explores the solution space, we report in Figs. 7 and 8 the complete trajectories, in the Gauss plane, of the excitation , during the synthesis process, for ten of the three hundred realizations with dif-

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Fig. 7. Real and imaginary part of the excitation a during the synthesis process for the first example for w w . Each black filled circle represents a starting point, while the corresponding black filled star represents the final point.

=

=0

Fig. 8. Real and imaginary part of the excitation a during the synthesis w . Each black filled circle process for the first example for w represents a starting point, while the corresponding black filled star represents the final point.

=

= 40000

ferent randomly selected starting points that have provided the results in Table III. More precisely, Fig. 7 is referred to the first , while Fig. 8 is still represented example for ferred to the first presented example but for . Each trajectory, which is depicted with a different marker and a different color (color only appears in the electronic version of this paper), moves from the starting point, denoted by a black filled circle, and terminates in the final point, denoted by a black filled star. We can observe that, when no requirements are imposed on the DRR (Fig. 7), the final points are mainly concentrated in a neighborhood of the center of the unit circle, centered at the origin of the Gauss plane. For other excitations, different from , the points are still mainly concentrated in a region internal to this circumference, but not necessarily close to the center. Instead, when a strong requirement is imposed on

Fig. 9. Cost function as a function of the iteration step for the first example, for w , and for the ten considered realizations. w

=

=0

Fig. 10. Final value of the cost function for the first example, for w

0, and for the ten considered realizations.

=w =

the DRR (Fig. 8), the trajectories tend to move toward the circumference of radius one, approaching it until condition (17) is satisfied. This means that, when a high value of is selected, the algorithm tries to provide an excitation vector with DRR as close as possible to unity. To maintain the readability of the figures, only ten trajectories of the first component of the excitation vector have been plotted. The other realizations and excitations confirm the behavior of these ten trajectories. The of the iteration behavior of the cost function as a function step for , for the ten considered realizations, is reported in Fig. 9, while Fig. 10 shows the final value of at the step in which (17) is satisfied. Similarly, Figs. 11 and 12 show the evolution of and its final value, refor spectively, for the ten considered realizations. Since the number of iterations can be different for each starting point, the curves in Figs. 9 and 11 can have different lengths. A direct comparison of Figs. 10 and 12 reveals that the final value of is larger when

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TABLE IV NUMBER OF EXECUTIONS WITH DIFFERENT STARTING POINTS REQUIRED TO REACH THE FAR-FIELD PATTERN REQUIREMENTS

Fig. 11. Cost function as a function of the iteration step for the first example, w , and for the ten considered realizations. for w

=

= 40000

more times, for different randomly generated starting points, until the far-field pattern requirements are satisfied. Table IV reports the maximum, minimum, and average number of executions of the proposed method, together with the average total CPU time, required to match the desired requirements on the far-field by considering one hundred realizations for each of the two considered examples. The values in the table reveal that usually less than two executions, and in some cases just one execution, are sufficient to obtain the required far-field, while maintaining an acceptable total CPU time, which is given by the sum of the CPU times of the executions. Thus, even if the final excitation vector depends on the starting point, the proposed method is able to provide satisfactory results in really few attempts. In the considered examples the requirements on the far-field amplitude pattern were not so stringent and hence many local solutions were available. Of course, the application of the proposed algorithm to more challenging problems with more stringent requirements might provide different performances and might require more attempts. V. CONCLUSION

Fig. 12. Final value of the cost function for the first example, for w , and for the ten considered realizations.

40000

=w =

more requirements are imposed in the synthesis process. In fact, the cost function in (5) contains only the term corresponding to , while also the terms the far field pattern for corresponding to the near-field and to the DRR are present for . Besides, in this second case the final value of is more sensitive to the starting point, being the heights of the bars in Fig. 10 more variable than those in Fig. 12. These results, in conjunction with those reported in Table III, reveal, for the proposed method, a considerable dependence of the final excitation vector from the starting point, since the algorithm does not perform a global search of the optimum solution in the entire solution space, but the results show, at the same time, a much more limited dependence of the performances (SLL, near-field amplitude in the region , DRR) from the starting point itself. Thus, many different local solutions to the same problem may be obtained by the proposed algorithm, but in general these solutions satisfy the synthesis requirements. If the requirements must be strictly satisfied, the proposed algorithm can be run for

A simple algorithm to solve 3D power pattern synthesis problems for conformal antenna arrays has been presented. The algorithm, which adopts an auxiliary phase function to reduce the power synthesis problem to a problem of field synthesis, allows to reduce the DRR of the array excitations and the electric field amplitude in a given region of interest close to the antenna. The results showed that reductions in the order of 99% for the DRR and in the order of 20 dB for the near-field can be obtained by properly selecting the weights involved in the cost function. Such performances can be achieved while maintaining satisfactory far-field patterns and acceptable CPU times. Besides, the results show that multiple requirements can be imposed on the far-field pattern, for example a SLL control and a strong reduction of the radiation pattern amplitude in an angular region. Considerable reductions of the electric field have been also obtained in different separate regions close to the antenna. Furthermore, the obtained results show that significant improvements of the performance, in order to satisfy particular requirements on the DRR, on the far-field pattern or on the near-field, can be often obtained by simply modifying the order of magand . This is an interesting charnitude of the weights acteristic of the presented method, as the choice of the values assigned to these weights is usually not critical, thus only few attempts can be sufficient to match the desired specifications. Since the presented algorithm does not perform a global search of the optimum solution, the dependence of the performances on

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the starting point have been investigated. This dependence can be strongly reduced by running the algorithm for more times with different randomly generated starting points in order to better explore the solution space. In the considered examples really few executions were sufficient to satisfy the far-field pattern requirements, even if a larger number of attempts may be necessary for solving more challenging synthesis problems. REFERENCES [1] Y. I. Choni, “Synthesis of an antenna according to a given amplitude radiation pattern,” Radio Eng. Electron. Phys., vol. 16, pp. 770–778, May 1971. [2] J. R. Mautz and R. F. Harrington, “Computational methods for antenna pattern synthesis,” IEEE Trans. Antennas Propag., vol. 23, no. 4, pp. 507–512, Jul. 1975. [3] O. M. Bucci, G. D’Elia, and G. Romito, “Power synthesis of conformal arrays by a generalised projection method,” IEE Proc. Microw., Antennas Propag., vol. 142, no. 6, pp. 467–471, Dec. 1995. [4] A. D. Khzmalyan and A. S. Kondrat’yev, “Phase-only synthesis of antenna array amplitude pattern,” Int. J. Electron., vol. 81, no. 5, pp. 585–589, 1996. [5] J. A. Rodríguez, L. Landesa, J. L. Rodíguez, F. Obelleiro, F. Ares, and A. García-Pino, “Pattern synthesis of array antennas with arbitrary elements by simulated annealing and adaptive array theory,” Microw. Opt. Technol. Lett., vol. 20, no. 1, pp. 48–50, Dec. 12, 1999. [6] M. Comisso and R. Vescovo, “Exploitation of spatial channel model for antenna array synthesis,” Electron. Lett., vol. 42, no. 19, pp. 1079–1080, Sep. 14, 2006. [7] M. Comisso and R. Vescovo, “Multi-beam synthesis with null constraints by phase control for antenna arrays of arbitrary geometry,” Electron. Lett., vol. 43, no. 7, pp. 374–375, Mar. 29, 2007. [8] L. I. Vaskelainen, “Constrained least-square optimization in conformal array antenna synthesis,” IEEE Trans. Antennas Propag., vol. 55, no. 3, pp. 859–867, Mar. 2007. [9] R. Vescovo, “Consistency of constraints on nulls and on dynamic range ratio in pattern synthesis for antenna arrays,” IEEE Trans. Antennas Propag., vol. 55, no. 10, pp. 2662–2670, Oct. 2007. [10] M. Comisso and R. Vescovo, “Fast iterative method of power synthesis for antenna arrays,” IEEE Trans. Antennas Propag., vol. 57, no. 7, pp. 1952–1962, Jul. 2009. [11] L. Landesa, F. Obelleiro, J. L. Rodriguez, J. A. Rodriguez, F. Ares, and A. G. Pino, “Pattern synthesis of array antennas with additional isolation of near field arbitrary objects,” Electron. Lett., vol. 34, no. 16, pp. 1540–1542, Aug 6, 1998.

[12] O. M. Bucci, A. Capozzoli, and G. D’Elia, “Power pattern synthesis of reconfigurable conformal arrays with near-field constraints,” IEEE Trans. Antennas Propag., vol. 52, no. 1, pp. 132–141, Jan. 2004. [13] R. Vescovo, “Power pattern synthesis for antenna arrays with null constraints in the near-field region,” Microw. Opt. Technol. Lett., vol. 44, no. 6, pp. 542–545, Mar. 20, 2005. [14] H. Steyskal, “Synthesis of antenna patterns with imposed near-field nulls,” Electron. Lett., vol. 42, no. 19, pp. 1079–1080, Sep. 14, 2006. [15] M. Comisso and R. Vescovo, “Fast power pattern synthesis with nearfield control for antenna arrays,” in Proc. IEEE Antennas Propag. Soc. Int. Symp. (APS), Jun. 1–5, 2009. [16] R. Vescovo, “Null formation with excitation constraints in the pattern synthesis for circular arrays of antennas,” Electromagn., vol. 21, no. 3, pp. 213–230, Apr. 2001.

Massimiliano Comisso (M’08) was born in Trieste, Italy. He received the “Laurea” degree in electronic engineering and the Ph.D. degree in information engineering from the University of Trieste. He was with Alcatel working in the field of optical communication systems and for Enteos in the field of microstrip antenna design. Currently, he is Postdoctoral Researcher in information technology with the Department of Industrial Engineering and Information Technology, University of Trieste. His research studies involve smart antennas, distributed wireless networks, antenna array synthesis, and small antennas. Dr. Comisso has been a Best Student Paper Award Finalist at Globecom 2006 and received the Best Paper Award at CAMAD 2009.

Roberto Vescovo (M’92) was born in Verona, Italy. He received the Laurea degree (summa cum laude) in electronic engineering from the University of Trieste, Italy, in 1982, and the Ph.D. degree in electronics and information engineering from the University of Padova, Italy, in 1987. He served as an Assistant Professor with the Dipartimento di Elettrotecnica Elettronica ed Informatica, University of Trieste, where he is currently an Associate Professor of electromagnetic fields. His main research interests are the synthesis of antenna arrays, the electromagnetic theory, and the electromagnetic scattering with related theoretical and computational aspects.

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Adaptive Wideband Beamforming With Frequency Invariance Constraints Yong Zhao, Wei Liu, Senior Member, IEEE, and Richard J. Langley, Member, IEEE

Abstract—A response variation ( ) element is introduced to control the consistency of an adaptive wideband beamformer’s response over the frequency range of interest. By incorporating the element into the linearly constrained minimum variance (LCMV) beamformer, we develop a novel linearly constrained beamformer with an improved output signal-to-interference-plus-noise ratio (SINR), compared to both the traditional formulation and the eigenvector based formulation, due to an increased number of degrees of freedom for interference suppression. In addition, two novel wideband beamformers robust against look direction estimation errors are also proposed as a further element. One is designed by imposing a application of the element and simultaneously limiting the constraint on the magnitude response of the beamformer within a pre-defined angle range at a reference frequency; the other one is obtained by combining the element and the worst-case performance optimization method. Both of them are reformulated in a convex form as the second-order cone (SOC) programming problem and solved efficiently using interior point method. Compared with the original robust methods, a more efficient and effective control over the beamformer’s response at the look direction region is achieved with an improved overall performance. Index Terms—Convex optimization, frequency invariance, look direction error, robust beamformer, wideband beamforming.

I. INTRODUCTION

I

N adaptive wideband beamforming [1], given the direction of arrival (DOA) information of the signal of interest, many traditional beamforming techniques can work effectively and achieve a satisfactory output signal-to-interference-plus-noise ratio (SINR) [2], [3]. One of the most well-known beamformers is the linearly constrained minimum variance (LCMV) beamformer or the Frost beamformer [4], [5], which minimizes its output power while preserving a unity gain at the look direction or subject to some more complicated constraints. Suppose the signal of interest comes from the broadside of the array, then a simple formulation of the constraints can be obtained. However, the problem with this simple formulation is that the beamformer will be over-constrained when we are not interested in the full range of normalized frequency. Moreover, we may not need to constrain the beamformer response over the frequency range of interest to be exactly unity and some variation can be Manuscript received June 15, 2010; revised August 17, 2010; accepted September 16, 2010. Date of publication February 04, 2011; date of current version April 06, 2011. The authors are with Communications Research Group, Department of Electronic & Electrical Engineering, University of Sheffield, Sheffield S1 3JD, U.K. (e-mail: [email protected]; [email protected]; [email protected]). Digital Object Identifier 10.1109/TAP.2011.2110630

allowed so that more freedom can be allocated for suppressing interferences. To design a wideband beamformer over a specified frequency range, conventionally we formulate a constraint matrix by sampling the frequency range of interest and constrain the response of the beamformer at those frequency points to be the desired ones. However, the large number of constraints resulted from this approach reduces the number of degrees of freedom in minimizing the output power of the beamformer. To reduce the number of constraints, an eigenvector constraint approach was developed based on the low rank representation of wideband signals [6]. To have a more efficient use of the degrees of freedom of the array, we propose a new method for wideband minimum variance beamforming, with a much less number of constraints, and resulting in a higher output SINR. In this method, a response element will be introduced to control the frevariation quency response of the beamformer at the look direction. Based on the new formulation, an online LMS-type (least mean square) adaptive solution is then derived. The performance of the above beamformers is very sensitive to array calibration errors, and especially the error in the DOA angle of the signal of interest. If the desired signal does not come exactly from the designed look direction, it will be considered as an interference and the beamformer will tend to null out the desired signal at its output. Many methods have been proposed to improve the robustness of the beamformer against DOA angle mismatch error [7], [8]. For example, we can impose additional derivative constraints to the beamformer so that a wider main beam can be obtained to cover all the possible directions of the signal of interest [9]–[11]. Another choice is the diagonal loading method which improves the robustness of the beamformer by constraining the norm of its weight vector [8], [12]. Moreover, we can employ inequality constraints to control the magnitude response of the beamformer over a specified DOA range. Such an idea was initially proposed for robust narrowband beamforming [13]–[15]. In this paper, we will extend it to the wideband case and design a robust wideband beamformer based on convex optimization. In the proposed method, the element is first constrained to attain a frequency invariant main beam, after which we only need to impose one single soft magnitude constraint at each sampled angle point. A class of robust beamformers based on worst-case performance optimization using convex optimization techniques have also been proposed for both narrowband and wideband arrays [15]–[17]. In [17], a group of constraints are imposed on sampled frequency points over the frequency range of interest to

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prevent the mismatched desired signal from being filtered out by the beamformer. There are two problems with the approach. One is its relatively high computational complexity due to its constraints imposed on a large number of sampled frequency points; the other one is that there is no mechanism to control the response consistency to the mismatched desired signal so that a potentially intolerable distortion to the desired signal may happen. To address the above two problems, we first apply the element to the array to ensure a good response consistency in the robust DOA region, and then impose just one constraint on the reference frequency point at the look direction. Compared to the approach in [17], both the system’s efficiency and the frequency response consistency to the desired signal are improved significantly. This paper is organized as follows. The wideband beamforming structure with tapped delay-lines (TDLs) is reviewed briefly in Section II, including the Frost beamformer with its two solutions. The traditional adaptive wideband beamformer with added frequency invariance constraints is proposed in Section III. The two wideband beamformers robust against look direction estimation errors are proposed in Section IV. Simulation results are provided in Section V, where the performances of both the proposed methods and the conventional methods are compared in details. Conclusions are drawn in Section VI. II. WIDEBAND BEAMFORMING A general structure for wideband beamforming is shown in Fig. 1, where is the number of taps associated with each of the sensor channels. The beamformer obeying this architecture samples the propagating wave field in both space and time. Its response as a function of the angular frequency and DOA can be expressed as

Fig. 1. A general wideband beamforming structure.

with and

. Then we obtain the response as a function of (6)

with

(7) In our simulations,we choose the middle point of the array as . the zero-phase reference point so that Suppose the signal of interest comes from the broadside of the . Then the Frost beamformer can be formulated as array follows: (8) where received array signal with

is the covariance matrix of the

(1) (9) where is the unit delay in the TDL or sampling period, is the delay between the th sensor and the zero-phase reference point. In a vector form, we have

and

is the coefficient vector defined as (3)

and

is the

steering vector given by

constraint matrix

.. .

(2) where

is a

.. .

..

.

.. .

where and are the column vectors containing ones and zeros, respectively. is the response vector with one entry being 1 and all the others being zero. The solution to the problem in (8) can be obtained by the method of Lagrange multipliers, given by [4] (11)

(4) where trix

can be approximated by the sample covariance ma-

For a uniform linear array (ULA) with an inter-element spacing , . With the normalized , we have angular frequency (5)

(10)

(12) with

being the number of samples available.

ZHAO et al.: ADAPTIVE WIDEBAND BEAMFORMING WITH FREQUENCY INVARIANCE CONSTRAINTS

Alternatively, an online LMS-type solution to the problem in (8) can be obtained as [4] (13) with

(14) where

is a real-valued step size.

III. ADAPTIVE WIDEBAND BEAMFORMER WITH FREQUENCY INVARIANCE CONSTRAINTS Given the constraints of the Frost beamformer in (8), the unity gain is preserved at the broadside direction over all possible frequencies. However, in many cases, the frequency range of interest is not the entire normalized frequency band and it is not necessary to maintain an exact unity gain over the frequency range of interest either. Applying the constraints only to the frequency range of interest and simultaneously reducing the consistency of the beamformer’s response at the look direction over the operating frequency range will leave more degrees of freedom for the beamformer to suppress the interfering signals. To design a wideband beamformer with a specified frequency range, conventionally we can formulate the constraint matrix by sampling the frequency range of interest and constrain the response of the beamformer to those frequency points to be the desired ones, which are usually some pure delays or zeros if we want to null out this signal. In this case broadside arrival of the signal of interest is unnecessary so that can be any angle within (for a linear array). Suppose the frequency range of and we uniformly sample with frequency interest is . The corresponding desired repoints , sponse of the beamformer for the frequency point with the . Then the constraint for the direction is given by frequency points can be formulated as (15) where

is an

constraint matrix

1177

and denote the real part and imaginary part where of their variables, respectively. Note that the constraints are assumed to be linearly independent so that has a rank . This is not an efficient way to utilize the degrees of freedom of the array, since each linear constraint uses one degree of freedom in the weight vector . With condegrees of freedom available straints there are only for minimizing the output power of the array. A more efficient representation is the eigenvector constraint design approach developed based on the low-rank representation of wideband source signals with singular value decomposition (SVD) [6]. As shown in Appendix I, the constraints in (18) can be approximated by (20) Combining the eigenvector constraints in (20) along with the minimization of output power of the beamformer, we have the following new LCMV formulation (21) with its LMS-type solution as (22) where

(23) By (20), the number of constraints is reduced compared to that in (18), However as shown in Appendix II, when both the frequency range of interest and the dimension of the array is large, this approach still leads to a large number of constraints. Furthermore, these constraints are “hard” and each of them will take exactly one degree of freedom from the system. In the next, we propose a “soft” approach to the design of constraints by introducing a new element to control the beamformer’s response over the frequency range of interest at the look direction, which is called response variation (RV). In a general form, it is defined as [18]–[20]

(16) and

is an

response vector (24) (17)

Note that and are complex-valued and we can change the complex constraints into real ones as follows (for a real-valued ) (18) with

(19)

(25) where shows the DOA range over which the parameter is the reference frequency and we have assumed is measured, that is real-valued. is a measurement of the Euclidean disand that at all the other opertance between the response at ating frequencies over a range of directions over which the is measured. When is zero, the beamformer has a consistent frequency invariant response over the frequency range and . the DOA range

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Since we only consider the look direction , is reduced to a single DOA angle point. Then (24) and (25) change to

(26)

Compared to the eigenvector constraint approach in (21), the number of ‘hard’ constraints is reduced significantly to only two and the consistency of the beamformer’s response at the look direction is represented by part of the cost function to be minimized. Thus the output SINR is expected to be improved due to the reduced number of ‘hard’ constraints.

and IV. ROBUST WIDEBAND BEAMFORMER AGAINST LOOK DIRECTION ESTIMATION ERROR (27) respectively. is calculated numerically by uniformly In the simulation, into grid points. To control the consistency discretizing of the frequency response of the beamformer at and also make sure the beamformer reaches the desired response, we can minand simultaneously constrain the beamformer’s reimize to the desired response , given by sponse at (28) Then the complete formulation for the proposed minimum variance beamformer can be obtained by combining (26) and (28) along with minimizing output power of the beamformer

(29) where is a real-valued trade-off parameter between the frequency invariant property at the look direction and the output power of the beamformer. A large will increase the consistency of the resultant beamformer’s response over the frequency range of interest at the look direction. Note that the constraint in (28) is complex-valued and we can change the single complex constraint into two real ones as follows: (30) with

(31) Then we can change (29) to

When the desired signal does not come exactly from the designed look direction , the beamformer will tend to suppress it as an interference. All the approaches introduced in the last section are very sensitive to this error. To improve the robustness of the system against the look direction estimation error, we next propose two methods based on convex optimization. A. Robust Wideband Beamformer With Frequency Invariance Constraints The first approach is based on a previously proposed method for robust narrowband beamforming, where inequality constraints on the magnitude response of the beamformer over a specified DOA range were introduced [13]–[15] (35) where is the response of a narrowband beamformer, and are the lower and upper limits of the magnitude response, is the DOA range where the magnitude constraints and are imposed. This idea can be extended to the wideband case directly as follows: (36) To represent the frequency-angle constraints in (36), we have to sample both the angle and the frequency ranges by a sufficiently large number of points. Although we can employ the approach in (20) for a more effective representation, it still demands a large number of constraints over the DOA range. An efficient element into the constraint set solution is to incorporate the to achieve a frequency invariant main beam so that we only need to impose one magnitude constraint at each sampled angle point corresponding to the reference frequency . We first limit the element in (24) to a very small value by imposing the following constraint: (37)

(32) Similar to the Frost beamformer in (13), we can easily derive an online LMS-type algorithm for the new problem in (32), as given in the following:

(33) with (34)

which can be simplified into (38) where , with being a diagonal matrix inbeing the eigenvector cluding all the eigenvalues of , and matrix containing the corresponding eigenvectors. Note here is calculated for the range . In the simulation, is computed numerically and and are and grid points, respectively. uniformly discretized into

ZHAO et al.: ADAPTIVE WIDEBAND BEAMFORMING WITH FREQUENCY INVARIANCE CONSTRAINTS

With this constraint, we now only need to impose the magnitude constraints at , given by

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Then the constraint in (46) can be rewritten as

(39) Note that the formulation in (39) is not convex due to the . To convert it into a convex form, as an part approximation, we use

(47) Applying the Cauchy-Schwartz inequalities along with the in, , we have equality

(40) Similarly we have

(48) (41) Thus, to satisfy the constraint in (46), we can let

where , with being the diagonal matrix in, and being the correcluding all the eigenvalues of sponding eigenvector matrix. To avoid a large noise gain, we also need to constrain the norm of [21]

(49) (50) Then the RB-WC problem in (46) changes to

(42) where is a positive real-valued constant. Then, a complete formulation for the first robust wideband beamformer with a frequency invariance constraint (RB-FI) is obtained as

(43) B. Robust Wideband Beamformer With Frequency Invariance Constraints and the Worst-Case Performance Optimization In this section, we will propose another wideband beamformer robust against the look direction estimation error based on the worst-case performance optimization. In the presence of look direction estimation error, the actual steering vector differs from the ideal one by an error vector

(51) A potential problem with the above method is that the magnitude response of the resultant beamformer at the desired direction can vary for different frequencies and this kind of variation could be out of control and cause very large distortions to the desired signal. Moreover, as in the previous case, we also need to sample the frequency range of interest by a sufficiently large number of points, which will inevitably increase the computational complexity of the system. As a remedy, we can apply the constraint in the same way as before, which will improve the frequency invariance property of the resultant beamformer at the desired direction and also reduce the computational comconstraint is given by plexity of the system. In this case the (52) where is a small positive value. Then the formulation in (46) can be simplified to

(44) (53) where is the real steering vector of the desired signal is the steering vector correfrom direction , and sponding to the designed look direction . The worst-case performance optimization approach tries to eliminate the uncertainty included in the steering vector by , given by upper bounding the norm of the error vector (45) where is a small positive value. In [17], a robust wideband beamforming design method based on the worst-case performance optimization (RB-WC) has been proposed with the following formulation

(46)

Similarly, the second constraint in (53) can be replaced by (54) Then we obtain the following convex formulation for designing the robust wideband beamformer with the frequency invariance constraints and the worst-case performance optimization (RB-FI-WC)

(55) Since there are only two constraints imposed in (55), the computational complexity of the system will be reduced significantly compared to the method in (51).

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C. Algorithm Implementation Next we will develop a second-order cone (SOC) formulation of (55) and then solve it using the optimization tool SeDuMi [22]. Following [16], by introducing a new non-negative vari. Equation (55) can be converted to able , we let

(56) which can be further transformed to the following canonical dual form of the SOC programming problem as

(57) is the quadratic cone where is the variable vector, corresponding to the th inequality of the dimension , 2, 3), and constraint in (56) (

(58)

Fig. 2. Comparison of the Frost beamformer in (13) and the RV approach in (33) with , 1 and 0.1: (a) convergence of the output SINR; (b) frequency responses at the look direction.

= 10

V. SIMULATIONS AND RESULTS To show the effectiveness of the proposed methods, simulations are performed based on a ULA with the following specifications. and ; • We consider a ULA with and ; • the array spacing is assumed to be half the wavelength corresponding to the maximum normalized signal frequency so that ; • the desired signal has a signal-to-noise ratio (SNR) of 10 dB and two wideband interferences have a signal-to-interference ratio (SIR) of 10 dB. The directions of arrival for the three signals vary in different design examples; . • desired response A. The LMS-Type Adaptive Beamformers

More specifically

(59) and

(60) where

(61) Note the only desired weights in after optimization is . The methods in (43) and (51) can be transformed to the canonical dual form of the SOC programming problem in the same way. Using the primal-dual reduction method, the computaand that tional complexity of the RB-FI-WC is [23]. Since , our proof the RB-WC is posed RB-FI-WC has a substantially lower computational complexity than that of RB-WC.

apFirst we compare the performance of the proposed proach in (33) and the LMS-type Frost beamformer in (13). It is assumed that the desired signal comes from the broadside direction and two interferences arrive from the directions and 20 , respectively. The step size is set to 0.000004 for both cases and three values of the trade-off parameter are used with 10, 1 and 0.1, respectively. Fig. 2(a) shows the learning curves for the output SINR versus the iteration number for both the Frost beamformer and the proposed one, which is obtained by averaging 200 simulation based one has results. We can see clearly that the proposed led to an improved output SINR compared to the Frost beamformer. Moreover, with decreasing, a higher output SINR has been achieved, which can be explained by the fact that more degrees of freedom are released for interference suppression by relaxing the consistency constraint at the look direction. The resultant frequency responses at the look direction by the Frost beamformer and the proposed one are shown in Fig. 2(b), where we can see that the Frost beamformer has exactly a unity response over all frequency components at the look direction, while with a decreasing , the frequency response consistency of the proposed approach becomes poor, as expected. Next we comapproach in (33) and pare the performances of the proposed the eigenvector constraint approach in (22). Since both can be directly used to design beamformers with an off-broadside main beam, we assume the desired signal comes from

ZHAO et al.: ADAPTIVE WIDEBAND BEAMFORMING WITH FREQUENCY INVARIANCE CONSTRAINTS

Fig. 3. Comparison of the RV approach in (33) and the eigenvector constraint approach in (22): (a) convergence of the output SINR; (b) output SINR versus input SNR.

and the two interferences arrive from and 0 , respectively. The step size is set to 0.000004 and 0.000006 for approach and the eigenvector constraint approach, rethe , spectively. For the eigenvector constraint approach, as , and the frequency range of interest , we have . Therefore, we choose . The learning curves for the output SINR versus the iteration number are shown in Fig. 3(a), obtained by averaging 200 approach achieves a better simulation results. The proposed performance in terms of interference suppression than that of the eigenvector constraint approach. Additionally, in this case the resultant variance of the magnitude responses at the direction for the proposed approach and the eigenvector constraint approach is 0.0084235 and 0.0091174, respectively, which indicates that the former one leads to a slightly better consistency at the look direction than the latter one. Thus we approach can achieve an can conclude that the proposed improved performance compared to the eigenvector constraint approach under the condition that they have similar consistency of responses at the look direction. The improvement in performance arises from a larger number of freedom for interference approach. For this case, the apsuppression by the degrees of freedom available for proach has minimizing the cost function of the beamformer; in contrast, the deeigenvector constraint approach has only grees of freedom available. We also give the output SINR result versus the input SNR for both approaches in Fig. 3(b). It can be observed that the approach can always achieve a better output SINR for any given value of the input SNR. B. Robust Beamformers Against Look Direction Error In this section, we perform simulations for the two proposed robust beamformers: the RB-FI beamformer in (43) and the RB-FI-WC beamformer in (55). In addition, the results based on the previously proposed RB-WC beamformer in (51) and the Frost beamformer with the Lagrange multipliers solution in (11) are also provided as a comparison. The desired signal comes and two interferences arrive from from the direction and 20 , respectively. In the following, the designed look direction is always 0 , which gives an angle estimation error of 10 .

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Fig. 4. The resultant beam pattern based on the RB-FI beamformer with the RV constraint applied to [ 10 ; 10 ].

0

Fig. 5. Output SINR versus input SNR for the RB-FI-WC beamformer, the RB-WC beamformer and the Frost beamformer in (11) with an angle estimation error of 10 .

First we perform one simulation based on the RB-FI beamconstraint is applied to the range . former. The The values of , , and are chosen. The resultant beam pattern is shown in Fig. 4, which shows a very good performance in terms of frequency invariant angle range, response ripple control and property over the interference suppression. is and the For RB-FI-WC, the range of and are chosen; for RB-WC, we set values of . Fig. 5 shows the output SINR versus the input SNR for the RB-FI-WC beamformer, the RB-WC beamformer and the Frost beamformer, obtained by averaging 200 simulation results. Obviously RB-FI-WC and RB-WC are much more effective in coping with the look direction mismatch problem. Moreover, for this case with an estimation error of 10 , the proposed RB-FI-WC provides a better performance than that of RB-WC, especially for high SNRs. In the next, we study their performances in term of output SINR versus angle estimation error, and the result is shown in Fig. 6(a). It can be seen that the RB-FI-WC beamformer outperforms the RB-WC beamformer when the error is greater than 6 . As we mentioned in the last section, one potential problem with the RB-WC beamformer is that it may have the frequency response consistency to the mismatched desired signal out of control, leading to an intolerable distortion. The variance of responses at the directions where the mismatched desired signal comes from versus DOA angle is shown in Fig. 6(b), where

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Fig. 6. Comparison of the RB-FI-WC beamformer and the RB-WC beamformer: (a) output SINR versus DOA of the mismatched desired signal with an SNR of 10 dB; (b) variance of responses at the direction where the mismatched desired signal comes from.

Fig. 8. Comparison of the RB-FI-WC beamformer and the RB-WC beamformer: (a) output SINR versus DOA of the mismatched desired signal with an SNR of 10 dB; (b) variance of responses at the direction where the mismatched desired signal comes from.

and . The variance of responses comes from between at the directions where the mismatched desired signal comes from is shown in Fig. 8(b), where the variances for RB-FI-WC represent a very good frequency consistency from DOA of to . For the RB-WC beamformer, it again causes a too large and intolerable variance of the responses when a large estimation error happens. The variance is much larger than the simulation results shown in Fig. 6(b) with a smaller problem size. VI. CONCLUSION Fig. 7. Convergence of the output SINR for the RV approach in (33) and the eigenvector constraint approach in (22).

the variances for the RB-FI-WC beamformer represent a very good frequency consistency from DOA of 1 to 10 . For the RB-WC beamformer, the variances have a dramatic rise and reach 8.5576 at 10 , causing a too large and intolerable variance of the responses. Finally we perform some simulations with a larger problem and size. The dimensions of the array are increased to and it is assumed that there are four interferences with , 0 and 30 , 60 , respecan SIR of 10 dB coming from tively, and one desired signal with an SNR of 10 dB coming . For the approach in (33) and the eigenvector from constraint approach in (22), the step size is set to 0.000004 and , and 0.000006, respectively. Based on (69), we have therefore we choose . is set to be 2.5. Fig. 7 shows the learning curves for the output SINR versus the iteration number , obtained by averaging 200 simulation results. The approach again has achieved a better performance. The resultant variance of the magnitude responses at the direction for the proposed approach and the eigenvector constraint approach is 0.0087 and 0.0095, respectively. For RB-FI-WC, is and the values of and the range of are chosen; for RB-WC, we set . The Output SINR versus DOA of the mismatched desired signal is shown in Fig. 8(a), in which the RB-FI-WC beamformer outperforms the RB-WC beamformer when the mismatched desired signal

A response variation (RV) element has been introduced to control the frequency invariant property of the adaptive wideband beamformer at the look direction region over the frequency range of interest. By adding it into the cost function of the LCMV beamformer, a new linearly constrained beamformer has been derived with a trade-off between frequency response consistency at the look direction and output power minimization. Due to the increased number of degrees of freedom for interference suppression, compared to the original Frost beamformer and the eigenvector constraint approach, an improved SINR is achieved. In addition, two novel wideband beamformers robust against look direction estimation errors are proposed with their solutions based on the convex optimization technique. One is designed by imposing an constraint on element and simultaneously limiting the magnitude the response of the beamformer within a pre-defined angle range at a reference frequency; the other one is obtained by combining element and the worst-case performance optimization the method. Compared with the original robust methods, a more efficient and effective control over the beamformer’s response at the look direction region has been achieved with an improved overall performance, as shown by our simulations. APPENDIX I In the eigenvector constraint approach, the constraint matrix is decomposed into the product of three matrices with a singular value decomposition (SVD) operation given by (62)

ZHAO et al.: ADAPTIVE WIDEBAND BEAMFORMING WITH FREQUENCY INVARIANCE CONSTRAINTS

where is an diagonal matrix containing the singular unitary values of in a descending order, is an unitary matrix. matrix and is a To find a rank approximation matrix to the matrix , we separate matrix into two parts as follows: (63) holds the first columns of , and holds its rewhere maining columns. Matrix is split in the same way as (64) Then

is given by (65)

after which the original constraint formulation changes to (66) It can be further simplified to (67) with

.

APPENDIX II The number of constraints in (67) depends on the value of . A large leads to a good response consistency at the look direction , but leaves less number of degrees of freedom for output power minimization. A detailed study has shown that a wideband signal can be represented accurately by [6]

(68)

is the temporal duration orthogonal basis functions, where for the signal to propagate through the beamformer to the output from the time it first reaches the array, given by ; for real-valued bandpass signals within the , , and frequency range is the ceiling function rounding its element to the next integer to towards infinity [6]. Then as a guideline, is chosen to be span the constraint space effectively. Based on the normalized , (68) changes to angular frequency and

(69) where the frequency range of interest

.

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REFERENCES [1] W. Liu and S. Weiss, Wideband Beamforming: Concepts and Techniques. Chichester, U.K.: Wiley, 2010. [2] E. W. Vook and R. T. Compton, Jr., “Bandwidth performance of linear adaptive arrays with tapped delay-line processing,” IEEE Trans. Aerosp. Electron. Syst., vol. 28, no. 3, pp. 901–908, July 1992. [3] N. Lin, W. Liu, and R. J. Langley, “Performance analysis of an adaptive broadband beamformer based on a two-element linear array with sensor delay-line processing,” Signal Processing, vol. 90, pp. 269–281, Jan. 2010. [4] O. L. Frost, III, “An algorithm for linearly constrained adaptive array processing,” Proc. IEEE, vol. 60, no. 8, pp. 926–935, Aug. 1972. [5] B. D. Van Veen and K. M. Buckley, “Beamforming: A versatile approach to spatial filtering,” IEEE Acoust., Speech, Signal Processing Mag., vol. 5, no. 2, pp. 4–24, April 1988. [6] K. M. Buckley, “Spatial/spectral filtering with linearly constrained minimum variance beamformers,” IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-35, no. 3, pp. 249–266, Mar. 1987. [7] ,M. S. Brandstein and D. Ward, Eds., Microphone Arrays: Signal Processing Techniques and Applications. Berlin, Germany: Springer, 2001. [8] ,J. Li and P. Stoica, Eds., Robust Adaptive Beamforming. Hoboken, NJ: Wiley, 2005. [9] M. H. Er and A. Cantoni, “Derivative constraints for broadband element space antenna array processors,” IEEE Trans. Antennas Propag., vol. AP-31, no. 6, pp. 1378–1393, Dec. 1983. [10] K. M. Buckley and L. J. Griffith, “An adaptive generalized sidelobe canceller with derivative constraints,” IEEE Trans. Antennas Propag., vol. 34, no. 3, pp. 311–319, Mar. 1986. [11] K. C. Huarng and C. C. Yeh, “Performance analysis of derivative constraint adaptive arrays with pointing errors,” IEEE Trans. Antennas Propag., vol. 40, pp. 975–981, Aug. 1992. [12] B. D. Carlson, “Covariance matrix estimation errors and diagonal loading in adaptive arrays,” IEEE Trans. Aerosp. Electron. Syst., vol. 24, pp. 397–401, Jul. 1988. [13] B. D. Van Veen, “Minimum variance beamforming with soft response constraints,” IEEE Trans. Signal Processing, vol. 39, no. 9, pp. 1964–1972, Sep. 1991. [14] M. H. Er, “On the limiting solution of quadratically constrained broadband beam formers,” IEEE Trans. Signal Processing, vol. 43, no. 1, pp. 418–419, Jan. 1993. [15] Z. L. Yu, W. Ser, M. H. Er, Z. H. Gu, and Y. Q. Li, “Robust adaptive beamformers based on worst-case optimization and constraints on magnitude response,” IEEE Trans. Signal Processing, vol. 57, pp. 2615–2628, Jul. 2009. [16] S. A. Vorobyov, A. B. Gershman, and Z.-Q. Luo, “Robust adaptive beamforming using worst-case performance optimization: A solution to the signal mismatch problem,” IEEE Trans. Acoust., Speech, Signal Processing, vol. 51, pp. 313–324, Feb. 2003. [17] M. Rubsamen and A. B. Gershman, “Robust presteered broadband beamforming based on worst-case performance optimization,” in Proc. IEEE Workshop on Sensor Array and Multichannel Signal Processing, Darmstadt, Germany, Jul. 2008, pp. 340–344. [18] H. Duan, B. P. Ng, C. M. See, and J. Fang, “Applications of the SRV constraint in broadband pattern synthesis,” Signal Processing, vol. 88, pp. 1035–1045, Apr. 2008. [19] Y. Zhao, W. Liu, and R. J. Langley, “Subband design of fixed wideband beamformers based on the least squares approach,” Signal Processing, vol. 91, pp. 1060–1065, Apr. 2011. [20] Y. Zhao, W. Liu, and R. J. Langley, “An application of the least squares approach to fixed beamformer design with frequency invariant constraints,” IET Signal Processing, vol. 5, 2011. [21] D. P. Scholnik and J. O. Coleman, “Formulating wideband array-pattern optimizations,” in Proc. IEEE Int. Conf. on Phased Array Systems and Technology, Dana Point, CA, May 2000, pp. 489–492. [22] J. K. Sturm, “Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones,” Optimiz. Meth. Software, vol. 11–12, pp. 625–653, Aug. 1999. [23] M. Lobo, L. Vandenberghe, S. Boyd, and H. Lebret, “Applications of second-order cone programming,” Linear Alg. Its Applicat., vol. 284, pp. 193–228, Nov. 1998.

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Yong Zhao was born in Liaoning, China, in May 1983. He received the B.Eng. degree in communication engineering from Tianjin University, China, in 2006 and the M.Sc. degree in communications and signal processing from Imperial College London, U.K., in 2007. Since December 2007, he has been working toward the Ph.D. degree at the University of Sheffield, Sheffield, U.K. His research interests include array signal processing and multirate systems.

Wei Liu (S’01–M’04–SM’04) was born in Hebei, China, in January 1974. He received the B.Sc. degree in space physics (minor in electronics) and the L.L.B. degree in Intellectual property law both from Peking University, China, in 1996 and 1997, respectively, the M.Phil. degree from the University of Hong Kong, in 2001, and the Ph.D. degree from the University of Southampton, Southampton, U.K., in 2003. He then worked as a Postdoctoral Researcher in the Communications Research Group, School of Electronics and Computer Science, University of Southampton, and later in the Communications and Signal Processing Group, Department of Electrical and Electronic Engineering, Imperial College London, U.K. Since September 2005, he has been with the Communications Research Group, Department of Electronic and Electrical Engineering, University of Sheffield, Sheffield, U.K., as a Lecturer. His research interests are mainly in array signal processing, blind source separation/extraction, and multirate signal processing. He has now authored and coauthored about 90 journal and conference publications, and a monograph about wideband beamforming, Wideband Beamforming: Concepts and Techniques (Wiley, March 2010).

Richard J. Langley (M’85) received the B.Sc. and Ph.D. degrees from the University of Kent, Kent, U.K. After spending some time working on communications satellites at Marconi Space Systems in the 1970s he became a Lecturer at the University of Kent in 1979. He was promoted to a personal Chair in Antenna Systems in 1994. In 1997 he founded the European Technology Centre for Harada Industries Japan, the world’s largest supplier of automotive antennas. The center researches and develops advanced hidden antenna systems for the global automotive market including radio, telephone and navigation systems. After successfully building up the technology and business he rejoined academic life in 2003. He is currently Head of the Communications Research Group, University of Sheffield, Sheffield, U.K. His main research is in the fields of automotive antennas, propagation in the built environment, frequency selective surfaces, electromagnetic band gap materials and applications, multi-function antenna systems and reconfigurable antennas. He has published over 250 papers in international journals and conferences Prof. Langley was Honorary Editor of the Inst. Elect. Eng. Proceedings—Microwaves, Antennas and Propagation from 1995 to 2003. In 2009, he initiated the setting up of the Wireless Friendly Building Forum to address the problems of wireless signal propagation in buildings and the built environment. He is currently Chair of the IET Antennas and Propagation Professional Network.

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A 4-Element Balanced Retrodirective Array for Direct Conversion Transmitter Leung Chiu, Member, IEEE, Quan Xue, Fellow, IEEE, and Chi Hou Chan, Fellow, IEEE

Abstract—A phase conjugated array capable of sending two individually modulated signals towards the direction of an interrogator is proposed. Each of the array elements is a dual-fed patch with one input port connected to a 90 phase shifting element and a resistive field effect transistor (FET) mixer and the other port just the mixer alone, making the array a balanced structure. A 4-element working prototype was designed at the center frequency of 5.8 GHz. A linearly polarized retrodirectivity has been experimentally confirmed by measuring bi-static radiation patterns with the interrogating signal coming at 0 , , , , and . A 8.6-dB peak power variation of re-radiation within normal to plane of the array has been shown in the measured mono-static radiation patterns. To test the ability of direct conversion, an active integrated direct conversion receiver employing the same array has been designed to receive and demodulate the response signals from the phase conjugated array. A 5 MHz square and 8 MHz sinusoidal waves as two base-band signals were carried through the phase conjugated array and successfully recovered by the direct conversion receiver.

+30 +60

15

60

45

Index Terms—Direct conversion, dual-fed patch antenna, resistive FET mixer, retrodirective array.

I. INTRODUCTION

A

RETRODIRECTIVE array with additional feature of direct conversion is proposed for a low-cost transceiver architecture. The rapid growths in wireless communications systems that permit high data-rate transmissions have demanded the spectrum to be shared more efficiently. For this reason, various digital modulation techniques have been proposed. However, the crowded spectrum results in serious interference problem. Another solution is to raise the operation frequency to millimeter-wave and even higher frequency bands. However, conventional heterodyne transceiver architecture at these frequency ranges is too expensive for mass production. Direct conversion technique as an alternative front-end architecture has been proposed for low-cost solutions. This technique is well known with various benefits in comparison to the conventional heterodyne transceiver solution [1].

Manuscript received January 09, 2010; revised June 15, 2010; accepted July 31, 2010. Date of publication January 28, 2011; date of current version April 06, 2011. This work was supported by the Shenzhen Science and Technology Planning Project for the Establishment of Key Laboratory in 2009 at Project CXB200903090021A. L. Chiu is with the Advanced Research and Development Centre, Telefield Limited, Hong Kong SAR, China (e-mail: [email protected]). Q. Xue and C. H. Chan are with the State Key Laboratory of Millimeter Waves, City University of Hong Kong, Kowloon, Hong Kong SAR, China (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.2011.2109355

Fig. 1. Phase conjugated array with modulated signal input.

A phase conjugated array exhibits retrodirectivity by heterodyne technique, and therefore the output is one of the products generated from the non-linear mixing. Recently, phase conjugating circuits basing phase locked loop were reported in [2], [3]. The array responds to a signal source by conjugating the phase of the incoming signal received by each antenna element and re-sending modulated signals towards the source direction, leading to a wide range of applications [4]. Apart from retrodirecitvity, the arrays can also exhibit variously additional functions such as multipath reduction for secure data transmission [5], hardware reduction for low-cost designs [6], [7], power management for DC power saving [8], and reconfigurable array system for retrodirectivity/direct conversion receivers [9]. The design reported in [9] achieves retrodirectivity or direct conversion receiver, which is controlled by the use of time-division duplexing and an improved base-band circuitry. In this paper, the architecture of the balanced phase conjugated array for direct conversion is introduced. Next, active integrated antenna for direct conversion receiver and its results are reported. Finally, measured results of retrodirectivity as well as dual-channel transmission are presented. II. SYSTEM OVERVIEW For a conventional phase conjugated array as shown in Fig. 1. The base-band IF signal is up-converted as a response signal by the phase conjugating mixer. The response signal is sent to the source direction. In our pervious work [10], a simple base-band 1 kHz ON-OFF keying signal was applied to the IF port and up-converted to 11.61 GHz pumping signal for the phase conjugated array. The pumping signal was then down converted

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to 5.81 GHz response signal. By receiving the response of the phase conjugated array, the base-band signal was successfully retrieved. The existence of the power amplifier has no effect on the retrodirectivity and pattern deformation. However, the power level of the pumping signal, which determines the conversion loss/gain of the phase conjugating mixer as well as the link gain, is controlled by the power amplifier. It is obvious that the data rate is limited by the bandwidths of both the antenna and phase conjugating mixer. In order to share the spectrum within a limited bandwidth more effectively, direct conversion technique is firstly introduced to the phase conjugated array. For the direct conversion circuitry, a balanced structure with 90 delay lines is required. The proposed design mainly consists of two identical phase conjugated arrays, Array 1 and Array 2, and , are is shown in Fig. 2(a). Two base-band signals, up-converted as two pumping signals with 90 phase difference, a critical step for direct conversion operation. The two pumping signal are amplifier by the two power amplifiers to maintain the link gain. The phase difference is achieved by a simple delay line, which is suitable for demonstration as a single-frequency and pump the Array operation. These two up-converted 1 and the Array 2, respectively. An additional 90 phase delay at the array’s working frequency should be installed in one of the arrays. As shown in Fig. 2(b), in Array 1, the delay line between the antenna and the phase conjugating mixer introduces an additional 180 phase delay for reflected RF signal. This delay line can be used to cancel the leakage RF signal generated in the phase conjugation, while this delay line has no effect on the phase conjugated signals as well as retrodirectivity. In our study, Array 1, Array 2, and RF signal in Fig. 2 are designed at the center frequency of 5.8 GHz, while the pumping signal, LO in Fig. 2, is at the center frequency of 11.6 GHz. III. BALANCED PHASE CONJUGATED ARRAY The proposed phase conjugated array is shown in Fig. 3. It mainly consists of two parts, namely, the circuit generating pumping signal and the balanced phase conjugated array. Two coaxial cables with equal length are used to connect the circuit and the array. For the circuit generating pumping signal, two base-band sigand , are up-converted to the center frequency of nals, 11.6 GHz as the pumping signals for phase conjugation. A 90 phase difference between the two pumping signals is required for direct conversion purpose. It is done by dividing the LO signal into two paths equally in magnitude but with the 90 phase shift. The output of the power amplifier determines the power level of the pumping signal, which is critical for conversion efficiency of the phase conjugation but not for the normalized radiation pattern deformation of the array. Therefore the existence of power amplifiers is optional if only retrodirectivity is tested. For the balanced phase conjugated array, 4 dual-fed patch elements and 2 sets of 4-element resistive FET mixers with two common pumping signal ports was fabricated on the rectangular printed circuit board with the size of 110 mm 240 mm. As shown in upper part of Fig. 3, the 4 microstrip lines connecting the antennas and the mixers at right hand side have additional 90 phase difference. The dual-fed patch antenna reported in

Fig. 2. (a) Dual-array architecture for direct conversion transmitter. (b) The phase delays of RF signals towards and reflected from the phase conjugating mixers in arrays 1 and 2.

[11] was used as a common load for two mixers at both sides. One 4-element dual-fed patch antenna array is used for radiating element instead of 2 4-element single-fed antenna arrays to save circuit area and provide isolation. The dual-fed patch antennas in Fig. 3 were designed at the center frequency of 5.8 GHz. For the phase conjugating mixer in Fig. 3, the frequencies of the input signal, phase conjugated signal, and pumping signal are 5.8 GHz, 5.8 GHz, and 11.6 GHz, respectively. The signal from the integrator, namely the RF, is chosen at 5.801 GHz, and hence the phase conjugated signal is 5.799 GHz. These two signals are chosen at different frequency to be easily observed and differentiated in the measurement. These frequencies can be the same in practical applications. The 90 delay line introduces an extra 180 phase difference for the reflected RF signal at the input port when compared with that of the other port of the dual-fed element; therefore the RF leakage are suppressed by about 35 dB in the bore-sight direction as shown in Fig. 4, where the importance of this signal suppression is reported in [12]. IV. DIRECT CONVERSION RECEIVER To test the direct conversion ability of the proposed array, a direct conversion receiver should be employed. The architecture of an active integrated antenna achieving direct conversion is proposed and its schematic diagram and layout are shown in Fig. 5. The working prototype was fabricated on the rectangular printed circuit board with the size of 110 mm 150 mm.

CHIU et al.: A 4-ELEMENT BALANCED RETRODIRECTIVE ARRAY FOR DIRECT CONVERSION TRANSMITTER

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Fig. 3. The layout of the proposed balanced phase conjugated array with the schematic diagram of the circuit for pumping signal.

Fig. 4. Comparison of output responses of balanced phase conjugated array with and without the 90 delay line.

Two active second sub-harmonic pumped mixers were employed for the down-converter. For the second harmonic mixing, large difference in RF and LO frequencies results in ease of achieving high isolation. Active mixing results in positive conversion gain. The structure of employed mixers is single-ended using FET as a core biased at 2 V and 10 mA and with RF and

LO center frequencies of 5.801 GHz and 2.9 GHz, respectively. RF and LO were injected into the gate terminal through a simple microstrip frequency diplexer and IF was generated by the time varying transconductance of the FET and extracted from the drain terminal through a low-pass filter. For the mixer measurement, 5.801-GHz RF and 2.9-GHz LO frequencies were chosen, where the 1-MHz IF power level was measured by a spectrum analyzer. The conversion gain and RF-LO isolation of the mixers with different RF frequencies were measured as shown in Fig. 6. The mixer provides about 5 dB conversion gain and 40 dB RF-LO isolation with 0 dBm LO and 20 mW DC power consumptions. For the direct conversion circuit, two divided LO signals with equal magnitude and 90 phase shift are required. For this paper, second-harmonic mixer is used; hence, the LO signal is divided into two paths for the two mixers with equal magnitude and 45 phase shift, while the RF single equally distributes to each active mixer. The proposed dual-fed antenna serves both the functions of receiving RF signal and an equal power divider with port-to-port isolation. The two IF ports serve as the I- and Q-channels for direct conversion receiver. The measurement setup for the receiver is shown in Fig. 7. A 5.801 GHz pure sinusoidal RF signal was transmitted through a standard horn antenna to the proposed receiver. The distance between the horn antenna and proposed design was kept at the far field region ( 2 m in our measurement). The two IF port were connected

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Fig. 7. The measurement of direct conversion with the fabrication layout of the integrated direct conversion receiver.

Fig. 5. (a) The schematic diagram of the integrated direct conversion receiver and the role of the proposed dual-fed antenna. (b) The layout of the direct conversion receiver for fabrication. Fig. 8. Voltage waveforms of the I- and Q-channels of the intergraded direct conversion receiver measured by digital oscilloscope.

V. EXPERIMENTS A. Retrodirectivity

Fig. 6. Measured conversion gain and RF-LO isolation of the active mixer for direct conversion receiver.

to the digital oscilloscope. Two pure sinusoidal waves with 90 phase difference were successfully obtained as shown in Fig. 8.

The re-radiation patterns of the proposed phase conjugated array should be measured to experimentally confirm the retrodirectivity. By following the experiment setup as mentioned in [4], the measurements of mono- and bi-static radiation patterns were carried out to demonstrate retrodirectivity. 11.6 GHz signals with in-phase and equal magnitude are applied to the two pumping signal ports. Applying an RF signal at 5.801 GHz results in an IF signal of 5.799 GHz and both signals are displayed on the spectrum analyzer. Fig. 9(a) shows the normalized bi-static radar cross-section patterns. 5 radiation patterns are measured while the transmitted horn is located at angles of 0 , , , , and from the normal to the plane of the array. The peak radiation always pointing to the source direction is experimentally confirmed. The measured mono-static radiation pattern shows that the peak radiation power variation is about 8.6 dB within a 120 scanning range as shown in Fig. 9(b).

CHIU et al.: A 4-ELEMENT BALANCED RETRODIRECTIVE ARRAY FOR DIRECT CONVERSION TRANSMITTER

Fig. 9. (a) Measured bi-static radiation patterns of the balanced phase conjugated array with interrogator at 0 , , , , and . (b) Measured mono-static radiation patterns.

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Fig. 11. (a) Voltage waveforms of dual-channel signal transmitted from phase conjugated array. (b) Voltage waveforms of dual-channel signal received from integrated direct conversion receiver.

base-band signals. In our demonstration, 5 MHz square and 8 MHz sinusoidal waves were generated as the two base-band signals as shown in Fig. 11(a). These signals were up-converted to the center frequency of 11.6 GHz and used to pump the two sets of mixers in the balanced phase conjugated array. To stimulate the phase conjugated array by sending a 5.801 GHz pure sinusoidal wave towards the array, the response signal at the center frequency of 5.799 GHz was generated. The response signal carrying the two base-band signals was received by the proposed direct conversion receiver. The distance between the proposed designs should be kept at about 3 m to ensure that they lie in the far-field region to each other. The two waveforms of the two base-band signals were successfully recovered and displayed on the digital oscilloscope as shown in Fig. 11(b). Fig. 10. (a) Measurement setup for dual-channel transmission ability of the 5.8 GHz balanced phase conjugated array.

VI. CONCLUSION B. Direct Conversion To test the ability of direct conversion, the measurement setup shown in Fig. 10 is required. All antennas in Fig. 10 were designed at the center frequency of 5.8 GHz. Firstly, two arbitrary waveform generators were used to generate two individual

A 4-antenna-element balanced phase conjugated array for direct conversion has been presented. Distinct linearly polarized retrodirectiveity has been experimentally confirmed by measuring the bi- and mono-static radiation patterns. To test the direct conversion ability, two individual base-band signals were

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up-converted as pumping signals for phase conjugation and carried through the response signal of the phase conjugated array. An active integrated direct conversion receiver has been designed and used to successfully recover the two individual baseband signals. REFERENCES [1] A. A. Abidi, “Direct-conversion radio transceivers for digital communications,” IEEE J. Solid-State Circuits, vol. 30, no. 12, pp. 1399–1410, Dec. 1995. [2] V. F. Fusco, C. B. Soo, and N. B. Buchanan, “Analysis and characterization of PLL based retrodirective arrays,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 2, pp. 730–738, Feb. 2005. [3] N. B. Buchanan and V. F. Fusco, “Mirror image sawtooth phase conjugator circuit for retrodirective antenna applications,” IEEE Antennas Wireless Propag. Lett., vol. 6, pp. 503–505, 2007. [4] R. Y. Miyamoto and T. Itoh, “Retrodirective arrays for wireless communications,” IEEE Microw., pp. 71–79, Mar. 2002. [5] V. F. Fusco and N. B. Buchanan, “Retrodirective antenna spatial data protection,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 490–493, 2008. [6] D. S. Goshi, K. M. K. H. Leong, and T. Itoh, “A sparse retrodirective transponder array with a time shared phase-conjugator,” IEEE Trans. Antennas Propag., vol. 55, no. 8, pp. 2367–2372, Aug. 2007. [7] D. S. Goshi, K. M. K. H. Leong, and T. Itoh, “A sparsely designed retrodirective transponder,” IEEE Antennas Wireless Propag. Lett., vol. 5, no. 1, pp. 339–342, Dec. 2006. [8] S. Lim, K. M. K. H. Leong, and T. Itoh, “Adaptive power controllable retrodirective array system for wireless sensor server applications,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 11, pp. 3735–3743, Dec. 2005. [9] R. Y. Miyamoto, K. M. K. H. Leong, S. S. Jeon, Y. Wang, Y. Qian, and T. Itoh, “Digital wireless sensor server using an adaptive smartantenna/retrodirective array,” IEEE Trans. Vehicular Tech., vol. 52, no. 5, pp. 1181–1188, Sep. 2003. [10] K. W. Wong, L. Chiu, and Q. Xue, “2D phase-conjugated retrodirective array with information carrying capability,” IET Electron. Lett., vol. 43, no. 12, pp. 653–654, Jun. 2007. [11] L. Chiu and Q. Xue, “Dual-fed microstrip patch with higher-order radiating mode achieving port-to-port isolation,” in Asia-Pacific Microwave Conf., 2007, pp. 1361–1364. [12] R. Y. Miyamoto, Y. Qian, and T. Itoh, “An active integrated retrodirective transponder for remote information retrieval-on-demand,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 9, pp. 1658–1662, Sep. 2001.

Leung Chiu (S’05–M’08) received the B.Eng. and Ph.D. degrees in electronic engineering from the City University of Hong Kong, Hong Kong, in 2004 and 2008, respectively. He is currently with the Advanced Research and Development Centre, Telefield Limited, Hong Kong SAR, China. His research interests include microwave circuits and antenna arrays.

Quan Xue (M’02–SM’04–F’11) received the B.S., M.S., and Ph.D. degrees in electronic engineering from the University of Electronic Science and Technology of China (UESTC), Chengdu, in 1988, 1990, and 1993, respectively. In 1993, he joined the UESTC, as a Lecturer, where he became an Associate Professor in 1995 and a Professor in 1997. From October 1997 to October 1998, he was a Research Associate and then a Research Fellow with the Chinese University of Hong Kong. In 1999, he joined the City University of Hong Kong where he is currently a Professor and serves as the Director of the Information and Communication Research Center, Deputy Director of State Key Laboratory (Hong Kong) of Millimeter-waves of China, and the Assistant Head, Department of Electronic Engineering. Since May 2004, he has been the Principal Technological Specialist of the State Integrated Circuit (IC) Design Base, Chengdu, China. He has authored or coauthored over 180 internationally referred journal papers and over 70 international conference papers His current research interests include microwave passive components, active components, antenna, microwave monolithic integrated circuits (MMIC), RFID and radio frequency integrated circuits (RFIC), etc. Dr. Xue is an Associate Editor of the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, and the Editor of the International Journal of Antennas and Propagation. He is an elected member of IEEE MTT-S AdCom, the Chair of IEEE Hong Kong Section AP/MTT Chapter. He was elected as an IEEE Fellow (2011) for contributions in microwave transmission line structures and integrated circuits.

Chi Hou Chan (S’86–M’86–SM’00–F’02) received the Ph.D. degree in electrical engineering from the University of Illinois at Urbana-Champaign, in 1987. In 1996, he joined the Department of Electronic Engineering, City University of Hong Kong, and was promoted to Chair Professor of Electronic Engineering in 1998. From 1998 to 2009, he was first Associate Dean, and then Dean of the College of Science and Engineering. He also served as Acting Provost of the university from July 2009 to September 2010. His research interests cover computational electromagnetics, antennas, microwave and millimeter-wave components and systems, RFICs and Terahertz devices and applications. Dr. Chan received the U.S. National Science Foundation Presidential Young Investigator Award in 1991 and the Joint Research Fund for Hong Kong and Macao Young Scholars, National Science Fund for Distinguished Young Scholars, China, in 2004. He received outstanding teacher awards from the EE Department, CityU, in 1998, 1999, 2000, and 2008. Students he supervised received numerous awards including one of the 22 Special Awards in the 2003 National Challenger’s Cup in China, the Third (2003) and First (2004) Prizes in the IEEE International Microwave Symposium Student Paper Contests, the IEEE Microwave Theory and Techniques Graduate Fellowship for 2004–2005, Undergraduate/Pre-Graduate Scholarships for 2006–2007 and 2007–2008, and the 2007 International Fulbright Science and Technology Fellowship offered by the U.S. Department of State.

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Dual Grid Array Antennas in a Thin-Profile Package for Flip-Chip Interconnection to Highly Integrated 60-GHz Radios Y. P. Zhang, Fellow, IEEE, M. Sun, Member, IEEE, Duixian Liu, Fellow, IEEE, and Yilong Lu, Member, IEEE

Abstract—We examine the current development of highly integrated 60-GHz radios with an interest in antenna-circuit interfaces. We design and analyze grid array antennas with special attention to the differential feeding and the patterned ground plane. More importantly, we integrate two grid array antennas in a package; propose the way of assembling it to the system printed circuit board; and demonstrate a total solution of low cost and thin profile to highly integrated 60-GHz radios. We show that the package in low temperature cofired ceramic (LTCC) technology measures only 13 2 13 2 0:575 mm3 ; can carry a 60-GHz radio die of current and future sizes with flip-chip bonding; and achieves good antenna performance in the 60-GHz band with maximum gain of 13.5 and 14.5 dBi for the single-ended and differential antennas, respectively. Index Terms—Ball grid array package, grid array antenna, low temperature cofired ceramic (LTCC), 60-GHz radio. Fig. 1. Illustration of the (a-b) single-end and (c-d) differential antenna-circuit interface in current highly integrated 60-GHz radios.

I. INTRODUCTION RADITIONAL commercialized 60-GHz radios have been designed as an assembly of several microwave monolithic integrated circuits (MMICs) in gallium arsenide (GaAs) semiconductor technology. They have been used for Gigabit Ethernet (1.25 Gb/s) bridges between local area networks [1], [2]. Recently, integrated transmitter (Tx) and receiver (Rx) GaAs pHEMT and mHEMT processes MMICs in 0.15have been realized to support data rates of several Gb/s for 60-GHz short-range applications [3], [4]. However, the 60-GHz radios in GaAs MMICs are expensive and bulky. In order for 60-GHz radios to have mass deployment and meet consumer marketplace requirements, the cost and size of any solution must be low and compact. That implies silicon, not GaAs as the better technology choice. In fact, designs towards low-cost highly integrated 60-GHz radios have been realized in silicon technologies. For example, Floyd, et al. have demonstrated a

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Manuscript received March 26, 2010; revised August 05, 2010; accepted August 30, 2010. Date of publication January 28, 2011; date of current version April 06, 2011. Y. P. Zhang and Y. L. Lu are with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798, Singapore (e-mail: [email protected]; [email protected]). M. Sun is with the Institute for Infocomm Research, Singapore 138623 (e-mail: [email protected]). D. Liu is with the IBM T. J. Watson Research Center, Yorktown Heights, NY 10598 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.2011.2109358

60-GHz Tx and Rx chipset in a 0.13silicon-germanium (SiGe) technology [5] and Tanomura, et al. in a 90-nm complementary metal oxide semiconductor (CMOS) technology [6]. An examination of the above works and many other reported 60-GHz highly integrated radios in SiGe and CMOS reveals that two types of antenna-circuit interfaces as shown in Fig. 1 can be identified in the current two-chip solutions. The first type features the 50- single-end and the second type the 100- differential antenna-circuit interfaces. For the first type, the 60-GHz on-chip input/output pads are designed as the ground-signal-ground (GSG) pads; while for the second type as the ground-signal-ground-signal-ground (GSGSG) pads. The GSG pads are bonded to an off-chip but in-package single-end antenna; while the GSGSG pads a differential antenna with either flip-chip or wire-bonding techniques [7]–[10]. A single-chip solution of a 60-GHz radio transceiver (TRX) in CMOS has been attempted [11], where differential Tx and Rx are integrated on the same die. It is known that CMOS scaling improves amplifier noise performance and gain but exacerbates the difficulty of generating sufficient output power by the power amplifier (PA) at 60 GHz [12]. Theoretically, a differential PA yields 3 dB more output power than a single-end one does. Hence, the differential antenna-circuit interface in Fig. 1(c) is preferred to the single-end antenna-circuit interface in Fig. 1(a) for the Tx integration of the TRX. Furthermore, the differential antenna-circuit interface in Fig. 1(d) is the better choice than the single-end antenna-circuit interface in Fig. 1(b) for the Rx integration of the TRX because the differential low noise amplifier

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(LNA) can achieve higher linearity, lower offset and better immunity to common-mode noise due to power supply variations or substrate coupling than the single-end LNA does [13]. The advantage of the fully differential architecture from the antenna to the circuit has been well understood for modern single-chip solutions of radio transceivers [14]. Regardless of the antenna-circuit interfaces; two antennas, one for transmission and the other for reception, are required for current highly integrated 60-GHz radios. Use of two antennas is not a problem as the antenna form factor at 60 GHz is on the order of millimeters. However, it may become a problem for the highly integrated 60-GHz radio that employs multiple antennas for beam steering to search the available path to enhance the link quality [15]. This is because multiple antennas not only require corresponding multiple electrostatic discharge circuits that consume substantial die area but also makes the whole radio bulky. It is known that the number of multiple antennas can be reduced to half by using transmit/receive (T/R) switches or circulators. Unfortunately, the T/R switches in CMOS in the 60-GHz band are still too lossy to be used [11], [16] and information on circulators for highly integrated 60-GHz radios is unknown. The remainder of the paper is organized as follows: Section II presents the design and analysis of grid array antennas with special attention to the differential feeding and the patterned ground plane. Section III describes the integration of the grid array antennas into a chip package in LTCC for highly integrated 60-GHz radio chipsets. As an example, a dual-feed grid array antenna is integrated for the differential Tx antenna-circuit interface and another single-feed grid array antenna for the single-end Rx antenna-circuit interface. Finally, Section IV concludes the paper.

II. DESIGN AND ANALYSIS OF GRID ARRAY ANTENNAS The grid array antenna was first proposed by Kraus in 1964 [17]. Since then, there have been some studies but all conducted at lower microwave frequencies [18]–[23]. Fig. 2 shows the basic grid arrangement and its variations in microstrip technology. The basic structure shown in Fig. 2(a) consists of rectangular meshes of microstrip lines on a dielectric substrate backed by a metallic ground plane and fed by a metal via through an aperture on the ground plane. Depending on the electrical length of the sides of the meshes, the grid array antenna may be resonant or nonresonant. For a resonant grid array antenna, the sides of the meshes should be one wavelength by a half-wavelength in the dielectric and the instantaneous currents would be out of phase on the long sides of the meshes and in phase on the short sides of the meshes, respectively. As a result, the long sides of the meshes behave essentially as microstrip line elements and the short sides act as both radiating and microstrip line elements producing the main lobe of radiation in the boresight direction. While for a nonresonant grid array antenna, the length of the short side of the meshes can be slightly more than one-third wavelength and the length of the long side of the meshes should be two times longer but three times shorter than the length of the short side of the meshes in the dielectric. Assuming that it is fed from one end, the currents in the short sides of the meshes

Fig. 2. The basic grid array antenna (a) and its variations (b-g).

follow a phase progression producing the maximum radiation in a backward angle-fire direction [17]. The grid array antenna was temporarily revived by Conti, Dowling, and Weiss in 1981 [18]. Fig. 2(b) shows their methods of amplitude control through control of microstrip line impedances to lower the first sidelobe. The grid array antenna has caught considerable attention of Nakano and his associates [20]–[23]. Since the middle of 1990s, they have reported the design and analysis of various grid array antennas. Fig. 2(c)–(e) show their miniaturized grid array antenna by meandering the long sides of the meshes, dual-linearly polarized grid array antenna by crossing the meshes, circularly polarized grid array antenna by modifying the short sides of the meshes, respectively. Fig. 2(f) and (g) shows our new 45 linearly polarized grid array antenna by adjusting the angle between the long and short sides of the meshes and miniaturized grid array antenna by meandering the long sides and bending the short sides of the meshes in a multi-layer metal structure, respectively, [24]. The bending makes the large part of the short sides of the meshes further away from the ground plane, which may improve the radiation. Although both resonant and nonresonant grid array antennas are useful for many applications, we only focus on the design and analysis of the resonant grid array antenna in this work at millimeter-wave frequencies. The design determines the dielectric substrate dimensions, the number of meshes, the microstrip

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Fig. 3. The dual-feed grid array antenna: (a) top and (b) bottom views.

line impedances, and the excitation location with the associated diameters of the metal via and the aperture through the analysis of the HFSS simulations. We demonstrate that the grid array antenna is easy to design, more importantly; the grid array antenna is particularly suitable for fabrication in LTCC as its meshed structure avoids micro fractures or warpage. Based on the A6 LTCC from Ferro, we design grid array antennas to operate in the 60-GHz band. The A6 ceramic type has the dielectric constant 5.9 and loss tangent 0.0015 and after firing and metallic paste is either silver or gold with good conductivity. A. Basic Dual-Feed Structure A basic dual-feed grid array antenna is designed, which targets the specifications at 61.5 GHz with the maximum gain of 15 dBi, the impedance and radiation bandwidth of 7 GHz, and the efficiency of 80%. Considering the specified gain value of 15 dBi and the various losses, one can find the required number of meshes to be at least 14, which leads to an estimation of the length and width of the substrate as 11.5 mm by 5 mm, respectively. The thickness of the substrate should be chosen to avoid mode surface wave. Finally, a body the excitation of the is determined by also taking the size of LTCC layout rule into account. For low cost and easy fabrication, the width and thickness of the microstrip lines are kept uniform as 0.15 and 0.01 mm, respectively. The optimized mesh dimensions and the location of dual feeds as well as the associated diameters of the metal vias and the apertures on the ground plane , , , are , , , , . and The grid array antenna has dual feeds as shown in Fig. 3. It is excited for differential and single-end operations, respectively. The differential feeding scheme here is different from those in [17], [23] where a gap is made on the short side of a mesh to connect to the differential source. Fig. 4(a) and (b) shows the simulated current distributions on the grid at 61.5 GHz for both excitations. Note that the instantaneous currents do not distribute as shown in [17]–[19], that is, they are out of phase on the long sides of the meshes and in phase on the short sides of the meshes, respectively. Rather, they are only truly out of phase on the long sides of the meshes and in phase on the short sides of the meshes near the feeding points. This is because at such a high frequency a slight mesh dimension change will cause a big change in signal phase over transmission, for example at 60 variation in will cause signal phase difGHz a 70ference, thus making the control over phase synchronization of

Fig. 4. The simulated results of the dual-feed grid array antenna: current distributions at 61.5 GHz for (a) differential and (b) single-end operations, (c) return loss, (d) E or xz - and (e) H or yz -plane patterns at 61.5 GHz, and (f) peak realized gain for both single-end and differential operations.

the far meshes from the source more difficult. The grid array antenna excited for differential operation has two source points, so it has more meshes of desirable current distributions than that

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Fig. 5. The ground planes: (a) meshed and (b) patterned.

excited for single-end operation. Fig. 4(c) compares the simulated antenna return losses. The impedance bandwidth is 10 GHz (16.7%) from 53 to 63 GHz for the single-end and 8 GHz (13%) from 57 to 65 GHz for the differential operations, respectively. The wider impedance bandwidth for the single-ended antenna is the result of less number of radiating elements that have in-phase currents. Fig. 4(d) and (e) compares the simulated antenna radiation patterns in the E-and H-planes at 61.5 GHz. As expected, the main lobe and deep null of radiation appear in the boresight direction for the co-and cross-polarization fields, respectively. The differential excitation yields a sharper main beam in the E-plane and a similar main beam in the H-plane and much weaker cross-polarization field in both planes as compared with the single-end operation, due to the current distributions shown in Fig. 4(a) and (b). The front-to-back ratio is 21 dB. Fig. 4(f) shows the simulated antenna peak gain. They are 16 and 13.5 dBi at 61.5 GHz for differential and single-end operations, respectively. The 3-dB gain bandwidths are enough. The simulated efficiency is better than 90% for both cases. B. Patterned Ground Plane The meshed ground plane is required in LTCC from the mechanical perspective. A patterned ground plane is created in this work which not only meets the requirement of mechanical reliability but also reduces the gain penalty. Fig. 5 shows the patterned ground plane of the grid array antenna. Note that the large metal patches are formed on the meshed ground plane below the radiating elements, which reduce radiation through the ground plane meshes and therefore help to reduce the gain penalty. Fig. 6 compares the simulated results of the grid array antenna between the conventional meshed and patterned ground planes for the differential operation. It is evident that both meshed and patterned ground planes shift down the resonant frequencies. They perturb the current return path and lead to the excitation of an electric field across the rectangular openings, and the reactive energy stored near the discontinuities is responsible for the downward frequency shift. Also, it can be observed that both meshed and patterned ground planes enhance the impedance bandwidth but degrade the radiation characteristics such as gain, side-lobe level, and front-to-back ratio. Nevertheless, as confirmed in Fig. 6(d), the patterned ground plane reduces the gain penalty by 0.7 dBi at 60 GHz compared with the case of the meshed ground plane. C. Dual Grid Array Antennas As previously discussed, current highly integrated 60-GHz radios require dual antennas. Nakano et al. designed dual grid

Fig. 6. The simulated results of the dual-feed grid array antenna for differential operation with the meshed and patterned ground planes: (a) return loss, (b) E or xz - and (c) H or yz -plane patterns at 61.5 GHz, and (d) peak realized gain.

array antennas in a double-layer structure [20]. A perpendicular orientation was arranged for the upper and lower grid array antennas. In this way, the upper grid array antenna radiates a horizontally polarized wave; while the lower grid array antenna does a vertically polarized one. A high isolation between both centrally located feeding terminals can be guaranteed. Fig. 7 shows our dual grid array antennas in a single-layer structure. It is formed by two grid array antennas of the basic structure studied in earlier in this section and has a body size of . A parallel orientation is arranged for the dual grid array antennas. Both grid array antennas radiate the wave of the same polarization. The single-layer structure simplifies the fabrication process. The parallel orientation reduces the outrage probability of 60-GHz radio links, which are usually deployed in line-of-sight environments, due to the polarization loss. Fig. 8 compares the simulated results of the grid array antenna with the dual feeds for differential operation and the grid array antenna with a single feed for single-end operation. As this is a transitional step in our development, no performance enhancement is made and no patterned ground plane is used. When one grid array antenna is excited, the other grid array antenna acts as a parasitic element, and vice versa. Fig. 8(a) shows

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Fig. 7. The dual grid array antennas: (a) top and (b) bottom views.

that the impedance bandwidth is 5.3 GHz (8.6%) from 59 to 64.3 GHz for the differential antenna and 9.7 GHz (15.8%) from 52.5 to 62.2 GHz for the single-end antenna. With reference to Fig. 4(d)–(f), one can find from Fig. 8(b)–(d) that the radiation patterns and the maximum gain of the single-end antenna are improved more noticeably than those of the differential antenna by the parasitic element. For example, the beamwidth in the E-plane is narrowed and the radiation of cross-polarization in the H-plane is suppressed, so the gain is improved to 15 dBi for the single-end antenna. Fig. 8(e) shows the simulated isolation level between the dual grid array antennas. For the sake of simulation simplicity, both grid array antennas are fed for single-end operation. Note the isolation is high because of the large physical separation between the two feeding terminals. III. INTEGRATION OF ARRAY ANTENNAS IN PACKAGE The integration of the grid array antenna in a wirebond package has been realized [25], [26]. In this Section, the integration of dual grid array antennas in a flip-chip package is described. Fig. 9 illustrates the integration. Note that the package features standard flip-chip bonding and there are three cofired laminated ceramic layers for the package. The 1st ceramic layer is 0.385 mm thick, the second to the third ceramic layers are both 0.095 mm thick. There are four metallic layers for the package. The top layer provides the metallization for the dual grid array antennas, the 1st buried layer metallization for the patterned ground plane, the second buried layer the metallization for the antenna feeding traces, and the bottom exposed layer the metallization for the signal traces. The package has 48 input/outputs with a JEDEC standard pitch of 0.25 mm. The . size of the whole package is Fig. 9 also illustrates the zoom-in view of the feeding networks of the dual grid array antennas. For the dual-feed one, it consists of such packaging elements as two quasi-coaxial cables cascaded first with two striplines, then another two quasicoaxial cables, and finally vias through two apertures on the ground plane in a GSGSG arrangement. It is interesting to note that the differential feeding ports in Fig. 9 are brought closer to each other as compared with those in Fig. 3, due to the requirement of flip-chip bonding to the on-die GSGSG pads. The radiating element between the differential feeding ports is removed to enhance their isolation. For the single-feed one, it consists of a quasi-coaxial cable cascaded with via through one aperture on the ground plane in a GSG arrangement. It is known that

Fig. 8. The simulated results of the dual grid array antennas with one for differential and the other for single-end operations: (a) return loss, (b) E or xz - and (c) H or yz -plane patterns at 61.5 GHz, (d) peak realized gain, and (e) isolation.

the GSG and GSGSG arrangements not only minimize potential electromagnetic interference but also improve the feeding performance. The GSG and GSGSG feeding networks are designed together with the grid array antennas. Both GSG and GSGSG pads have a pitch of 0.25 mm. Fig. 10 shows the bottom view of the package without the signal traces but with the integrated balun for testing the differential grid array antenna with the single-ended equipment. The package with the dual microstrip grid array antennas were fabricated in FERRO A6 LTCC in Singapore Institute of

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Fig. 10. Illustration of the integrated balun for testing the differential grid array antenna with the single-ended equipment.

Fig. 11. Photos of the dual grid array antennas in the package on the fixture for testing: (a) top view. (b) Bottom view with signal traces. (c) Bottom view with the integrated balun.

Fig. 9. Illustration of integration of dual grid array antennas in a package: (a) top view. (b) Explored view. (c) Bottom view, as well as the zoom-in view of the feeding networks of the dual grid array antennas.

Manufacturing Technology. Fig. 11 illustrates the test fixture to hold the package for testing.

Fig. 12 compares the simulated and measured results of the single-end antenna for sample A. The return losses agree very well from 56 to 58.5 GHz. The agreement becomes poor for higher frequencies, due to the following reasons: dimension tolerance control, material property variation, and the difference between the wave-port excitation in simulation and the probe excitation in measurement. The measured return loss is higher than

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Fig. 12. The simulated and measured results of the single-end antenna in the package: (a) return loss. (b) E or xz - and (c) H or yz -plane patterns at 61.5 GHz. (d) Peak realized gain.

Fig. 13. The simulated and measured results of the differential antenna in the package: (a) return loss. (b) E or xz - and (c) H or yz -plane patterns at 61.5 GHz. (d) Peak realized gain.

10 dB from 56.4 to 61.7 GHz and 8 dB from 55 to 63.4 GHz indicating acceptable matching to 50- sources at these frequencies. The simulated and measured radiation patterns are in close agreement at 61.5 GHz. The measured maximum gain is 13.5 dBi at 59 GHz with 3-dB gain bandwidth of 4.5 GHz. The simulated radiation efficiency is better than 85%. Fig. 13 compares the simulated and measured results of the differential antenna for sample B. The discrepancies between

the simulated and measured return losses are due to the same reasons explained above for single-end antenna. The measured return loss is higher than 8 dB from 56.2 to 63.2 GHz indicating acceptable matching to 50- sources at these frequencies. Again, the simulated and measured radiation patterns at 61.5 GHz are in close agreement. The larger side lobes in the back side are caused by the balun. The measured maximum gain is 13.5 dBi at 57.5 GHz with 3-dB gain bandwidth of 5.3 GHz.

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Fig. 14. The concept of assembling the highly integrated 60-GHz radio in the package to the system PCB.

If the additional loss of 1 dB from the integrated balun is de-embedded, the maximum gain of the differential antenna becomes 14.5 dBi. The simulated radiation efficiency is also better than 85%. Generally, the measured results confirm the simulated ones. For instance, the differential grid array antenna exhibits a sharper radiation beam in the E-plane, a similar radiation pattern in the H-plane, and a higher gain as compared with those of the single-end grid array antenna. Fig. 14 illustrates our concept of assembling the highly integrated 60-GHz radio in the chip-scale package to the system printed-circuit board (PCB). A cavity or even an opening needs to be created in the PCB to house and protect the radio die. The lands on the chip package are soldered to the PCB to finish interconnect from the chip to the PCB through the package. This is believed to be a very cheap solution to highly integrated 60-GHz radios with a rather thin profile. IV. CONCLUSION The examination of the reported highly integrated 60-GHz radios was made. It was found that both 50- single-end and 100- differential antenna-circuit interfaces were used in current two-and single-chip solutions. A dual-feed grid array antenna was designed in LTCC and analyzed for both single-end and differential feedings. It was shown that the differential feeding resulted in higher gain but narrower impedance bandwidth and the patterned ground plane reduced the gain penalty. Dual grid array antennas in a single-layer structure were also designed and integrated in a package with one antenna for single-end driving and the other for differential feeding. The results showed that the package achieved good antenna performance in the 60-GHz band with maximum gain of 13.5 and 14.5 dBi for the single-ended and differential antennas, respectively. Furthermore, the novel concept of assembling a highly integrated 60-GHz radio in the package to the system PCB was disclosed, which was believed to be a very cheap system solution to highly integrated 60-GHz radios with a rather thin profile. ACKNOWLEDGMENT The authors would like to thank K. M. Chua and L. L. Wai of Singapore Institute of Manufacturing Technology for their support in fabrication of the package in LTCC. REFERENCES [1] K. Maruhashi and M. Ito et al., “60 GHz-band flip-chip MMIC modules for IEEE1394 wireless adapters,” in Proc. 31st Eur. Microw. Conf., Sep. 2001, vol. 1, pp. 407–410. [2] K. Ohata et al., “1.25 Gbps wireless gigabit Ethernet link at 60 GHzband,” in IEEE MTT-S Int. Microw. Symp. Dig., Philadelphia, PA, Jun. 8–13, 2003, pp. 373–376.

[3] S. E. Gunnarsson et al., “Highly integrated 60 GHz transmitter and receiver MMICs in a GaAs pHEMT technology,” IEEE J. Solid-State Circuits, vol. 40, no. 11, pp. 2174–2186, Nov. 2005. [4] S. E. Gunnarsson et al., “60 GHz single-chip front-end MMICs and systems for multi-Gb/s wireless communication,” IEEE J. Solid-State Circuits, vol. 42, no. 5, pp. 1143–1157, May 2007. [5] B. Floyd, S. Reynolds, U. Pfeiffer, T. Beukema, J. Grzyb, and C. Haymes, “A silicon 60 GHz receiver and transmitter chipset for broadband communications,” in Proc. ISSCC Dig. Tech. Papers, San Francisco, CA, Feb. 4–9, 2006, pp. 184–185. [6] M. Tanomura, Y. Hamada, S. Kishimoto, M. Ito, N. Orihashi, K. Maruhashi, and H. Shimawaki, “TX and RX front-Ends for 60 GHz band in 90 nm standard bulk CMOS,” in ISSCC Tech. Dig. Papers, San Francisco, CA, Feb. 8–12, 2008, pp. 558–559. [7] U. Pfeiffer, J. Grzyp, D. Liu, B. Gaucher, T. Beukema, B. Floyd, and S. Reynolds, “A chip-scale packaging technology for 60-GHz wireless chipsets,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 8, pp. 3387–3397, Aug. 2006. [8] Y. P. Zhang, M. Sun, K. M. Chua, L. L. Wai, and D. Liu, “Antenna-in-package design for wireband interconnection to highly integrated 60-GHz radios,” IEEE Trans. Antennas Propag., vol. 57, no. 10, pp. 2842–2852, Oct. 2009. [9] Y. P. Zhang, M. Sun, K. M. Chua, L. L. Wai, and D. Liu, “Integration of slot antenna in LTCC package for 60-GHz radios,” Electron. Lett., vol. 44, no. 5, pp. 330–331, Mar. 2008. [10] M. Sun, Y. P. Zhang, K. M. Chua, L. L. Wai, D. Liu, and B. Gaucher, “Integration of Yagi antenna in LTCC package for differential 60-GHz radio,” IEEE Trans. Antennas Propag., vol. 56, no. 8, Aug. 2008. [11] A. Tomkins, R. A. Aroca, T. Yamamoto, S. T. Nicolson, Y. Doi, and S. P. Voinigescu, “A zero-IF 60 Ghz 65 nm CMOS transceiver with direct BPSK modulation demonstrating up to 6 Gb/s data rates over a 2 m wireless link,” IEEE J. Solid-State Circuits, vol. 44, no. 8, pp. 2085–2099, Aug. 2009. [12] C. H. Doan, S. Emami, D. A. Sobel, A. M. Niknejad, and R. W. Brodersen, “Design considerations for 60 GHz CMOS radios,” IEEE Commun. Mag., vol. 42, pp. 132–140, Dec. 2004. [13] D. Huang, R. Wong, Q. Gu, N. Y. Wang, T. W. Ku, C. Chien, and M.-C. F. Chang, “A 60 GHz CMOS differential receiver front-end using on-chip transformer for 1.2 volt operation with enhanced gain and linearity,” in Proc. Symp. VLSI Circuits, Honolulu, HI, Jun. 15–17, 2006, pp. 144–145. [14] Y. P. Zhang, J. J. Wang, Q. Li, and X. J. Li, “Antenna and transmit/receive switch for single-chip radio transceivers of differential architecture,” IEEE Trans. Circuits Syst. I, vol. 55, no. 11, pp. 3564–3570, Dec. 2008. [15] Y. P. Zhang and D. Liu, “Antenna-on-chip and antenna-in-package solutions to highly integrated millimeter-wave devices for wireless communications,” IEEE Trans. Antennas Propag., vol. 57, no. 10, pp. 2830–2841, Oct. 2009. [16] J. He and Y. P. Zhang, “Design and analysis of SPST and SPDT switches for 60-GHz applications in 65-nmCMOS,” in Proc. APMC 2008, Hong Kong and Macau, China, Dec. 16–20, 2008. [17] J. D. Kraus, “A backward angle-fire array antenna,” IEEE Trans. Antennas Propag., vol. 12, no. 1, pp. 48–50, Jan. 1964. [18] R. Conti, J. Toth, and T. Dowling, “The wire grid microstrip antenna,” IEEE Trans. Antennas Propag., vol. 29, no. 1, pp. 157–166, Jan. 1981. [19] K. D. Palmer and J. H. Cloete, “Synthesis of the microstrip wire grid array,” in Proc. Inst. Elect. Eng. 10th Conf. Antennas Propag., Edinburgh, U.K., Apr. 14–17, 1997, pp. 114–118. [20] H. Nakano, I. Oshima, H. Mimaki, K. Hirose, and J. Yamauchi, “Center-fed grid array antennas,” in Proc. IEEE AP-S Int. Symp., 1995, pp. 2010–2013. [21] H. Nakano, T. Kawano, and J. Yamauchi, “Meander-line grid array antenna,” Proc. Inst. Elect. Eng. Microw. Antennas Propag., vol. 145, no. 4, pp. 309–312, Aug. 1998. [22] T. Kawano and H. Nakano, “Cross-mesh array antennas for dual LP and CP waves,” in Proc. IEEE AP-S Int. Symp., 1999, pp. 2748–2751. [23] H. Nakano, T. Kawano, Y. Kozono, and J. Yamauchi, “A fast MoM calculation technique using sinusoidal basis and testing functions for a wire on a dielectric substrate and its application to meander loop and grid array antennas,” IEEE Trans. Antennas Propag., vol. 53, no. 10, pp. 3300–3307, Oct. 2005. [24] Y. P. Zhang and M. Sun, Grid Array Antennas and an Integration Structure PCT filed on Dec. 12, 2008. [25] M. Sun, Y. P. Zhang, Y. X. Guo, K. M. Chua, and L. L. Wai, “Integration of grid array antenna in chip package for highly integrated 60-GHz radios,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 1364–1366, 2009. [26] M. Sun and Y. P. Zhang, “Design and integration of 60-GHz grid array antenna in chip package,” in Proc. Asia-Pacific Microw. Conf., Hong Kong and Macau, China, Dec. 16–20, 2008.

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Y. P. Zhang (M’03–SM’07–F’10) received the B.E. and M.E. degrees from Taiyuan Polytechnic Institute and Shanxi Mining Institute of Taiyuan University of Technology, Shanxi, China, in 1982 and 1987, respectively, and the Ph.D. degree from the Chinese University of Hong Kong, Hong Kong, in 1995, all in electronic engineering. From 1982 to 1984, he was with the Shanxi Electronic Industry Bureau; from 1990 to 1992, the University of Liverpool, Liverpool, U. K.; and from 1996 to 1997, City University of Hong Kong. From 1987 to 1990, he taught at the Shanxi Mining Institute and from 1997 to 1998, the University of Hong Kong. He was promoted to a Full Professor at Taiyuan University of Technology in 1996. He is now an Associate Professor and the Deputy Supervisor of Integrated Circuits and Systems Laboratories with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore. He has broad interests in radio science and technology and published widely across seven IEEE societies. Dr. Zhang received the Sino-British Technical Collaboration Award in 1990 for his contribution to the advancement of subsurface radio science and technology. He received the Best Paper Award from the Second International Symposium on Communication Systems, Networks and Digital Signal Processing, July 18–20, 2000, Bournemouth, U.K., and the Best Paper Prize from the Third IEEE International Workshop on Antenna Technology, March 21–23, 2007, Cambridge, U.K. He has organized/chaired dozens of technical sessions of international symposia. He was awarded a William Mong Visiting Fellowship from the University of Hong Kong in 2005. He has delivered scores of invited papers/keynote addresses at international scientific conferences. He was a Guest Editor of the International Journal of RF and Microwave Computer-Aided Engineering and an Associate Editor of the International Journal of Microwave Science and Technology. He serves as an Editor of ETRI Journal, an Associate Editor of the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, and an Associate Editor of the International Journal of Electromagnetic Waves and Applications. He also serves on the Editorial Boards of a large number of Journals including the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES and IEEE MICROWAVE AND WIRELESS COMPONENTS LETTERS.

M. Sun (M’09) received the B.E. and M.E. degrees from the Hunan University and Beijing Institute of Technology, China, in 2000 and 2003, respectively, and the Ph.D. degree from the Nanyang Technological University (NTU), Singapore, in 2007, all in electronic engineering. She became a Research Associate with NTU in 2006 and subsequently converted to Research Fellowship in 2007. In 2009, she joined the Institute for Infocomm Research, Singapore, as a Research Fellow. Her research interests include millimeter-wave and Terahertz antenna design. Dr. Sun was a recipient of the Best Paper Prize from the Third IEEE International Workshop on Antenna Technology, March 21–23, 2007, Cambridge, U.K.

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Duixian Liu (S’85–M’90–SM’98–F’10) received the B.S. degree in electrical engineering from Xidian University, Xi’an, China, in 1982, and the M.S. and Ph.D. degrees in electrical engineering from The Ohio State University, Columbus, in 1986 and 1990, respectively. From 1990 to 1996, he was with Valor Enterprises, Inc., Piqua, OH, initially as an Electrical Engineer and then as the Chief Engineer, during which time he designed an antenna product line ranging from 3 MHz to 2.4 GHz for the company, a very important factor for the prestigious Presidential “E” Award for Excellence in Exporting in 1994. Since April 1996, he has been with the IBM T. J. Watson Research Center, Yorktown Heights, NY, as a Research Staff Member. He has 28 patents issued and 15 patents pending. His research interests are antenna design, EM modeling, digital signal processing, and communications technology. He has served as external Ph.D. examiner for several universities and external examiner for some government organizations on research grants. His research interests are antenna design, chip packaging, electromagnetic modeling, digital signal processing, and communications technology. He has authored or coauthored approximately 60 journal and conference papers. Dr. Liu is an Associate Editor for the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION and was a Guest Editor for the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION Special Issue on “Antennas and Propagation Aspects of 60–90 GHz Wireless Communications” (July 2009). He has received three IBM’s Outstanding Technical Achievement Awards and one Corporate Award, the IBM’s highest technical award. He was named Master Inventor in 2007. He has been the Organizer or Chair for numerous international conference sessions or special sessions and also a Technical Program Committee member. He was the General Chair of the 2006 IEEE International Workshop on Antenna Technology: Small Antennas and Novel Metamaterials, White Plains, NY.

Yilong Lu (S’90–M’92) received the B.Eng. degree from Harbin Institute of Technology, China, in January 1982, the M. Eng. degree from Tsinghua University, China, in November 1984, and the Ph.D. degree from University College London, U.K., in November 1991, all in electronic engineering. From November 1984 to September 1988, he was with the Department of Electromagnetic Fields Engineering, University of Electronic Science and Technology of China, Chengdu, China, as a lecturer in the Antenna Division. In December 1991, he joined the School of Electrical and Electronic Engineering, Nanyang Technological University (NTU), where he is currently a full Professor in the Communication Engineering Division and is also the leader of Radar Research Group, the Coordinator of the Microwave Circuits, Antennas and Propagation Research Group, and Deputy Director of Centre for Modeling and Control of Complex Systems, in NTU. He was a Visiting Academic with the University of California—Los Angeles from October 1998 to June 1999. His research interests include antennas, array based signal processing, radar systems, computational electromagnetics, and evolutionary computation for optimization of complex problems. He is a member of Editorial Board for IET Radar, Sonar and Navigation.

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Evaluation of a New Wideband Slot Array for MIMO Performance Enhancement in Indoor WLANs Jorge R. Costa, Senior Member, IEEE, Eduardo B. Lima, Carla R. Medeiros, and Carlos A. Fernandes, Senior Member, IEEE

Abstract—A new wideband compact slot antenna array for indoor WLAN access points (AP) is described, covering several wireless communication services from 2.4 to 4.8 GHz, that is especially designed to enhance MIMO system capacity. The array topology provides both spatial and polarization diversity. Despite very close packing of the array elements, these exhibit very low mutual coupling and low cross-polarization, greatly favoring MIMO diversity gain. A detailed MIMO performance comparison is conducted against a common array of patches in indoor environment, based both on simulation and indoor measurements: the new antenna shows a clear improvement in terms of channel capacity. Index Terms—Multiple input multiple output (MIMO) array, printed wideband antenna, spatial and polarization diversity, tapered slot antenna, wireless local area network (WLAN).

I. INTRODUCTION

T

HE benefits from using multiple input and multiple output antennas (MIMO) in wireless communications have been widely discussed in the literature. Spatial diversity improves system reliability by decreasing the sensitivity to fading (diversity gain) without requiring additional bandwidth, unlike the case of frequency or time diversity. Spatial multiplexing can be implemented as an alternative to spatial diversity. In spatial transmitter antenna sends an independent multiplexing each data stream, unlike the case of spatial diversity where each antenna sends correlated data (e.g., Alamouti code [1]). With spatial multiplexing the spectral efficiency can be increased by a [2]. factor Both for spatial diversity and for spatial multiplexing strategies, it is required that the channels between the multiple transmitting (Tx) and receiving (Rx) antennas are uncorrelated to maximize data throughput and for successful data stream decoding. Correlation may result from poor multipath richness of the scenario and from mutual coupling between antenna array elements [3]. Therefore, one important design specification for MIMO antenna arrays is to minimize inter-element coupling. Manuscript received March 19, 2010; revised August 20, 2010; accepted September 02, 2010. Date of publication January 31, 2011; date of current version April 06, 2011. This work was supported in part by the European Union under project FP7 ICT-2007-216715 (NewCom++). J. R. Costa is with the Instituto de Telecomunicações, IST, 1049-001 Lisboa, Portugal and also with the Departamento de Ciências e Tecnologias da Informação, ISCTE—Lisbon University Institute, 1649-026 Lisboa, Portugal (e-mail: [email protected]). E. B. Lima, C. R. Medeiros, and C. A. Fernandes are with Instituto de Telecomunicações, IST, 1049-001 Lisboa, Portugal. 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.2011.2109685

Polarization diversity was found to be as significant as spatial diversity at improving diversity gain, and hence MIMO system capacity [4]. Therefore, MIMO array elements should also present pure and frequency stable polarization to obtain full benefit from polarization diversity. The present study aims to present and evaluate a 2 2 printed array antenna intended to enhance MIMO performance in indoor access points (fixed terminal), with the following cumulative characteristics: low inter-element coupling, pure linear polarization from each array element, good front-to-back ratio for wall mounting, high radiation efficiency, and stability of these characteristics over a large bandwidth to accommodate multiple wireless standards such as WiFi, LTE, WiMAX and part of the UWB band. Furthermore, the four array elements are to be arranged in a compact layout with space and polarization diversity. The proposed array elements are based upon the linear polarized crossed exponentially tapered slot (XETS) antenna developed by the authors for ultrawideband (UWB) systems [5], [6] and modified in [7] to include a back cavity to change the inherently bidirectional radiation pattern into a unidirectional one. This new configuration is referred as cavity back crossed exponentially tapered slot (CXETS) antenna and presents 67% bandwidth. The antenna copes with all the specifications listed above and confirmed MIMO improved performance when compared by simulation and measurement to a reference printed patch array with the same element layout and diversity polarization scheme. To the authors’ knowledge, no other antenna reported in the literature copes simultaneously with all the above specifications, which makes the CXETS array a novel alternative for MIMO fixed terminal. Many MIMO arrays were proposed in the literature, but in all cases they match only part of the above requirements [8]–[13], and clearly no solution cumulatively presents comparable characteristics. This paper is organized as follows. The new CXETS array and the array of patches that is used for comparison are both presented in Section II. Section III describes both the MIMO setup for channel transfer matrix measurements and the performance evaluation of several MIMO configurations with different number of array elements based upon channel capacity estimation. Numerical and experimental results are given and discussed throughout. Conclusions are drawn in Section IV. II. ARRAY DESCRIPTION Both the XETS and rectangular patch arrays were designed using the transient solver of CST Microwave Studio™, based on Finite Integration Method [14]. To ensure fair performance

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TABLE I CXETS ANTENNA PARAMETER VALUES IN MILLIMETERS

Fig. 2. Measured amplitude of the CXETS array scattering matrix elements. Fig. 1. (a) Photograph of the assembled CXETS array prototype, showing the antenna top face; (b) Disassembled antenna showing the FR4 cavity back and the feeding coaxial cables soldered on the XETS back petals.

comparison between both arrays, they were designed to have similar aperture area. A. CXETS Array The crossed exponentially tapered slot (XETS) antenna configuration was developed by the authors for UWB systems [5], [6]. It is compact, with stable radiation pattern, low cross-polarization and low mutual coupling when packed in a closed array configuration. However, the radiation pattern is bidirectional. To produce a unidirectional radiation pattern, a single antenna was redesigned to include a grid back cavity [7]. This new configuration is onwards referred as the CXETS. The XETS elements are printed on both sides of DUROID and thickness 5880 substrate, mm. The top face of the four radiating elements is shown in Fig. 1(a) and the back face, with the feeding coaxial cables, is shown in Fig. 1(b). These are small diameter EZ-47 semi-rigid coaxial cables (1.19 mm diameter), soldered between two replicas of opposing petals from the front face of each XETS, Fig. 1(b). The RF signal is capacitively coupled to the corresponding front petals of each antenna. The fed petals define the E-plane of each XETS. It was verified that common-mode currents resulting from the unbalanced feeding are low in this assembly and do not perturb significantly the expected balanced antenna performance. It is seen in Fig. 1(b) that adjacent elements have orthogonal orientation, so, orthogonal polarization. The meshed cavity walls are printed on FR4 substrate, and thickness mm. The mesh is uniform, with 6 mm step. This is a compromise value found through CST simulation that allows retaining a reasonable part of the original antenna bandwidth (now 2.4 to 4.8 GHz) and at

the same time enables reaching better than 10 dB front-to-back ratio (f/b). Each array element has its own cavity; the two walls separating each cavity can be observed in Fig. 1(b). The metallic mesh is also printed on the internal walls. In the assembled antenna, the cavities are filled with low density Styrofoam to provide physical support to the thin 10 mils DUROID substrate. The Styrofoam can be observed in one of the cavities of the disassembled antenna in Fig. 1(b). The remaining parameter values are indicated in Table I (the same parameter naming is used as in [5]). The overall size of the array (Fig. 1(a)) is 114 114 21 mm where is the wavelength at 2.4 GHz, the lower operating frequency. Smaller MIMO antenna assemblies have been reported for handheld communication devices, but none of those antennas offer 1:2 bandwidth. The proposed antenna is intended specifically for WLAN access points (fixed terminal). Fig. 2 presents the measured scattering matrix of the CXETS array. The input reflection coefficient of all elements is below dB across the interval from 2.4 to 4.8 GHz. This covers WiFi, LTE, WiMax and the lower channels of UWB systems. The corresponding assigned bands are indicated in Fig. 2. It is seen from Fig. 2 that mutual coupling between array elements dB across the entire antenna bandwidth. It is mostly below is noted that the original XETS antenna presents very low coudB) between adjacent cross-polarized array pling (about elements [5]. It is the addition of the back cavity that increases dB, but this value is still attractive for mutual coupling to the present application. It is known that a N-port antenna efficiency is better characterized by the total active reflection coefficient (TARC) [13], [15] than by the usual inspection of the -matrix elements. TARC is the square root of the ratio between total reflected power at the array ports and total power fed to the ports. It is calculated by applying a unity magnitude voltage with random phase at each port of the array. In this way, the reflected power translates

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Fig. 5. (a) Photograph of the rectangular patch array prototype; (b) photograph of the air subtract and coaxial feeding.

Fig. 3. Measured TARC for the CXETS array prototype.

Fig. 6. Measured radiation pattern of the first element of the patches array at 2.6 GHz.

B. Rectangular Patch Array Fig. 4. Measured radiation pattern of the first element of the CXETS array at 2.6 GHz.

different combinations of the excitation signals, which is more appropriate to describe the real conditions of a MIMO array operation than the usual single port excitation strategy. It accounts for both coupling and random signal combining between each port signal [13]. The measured scattering matrix of the CXETS was used to compute the antenna TARC. The results for different combinations of the input signals are presented in Fig. 3. Since the CXETS elements present low mutual coupling, TARC level is dB across the bandwidth. almost always below The measured main planes radiation pattern of array element #1 is presented in Fig. 4, with the other CXETS terminated by matched loads. Vertical scale refers to antenna gain. Cross-podB. The other three array elements larization level is about present similar radiation patterns, apart from plane interchange for the 90 rotated antennas. The measured front-to-back ratio across the bandwidth varies between 11 and 20 dB while gain ranges from 5.5 to 8 dBi. dB at Cross-polarization improves with frequency from dB at 4.8 GHz. CST simulations indicate 2.4 GHz to about that the array radiation efficiency is always above 80%.

The rectangular patch array prototype is printed on , thickness DUROID 5880, mm, see Fig. 5. It was designed to operate in the WiMax or LTE band from 2.5 to 2.7 GHz. The desired bandwidth was achieved through the use of an air gap mm. between the dielectric layer and the ground plane Of course it is not possible to approach the wide CXETS bandwidth with a conventional patch antenna. Patch dimensions are 33.4 mm 40.1 mm . Each patch is probe fed by an EZ-141 coaxial cable (3.58 mm diameter). The outer conductor of the feeding coaxial cable is soldered to the ground plane and extended up to the dielectric; this arrangement reduces the excitation of surface waves that can propagate along the air gap and therefore lowers mutual coupling between elements as will be seen next. Patches are rotated by 90 in relation to the adjacent one, Fig. 5. Therefore, the polarization from the patches in one diagonal of the array is orthogonal with respect to the polarization from the other two patches. The ground plane extends by 15.9 mm from the edges of the patches. With 7.3 mm gap between adjacent elements, the . overall array size is 112.3 mm 112.3 mm Fig. 6 presents the measured radiation pattern at 2.6 GHz from array patch #1, with the other patches terminated by matched loads. Vertical scale refers to antenna gain. Cross polarization is high as expected. Very similar radiation patterns

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Fig. 9. MIMO measuring setup. Fig. 7. Measured magnitude of the patch array scattering matrix elements.

Fig. 10. MIMO measuring environment: (a) transmitting array; (b) receiving array.

Fig. 8. Measured TARC for the patch array prototype.

were measured for the other three array elements, apart from plane interchange for orthogonal patches. Fig. 7 presents the measured magnitude of the array scattering matrix elements. Grey shading represents the WiMax or LTE dB bands. Coupling between adjacent patches reaches within the WiMax or LTE band while coupling between diagdB level. onal patches is almost constant across the band at This is an improvement with respect to the usual configuration where the outer conductor of the feeding coaxial cable is terminated at the ground plane. In fact, CST simulations indicate dB in this case. mutual coupling level of Fig. 8 presents TARC curves computed from the measured S-matrix, for random phase and unit amplitude at the excitation ports. It can be observed that the worst value within the WiMax dB. The large dispersion of the TARC or LTE bands is about curves denotes a strong coupling between antenna elements, unlike the CXETS. III. MIMO PERFORMANCE EVALUATION A. Measurement Setup In order to fully test the non-ideal antenna effects in real MIMO environments, two identical arrays of each type were manufactured and tested using a channel sounder in a 4 4 link configuration ( and ), with equal Tx and Rx antennas. The setup is indicated in Fig. 9. A Vector Network

Analyzer (Agilent E8361A) is used for measuring the MIMO channel transfer matrix (with elements). The two ports from the vector network analyzer (VNA) are electronically switched through all elements of the Tx and Rx antenna arrays. An in-house LabView application is used to remotely control the measurement set-up and data logging. The system enables to sequentially retrieve up to 16 channel responses (corresponding to every combination of Tx and Rx array elements, up to four elements in each array). Total acquisition time is of the order of 30 s. An indoor laboratory was chosen as the measurement environment, Fig. 10. The Rx array was kept stationary at a fixed position, while the Tx array scanned a 1 m 1 m area with 10 cm mesh size, yielding measurement locations. Distance between Rx and Tx antennas was in the order of 4 m. The channel frequency responses corresponding to each of the 16 Rx/Tx array element combinations were measured in the 2.5 to 2.7 GHz interval (WiMax and LTE bands) taking frequency samples. To force non-line-of-sight link (NLOS) conditions, a metallic cabinet was used to block the direct ray path. It is well known that MIMO performance improvement is best evidenced for NLOS scenarios, where diversity gain is the highest, corresponding to most uncorrelated transmission channels [2]. In order to enhance multipath propagation, some artificial scatters were placed in the room, made from objects covered with ruffled aluminum foil. Antenna coordinates and orientation, as well as cables, calibration procedures and scenario were exactly the same for both the CXETS array and the patch array. The environment was kept rigorously stationary throughout the whole measurement campaign, including the use of remote data logging to avoid operator influence on the scenario.

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2

Fig. 11. Measured correlation between channels in a 2 2 MIMO setup using: (a) two cross-polarized patches (#1 and #2); (b) two co-polarized patches (#1 and #3); (c) two cross-polarized CXETS (#1 and #2); (d) two co-polarized CXETS (#1 and #3).

Fig. 12. Measured and simulated capacity for a MIMO set-up with SNR=10 dB at Rx and Tx, assuming power control : (a) two co-polarized array elements (b) two cross-polarized array elements; (c) three array elements; (d) all four array elements.

Measurements were performed with equal Tx and Rx antennas, instead of using a typical antenna at the personal terminal, because this allows isolating the contribution from the proposed CXETS antenna, without being limited or degraded by the characteristics of the personal terminal antenna.

cross-polarized antenna elements are used. This confirms that diversity gain increases when orthogonal polarized antennas are used [4]. It is noted that the decrease in channel correlation is achieved despite the co-polarized antennas being farther from each other as they are laid along the diagonal of the array and not adjacent to each other as with the cross-polarized ones, see Figs. 1 and 5. Consider the case where only the receiver has channel information; the maximum achievable channel capacity is given by Shannon’s equation [16]

B. Capacity Estimation A first analysis of the measurements is performed using data from only two elements of the array, as in a 2 2 MIMO configuration ( and ). Either two cross-polarized array elements in both Tx and Rx are selected (data from antennas #1 and #2), or two co-polarized elements are selected (data from antennas #1 and #3). This analysis allows confirming the benefit from using polarization diversity and to evaluate how the CXETS and the patch arrays compare with respect to channel capacity in view of their different coupling characteristic. Maximum MIMO performance requires parallel uncorrelated channels [16]. For the 2 2 configuration there are 4 parallel channels and therefore, 6 correlation coefficients can be determined between them

(2) identity matrix, SNR is the signal-to-noise where is a ratio at the receiver and represents the Hermitian operator. To exclude the effect of path loss variation on the received power, is normalized so that the average power is unitary [10], [17], [18] (3)

(1) where corresponds to the expected value calculated represents the complex over the measured bandwidth, conjugate, and and represent measured channel responses between any two antenna pairs. All six correlation coefficients are determined for each of the measured positions inside the room. The corresponding cumulative distribution function (CDF) is represented in Fig. 11 for the CXETS array link and for the patch array link, considering either the crossor co-polarized antennas. Therefore, the plot vertical axis indicates the probability of finding a position in the room with channel correlation coefficient below the abscissa value. The first observation is that for both polarization configurations, correlation is much lower for the CXETS, owing to its better inter-element isolation and polarization purity. Anyway, for both array types, the correlation coefficient is lower when

Although this normalization removes path loss, it does not hide the effect of propagation-induced correlation or array element coupling-induced correlation. As previously mentioned, the non-normalized channel transfer matrix corresponds directly to the -matrix measured at the Tx and Rx array ports, using a vector network analyzer [17]. Capacity values are calculated for the CXETS and patch arrays at dB for each frequency and position sample in different MIMO configurations, using (2) and the measured channel matrix. Fig. 12 presents the corresponding cumulative distribution function. For reference, Fig. 12 also shows for each MIMO configuration the calculated capacity for a Rayleigh fading channel, that is, each element of the normalized channel transfer matrix is given by (4)

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TABLE II CAPACITY VALUES FOR 50% PROBABILITY

where is an independent and identically distributed (i.i.d.) value with Gaussian distribution function, zero mean and unit variance . This corresponds to totally uncorrelated channels. Capacity improvement from the CXETS array against the array of patches is more evident for higher order MIMO configurations (Fig. 12(c)–(d)). This is clearer when comparing median capacity values (dashed horizontal line in Fig. 12). The results for each type of array are indicated in Table II along with the ratio between the capacity obtained from measured channel data and the capacity calculated for completely independent channels (Rayleigh curves). Table II shows that due to antenna mutual coupling and channel correlation, capacity does not scale linearly with number of channels as would be expected if they were independent [16]. In fact, the detrimental effect of mutual coupling on capacity increases with the number of array elements [15]. However it is clear in all cases the importance of better isolation and polarization purity of the CXETS. These results assume that the system uses power control to fix SNR at constant 10 dB level. Otherwise, in environments with poor multipath richness, configuration #b will tend to receive in average more power than configuration #a. If we want to alternatively compare capacity for constant Tx power, we can consider for all cases of Table II the complete measured channel in the normalization (3) and transfer matrix extract afterwards the sub-matrixes from corresponding to each configuration [18]. For configuration #a, this leads instead to 4.57 bit/s/Hz for the patch array and 5.18 bit/s/Hz for the CXETS array, which is not very different from the fixed SNR case thanks to the richness of the tested scenario. As could be expected, the other array configurations capacity remains practically unchanged. For spatial multiplexed MIMO systems, it is relevant to further evaluate the eigenvalues of the measured . Each eigenvalue represents an orthogonal parallel channel where independent data stream can be transmitted simultaneously, thus increasing the bit rate of the system [2]. Calculations were made for the 4 4 MIMO configuration ( and ), using all transfer functions measured at each frequency and at each array location in the room. For each measurement, the eigenvalues were ordered such that . Fig. 13 presents the corresponding probability density functions considering 0.5 dB eigenvalue amplitude steps (bins). As the level of correlation increases from the ideal Rayleigh channel case to the measured CXETS and patch array cases, the mean value of the first eigenvalue increases while the mean of the other eigenvalues decreases. Such effect reflects the

Fig. 13. Measured and simulated eigenvalues of the 4

2 4 MIMO setup.

degradation of spatial multiplexed MIMO performance and the tendency to approximate the case of a single independent information channel, as in a SISO configuration. Again the CXETS array performance is much closer to the uncorrelated channel case (Rayleigh curves) than the array of patches due to much lower mutual coupling. Shannon’s (2) determines the maximum achievable capacity, but the actual throughput in a real system depends on further aspects like modulation and coding. A system emulator developed in Matlab was used to evaluate the simultaneous transmission of four independent and parallel bit streams using the measured channel transfer matrix in 4 4 MIMO configuration ( and ). For instance using binary phase shift keying (BPSK) modulation and maximum-likelihood (ML) detection at the receiver, the transmission between two CXETS arrays presents a bit error rate (BER) at dB that is almost 15 times lower than the equivalent configuration using two arrays of patches. IV. CONCLUSION The proposed CXETS antenna array, with its very low coupling between the closely packed array elements and with its pure linear polarization characteristics, is shown to clearly enhance the known benefits from using space and polarization diversity in MIMO systems, when compared to commonly used antennas, like the array of patches. The large operating bandwidth from 2.4 to 4.8 GHz with stable radiation characteristics allows using the CXETS array for multisystem access points covering services like WiFi, WiMax, LTE and UWB. The antenna cavity back increases the front to back radiation level above 12 dB allowing its mounting on metal surfaces or against a wall for WLAN access points. These cumulative characteristics make CXETS a unique and effective candidate for a very wideband MIMO array. In order to test its advantage for MIMO applications, basic MIMO performance comparison was conducted at 2.6 GHz (WiMax or LTE bands) between the CXETS array and a narrowband patch array with the same spatial and polarization configuration. Based upon measured channel transfer matrix using a MIMO channel sounder, it was shown that the new

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array increases channel capacity, better approaching the ideal limit for completely uncorrelated channels. Although all MIMO performance tests were conducted only at the lower fraction of the CXETS available bandwidth, it is expected that the overall performance will improve further for the rest of the band. In fact, while the antenna radiation characteristics remain almost unchanged across the whole band, for higher frequencies the separation between CXETS array elements becomes larger in terms of the operating wavelength and, therefore, the channels between transmit and receive antennas are expected to become more uncorrelated. ACKNOWLEDGMENT The authors acknowledge the collaboration from V. Fred and C. Brito for prototype construction, and A. Almeida for prototype measurements. REFERENCES [1] S. M. Alamouti, “A simple transmit diversity technique for wireless communications,” IEEE J. Sel. Areas Commun., vol. 16, pp. 1451–1458, Oct. 1998. [2] D. Tse and P. Viswanath, Fundamentals of Wireless Communication. Cambridge, U.K.: Cambridge Univ. Press, 2005. [3] R. Janaswamy, “Effect of element mutual coupling on the capacity of fixed length linear arrays,” IEEE Antennas Wireless Propag. Lett., vol. 1, pp. 157–160, 2002. [4] J. Valenzuela-Valdés, M. García-Fernández, A. Martínez-González, and D. Sánchez-Hernández, “The role of polarization diversity for MIMO systems under Rayleigh-fading environments,” IEEE Antennas Wireless Propag. Lett., vol. 5, pp. 534–536, 2006. [5] J. Costa, C. Medeiros, and C. Fernandes, “Performance of a crossed exponentially tapered slot antenna for UWB systems,” IEEE Trans. Antennas Propag., vol. 57, pp. 1345–1352, May 2009. [6] C. Medeiros, J. Costa, and C. Fernandes, “Compact tapered slot UWB antenna with WLAN band rejection,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 661–664, 2009. [7] C. Medeiros, E. Lima, J. Costa, and C. Fernandes, “Wideband slot antenna for WLAN access points,” IEEE Antennas Wireless Propag. Lett., vol. 9, pp. 79–82, 2010. [8] R. Bhatti, J. Choi, and S. Park, “Quad-band MIMO antenna array for portable wireless communications terminals,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 129–132, 2009. [9] H. Li, J. Xiong, and S. He, “Extremely compact dual-band PIFAs for MIMO application,” Electronics Lett., vol. 45, pp. 869–870, Aug. 2009. [10] A. Rajagopalan, G. Gupta, A. Konanur, B. Hughes, and G. Lazzi, “Increasing channel capacity of an ultrawideband MIMO system using vector antennas,” IEEE Trans. Antennas Propag., vol. 55, pp. 2880–2887, Oct. 2007. [11] S. Zhang, Z. Ying, J. Xiong, and S. He, “Ultrawideband MIMO/diversity antennas with a tree-like structure to enhance wideband isolation,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 1279–1282, 2009. [12] S. Zhang, P. Zetterberg, and S. He, “Printed MIMO antenna system of four closely-spaced elements with large bandwidth and high isolation,” Electron. Lett., vol. 46, pp. 1052–1053, Jul. 2010. [13] M. Manteghi and Y. Rahmat-Samii, “Multiport characteristics of a wide-band cavity backed annular patch antenna for multipolarization operations,” IEEE Trans. Antennas Propag., vol. 53, pp. 466–474, Jan. 2005. [14] CST [Online]. Available: www.cst.com [15] D. Browne, M. Manteghi, M. Fitz, and Y. Rahmat-Samii, “Experiments with compact antenna arrays for MIMO radio communications,” IEEE Trans. Antennas Propag., vol. 54, pp. 3239–3250, Nov. 2006. [16] G. Foschini and M. Gans, “On limits of wireless communications in a fading environment when using multiple antennas,” Wireless Personal Commun., vol. 6, pp. 311–335, 1998. [17] M. Herdin, H. Ozcelik, H. Hofstetter, and E. Bonek, “Variation of measured indoor MlMO capacity with receive direction and position at 5.2 GHz,” Electron. Lett., vol. 38, pp. 1283–1285, Oct. 2002. [18] J. Wallace, M. Jensen, A. Swindlehurst, and B. Jeffs, “Experimental characterization of the MIMO wireless channel: Data acquisition and analysis,” IEEE Trans. Wireless Commun., vol. 2, pp. 335–343, Mar. 2003.

Jorge R. Costa (S’97–M’03–SM’09) was born in Lisbon, Portugal, in 1974. He received the Licenciado and Ph.D. degrees in electrical and computer engineering from the Instituto Superior Técnico (IST), Technical University of Lisbon, Lisbon, Portugal, in 1997 and 2002, respectively. He is currently a Researcher at the Instituto de Telecomunicações, Lisbon, Portugal. He is also an Assistant Professor at the Departamento de Ciências e Tecnologias da Informação, Instituto Superior de Ciências do Trabalho e da Empresa. His present research interests include lenses, reconfigurable antennas, MEMS switches, UWB, MIMO and RFID antennas. He is the coauthor of four patent applications and more than 50 contributions to peer reviewed journals and international conference proceedings. More than ten of these papers have appeared in IEEE Journals. Prof. Costa is currently serving as an Associate Editor for the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION.

Eduardo B. Lima was born in Viseu, Portugal, in 1978. He received the Licenciado and M.Sc. degrees in electrical and computer engineering from the Instituto Superior Técnico (IST), Lisbon, Portugal, in 2003 and 2008, respectively. He is a Researcher and also the software developer for antenna measurements control at the Instituto de Telecomunicações, Lisbon, Portugal. From 2004 to 2007, he was involved on the ESA/ESTEC project ILASH (Integrated Lens Antenna Shaping). He is the coauthor of one patent application and twelve technical papers in international journals and conference proceedings in the area of antennas. His present research interests include dielectric lens antennas and MIMO.

Carla R. Medeiros was born in Ponta Delgada, Açores, Portugal, in 1982. She received the Licenciado and M.Sc. degrees in electrical and computer engineering from the Instituto Superior Técnico (IST), Technical University of Lisbon, Lisbon, Portugal, in 2006 and 2007, respectively. Since 2006, she has been a Researcher at the Instituto de Telecomunicações (IT), focusing her work on antenna for wireless communications. She collaborates in national research projects. Her current research interests are in the areas of reconfigurable, RFID, MIMO and UWB antennas.

Carlos A. Fernandes (S’86–M’89–SM’08) received the Licenciado, M.Sc., and Ph.D. degrees in electrical and computer engineering from Instituto Superior Técnico (IST), Technical University of Lisbon, Lisbon, Portugal, in 1980, 1985, and 1990, respectively. He joined the IST in 1980, where he is presently Full Professor at the Department of Electrical and Computer Engineering in the areas of microwaves, radio wave propagation and antennas. He is a Senior Researcher at the Instituto de Telecomunicações and member of the Board of Directors. He has been the Leader of antenna activities in National and European Projects as RACE 2067—MBS (Mobile Broadband System), ACTS AC230—SAMBA (System for Advanced Mobile Broadband Applications) and ESA/ESTEC—ILASH (Integrated Lens Antenna Shaping). He has coauthored a book, a book chapter, and more than 120 technical papers in peer reviewed international journals and conference proceedings, in the areas of antennas and radiowave propagation modeling. His current research interests include dielectric antennas for millimeter wave applications, antennas and propagation modeling for personal communication systems, RFID antennas, artificial dielectrics and metamaterials.

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Parameter Estimation of Damped Power-Law Phase Signals via a Recursive and Alternately Projected Matrix Pencil Method Khaled Chahine, Vincent Baltazart, and Yide Wang, Associate Member, IEEE

Abstract—We propose a novel algorithm based on the matrix pencil method to estimate the parameters of a class of signals modeled as damped power-law phase signals. This class arises primarily from the electromagnetic probing of dispersive geological and civil engineering materials as a consequence of the universal dielectric response. When stratified media are considered, direct application of conventional matrix-shifting methods is hindered not only by the nonlinear frequency dependency which destroys the desired shift-invariance property of the data matrix, but also by the stratified structure of the medium which introduces a cumulative effect. In this regard, the proposed algorithm restores recursively the Vandermonde structure of one mode vector at a time by means of a spline-interpolation technique and then orthogonally projects it to filter out its contribution before passing to another. The algorithm is tested on simulated and experimental data resulting from the probing of a stratified dispersive medium, and its performance is assessed against the Cramér-Rao lower bound. For the example of experimental data, collected from concrete cores by means of a cylindrical transition line, the permittivities at the reference frequency and the dispersion indices are determined using the new algorithm and compared with those of a nonlinear optimization scheme. Index Terms—Cramér-Rao lower bound (CRLB), depth-variant model, electromagnetic characterization, frequency power law, time delay estimation, Vandermonde structure.

I. INTRODUCTION

I

N many signal processing applications involving remote sensing, it is required to estimate signal parameters from measurements collected by one or more sensors. The key to successful parameter estimation resides in the integration of robust physical phenomenology into the design of signal processing techniques. This enables them to exploit or account for physical model-based phenomena in real data. The parameters are then related to the quantitative and qualitative properties of the medium or the object of interest (e.g., [1]). Examples of such applications include wavefront extraction from swept-frequency Manuscript received March 31, 2010; revised August 10, 2010; accepted November 10, 2010. Date of publication January 28, 2011; date of current version April 06, 2011. This work was supported in part by the Regional Council of the “Pays de la Loire.” K. Chahine and V. Baltazart are with the Reconnaissance and Geophysics Group, Laboratoire Central des Ponts et Chaussées, 44341 Bouguenais Cedex, France (e-mail: [email protected]). Y. Wang is with the École Polytechnique de l’Université de Nantes, IREENA, La Chantrerie, 44306 Nantes Cedex 3, France. 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.2011.2109359

scattering data and material characterization, for which several existing model-based estimation methods have been adapted by incorporating frequency-dependent features into the signal model [2]–[4]. Such an incorporation has been shown to result in improved performance. In this context, this paper deals with the electromagnetic characterization of stratified dispersive media in civil engineering using model-based parameter estimation. Ground penetrating radar is an example of active remote sensing techniques which provide access to subsurface information that can be analyzed mainly for identification, control or characterization purposes [5]–[7]. For such techniques, if the probed medium is nondispersive, the received signal model consists of delayed and scaled replicas of the transmitted signal giving rise to the widely encountered damped/undamped exponential model [8]. Owing to the rank deficiency of its Hankel prediction matrix, specific estimation methods have been developed for this model such as MPM, KT, and MKT [9]–[11]. However, if the medium is lossy and dispersive, the arising signal model can no more be assimilated to the aforementioned model as it introduces additional parameters to account for dispersion. The dispersive behavior stems generally from the frequency dependency of the dielectric response of the medium and is modeled subsequently by a particular variant of the universal dielectric response known as the constant- model. This model results in a class of signals expressed as damped power-law phase signals for which no suitable model-based estimation method readily exists. A direct consequence of the dispersive signal model is that the bijection, recalled in [11], between the number of sources in a data sequence and the rank of its Hankel prediction matrix is not maintained [12]. As a result, estimation methods based on the rank deficiency of the prediction matrix are not directly applicable. In [13], a sensitivity study of some conventional methods, based on a band-limited approximation of the constant- model, quantified their performance degradation and derived an expression for the time delay bias in terms of the dispersion index. In this paper, we propose a novel algorithm tailored to the estimation of the time delays and dispersion indices of a stratified dispersive medium. After introducing the notion of equivalent medium, the proposed algorithm restores recursively the Vandermonde structure of one mode vector at a time by means of a spline-interpolation technique and then orthogonally projects it to filter out its contribution before passing to another. In contrast to conventional MPM, the proposed method provides unbiased estimates and is found to approach the Cramér-Rao lower bound (CRLB). The experimental validation of the algorithm is

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then conducted by fitting complex frequency-dependent dielectric data of two concretes collected by a cylindrical transition line. The rest of the paper is organized as follows. In the next section, the frequency power law for the dielectric permittivity is presented and its corresponding signal model is derived. In Section III, the principle of MPM is recalled and the proposed algorithm is developed. Section IV examines the performance of the algorithm on simulated and experimental data. Finally, Section V provides the summary and conclusion. II. DATA MODEL A. Constant-

Model

To model radar wave propagation in a constant- medium, [14] used a complex power law of frequency for the effective dielectric permittivity of the form (1) in which and are constants, is an arbitrary reference . The dispersion index frequency and is equal to for ( ) is related to the quality factor as follows: (2) . Considering only positive frequencies and where substituting (1) in the expression of the complex wavenumber where is the magnetic permeability, we obtain (3) with (4) The dispersion and absorption terms are introduced by the real and , and imaginary parts of the wavenumber , with the phase velocity and the absorption coefficient, respectively, given by (5) with (6)

and (7) The case corresponds to lossless propagation ( , is real). From (5), it can be i.e., no attenuation or is simply the phase velocity at the arbitrary refseen that erence frequency , and since is slightly dependent on frequency, the absorption coefficient in (7) obeys a frequency power law. The result in (5) is the dispersion relationship proposed for the first time by Kjartansson for the case of mechanical losses in solids [15]. The phase velocity, which also obeys a frequency power law, increases with frequency. In [16], Bickel showed that for constant- media the group velocity is always greater than the phase velocity as a consequence of being less than one. The complex power law for the dielectric constant in (1) is easy to use in the frequency domain and is valid for positive values of . The wave propagation properties of materials can be described completely by only two parameters, and the phase velocity at an arbitrary reference frequency . This simplicity makes it practical to use in the inverse imaging technique and in any inversion scheme. This approach has been successfully used in numerical 3-D forward modeling of GPR data [17], in the estimation of water content of saturated rocks [18], and more recently for the characterization of the dielectric permittivity of concrete [19]. B. Stratified Medium For a horizontally stratified medium, the backscattered complex signal can be modeled as a linear combination of echoes each of which emanates from the interface between two horizontally superposed layers under normal incidence. Each layer is considered to be homogeneous and characterized by a thickness , a constant quality factor (or, equivalently, a dispersion index ), and a dielectric constant . To this end, the geophysical literature draws a distinction between depth-variant and depth-invariant models [20]. In contrast to the former, the latter model assumes that does not vary from one layer to another. Parameter estimation in the context of the depth-invariant structure has been previously addressed in [12]. In this paper, we propose a general estimation scheme that can handle both cases. Upon substituting for each layer the expression of the corresponding wavenumber in the equation of a plane wave , the backscattered as propagating along the –axis, signal given in (8), shown at the bottom of the page, is obtained. is the frequency-independent (see where echo, is the [21]) complex amplitude of the layer, and is complex time delay corresponding to the . After white Gaussian noise with zero mean and variance

(8)

CHAHINE et al.: PARAMETER ESTIMATION OF DAMPED POWER-LAW PHASE SIGNALS VIA A RAP-MPM

sampling, the frequency variable, , is replaced by , is the sampling period or the frequency shift. The where backscattered signal becomes

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The superscript denotes the transpose operator. Note that because of the depth-variant structure, the elements of the mode given by the product of the poles in (17) can not be vector in (12). However, introducing the noput under the form of tion of “equivalent medium” allows to alleviate this problem. C. Equivalent Medium

(9) where the pole of the

layer is given by

(10) The product in the signal model represents the cumulative eftraversed layers of the stratified medium on fect of the echo. Assuming a depth-invariant model allows to the rewrite (9) in the following compact form: (11)

The forward filtering effect experienced by each of the echoes of a depth-variant structure in a horizontally stratified earth can be approximated by the effect of a one-layered . medium characterized by its equivalent parameters In this paper, the equivalent parameters are determined from the following system of equations:

(20)

where and are the real and imaginary parts, respectively. yields the expression The solution of the above system at and the following expression for : found in (13) for

where (21)

(12) and

As a result, the frequency response of the parallel-cascade configuration of the constituent layers combined by addition and multiplication (see (9)) reduces to a parallel configuration of effective one-layered media combined only by addition (see (11)). This allows us to write (22)

(13) where is the equivalent time delay of the echo (note that, compared with (9), the subscript of disappears in (11) as a consequence of the assumed depth-invariant model). From (9) and (11), we can see that the depth-invariant case is merely a special case of the more general depth-variant case. In what follows, we only consider the problem formulation under the depth-variant assumption. Under matrix form, the signal model is expressed by (14) with the following notational definitions: (15) (16) (17) (18) (19)

(23) and

(24) It is worth mentioning that the above approximation becomes . Note also exact for a depth-invariant structure, i.e., that upon setting , the above model reduces to the undamped exponential model. Therefore, the performance of conventional algorithms is expected to improve with increasing values of . The parameter estimation problem can now be stated as folestimate the time delows. Given the data sequence lays and the dispersion indices of the stratified medium.

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III. MATRIX PENCIL METHOD

B. Recursive and Alternately Projected MPM

A. Conventional MPM For the sake of self-containment, we recall briefly the principle of MPM. Starting from an undamped exponential data sequence of length obtained from the model given in (11) upon setting the value of the index to 1, MPM chooses a free param. eter, , known as the pencil parameter such as The proper choice of results in significant robustness against noise. The next step is to construct a Hankel data matrix

.. .

.. .

..

.. .

.

(25)

The proposed algorithm is developed for the case of two is straightforward; however, echoes. The extension for it requires more effort and notation and is not considered in and are known ( this work. Suppose, for now, that is practically equal to ) and consider the application of a nonlinear transformation to the uniform frequencies such that

Setting with signal:

(35) in the above equation and probing the medium give rise to the following discrete backscattered

(36) Two matrices are then obtained by removing the last and first columns of . In MATLAB notation, they are given as follows: (26) (27) The matrix pencil for the two matrices and is defined as their linear combination , with a scalar parameter. In the absence of noise and owing to the assumed signal model, it is easily verified that and admit the following Vandermonde decomposition: (28)

Nevertheless, from a practical point of view, as it is infeasible to probe the medium with nonuniform frequencies, the data sequence in (36) is obtained by cubic spline interpolation. This is achieved by determining the spline interpolant from the uniform data sequence provided by the measurement device, and then evaluating the spline interpolant at the given nonuniform fre. Neglecting interpolation errors, the data model quencies in matrix form becomes (37) with the following notational definitions: (38)

(29)

(39) where

(40) .. .

.. .

.. .

.. .

..

.

..

.. .

.

.. .

(30) (41) (42) (31)

(32) (33) revealing the fundamental shift-invariance property in the column and row spaces. The matrix pencil can then be written as

It is readily verified from (40) that the transformed mode vector of the first echo admits a Vandermonde structure, and so its data matrix becomes of rank one. Consequently, the matrices and

admit the following decomposition: (43) (44)

where

(34) is a rank-reducing number of Hence, each value of are, therefore, the generalized the pencil. The estimates of . For noisy data, eigenvalues (GEVs) of the matrix pair total least squares matrix pencil (TLSMP) is usually preferred in which the singular value decomposition is used to prefilter the complex signals, and then conventional procedures follow. For more details, the reader can refer to [9].

(45) (46) The corresponding matrix pencil can then be written as (47)

CHAHINE et al.: PARAMETER ESTIMATION OF DAMPED POWER-LAW PHASE SIGNALS VIA A RAP-MPM

Similar to conventional MPM, the value of reduces the rank of the pencil in (47) to zero and ought to be a . However, due to the depthGEV of the total matrix pair variant structure of the dispersive medium, the Vandermonde structure can be restored for one mode vector at a time. In such a case, estimating as the GEV of the total matrix pair is valid provided that the contribution of other echoes is filtered out after each restoration. As a result, will be first estimated coarsely as a GEV of the total matrix pair. The estimation will be refined and can in later steps of the algorithm. The estimates of then be deduced from the value of as follows:

(48)

(49)

where and are the real and imaginary parts, respectively. Now that we know a rough estimate of the pole, we can filter out the contribution of the associated mode vector to the Hankel via orthogonal projection as in [22]. data matrix However, contrary to [22] that used a QR decomposition of the known mode vector to determine the orthogonal subspace, we carry out a singular value decomposition of the Hankel data maas . Using MATLAB trix constructed from notation, the orthogonal complement is then given by . The definition of reflects the or, equivalently, would be overesfact that the rank of admits a Vandermonde structure guaranteed timated, unless in (35). More importantly, such a definiby setting enables, once and are estimated, to filter tion of out the contribution of even if the rank is overestimated or, . To recapitulate, the Vandermonde strucequivalently, ture of the mode vector is essential only to parameter estimation as seen in (48) and (49) but not to filtering as shown by the definition of the orthogonal complement. Left multiplying by the gives conjugate transpose of

(50) The superscript denotes the conjugate-transpose operator. As pointed out in [23], the multiplication from the left destroys the shift-invariance property in the column space but maintains it can still be used to esin the row space. This means that timate the parameters of the second echo. When the processed data sequence is noisy, the orthogonal projection is only an approximation due to the perturbation introduced by noise in the signal subspace, and thus only the principal left singular vectors are to be considered when determining the orthogonal complement. The number of these vectors must be chosen in a way that ensures the best orthogonal projection on the one hand and

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maximum filtering of the first mode vector on the other hand. Obviously, the aforementioned requirements are contradictory (orthogonal projection improves with decreasing number of singular vectors since the largest singular values are less affected by noise, while the filtering effect improves with increasing number of vectors), and so the optimum number of principal singular vectors as to minimizing estimation errors is found heuristically . to be two, i.e., in (35) restores the VanderSimilar to (36), setting monde structure, only this time for the second mode vector. The and are then determined from the GEV of estimates of using (48) and (49) mutatis mutandis. the matrix pair The following steps of this alternately projected approach is to filter out the contribution of the second mode vector and re-estimate the parameters of the first and so on until “practical convergence” is achieved (to be discussed afterwards). Once and are determined, the complex amplitudes can be estimated using a least squares fit having the following solution:

(51) Note that the development of the algorithm has hitherto been and . In practice, based on the a priori knowledge of however, the values of the indices are unknown and need to be estimated. To solve this problem, we propose a recursive scheme of the previously described approach. The Recursive and Alternately Projected Matrix Pencil Method (RAP-MPM) can now be summarized in the following steps for the case of . 1) Construct a Hankel matrix from a data sequence selected at frequencies given in (35). If the model order ( is known), left multiply by the conjugate transpose of . We start with and as initial value which corresponds to the exact uniform data sequence provided by real measurements. and by running MPM 2) Determine and using (48) and (49). with the value used 3) Compare the estimated value of in step 1. If they are different, i.e., , substitute the in (35) to obtain a new set of nonuninew estimate of form frequencies at which the spline interpolant is to be evaluated. Repeat steps 1, 2 and 3 until convergence, i.e., . This termination condition might never be satisfied for noisy data, and so it will be replaced by a prede. The convergence of fined practical threshold, e.g., this recursive scheme was established in [12]. and of the last iteration 4) Retain the estimates of is to be determined from and use them each time for a given value of . the mode vector to and repeat steps 1 to 4 5) Set the model order until the relative change of the estimates between two consecutive alternated projections is smaller than a predefined value. Two to three projections were found to be sufficient (see Fig. 1). and to estimate 6) Use the estimates of the complex amplitudes using (51).

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= 0 87

= 3 2 ns

= 0 83

Fig. 1. The relative error for two poles (n : , : ,n : , : ) as a function of the projection index in the absence of noise. The  bottom and left axes are for the first pole, while the top and right axes are for the second. As can be seen, error stabilizes after two alternate projections. Note also that despite the absence of noise, the relative error is not zero because of errors introduced by interpolation and the equivalent-medium approximation.

= 1 2 ns

Fig. 2. The squared relative error for five different values of n in the absence of noise. Each relative error function has a zero at n n.

=

IV. PERFORMANCE EVALUATION In this section, the validity of the equivalent-medium approximation is examined and the mean squared error (MSE) of the proposed method is compared with the CRLB for three different scenarios. The bounds were derived in a manner similar to that in [12]. However, the elements of the Fisher information matrix could not be simplified to a rather concise and elegant form and were omitted from this presentation. In our simulations, from a bandwidth of 2 GHz sampled at and centered at were used. The signal-to-noise ratio (SNR) was defined as the average SNR over the data interval (52)

A. The Validity of the Equivalent Medium To study the validity of the equivalent-medium approximation, Fig. 2 shows the squared relative error obtained as the squared ratio of the difference between the true and equivalent models to the true model as a function of for , , and (0.5 and 1 are the limit values of for civil engineering materials [21]). As expected, each of the relative error functions which indicates that the approximareveals a zero at

Fig. 3. The MSEs of  a function of n .

= 3:2 ns and  = 1:2 ns along with the CRLBs as

Fig. 4. The MSE of n a function of n .

= 0:83 and 0:5  n  1 along with the CRLBs as

tion becomes exact for the depth-invariant structure. However, departs in either direction from , error increases almost as until reaching a symmetrically with respect to the value of , 0.75, maximum of 2.5%, 1%, 3%, 19% and 15% for 0.83, 0.5 and 1, respectively. This result leads to the conclusion that the equivalent-medium approximation is rather satisfactory . except for extreme cases having B. MSE Versus This simulation examines the influence of the difference beand on the performance of RAPtween the two indices MPM. The estimates were computed for 200 independent trials by setting and at noise variance . The values of the other parameters were varying , , , chosen to be and . For and , the MSEs show, owing to the constancy of , a slight variation and approach their CRLBs. and show a greater variaIn comparison, the MSEs of that modifies the SNR of the tion related to the variation of second echo (see Figs. 3 and 4). They also approach the CRLBs over a satisfactory range of values in the neighborhood of 0.83 (the value that renders the approximation exact). These results indicate that RAP-MPM is robust to the model discrepancy introduced by the equivalent-medium approximation. C. MSE Versus SNR In this simulation, the estimates of , , and were computed for 200 independent trials at different values of SNR.

CHAHINE et al.: PARAMETER ESTIMATION OF DAMPED POWER-LAW PHASE SIGNALS VIA A RAP-MPM

= 3 2 ns

= 1 2 ns

Fig. 5. The MSE of  : and  : along with the CRLBs for n : . the depth-invariant case as a function of SNR at n

= 0 83

=

= 0 83

= 0 83 along with the CRLBs for the

Fig. 6. The MSE of n : and n : depth-invariant case as a function of SNR.

= 3 2 ns

= 1 2 ns along with the CRLBs for = 0:87 and n = 0:83.

Fig. 7. The MSE of  : and  : the depth-variant case as a function of SNR at n

Two cases representing the depth-invariant ( ) ) structures were conand depth-variant ( sidered. The values of the parameters common to both cases , , , were chosen to be . The MSEs along with the corresponding and CRLBs are plotted in Figs. 5–8. It is seen from these figures that the MSEs of RAP-MPM are quite close to the CRLBs starting for the depth-invariant case (Figs. 5 and from for the depth-variant case (Figs. 7 and 8). 6) and These results indicate that RAP-MPM is not biased. Note that in the case of the depth-variant structure, the lower noise threshold (0.87 vs. 0.83). is due to the higher value of

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= 0 87

Fig. 8. The MSE of n : and n depth-variant case as a function of SNR.

= 0:83 along with the CRLBs for the

= 3:2 ns along with the CRLBs for the depth-invariant = 0 83 and the depth-variant (DV) case at n = 0:87 as a

Fig. 9. The MSE of  : (DI) case at n function of  .

B

Fig. 10. The MSE of n = 0:83 for the depth-invariant (DI) case and n = 0:87 for the depth-variant (DV) case along with the CRLBs as a function of B .

D. MSE Versus Similar to the previous simulation, both structures were considered and the estimates were computed for 200 independent . was varied trials, only this time at different values of between 0.2 ns and 1.5 ns while was kept fixed at 3.2 ns. The variance of noise was . As can be seen from decrease Figs. 9–12, for both structures, the MSEs of and , whereas for and the monotonically with increasing MSEs show two variations resulting from two opposing effects. and the SNR of the second echo vary Bearing in mind that

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= 1:2ns along with the CRLBs for the depth-invariant = 0 83 and the depth-variant (DV) case at n = 0:87 as a

Fig. 11. The MSE of  : case (DI) at n function of  .

B

= 0 83

=

Fig. 12. The MSE of n : for the depth-invariant (DI) case and n : for the depth-variant (DV) case along with the CRLBs as a function of  .

0 83 B

in opposite manners with increasing values of , the MSEs debeyond crease until they reach a threshold value at which the effect of low SNR dominates and the MSEs tend to increase (Figs. 11 and 12). Moreover, the MSEs of all paramvalues below the threshold eters approach the CRLBs for (except for the first one or two points) and are quite close to the CRLBs for values above the threshold. These results indicate that RAP-MPM affords a high resolution capacity. E. Experimental Results on the Characterization of Concrete The experimental setup shown in Fig. 13 was used to measure the complex dielectric permittivity of concrete mixtures at radar frequencies. It consisted of a vector network analyzer (VNA) controlled by a central processing unit and connected to a cylindrical transition line via a coaxial cable. The transition line enables permittivity measurement of cylindrical samples of heterogeneous materials with large aggregate dimensions (up to 20 mm) over a frequency range from 50 MHz to 1.5 GHz in air [24]. Data were limited to the bandwidth [50 MHz, 600 MHz] to avoid the effects of electromagnetic (EM) resonance related to the geometries of the cylindrical samples. The concrete mixtures, denoted B1 and B2, were characterized by a water-to-cement ratio of 35%, two values of relative humidity of 13% and 18%, and two maximum aggregate sizes of 10 mm and 5 mm, respectively. For each mixture, a cylindrical sample of 75 mm diameter and 70 mm length was machined. The relation between

Fig. 13. The experimental setup consisting of a VNA, a coaxial cable, a cylindrical transition line, and a processing unit.

Fig. 14. The measured and fitted real and imaginary parts of the dielectric permittivity of B1.

the complex permittivity and the reflection coefficient at the cylindrical transition was obtained from a mode-matching technique including axisymmetric higher order modes excited at the transition. The line was calibrated using a specific calibration kit and complex permittivities of the mixtures were retrieved separately using an iterative optimization procedure. These permittivities were then plugged into the expression of the complex wavenumber to obtain the following signal model: (53) is the two-way travel distance. The algorithm was where and . Then, and were obused to determine , , tained from (6). This allowed the reconstruction of the complex dielectric permittivities by using (1) and their comparison with real measurements as shown in Figs. 14 and 15. In addition, to gain more insight into the performance of the algorithm, the estimates of and were compared with those obtained by a nonlinear least squares (NLS) optimization scheme applied separately on each permittivity measurement. Table I gathers the parameters obtained by both approaches along with the corresponding residues. It can be seen that the proposed algorithm yields comparable estimates and affords a lower fitting error than the NLS scheme. Therefore, considering the EM characterization of concrete as an example confirms not only the promising performance of RAP-MPM but also the validity of the constant- model for such an application.

CHAHINE et al.: PARAMETER ESTIMATION OF DAMPED POWER-LAW PHASE SIGNALS VIA A RAP-MPM

Fig. 15. The measured and fitted real and imaginary parts of the dielectric permittivity of B2.

TABLE I THE PARAMETERS OBTAINED BY NLS AND RAP-MPM ALONG WITH THE : = IS THE DIELECTRIC CORRESPONDING RESIDUES. " PERMITTIVITY OF FREE SPACE

= 8 854 2 10

Fm

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[6] J. Li and R. Wu, “An efficient algorithm for time delay estimation,” IEEE Trans. Signal Processing, vol. 46, pp. 2231–2235, Aug. 1998. [7] S. M. Shrestha and I. Arai, “Signal processing of ground penetrating radar using spectral estimation techniques to estimate the position of buried targets,” EURASIP J. Appl. Signal Processing, vol. 12, pp. 1198–1209, 2003. [8] E. K. Miller, “Model-based parameter estimation in electromagnetics. I. Background and theoretical development,” IEEE Antennas Propag. Magazine, vol. 40, no. 1, pp. 42–52, Feb. 1998. [9] Y. Hua and T. K. Sarkar, “Matrix pencil method for estimating parameters of exponentially damped/undamped sinusoids in noise,” IEEE Trans. Acoust., Speech Signal Processing, vol. 38, pp. 814–824, May 1990. [10] R. Kumaresan and D. Tufts, “Estimating the parameters of exponentially damped sinusoids and pole-zero modeling in noise,” IEEE Trans. Acoust., Speech Signal Processing, vol. ASSP-30, pp. 833–840, Dec. 1982. [11] Y. Li, K. Liu, and J. Razavilar, “A parameter estimation scheme for damped sinusoidal signals based on low-rank Hankel approximation,” IEEE Trans. Signal Processing, vol. 45, pp. 481–486, Feb. 1997. [12] K. Chahine, V. Baltazart, and Y. Wang, “Interpolation-based matrix pencil method for parameter estimation of dispersive media in civil engineering,” Signal Processing, vol. 90, pp. 2567–2580, Aug. 2010. [13] K. Chahine, V. Baltazart, Y. Wang, X. Dérobert, and C. Le Bastard, “Effects of frequency-dependent attenuation on the performance of time delay estimation techniques using ground penetrating radar,” in Pro. 17th Eur. Signal Processing Conf., Glasgow, Scotland, Aug. 2009, pp. 749–753. [14] M. Bano, “Modelling of GPR waves for lossy media obeying a complex power law of frequency for dielectric permittivity,” Geophys. Prospect., vol. 52, pp. 11–26, 2004. [15] E. Kjartansson, “Constant -wave propagation and attenuation,” J. Geophys. Res., vol. 84, pp. 4737–4748, 1979. [16] S. H. Bickel and R. R. Natarajan, “Plane-wave deconvolution,” Geophysics, vol. 50, pp. 1426–1439, 1985. [17] A. Bitri and G. Grandjean, “Frequency-wavenumber modelling and migration of 2D GPR data in moderately heterogeneous dispersive media,” Geophys. Prospect., vol. 46, pp. 287–301, 1998. [18] M. Bano and G. F. Girard, “Radar reflections and water content estimation of aeolian sand dunes,” Geophys. Res. Lett., vol. 28, pp. 3207–3210, 2001. [19] T. Bourdi, J. E. Rhazi, F. Boone, and G. Ballivy, “Application of Jonscher model for the characterization of the dielectric permittivity of concrete,” J. Phys. D: Appl. Phys., vol. 41, Oct. 2008. [20] Y. Wang, “Inverse -filter for seismic resolution enhancement,” Geophysics, vol. 71, no. 3, Jun. 2006. [21] G. Turner and A. Siggins, “Constant- attenuation of subsurface radar pulses,” Geophysics, vol. 59, pp. 1192–1200, 1994. [22] H. Chen, S. Van Huffel, and J. Vandewalle, “Improved methods for exponential parameter estimation in the presence of known poles and noise,” IEEE Trans. Signal Processing, vol. 45, pp. 1390–1393, May 1997. [23] H. Chen, S. Van Huffel, A. Van Den Boom, and P. Van Den Bosch, “Subspace-based parameter estimation of exponentially damped sinusoids using prior knowledge of frequency and phase,” Signal Processing, vol. 59, pp. 129–136, May 1997. [24] M. Adous, P. Quéffélec, and L. Laguerre, “Coaxial/cylindrical transition line for broadband permittivity measurement of civil engineering materials,” Meas. Sci. Technol., vol. 17, pp. 2241–2246, 2006.

Q

V. CONCLUSION Incorporating dispersion effects into the formulation of model-based estimation methods allows for a more accurate characterization of dispersive materials. To this end, we proposed RAP-MPM, a novel algorithm based on the matrix pencil method, to estimate the parameters of damped power-law phase signals arising from the incorporation of the constantmodel. RAP-MPM first unifies the form of the mode vectors through an equivalent-medium approximation and then uses a recursive and alternately projected scheme to restore their Vandermonde structures without the a priori knowledge of the dispersion indices. The algorithm was found to approach . In the the CRLB for scenarios of varying , SNR, and experimental part, it was satisfactorily used to characterize two concrete mixtures. An avenue for further research would be to test the algorithm on wideband data collected in situ by a step-frequency radar. REFERENCES [1] U. Spagnolini, “Permittivity measurements of multilayered media with monostatic pulse radar,” IEEE Trans. Geosci. Remote Sensing, vol. 35, pp. 454–463, 1997. [2] A. Moghaddar, Y. Ogawa, and E. K. Walton, “Estimating the timedelay and frequency decay parameter of scattering components using a modified MUSIC algorithm,” IEEE Trans. Antennas Propag., vol. 42, pp. 1412–1418, Oct. 1994. [3] M. McClure, R. C. Qiu, and L. Carin, “On the superresolution identification of observables from swept-frequency scattering data,” IEEE Trans. Antennas Propag., vol. 45, pp. 631–641, Apr. 1997. [4] F. Sagnard and G. E. Zein, “In situ characterization of building materials for propagation modeling: frequency and time responses,” IEEE Trans. Antennas Propag., vol. 53, pp. 3166–3173, Oct. 2005. [5] C. Le Bastard, V. Baltazart, Y. Wang, and J. Saillard, “Thin pavement thickness estimation using GPR with high and super resolution methods,” IEEE Trans. Geosci. Remote Sensing, vol. 45, pp. 2511–2519, 2007.

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Khaled Chahine was born in Beirut, Lebanon, on March 6, 1984. He received the electronics and electrical engineering degree from the Lebanese University, Lebanon, in 2007 and the master’s degree in electronic systems from the École Polytechnique de l’Université de Nantes, France, in 2007. He is currently working toward his Ph.D. degree at the Laboratoire Central des Ponts et Chaussées, Nantes, France. His research interests include spectral analysis and blind source separation.

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Vincent Baltazart received the Ph.D. degree in signal processing from the University of Rennes, France, in 1994. From 1992 to 1993, he worked on ionospheric modeling and propagation at IPS Radio and Space Services, Sydney, Australia. From 1994 to 1996, he worked on microwave remote-sensing techniques at the Université Catholique de Louvain, Louvain-la-Neuve, Belgium. In 1996, he joined the Laboratoire Central des Ponts et Chaussées, France, as a Researcher in the field of optical sensors. He is currently involved in nondestructive testing and evaluation techniques for civil engineering applications.

Yide Wang (M’04–A’04) received the B.S. degree in electrical engineering from the Beijing University of Post and Telecommunication, Beijing, China, in 1984 and the M.S. and Ph.D. degrees in signal processing from the University of Rennes, France, in 1986 and 1989, respectively. He is now a Professor with the École Polytechnique de l’Université de Nantes, France. His research interests include array signal processing, spectral analysis, and mobile wireless communication systems.

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Lenses for Circular Polarization Using Planar Arrays of Rotated Passive Elements Rudi H. Phillion, Student Member, IEEE, and Michal Okoniewski, Fellow, IEEE

Abstract—A planar array of passive lens elements can be phased to approximate the effect of a curved dielectric lens. The rotational orientation of each element can provide the required phase shift for circular polarization. The array elements must be designed so that the hand of circular polarization changes as the electromagnetic wave passes through the lens. An element is presented that is based on an aperture-coupled microstrip patch antenna, and two lenses are designed. Each lens has a diameter of 254 mm and contains 349 elements. The elements have identical dimensions but the rotational orientation of each element is selected to provide a specific lens function. The first lens is designed to collimate radiation from a feed horn into a beam pointing 20 from broadside. At 12.9 GHz the aperture efficiency is 48%. The second lens acts as a Wollaston-type prism. It splits an incident wave according to its circular polarization components. Index Terms—Array lens, artificial lens, phased-arrays, reflectarrays, transmit-arrays.

I. INTRODUCTION

L

ENSES can be used to redirect, converge, or diverge electromagnetic radiation. For microwave applications, lenses are often used to collimate radiation from a feed as part of an electrically-large aperture antenna. Dielectric lenses are widely used passive structures where the curvature of the dielectric material creates the lens effect. Phased-array lenses consist of an array of active elements and each element contains an inner- and outer-surface radiator connected by a phase shifter [1]. The lens effect is created by the phase shifters, which collimate radiation from the feed and steer the resulting beam in the desired direction. More recent contributions to phased-array lenses combine planar antenna elements with amplifiers for spatial power combining [2]–[4]. Some of the current phased-array lens research is directed towards optimization of individual phase-agile elements for beam-forming antennas [5], [6]. For a broad range of applications, either a dielectric or a phased array lens may be a good choice. However, for fixed beam applications requiring a very large aperture, the weight of a dielectric lens may be prohibitive. Space-borne use can also require that the antenna be deployable in orbit. For these lens applications, a planar array of passive phase-shifting elements Manuscript received May 28, 2010; revised August 20, 2010; accepted September 17, 2010. Date of publication January 31, 2011; date of current version April 06, 2011. This work was supported in part by the NSERC and in part by Alberta Ingenuity. The authors are with the Department of Electrical and Computer Engineering, Schulich School of Engineering, University of Calgary, Calgary, AB, T2N 1N4, 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.2011.2109694

can be preferable. This type of antenna is known simply as an array lens. The fixed phase shift of each element creates a phase distribution across the aperture of the array. This distribution of phase shift has a lensing effect if it approximates the propagation delay due to the shape of a dielectric lens. A Fresnel zone plate is an example of a planar passive lens, but an array lens typically refers to a two-dimensional lattice of elements. Similar to how a reflectarray approximates a curved reflector, an array lens approximates a curved dielectric material. It is for this reason that some authors refer to an array lens as a transmit-array. The trade-offs between a reflectarray and an array lens are similar to those between a curved reflector and a dielectric lens. A reflectarray is conceptually simpler, has low loss, and is easier to construct mechanically if it can be supported from behind. On the other hand, an array lens eliminates the feed blockage that is typical of reflectarrays; this allows the use of large feed antennas or multiple feeds without compromising the radiation aperture. Of particular importance is the array lens’s improved tolerance to surface errors. Small deviations in the flatness of a millimeter-wave reflectarray can lead to significant phase errors. On the other hand, surface deviations in array lenses have a smaller effect on the total path length; and hence, the resulting phase errors are smaller ([7], p. 22). This feature will simplify the mechanical structure needed to maintain a flat array surface. Another advantage is that the feed is physically covered by the array. In some applications it is important to conceal the feed, e.g., for aesthetic or aerodynamic reasons. In addition to array lenses and reflectarrays, other antenna configurations that excite a passive array from a single element are currently being developed. In many of these, the feed element is placed in a cavity between the array and a reflective surface. This array configuration can be considered as parasitic loading [8], or as a discretized Fabry-Perot resonator [9]. In the 1960s, passive array lens elements consisting of two antennas connected by a coaxial transmission line were presented [10]. Although the arrays were planar, the elements were not low profile. More recently, element concepts based on microstrip patch antennas [11], [12], and a lens prototype using multiple layers of cross elements [13] were contributed. Concurrent to this work, other array lenses are being developed using a wide variety of elements: double square rings [14], patch antennas with perpendicular transmission lines [15], and patch antennas that transmit a rotated polarization [16]. These designs achieved a peak aperture efficiency between 40%–47% if the feed pattern was well suited to the size of the array. In this work, element rotation is used to phase the elements. This technique is applicable to circular polarization systems and

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Fig. 1. An array lens for circular polarization. The array consists of rotated passive elements (Front View). The multi-layer structure consists of metal patterned on dielectric substrates (Side View). Each element consists of stacked microstrip patches on both sides of an aperture in the ground plane (Single Element).

was first used as a phase shifter for cylindrical waveguide [17], [18]. It was used in reflectarrays as early as 1976 [19]; but, it has recently received more attention since a reflectarray design using microstrip patch elements with delay stubs was presented [20]. Since then, various passive reflectarray elements have been proposed for this technique: shorted ring slots [21], dipoles over a ground plane [22], edge-shorted disks [23], ring elements [24], loaded ring slots [25], and split square rings [26]. As well, some actively rotating reflectarray elements have been presented: ring slots with pin-diode-switched radial stubs [27], bow-tie elements with MEMS switches [28], edge-shorted disks using electric motors [29], and dipoles over a ground plane using MEMS motors [30]. Element rotation was suggested for an array lens [10], [31] but it required two antennas connected by a transmission line. The element presented in this work uses stacked microstrip patches as the inner- and outer-surface radiators. A conceptual drawing of the lens is shown in Fig. 1. Since aperture coupling links the inner- and outer-surface radiators, no transmission line is required. This elemental antenna is described in detail in Section III; however, we start the paper by analyzing the phase shift of a generalized array element in Section II. Two prototype lenses, their design, and measurements are presented in Section IV. II. USING ROTATION TO ALTER TRANSMISSION PHASE Under specific conditions, the phase of a wave transmitted through a lens element can be shifted by simply rotating the element in the plane of the array. The incident wave must have circular polarization (CP), and the hand of polarization must change as it passes through the element. These conditions are derived and discussed in this section. A. Scattering Parameters of a Rotated Element An infinite planar structure composed of periodic elements in free space is shown in Fig. 2. The reflection and transmission properties of this structure can be represented as a scattering matrix. This representation considers each side of the element as a port through which wave energy can enter or exit the system; furthermore, at each port the wave energy is typically decomposed into a set of modes (often TE and TM modes). The modal

Fig. 2. The coordinate system of the periodic array. In simulation, a single element is surrounded by Floquet boundaries.

scattering matrix relates the modal amplitudes leaving each port to the modal amplitudes incident on each port. For a typical lens, the array is far enough from the feed that only the scattering parameters of propagating modes are required. If the periodicity of the infinite array is less than one-half wavelength, one propagating TE and one propagating TM mode can be scattered from each port [32]. Initially, incidence will be restricted to plane wave modes propagating along the -axis. For this excitation, both modes scattered from each port are TEM; thus for clarity they are labeled as -polarized and -polarized wave modes. The equation relating these waves is (1) where is a vector representing the incident waves, is is the scata vector representing the scattered waves, and tering matrix. The superscript reinforces that this scattering matrix relates linearly polarized wave modes. If the element is symmetric about all three principle planes, the expanded form of (1) is

(2)

The superscript of each vector component labels the mode as either an - or -polarized wave. The subscript of each vector port, and 2 is the component indicates the port; 1 is the port of the element. In the scattering matrix, and are the and reflection coefficients to - and -polarized waves, and are the transmission coefficients. The zeros in the scattering and matrix are due to the symmetry of the element about the planes; no energy is scattered into cross-polarized modes for ideal, symmetric elements. The reflection coefficients at Port 1 plane. and Port 2 are equal due to the symmetry about the When the entire structure is rotated about the -axis by an angle , the transformation to the scattering matrix is (3)

PHILLION AND OKONIEWSKI: LENSES FOR CIRCULAR POLARIZATION USING PLANAR ARRAYS OF ROTATED PASSIVE ELEMENTS

where

is a rotation matrix

(4)

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shifted components are all due to the polarization anisotropy of the element; they are proportional to the difference between and , or and . The scattering parameters that are unchanged by rotation are all equal to the average of and , and . or B. Ideal Phase-Shifting Element

Another scattering matrix can be defined so that it relates circularly-polarized wave modes (5) In expanded form, the vector components are labeled with a superscript that indicates the hand of polarization; indicates left-hand circular polarization (LCP) and indicates right-hand circular polarization (RCP)

(6)

To create a CP lens using an array, each element is required to alter the phase of the transmitted CP modes. The ideal passive lens element will have no reflection (10) and lossless transmission (11) As well, the polarization anisotropy will eliminate the scattered fields that are unchanged by rotation (12)

For mathematical convenience, the waves in each vector are orto predered so that the component that rotates from cedes the component that rotates from to . Thus, the and seem out of order when components of vectors labeled as and . The superscript of each CP scattering parameter is labeled in a similar fashion to the subscript: for example relates the left-hand component scattered at Port 2 to the right-hand component incident on Port 1. The scattering matrix for linearly-polarized modes can be transformed into the matrix for circularly-polarized modes by: (7) is the coordinate transformation from Cartewhere sian to circular unit vectors (8)

By combining (3) and (7), the values of each CP scattering parameter are determined (9a) (9b) (9c) (9d) (9e) (9f) Note that the above formulas indicate that when the elemental antenna is rotated, four of the above parameters are phase advanced by twice the rotation angle, four are phase delayed by twice the rotation angle, and eight remain unchanged. The phase

If the element properties conform to (10)–(12) then the scattering matrix is

(13)

Circular polarization incident from one port of this element will be scattered from the other port as circular polarization of the opposite hand and phase shifted by an amount proportional to the element’s rotation angle. When many elements are combined in an array, the rotation angle of each element can be selected to provide a phase shift distribution and thus the array functions as a lens. considers an element contained in The derivation of an infinite array of identical elements, and it requires that the array lattice rotate along with the element. It also considers only broadside incidence. When used in a lens, these ideal conditions do not exist. Although the array contains identical elements, the rotation angle of neighboring elements will change their mutual coupling. Therefore, the ideal element is one that can be placed in a fixed array lattice, and it will provide phase shift that is proportional to its rotation angle regardless of the rotation of its neighbors. It will maintain acceptable performance across the required bandwidth and for the necessary range of incidence angles. III. APERTURE-COUPLED PATCH ELEMENT The previous section showed how a rotated element can act as a phase shifter if it satisfies (10)–(12). An element that is designed to meet these criteria is shown in Fig. 3. It consists of five metal layers patterned on low-loss dielectric materials. The middle layer is a cross-shaped aperture in a ground plane. A pair of metal rectangles on each side of the ground plane acts as an aperture-coupled stacked-patch antenna [33]. Aperture coupling is widely used in reflectarray elements, where the patch is coupled to a strip-line or microstrip transmission line [34], [35]. Here, the aperture does not couple to a transmission

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Fig. 4. Simulation results for broadside incidence. Variation of linear polarization reflection and transmission coefficient magnitudes with frequency.

Fig. 3. Geometry of the array lens element. Parameter values for operation at 12.4 GHz are given in Table I. For these values, the patches and aperture in the image are to scale but the dielectric thicknesses have been exaggerated by a factor of 5. TABLE I PARAMETER VALUES OF THE LENS ELEMENT AT 12.4 GHZ Fig. 5. Simulation results for broadside incidence. Variation of linear polarization transmission coefficient phases with frequency. Also shown is the difference in phase between the two components (1).

line; instead, it simply couples the stacked patch on one side of the ground plane to the stacked patch on the other side. With this geometry, one side of the element can be considered as the inner-surface radiator and the other side as the outer-surface radiator. To provide phase change, all four patches and the aperture must be rotated together. The geometrical parameters for an element designed to operate at 12.4 GHz are listed in Table I. To approximate ideal elements, as expressed by (10) and (11), the element should have magnitudes of transmission coefficients and close to 1, but magnitudes of reflection coefficients and close to 0. This is achieved by using patches close to half-wave resonance. This structure was simulated in HFSS using Floquet excitation and boundaries. The resulting coefficients for broadside incidence are shown in Fig. 4. The reflection coefficient nulls are centered around 12.4 GHz and they dB across a 1.2 GHz bandwidth. This both remain below is the element transmission bandwidth, as most of the incident power passes through the lens. The challenging criteria for this element is expressed by (12), and . which requires a phase difference of 180 between This is achieved by using rectangular, instead of square, patches. As the aspect ratio of the rectangle increases, the resonant frequencies of each patch edge move further apart and the phase and increases. With a single patch difference between on each side of the ground plane, the resonant frequency shifts

required to obtain a 180 difference are too large and the magand are reduced. Using stacked patches and nitudes of allow the relative phase difference to reach a low value of 180 . The simulated transmission coefficient phases, along with the phase difference, are shown in Fig. 5. Numerical optimizaand phase diftion of the geometry is used to keep the ference close to 180 over the transmission bandwidth. However, only the metallic dimensions are optimized as the dielectric properties are selected to be readily available materials and thicknesses. Although deviations in element size from resonance are used to attain the 180 phase difference, this element does not use variable sizes to create the phase shift distribution across the array. Only one size of element is designed; when rotated, it can provide any CP phase shift between 0 and 360 . From these simulation results, the scattering matrix of this element can be computed. Magnitudes of four of the CP scattering parameters are shown in Fig. 6. These four parameters all relate scattered fields due to RCP incidence on Port 1 of the element. Between 11.7 and 13.2 GHz the magnitude of is close to 0 dB and the the magnitudes of the other three dB. The phase of the parameter parameters are below will be proportional to rotation; thus, in this frequency band the element acts as a phase shifter to broadside incidence. However, additional simulations are required to determine the with incidence angle. Fig. 7 shows the magnivariation of tude variation; the polar angle is the incidence angle from broadplane. side, the azimuth angle is measured from to in the Due to the symmetry of the element, azimuth angles beyond

PHILLION AND OKONIEWSKI: LENSES FOR CIRCULAR POLARIZATION USING PLANAR ARRAYS OF ROTATED PASSIVE ELEMENTS

Fig. 6. Simulation results for broadside incidence. Variation of the magnitude of four scattering parameters with frequency.

Fig. 7. Simulation results at 12.4 GHz. Variation of the magnitude of S incidence angle.

with

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Fig. 9. Simulation results for 25 polar incidence. Variation of the magnitude and phase of S due to fabrication errors. Two error types are shown: the misalignment in the x direction of the patch on Metal 1, and an air gap between the two Dielectric 1 layers.

The variations due to layer misalignment in the plane are quite small compared with those due to layer separation in the direction. The aperture is designed to have the same dielectric material on both sides; however, an air gap can form between the two Dielectric 1 layers during bonding. As shown in Fig. 9, this gap can cause a significant variation in both the magnitude ; an air gap of 0.5 mm alters the phase by 28 . and phase of The simulation results presented in this section confirm that the aperture-coupled stacked-patch element is a good candidate for a CP array lens. Acceptable properties are maintained over a 1.2 GHz bandwidth and for incidence angles up to 40 . IV. ARRAY LENS PROTOTYPES

Fig. 8. Simulation results at 12.4 GHz. Error between the phase of S linear phase shift relationship.

and the

180 are not simulated. Note that the transmission magnidB for polar angles less than 40 . tude is better than Further simulations are performed to investigate the phase shift from element rotation in a fixed array lattice. Over a range of incidence angles, the phase should be advanced by twice the rotation angle. Fig. 8 shows the transmission phase error as the element is rotated and the incidence angle varies. For near broadside incidence (polar angle of 5 ), the relationship between phase and rotation is very close to linear, with a maximum phase error of 3 . The relationship at a 35 polar incidence angle fluctuates from linear, but the maximum phase error is only 16 . Another set of simulations examines the tolerance of the element to fabrication errors. When the dielectric layers are assembled, misalignment can occur between the patches and the aperture. Alignment errors up to 2 mm are examined for each patch occurs when the in both directions. The largest variation in patch on Metal 1 is misaligned in the direction. For a 25 polar incidence angle, its magnitude and phase variation are shown in Fig. 9; misalignment by 1 mm alters the phase by an additional 4 .

Two lenses are designed using planar arrays of aperture-coupled stacked-patch elements. The first lens functions as a modified convex lens; it focuses radiation from a feed antenna into a high-gain beam. The second lens functions as a Wollaston-type prism; it splits an incident wave into two beams according to the ratio of its CP components. Both arrays are designed for operation at 12.4 GHz and their elements have the geometrical parameters listed in Table I. The total thickness of the six dielectric layers is 7.1 mm. The materials used to assemble the lenses are: Rogers RT/duroid 5880 for Dielectric 1 and Dielectric 3, and Emerson and Cuming Eccostock PP-4 foam for Dielectric 2. With these materials, the mass of each lens is 0.91 g/cm . A. Modified Convex Lens for Use in a High-Gain Antenna This lens is designed to focus the RCP radiation from a horn antenna into a high-gain LCP beam. The array elements are spaced by a half wavelength (12.08 mm) in a square lattice. The boundary of the array is a staircase approximation of a circle; 349 square elements are used to fill the 254 mm diameter aperture. Each element’s rotation is selected to provide a phase shift distribution across the array. For this specific lens, the phase shift has two functions: firstly, it compensates for the spatial delay between the feed horn and the array surface; and secondly, it creates a linear phase gradient which directs the major lobe to 20 from broadside. To accomplish this, the element rotations are set to (14)

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Fig. 10. The entire Metal 2 layer of each lens. The element rotations are set according to (14) and (15). The diameter of both lenses is 254 mm.

Fig. 12. Simulation of the lens when excited by the feed horn. Each pixel represents one of the 349 elements. (a) Amplitude of a . (b) Phase of a . (c) Magnitude of S . (d) Phase of S . (e) Amplitude of b . (f) Phase of b .

Fig. 11. The modified convex lens and feed horn are mounted in a plastic frame. Detail of the Metal 0 and Metal 2 layers are shown. The ruler measures centimeters.

where is the rotation of an element at position , the free-space wavenumber is , and is the lens focal length. The 1/2 multiplier compensates for the 2:1 relationship between phase shift and rotation. The left image in Fig. 10 shows the geometrical pattern of the Metal 2 layer for this array. Any phase shift distribution can be synthesized by selecting the appropriate element rotations; this selection demonstrates the ability of an array lens to provide a specialized fixed-beam pattern. The feed antenna is a pyramidal horn with a gain of 15 dB at 12.4 GHz. It is placed such that it is in line with the center of the array and the focal-length-to-diameter ratio (F/D) of the complete antenna is 0.81. Note that the horn was not designed as a feed antenna but that the lens is designed specifically for use with this horn. As shown in Fig. 11, a rigid plastic frame is used to mount both the lens and the feed for anechoic chamber measurements. To facilitate mounting, the dielectric material is rectangular; however, the metalization of the lens (including the ground plane) is contained within the staircase approximation of a circle. Using phase delay to achieve the desired beam pattern would require delays up to 1330 . Element rotation can provide a relative phase shift of 0 to 360 ; thus, the phase shift provided by the array is similar to that of a zoned lens. 1) Antenna Simulation: The array lens is an electrically large antenna but each element has fine geometrical details. This range of sizes makes it difficult to perform a computational EM simulation of the entire antenna. In this work, the analysis is broken down into smaller tasks by assuming local periodicity for each element.

The elements are considered to be in their own infinite array environment and they are analyzed individually. First, radiation from the feed horn is modeled to determine the incident fields on each element. Then, a scattering matrix is calculated for each element. Finally, the scattered fields from each element are combined to determine the overall radiation pattern. Full-wave commercial software packages are used to simulate both the feed and the array elements. These two tasks require two different types of analysis: simulation of the feed models a source radiating into free space; whereas simulation of each element models plane wave excitation of a periodic surface. Many software packages can perform both of these tasks; however, in this work a different package is chosen for each task, as our group has more experience in using these two packages in the two respective types of problems. The FEKO software package is used to simulate the feed horn transmitting into free space. Electric and magnetic field components are determined in the plane of the array, specifically at the 349 points that will be at the center of each element. The Poynting vector is computed at these points and its direction is used as the element’s incidence angle. The fields at each point are translated into a coordinate system aligned with the Poynting vector at that point. In this coordinate system, the incident elec,a tric fields are decomposed into a transverse RCP wave , and a radial field component parallel transverse LCP wave to the Poynting vector. The radial field component is always at least 30 dB smaller than the other components confirming that the array is placed outside of the horn’s near field region. The amplitude and phase of are illustrated in Fig. 12(a)–(b). These patterns are typical of a horn antenna. is determined for each element A scattering matrix assuming local periodicity in Ansoft HFSS. Each matrix is different as each element has a different incidence angle and has been rotated to provide a different phase shift. In a functional application, this type of antenna can have many thousands of elements and so a method is devised to approximate each matrix

PHILLION AND OKONIEWSKI: LENSES FOR CIRCULAR POLARIZATION USING PLANAR ARRAYS OF ROTATED PASSIVE ELEMENTS

using a small number of simulations. The first approximation is to consider the effect of counter-clockwise element rotation to be equal to the effect of clockwise change in azimuth incidence angle. To determine each scattering matrix, the difference between the actual azimuth incidence angle and the actual element rotation angle is used as the simulated azimuth incidence angle, and the simulated rotation angle is set to 0 . This is a good approximation; however, the coupling to neighboring elements is slightly different between the lens and the simulation. The second approximation is to simulate only specific incidence angles and interpolate the results. Polar angles from 0 to 35 in 5 steps, and azimuth angles from 0 to 165 in 15 steps requires 96 simulations. Due to the geometrical symmetry, azimuth angles beyond 180 need not be simulated, as they are equivalent to those below 180 . From these 96 simulations, a scattering matrix for every element can be determined. parameter are illustrated in Interpolated values of the is close Fig. 12(c)–(d). For most elements, the magnitude of to 0 dB; however for some elements near the edge of the array it dB. The interesting phase shift pattern is required is below to steer the resulting beam to 20 off of broadside. A custom code is used to combine the full-wave simulation results into the overall radiation pattern. Substituting the inciand the scatdent field from the feed antenna simulation into tering matrices from the single element simulations (5) results in the scattered field on both sides of the array lens . Values of are illustrated in Fig. 12(e)–(f). The amplitude distribution is similar to that of the feed horn and the horizontal phase gradient will steer the major lobe to 20 from broadside. The scattered fields are transformed into a radiation pattern. Cuts from this calculated pattern are compared to measurements in Section IV.A.3. The analysis technique presented in this section assumes local periodicity to model each lens element, and uses two approximations to reduce the number of single element simulations. These simplifications introduce some error; however, they allow the full-wave simulation results to be reused for any lens configuration. A single element is simulated for 96 incidence angles, and the results can be used to model any set of element rotations. Specific full-wave simulation codes, such as [36], are being developed to analyze entire array lens and reflectarray structures. For layered, planar structures, these codes will be simple enough to analyze large antennas with fine geometrical details. The assumption of local periodicity will no longer be required and the lens can be more accurately modeled. For the prototype antenna, it could have been possible to simulate all 349 elements, each with their individual incidence angle and rotation. This would remove the approximations relating incidence angle and rotation, but would still assume local periodicity. This technique is not used for the prototype because it is intended to demonstrate both the design method and the proposed simulation of a large-scale array. 2) Aperture Efficiency Calculation: This antenna is designed to have high gain but there are many factors that decrease its aperture efficiency. The main factors are: power spillover from the feed, tapered amplitude distribution, scanning loss, and nonideal transmission through the lens. The existing simulation results are reused to predict these losses.

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TABLE II CALCULATED CONSTITUENTS OF APERTURE EFFICIENCY

The spillover efficiency is obtained from the feed simulation by comparing the power incident on the array to the total power radiated by the horn. The transmission efficiency is determined from the 349 elemental scattering matrices by comparing the total power scattered as and components to the incident power. The polarization efficiency considers only the power scattered as . The phase error and amplitude taper efcomponents according to ficiencies are calculated from the ([37], p. 138). Note that this definition of phase error includes the 20 scanning loss. A summary of the calculated constituents for this antenna are listed in Table II. The F/D ratio of 0.81 was chosen as a tradeoff between the spillover and taper efficiencies; together, they reduce the potential gain by 1.7 dB. This is quite high as a feed for a comparable reflectarray reduces the potential gain by only 1.1 dB [26]. Reflection and loss in the lens material reduce the power transmitted through the lens by 0.3 dB. Cross-polarization transmission causes a further reduction of 0.2 dB. The total phase error loss is 0.3 dB; but choosing to scan the major lobe to 20 reduces the effective area to 94% of the physical area. This reduction constitutes the majority of the phase error efficiency. The sum of all the losses is 2.5 dB, which corresponds to an aperture efficiency of 56%. 3) Antenna Measurements: The lens assembly is mounted in a 3.4 m far-field anechoic chamber that is available to us at the University of Calgary; however, for this antenna, the boundary between the radiating near-field region and the far-field region occurs at 5.3 m [38, p. 33]. To compare the measured radiation pattern with the computational analysis presented in Section IV.A.1, the fields scattered from the array in the computational analysis are transformed into a radiation pattern that is also calculated at a 3.4 m distance. Azimuth and elevation cuts of the radiation pattern are shown in Figs. 13 and 14. The angular coordinate system of the measurement is selected so that both radiation pattern cuts pass through the major lobe. The azimuth pattern cut is in the plane and contains the broadside direction. The elevation cut is perpendicular to the azimuth cut but it is not in any principle plane; it passes through the major lobe but not through broadside. In this coordinate system, the desired beam direction is at azimuth angle and a 0 elevation angle. a For both cuts, the LCP component shows the major lobe pointing in the desired direction. The measured 7 half-power beamwidth matches that predicted from simulation; however, the side-lobe levels are higher. This is potentially due to two types of errors. First, the assembled lens may deviate from the simulated lens geometry. Misalignment of the dielectric layers, and air gaps between layers have negative effects. Secondly,

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Fig. 15. Reflection coefficient measured at the coax connector to the horn.

Fig. 13. Azimuth cut of the radiation pattern. (a) LCP component, (b) RCP component.

Fig. 16. Measured aperture efficiency and axial ratio in the beam pointing direction.

Fig. 14. Elevation cut of the radiation pattern. The cut is located so that it passes through the major lobe. (a) LCP component, (b) RCP component.

the infinite periodicity simulations may not correctly model the perimeter elements. The cross-polarization component (RCP) is 6 dB higher than predicted in the beam pointing direction, and is even further from predicted in other directions. However, across both pattern cuts it is at least 20 dB below the level of the major lobe. Low RCP levels are predicted for this antenna as the required scatand ) do not apply the phase shifts tering parameters ( necessary to compensate for the spatial delay. The reflection coefficient at the input to the horn was measured both with and without the lens. It is shown in Fig. 15 and

includes the effect of the coax-to-waveguide adapter. The additional nulls confirm that a wave does reflect from the lens back into the horn. Although the major lobe points to an azimuth angle of at 12.4 GHz, that angle will have slight variation with frequency. At 11.0 GHz it points to and at 13.5 GHz it points to . For frequencies in this range, the aperture efficiency and axial ratio are calculated in the direction of the major lobe. As shown in Fig. 16, the aperture efficiency at 12.4 GHz is 0.42 and the maximum aperture efficiency of 0.48 is reached at 12.9 GHz. The axial ratio is below 2 dB in the 12.4–13.3 GHz bandwidth (7%) where the aperture efficiency is above 0.40. The aperture efficiency at 12.4 GHz is 1.2 dB below the efficiency predicted in Table II. Some of this discrepancy is associated with the local periodicity assumption of the computational analysis. Additional losses are likely due to the prototype construction, most notably from misalignment of the metal layers, an air gap at the aperture, misalignment of the horn, and from the mounting structure that is located near the elements at the edge of the array. This prototype antenna shows the potential for array lenses to be used as thin, lightweight, high-gain antennas. The maximum measured aperture efficiency is 48%, corresponding to a gain reduction of 3.2 dB. At 12.4 GHz, a loss of 1.7 dB is associated with the feed horn; 0.3 dB is lost to direct the major lobe off broadside; 0.6 dB is lost from the lens; and 1.2 dB is lost to other minor factors. Simple improvements can be made by using a more suitable feed and by focusing the beam at broadside. More importantly, the reduction in gain attributed to the lens and other minor factors is less than 2 dB.

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Fig. 17. Beam splitting action of: (a) the array lens described in this paper, and (b) the Wollaston prism.

Fig. 18. Ray diagram representation of the waves transmitted through an infinite array lens configured to split circular polarization. Scattering associated with: (a) S , (b) S , (c) S , (d) S .

Fig. 19. The Wollaston prism array lens mounted in the anechoic chamber. For the picture, part of the lens is removed so that elements are visible in the aperture.

B. Wollaston-Type Prism The second lens prototype is designed to split an incident wave according to its CP components. The effect is similar to a Wollaston prism, which separates an incident wave into two linearly polarized beams. As shown in Fig. 17, the array lens decomposes an incident wave into its CP components and redirects the power from each component into two separate beams. The Wollaston prism is typically used at optical frequencies and consists of two triangular pieces of birefringent material. Again, this prototype lens contains 349 elements and their rotation is selected to provide a phase shift distribution across and have the array. The prism effect occurs because equal magnitudes but opposite phase shifts. Element rotations are set to (15) which results in the geometrical pattern shown in the right image of Fig. 10. This pattern creates a linear phase shift gradient across the array; however, the gradient will have opposite signs for the and . When LCP and scattered fields associated with RCP modes are incident from the same direction, they will scatter into two beams in different directions. Plane waves incident from broadside on Side 1 will scatter into two beams azimuth angles. Fields scattered in the on Side 2 at beam are mostly associated with the parameter; fields in the beam are mostly associated with the parameter. Because the element is not ideal, scattered fields on Side 2 will and parameters. The phase also be associated with the of these parameters is unaffected by element rotation and the fields they scatter will always form beams in the same direction as the incident wave. Ray diagrams associated with these four scattering parameters are shown in Fig. 18. 1) Radiation Pattern Measurement: The prism effect is demonstrated by illuminating the lens with a linearly polarized plane wave and measuring the radiation pattern. To approximate plane wave incidence, the lens is placed in front of a large reflector antenna as shown in Fig. 19. A metal sheet with a large circular aperture surrounds the lens to prevent spillover of the incident field from appearing in the measured radiation pattern.

Fig. 20. Circular polarization components of the radiation pattern measured with the lens (RCP-L and LCP-L) and with no lens (RCP-N and LCP-N).

The radiation pattern is measured with the array lens in the aperture, and then again with an empty aperture. CP radiation patterns are shown in Fig. 20. Since the incident field is linearly polarized, the LCP and RCP patterns with no lens have identical major lobes at broadside. The lens successfully redirects these azimuth angles; however, the gain in these direclobes to tions is 3.0 dB lower than the gain through the empty aperture, and the half-power beamwidth is increased from 6 to 7 . Part of this loss can be calculated using the analysis from Section IV.A.2; 0.3 dB is lost by scanning to 20 , 0.3 dB is lost in transmission through the lens elements, and 0.2 dB is lost to polarization errors. 2.2 dB is lost to other factors, which is 1 dB more than for the modified convex lens. Part of this increase can be attributed to the measurement setup: 15% of the elements lie on the perimeter of the array and are partially covered by the metal screen. If these elements have poor transmission it would explain the increased loss and the change in beamwidth. Excitation from a linearly polarized plane wave demonstrates the beam splitting effect of the prism lens. Only one beam results when the incident plane wave is circularly polarized. The two beams have different amplitudes when it is elliptically polarized: the major lobe amplitude of the RCP pattern is proportional to the LCP component of the incident field; and conversely, the major lobe amplitude of the LCP pattern is proportional to the RCP component of the incident field. For broadside incidence, the scattered beams are always pointing in equal but opposite azimuth directions. If the incidence is from a different azimuth angle, the phase gradient of the incident field projected on the array is added to the gradient

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provided by the element rotations. For example, incidence at 20 on the prototype lens will create beams at 0 and 43 . By increasing the incidence angle, it may be possible to push one of the beams to an angle beyond the scanning capability of the array element while maintaining the other beam. In this configuration the prism would act as a circular polarization filter. V. CONCLUSION Lenses for circular polarization can be fabricated using planar arrays of rotated elements. A novel elemental antenna for this type of array has been introduced. The phase shift it provides is proportional to its rotation angle in the plane of the array. Design procedures for the elemental antenna as well as for the entire array have been presented, including the description of approximations and simplifications used. Two prototypes were fabricated; a high-gain antenna and a beam-splitting prism. The radiation pattern of the high-gain antenna was calculated assuming local periodicity. It predicts the major lobe of the measured radiation pattern, but not the sidelobe level. The gain of each array has been measured experimentally and predicted by calculating constituents of aperture efficiency. An explanation for the observed differences was offered. Full-wave simulation of the entire array should improve the agreement between the predicted and measured results. Overall, the aperture efficiency of the high gain antenna was comparable to that of other reflectarrays and array lenses. Both prototypes demonstrate that circular polarization lenses can be composed of planar arrays of rotated passive elements. Additional developments in this area could include more compact elements on fewer dielectric layers, or actively rotating phase agile elements. ACKNOWLEDGMENT They would also like to thank Emerson and Cuming as well as Rogers Corporation for providing material samples that were used in preliminary stages of this work. REFERENCES [1] L. Schwartzman and L. Topper, “Analysis of phased array lenses,” IEEE Trans. Antennas Propag., vol. 16, no. 6, pp. 628–632, Nov. 1968. [2] F.-C. E. Tsai and M. E. Bialkowski, “Investigations into the design of a spatial power combiner employing a planar transmitarray of stacked patch antennas,” in Proc. MIKON-2004 Microwaves, Radar and Wireless Communications 15th Int. Conf., 2004, vol. 2, pp. 509–512. [3] S. Ortiz, J. Hubert, L. Mirth, E. Schlecht, and A. Mortazawi, “A highpower Ka-band quasi-optical amplifier array,” IEEE Trans. Microwave Theory Tech., vol. 50, no. 2, pp. 487–494, 2002. [4] S. Hollung, A. Cox, and Z. Popovic, “A bi-directional quasi-optical lens amplifier,” IEEE Trans. Microwave Theory Tech., vol. 45, no. 12, pp. 2352–2357, 1997. [5] J. Lau and S. Hum, “A low-cost reconfigurable transmitarray element,” in Proc. IEEE Antennas and Propagation Society Int. Symp. APSURSI’09, 2009, pp. 1–4. [6] A. Munoz-Acevedo, P. Padilla, and M. Sierra-Castaner, “Ku band active transmitarray based on microwave phase shifters,” in Proc. 3rd Eur. Conf. on Antennas and Propagation EuCAP, 2009, pp. 1201–1205. [7] J. Huang and J. A. Encinar, Reflectarray Antennas. Hoboken, NJ: Wiley, 2008. [8] M. Alvarez-Folgueiras, J. A. Rodriguez-Gonzalez, and F. Ares-Pena, “Pencil beam patterns obtained by a planar array of parasitic dipoles fed by only one active element,” presented at the 4th Eur. Conf. on Antennas and Propagation EuCAP, 2010.

[9] Z.-H. Wu and W.-X. Zhang, “Broadband printed compound air-fed array antennas,” IEEE Antennas Wireless Propag. Lett., vol. 9, pp. 187–190, 2010. [10] R. Brown and R. Dodson, “Parasitic spiral arrays,” in IRE Int. Convention Record, Mar. 1960, vol. 8, pp. 51–66. [11] D. Pozar, “Flat lens antenna concept using aperture coupled microstrip patches,” Electron. Lett., vol. 32, no. 23, pp. 2109–2111, 1996. [12] K.-W. Lam, S.-W. Kwok, Y. Hwang, and T. K. Lo, “Implementation of transmitarray antenna concept by using aperture-coupled microstrip patches,” in Proc. Asia-Pacific Microwave APMC’97, 1997, vol. 1, pp. 433–436. [13] S. Datthanasombat, J. Prata, A. L. R. Arnaro, J. A. Harrell, S. Spitz, and J. Perret, “Layered lens antennas,” in Proc. IEEE Antennas and Propagation Society Int. Symp., 2001, vol. 2, pp. 777–780. [14] C. G. M. Ryan, M. R. Chaharmir, J. Shaker, J. R. Bray, Y. M. M. Antar, and A. Ittipiboon, “A wideband transmitarray using dual-resonant double square rings,” IEEE Trans. Antennas Propag., vol. 58, no. 5, pp. 1486–1493, 2010. [15] P. Padilla, A. Munoz-Acevedo, and M. Sierra-Castaner, “Passive microstrip transmitarray lens for ku band,” presented at the 4th Eur. Conf. on Antennas and Propagation EuCAP 2010, 2010. [16] H. Kaouach, L. Dussopt, R. Sauleau, and T. Koleck, “X-band transmitarrays with linear and circular polarization,” presented at the 4th Eur. Conf. on Antennas and Propagation EuCAP, 2010. [17] A. Fox, “An adjustable waveguide phase changer,” Proc. Inst. Radio Engineers, vol. 35, pp. 1489–1498, Dec. 1947. [18] A. Martynyuk, N. Martynyuk, S. Khotiaintsev, and V. Vountesmeri, “Millimeter-wave amplitude-phase modulator,” IEEE Trans. Microwave Theory Tech., vol. 45, pp. 911–917, Jun. 1997. [19] H. Phelan, “Spiraphase—A new, low cost, lightweight phased array,” Microw. J., vol. 19, pp. 41–44, Dec. 1976. [20] J. Huang and R. Pogorzelski, “A Ka-band microstrip reflectarray with elements having variable rotation angles,” IEEE Trans. Antennas Propag., vol. 46, pp. 650–656, May 1998. [21] A. Martynyuk and J. M. Lopez, “Reflective antenna arrays based on shorted ring slots,” in Proc. IEEE Int. Microwave Symp. Digest, MTT-S, May 2001, vol. 2, pp. 1379–1382. [22] B. Subbararo, V. Fusco, and R. Cahill, “Spatial phase shifter for reflect array (circular polarisation) applications,” in Proc. 7th IEEE High Frequency Postgraduate Student Colloq., 2002, p. 6. [23] B. Subbararo, V. Fusco, and R. Cahill, “Rotary spatial phase shifter for reflectarray beamsteering,” in Proc. Int. Conf. on Antennas and Propagation, ICAP, Exeter, U.K., 2003, pp. 341–344. [24] B. Strassner, C. Han, and K. Chang, “Circularly polarized reflectarray with microstrip ring elements having variable rotation angles,” IEEE Trans. Antennas Propag., vol. 52, pp. 1122–1125, Apr. 2004. [25] A. Martynyuk, J. Lopez, and N. Martynyuk, “Spiraphase-type reflectarrays based on loaded ring slot resonators,” IEEE Trans. Antennas Propag., vol. 52, pp. 142–153, Jan. 2004. [26] A. Yu, F. Yang, A. Elsherbeni, and J. Huang, “An X-band circularly polarized reflectarray using split square ring elements and the modified element rotation technique,” in Proc. IEEE Antennas and Propagation Society Int. Symp. AP-S, 2008, pp. 1–4. [27] A. Martynyuk, J. M. Lopez, and N. Martyunuk, “Reflective passive phased array with open polarization phase shifters,” in Proc. IEEE Int. Symp. on Phased Array Systems and Technology, Oct. 2003, pp. 482–487. [28] H. Legay, B. Pinte, M. Charrier, A. Ziaei, E. Girard, and R. Gillard, “A steerable reflectarray antenna with MEMS controls,” in Proc. IEEE Int. Symp. on Phased Array Systems and Technology, Oct. 2003, pp. 494–499. [29] V. Fusco, “Mechanical beam scanning reflectarray,” IEEE Trans. Antennas Propag., vol. 53, pp. 3842–3844, Nov. 2005. [30] R. Phillion, “A Microfabricated Rotating Element for Reconfigurable Reflectarray Antennas,” Master’s thesis, Univ. Calgary, , 2007. [31] D. Nakatani and J. Ajioka, “Lens designs using rotatable phasing elements,” in Proc. Antennas and Propagation Society Int. Symp., 1977, vol. 15, pp. 357–360. [32] A. K. Bhattacharyya, Phased Array Antennas. Hoboken, NJ: Wiley, 2006. [33] S. Targonski and D. Pozar, “Design of wideband circularly polarized aperture-coupled microstrip antennas,” IEEE Trans. Antennas Propag., vol. 41, no. 2, pp. 214–220, Feb. 1993. [34] E. Carrasco, M. Barba, and J. A. Encinar, “Reflectarray element based on aperture-coupled patches with slots and lines of variable length,” IEEE Trans. Antennas Propag., vol. 55, no. 3, pp. 820–825, 2007. [35] M. Riel and J.-J. Laurin, “Design of an electronically beam scanning reflectarray using aperture-coupled elements,” IEEE Trans. Antennas Propag., vol. 55, no. 5, pp. 1260–1266, 2007.

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[36] P. De Vita, F. De Vita, A. Di Maria, and A. Freni, “An efficient technique for the analysis of large multilayered printed arrays,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 104–107, 2009. [37] T. A. Milligan, Modern Antenna Design, 2nd ed. Hoboken, NJ: Wiley, 2005. [38] C. Balanis, Antenna Theory: Analysis and Design, 2nd ed. New York: Wiley, 1997.

Rudi H. Phillion (S’03) received the B.Eng. degree in aerospace engineering from Carleton University, Ottawa, ON, Canada, in 2004, and the M.Sc. degree in electrical engineering from the University of Calgary, Calgary, AB, Canada, in 2007, where he is currently working towards the Ph.D. degree. His current research interests include array lens and reflectarray antennas, as well as RF micromachined devices. Mr. Phillion was awarded the Carleton University Senate Medal in 2004, the Alberta Ingenuity Studentship in 2004, the Bell 125th Anniversary Scholarship in 2005, the IEEE Antennas and Propagation Society Pre-Doctoral Research Award in 2006, and the Natural Sciences and Engineering Research Council of Canada Scholarship in 2004 and 2007.

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Michal Okoniewski (F’09) is a Professor with the Department of Electrical and Computer Engineering, University of Calgary, Calgary, AB, Canada, where he is the Libin/Ingenuity Chair in biomedical-engineering and Canada Research Chair in applied electromagnetics. His research interests range from computational electrodynamics, to reflectarrays/transmitarrays, RF MEMS and RF micromachined devices, as well as hardware acceleration of computational methods. He is also involved in bioelectromagnetics and is interested with tissue spectroscopy, NMR micro-imaging, and other medical technologies. In 2004, he co-founded Acceleware Corp. Dr. Okoniewski is a Registered Professional Engineer (P.Eng.).

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Automated Analytic Continuation Method for the Analysis of Dispersive Materials Kivanc Inan and Rodolfo E. Diaz, Member, IEEE

Abstract—All dielectrics are dispersive. This frequency dependence of materials must be modeled in a well-defined way whenever microwave structures are expected to operate over broad bands of frequency. The well known analytic properties of the permittivity can be used to generate such models by fitting them to experimental data using non-linear optimizers. However, in that approach the questions of convergence to the true global solution and the sensitivity to experimental noise remain open. Here it is shown that an automated deterministic approach to generate such a model for the important case of multi-Debye relaxation materials can be implemented. The method is compared to a recently proposed alternate approach: hybrid particle swarm-least squares optimization method (PSO/LS) that was demonstrated on idealized data sets with bandwidths in excess of 10 000:1. In our case no arbitrary parameters need be set to guarantee convergence nor need any constants be added after the fact to match the data. The case of materials with DC conductivity (imaginary permittivity growing to infinity at DC) is as easily dealt with as the conventional pure Debye case. Physically realizable results are generated even when the data is realistically noisy and spans a frequency bandwidth as small as 18:1. Index Terms—Analytic continuation, deterministic solution, dispersive materials, least squares methods, particle swarm optimization, permittivity.

urations often disagree at their respective band edges. Furthermore, the properties of manufactured materials may vary from batch to batch and even within a single sample. Therefore, a realistic experimental data set gathered from different samples or different pieces of the same sample appears inevitably as a probabilistic distribution of data points. The idealized smooth data often assumed when this problem is addressed is a fantasy which the engineer never sees. Any realistic approach to fit experimental data to an analytic model must recognize and deal with such sources of uncertainty. There have been several studies in the literature where Debye function expansions of complex permittivity are considered. The basic model is the single Debye equation([1, Sec. 2.8.1] (1) is the permittivity at infinite where is the DC permittivity, frequency and is the relaxation time. However a single Debye relaxation is usually not enough to model most materials over two or more decades of frequency. Therefore the more general form of a (potentially infinite) sum of Debye terms is used by many researchers to approximate the complex permittivity

I. INTRODUCTION

(2)

A

LL materials possess a frequency-dependent permittivity. In order to design realistic electromagnetic structures that operate over broad bandwidths of frequency, a compact and complete analytic model of dielectrics is needed. It is known that the analytic function properties of the permittivity lead to analytic formulas with which the behavior of any physically realizable dielectric can be analyzed. Most natural dielectric materials used in the radio frequency range and almost all artificial materials consisting of electrically small resistive inclusions in a host medium or structure act as multi-Debye relaxation materials. This frequency dependence must be modeled over a broad range of frequencies to enable the design of broadband microwave structures. However, it is well known that the experimental data gathered on such materials over broad bands of frequencies, in addition to being noisy, suffer from nonphysical discontinuities since the results from different test config-

Manuscript received March 31, 2009; revised July 13, 2010; accepted September 22, 2010. Date of publication January 28, 2011; date of current version April 06, 2011. K. Inan was with the Department of Electrical Engineering, Arizona State University, Tempe, AZ 85287 USA. He is now with Aselsan Inc., TR-06370 Ankara, Turkey (e-mail: [email protected]). R. E. Diaz is with the Department of Electrical Engineering, Arizona State University, Tempe, AZ 85287 USA. Digital Object Identifier 10.1109/TAP.2011.2109345

where is the permittivity at infinite frequency, is the DC conductivity, is the weight and is the relaxation frequency of the th term in the sum with the relation between the permittivity and conductivity expressed as (3) In [2] and [3] two Matlab procedures are applied to the permittivity data to produce a partial fraction expansion that can be converted into the multi-Debye function expansion as given in (2). However, this approach sometimes leads to a nonphysical dielectric with the weights becoming negative as illustrated in [4] and [5]. A different approach where the matrix form of an equation similar to (2) is inverted to find the corresponding weight parameters and relaxation frequencies is presented in [6]. A similar technique that expresses the matrix representation in terms of Taylor series is given in [7]. Methods based on the least squares minimization of the expansion error are presented in [8] and [9]. All these approaches suffer from the possibility of generating negative weights and they provide no clear way to choose the corresponding relaxation frequencies. An effective method, which avoids the previous instabilities, is presented in [10]. The authors describe the Debye function expansions of permittivity using a hybrid particle swarm-least

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Fig. 1. Circuit representation of the permittivity of the materials.

squares optimization (PSO/LS), which is a combination of constrained least squares approximation and particle swarm optimization [11]. In that paper the authors assume that the data to be fit is error free and spans 4 decades of frequency. The optimization leads to good agreement with the original data. However the question of the applicability of the method to the case where the measured data is noisy and can only be obtained in some portions of the spectrum has not been presented in detail. In addition to the previous approaches, and following the idea in [12]–[14], we expand the analytic properties of the permittivity function into an infinite sum of series RC circuits. We recognize that since Debye materials can be represented as a sum of special complex functions reducible to analytic circuits, the properties of those circuits can be exploited to deterministically extract the parameters of the model. The sum is not simply a “fit” to the data; it is an analytic function that can be used to match the measured data. In general the circuit representation of the permittivity of the materials (corresponding to (2)) is demonstrated in Fig. 1. (infinite permittivity) summarizes all the Debye In Fig. 1, terms above the end of the frequency band of interest (not available in the experimental data). All RC branches that have relaxation frequencies above the measured frequency range only look like capacitors since the measured frequency never reached their relaxation frequency. All those capacitors add together to . Similarly (DC conductivity) summathe term we call rizes all the Debye terms below the start of frequency of the band. This is because, relative to the relaxation frequency of those circuits, the data was taken at such a high frequency that the capacitors were essentially shorted and all the resistors add in parallel to a single value. It therefore must be emphasized we find is not necessarily the true but the that the sum of the conductivities of all these circuit branches. The effects of these two terms can be seen as trends to infinity in the Grant plane ( vs. ) and the Cole-Cole plot ( vs. ) respectively. In each RC circuit branch in the measured frequency and where range relate the permittivities and conductivities to the capacitances and the conductances, respectively. These individual series RC branches appear as semicircles in these planes, arranged side by side in order of their relaxation frequencies [12]. Therefore the aim of the analytic function fit is to identify all the parameters in the circuit, so the best approximation to is derived. In Section II of this paper, the key procedure enabling the automated analytic continuation method, the projection algorithm approximation, is presented. In Section III, the method is illustrated with an example. The method is then compared to the PSO/LS with some additional examples in Section IV, followed by the conclusion in Section V.

Fig. 2. Cole-Cole plot of the Debye material.

Fig. 3. Grant plane of the Debye material.

II. PROJECTION ALGORITHM APPROXIMATION From the introduction, one of the best approximations of is given by (2). To understand Grant’s [12] insight into the meaning of the terms of this expansion and the circuit elements of Fig. 1, consider the assumed Debye material

Fig. 2 shows the Cole-Cole plot of this material while Fig. 3 shows the Grant plane. In the Cole-Cole plot ascending frequency traces the curve from right to left, while in the Grant plane it traces the curve from left to right. Note that in the Grant plane the DC conductivity is easy to see because it is simply the low frequency horizontal axis intercept of the first semicircle, while in the ColeCole plot, infinite permittivity is easy to see because it is the high frequency horizontal axis intercept of the last semicircle. In the Cole-Cole plot the DC conductivity looks like a tail ) at the low end of the data. In the growing to infinity (as Grant plane the infinite permittivity looks like a tail growing to infinity (as ) at the high end of the data. The two relaxations in the assumed material create a pair of “blended” semicircles. In the Cole-Cole plot the diameters of the semicircles correspond to the capacitances of the individual circuit branches. In the Grant plane the diameters of the semicircles correspond to the conductances of the individual circuit branches. In both planes the point corresponding to the relaxation frequency of a given circuit branch appears usually directly above the center of the corresponding semicircle.

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The deterministic approach begins with initially identifying and . As pointed out these are easy to lothe values of cate in their respective planes. All we need to do is extend (extrapolating) the data to DC in the Grant plane and to infinite frequency in the Cole-Cole plot. This extrapolation is implemented by assuming that the band edge data points belong to a single semicircle. Given any finite frequency range of Cole-Cole data, it follows that if we approximate the last two data points (high end) as being members of a semicircle and the first two data points (low end) as also being members of a semicircle we will obtain a physically realizable, analytic, representation of the material—over that frequency range. (Note that two data points and the knowledge that the semicircle has its center on the real axis are enough to determine the radius and center of the circle, if we assume the data has no noise. In the presence of noise, a few data points near the end in question must be used to obtain the best possible guess as to the radius and center of these ending circles.) Thus, in the Grant plane, the data are extended with respect to the initial slope of the data. By simple geometry the radius and center of the semicircle that best matches the low frequency data is found and this semicircle is then extended to the horizontal axis. The intersection of the extended data with the real axis of . the conductivity defines Then we can define

Fig. 4. The resultant circuit after  and " removal and an example of a Cole-Cole plot with possible circuit semicircles.

(4) (and its associated infinite tail) from the perthus removing mittivity data. It is again emphasized that is not necessarily the true DC conductivity but rather it just summarizes all the Debye terms below the low end of the frequency band of our data. In a similar manner, within the Cole-Cole plot, the highest frequency data is assumed to lie on a single semicircle and extended. The intersection of the extended data with the real axis . Thus of the permittivity gives (5) generates a new data set from which , containing all the Debye terms above the end of the frequency band, has been subtracted. This removes the extension to infinity from the Grant plane. and , only the Debye terms, each After removal of of which represents a semicircle, are left as illustrated in Fig. 4. in the As mentioned, the diameters of these semicircles are in the Grant plane. This means that the Cole-Cole plot and sum of all diameters of the semicircles is the total capacitance in the Cole-Cole plot and the total conductance in the Grant plane. (In Fig. 4, the total capacitance is seen to be approximately 1.1 Farads/m for the chosen case.) Now the questions are: how many semicircles are there? (As suggested in Fig. 4, the same “blended” data may be obtained from two, three or more semicircles.) How do we separate them? And what are their individual relaxation frequencies? The projection algorithm automatically answers these questions. The projection algorithm arises from the realization that in the case of a single Debye relaxation semicircle the angular position

Fig. 5. Demonstration of the projection algorithm. (a) At every position along the Cole-Cole plot, the center of a “local” semicircle can be identified. (b) The location of that center and the knowledge of the frequency of the point from where that center was estimated yield the corresponding local relaxation frequency.

of a frequency data point as measured from the center of the semicircle uniquely identifies the relaxation frequency of that semicircle. (The point at the peak is at the relaxation frequency. In the Cole-Cole plot the points to the right of the peak are lower in frequency while the points at the left are higher in frequency). The detailed derivation is given in the Appendix. Armed with this knowledge and assuming that the data are smoothed, we then “walk” along the data in the Cole-Cole plot pretending that the local data belongs to a single semicircle. The perpendicular to the slope of that local data identifies the “local” center of the semicircle and the angle relative to that center identifies the “local” relaxation frequency. This procedure is illustrated in Fig. 5. on a Debye semicircle Since the frequency of a point is given by relative to the relaxation frequency (6) is the angle from the normal—positive to the right, (where negative to the left), each data point in the Cole-Cole plot reveals

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Fig. 6. Typical plot of relaxation frequency distribution versus real permittivity obtained by the projection algorithm.

the relaxation frequency of its own semicircle according to the following: (7)

It is easy to see that in the case of only one semicircle, all points identify only one center and therefore the procedure is exact. In the case of multiple blended semicircles the procedure gives an excellent approximation to the real relaxation frequency distribution of the data, as demonstrated below. Thus the procedure is as follows: First, determine the center of the local semicircle that is tangent to the observation frequency point and is centered on the real axis. Then the angle from the normal is easily determined and (7) is applied. Recording this relaxation frequency as the curve is traversed of each we generate a plot of relaxation frequency vs. observation point. A typical example is shown in Fig. 6. The projection algorithm interprets the Cole-Cole plot as arising from a continuous distribution of relaxation frequencies. This continuum can be approximated by any large number of terms as long as at least one Debye branch per frequency decade is assumed. Choosing to divide the permittivity axis of Fig. 6 (which in our assumed material example has a total relative permittivity range of 1.1) into n equal intervals, we get the following approximate model of the assumed experimental data: (8) for this specific problem and is the number where of Debye semicircles that will be used in the model. The number of the semicircles can be large and as the number of the semicircles assumed increases, then the data fitting becomes more and more accurate. However, since each Debye relaxation term spans approximately a decade of frequency, there is a point of diminishing returns where assuming more terms affords no greater accuracy in the fit. Considering the fact that there might be more than one relaxation frequency in each decade in the actual complex permittivity data, assuming a couple of relaxation frequencies for each decade will increase the accuracy.

Fig. 7. (a) Real part of the permittivity versus frequency of the original data. (b) Imaginary part of the permittivity versus frequency for the original data. and " are removed and the (c) Cole-Cole plot of the original data after  extended data. (d) The original extended data and its approximation after the projection algorithm.

Let us consider another example, this time with five Debye relaxations. Assume the material has a permittivity as described by (2), where and GHz. The frequency interval is assumed to be 1–36 GHz. In Fig. 7(a) and (b), real and imaginary parts of the original data versus frequency are shown, respectively. Fig. 7(c) shows the Cole-Cole plot of the original data and the extended data and and extended over a wide freafter removal of quency range with respect to the relaxation frequencies of the edge data. In Fig. 7(d), the comparison of the approximate rebuilt data from the projection algorithm and the original extended data in the Cole-Cole plot is demonstrated where the guess relaxation frequencies are calculated via (8). The comparison of the exact data with the approximate data in the frequency domain is given in Fig. 8. Fig. 8(a) and (b) compare the exact and approximate data in terms of real and imaginary parts, respectively. In Fig. 8(c) and (d), the percent errors in real and imaginary parts of the data of the projection algorithm are given. Clearly, this method yields a close approximation to the right answer but is not exact. However it serves as a suitable first guess for nonlinear optimizers or as given in the next section, for a deterministic algorithm that can approach the exact answer asymptotically. III. THE AUTOMATED ANALYTIC CONTINUATION METHOD The algorithm for the automated analytic continuation of dielectric data proceeds as follows. A. Data Smoothing If the data includes noise, then it has to be smoothed before starting the process. In our case, two different smoothing algorithms are used. One of them is polynomial fitting and the other is Gaussian smoothing.

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to guide the result towards the best fit. In this sense it could be construed to be akin to a steepest descent or conjugate gradient method in the way it navigates towards the solution guided by the function.) The idea is to define a set of functions, which perturb the relaxation frequencies so that the error between the smoothed experimental data (called the exact data in the following) and the approximate solution gets smaller. These functions are chosen from the requirement that as the approximation approaches the exact solution, not only should the error (difference between the exact and approximate solution) be reduced but also the slope of the error curve as a function of (logarithmic) frequency should tend to zero [15]. The latter requirement prevents local error corrections from creating large errors at other frequency points. The iterative operations are then given by Fig. 8. (a) Comparison of " for original data with the approximate data. (b) Comparison of " for original data with the approximate data. (c) Percentage error in " . (d) Percentage error in " .

(9) (10)

1) Polynomial Fitting: This method fits discrete data in a least squares sense by polynomials in one variable. According to the total error (between exact and fitted data) defined as acceptable, the order of the polynomial is chosen automatically. The maximum degree of the polynomial is input initially. 2) Gaussian Smoothing: This method fits discrete data using a running Gaussian average. The Gaussian width which varies with the frequency bandwidth of the data has to be input initially for this case. The crucial point of this method is that as the permittivity data is smoothed, the corresponding frequency data is also smoothed. This avoids non-physical distortion of the data near the ends of the data set and reduces the bandwidth of the data set, acknowledging that some information is being lost by the smoothing process. Data at the original frequency points are calculated by interpolation at the end of the smoothing process. B. Data Extension and , the end of the data is extrapoAfter removing lated to the horizontal axis by fitting its end to semicircles just as and . This adds fictitious it was done in the search for extrapolated frequency domain data to both ends of the spectrum that simplifies the automation. C. Projection Algorithm To obtain the first guess distribution of Debye terms, the projection algorithm is applied. The weight of each Debye term is simply the total capacitance divided by the assumed number of Debye terms and the initial guess of relaxation frequencies are the ones derived via (8). D. Fine Tuning the Relaxation Frequency Distribution Starting with the projection algorithm solution, fine tuning of the relaxation frequency distribution is achieved in a deterministic way from (8). (We call this procedure deterministic because it is not a random search or an optimized search through the entire space of complex permittivity to find a solution that minimizes some “cost function”. Instead it follows a precise (numerical) recipe that uses the properties of the function at every step

(11) (12) where is the next guess of the th relaxation frequency th successive iteration in the fine tuning procedure at is the th sample frequency where the number of samand ples are defined by the user. Defining the error function for all sample frequencies will increase the accuracy. is a “damping” constant that defines the rate of change in the next guesses of the relaxation frequencies. Choosing to be small increases the number of iterations to get a smaller error but also increases the , probability of getting the smallest error. We choose which helps to get the smallest error in fewest number of iterations. One important detail for (12) is that the term has to be calculated at the point . This can be accomplished by linear interpolation. The iterative procedure continues until the error stops getting smaller. As seen from (9)–(10), it is the error in the real part of the permittivity that is minimized through the iterative process. In order to decrease the error of the imaginary part of the permittivity, a similar error reduction algorithm is applied in the Grant plane. Thus the same procedure is carried out for instead of . Equivalent branch conductances are found via where the subscript refers to the th branch of the equivalent circuit. This time (9)–(12) are applied for . Reducing the error in decreases the error in in the Cole-Cole plot. Since the first fine tuning cycle adjusts the relaxation frequencies it has the effect of adjusting the conductivities of the Debye circuit branches without altering the capacitances. However, the “second pass” using the real conductivity changes the capacitances while keeping the conductivities at their last set of values. This means that with realistic “imperfect” data, the acceptable error in can increase while the error in decreases. Next, an example demonstrating the deterministic fine tuning algorithm is presented. Let us consider the same 5-Debye-term and , the problem as in Section II. After subtracting extended data, the data found after the projection algorithm and the data after fine tuning in the Cole-Cole plot and fine

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Fig. 11. Cole-Cole plot of the Cole-Davidson data. Fig. 9. Cole-Cole plots obtained after each successive approximation (with  and " subtracted).

Fig. 12. (a) Comparison of " for original data with the approximate data. (b) Comparison of " for original data with the approximate data. (c) Error in " in percentages. (d) Error in " in percentages. Fig. 10. (a) Comparison of " for original data with the approximate data (b) Comparison of " for original data with the approximate data (c) Error in " in percentages (d) Error in " in percentages.

tuning in the Grant plane are given in Fig. 9. The agreement with the extended data gets better from the projection algorithm to tuning in the Cole-Cole plot to tuning in the Grant plane. The final values of the parameters obtained via the automated analytic continuation method are and GHz. The original data is compared to the reconstructed data in Fig. 10. In Fig. 10(a) and (b), the real and imaginary parts of the permittivity data are compared. In Fig. 10(c) and (d), the errors of these data are given respectively. As seen from the plots, the deterministic solution and the exact data agree very well. Finally, we want to demonstrate the application of the proposed method to the extreme case of Cole-Davidson data as illustrated in Fig. 11, where the Cole-Cole plot shows a curve that approaches the real axis at an angle less than 90 degrees. The equation of the Cole-Davidson is similar to Debye equation given in (1). The denominator in (1) has a power of for

, Cole Davidson expresthe Cole-Davidson case. For sion reduces to Debye equation. Let us assume a Cole Davidson based material with the following parameters: GHz and . The frequency interval is assumed to be 1 GHz–100 THz. In Fig. 12(a) and (b), the comparison of the real and imaginary parts of the original data with the approximate data after applying the automated analytic continuation procedure is presented where 15 circuit branches to increase the accuracy are assumed. In Fig. 12(c) and (d), the errors of these data are given respectively. As seen from the plots, the deterministic solution and the exact data still agree for such a case that appears to be “nonDebye.” The final weights calculated via automated analytic continuation method are; 0.0618, 0.0637, 0.0643, 0.065, 0.0651, 0.065, 0.0649, 0.0649, 0.0649, 0.0649, 0.0649, 0.065, 0.065, 0.0649, 0.065 while the corresponding relaxation frequencies are found as 1.83, 4.53, 9.94, 18.32, 26.56, 39.13, 59.74, 97.86, 171.94, 323.09, 654.23, 1502.96, 4404.96, 17592.3, 91733.36 . We have GHz, respectively with not found any realistic RF material data that cannot be fitted with this procedure.

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Fig. 13. Real and imaginary parts of the noisy and smoothed original data assumed to be measured over the range 1–18 GHz.

IV. COMPARISON OF THE AUTOMATED ANALYTIC CONTINUATION METHOD WITH PSO/LS METHOD As seen in the previous section, the automated analytic continuation method works well enough for typical RF material data fitting problems. The only variable selected in an ad hoc manner is the “damping” constant of the fine tuning iteration. However, this number can be chosen to be small enough (0.1–0.5) to always work, regardless of the data, at a negligible computational cost. We also considered applying the PSO/LS algorithm to the same realistic problems where the frequency range of data is around a decade and the data is noisy. The PSO/LS algorithm also works fine. However, the CPU time required increases substantially for this approach, since the optimization in the method takes time. For reasonable calculation time, the number of Debye terms is chosen to be 5 for PSO/LS method. Using realistic noisy data and the mandatory smoothing we find that both approaches usually lead to similar error percentages, but the automated analytic continuation method proposed here is faster than PSO/LS. Another inconvenience with the PSO/LS method is the lack of guidelines for selecting the constants that drive the convergence of the algorithm. Let us apply the PSO/LS method as given in [10] to another example. Let us assume a Debye material that has permittivity of (2), where and GHz. The measured data is assumed to be from 1 GHz to 18 GHz. There is also random noise added to the measurement, which is 10% of the original data. The corresponding noisy data and its smoothed version are given in Fig. 13 for real and imaginary parts of the data, respectively. Now, both the analytic continuation method and PSO/LS will be applied to this data in order to fit to the actual complex permittivity. In PSO, the velocity of the particles is adjusted according to their distances from the locations and using (13) where, is the velocity component in the th dimension; is the th coordinate of the particle’s current position (i.e., its

Fig. 14. (a) Real parts of the five-term Debye function expansions compared to those of the original data. (b) Imaginary parts of the five-term Debye function expansions compared to those of the original data. (c) Percentage error of real parts. (d) Percentage error of imaginary parts.

Fig. 15. (a) Real parts of the five-term Debye function expansions compared to those of the original data. (b) Imaginary parts of the five-term Debye function expansions compared to those of the original data. (c) Percentage error of real parts. (d) Percentage error of imaginary parts.

current nth relaxation frequency); is the th coordinate of is the th coordinate of gBest; and is a random number generator with a uniform distribution over [0, 1]. The constants and are weights used to control the progression of the algorithm. The results obtained using PSO/LS when the number of particles is set to 30 and the weights are set to and , as given in [10], are shown in Fig. 14. PSO/LS did worse than the proposed analytic continuation method. The CPU time for PSO/LS is 29.468 seconds to get this result where it is only 1.077 seconds for the automated analytic continuation method. (The corresponding PC used for the comparison has a Pentium IV 3.0 GHz CPU and 1.0 GB of RAM.) When the weights are adjusted to and , PSO/LS method converges with a similar amount of error as our proposed method as shown in Fig. 15. However, the computation time of 7.599 seconds for PSO/LS in the second case

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TABLE I RELAXATION FREQUENCIES AND WEIGHTS OF THE FIVE-TERM EXPANSIONS USED TO APPROXIMATE THE DEBYE EQUATION WITH THE CORRESPONDING CONSTANTS

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weights in the adjustment of the particle velocities have to be chosen carefully (usually by trial and error) in order to prevent local minima, and the number of the assumed Debye terms has to be kept low enough for reasonable computation times. A hybrid method that uses the projection algorithm for the initial guesses of weights and relaxation frequencies and the PSO/LS as an optimization for these parameters (as opposed to the deterministic fine tuning procedure described in the text) may lead to better agreements between measured and modeled data and will be a subject of future research. APPENDIX and Consider a single Debye relaxation distribution after are removed. The permittivity of the Debye material at frequency is related to its single Debye relaxation frequency as given in Fig. 16 (A1) where corresponds to the radius of the semicircle as well as the weight of the single Debye term. (This is simply the result of fitting the Cole-Cole plot of a single Debye term to the equation of a semicircle.) Rewriting (A1) after cancelling out (A2) leads to

Fig. 16. Cole-Cole plot of a single Debye relaxation.

(A3) is still worse than the proposed method. The weights, relaxand for the original Debye equaation frequencies, tion, PSO/LS method and the automated analytic continuation method are given in Table I.

If we define,

and

, then we get (A4)

Further simplifying (A4) results in V. CONCLUSION The automated analytic continuation method provides a simple way to deduce the constitutive parameters of a material in terms of a sum of analytic functions where any nonphysical behavior due to test error can be eliminated. In addition, a reasonable broadband frequency dependent model of the material under test is obtained. The proposed method converges very quickly and does not rely on any optimization processes. Once the approximate relaxation frequencies are found via a novel projection algorithm, they are tuned to decrease the error between the original data and the calculated data by an asymptotic approach. The efficiency of this approach can be attributed to the fact that it exploits the analytic circuit properties of the multi-Debye model for the material. The resulting algorithm is automatic and deterministic in contrast to previously proposed “optimization” approaches. Both PSO/LS and the automated analytic continuation method provide an effective, accurate means of expanding an arbitrary complex permittivity function in terms of a weighted sum of Debye functions. However there are some constraints that have to be considered for a PSO/LS based approach. The

(A5) Finally we get the corresponding single Debye relaxation as (A6)

REFERENCES [1] C. A. Balanis, Advanced Engineering Electromagnetics. New York: Wiley, 1989. [2] M. Mrozowski and M. A. Stuchly, “Parameterization of media dispersive properties for FDTD,” IEEE Trans. Antennas. Propag., vol. 45, pp. 1438–1439, Sept. 1997. [3] M. A. Eleiwa and A. Z. Elsherbeni, “Debye constants for biological tissues from 30 Hz to 20 GHz,” J. Appl. Comput. Electromagn., vol. 16, no. 3, pp. 202–213, Nov. 2001. [4] P. G. Petropoulos, “Stability and phase error analysis of FD-TD in dispersive dielectrics,” IEEE Trans. Antennas. Propag., vol. 42, pp. 62–69, Jan. 1994. [5] J. L. Young, A. Kittichartphayak, Y. M. Kwok, and D. Sullivan, “On the dispersion errors related to (FD)2TD type schemes,” IEEE Trans. Microw. Theory Tech., vol. 43, pp. 1902–1910, Aug. 1995.

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[6] D. R. Uhlmann and R. M. Hakim, “Derivation of distribution functions from relaxation data,” J. Phys. Chem. Solids, vol. 32, pp. 2652–2655, 1971. [7] P. Lindon, “Obtaining relaxation spectra from experimental data,” Proc. IEEE, vol. 58, pp. 1389–1390, Sep. 1970. [8] W. D. Hurt, “Multiterm Debye dispersion relations for permittivity of muscle,” IEEE Trans. Biomed. Eng., vol. BME-32, no. 1, pp. 60–64, Jan. 1985. [9] R. J. Sheppard, “The least-squares analysis of complex weighted data with dielectric applications,” J. Phys. D: Appl. Phys., vol. 6, pp. 790–794, 1973. [10] D. F. Kelley, T. J. Destan, and R. J. Luebbers, “Debye function expansions of complex permittivity using a hybrid particle swarm-least squares optimization approach,” IEEE Trans. Antennas. Propag., vol. 55, pp. 1999–2005, July 2007. [11] J. Robinson and Y. Rahmat-Samii, “Particle swarm optimization in electromagnetics,” IEEE Trans. Antennas. Propag., vol. 52, pp. 397–407, Feb. 2004. [12] F. A. Grant, “Use of complex conductivity in the representation of dielectric phenomena,” J. Appl. Phys., vol. 29, pp. 76–80, 1958. [13] R. E. Diaz, “The Analytic Continuation Method for the Analysis and Design of Dispersive Materials,” Ph.D. dissertation, University of California, Los Angeles, 1992. [14] R. E. Diaz and N. G. Alexopoulos, “An analytic continuation method for the analysis and design of dispersive materials,” IEEE Trans. Antennas. Propag., vol. 45, no. 11, pp. 1602–1610, Nov. 1997. [15] S. Samarasinghe, Neural Networks for Applied Sciences and Engineering: From Fundamentals to Complex Pattern Recognition. Boca Raton, FL: CRC Press, 2006, ch. 4, pp. 113–134.

Kivanc Inan received the B.S. and the M.S. degrees from Bilkent University, Ankara, Turkey, in 2002 and 2005, respectively, and the Ph.D. degree from Arizona State University (ASU), Tempe, in 2009. During his M.S. and Ph.D. studies, his research interests included analytic continuation of dispersive materials, rough surface scattering, high-frequency and numerical techniques in electromagnetic scattering and diffraction, propagation modeling and simulation and computational electromagnetics. He is currently a senior expert Systems Engineer with Aselsan Inc., Ankara, Turkey, where he performs the configuration management of various military radar systems.

Rodolfo E. Diaz (M’00) received the B.S. degree in physics from Yale University, New Haven, CT, in 1978, and the M.S. degree in physics and Ph.D. degree in electrical engineering from the University of California at Los Angeles (UCLA), in 1980 and 1982, respectively. During 20 years in the aerospace industry, his research has spanned many of the disciplines comprising modern electromagnetic engineering from lightning protection, electromagnetic compatibility, and electromagnetic radiation safety on the space shuttle, through design of broadband missile antennas and radomes, to the design, evaluation, and prototyping of electromagnetic composite materials for low observable (stealth) applications. He is currently an Associate Professor with the Department of Electrical Engineering, Arizona State University (ASU), Tempe, where he directs the Material-Wave Interactions Laboratory, and performs research on optical, microwave, and acoustic interactions with natural and engineered materials. He holds 19 patents ranging from the design of broad band radomes to the amplification of magnetic fields.

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3D-Aggregate Quantitative Imaging: Experimental Results and Polarization Effects Christelle Eyraud, Jean-Michel Geffrin, and Amélie Litman, Member, IEEE

Abstract—We present reconstructions of an aggregate of small spheres from experimental scattered fields using a working frequency of 18 GHz. This target presents at the same time a complex 3D shape and a low-contrast permittivity. Concerted experimental and numerical efforts have enabled to obtain accurate reconstructions. In particular, we took into account the real random noise via a Bayesian framework. Reconstructions have been realized with scattered fields measured in different polarization cases: the results are compared and discussed. Index Terms—Aggregate, Bayesian formulation, microwave analogy, microwave imaging, 3D inverse scattering problem. Fig. 1. Views from different directions of the aggregate, composed of 74 – . (b)  – . spheres of 2.5 mm radius. (a) 

= 90

I. INTRODUCTION

E

LECTROMAGNETIC wave probing is a useful tool to obtain information from an object in a non destructive way. Indeed, the physical features of an unknown target (position, shape, size, complex permittivity) can be retrieved from its scattered field thanks to the resolution of an inverse problem. This implies accurate measurements of the scattered field associated with an efficient inversion algorithm. Nowadays, as several inversion procedures have been extended to 3D configurations (see [1]–[3] for example) and as scattered fields of 3D-objects can be precisely measured [4], [5], the characterization of complex shaped 3D-targets can thus be realized. In particular, different inversion algorithms were recently tested against the same experimental dataset (3D Fresnel database) measured with the Institut Fresnel setup and the results were published in a special section of the Inverse Problems review [6]. In the present work, we are interested in a specific target which presents a particularly complex geometry including small details: an aggregate of spheres. To our knowledge, it is the first time that a target with such a complex geometry has been reconstructed with such a high frequency (18 GHz). To give an example, in the special section of the 3D Fresnel database [5], [6], the highest frequency was 8 GHz. To reach such a goal, we took profit of a joint work between the inversion developments and the experimental ones. In particular, (i) we have optimized the quality of the data (knowledge and correction of the experimental errors), (ii) their quantity (number of measurements) and Manuscript received December 04, 2009; revised March 29, 2010; accepted November 10, 2010. Date of publication February 04, 2011; date of current version April 06, 2011. The authors are with the Institut Fresnel UMR CNRS 6133, Université Paul Cézanne Aix-Marseille III, Ecole Centrale de Marseille, Université de Provence Aix-Marseille I, Marseille France (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.2011.2109353

=0

= 90

= 270

(iii) we have based our inversion procedure on a physical description of the problem using a Bayesian approach and a positivity constraint. The scattered field of this target has been measured in different polarization cases. Indeed, polarization of the illuminating wave and the choice of the polarization of the receiving antenna are two important parameters as geophysical studies or radar studies point it out ([7], [8] for example). In inversion procedures, the different polarization cases are very often considered to contain the same quantity of information [6], [9], [10]. Here, we present reconstructions with different polarization cases and we discuss the interest and relevance of each case either taking each one separately or using various combinations. This paper is organized as follows. In Section II, the aggregate and its specific characteristics are presented. Sections III and IV explain the experimental study and the inversion procedure, each one being influenced by the other. The results of the reconstructions are shown in Section V with a discussion on the polarization inference on the reconstructions. Finally, some concluding remarks are provided in Section VI. II. THE AGGREGATE The studied target is an aggregate of spheres (Fig. 1), composed by 74 dielectric spheres of 2.5 mm radius. It has a fractal dimension of 1.7 [11]. The spheres are made of polyacetal and the relative permittivity of this material was measured with the commercial kit EpsiMu [12], [13]. The relative permittivity value was then refined comparing diffraction measurements of a single 50 mm diameter sphere made of the same material and Mie computations [14], [15] with a minimization process. The . final value was found to be purely real and equal to More details concerning this object and accurate comparisons between simulated and experimental scattered fields can be

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Fig. 2. Measurement setup: (a) picture and (b) sketch.

found in [16] and on this website: http://www.fresnel.fr/3Ddirect/database.php. This aggregate presents several singular characteristics. Its geometry is entirely non-symmetrical (no symmetry axis or plane). This implies that this target necessitates measurements for several incident and receiver directions to provide the possibility of a good characterization. It is a depolarizing target [17] and thus that measurements of different polarization cases are interesting. These measurements can be realized with our experimental setup almost all around the 2 m radius sphere enclosing the object [5]. This target has a very low density, i.e., the volume of the material represents less than 3% of the volume of the minimum sphere including the target, so it is globally a low-contrasted object. This involves that high accuracy on the measured scattered fields is definitely needed. Taking profit of our previous works [4], we are able to eliminate the systematic errors, to correct for the drift errors [18] and to estimate the remnant noise (random noise) on each measured point [19]. This object includes small details compared to its size, i.e., each sphere diameter represents 1/20 of the diameter of the minimum sphere enclosing the target. As these spheres have small dimensions, we chose to realize the inversion process using scattered fields measured at a frequency such that the individual (18 GHz) to be able to resphere has a diameter of about trieve such a fine geometry. On the other hand, the perturbations disturbing the scattered field are becoming more important and all the geometrical imprecisions are more sensitive when the frequency increases. In particular, the random noise increases, thus it is specially important to take this noise into account in the inversion process. III. EXPERIMENTAL STUDY A. System Overview The measurements were performed in the faradized anechoic chamber (14.5 m 6.5 m 6.5 m) of the C.C.R.M. (Centre Commun de Ressources en Microondes) in Marseille (Fig. 2), already described in various papers [10], [20]. The equipment is based on a Vector Network Analyzer (HP 8510 B), used in a multiple sources configuration with two synthesizers and two external mixers. This setup is represented on Fig. 2(a). The transmitting antenna can be moved along the vertical arch in a vertical plane (

angle). The target – placed at the top of a cylindrical expanded polystyrene mast at the position (0, 0, 0) – can rotate on itself around the vertical –axis ( angle) which allows to increase the number of incident directions. ( , ) are the basis vectors of a spherical coordinate system (Fig. 2(b)), where the conventions for the polar and azimuthal angle notations ( , ) defined in [21]. To reduce the measurement time, two antennas (in orthogonal polarization) are placed in two different positions on the vertical arch and are selected alternatively as sources using a microwave switch, i.e., for each receiver position, the field is measured successively from two different illuminations without mechanical movements. The receiving antenna is positioned on around the an arm and can rotate in the horizontal plane target ( angle). The distance between the source and the object (resp. between the receiver and the object) is (resp. ). The transmitting and the receiving antennas are two high gain pyramidal horns (ARA MWH1826B) – to have a scattered field signal as high as possible – with a working frequency between 18 GHz and 26 GHz. The cable lengths and the power levels have been optimized so that the level of the forward scattering signal is just below the mixer saturation. These antennas are linearly polarized and can be rotated on their support to change or ). the polarization direction (along B. Choice of the Experimental Parameters As this target is a depolarizing one, its scattered field was and ( measured in the four polarization cases: , , means that the emitter has the polarization and the receiver is sensitive to the polarization ). The angular step of the transmitting antenna movement and of the receiving one were chosen in accordance with the maximum spatial frequency of the scattered field at 18 GHz. Indeed, of bounded induced sources presents the the scattered field property of having a limited spectral bandwidth in a spherical configuration when this field is observed far enough from the target [22]. A minimum spatial sampling angle can thus be deduced for the emitter and for the receiver allowing to measure all available information on the target. Indeed, the fields were meaon 53 equidistant points placed on sured with the step – 100 are excluded due to a circle in the horizontal plane mechanical impossibilities. The aggregate was illuminated with 117 incident directions. For this, the transmitting antenna was in the vertical plane realizing moved with the step 13 incidences and furthermore, for each of those positions, 9 more incidences were created by rotating the target around the -axis with the step . This last step is larger than the maximum sampling criterion (11 ), but this choice was made to keep a reasonable measurement time. Finally, a calibration process, using a single reference target case only for sake of (a metallic sphere), estimated in the simplicity [4], [23], was applied to every scattered field keeping the same coefficient in each polarization case. IV. INVERSION PROCEDURE The experimental scattered fields, measured on the antennas sphere (far away from the target), are allowing to deduce the

EYRAUD et al.: 3D-AGGREGATE QUANTITATIVE IMAGING: EXPERIMENTAL RESULTS AND POLARIZATION EFFECTS

3D relative permittivity map in the domain (box including the target) thanks to an appropriate inversion procedure. A Bayesian formulation was used to balance each data according to its accuracy and to describe all available information. To solve this non-linear inverse problem, an iterative scheme is adopted and this necessitates an accurate forward problem. A. Forward Problem We calculate the scattering of a 3D inhomogeneous structure in the frequency domain using a volume integral formulation [24]. The field inside the imaging domain is computed with a biconjugated gradient stabilized FFT method [25] based on a 1D FFT to improve the calculation speed [26]. The computation and the memory requirement complexity is then of . The free space dyadic Green tensor is used. is B. Inversion Algorithm We chose to introduce information on the effective random noise which disturbs the scattered field measurements in the inversion procedure using a Bayesian formulation. The experimental noise was characterized on the measured fields (the total and incident fields) as explained in a previous study [19]. The results established that the deduced noise on the scattered field can be described by an additive complex random variable whose real and imaginary parts are normally distributed and with associated standard deviations depending on the total field magnitude, on the incident field magnitude and on the frequency. The noise disturbing the scattered field data, for a given frequency, depends thus strongly on the incident direction and on the receiver position. Indeed, the standard deviations can vary by a factor 10 depending on the considered measured point. The Bayesian formulation allows to introduce this knowledge in the inversion procedure. The determination of the unknown relative with the knowledge of the scattered field permittivity map measurement can be described by the density of probability [27]–[29] (1) We chose to introduce no statistical information at all on the relative permittivity map, i.e., was assumed to follow an uniform law. Maximizing the distribution to obtain the unknown with the knowledge of the measured scattered field , the solution corresponds then to the Maximum Likelihood. Thus, as there is no noise correlation from one measurement point to another due to the measurement protocol, the involved costfunctional to minimize can be written as [30]

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(incident direction, receiver direction), the comparison of the calculated scattered field and of the measured one is balanced with a weighting coefficient – depending on the potential noise disturbing the measurement on this point. The 3D-complex relative permittivity map of the considered domain was obtained through an iterative procedure which of (2). The minimization is minimizes the cost-functional realized with a conjugate gradient algorithm using a positivity constraint and with the standard Polak-Ribiere coefficients [31], [32]. More details about this algorithm can be found in [30]. C. Reconstruction Error Quantification To give a criterion of the reconstruction quality, we have considered different error functions comparing the reconstructed 3D relative permittivity maps and the true one (due to the difficulty to extract a significative one). It can be noticed that if there is a difference – even weak – between the expected position of the aggregate (considered in the following criteria calculations) and its real position in the experiments, the influence on the criteria can be dramatically strong. Thus, the first step consists in detecting this possible geometrical shift. 1) Estimation of the Real Target Position: We have determined the real position of the center of mass of the aggregate using a 3D cross correlation between the re) and the expected one constructed contrast map ( (3) where is the true relative permittivity value, is the reconstructed value at the end of the inversion process and describes the position of the considered cell in the domain. It can be noticed that the cross correlation has been calculated on ) maps (and not on the maps) in order to compare the ( only the voxels where some material can be found. 2) Relative Permittivity Map Error Criteria: Once this operation has been realized, we can evaluate the similarity between the two maps. We have considered here four different criteria (the first and the second are standard ones and we propose also two other ones). The first one (resp. second), is a comparison between the two relative permittivity maps with a relative L2 norm (resp. L1 norm) (4) The third criterion , corresponds to the expression of the cross correlation normalized on the contrast maps – for the best (previously determined) position (5)

(2) where (resp. ) is the total number of sources (resp. receivers), for the transmitter and the receiver , is the simulated scattered field, and are the noise standard deviations on the real part and on the imaginary part. With the so-defined cost-functional, for each couple

Its interest is to provide a value linked to the similarity between the two maps comparing only the voxels containing some material with respect to the true map. To consider the reconstructed relative permittivity also for the which correempty voxels, we consider a fourth criterion, sponds to the correlation between the two maps (6)

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where is the covariance between and and the variance of . All those parameters have been estimated for each of the reconstructions presented below. The more similar the two maps are, (i) the closer to 0 the and are, (ii) the closer to 1 the criteria criteria and are. D. Inversion Parameters The initial guess of the relative permittivity map was chosen to be equal to the one of the background medium (more precisely, a small constant value was added to avoid the case where the gradients vanish due to the positive constraint expression). The inversion process was stopped when the cost functional is ( being the average of the measured noise standard below deviation values). For the following reconstructions, the convergence was reached at (or before) the 3rd iteration. An invescentered at ( , tigated domain of (75 50 75) , ) mm was considered with a cubic lattice spacing ). equal to 1.6 mm (

Fig. 3. Real aggregate representation with an iso-surface threshold function.

Fig. 4. Reconstruction result at 18 GHz using only the scattered field in the  polarization and without taking into account the random noise.

V. RESULTS The inversions were performed in three cases: (i) with or without any information on the real random noise, (ii) separately for different polarization cases of the measured scattered fields and (iii) for several combined polarization cases of the measured scattered fields. The different reconstructions are presented and discussed in this section. Note that, for every reconstruction, the real position of the object was found with the cross-correlation criterion.

Fig. 5. Reconstruction result at 18 GHz using only the scattered field in the  polarization and taking into account the random noise.

A. Target Position The cross correlation criterion (3) allows to determine the real position that the aggregate had during the experiments. For all the polarization cases, the same position was found: . This positioning error is very small considering the fact that the target has to be placed at the center (0,0,0) mm of our setup, a 4000 mm diameter sphere. B. Single Polarization Cases 1) Polarization: In this section, we present the reconstruction results obtained using the scattered fields in the polarization case only. The first reconstruction was realized without any use of the random noise knowledge, i.e., which amounts to taking in (2). This reconstruction is presented with a 3D visualization in Fig. 4. For this figure (and all the similar figures in the following), the iso-surface threshold value is taken when the relative permittivity is different from the free space one and the angles plotted at the top of each view correspond to the (azimuth, elevation) coordinates of the viewpoint. This figure has to be compared with the one of the expected aggregate (Fig. 3). This inversion allows to reconstruct a 3D structure but the fine geometry of the aggregate is not really well obtained, i.e., the outlines are not distinct. Moreover, the maximum relative permittivity value of the reconstructed map is weak: 1.94 (the true one being 2.85). Secondly, we took into account the experimental random noise in the inversion process with the formulation explained in

Fig. 6. Relative permittivity maps reconstructed in the x0y plane at 18 GHz and at several altitudes z using scattered fields in the  polarization and taking into account the random noise (the colorbar/grayscale is the same for the three plots). (a) z = 20 mm. (b) z = 15 mm. (c) z = +21 mm.

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Section IV-B. As it can be seen on Fig. 5, the reconstruction is strongly improved and now the form of the reconstructed object is very close to the real one even if parts of the target (4 spheres) are still missing. The relative permittivity of the target is also higher — the maximum value is now 2.9 — and is similar to the true one (2.85). To study more precisely this result and in particular to visualize the value of the reconstructed relative permittivity, we have plotted the relative permittivity maps at several altitudes (Fig. 6). On these maps, no interpolation has been done, i.e., the value of each voxel represents the reconstructed value. The voxels with material – considering the true 3D map and the same discretization – are over plotted (overlayed voxels). As it can be seen, the spheres are reconstructed at the proper positions and with a satisfactory relative permittivity value. Each of the associated criteria defined to

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TABLE I CRITERIA VALUES FOR THE DIFFERENT RECONSTRUCTION CASES (RAND: WITH EFFECTIVE RANDOM NOISE INFORMATION AND NO: WITHOUT RANDOM NOISE INFORMATION)

Fig. 9. Reconstruction result at 18 GHz using the scattered field in the  and in the  polarizations.

Fig. 7. Reconstruction result at 18 GHz using only the scattered field in the  polarization.

Fig. 8. Reconstruction result at 18 GHz using the scattered field in the  and in the  polarizations.

quantify the reconstruction error is improved (Table I, line 2). This means that both permittivity estimations of the voxels with and without material are better. Thus, all the other reconstructions presented in this paper, were obtained taking into account the random noise in the procedure as this knowledge of the real noise does improve the results. 2) Polarization: We now only consider the data measured in the polarization case. As it can be seen in Fig. 7, the horizontal zones are preferentially and almost solely reconstructed. The maximum value of the reconstructed relative permittivity and criteria is weak (1.4). For these two reasons, the are low (Table I, line 3). The surprising values of the and criteria are due to the fact that these criteria are mainly sensitive to the great number of well reconstructed empty voxels. Physical reasoning is derived in Section V-D to understand the obtained results. C. Two Polarization Cases In this paragraph, we consider two polarization cases together in the inversion process. As the polarization brings more information than the one, we have kept this polarization and we have tried to add an other polarization to see if we can improve the results. and Polarizations: In this case, the scattered field in 1) polarization is used in combination with the field meathe sured in the case. As it can be seen on Fig. 8, the recon-

Fig. 10. Relative permittivity maps reconstructed in the x0y plane at 18 GHz and at several altitude z using scattered fields in the  and in the  polarizations (the colorbar/grayscale is the same for the three plots). (a) z = 20 mm. (b) z = 15 mm. (c) z = +21 mm.

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struction is very close to the one obtained considering only the polarization. The cross correlation, the scattered field in the correlation, L2-norm and the L1-norm criteria are also similar case (Table I). to the ones obtained in the 2) and Polarizations: In this case, the scattered field polarization is used in combination with the meain the case. The reconstruction is presented in the sured field in the Figs. 9–10. As it can be seen, the voxels including material are polarmore precisely reconstructed (in comparison with the ization case) and the relative permittivity value is also higher in is also improved these zones. The cross correlation criterion polarization case (Table I, line 5). It can be compared to the noticed that some artefacts did appear, but their values are very weak (the relative permittivity values of these voxels being less than 1.2) and they are emphasized by the threshold representa, tion. Nevertheless, their influences are also visible on the and criteria which are worse in this case. polarization with the one is The combination of the not presented here because this is not an interesting case as explained in Section V-D. D. Discussion of the Polarization Interest We propose here two tools to analyze and understand the reconstructions obtained with the various polarizations. 1) First Tool: The first interpretation (Fig. 11) is based on a representation of the different incident field vectors (in black) for all the source positions. To clarify the representation, only the end of these vectors are plotted (with the marker) – each vector being brought back at the center of the setup. The projection of the scattered field vector which is measured by the receiving antenna is overlaid (in cyan/gray) for all the receiver positions. These vectors are normalized in order to focus on information provided by the projection angle. This representation was realized for the four polarization cases. 2) Second Tool: The second tool concerns the induced currents in the zone. It therefore takes into account the depolar-

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Fig. 11. Representation for ,  ,  and  polarizations of the incident vectors (black) and of the received scattered vectors (cyan/gray) for all the source and receiver positions – each vector being brought back at the center of the setup – (for the incident vectors, only the end of these vectors are plotted with the marker ).

+

Fig. 12. Histogram of the normalized induced currents projected on the receiver polarization direction for the four polarization cases: I , I , I and I in (7).

izing effects introduced by the target. Indeed, we have numerically calculated the associated direct problem in the configuration of the experiments. We have then observed the values of the projected induced currents on the receiver polarization di, , , rections (7) for the four polarization cases and we have plotted them in terms of histograms (Fig. 12) (7) where is the value of the induced current in for is the unity a source positioned at with polarization and vector of the measured field for a receiver with polarization . 3) Discussion: We can first observe that the incident polarization along allows to double the number of incident polarization directions (Fig. 11(a)) – compared to the incident polarcase is also ization along (Fig. 11(d)). Furthermore, the the case containing the highest values of the induced currents (Fig. 12) and thus the signal level is higher and less sensitive to noise [19]. In consequence, the most interesting polarization case is . This is in agreement with the quality of the reconstruction obtained using this polarization (Section V-B.I).

case (Fig. 11(d)–(f)), the incident field Looking to the plane for all the source positions. vector is contained in the Moreover, all the receivers are only sensitive to the fields conplane. Thus, it seems natural that the horizontal tained in the geometrical zones of the object are preferentially reconstructed (Section V-B.II). In comparison, it can be noticed that in the case, even if the scattered field is always measured along the axis, the incident vector has both components in the plane and components along the axis (Fig. 11(a)–(c)). and the Considering now the reconstruction with the polarizations, we can first note that this combination does not case add new incident direction in comparison with the (Fig. 11(a)–(f)). Looking then to the induced currents (Fig. 12), contains we can see that, contrarily to our first intuition, the only middle range values. These two points can explain why we did not see real improvements in the reconstructions when fields to the ones. adding the and the Considering the reconstructions obtained with the measurements, it appears that the data are bringing new – non redundant – information on the object (Fig. 11(g)–(i)), with measurement (Fig. 11(a)–(c)). Indeed, in the respect to the present case, for the incident wave polarization, the scattered field is measured along the two orthogonal polarizations and – instead of only the one in the case – thus the scattered vector is entirely measured on the receivers for a given illumination. This can explain the improvement of the reconstruction. It can be furthermore noticed that all the incident vector direccase are contained in the case as well as all tions in the the receiving vector projections. polarization Finally, looking to the Fig. 11(j)–(l), in the case, (i) the incident and the receiver field vector are always orthogonal, (ii) the target is illuminated with polarization vectors polarization and the field is received already contained in the case. Morealong the same polarization vector than in the over, considering the induced currents, this polarization contains only very few non negligible values (Fig. 12). So, for targets which are not highly depolarizing objects – as the present polarization measurement is not interesting and can one, the be omitted. This can allow to remove a quarter of the measurement time. To summarize this part, we can remark that these results are not intuitive. Indeed, in our configuration, due to the sources and and are not receivers displacements, the co-polarizations equivalent at all. Moreover it is more useful to add the crossin this polarization , than the co-polarization , to the case. VI. CONCLUSION In this paper, our work is based on a good complementarity between the numerical part (positive constraint, Bayesian approach) and the experimental part (number of measurements, calibration, precision of the measurements). This allows to realize faithful reconstructions from measured data of a 3D complex shape target which is, in the present case, an aggregate of 74 spheres. We have also performed similar reconstructions of this object from synthetically generated data – using the same positions of the transmitters/receivers than the ones used in the

EYRAUD et al.: 3D-AGGREGATE QUANTITATIVE IMAGING: EXPERIMENTAL RESULTS AND POLARIZATION EFFECTS

experiments – and as the results are very close to the ones obtained with the measurements, this tends to demonstrate that the concerted experimental and numerical efforts have been properly made. The evaluation of the reconstructions is not a trivial problem. Four criteria have been considered to evaluate the results. The usually adopted criteria (relative L1 and L2-norms) are not very adapted to evaluate the quality of the reconstruction of the present target. Indeed, their values are particularly dependent on the reconstruction of the empty voxels which are not the most interesting voxels. For these two reasons, we have proposed and the correlation two other criteria: the cross-correlation . Their variations are more significant and in particular the allows to compare only the voxels within the material. In this study, inversions have been presented using experimental scattered fields, measured with different polarizations. The results have shown that the choice of the polarizations has a great influence on the reconstruction results and the more the polarizations of the emitter and of the receiver are describing a diversity of directions, the better the results are. With our setup, polarization is the most interesting case as it contains a the larger diversity of polarization directions of the incident wave and because the measured values are relatively high (compared to the other polarization cases). If a second polarization case can be added, the most favorable one seems to be the polarization case even if it corresponds generally to weaker measurement values. It would be interesting, in future work, to corroborate these conclusions with the inversion results obtained with other complex-shape targets. ACKNOWLEDGMENT The authors would like to thank R. Vaillon, B. Lacroix and O. Merchiers from the CETHIL laboratory, Lyon for the design and the realization of the aggregate. REFERENCES [1] S. Y. Semenov, A. E. Bulyshev, A. E. Souvorov, A. G. Nazarov, Y. E. Sizov, R. H. Svenson, V. G. Posukh, A. Pavlovsky, P. N. Repin, and G. P. Tatsis, “Three-dimensional microwave tomography: Experimental imaging of phantoms and biological objects,” IEEE Trans. Microw. Theory Tech., vol. 48, pp. 1071–1074, Jun. 2000. [2] A. Abubakar and P. M. van den Berg, “Iterative forward and inverse algorithms based on domain integral equation for three-dimensional electric and magnetic objects,” J. Comput. Phys., vol. 195, pp. 236–262, Mar. 2004. [3] J. D. Zaeytijd, A. Franchois, and J.-M. Geffrin, “A new value picking regularization strategy-application to the 3-d electromagnetic inverse scattering problem,” IEEE Trans. Antennas Propag., vol. 57, pp. 1133–1149, Apr. 2009. [4] C. Eyraud, J.-M. Geffrin, P. Sabouroux, P. Chaumet, H. Tortel, H. Giovannini, and A. Litman, “Validation of a 3D bistatic microwave scattering measurement setup,” Radio Sci., vol. 43, p. RS4018, 2008. [5] J.-M. Geffrin and P. Sabouroux, “Fresnel database continuation: Experimental setup and improvements for 3D scattering measurements,” Inverse Prob., vol. 25, p. 024001, Feb. 2009. [6] A. Litman and L. Crocco, “Guest editor introduction,” Inverse Prob., vol. 26, p. 020201, Feb. 2009. [7] Y. Wang and J. Saillard, “Characterization of the scattering centers of a radar target with polarization diversity using polynomial rooting,” in Proc. IEEE Int. Conf. on Acoustics, Speech, and Signal Processing, Salt Lake City, UT, May 2001, pp. 2893–2896.

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[8] A. Freeman and S. L. Durden, “A three-component scattering model for polarimetric sar data,” IEEE Trans. Geosci. Remote Sensing, vol. 36, pp. 963–973, 1998. [9] K. Belkebir and M. Saillard, “Special section on testing inversion algorithms against experimental data: Inhomogeneous targets,” Inverse Prob., vol. 21, pp. S1–S3, 2005. [10] J.-M. Geffrin, P. Sabouroux, and C. Eyraud, “Free space experimental scattering database continuation: Experimental setup and measurement precision,” Inverse Prob., vol. 21, pp. S117–S130, Nov. 2005. [11] O. Merchiers, J.-M. Geffrin, R. Vaillon, P. Sabouroux, and B. Lacroix, “Microwave analog to light scattering measurements on a fully characterized complex aggregate,” Appl. Phys. Lett., vol. 94, p. 94:181107, May 2009. [12] D. Ba and P. Sabouroux, “Epsimu, a toolkit for permittivity and permeability measurement in microwave domain at real time of all materials: Applications to solid and semisolid materials,” Microw. Opt. Technol. Lett., vol. 52, pp. 2643–2648, Dec. 2010. [13] E. Vanzura, J. Baker-Jarvis, J. Grosvenor, and M. Jasenik, “Intercomparison of permittivity measurements using the transmission/reflection method in 7-mm coaxial transmission lines,” IEEE Trans. Microw. Theory Tech., vol. 42, pp. 2063–2070, Nov. 1994. [14] M. Born and E. Wolf, Principles of Optics. Oxford, U.K.: Pergamon Press. [15] H. Tortel, “Localization and derivation of an optimal sphere for 3d perfectly conducting objects,” J. Electromagn. Waves Applicat., vol. 16, pp. 771–791, Jun. 2002. [16] O. Merchiers, C. Eyraud, J.-M. Geffrin, R. Vaillon, B. Stout, P. Sabouroux, and B. Lacroix, “Microwave measurements of the full amplitude scattering matrix of a complex aggregate: A database for the assessment of light scattering codes,” Opt. Express, vol. 18, p. 121191, Jan. 2010. [17] C. Monzon, “A cross-polarized bistatic calibration device for rcs measurements,” IEEE Trans. Antennas Propag., vol. 51, pp. 833–839, 2003. [18] C. Eyraud, J.-M. Geffrin, A. Litman, P. Sabouroux, and H. Giovannini, “Drift correction for scattering measurements,” Appl. Phys. Lett., vol. 89, p. 244104, Dec. 2006. [19] J.-M. Geffrin, C. Eyraud, A. Litman, and P. Sabouroux, “Optimization of a bistatic microwave scattering measurement setup: From high to low scattering targets,” Radio Sci., vol. 44, p. RS003837, Mar. 2009. [20] P. Sabouroux, J.-M. Geffrin, and C. Eyraud, “An original microwave near-field/ far-field spherical setup: Applications to antennas and scattered field measurements,” in Proc. Antenna Measurement Tech. Assoc. Symp., Newport, RI, Nov. 2005, pp. 292–296. [21] D. Zwillinger, Crc Standard Mathematical Tables and Formulae, 31st ed. London, U.K.: Chapman and Hall/CRC, 2002. [22] O. M. Bucci and T. Isernia, “Electromagnetic inverse scattering: Retrievable information and measurement strategies,” Radio Sci., vol. 32, pp. 2123–2137, 1997. [23] P. M. van den Berg, M. G. Coté, and R. E. Kleinman, “Blind shape reconstruction from experimental data,” IEEE Trans. Antennas Propag., vol. 43, pp. 1389–1396, 1995. [24] J. A. Kong, “Electromagnetic wave theory,” EMW, 2000. [25] H. A. van der Vorst, Iterative Krylov Methods for Large Linear Systems. Cambridge, U.K.: Cambridge Univ. Press, 2003. [26] B. E. Barrowes, L. F. Teixeira, and J. A. Kong, “Fast algorithm for matrix-vector multiply of asymmetric multilevel block-Toeplitz matrices in 3-D scattering,” Microw. Opt. Technol. Lett., vol. 31, pp. 28–32, 2001. [27] G. Demoment and J. Idier, “Problèmes inverses : De l’expérimentationà la modélisation: Approche bayésienne pour la résolution des problèmes inverses en imagerie,” Observatoire Francais des Techniques Avancées, vol. 22, pp. 59–77, 1999. [28] A. Baussard, D. Prémel, and Venard, “A Bayesian approach for solving inverse scattering from microwave laboratory-controlled data,” Inverse Prob., vol. 17, pp. 1659–1669, 2001. [29] O. Féron, B. Duchêne, and A. Mohammad-Djafari, “Microwave imaging of inhomogeneous objects made of a finite number of dielectric and conductive materials from experimental data,” Inverse Prob., vol. 21, pp. S95–S115, Nov. 2005. [30] C. Eyraud, A. Litman, A. Hérique, and W. Kofman, “Microwave imaging from experimental data within a Bayesian framework with realistic random noise,” Inverse Prob., vol. 26, p. 024005, Feb. 2009. [31] E. Polak, Computational Methods in Optimization. New York: Academic Press, 1971. [32] R. E. Kleinman and P. M. van den Berg, “Two-dimensional location and shape reconstruction,” Radio Sci., vol. 29, pp. 1157–1169, 1994.

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Christelle Eyraud was born in France in 1980. She received the Ph.D. degree in physics from the Université Aix Marseille I, France, in 2006. She worked on measurement errors analysis for microwave scattering phenomena. After a postdoctoral position at the laboratory PALMS, Université Rennes 1, France, she worked, as a Postdoctoral Researcher, at the Laboratoire de Planétologie de Grenoble, France, on a 3D inverse algorithm taking into account the measurement uncertainties. In 2008, she joined the Institut Fresnel, Université Aix-Marseille, CNRS, Ecole Centrale Marseille, France, as an Assistant Professor. Her research interests include inverse scattering problems and measurement techniques.

Jean-Michel Geffrin received the Ph.D. degree in physics from the University of Paris XI, France, in 1993. He worked for ten years as a Research Engineer in France, where he developed specific antennas and experimental setups for measuring targets radiation pattern. In 2002, he joined the Institut Fresnel, Université Aix-Marseille, CNRS, Ecole Centrale Marseille, France, to reinforce the hyperfrequency experimentalist team. He has contributed to the constitution of the second and third databases of scattered fields proposed by the Institut Fresnel to the inverse problem community http://www.fresnel.fr/3Ddatabase/database.php and to the direct database proposed to the (light) scattering community http://www. fresnel.fr/3Ddirect/database.php.

Amélie Litman (M’97) was born in France in 1972. She received the Ph.D. degree in applied mathematics from the University of Paris XI, Paris, France, in 1997. During 1997–1998, she was with the Eindhoven University of Technology, Eindhoven, The Netherlands, working in a postdoctoral position. From 1998 to 2002, she was with Schlumberger, France, where she worked on the development of inversion algorithms for oil prospecting. In October 2002, she joined Institut Fresnel, Université Aix Marseille, CNRS, Ecole Centrale Marseille, France, as an Assistant Professor. Her research interests include forward and inverse scattering techniques.

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Characterization of Metamaterials Using a Strip Line Fixture Leila Yousefi, Member, IEEE, Muhammed Said Boybay, Member, IEEE, and Omar M. Ramahi, Fellow, IEEE

Abstract—A method is introduced to measure the effective constitutive parameters of metamaterials having negative permittivity, negative permeability, or negative permeability and negative permittivity simultaneously. The method is based on the strip line topology, thus offering low cost and low setup complexity in comparison to other methods. The method proposed here is validated by numerically simulating the measurement setup while using different types of metamaterials. To validate the method experimentally, a metamaterial having negative permeability over a band of frequencies is characterized. Good agreement is obtained between the experimental and numerical results. Index Terms—Artificial magnetic materials, characterization, fractal Hilbert curves, metamaterials, permeability, permittivity, strip line fixture.

I. INTRODUCTION ETAMATERIALS are artificial materials engineered for specific electric and magnetic responses [1]–[5]. Since the first attempts for designing metamaterials [6], [7], new applications of such materials have been proposed. In addition to the extraordinary properties and applications of metmaterials such as superlensing [5] and cloaking [8]; applications related to antenna technologies [9]–[12], near field characterization methods [13] and sub-wavelength resonators [14], have been reported. In order to efficiently realize these applications, new metamaterial designs have been proposed to increase the bandwidth and reduce the loss or the size of the unit cell [15]–[18]. Characterization of the electric and magnetic properties of metamaterials is crucial in the design and fabrication cycle. For design verification, the electrical and magnetic properties of the structure need to be measured. Since metamaterials are typically inhomogeneous and anisotropic (except for some designs as in [19], [20]), their characterization presents several challenges. Several experimental methods have been reported for retrieval of the constitutive parameters of metamaterials such as the resonator method [21], [22], the free-space method [23]–[25], the waveguide method [26], [27], and the microstrip line method

M

Manuscript received March 30, 2010; revised August 05, 2010; accepted September 27, 2010. Date of publication January 28, 2011; date of current version April 06, 2011. The authors are with the Electrical and Computer Engineering Department, University of Waterloo, Waterloo ON N2L 3G1, Canada (e-mail: [email protected]; [email protected]; oramahi@ece. uwaterloo.ca). 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.2011.2109360

[28]. Each of these methods has its advantages and disadvantages [29]. The resonator method provides high accuracy but it is inherently narrowband, and an individual measurement setup should be prepared for retrieval of the constitutive parameter at each single frequency; therefore, it is not a good candidate for characterization of metamaterials which are dispersive in nature. The free-space approach, on the other hand, provides good accuracy, however, at the cost of an expensive setup that involves two horn antennas combined with lens assemblies to generate plane waves [24]. Furthermore, in the free-space method, since standard horn antennas have limited frequency bandwidth, different setups are needed to test metamaterials operating at different frequency bands (for example, an antenna used for testing a structure which operates at 2 GHz cannot be used to test another structure which operates at 3 GHz). In the waveguide method, the sample of the metamaterial is placed at the cross section of a waveguide and its constitutive parameters are calculated from the reflected and transmitted waves [26], [27]. The setup needed for this method is less costly when compared to the free-space method, but to test metamaterials operating at different frequency bands, different setups are needed (a disadvantage shared with the free-space method). Another severe constraint on the waveguide method is that a large metamaterial sample is required to fill the entire cross section of the waveguide. This would be costly when testing metamaterials that operate at lower microwave frequencies, (as an example, to test a metamaterial operating at 500 MHz, the sample size would be approximately 0.5 m 0.2 m). The microstrip line method which was reported in [28], [30] has the advantage of lower cost setup in comparison to the freespace or the waveguide methods, while having the capability to extract the permeability and permittivity over a wide band of frequencies (in comparison to the resonator method). However, since the microstrip line method supports quasi-TEM mode, approximate equations based on conformal mapping techniques are required for calculation of its characteristic impedance. As shown in [30], these approximate formulas impose restrictions when extracting negative permittivity or permeability values. In this paper, a new method based on strip line topology is presented for characterization of metamaterials. The proposed method is suitable for characterization of all types of metamaterial structures including single negative and double negative media. Comparing this method to the free-space method [23]–[25], the method presented here has the advantages of smaller sample size and inexpensive setup requirement. Unlike the free-space method which needs an expensive setup of two horn antennas combined with lens assemblies, the strip line setup is simple and inexpensive. In this method a small sample

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of the metamaterial is required, while the free-space approach needs a sample of the size of several wavelengths [23]–[25] to be able to perform plane wave measurement. In comparison with the rectangular waveguide method [26], [27], the method presented here has the advantages of smaller sample size requirement, inexpensive setup, and the capability of TEM mode excitation. In the waveguide method, the TEM mode is not supported leading to a non-uniform field in the cross section, which consequently complicates the retrieval process [26], [27]. In the rectangular waveguide method, the size of the metamaterial sample should be at least half of the wavelength at the resonant frequency due to the cutoff frequency restriction. Additionally, the rectangular waveguide method requires standard coaxial to waveguide adaptors which add to the setup cost and complexity. The parallel plate waveguide method [31] supports the TEM mode and therefore provides more flexibility in the sample size and easier retrieval process when compared to the rectangular waveguide method. However; to be able to excite a parallel plate waveguide with coaxial ports, a precise tapering is required [31] which makes fabrication of the setup both complex and expensive. In comparison to the microstrip line method, the method presented here has the advantage of supporting a TEM mode which avoids restrictions introduced by the quasi-TEM nature of the fields in the microstrip line setup [30]. Various microstrip and strip line-based retrieval methods with different configurations were reported in the literature for characterization of natural materials [32]–[39], but to the authors’ knowledge, no strip line-based method is reported for characterization of metamaterials which are typically anisotropic and dispersive. In the following sections, first the retrieval method is explained. Then in Section III, using numerical full wave analysis, the accuracy of the method is verified for various types of metamaterials. Finally in Section IV, the method is used for experimental characterization of magnetic metamaterials with unit cells of 3rd order fractal Hilbert configuration.

Fig. 1. The setup configuration for the strip line fixture used for extraction of the permittivity and permeability of the metamaterial media.

Fig. 2. Transmission line model of the setup configuration shown in Fig. 1.

The transmission line model shown in Fig. 2 is used to analyze the behavior of the field in the substrate. In this model, three transmission lines are used to represent the three regions of the strip line fixture shown in Fig. 1. According to this model, the voltage and current in all three regions are formulated by (1)

II. RETRIEVAL METHOD The setup configuration is shown in Fig. 1. The setup consists of a two-port strip line fixture. The substrate of the strip line includes three parts: two double positive dielectric with known constitutive parameters at sides next to the excitation ports, and the metamaterial to be characterized placed in the middle. By measuring the scattering parameters of this two-port strip line fixture, the permittivity and permeability of the metamaterial under test are extracted. Based on the coordinate system presented in Fig. 1, the component of the magnetic field and the component of the electric field are the dominant field components in the strip line structure. Therefore this configuration can be used for retrieval of and . The method used here shares the theoretical foundation with the free-space approach [23]–[25] in the sense that the two methods use reflection and transmission of waves from the metamaterial sample to extract the constitutive parameters. However, as explained in the introduction section, the method presented here has several benefits over the free-space method from the fabrication and measurement point of view.

(2) (3) (4) (5) (6) where and are the propagation constant and characteristic impedance in the known dielectric (Regions I, and III in Fig. 2), respectively. Since the known dielectric is nonmagnetic and can be written as [40] and isotropic, (7) (8) (9)

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where is the width of the strip line, is the total height of the substrate and is the relative permittivity of the known diand are the propagation conelectric. In (3) and (4), stant and characteristic impedance in the unknown metamaterial sample, respectively. Since the metamaterial sample is in genfields should be eral anisotropic, the direction of the and considered when deriving , and . In the stripline topology shown in Fig. 1 the dominant component is in the direction and the dominant component is in the direction. Therefore , and are formulated as (10) (11) (12) Fig. 3. The constitutive parameters of the sample under test (Frequency-dependent case).

where is the permeability in the direction, and is the permittivity in the direction. In order to express the permittivity and permeability of the metamaterial in terms of the measured S-parameters, (1)–(6) are solved by applying the boundary conand yielding the following relationships: ditions at

(13)

III. FULL WAVE NUMERICAL ANALYSIS To investigate the accuracy of the method and to analyze the effect of having anisotropic samples, full wave numerical simulation is used. Ansoft HFSS10, a commercial simulation tool based on the three-dimensional finite element method, is used for numerical analysis. A. Isotropic Metamaterial Sample

(14) (15) (16) (17)

The formulas presented for (11) and (8) are approximate [40]. Notice that the bracketed expressions in (11) and (8) are identical since the width of the strip line and the height of the substrate are the same for the dielectric part and metamaterial part. Therefore, when calculating the extracted parameters, and , the bracketed parts of the expressions in and , cancel out. As a result, the approximations embedded in and that relate to the line width and substrate height will not affect the extracted parameters , and thus, the accuracy of the method. In the above equations, only the dominant electric and magnetic fields are considered. However, in the strip line structure, the component of as well as the component of which are ignored in (10) and (11) are present in the field distribution. In the case of isotropic sample, the fact that the permittivity and permeability are the same in all the directions does not affect the accuracy of the results. However, in the case of anisotropic metamaterials, neglecting the non-dominant field components is expected to affect the accuracy of the extracted permittivity and permeability. In Section III, using full wave numerical analysis, the effect of anisotropy is investigated and a solution is proposed.

To test the method for metamaterial samples with frequencydependent constitutive parameters, a metamaterial with constitutive parameters shown in Fig. 3 is used as the sample under test. In this section, we assume that the sample is isotropic. In the next section we consider anisotropic samples. The parameters shown in Fig. 3 is generated using the Lorentz model [41] for both permittivity and permeability (18) (19) where the parameters in the above equations are selected as: MHz, MHz, MHz, MHz, and MHz. The constitutive parameters data shown in Fig. 3 includes all possible cases: double positive, -negative, -negative, and double negative. Therefore; by characterizing this sample, the accuracy of method will be tested for all types of metamaterials. Using numerical simulations, the S-parameters are generated for the test material, and using (13)–(17), the constitutive parameters are calculated. In Figs. 4 and 5, the superimposed plots of the extracted parameters and the actual assigned parameters are presented. A strong agreement is observed between the extracted parameters and the actual data. In this simulation, parameters of the strip line fixture (see Fig. 1) are chosen as folcm, cm, mm, mm, the lows: host dielectric is Rogers RT/duroid 5880 with , and .

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Fig. 6. Results of simulation for anisotropic sample. Extracted permittivity ( ) is plotted versus actual value of permittivity ( ).

Fig. 4. Extracted permittivity for the data shown in Fig. 3.

Fig. 7. Results of simulation for anisotropic sample. Extracted permeability ( ) is plotted versus actual value of permeability ( ).

Fig. 5. Extracted permeability for the data shown in Fig. 3.

B. Anisotropic Metamaterial Sample and Curve Fitting Usually metamaterial designs and applications involve anisotropy in the structure. Therefore characterization of anisotropic samples using the proposed method has a practical importance. The equations used for extraction of permittivity and are derived by neglecting and permeability from the effect of non-dominant field components on the characteristic impedance and propagation constant. Therefore, we expect some deviation between the extracted parameters and the actual parameters when the sample is anisotropic. To present a quantitative study on this deviation, we assume a sample with the following permittivity and permeability tensors: (20)

(21)

First we assume a frequency-independent sample with conand . Using the extraction procedure stant values for used in Section III.A, the constitutive parameters are calculated. In Figs. 6 and 7, assigned values of constitutive parameters are plotted as a function of extracted values. In this simulation, the topological dimensions of the strip line fixture and the host dielectric are as in Section III.A. The results presented in Figs. 6 and 7 show that the extracted and . values and are not equal to the assigned values Notice that if the extracted and assigned values were equal, the two curves in Figs. 6 and 7 would be straight lines with unity slope. The reason behind this deviation for the anisotropic samples, as explained in Section II, is due to neglecting the nondominant components of and when deriving (10)–(17). In order to address this problem, we propose a post-processing solution to compensate for the effect of anisotropy. In this solution, first we obtain a function to describe the relation between the extracted and assigned parameters. Then, we apply the derived function to the measurement results to compensate the effect of anisotropy. Using curve fitting tools in MATLAB, polynomial functions that represent the numerical results presented in

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Figs. 6 and 7 are derived. The fitted functions are also plotted in Figs. 6 and 7. The formulas of the resultant polynomials are

(22)

(23) The equations presented in (22) and (23) are not universal equations and are dependent on the geometrical parameters of the strip line fixture. We have performed a comprehensive numerical study to investigate the dependence of the derived equations on different geometrical and electrical specifications of the designed strip line fixture. Our study shows that the results of the extracted and are independent of each other. This implies that, for a specific value of permeability, changing the permittivity does not affect the extracted value of permeability and vise versa. In addition our study shows that the extracted and are independent of the value of the constitutive parameters in and . On the other hand, the exthe propagation direction, tracted and depend on the value of constitutive parameters on the cross section of the strip line fixture, and , and on the geometrical parameters of the strip line, and . Since and are known, they all can be included in the numerical simulation to update the fitting formulas (22) and (23) for any strip line fixture. When deriving the fitting formulas of (22) and (23), the simulation results for a homogeneous sample are used. Since small unit cells constitute the metamaterial samples, their inhomogeneity is unavoidable; however, what of interest here is the macroscopic properties of metamaterial structure which are the average values of and [1]–[5]. Since the size of the unit cells is much smaller than the wavelength, it is expected that the geometry of the unit cell does not change the non-dominant field effects. Therefore, although the fitting formulas of (22) and (23) are derived using a homogeneous sample, the formulas are expected to give a reasonable accuracy when used for the effective permittivity and permeability of metamaterials with different geometries. The good accuracy illustrated in the next section for experimental characterization of metamaterials with fractal Hilbert geometry validates this conclusion. It should be noted that the fitting formulas of (22), (23) were derived based on the special anisotropic case for metamaterials as illustrated in (20), (21). In this type of anisotropy, the values and have been assumed to be equal to those of the of host dielectric on which the metamaterial structure is fabricated (which are known frequency-independent constants). This assumption is valid for most metamaterial structures fabricated by stacking planar printed circuit boards to provide three-dimensional substrates. For the three-dimensional isotropic metamaterial structures which provide the same frequency-dependent

Fig. 8. Results of simulation for anisotropic sample. Real part of extracted permeability before and after fitting is plotted and compared with the actual data.

Fig. 9. Results of simulation for anisotropic sample. Imaginary part of extracted permeability before and after fitting is plotted and compared with the actual data.

parameters in all directions (reported in earlier works such as [19], [20]), the stripline method reported here is expected to work. However, if a three-dimensional metamaterial is designed and but in such a way that provides frequency-dependent with different values from and , then the fitting method reported here will not be suitable. The values of the constitutive and , however, do parameters in the propagation direction, not affect the fitting formulas of (22), (23). Therefore, it is of no consequence if they are assumed to be constant or frequency dependent. As shown in the section on numerical and experimental validation, despite being a frequency-dependent parameter, the fitting method results in a good accuracy. To verify the accuracy of the proposed fitting solution, we consider an anisotropic sample with the constitutive parameters shown in Fig. 3. The parameters shown in Fig. 3 are selected as , and . The extracted results before and after applying the fitting formulas are shown in Figs. 8–11, and compared with the actual data. As shown in these figures, after applying the fitting solution, the extracted parameters have acceptable agreement with the actual data.

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Fig. 12. Fractal Hilbert3 inclusion used for constructing magnetic metamaterial. l mm, y : mm, x z mm.

=8

1 = 1 57

1 = 1 = 11

Fig. 10. Results of simulation for anisotropic sample. Real part of extracted permittivity before and after fitting is plotted and compared with the actual data.

Fig. 13. A single strip containing 2 unit cells of inclusions fabricated using printed circuit board technology.

Fig. 11. Results of simulation for anisotropic sample. Imaginary part of extracted permittivity before and after fitting is plotted and compared with the actual data.

IV. EXPERIMENTAL VALIDATION USING FRACTAL HILBERT3 INCLUSIONS The accuracy of the proposed method is tested by experimentally characterizing an anisotropic magnetic metamaterial. A metamaterial structure based on Fractal Hilbert3 inclusions is designed to achieve a magnetic response with negative permeability [17], [42]. The unit cell of the structure is shown in Fig. 12. The inclusion consists of a conducting trace having a mm and separation between the traces is width of mm. This metamaterial was fabricated and characterized using the strip line fixture method. Using printed circuit technology, strips with 2 unit cells of the Fractal Hilbert3 inclusions were fabricated as shown in Fig. 13. The substrate material is Rogers RT/duroid 5880 with and . A three-dimensional metamaterial substrate was assembled by stacking 33 of the fabricated strips in the direction. Due to the thickness of the metal strips and imperfection in the procedure used to stack the strips, an average air gap of 50 m develops between the strips. The air gap while unavoidable in the fabrication process is nevertheless measurable. Therefore the effect of the air gap can be easily included in the design. The fabricated metamaterial substrate has dimen, and directions, sions of 5.5 cm, 4 cm, and 1.1 cm in the

Fig. 14. The fabricated strip line fixtures. (a) without the metamaterial sample (this fixture is used as a reference), (b) with the metamaterial sample to be measured.

respectively. The strip line fixture has dimensions of cm, cm, mm, mm. (see Fig. 1) The fixtures used for characterization of the metamaterial substrate are shown in Fig. 14(a), (b). The fixture shown in Fig. 14(a) is used to measure the properties of the strip line without the metamaterial sample. These measurements are used to determine the phase reference plane for the measurement results of the fixture with the meetamaterial sample. Using a vector network analyzer, the S-parameters of the fixture shown in Fig. 14(a) were measured. These parameters are presented in Figs. 15 and 16. The fabricated strip line fixture has the return loss of less than dB and insertion loss better than 0.1 dB when the metamaterial sample is not present. Therefore, the transitions between the connectors and the strip line provides sufficient accuracy needed for extraction of constitutive parameters of the metamaterial substrate. Fig. 16 shows the phase of the measured along with the phase shift expected in the case of a transmission line with a length of 20 cm. These results show that the strip line without the sample can be modeled as a transmission line with a physical length of 20 cm. The actual length of the strip line fixture without the metamaterial sample is 18 cm. The

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Fig. 15. Magnitude of the measured S parameters of the reference fixture (see Fig. 14(a)).

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Fig. 18. Phase of the measured S parameters of the fixture with the metamaterial sample (see Fig. 14(b)).

Fig. 19. The Extracted measured permeability after fitting (using (22)) is compared with numerical simulation results. Fig. 16. Phase of the measured S21 of the reference fixture (see Fig. 14(a)).

Fig. 20. The Extracted measured permittivity after fitting (using (23)) is compared with numerical simulation results. Fig. 17. Magnitude of the measured S parameters of the fixture with the metamaterial sample (see Fig. 14(b)).

extra phase delay is provided by the N-type connectors. The data shown in Fig. 16 is used as a reference to determine the phase reference plane for the measurement results when the metamaterial sample is placed in the middle of the strip line. Next, the S-parameters of the fixture with the metamaterial sample as shown in Fig. 14(b) is measured. The magnitude, and phase of the measured S-parameters are presented in Figs. 17 and 18, respectively.

Using the measured S-parameters and the extraction method explained in Section II along with the fitting formulas in (22) and (23), the constitutive parameters of the metamaterial sample are extracted as shown in Figs. 19 and 20. In this extraction, the of the reference fixture (shown in phase of the measured Fig. 16) is subtracted from the phase of the measured and after inserting the metamaterial sample. The subtraction is necessary to eliminate the phase delay due to the two transmission lines before and after the sample. The results are compared with the constitutive parameters extracted numerically.

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The numerical results are obtained using Ansoft HFSS10, and the numerical setup reported in [17]. In the numerical setup, a unit cell of the artificial material combined with periodic boundary conditions are used to mimic an infinite slab of artificial materials. For numerically extraction of constitutive parameters, plane wave analysis is used, and parameters are extracted from the reflected and transmitted waves from the unit cell [17]. The 50 m air gap was also included in the simulation. As shown in Figs. 19 and 20, good agreement is observed between the simulation and measurement results. It should be noted that in the numerical analysis periodic boundary conditions are used to mimic an infinite number of unit cells. However, in practice we can realize only finite number of unit cells. For example in the setup used in this work (see Fig. 14), the fabricated substrate contains 33 unit cells of inclusions in the direction, and only two unit cells in the direction. Increasing the number of unit cells provides higher homogeneity in the fabricated substrate, thus expected to yield better agreement with measurements. However; on the other hand, in a wide class of applications such as antenna miniaturization, only few unit cells is used in the direction to avoid high profile substrates [43], [44]. V. CONCLUSION This work presented a new method for metamaterial characterization. The sample under test is used as the substrate of a strip line structure and the permittivity and permeability of the sample are extracted from the measured S-parameters. The method is inexpensive, easy to build and does not require a large sample. The method and the extraction theory are verified numerically for isotropic single and double negative materials. The method is also applied to the characterization of anisotropic metamaterials by employing a fitting function that compensates for the anisotropic behavior of the sample under test. To validate the method experimentally, an anisotropic sample is designed and fabricated. The strip line structure extracted the permittivity and permeability of the fabricated sample with less then 3% shift in the resonance frequency in comparison with the numerically extracted parameters. REFERENCES [1] V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of  and ,” Soviet Phys. Usp., vol. 10, pp. 509–514, 1968. [2] N. Engheta and R. W. Z. , Metamaterials: Physics and Engineering Explorations. Hoboken-Pistacatway, NJ: Wiley-IEEE Press, 2006. [3] G. V. Eleftheriades, Negative-Refraction Metamaterials. Hoboken, NJ: Wiley, 2006. [4] A. Kabiri, L. Yousefi, and O. M. Ramahi, “On the fundamental limitations of artificial magnetic materials,” IEEE Trans. Antennas Propag., vol. 58, no. 7, pp. 2345–2353, Jul. 2010. [5] J. B. Pendry, “Negative refraction makes perfect lens,” Phys. Rev. Lett., vol. 85, pp. 3966–3969, Oct. 2000. [6] J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Low frequency plasmons in thin-wire structures,” J. Phys.: Condens. Matter, vol. 10, pp. 4785–4809, June 1998. [7] J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech., vol. 47, pp. 2075–2084, Nov. 1999. [8] D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science, vol. 314, pp. 977–980, Nov. 2006.

[9] R. W. Ziolkowski and A. D. Kipple, “Application of double negative materials to increase the power radiated by electrically small antennas,” IEEE Trans. Antennas Propag., vol. 51, no. 10, pp. 2626–2640, Oct. 2003. [10] L. Yousefi, B. Mohajer-Iravani, and O. Ramahi, “Enhanced bandwidth artificial magnetic ground plane for low-profile antennas,” IEEE Antennas Wireless Propag. Lett., vol. 6, pp. 289–292, 2007. [11] Y. Lee, W. Park, J. Yeo, and R. Mittra, “Directivity enhancement of printed antennas using a class of metamaterial superstrates,” Electromagnetics, vol. 26, pp. 203–218, Apr. 2006. [12] H. Attia, L. Yousefi, M. Bait-Suwailam, M. Boybay, and O. M. Ramahi, “Enhanced-gain microstrip antenna using engineered magnetic superstrates,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 1198–1201, 2009. [13] M. S. Boybay and O. M. Ramahi, “Near-field probes using double and single negative media,” Phys. Rev. E, vol. 79, p. 016602, Jan. 2009. [14] C. Holloway, D. Love, E. Kuester, A. Salandrino, and N. Engheta, “Subwavelength resonators: On the use of metafilms to overcome the =2 size limit,” IET Microw. Antennas Propag., vol. 2, no. 2, pp. 120–129, Feb. 2008. [15] R. Marques, F. Medina, and R. Rafii-El-Idrissi, “Role of bianisotropy in negative permeability and left-handed metamaterials,” Phys. Rev. B, vol. 65, p. 144440, Apr. 2002. [16] A. Erentok, P. Luljak, and R. Ziolkowski, “Characterization of a volumetric metamaterial realization of an artificial magnetic conductor for antenna applications,” IEEE Trans. Antennas Propag., vol. 53, no. 1, pp. 160–172, Jan. 2005. [17] L. Yousefi and O. Ramahi, “Artificial magnetic materials using fractal Hilbert curves,” IEEE Trans. Antennas Propag., vol. 58, no. 8, pp. 2614–2622, Aug. 2010. [18] A. Erentok, R. W. Ziolkowski, J. A. Nielsen, R. B. Greegor, C. G. Parazzoli, M. H. Tanielian, S. A. Cummer, B.-I. Popa, T. Hand, D. C. Vier, and S. Schultz, “Low frequency lumped element-based negative index metamaterial,” Appl. Phys. Lett., vol. 91, no. 18, p. 184104, Nov. 2007. [19] E. Verney, B. Sauviac, and C. R. Simovski, “Isotropic metamaterial electromagnetic lens,” Phys. Lett. A, vol. 331, no. 3–4, pp. 244–247, 2004. [20] J. D. Baena, L. Jelinek, and R. Marqués, “Towards a systematic design of isotropic bulk magnetic metamaterials using the cubic point groups of symmetry,” Phys. Rev. B, vol. 76, no. 24, p. 245115, Dec. 2007. [21] L. Chen, C. K. Ong, and B. T. G. Tan, “Cavity perturbation technique for the measurement of permittivity tensor of uniaxially anisotropic dielectrics,” IEEE Trans. Instrum. Meas, vol. 48, pp. 1023–1030, Dec. 1999. [22] K. Buell and K. Sarabandi, “A method for characterizing complex permittivity and permeability of meta-materials,” in Proc. IEEE Antennas and Propagation Society Int. Symp., Jun. 2002, vol. 2, pp. 408–411. [23] R. B. Greegor, C. G. Parazzoli, K. Li, B. E. C. Koltenbah, and M. Tanielian, “Experimental determination and numerical simulation of the properties of negative index of refraction materials,” Opt. Expr., vol. 11, pp. 688–695, Apr. 2003. [24] A. F. Starr, P. M. Rye, D. R. Smith, and S. Nemat-Nasser, “Fabrication and characterization of a negative-refractive-index composite metamaterial,” Phys. Rev. B, vol. 70, p. 113102, Sep. 2004. [25] D. R. Smith, D. Schurig, and J. J. Mock, “Characterization of a planar artificial magnetic metamaterial surface,” Phys. Rev. E, vol. 74, p. 036604, Sep. 2006. [26] N. J. Damascos, R. B. Mack, A. L. Maffett, W. Parmon, and P. L. E. Uslenghi, “The inverse problem for biaxial materials,” IEEE Trans. Microw. Theory Tech., vol. 32, no. 4, pp. 400–405, Apr. 1984. [27] H. Chen, J. Zhang, Y. Bai, Y. Luo, L. Ran, Q. Jiang, and J. A. Kong, “Experimental retrieval of the effective parameters of metamaterials based on a waveguide method,” Opt. Expr., vol. 14, no. 26, pp. 12 944–12 949, Dec. 2006. [28] L. Yousefi, H. Attia, and O. M. Ramahi, “Broadband experimental characterization of artificial magnetic materials based on a microstrip line method,” J. Progr. Electromagn. Res. (PIER), vol. 90, pp. 1–13, Feb. 2009. [29] L. F. Chen, C. K. Ong, C. P. Neo, V. V. Varadan, and V. K. Varadan, Microwave Electronics Measurement and Materials Characterization. Hoboken, NJ: Wiley, 2004. [30] M. S. Boybay, S. Kim, and O. M. Ramahi, “Negative material characterization using microstrip line structures,” in Proc. IEEE AP-S Int. Symp. Antennas Propagation, Jul. 2010, vol. 1B, pp. 1–4.

YOUSEFI et al.: CHARACTERIZATION OF METAMATERIALS USING A STRIP LINE FIXTURE

[31] A. Erentok, R. W. Ziolkowski, J. A. Nielsen, R. B. Greegor, C. G. Parazzoli, M. H. Tanielian, S. A. Cummer, B.-I. Popa, T. Hand, D. C. Vier, and S. Schultz, “Low frequency lumped element-based negative index metamaterial,” Appl. Phys. Lett., vol. 91, pp. 1 841 041–1 841 043, Nov. 2007. [32] J. Baker-Jarvis, E. J. Vanzura, and W. A. Kissick, “Improved technique for determining complex permittivity with the transmission/reflection method,” IEEE Trans. Microwave Theory Tech., vol. 38, no. 8, pp. 1096–1103, Aug. 1990. [33] P. Queffelec, P. Gelin, J. Gieraltowski, and J. Loaec, “A microstrip device for the broad band simultaneous measurement of complex permeability and permittivity,” IEEE Trans. Magn., vol. 30, no. 2, pp. 224–231, Mar. 1994. [34] Y. Heping, K. Virga, and J. Prince, “Dielectric constant and loss tangent measurement using a stripline fixture,” IEEE Trans. Adv. Packag., vol. 21, pp. 441–446, Nov. 1999. [35] J. Hinojosa, L. Faucon, P. Queffelec, and F. Huret, “S-parameter broadband measurements of microstrip lines and extraction of the substrate intrinsic properties,” Microw. Opt. Technol. Lett., vol. 30, no. 1, pp. 65–69, Jul. 2001. [36] V. Bekker, K. Seemann, and H. Leiste, “A new strip line broad-band measurement evaluation for determining the complex permeability of thin ferromagnetic films,” J. Magnetism Magn. Mater., vol. 270, no. 3, pp. 327–332, 2004. [37] W. Davis, C. Bunting, and S. Bucca, “Measurement and analysis for stripline material parameters using network analyzers,” in Proc. Instrumentation and Measurement Technology Conf. IMTC-91., May 1991, pp. 568–572. [38] W. Davis, C. Bunting, and S. Bucca, “Measurement and analysis for stripline material parameters using network analyzers,” IEEE Trans. Instrum. Meas., vol. 41, no. 2, pp. 286–290, Apr. 1992. [39] J. Hinojosa, “Permittivity characterization from open-end microstrip line measurements,” Microw. Opt. Technol. Lett., vol. 49, no. 6, pp. 1371–1374, 2007. [40] D. M. Pozar, Microwave Engineering, 2nd ed. New York: Wiley, 1998. [41] R. W. Ziolkowski, “Design, fabrication, and testing of double negative metamaterials,” IEEE Trans. Antennas Propag., vol. 51, no. 7, pp. 1516–1529, Jul. 2003. [42] L. Yousefi and O. M. Ramahi, “Miniaturised antennas using artificial magnetic materials with fractal Hilbert inclusions,” Electron. Lett., vol. 46, no. 12, pp. 816–817, 2010. [43] 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. [44] P. M. T. 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. Antenna Propag., vol. 54, pp. 1654–1662, Jun. 2006. Leila Yousefi (M’09) was born in Isfahan, Iran, in 1978. She received the B.Sc. and M.Sc. degrees in electrical engineering from Sharif University of Technology, Tehran, Iran, in 2000, and 2003 respectively, and the Ph.D. degree in electrical engineering from University of Waterloo, Waterloo, ON, Canada, in 2009. Currently she is working as a Postdoctoral Fellow at the University of Waterloo. Her research interests include metamaterials, miniaturized antennas, electromagnetic bandgap structures, and MIMO systems.

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Muhammed Said Boybay (S’07–M’09) received the B.S. degree in electrical and electronics engineering from Bilkent University, Turkey, in 2004 and the Ph.D. degree in electrical and computer engineering from the University of Waterloo, Waterloo, ON, Canada, Canada, in 2009. From 2004 to 2009, he was a Research and Teaching Assistant in the Mechanical and Mechatronics Engineering, and Electrical and Computer Engineering Departments, University of Waterloo. Currently, he is a Postdoctoral Fellow in the Department of Electrical and Computer Engineering, University of Waterloo. His research interests include double and single negative materials, near field imaging, electrically small resonators, electromagnetic bandgap structures and EMI/EMC applications.

Omar M. Ramahi (F’09) was born in Jerusalem, Palestine. He received the B.S. degrees in mathematics and electrical and computer engineering (summa cum laude) from Oregon State University, Corvallis, and the M.S. and Ph.D. in electrical and computer engineering from the University of Illinois at Urbana-Champaign. He held a visiting fellowship position at the University of Illinois at Urbana-Champaign and then worked at Digital Equipment Corporation (presently, HP), where he was a member of the Alpha Server Product Development Group. In 2000, he joined the faculty of the James Clark School of Engineering, University of Maryland at College Park, as an Assistant Professor and later as a tenured Associate Professor. At Maryland, he was also a faculty member of the CALCE Electronic Products and Systems Center. Presently, he is a Professor in the Electrical and Computer Engineering Department and holds the NSERC/RIM Industrial Research Associate Chair, University of Waterloo, Waterloo, ON, Canada. He holds cross appointments with the Department of Mechanical and Mechatronics Engineering and the Department of Physics and Astronomy. He served as a consultant to several companies and was a co-founder of EMS-PLUS, LLC and Applied Electromagnetic Technology, LLC. He has authored and coauthored over 240 journal and conference papers. He is a coauthor of the book EMI/EMC Computational Modeling Handbook, (Springer-Verlag, 2001). Dr. Ramahi presently serves as an Associate Editor for the IEEE TRANSACTIONS ON ADVANCED PACKAGING and as the IEEE EMC Society Distinguished Lecturer.

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LF Ground-Wave Propagation Over Irregular Terrain Lili Zhou, Xiaoli Xi, Jiangfan Liu, and Ningmei Yu

Abstract—Traditional methods used to predict the ground-wave field strength at low frequency are not applicable for terrains with serious irregularities because of the analytical approximations. In this paper, the two dimensional finite-difference time-domain (FDTD) algorithm is applied to calculate the field strength of the low frequency (LF) ground wave propagating over irregular terrains. The propagation characteristics are studied as functions of the mountain’s gradient, height, and width, respectively. We also focus on studying the cases with multiple mountains in the path. Moreover, the error of the traditional integral equation method is analyzed by comparisons. The results show that when the mountain’s gradient and height are high, additional oscillations in the field strength will appear in front of and in the mountain region due to the wave reflection and scattering. At last, measurements of the Loran-C signals are taken along a real path between Pucheng and Tongchuan in Shaanxi province, China. It is found that most of the measured and FDTD results have good agreements while some still have big errors owing to the model approximation. The FDTD method gets better precision than the integral equation method in the irregular terrain. Index Terms—Finite-difference time-domain (FDTD) methods, ground-wave propagation, irregular terrain, low frequency.

I. INTRODUCTION OW frequency ground wave has been widely used in navigation, positioning, timing and frequency dissemination systems. However, since the signals propagate over paths of varying conductivity, topography and atmosphere refraction index, it is difficult to predict them precisely, which may lead to low positioning or timing precision. In order to obtain high accuracy, it is essential to learn the characteristics of the ground wave, especially when propagating over irregular terrains [1], [2]. The investigation of LF ground-wave propagation can be traced back to the beginning of last century. The early contributions of Zenneck and Sommerfeld [3], [4] were based on a flat-earth model, where the earth surface was assumed to be a flat plane with permittivity and finite conductivity. Other pioneers such as Fock, Millington, Wait, Hufford et al., further studied the spherical smooth-earth model, stratified-ground model, mixed-path model and irregular-earth model [5], [6]. For irregular terrains, two theoretical methods are usually

L

Manuscript received May 29, 2010; revised September 04, 2010; accepted September 28, 2010. Date of publication January 31, 2011; date of current version April 06, 2011. This work was supported in part by the National Natural Science Foundation of China (No. 60671035) and in part by Shaanxi Provincial Project of Special Foundation of Key Disciplines. L. Zhou, X. Xi, and N. Yu are with the Electrical Engineering Department, Xi’an University of Technology, Xi’an 710048, China (e-mail: cimeiyuer@126. com; [email protected]; [email protected]; [email protected]). J. Liu is with the Electrical Engineering Department, Northwestern Polytechnical University, Xi’an 710072, China (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2011.2109693

used to predict the ground-wave propagation, namely, the Millington algorithm [7] and the integral equation method [8]. The Millington method approximates the topographic effect using an equivalent conductivity. It is simple, but the calculation error is big in the mountain region. The integral equation method considers the real topographic effect, but the accuracy is also poor when the parameters and topography change greatly along or around the paths [8]. As only a limited number of highly idealized ground-wave propagation problems have mathematically exact solutions, semianalytical/numerical and pure numerical simulation methods are appropriate ways to handle realistic ground-wave propagation problems [9]. Recently, the numerical methods including the finite element method (FEM), method of moment (MoM), finite-difference time-domain (FDTD) and parabolic-equation (PE) methods, are often applied in the research of wave propagation along the Earth’s surface [9]–[20]. Former researches were more concerned in predicting the transmission losses of ground-wave in MF/HF bands or calculating the VLF-LF propagation in the Earth-Ionosphere waveguide without considering the terrain impacts. In fact, in LF band, terrain impacts are not negligible when the terrain relief is close to the wavelength. In this paper, a two-dimensional cylindrical coordinate FDTD algorithm is used to simulate the LF ground-wave propagation. The purpose is to analyze the terrain effects on the wave propagation characteristics and give the variation trends of the ground wave propagating over the paths with extremely serious irregularities. Comparing the results to those of the traditional integral equation method, the error and applicable conditions of the integral equation method is analyzed. II. COMPUTATIONAL MODEL AND LAGORITHMS For a flat and homogenous path, ground-wave propagation problems can be solved analytically by the flat-earth formula in a cylindrical coordinate system, detailed equations can be found in [21], it is also listed in the Appendix A for the convenience of the reader. For more complicated propagation path with varying terrain and electrical parameters, the integral equation method can be applied. The electrical field strength can be expressed as [8], [22], as a function of the ground wave attenuation factor given by (1) Detailed equation for can be found in the Appendix B. The above two methods are used in the following sections as references. We now focus on the numeric method, i.e., the FDTD method, for the sake of being able to capture the real topographic effects. The FDTD method has been well documented in the literature [23] therefore details are omitted in this paper. In the

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Fig. 1. Computational model.

FDTD simulation, the perfectly matched layer (PML) absorbing boundary condition (ABC) [24], [25] is very important for terminating the simulation domain without generating fake reflections. Implementation of the PML can be done by mapping the Maxwell’s equations onto a stretched-coordinate-system and including lossy factors. In the cylindrical coordinate system, the coordinate stretching is defined by

Fig. 2. Comparison of FDTD and theoretical results, E as a function of frequencies.

III. SIMULATION AND RESULT ANALYSIS Algorithm Verification

(2)

Fig. 2 shows a comparison of the FDTD and flat earth formula results. The source is a Gaussian pulse, given by

(3) (7) , are the PML where . For a lossy material as parameters. Outside PML, the earth, the Maxwell’s equation in the PML can be described in the cylindrical coordinate system as

(4) (5) where and earth and

are relative permittivity and conductivity of the

(6)

Following the standard finite-difference approach, the field update equations can be derived. Assuming the terrain being consistent in direction, the ground-wave propagation problem can be solved in a 2-D cylindrical coordinate system using the FDTD method. Fig. 1 shows the FDTD computational model. The computational domain is defined with the size in and directions of 100 and 5.625 km, respectively. An electric dipole is located at coordinate zero. The spatial steps along direction ) and direction are both 18.75 m. The (denoted as time step is set to be 31.25 ns according to the Courant stability limit. The computational domain is meshed by with 10 cells of PML on the right and top. The left boundary is an axisymmetric boundary and a lossy ground termination is used at the bottom. The dipole source and receivers are located at one cell above the ground.

is calculated as a function of frequency. The field strength The receivers are placed at , 10, 50, and 100 km, re, spectively. Parameters of the earth are given by . As shown in Fig. 2, the results of the two methods agree well at low frequency range. Due to numerical dispersion, larger deviation is observed at higher frequency and longer propagation distance. Better accuracy can be achieved using smaller grid cell size. A. A Single Mountain in the Propagation Path Using the FDTD method, the ground-wave propagation characteristics over a single mountain with Gaussian shape are studied. The terrain function of the Gaussian-shaped mountain is given by (8) where is the distance, is a parameter for controlling the mountain width, is the height, and is the distance form the antenna to the center of the mountain. An average slope factor is used to express the steepness of a mountain, given by (9) where is the real width of a mountain and . Fig. 3(a) shows the electric field strength at 100 kHz over a Gaussian mountain with a fixed width but different heights. The result from the flat earth model is taken as reference. The pa; rameters used in the simulation are given as follows: ; ; , 0.5 km, 0.75 km, 1

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Fig. 3. Filed strength distribution over the propagation paths with a Gaussianshaped mountain of the same width but different heights: (a) jE j at f calculated by FDTD method. (b) Comparison of jE j between FDTD and integral equation results. (c) jE j distribution at different frequencies .

100 kHz 16 )

= ( =

km, 1.5 km, and 2.5 km leading to an average slope of 3 , 6 , 8 , 11 , 16 , and 26 , respectively. The integral equation method is the only theoretical method which is capable of taking the terrain effects into account. In order to study the applicable conditions of the integral equation method, its results are compared to the FDTD ones, shown in

Fig. 3(b). We may note that the integral equation results are calculated in a 2-D Cartesian coordinate system, which represents a uniform terrain in the perpendicular direction. This is acceptable when the distance between the mountain and transmission antenna is sufficiently far. Fig. 3(c) shows the electric field strength , corat different frequencies over a mountain with responding to an average slope of 16 . The frequencies are 60, 90, 120, 150, and 180 kHz, respectively. As shown in Fig. 3(a), the flat-earth model gives a good prediction before the mountain. In the mountain region, the amincreases first while the maxplitude of the electric field imum value appears when approaching the mountain top. After falls rapidly as the topographic height decreasing. The that, minimum value appears when reaching the mountain foot. In the end, the field strength increases again and the variation graduwavelength from the mountain. It ally tends to be stable at , the is observed that when the mountain is lower than 1 km difference between FDTD and the flat-earth model result is less than 0.2 dB. This value increases as increasing mountain , the difference is 1.3 dB. The height , at results in Fig. 3(b) show that when the steepness of the mountain is small, the field strength calculated by the two methods match well. In the flat areas in front and after the mountain, the difference is less than 0.6 dB. The discrepancy mainly exists in the mountain region during the terrain rising and higher mountain leads to bigger difference. This is caused by the approximation of the integral equation method, where the terrain change after the receiver position is not considered, so that the backscattered , fields are neglected. As shown in Fig. 3(b), when , it is up to the difference is less than 1 dB; when , the difference increases to 4 dB. As 1.5 dB; when expected, similar phenomenon is found in Fig. 3(c). At different frequencies in LF band, the higher the frequency is, the greater change is observed in the mountain region. at 100 kHz over Fig. 4(a) shows the electric field strength a mountain with the same height but different widths. The flat earth model result is also shown as reference. The parameters ; used in the simulation are given as follows: ; , 5 km and 15 km leading to an average slope of 49 , 30 , and 11 , respectively. Some of the results are compared in Fig. 4(b) with those from the integral equation method, too. As shown in Fig. 4(a), steeper mountain slope leads to larger field attenuation after the mountain. When the average slope is greater than 30 , in front of the mountain there are significant oscillations stemming from the reflection and scattering effects of the mountain. Refining the FDTD mesh does not change the result, which proves that the simulation has been converged and the oscillation is not artificial. The distances between the nodal/antinodal points in the oscillation are around half wave length of the source, which also proves that the oscillation is caused by the reflection. As shown in Fig. 4(b): when the mountain is very steep, in front of the mountain, there is a big differcalculated by the FDTD method and the ence between the integral equation method. The integral equation method cannot catch the reflection because the terrain change after the observation point is not taken into consideration. Therefore no oscillation is observed in the integral equation results. This tells us

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equation method. It is also noted that at lower frequency, the attenuation after the mountain is smaller, while the oscillation amplitude in front of the mountain is larger. B. Multiple Mountains in the Propagation Path In order to study the effects on the ground wave propagation over multiple mountains, several 100-km-long propagation paths with two or three mountains of different center distances are taken as examples. The mountain is 1.5 km high, 10 km wide and the center distances are 5 km, 10 km and results calculated by the FDTD 30 km respectively. The method and the integral method are shown in Fig. 5(a)–(f). It is noted that, when there are multiple mountains with the same shape in the propagation path and the mountains do not overlap with each other, the characteristics of ground wave propagation over each mountain is basically the same. After certain distance (for example, 80 km), all the results are more or less identical. In addition, the results of two methods agree well and the calculation error mainly exists in the mountain region during the terrain rising. However, if the mountains overlap with each other, oscillation is observed in the hollow or in front of the mountains, caused by (multiple) wave reflection/scattering. This oscillation cannot be predicted by the integral equation method. C. Comparison Between FDTD and Measurement Results

Fig. 4. Filed strength distribution over the propagation paths with a Gaussianshaped mountain of the same height but different widths: (a) jE j at f calculated by FDTD method. (b) Comparison of jE j between FDTD and integral equation results. (c) jE j distribution at different frequencies .

100 kHz 49 )

= ( =

that when the topographical slope is extremely high, the accuracy of the integral equation method is so low that the method is no longer applicable. field at different frequencies, i.e., Fig. 4(c) shows the , 60, 120, and 180 kHz, over a mountain with . Similar propagation/reflection effects are observed in the FDTD results, which cannot be predicted by the integral

The field strength along a path between Pucheng (a Loran-C station) and Tongchuan is measured by a Locus 1030 receiver and it is calibrated by the data of Lintong monitoring station. The latitude and longitude are measured by a THC-1 GPS receiver. The transmitting power of Pucheng station is . The geographic information is obtained from a digital geographic information system. Among all the measured data, points 1 to 8 are located approximately along one propagation path. Fig. 6 shows the topography of experimental region and the distribution of measurement points. The field strength calculated by the FDTD method and the measured results are compared in Fig. 7. It can be seen that the tendency of FDTD result is consistent with the reference result obtained by the flat earth model. A long propagation path with overlapping mountains leads to oscillations at unequal amplitude as the terrain varies. The oscillation has a wavelength of half of the source’s wavelength and it is significant in the region with large terrain gradient. As shown in Fig. 7, the largest error between the FDTD result and the measurement happens at point 4 and 8, where the terrain changes dramatically in the cross sectional plane perpendicular to the propagation path. This tells us that the error is mainly due to the neglected 3-D topography changes. IV. CONCLUSION In this paper, a 2-D FDTD method is applied to calculate the electric field strength of LF ground wave propagating over irregular terrains. The propagation characteristics are investigated as functions of the mountain’s height, width and gradient. The cases with multiple mountains in a propagation path are also analyzed. The FDTD and measurement results show that when the gradient of the mountain is high, half-wavelength oscillations appear before or between the mountain regions. This is due to

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= 5 km = 30 km = 10 km

Fig. 5. Filed strength distribution over the propagation paths with multiple mountains of different center distances: (a) jE j at 100 kHz, d , double mountains along propagation path, (b) jE j at 100 kHz, d , double mountains along propagation path, (c) jE j at 100 kHz, d , double , three mountains along propagation path, (e) jE j at 100 kHz, d , three mountains mountains along propagation path, (d) jE j at 100 kHz, d along propagation path (f) jE j at 100 kHz, double different shape mountains along propagation path.

= 10 km = 5 km

the wave reflection and scattering, the higher the gradient (or lower frequency) is, the greater the oscillations are. These oscillations cannot be captured by the integral equation method, although it is the only theoretical method being able to take the terrain effect into account. Therefore, numerical method is the only candidate for calculating ground wave propagation over

irregular terrain with significant topographic change. The utilization of a 2-D cylindrical coordinate FDTD modeling leads to errors at the points where the terrain changes dramatically in the cross sectional plane perpendicular to the propagation path. However, at most of the observation points, the FDTD results agree well with the measurement.

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Fig. 8. Computational model.

where

,

Fig. 6. Topography of the experimental region and distribution of the measured points.

(A2) (A3) Integral Equation Method: The integral equation method [22] is the only theoretical method which takes the terrain effect in to consideration. For a simulation model shown in Fig. 8, the can be expressed as electric field strength (B1) where is the earth radius, is the great-circle angular distance, the dipole direction is set along the axis and the observation plane. In (B1), is the ground wave point is located at attenuation factor, given by

Fig. 7. Comparison of FDTD and measured results.

APPENDIX Flat Earth Formulation: If the distance between the transmitter and receiver is short and the wave propagation path is homogenous and flat, the ground-wave field strength can be solved by the flat-earth formula. In the cylindrical coordinate system, assuming the dipole source height is from the ground, the observation height is , the horizontal distance between the observation point and the source is , the vacuum permeability is , the earth wave number is much larger than the free space and both of the observation point and source wave number are not too far from the ground, then the theory expression of the vertical field strength is as follows [22]:

(A1)

(B2) Here, is the integration point along the path, denotes the distance from the source to the observation point, and denote the distances from the observation point to the source and , respectively. and represent the horizontal distances from to the source and , respectively. REFERENCES [1] G. Johnson, R. Shalaev, R. Hartnett, P. Swaszek, and M. Narins, “Can loran meet GPS backup requirements?,” IEEE Aerosp. Electron. Syst. Mag., vol. 20, no. 2, pp. 3–12, Feb. 2005. [2] D. Last and P. Williams, “New ways of looking at Loran-C ASFs,” presented at the 31st Int. Loran Assoc. Convent. Tech. Symp., Wash., DC, Oct. 2002. [3] J. Zenneck, “Propagation of plane EM waves along a plane conducting surface,” Ann. Phys., vol. 23, pp. 846–866, Sep. 1907. [4] A. N. Sommerfeld, “Propagation of waves in wireless telegraphy,” Ann. Phys., vol. 81, pp. 1135–1153, Dec. 1926. [5] J. R. Wait, “The ancient and modern history of EM ground-wave propagation,” IEEE Antennas Propag. Mag., vol. 40, no. 5, pp. 7–24, Oct. 1998. [6] W. Y. Pan, LF VLF ELF Wave Propagation. Chengdu, China: UEST, 2004. [7] N. DeMinco, “Propagation prediction techniques and antenna modeling (150 to 1705 kHz) for intelligent transportation systems (ITS) broadcast applications,” IEEE Antennas Propag. Mag., vol. 42, no. 4, pp. 9–33, Aug. 2000.

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[8] H. Gesny and O. Ravard, “Propagation over irregular terrain in the VHF band a review of integral equation models,” in Proc. IEE Nat. Conf. Antennas Propag., Apr. 1, 1999, pp. 61–64. [9] F. Akleman and L. Sevgi, “A novel MoM-and SSPE-based groundwave-propagation field-strength prediction simulator,” IEEE Antennas Propag. Mag., vol. 49, no. 5, pp. 69–82, Oct. 2007. [10] G. Apaydin and L. Sevgi, “FEM-based surface wave multimixed-path propagator and path loss predictions,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 1010–1013, 2009. [11] L. Sevgi, “Groundwave modeling and simulation strategies and path loss prediction virtual tools,” IEEE Trans. Antennas Propag., vol. 55, pp. 1591–1598, Jun. 2007. [12] W. Y. Pan, H. Y. Peng, and H. Q. Zhang, “Parabolic equation algorithm of wave attenuation along inhomogeneous smooth ground,” Chin. J. Radio Sci., vol. 21, no. 1, pp. 37–42, Feb. 2006. [13] L. L. Zhou, X. L. Xi, and N. M. Yu, “Comparison of three methods of calculating low frequency ground-wave propagation over irregular terrain,” Chin. J. Radio Sci., vol. 24, no. 6, pp. 1158–1163, Dec. 2009. [14] S. A. Cummer, “Modeling electromagnetic propagation in the earthionosphere waveguide,” IEEE Trans. Antennas Propag., vol. 48, pp. 1420–1429, Sep. 2000. [15] J. J. Simpson and A. Taflove, “Three-dimensional FDTD modeling of impulsive ELF propagation about the earthsphere,” IEEE Trans. Antennas Propag., vol. 52, pp. 443–451, Feb. 2004. [16] J. P. Berenger, “FDTD computation of VLF-LF propagation in the earth-ionosphere waveguide,” Ann. Telecommun., vol. 57, no. 11–12, pp. 1059–1090, 2002. [17] Y. Wang, H. G. Xia, and Q. S. Cao, “Analysis of ELF propagation along the earth surface using the FDTD model based on the spherical triangle meshing,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 1017–1020, 2009. [18] J. J. Simpson and A. Taflove, “A review of progress in FDTD Maxwell’s equations modeling of impulsive subionospheric propagation below 300 kHz,” IEEE Trans. Antennas Propag., vol. 55, pp. 1582–1590, Jun. 2007. [19] A. V. Popov and V. V. Kopeikin, “Electromagnetic pulse propagation over nonuniform earth surface: Numerical simulation,” Progr. Electromagn. Res. B, vol. 6, pp. 37–64, 2008. [20] G. Apaydin and L. Sevgi, “Numerical investigations of and path loss predictions for surface wave propagation over sea paths including hilly island transitions,” IEEE Trans. Antennas Propag., vol. 58, pp. 1302–1314, Apr. 2010. [21] R. W. P. King and S. S. Sandler, “The electromagnetic field of a vertical electric dipole over the earth or sea,” IEEE Trans. Antennas Propag., vol. 42, pp. 382–389, Mar. 1994. [22] T. S. M. Maclean and Z. Wu, Radiowave Propagation Over Ground. London, U.K.: Chapman and Hall, 1993. [23] A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method. Norwood, MA: Artech House, 2000. [24] F. L. Teixeira and W. C. Chew, “Systematic derivation of anisotropic PML absorbing media in cylindrical and spherical coordinates,” IEEE Microw. Guided Wave Lett., vol. 11, pp. 371–373, Nov. 1997. [25] L. L. Zhou, X. L. Xi, N. M. Yu, and Y. R. Pu, “Modeling of LF groundwave propagation at short distances based on 2-D cylindrical-coordinate FDTD method,” in Proc. 8th Int. Symp. Antennas, Propag. EM Theory, ISAPE, 2008, pp. 855–858.

Lili Zhou received the B.S. and M.S. degrees from Xi’an University of Technology, Xi’an, China, in 2004 and 2007, respectively, where she is currently working toward the Ph.D. degree.

Xiaoli Xi received the B.S. degree in applied physics from the University of Defense Technology, ChangSha, China, in 1990, the M.S. degree in biomedical engineering from the Fourth Military Medical University, Xi’an, China, in 1998, and the Ph.D. degree in electrical engineering from Xi’an Jiaotong University, Xi’an, China, in 2004. She is currently a Professor with the Department of Electric Engineering, Xi’an University of Technology, Xi’an. Her recent research interests include wave propagation, antenna design, and communication signal processing.

Jiangfan Liu received the B.S. and M.S. degrees from Xi’an University of Technology, Xi’an, China, in 2004 and 2007, respectively. He is currently working toward the Ph.D. degree at Northwestern Polytechnical University.

Ningmei Yu received the B.S. degree in electronic engineering from Xi’an University of Technology, Xi’an, China, in 1986 and the M.S. and Ph.D. degrees in electronic engineering from Tohoku University, Sendai, Japan, in 1996 and 1999, respectively. She is currently a Professor with the Department of Electric Engineering, Xi’an University of Technology, Xi’an.

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Ultrawideband Multi-Static Scattering Analysis of Human Movement Within Buildings for the Purpose of Stand-Off Detection and Localization Michael Thiel, Member, IEEE, and Kamal Sarabandi, Fellow, IEEE

Abstract—An efficient and accurate method for the analysis of human backscatter response to ultrawideband near-field UHF radar systems is presented. The method combines a full-wave analysis of a human body model with a PO-like indoor propagation code known as “brick-tracing.” This method allows for the analysis of radar detection and tracking of humans within a realistic building environment. Using this method multi-static radar imaging of a human in buildings is evaluated and certain special features are pointed out. The simulation results are validated with actual radar measurements of a mannequin in a lab environment. A viable method for real time detection and localization of moving human inside buildings is then applied to the simulated and measured data and the building effects on the detection and localization are discussed. Index Terms—Human detection, through-wall imaging, ultrawideband radar.

I. INTRODUCTION

T

HE detection and localization of objects enclosed in buildings or hidden behind walls via ultrawideband near-field radar has received increased attention in recent years [1][2]–[7]. A through-wall imaging radar system gives an opportunity to safely identify hidden objects without intrusion of or damage to the enclosing structure. One prominent application of such trough-wall radar systems is the detection of human subjects inside buildings [8], it can be used to locate humans inside burning or collapsing buildings or entrenched criminals for law enforcement agents. Use of millimeter-wave radar to detect unobscured humans based on body movement caused by breathing has been researched by several [9], [10]. These systems have one major disadvantage: While the attenuation through walls composed of thin plastic, glass, wood and drape is relatively low up to millimeter waves, the attenuation through actual building material like brick or concrete is extremely high due to the

Manuscript received August 31, 2009; revised July 15, 2010; accepted November 10, 2010. Date of publication January 28, 2011; date of current version April 06, 2011. M. Thiel was with the Electrical Engineering Department, University of Michigan, Ann Arbor, MI 48109-2122 USA. He is now with Schlumberger-Doll Research, Cambridge, MA 02139 USA (e-mail: [email protected]). K. Sarabandi is with the Electrical Engineering Department, University of Michigan, Ann Arbor, MI 48109 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.2011.2109349

high loss-tangent of the material and large electrical thickness of the walls [11]. Therefore high-frequency systems, despite their high resolution, have extremely poor wall penetration capabilities in most realistic scenarios [8], [12]. Consequently, the maximum frequency with some capability for penetrating through most wall types [12] is limited to low microwave frequencies. The current work on through-wall radar human detection focuses on the ultra-high frequency band (UHF, 0.5 to 3.0 GHz) [1], [4], [8], [13]. In order to get reasonable range resolution, a wide bandwidth is needed. Ultrawideband through-wall radar systems at UHF face two challenges. First, the interpretation of building radar returns for object detection is difficult because of the multipath nature of propagation inside buildings caused by wall reflections. Secondly, the radar return is dominated by the building itself because of strong direct specular wall reflections. Interpretation of the return gets even more difficult if a complex object like a human body is involved. Consequently, for arriving at a reliable human body detection and localization algorithm, one needs an accurate model to predict human scattering in buildings for phenomenological studies. While approximating humans with cylinders or evaluating human scattering behind a single wall can prove the concept of human detection and localization [3], [4] one needs more complex models to accurately estimate the human fully polarimetric scattering and examine all building effects on it. One solution is a complete full-wave analysis of the building including the human. But at this frequency band, the simulation domain of even a simple scenario gets already too large in terms of wavelength. Consequently full-wave methods require high computational resources and are not feasible for human backscattering analysis in larger buildings with standard computers [13]. This paper introduces a realistic and computationally efficient method to obtain the human response within a building to UWB through-wall radars without costly measurements over a wide bandwidth covering UHF. The method is based on a full-wave analysis of a human body model only which is then incorporated into a hybrid PO-like code for indoor propagation using the equivalence principle [14], [15] to calculate the radar response of the human inside the building (called “bricktracer”) [16], [17]. This hybrid method is computationally tractable compared to full-wave methods but still can predict indoor wave propagation accurately in the presence of inhomogeneous periodic walls and windows which are common in buildings [17]. The paper is structured in the following manner: First a simple but anatomically accurate human model that can be animated for

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any desired posture, derived from an actual human MRI scan, is utilized in conjunction with a full-wave electromagnetic model to generate the scattered field of a human in a building environment. Next, the combination of this model with the indoor propagation part is described. After that the human backscattering itself and the human radar return in buildings is characterized with this method. A viable strategy to detect and localize moving humans inside buildings is tested against the simulated data of the method and also verified with actual measurements of a mannequin behind walls in a lab environment. At the end, the feasibility of the localization and detection idea is shown for a realistic building scenario. II. HUMAN BODY MODEL The human body consists of many parts, most of them having a high dielectric constant of 40 to 50 with high conductivity of (for a frequency of 1 GHz) because of its high and water content. The only exceptions are bones with and fat with and [18]. The human body is covered with cloths which are usually thin, air filled (like cotton) that show low and loss tangent. Consequently, the cloths can be assumed to be transparent at the desired frequency range. While detailed human body models and models for electric properties of the body parts exist, it is not practical to use them for backscattering analysis in conjunction with large indoor propagation tools because of their complexity especially if movable joints are needed to emulate realistic body movement like walking. The human body structure and its dielectric properties may be suitable for a simplifying approximation of considering the human body as homogeneous dielectric material with an average dielectric constant of all body parts. Therefore, for a full male human MRI scan with 84 distinct body parts [19], full-wave finite-difference-time-domain (FDTD) RCS simulations are compared with the RCS simulations of the same model having an average dielectric constant [20]. Simulations were carried out for 0.5 GHz, 1.0 GHz and 1.5 GHz to cover the full frequency band and for various plane wave incidence angles and polarizations. The average dielectric and . Fig. 1 shows the properties are complete human model and compares azimuthal 2D cuts of the human RCS pattern for a plane wave incident from the front for the homogeneous and complete human models. It can be seen that the homogeneous model predicts the human backscattering very well. If one compares the full bistatic 3D RCS, it is found that the average relative error of the RCS in directions the human significantly scatters always stays between 1 and 2 dB for both co- and cross-polarizations over all bistatic angles and frequencies. This indicates that the homogeneous body assumption is tolerable for the type analysis we are pursuing in this article. III. INCORPORATION INTO INDOOR PROPAGATION CODE A freely available program is used to generate a surface mesh of an arbitrarily shaped human in arbitrary body position (like the person walking in Fig. 2) [21]. This model is meshed as a solid in a commercial full-wave 3D finite-difference-time-domain (FDTD) solver with the found average dielectric properties [20]. The body can now be excited with an arbitrary loaded

Fig. 1. Human body model of Semcad virtual family (take form Semcad X ) Manual) and bistatic RCS comparison (horizontal plane, inc. wave at  of exact model to homogeneous assumption for vertical polarized plane wave from the front at 1.0 GHz.

= 90

Fig. 2. Surface model of walking person.

field distribution on a surface enclosing it (frequency domain signal Huygens Source, see [20]). This is combined with an existing hybrid method for indoor propagation based on surface fields which can also include inhomogeneous periodic walls (bricktracer). The bricktracer is an iterative method based on physical optics and full-wave simulation. It can calculate the multipath interaction between the walls and floor/ceiling, including diffraction at corners, windows and doors [16], [17]. The full-wave analysis is incorporated into the hybrid method in the following manner (similar to [15]): First the scene is fully simulated with the bricktracer and the fields on the surface of the Huygens box are recorded which is then used as the incident field for the full-wave simulation of the human body model. After the scattered field is found and extracted in the frequency domain it then acts as a distributed transmitter in the bricktracer and is radiated through the scene to the receiver points. This first-order approximation is sufficient for accurate human scattering analysis in buildings as the second and higher order interactions (illumination of the Huygens box by the reflected primary scattered field from the walls) are small compared to the first order scattered field.

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The required CPU time for the full-wave simulation of a typical walking human ranges from a couple of minutes at 0.5 GHz to approximately 50 min at 1.5 GHz on an Intel Xeon 3.16 GHz workstation. Approximately 40,000 field points are needed to correctly transfer the incident field of the bricktracer to the FDTD solver for frequencies up to 1.5 GHz. A bricktracer simulation to find the incident on or scattered field of the human can take from a couple of seconds for a simple wall to up to 2 h for a complex scenario (like Fig. 14) on an AMD Phenom 2.5 GHz workstation. In order to retrieve the UWB time-domain response from the frequency domain simulations, the simulations have to be carried out for a number of discrete frequency points over the desired band, with appropriate frequency steps according to the maximum range of the scene to avoid aliasing. Once the received voltage at the desired receiving point is found for each frequency point it can then be inverse Fourier-transformed to get the time-domain radar response. IV. SIMULATION RESULTS A. Human Backscattering Initially we study the polarimetric backscatter response of different stationary human subjects in free-space above a ground plane at different look angles. It is found that both changing body pose or changing the look angle of one body pose can significantly alter the human return. In both cases, the co-polarized return varies within 5 dB. Because of the complex shape of the human body a noticeable amount of cross-polarization is generated which is depended on the pose and look angle. On average, the cross-polarized return is 10 dB below that of the co-pol. arised return with a variation of 5 dB. To illustrate this result, the co- and cross-polarized backscattering of a woman standing, walking and sitting on a groundplane 8 m away from the Tx (Tx height 1.3 m) is plotted in Fig. 3 (windowed IFFT of ). 0.5 GHz to 1.2 GHz simulation with Although scattering and reflection off principal plane walls or other building features can cause depolarization, the relative high cross-pol. scattering of the human can still be exploited for detection of humans inside buildings. B. Human Backscattering Behind Walls In order to extract the relative magnitude of the human body backscattering compared to walls, a human behind a simple wall on a ground plane was considered. The wall is chosen to be reinand and an forced concrete, a wall with arrangement of metal rods every 30 cm, having a height of 2.6 m and a length of 8 m. The Tx is placed 1.3 m above the ground, 3 m away from the wall and the human is 5 m away from the wall. Once again, simulations were carried out between 0.5 GHz and 1.5 GHz in 0.01 GHz steps and co-pol. backscatter was recorded at the transmitter point. Fig. 4 shows the IFFT of the co-pol. frequency response (windowed with Chebyshev function) of the total human and wall backscatter and the backscatter contribution from the human behind the wall alone. It can be seen that the human return is very weak compared to the wall return (40 dB lower). Also, the wall return is dispersed in the time domain because of the wall inhomogeneity and almost covering

Fig. 3. Time-domain co-pol. (b) and cross-pol. (c) backscattering of a woman sitting, standing and walking on a ground plane (a) 8 m away from the Tx/Rx (0.5 to 1.2 GHz), normalized to the co-pol. return of the walking woman.

Fig. 4. Time domain radar return of human (8 m away from Tx) behind wall reinforced concrete wall (3 m away from Tx) for co- and cross-polarized receiving antenna, 0.5 to 1.4 GHz, normalized on co-pol. wall return.

the human target 5 m away from it. Again the cross-polarized human return is about 10 dB lower than the co-pol. return but due to depolarizing ground reflections a significant cross-pol. return can be seen for the wall as well. It is obvious that if the human moves closer to the wall, both it’s co- and cross-polarized returns get indistinguishable from the wall return even for a homogeneous wall. C. Human Backscattering in Building and Detection In the following a more realistic building scenario consisting of two rooms is considered. The building dimension is 4 m by 8 m and is illuminated from outside by a vertically polarized Hertzian dipole (see Fig. 5). Each room has one window and door, and the walls are homogeneous dielectric slabs ( and ), as well as the floor and the ceiling.

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Fig. 5. Building layout (right) and top view (left) with transmitter, receiver and human position (floor and ceiling not shown).

Fig. 7. The time-difference time-domain backscattering of a human walking in the building (see Fig. 5 for building layout, Tx/Rx and human position), 0.5 to 1.4 GHz, normalized on single-view backscattering of Fig. 6.

Nevertheless both co- and cross-pol. return are affected by the multipath propagation inside the building which is the cause for the multiple sidelobes. D. Human Localization Fig. 6. The time-domain building backscattering with a human walking in the building (see Fig. 5 for building layout, Tx/Rx and human position), 0.5 to 1.4 GHz, normalized.

The human is positioned at the center of the front room and the transmitter is 2.0 m away from the front wall. The human is standing on the ground, the transmitter is 1.5 m above the ground (ceiling height 2.6 m). Simulations were carried out between 0.5 GHz and 1.5 GHz in 0.005 GHz steps and co-pol. and cross-pol. backscattering was recorded at the transmitter point. Fig. 6 shows the IFFT of the co-pol. frequency response (windowed with Hamming function) for both the building with the human and the human scattering only. It can be seen that the human backscattering is now completely obscured by the building response even for homogeneous walls because of the multipath propagation inside the building, making it impossible to detect the human within the building return without any further processing. One possibility to isolate the human backscatter response from the building is to take advantage of its movement. Basically by subtracting two temporally close backscatter measurements of the building and human from the same transceiver location, one can remove much of the stationary response from the building itself [8]. To demonstrate this concept we considered the two postures shown in Fig. 2 separated in position by 20 cm (human moved forward by 20 cm). Fig. 7 right shows the difference in co- and cross-polarized time-domain responses of the building and the moving human for the two positions and postures of the human in the building. The human can now be easily localized 4.2 m away from the transmitter for both polarizations. A strong shadow effect on the wall behind the human can be seen (at 6.5 m) for co-pol but not for cross-pol.

If one wants to localize and not only detect a human inside a building, further information has to be gathered. The next step is to keep a single Tx but collect the scattered field with an array of Rx (multistatic radar) centered around the Tx. This setup can be used to form a simple and instantaneous 2D image of the scene under investigation. It could be realized by mounting an Rx array (together with Tx) on a vehicle which then can be positioned in front of the building under investigation. The 2D image generation of the scene follows the standard SAR processing or multistatic radar image generation [1], [6]: Given a frequency modulated continuous wave radar system at the -th antenna and which outputs the received voltage -th frequency point, the reflectivity at a given point is found through focusing the array by (1) where the phase shift

is given by (2)

with wavenumber and the Tx position and position of the -th Rx antenna . The reflectivity is estimated on a grid at receiver height to find the 2D image. In order to reduce sidelobes of the image the input data is windowed along the array and frequency data. This has been simulated for the previous scene (Fig. 5). For the same Tx position, the scattered field was recorded along the front wall with a 6 m receiver array (receiver spacing 0.1 m). Fig. 8 shows the 2D image of the building with the human, generated at the Tx height for co-polarized receiving antennas. The

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Fig. 8. Multistatic radar image of building of Fig. 5 for co-polarized Rx antennas.

building is indicated with dashed black lines and the human is located at (2.0, 2.0) m. The image is normalized on the maximum reflectivity of the scene. Fig. 9 on top shows the time-difference image for co-pol., also normalized on the maximum reflectivity of the scene. While the human is still hidden within the building return for the single view image at co-pol., the time-difference image clears up the scene, one can now localize the human at (2.0, 2.0) m. Since the human scatters the incoming wave back to all Rx antennas (like a point target) the human return is now only 20 dB below the maximum of the wall return in the 2D image. But the image shows strong ghosts (caused by the walls to the side, e.g., at (2.0, 2.0) m) and shadows on the walls behind the human, e.g., at (4.0, 2.0) m) which could be wrongly identified as additional humans. The same is true for the time-difference cross-pol. image which is shown in the bottom part of Fig. 9: The human cross-pol. return is enhanced and only 30 dB below the wall return but as in the case of co-pol. images ghost images and wall shadows still exist. However the image is less cluttered and shows less wall shadows because the human itself generates most of the cross-pol. If one is able to reliably collect both co- and cross-polarized backscattering of the scene, they can be combined properly to form one enhanced difference image. One way to do so is a pixel-by-pixel multiplication [2] of the two (normalized) co- and cross-polarized images. While there is always a strong return at the actual human position for both polarizations, the return of the ghost and shadows varies significantly in position and magnitude. Consequently, the multiplication can eliminate much of the ghost and shadow targets. This is illustrated in Fig. 10 which shows the product of the normalized images of Fig. 9. V. MEASUREMENT OF HUMAN THROUGH-WALL RADAR RESPONSE The simulation results were tested against actual radar measurements. A stepped frequency radar system was set up in the lab environment using two double-ridge horn antennas, a vector network analyzer and offline data processing [22]. The antennas were placed above each other and a -table was utilized to emulate a receiver array for 2D measurements (Tx height of 1.5 m and Rx height of 1.2 m). Two walls (solid concrete and cinderblock with block length 0.4 m) were built up in front of

Fig. 9. Multistatic time-difference radar image of building of Fig. 5 for co-polarized (top) and cross-polarized (bottom) Rx antennas.

Fig. 10. Combined multistatic co- and cross-polarized time-difference radar image of Fig. 9.

the radar system (1.7 m away from the antennas) and the scene was terminated on the side by absorbers. A full body plastic mannequin was used as a human target. In order to emulate the high dielectric constant of the body, the mannequin was painted with graphite based conductive paint (transmittivity of less than 30 dB of one layer of paint with 10% absorption [23]). Fig. 11 shows the described measurement setup in which the Tx and Rx antenna in front of the cinderblock wall can be seen in the left picture and the mannequin standing 0.5 m behind the wall in the right picture. First a 2D image of the solid concrete wall (thickness 9 cm) with the mannequin 2.4 m behind it was created. The -table dimensions limited the array size to 1 m so 10 array points were

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Fig. 11. Measurment setup with antennas in front of cinderblock wall and mannequin behind cinderblock wall.

Fig. 13. Multistatic radar image of cinderblock wall with a human behind it (a) and time-difference radar image of the wall and human after the human is moved 20 cm (b) for co-polarization in dB, normalized to max. of wall return.

Fig. 12. Multistatic radar images of a solid wall with a mannequin 2.4 m behind it generated with measured (a) and simulated data (b) for co-polarization in dB, normalized to max. of wall return.

measured between 1.0 GHz and 2.0 GHz with 201 frequency points. The wall was 1.7 m away from the Tx and Rx. The same scene was recreated with the bricktracer, it consists of a 9 cm thick solid concrete wall (length 3 m, height 2 m with and assumed), two low dielectric walls to the side to the emulate the absorbers, a concrete floor and ceiling and one male human body model resembling the manand nequin (simulated as a lossy metal surface with ). Simulations were also carried out between 1.0 GHz and 2.0 GHz. Fig. 12 shows the measured and simulated images of the human subject behind the concrete wall. The wall return along with multi-path caused by the laboratory floor and ceiling are well predicted by the simulation results. However, additional multipath caused by the surrounding objects in the lab are not included in the simulation and thus are not visible. Because the simulation does not show this clutter, the mannequin can be identified in Fig. 12(b) at 4 m 20 dB below the wall return. However, as shown in Fig. 12(a) the measured human return is mixed with the additional clutter return. Finally, an analogous 2D image of a cinderblock wall (thickness 20 cm) with human 2.4 m behind it was created. Fig. 13 shows the image of the wall with the human behind and the

time-difference image with the human moved in -direction by 20 cm, both normalized on the maximum of the wall return and windowed by a Hamming function. In the single image, the specular front-wall and back-wall reflection stands out again at (1.7, 0.0) m but is surrounded by heavy clutter to both sides in -direction caused by the Bragg modes of the periodic wall (compare to the solid concrete wall of Fig. 12. Because of the higher attenuation through the cinderblock wall, the human cannot be identified within all the clutter behind the wall. However the difference image reduces the stationary scene return by more than 50 dB and the difference signal of the human appears at (4.5, 0.0) m 37 dB below the wall return (shifted back farther than behind the solid wall because of the thicker wall). It should be noted that the periodic wall does not degrade the imaging and localization of the human in this case. The direction and phase of the additional Bragg modes are frequency dependent and therefore add mostly incoherently at the Rx location. Only if the Tx or the target is very close the wall the Bragg modes of a finite wall distort the image of the target a significant amount because the further away the Tx or target from the wall, the smaller the opening angle under which the Tx/target sees the wall, leaving out some Bragg modes. In general periodic walls are not a concern of through-wall imaging if the frequency is kept low enough to minimize the Bragg modes. VI. REALISTIC DETECTION SCENARIO In order to prove the concept of human detection for realistic scenarios, a complex building setup with a smaller Rx array was considered. The buildings spans 11.0 by 6.5 m and consists of four rooms, each with a door to the hallway and a window on the opposite walls. All walls are homogeneous concrete ( and ). The receiver array is 4.0 m long with a Rx spacing of 0.1 m and Tx in the middle, sits 1.3 m above the ground and simulations were carried out from 0.5 to 1.4 GHz in 0.005 GHz steps. Fig. 14 shows the calculated radar image of the building overlaid with its layout and Tx and Rx positions

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Fig. 16. Combined normalized multistatic co- and cross-polarized time-difference radar image of Fig. 15. Fig. 14. Simulated co-polarized backscattering of building overlayed with the building footprint (black dotted line), normalized on its maximum. The solid black line indicates the Rx array with the Tx in the center.

Fig. 15. Simulated co-polarized (left) and cross-polarized (right) time-difference image of Fig. 14 normalized on the max. of the building return.

marked. The Rx array only records specular wall reflections from the wall at the center in front of it, otherwise the only major contribution of the building at the receivers are the building corners. Two walking humans are in the building, one in the upper right and one in the lower left room (both marked with a solid black line). The time-difference images eliminate the wall response and the humans appear 32 dB and 42 dB for co-pol., 35 dB and 51 dB for cross-pol. below the maximum building return. Because of the complex building layout, the signal delay through the walls is not equal for every Rx point, which distorts the return of both humans. Both humans are accompanied by various ghost images and the ghost images of the human in the front room are stronger than the return of the human in the rear room. The combined co- and cross-pol. images can eliminate some ghost images (Fig. 16). Nevertheless, without any knowledge of the building layout and the human path, these ghost images lead to false detections of additional humans. VII. CONCLUSION A hybrid simulation model of wideband polarimetric radar backscattering of humans within a building was introduced. The model is composed of a full-wave simulation tool that can accurately predict the scattering from a realistic human model, and an accurate indoor wave propagation and scattering model,

known as bricktracer. These two models are efficiently connected using a Huygen’s interface box that encloses the entire human body. Sample backscattering results for a single human, a human behind a reinforced concrete wall and a human walking in a realistic building were presented and the simulation results were confirmed by actual radar measurements of a mannequin behind a wall. It was shown that the backscatter response of human can be completely obscured by the building backscatter clutter. Timesequenced subtraction of the backscattered responses using a stationary transceiver was evaluated to detect a moving human and a multistatic radar imaging setup was proposed for fast localization of human movement inside buildings. The system’s ability to detect and localize human movement within buildings was verified through simulation and measurement. Furthermore, the high cross-polarization backscattering of the human body allows additional use of cross-pol. measurements to enhance human detection and localization. REFERENCES [1] M. Dehmollaian and K. Sarabandi, “Refocusing through building walls using synthetic aperture radar,” IEEE Trans. Geosci. Remote Sens., vol. 46, no. 6, pp. 1589–1599, 2008. [2] C. Debes, M. G. Amin, and A. M. Zoubir, “Target detection in singleand multiple-view through-the-wall radar imaging,” IEEE Trans. Geosci. Remote Sens., vol. 47, no. 5, pp. 1349–1361, 2009. [3] H. Yacoub and T. K. Sarkar, “A homomorphic approach for throughwall sensing,” IEEE Trans. Geosci. Remote Sens., vol. 47, no. 5, pp. 1318–1327, 2009. [4] K. M. Yemelyanov, N. Engheta, A. Hoorfar, and J. A. McVay, “Adaptive polarization contrast techniques for through-wall microwave imaging applications,” IEEE Trans. Geosci. Remote Sens., vol. 47, no. 5, pp. 1362–1374, 2009. [5] W. Genyuan and M. G. Amin, “Imaging through unknown walls using different standoff distances,” IEEE Trans. Signal Processing, vol. 54, no. 10, pp. 4015–4025, 2006. [6] Y.-S. Yoon and M. G. Amin, “High-resolution through-the-wall radar imaging using beamspace MUSIC,” IEEE Trans. Antennas Propag., vol. 56, no. 6, pp. 1763–1774, 2008. [7] P. C. Chang, R. J. Burkholder, J. L. Volakis, R. J. Marhefka, and Y. Bayram, “High-frequency em characterization of through-wall building imaging,” IEEE Trans. Geosci. Remote Sens., vol. 47, no. 5, pp. 1375–1387, 2009. [8] N. Maaref, P. Millot, P. Pichot, and O. Picon, “A study of UWB FM-CW radar for the detection of human beings in motion inside a building,” IEEE Trans. Geosci. Remote Sens., vol. 47, no. 5, pp. 1297–1300, 2009.

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[9] S. Z. Gilrbilzl, W. L. Melvin, and D. B. Williams, “Comparison of radar-basedhuman detection techniques,” in Proc. 41st Asilomar Conf. on Signals, Systems and Computers (ACSSC 2007), 2007, pp. 2199–2203. [10] A. G. Yarovoy, L. Ligthart, J. Matuzas, and B. Levitas, “Uwb radar for human being detection,” IEEE Aerosp. Electron. Syst. Mag., vol. 21, pp. 22–26, 2006. [11] L. M. Frazier, “Radar surveillance through solid materials,” in Proc. SPIE, 1997, vol. 2938, pp. 139–146. [12] J. David, D. Ferris, and N. C. Currie, “Survey of current technologies for through-the-wall surveillance (TWS),” in Proc. SPIE, 1999, vol. 3577, pp. 62–72. [13] T. Dogaru and C. Le, “Sar images of rooms and buildings based on FDTD computer models,” IEEE Trans. Geosci. Remote Sens., vol. 47, no. 5, pp. 1388–1401, 2009. [14] R. F. Harrington, Time-Harmonic Electromagnetic Fields. New York: McGraw-Hill, 1961. [15] N. Bliznyuk, N. Engheta, and A. Hoorfar, “An efficient hybrid computational technique for certain scattering and radiation problems in layered media,” in Proc. IEEE Aerospace Conf., 2004, pp. 1040–1052. [16] M. Thiel and K. Sarabandi, “An hybrid method for indoor wave propagation modeling,” IEEE Trans. Antennas Propag., vol. 56, no. 8, pp. 2703–2709, 2008. [17] M. Thiel and K. Sarabandi, “3D-wave propagation analysis of indoor wireless channels utilizing hybrid methods,” IEEE Trans. Antennas Propag., May 2009. [18] An Internet Resource for the Calculation of the Dielectric Properties of Human Body Tissues [Online]. Available: http://niremf.ifac.cnr.it/ tissprop/ 2007 [19] Virtual Family Models IT’IS Foundation, 2008. [20] SEMCAD X Schmid & Partner Engineering AG, 2008. [21] MakeHuman Project [Online]. Available: http://www.makehuman.org/ 2008 [22] B. Michael, W. Menzel, and A. Gronau, “A real-time close-range imaging system with fixed antennas,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 12, pp. 2736–2741, 2000. [23] [Online]. Available: http://www.lessemf.com/paint.html Michael Thiel (S’07–M’10) received the Dipl.-Ing. degree in electrical engineering from the University of Ulm, Ulm, Germany, in 2005 and the Ph.D. degree in electrical engineering from The University of Michigan, Ann Arbor, in 2010. Previously, he was a Graduate Student Research Assistant with the Radiation Laboratory, The University of Michigan. Since 2010, he is with Schlumberger-Doll Research, Cambridge, MA. His research interests are indoor wave propagation modeling, through-the-wall radar imaging and electromagnetic field theory and inversion.

Kamal Sarabandi (S’87–M’90–SM’92–F’00) received the B.S. degree in electrical engineering from Sharif University of Technology, Iran, in 1980, dual M.S. degrees in electrical engineering and mathematics, both in 1986, and the Ph.D. degree in electrical engineering in 1989 from The University of Michigan, Ann Arbor. He is Director of the Radiation Laboratory and a Professor in the Department of Electrical Engineering and Computer Science, University of Michigan. His research areas of interest include microwave and millimeter-wave radar remote sensing, Metamaterials, electromagnetic wave propagation, and antenna miniaturization. He has 22 years of experience with wave propagation in random media, communication channel modeling, microwave sensors, and radar systems and is leading a large research group including two research scientists, 12 Ph.D. and 2 M.S. students. He has graduated 30 Ph.D. and supervised numerous postdoctoral students. He has served as the Principal Investigator on many projects sponsored by NASA, JPL, ARO, ONR, ARL, NSF, DARPA and a larger number of industries. He has published many book chapters and more than 180 papers in refereed journals on miniaturized and on-chip antennas, metamaterials, electromagnetic scattering, wireless channel modeling, random media modeling, microwave measurement techniques, radar calibration, inverse scattering problems, and microwave sensors. He has had more than 420 papers and invited presentations in many national and international conferences and symposia on similar subjects. Dr. Sarabandi was appointed by the NASA Administrator as a member of the NASA Advisory Council. He also served as a Vice President of the IEEE Geoscience and Remote Sensing Society (GRSS) and as a member of the IEEE Technical Activities Board Awards Committee. He is serving on the Editorial Board of the IEEE Proceedings, and served as Associate Editor of the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION and the IEEE SENSORS JOURNAL. He is a member of Commissions F and D of URSI and is listed in American Men & Women of Science Who’s Who in America and Who’s Who in Science and Engineering. He was the recipient of the Henry Russel Award from the Regent of The University of Michigan. In 1999, he received a GAAC Distinguished Lecturer Award from the German Federal Ministry for Education, Science, and Technology given to about ten individuals worldwide in all areas of engineering, science, medicine, and law. He was also a recipient of the 1996 EECS Department Teaching Excellence Award and a 2004 College of Engineering Research Excellence Award. In 2005, he received two prestigious awards, namely, the IEEE GRSS Distinguished Achievement Award and the University of Michigan Faculty Recognition Award. He also received the best paper Award at the 2006 Army Science Conference. In 2008, he was awarded a Humboldt Research Award from The Alexander von Humboldt Foundation of Germany granted to scientists and scholars in all disciplines with internationally recognized academic qualifications. In the past several years, joint papers presented by his students at a number of international symposia (IEEE APS’95,’97,’00,’01,’03,’05–’07; IEEE IGARSS’99,’02,’07; IEEE IMS’01, USNC URSI’04–’06, AMTA ’06, URSI GA 2008) have received student paper awards.

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Analytical Propagation Modeling of BAN Channels Based on the Creeping-Wave Theory Thierry Alves, Benoît Poussot, and Jean-Marc Laheurte

Abstract—The first analytical model of Body Area Network channels is presented. The formulation includes the body morphology and the characteristics of the human tissues. The studied transmission paths are along curved parts like the waist or the head. The model is derived from the diffraction theory describing the attenuation of creeping waves along a circular path on a lossy dielectric surface. The model is validated by measurements performed with Planar Inverted-F Antennas on human subjects. Index Terms—Body area network (BAN), creeping waves, small antennas.

I. INTRODUCTION

W

IRELESS Body Area Networks (BAN) require body worn antennas with low power devices to limit biological interactions with microwaves and to save battery life. In order to optimize the power consumption, it is crucial to integrate all propagation mechanisms into appropriate BAN channel models for a proper estimation of the link budgets. A channel model includes statistical and deterministic parts. The statistical part is linked to variations of the arm/leg/head positions in walking/running scenarios or the displacements of antennas with respect to the body, due to breathing or slight displacement of clothes for instance. The statistical evaluation of the BAN channels is beyond the scope of this paper. The deterministic part of the channel model describes the path loss in a stationary environment obtained with a standing position and no movement in an anechoic chamber. It can be seen as the mean value of the power distribution in a dynamic environment [1]. This deterministic part can either be determined from electromagnetic considerations [2]–[5], extracted from numerical simulations [6] or measurement campaigns [7]–[9]. Several investigations [2], [3] are based on the resolution of the wave equation around a lossy circular cylinder for canonical sources (dipole, current line) but result in complex formulations requiring extra computations. In [4], [5], Fort and Keshmiri compute the path loss at different frequencies using a cylinder radius of 15 cm showing a good agreement with their own measurements. Other papers propose to extract a model from finite-difference time-domain (FDTD) simulations. In [6], Ryckaert developed a FDTD-based link budget model valid around

Manuscript received October 22, 2009; revised June 17, 2010; accepted July 06, 2010. Date of publication December 03, 2010; date of current version April 06, 2011. The authors are with the ESYCOM, Universite Paris-Est Copernic, Marne-laVallee cedex 2, 77454, France (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.2010.2096184

torso at 400, 900 and 2400 MHz. Simulations were carried out on numerical phantoms based on the Visual Human Project. In [8], the path gain around the waist is measured and a fitting model is extracted. However, fitting models extracted from [6] and [8] are only valid for one particular person and suffer from a lack of parametric dependence on the body morphology or the antenna/propagation characteristics. The goal of this paper is to develop an analytical formulation of the channel model that is easier to handle by the user than time consuming FDTD simulations. One approach could be based on the unified theory of diffraction (UTD) [10] which allows for any kind of source and can be applied to any convex body. A more limited one is based on the creeping-wave theory valid for propagation along quasi-circular paths. The creeping-wave theory can only be applied to cases where transmit and receive antennas are located at the same cross-section of the waist, the trunk or the head. This paper restricts the analysis to this configuration. The development of analytical path gain expressions valid for more complex curvatures are left to further studies based on the UTD. The assumption that cylinders can be used to model different body parts is validated by measurements performed with Planar Inverted-F Antennas (PIFA) on human subjects. The analytical formulation of the propagation path gain includes parameters such as the dielectric and conductive properties of the human tissues, the gain and polarization of the antenna and the perimeter of the body part under consideration. II. DETERMINATION OF THE PATH GAIN IN THE BAN CONTEXT A. Dominant Propagation Mode at Microwave Frequencies At microwaves frequencies, the human body is mainly a lowloss dielectric medium with . An approximation for the in-body attenuation constant is then given by (1) are respectively the free space impedance, the where body conductivity and the body relative permittivity. At 2.4 GHz, electric parameters commonly used to model a uniform body are around 50 for and 1.7 S/m for [8]. Let us consider the direct path across the body between two opposite points on the waist separated by 30 cm. Using (1), the attenuation is approximately 120 dB. On the other hand, the attenuation amplitude measured in an anechoic chamber, i.e., when possible reflections on the ground or walls are omitted, ranges from 50 to 60 dB [8]. This result shows that surface waves along the body are the only propagation mechanism to consider at 2.4 GHz.

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In [13], is expressed as a sum of modes known as creeping can be reduced to the dominant waves. In the BAN context, creeping mode

(4) where (5a) (5b)

Fig. 1. Path gain around the head for the vertical (plain) and horizontal (dashed) polarizations.

B. Propagation of Creeping Waves on the Body Surface In this part, the creeping-wave theory is rewritten in the context of the body surface at microwave frequencies. The creeping-wave theory initially developed for propagation models around the earth was applied to circular paths. Therefore, it can be transferred to the propagation around the circular section of a cylinder as pointed out by Wait in [11]. A cylinder can also support Norton wave modes (i.e., surface waves) in its longitudinal direction. Norton waves are beyond the scope of this paper but should be included if the communication is established between two devices at different cross-sections. Throughout this work, the wave polarization is called vertical (resp. horizontal) for an electric field normal (resp. tangential) to the body surface. The vertical component of the electric field with respect to the surface is much less attenuated than the horizontal component [12] as will be shown in Fig. 1. Therefore, we will focus the study on the vertical polarization only. In the following mathematical developments, all complex values are underlined. Let us assume a circular path around a cylinder characterized by its radius and its complex permittivity . The electric field at a distance from the transmitting antenna is expressed as [13]

is obtained by applying The simplified expression (4) for the following assumptions valid for the body surface: the perimeter fulfils the condition [14]. The minimum perimeter under consideration in the BAN cm. As context is the mean perimeter of adult heads cm, the ratio defined above is 5. It will be shown in the experimental section that this ratio is large enough to apply (4) successfully. the height of the antennas above the surface is small. is much larger than unity. and the gain of the Introducing the received power receiving antenna, it is then straightforward to show that the path gain is expressed as

(6) where the two terms in brackets correspond to the interference between the direct wave travelling clockwise and the indirect travelling counter-clockwise around the cylinder. is wave the attenuation factor in Np/m given by (7) The expression of the decaying factor

is in the form of:

(2) (8)

with (3) is the field created at a distance by the same antenna located is the antenna gain, above a perfectly conducting plane. is the feeding power in watts and is the attenuation function which depends both on the wave polarization and the is the wave number electric properties of the surface. in free space.

K-values are directly linked to the impedance of the body surface. For instance, K is equal to 0.36 for and 0.39 . Refinements in the determination of the surface for impedance can be based on a multilayered model of the body including the thickness and the electrical features of the skin and the fat [15]. In this case, K-values range from 0.39 to 0.52. One can note that the decaying factor decreases when the curvature radius increases. In other words, will be higher for a link around the head than a link around the waist.

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Fig. 2. Set of two PIFAs used in the measurement campaign.

Equation (6) is plotted in dB in Fig. 1. The interference between the direct and indirect paths is important around the antipode where the field amplitudes are equivalent. At the antipode, a small phase change can cause fadings as large as 5 dB or more. In practical situations, this phase variation can be due to breathing ( varies with the thoracic extensions). Therefore, it is preferable not to locate antennas at the antipodes. For the sake of completeness, a comparison is made with path losses calculated for horizontally polarized waves (E-field parallel to the body surface). Path loss formulas for the horizontal polarization are obtained by using mathematical developments given in [16]. Fig. 1 clearly shows the inefficiency of the horizontal polarization for on-body links, with differences of at least 40 dB observed when compared to the vertical polarization. This confirms that BAN antennas must be vertically polarized to optimize the link budget. III. ANTENNA DESIGN In order to match the characteristics of the creeping waves, the on-body antennas must be vertically polarized with an end-fire pattern along the body surface. A monopole antenna is suitable but unpractical in the BAN context because reduced heights are required. On the other hand, PIFAs are good candidates with an azimuthal radiation characterized by an omnidirectional pattern and ). and no dominant polarization (combination of A PIFA is equivalent to a parallel LC circuit with a small topand a short-circuited section (Fig. 2). loaded monopole The resonant frequency depends on the length of the upper plate and the width of the short-circuited section. The PIFAs realized to test the channel are 5 mm-thick with a reduced ground plane (25 mm 37 mm). The antennas are fed by a coplanar line to keep flat features. The coplanar line is ended by a vertical feeding probe connected to the upper plate. Matching is achieved by suitably locating the feeding probe. dB in a 5% bandThe measured return loss is lower than width around 2.4 GHz. The antenna matching is weakly affected by the body because of the masking effect of the ground plane which reduces the field intensity in the human tissues in comparison to a printed Inverted-F Antenna. IV. ON-BODY PATTERN SIMULATIONS AND GAIN ESTIMATION The experimental evaluation of the antenna gain is a difficult task in the presence of the body. If real human beings are used,

Fig. 3. (a) Free space PIFA. E' (dashed line) and E (plain line) components in the x0y plane. (b) PIFA located 5 mm above the head surface. E' (dashed line) and E (plain line) components in the x0y plane tangent to the head surface. The cylinder perimeter is 60 cm. (c) PIFA located 5 mm above the waist surface. E' (dashed line) and E (plain line) in the x0y plane tangent to the waist surface. The cylinder perimeter is 92 cm.

repeatability is not easily obtained. If liquid-filled phantoms are used, the constitutive parameters of the shell are very different from the human tissues and might alter the validity of the results. In any case, the exact direction of the azimuthal gain is difficult to determine experimentally and it was decided to extract it from HFSS (High Frequency Structure Simulator) [17] simulations corroborated by Empire simulations [18]. The head is modeled by a lossy dielectric cylinder while the waist phantom consists in a lossy dielectric flattened cylinder. The PIFA is placed 5 mm above the body surface.

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For the cartesian coordinates system of Fig. 2, the simulated radiation patterns are plotted in the x0y plane for three cases: — in free space (Fig. 3(a)); — on the head phantom (Fig. 3(b)) with the x0y plane tangent to the head surface; — on the waist phantom (Fig. 3(c)) with the x0y plane tangent to the waist surface. corresponds to the vertical poIn the x0y plane to the horizontal polarization. The azimuthal larization and which are the directions of the gains are given for direct and indirect paths between the facing PIFAs when placed on the body surface. and components is dominant. In free space, none of the pattern is almost uniform. The azimuthal gain is dBi. The The computed antenna efficiency is 97%. In presence of the body (head or waist), the dominant E-field which indicates that the PIFA is vertically component is polarized. This flows from the strong attenuation of the horizontally polarized fields on the body surface. In both cases, the antenna efficiency drops to 28% because a large part of the near-field is concentrated inside the lossy body. dBi and dBi. In the head case, the azimuthal gains are These values are close to those observed in the free-space case. dBi. The In the waist case, the maximum gain drops to gain is almost 3.5 dB higher for the head than for the waist. This difference is not due to variations in tissue parameters between the head and the waist, only to perimeter changes as proven through various parametric studies.

Fig. 4. On-body measurement setup.

V. MEASUREMENT CAMPAIGN A. Instrumentation Setup and Measurement Procedure For the signal acquisition, the Agilent network analyzer E8361C PNA is used as a narrow band receiver with an IF bandwidth set to 10 kHz. A set of 20001 samples is recorded during 60 s (1 sample each 3 ms). A continuous wave generator synchronized with the network analyzer is used as a transmitter. The transmitting power is set to 0 dBm. In order to adjust the antennas on the body, the transmitting PIFA is fixed on a large elastic belt while the receiving PIFA can slide on it. The transmitting PIFA is located on the belly button. Flexible cables RG-316, (1.5 dB/m loss at 2.4 GHz) are used to facilitate the body movements. Repeatability is assured by holding the cables with adhesive tapes. In small antennas with unbalanced feedings and limited ground planes, currents might flow along the outer parts of the coaxial cable and radiate. This will obviously affect measurements. Parasitic currents are greatly reduced by inserting a cylindrical ferrite rod on the cable, close to the SMA connector. Placing the TX and RX cables orthogonally and far from each other also reduces coupling (Fig. 4). The measurements are done in an anechoic chamber. The person under test is placed on a small plastic support with absorbers put on the ground to avoid reflections from it. During the measurement campaign, hands are kept away from the path link to avoid reflections and reduce interferences. Movements should

Fig. 5. Measured signal dynamic along the waist.

be reduced to a minimum. The distance between two consecutive positions of the receiving antenna along the belt is 5 cm. The PIFAs face each other with almost identical gains. B. Extraction of the Deterministic Path Loss To determine the deterministic path loss, post processing of measured data is required to eliminate fading resulting from small movements and breathing. For a given link, a set of N files is recorded corresponding to N antenna positions. Each file contains 20001 samples of received power. Fig. 5 shows that the signal dynamic (defined as the difference between the maximum and minimum value in a data file) increases with the distance between antennas and shows strong variations at the antipode. It has been observed that the dynamic is even larger for higher perimeters. All types of fading are filtered efficiently by averaging the samples for each antenna position. C. Comparison With the Theoretical Model Measurements are firstly presented for a perimeter value of 60 cm (head) and the path gain expression (6) expressed in dB.

ALVES et al.: ANALYTICAL PROPAGATION MODELING OF BAN CHANNELS BASED ON THE CREEPING-WAVE THEORY

Fig. 6. Measurements and path gain model for the head, p = 60 cm.

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Fig. 8. Measurements and path gain model for the waist, p = 92 cm.

head and the waist for different perimeters. The accuracy of the procedure is based on a proper estimation of the antenna gain which must be improved by using more sophisticated numerical phantoms. The path gain equation developed in this paper can be extended to any deterministic model of BAN channels around circular paths on the body. In case the communication occurs between different cross-sections of the body, refinements must be added by including the longitudinal surface modes in the study or by using the UTD [10]. REFERENCES

Fig. 7. Measurements and path gain model for the waist, p = 70 cm.

This expression can only be applied if the gain value in the direction of interest is known. Fig. 6 shows an excellent agreement between our model and measurements when the simulated azdB is used. It appears that the signal interferimuthal gain ence at the antipode is hardly measurable as spatial variations of the amplitude are in the order of the antenna dimensions. Two path gain measurements are also presented for the waist case. One is conducted on a female with a waist perimeter of 70 cm (Fig. 7). The second one is measured on a male with a perimeter of 92 cm (Fig. 8). Good agreement with theory is again observed by setting the antenna azimuthal gain to dBi which is the value obtained with HFSS simulations. VI. CONCLUSION A simple analytical path gain model for BAN was derived from the creeping-wave theory. It was shown that the dominant mode is sufficient for the curvatures under consideration in the BAN context. The strong attenuation of horizontally polarized creeping waves was first controlled to match the antennas polarization and pattern to the dominant vertically polarized creeping waves. Then, an experimental procedure was developed to check the validity of the model for two links around the

[1] F. F. Pérez and E. P. Mariño, Modeling the Wireless Propagation Channel. New York: Wiley, 2008. [2] A. Gupta and T. D. Abhayapala, “Body area networks: Radio channel modelling and propagation characteristics,” in Proc. 9th Australian Communications Theory Workshop, 2008, pp. 58–63. [3] D. Ma and W. X. Zhang, “Analytic propagation model for body area network channel based on impedance boundary condition,” in Proc. Eur. Conf. on Antennas and Propagation (EuCAP), Berlin, Mar. 23–27, 2009, pp. 974–978. [4] F. Keshmiri and A. Fort, “Analysis of wave propagation for BAN applications,” in Proc. Eur. Conf. on Antennas and Propagation (EuCAP), Berlin, Mar. 23–27, 2009, pp. 709–712. [5] A. Fort, F. Keshmiri, G. R. Crusats, C. Craeye, and C. Oestges, “A body area propagation model derived from fundamental principles: Analytical analysis and comparison with measurements,” IEEE Trans. Antennas Propag., vol. AP-58, pp. 503–514, Feb. 2010. [6] J. Ryckaert and P. De Doncker, “Channel model for wireless communication around human body,” Electron. Lett., vol. 40, no. 9, pp. 543–544, Apr. 2004. [7] E. Reusens and W. Joseph, “On-body measurements and characterization of wireless communication channel for arm and torso of human,” in Proc. 4th Int. Workshop on Wearable and Implantable Body Sensor Networks (BSN 2007), Aachen, Mar. 26–28, 2007, pp. 264–269. [8] P. S. Hall and Y. Hao, “Antennas and propagation for on-body communication systems,” IEEE Antennas Propag. Mag., vol. 49, no. 3, pp. 41–58, Jun. 2007. [9] H. Ghannoum, C. Roblin, and X. Begaud, “Investigation and modeling of the UWB on-body propagation channel,” in Wireless Personal Communications Springer, May 2010, vol. 52, no. 1, pp. 17–28. [10] P. H. Pathak, N. Wang, W. D. Burnside, and R. G. Kouyoumjian, “A uniform GTD solution for the radiation from sources on a convex surface,” IEEE Trans. Antennas Propag., vol. AP-29, no. 4, pp. 609–622, Jul. 1981. [11] J. R. Wait, “On the excitation of electromagnetic surface waves on a curved surface,” IRE Trans. Antennas Propag., pp. 445–448, Jul. 1960. [12] J.-C. Pelissolo, Propagation des ondes radioélectriques. Fasc.1: bases théoriques, rôle et influence du sol. Paris: Ecole Nationale Supérieure de Techniques Avancées, 1970.

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[13] J. R. Wait, “The ancient and modern history of EM ground-wave propagation,” IEEE Antennas Propag. Mag., vol. 40, no. 5, pp. 7–24, Oct. 1998. [14] T. L. Eckersley, “Direct-ray broadcast transmission,” Proc. IRE, vol. 20, pp. 1555–1579, Oct. 1932. [15] J. R. Wait, Electromagnetic Waves in Stratified Media. New York: Pergamon Press, 1962. [16] H. Bremmer, “Applications of operational calculus to ground-wave propagation, particularly for long waves,” IRE Trans. Antennas Propag., pp. 267–272, Jul. 1958. [17] 3D Full-wave Electromagnetic Field Simulation [Online]. Available: http://www.ansoft.com/products/hf/hfss/ [18] EMPIRE XCcel [Online]. Available: http://www.empire.de/

Benoît Poussot was born in Dijon, France, in 1979. He received the M.S. degree in microwave engineering and the Ph.D. degree in electrical engineering from the University of Paris-Est Marne-La-Vallée, France, in 2004 and 2007, respectively. He is a CNAM Engineer in “Conception et Production Industrielle.” Since 2008, he is a Research Assistant in the ESYCOM laboratory, University of Paris-Est Marne-La-Vallée. His research interests include parasitic antenna arrays analysis and modeling, diversity measurements, reconfigurable mm-wave antennas on silicon substrates, MEMs based active antennas and BAN antennas.

Thierry Alves was born in Bordeaux, France, in 1983. He received the M.S. degree in microelectronics from the University Bordeaux 1, France, in 2007. He is an ENSEIRB Engineer in electronics and radiocommunications. During a six-month training course in 2007, he was in charge of non-linearities measurements of RF power amplifiers in the INESC laboratory, Universidade do Porto, Portugal. Since December 2007, he is pursuing his Ph.D. degree at the ESYCOM laboratory, University of Paris-Est Marne-La-Vallée, where he is working on BAN antenna design and propagation channel characterization. Mr. Alves’ work is supported by the French ANR project BANET.

Jean-Marc Laheurte received the M.Sc. and Ph.D. degrees in electrical engineering and the Habilitation à Diriger les Recherches degree from the University of Nice, Nice, France, in 1989, 1992, and 1997, respectively. He was a Research Assistant in the EPF of Lausanne, Lausanne, Switzerland (1989–1990), a Postdoctoral Researcher at the University of Michigan at Ann Arbor (1992), and an Associate Professor at the University of Nice-Sophia Antipolis (1993–2002). Since 2002, he is a Professor at the University of Paris-Est Marne-La-Vallée where he is in charge of the antenna group of the ESYCOM laboratory and heads the Gaspard Monge Institute. His current research interests include the design and evaluation of diversity antennas for indoor communications, RFID systems, BAN antennas and BAN channel modelling. He is also active in the electromagnetic modeling of urban scatterers in mobile communications. He has authored or coauthored over 50 technical articles and 60 conference papers. He teaches in various European institutions in the frame of the European School of Antennas.

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Fast Optimization of Electromagnetic Design Problems Using the Covariance Matrix Adaptation Evolutionary Strategy Micah D. Gregory, Student Member, IEEE, Zikri Bayraktar, Student Member, IEEE, and Douglas H. Werner, Fellow, IEEE

Abstract—A new method of optimization recently made popular in the evolutionary computation (EC) community is introduced and applied to several electromagnetics design problems. First, a functional overview of the covariance matrix adaptation evolutionary strategy (CMA-ES) is provided. Then, CMA-ES is critiqued alongside a conventional particle swarm optimization (PSO) algorithm via the design of a wideband stacked-patch antenna. Finally, the two algorithms are employed for the design of small to moderate size aperiodic ultrawideband antenna array layouts (up to 100 elements). The results of the two electromagnetics design problems illustrate the ability of CMA-ES to provide a robust, fast and user-friendly alternative to more conventional optimization strategies such as PSO. Moreover, the ultrawideband array designs that were created using CMA-ES are seen to exhibit performances surpassing the best examples that have been reported in recent literature. Index Terms—Covariance matrix adaptation, evolutionary strategy, microstrip antennas, optimization algorithm, particle swarm optimization, self-adaptive, stacked patch antenna.

I. INTRODUCTION

W

HILE the computing platforms and simulation tools applied to electromagnetics (EM) design optimization have experienced steady advancement over the years, the optimization algorithms that are used have remained largely the same. Genetic algorithms (GAs), particle swarm optimization (PSO), differential evolution (DE) and related techniques dominate as the mainstream evolutionary strategies (ES), mostly due to widespread availability, understanding and user confidence [1]–[11]. These optimization techniques have enjoyed a great deal of success and have been applied in a wide variety of electromagnetic device design problems including antennas [12]–[19], antenna arrays [20]–[29], frequency selective surfaces [30]–[36], filters [37] and [38] and many others. The utility of the algorithms allows application to any EM design problem that can be reduced to a simple set of defining parameters.

Manuscript received September 27, 2009; revised May 29, 2010; accepted September 09, 2010. Date of publication January 28, 2011; date of current version April 06, 2011. The authors are with the Electrical Engineering Department, The Pennsylvania State University, University Park, PA 16802 USA (e-mail: mdg243@psu. edu; [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.2011.2109350

Whereas the electromagnetics community has been producing fruitful designs from these optimization tools, the evolutionary computation (EC) community has been creating more powerful, more efficient and more user-friendly algorithms. These new algorithms address issues such as self-adaptation of evolutionary settings and learning of complex variable dependency or inseparability. Many of the current difficulties with using GAs, PSO, DE and related techniques lie in the evolutionary settings of the algorithms. These settings play a vital role in the productivity of the optimization. For example, the user of a GA may experience quite different optimization results and variations in speed and success by choosing different crossover and mutation probabilities [28]. An ES that requires few user settings offers the advantage of avoiding any human choices that impact the performance of the algorithm. For this reason, the EC community has invested a great deal of research into self-adaptive strategies [39]–[41]. These algorithms automatically adjust their parameter settings based on available evolutionary information, avoiding any issues with human interaction. An additional advantage they offer is that the best parameter choices may change throughout the optimization and is accounted for in properly designed, self-adaptive strategies. Optimization problems with complex variable dependency (e.g., inseparable solution spaces) are difficult to solve for most evolutionary strategies. Modern ES tackle this issue by attempting to learn the dependencies between optimization parameters as the evolution progresses, enabling the algorithm to choose future solutions that are most likely to better solve the problem [42]. Although more computational overhead is typically associated with these advanced evolutionary strategies, for complex problems there is a significant benefit via a reduction in the number of function evaluations required to find a suitable solution. For electromagnetics problems, the function evaluation (e.g., full-wave simulation) time almost always trumps the computational time required by the evolutionary strategy, often by many orders of magnitude. Of the recent strategies proposed by the evolutionary computation community, the covariance matrix adaptation evolutionary strategy (CMA-ES), stands out as an excellent performing, self-adaptive algorithm with very few settings required by the user [43]–[46]. It has been shown to work well on a large variety of test problems and applications [47]–[49]. In an effort to reduce optimization time and enhance the quality of electromagnetic designs, it will be applied here to problems

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TABLE I INTERNAL AND EXTERNAL EVOLUTIONARY PARAMETERS AND THEIR DEFINITIONS FOR THE CMA-ES

Fig. 1. Operational diagram of the CMA-ES.

that are typically reserved for PSO [1]–[5] or a GA [6]–[9]. In the following, the CMA-ES will be compared to a standard PSO algorithm using several test functions, a stacked-patch antenna design problem and the optimization of ultrawideband (UWB) linear antenna array layouts. The proposed ultrawideband array design method, where each element spacing in the linear array is controlled by an optimizable parameter, allows for relatively compact arrays with performance to array size ratios not possible with other UWB design methods. This method of UWB array design requires a relatively large parameter set, however, it can lead to very high performance designs with a competent optimization tool such as the CMA-ES. When compared to several published examples of compact linear aperiodic arrays, this method offers much lower sidelobe levels for equivalent array sizes. II. CMA-ES ALGORITHM OVERVIEW The covariance matrix adaptation evolutionary strategy is a relatively new algorithm, introduced by Hansen and Ostermeier -ES and CSA-ES, it is a in 2001 [43]. Like the earlier population-based strategy that operates by moving the population, in the form of a multivariate normal distribution, around the search space [50] and [51]. The sampling distribution takes the form of a hyper-ellipsoid, where the edge of the hyper-ellipsoid represents equal likeliness of selection. The CMA-ES differs from the other Normal distribution-based algorithms through the way in which the distribution moves and is constantly reshaped (adaptive techniques) as well as how the search history is maintained and exploited. A simplified functional diagram of the CMA-ES is shown in Fig. 1. The terms and definitions listed in Table I are used to describe the functions of the algorithm. As mentioned, the CMA-ES functions by moving a search distribution about the parameter space. An example 2-D parameter space (horizontal and vertical variables) is shown in Fig. 3, where the algorithm is operating on the function shown in Fig. 2.

Fig. 2. Surface plot of the 2-D rotated hyper-ellipse test function used for demonstration of the algorithm in Figs. 3–5. The global function minimum lies at (x ; x ) = (0; 0).

The sampling distribution, shown by various size ellipses, is completely described by the multivariate Normal (Gaussian) distribution given in

(1) where can be broken apart into its eigenvectors and eigenvalues as in

(2) to allow for the numerical sampling of . The main purpose of the CMA-ES is productive control of the movement, orientation and size of this distribution in an effort to find the parametric location of lowest function value (or cost). For the example shown in Fig. 3, the strategy is moving this distribution almost directly to the function minimum while simultaneously refining the search (shrinking the search ellipse ). The algorithm uses knowledge of the function landscape

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Fig. 3. The movement of the mean and transformation of the search distribution as the CMA-ES operates with a large population on a two-dimensional rotated hyper-ellipse test function in Fig. 2 (log scaled). The initial mean is configured to (30, 30). Each ellipse represents the single- isocontour of the bi-variate Gaussian distribution at each iteration, giving insight into where the algorithm is searching for solutions. The dashed lines leading from the mean to their respective ellipses are the principle components of the covariance matrix. Small populations lead to more sporadic movement of the mean due to a less reliable estimation of the search space, however, the result is often fewer total function evaluations.

to best determine where to move the distribution mean and how to adjust the search scope and direction (ellipse shape). An ellipse is used to describe the Normal distribution since it can represent a contour of equal likelihood of selection (e.g., a standard deviation) in any direction from the mean. For three dimensions, it takes the form of an ellipsoid; a hyper-ellipse for more than three dimensions. A special feature of the CMA-ES is that, unlike many other distribution-based algorithms, the axes of the ellipse are not necessarily in fixed alignment with the parameters (i.e., non-axis-aligned ellipse) through use of a full covariance matrix rather than a simple variance vector (or covariance matrix with only diagonal terms) [43]. The CMA-ES is initialized mainly by choosing a population . The initial distribution mean size and setting , is usually selected at random in the parameter space unless the user has prior knowledge of the problem and wishes to exploit it. Typically, the initial search standard deviation is of the range of each parameter. Given the optiset to , , set to mizable parameter ranges . Then, set the diagonal elements of to . An example is given below for a three dimensional case

(3)

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Fig. 4. Illustration of the selection process used in the CMA-ES. Here, a popmembers (with the  highlighted children) is used ulation of  on the function in Fig. 2. The population is created through (1), with the dotted line indicating the single- (standard deviation) isocontour (defined by  ) . Then, the best  members are selected surrounding the initial mean, h i as shown by the filled circles. Their parameter values are used to form the new mean in (4), where relative weights w are illustrated by the size of the concentric circles surrounding each child.

= 50

= 25

x

( )

(x )

C

The case shown in Fig. 1 for applies for the simplest and . Then, scenario where , the initial search distribution appears as an since axis-aligned ellipse with the center located at and having axes sizes equal to 1/3 of the parameter ranges. In Fig. 3, the parameter ranges are equal and the initial distribution appears as a circle. With the initial mean and covariance matrix defined, the algorithm is ready to begin with the first iteration of sampling and evaluation. From (1), points are sampled and evaluated by way of calculations, simulations, or other methods. A numerical cost (or function) value is returned and the sampled points are arranged in order of cost from lowest to highest. The members (whose parameters are represented by ) are sorted according to cost and the best are used in

(4) to form the mean of the next iteration. Fig. 3 illustrates the first iteration of sampling and selection for the simple two-dimensional rotated hyper-ellipse test function. For the formation of the new mean in (4), the recombination weights, , must be filled according to

(5) This weighted average gives the most proportion of the new mean to the best performing members of the sample as shown in Fig. 4.

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The variance effective selection mass (effective number of children accounting for the weighted mean), required for proper covariance matrix normalization of the weighted sample in (5) is defined in

(6) ), then . If no weighting scheme is used, (i.e., After the new mean is found, the conjugate evolution path

less sporadic due to sufficient sampling) and according to (9), and is composed entirely of recent mean-movement . Note that in (7), practical evaluation information of is done through the identity

(12) rather than actual matrix inversion. Since is always diagonal-only, inversion of this matrix is straightforward. The eigen-decomposition of is already re), therefore minimal quired for sampling in (1) (except at additional computational effort is needed. Next, the evolution path is updated according to

(7) and step-size

(8) can be updated using the step-size learning rate

(13) Like the conjugate evolution path in (7), some historical path movement information is retained according to the learning rate

(14) (9)

and step-size damping factor

Now the covariance matrix can be updated using three separate portions of information. The first is a historical contribution, which dampens the rate of change of the covariance matrix based on the learning rate

(10) The conjugate evolution path in effect keeps a record of where the mean has traveled in past iterations. Movement is (since movenormalized according to the step-size and ment in different directions requires different normalization according to the search ellipse). The purpose of keeping a travel is to determine if the step-size is too large or too record in small. An expected travel indicator of the multivariate normal distribution

(11) is used and compared to to adjust the step-size in (8). When the distribution movement is less than expected, the step size decreases, while for movement more than expected, the step-size increases. The learning rate and damping factor in (9) and (10) are used to smooth the changes in step-size. They impact the speed and behavior of the algorithm and were chosen to successfully work on a large variety of problems (changes are not recommended) [43]. Rather than base the conjugate evolution path solely on the movement from the last and present iterations, some historical information is retained by way of the learning rate . For small populations ( and are small), a significant portion of historical information is retained, however, large populations do not require this (mean movement is

(15) The amount of historical information retained is based on the problem dimension and population size. The second portion is called the rank-one update, where the search ellipse is elongated along the direction of mean movement through use of the evolution path, in (13). The last contribution is called the rankupdate, where the ellipse is formed from the distribution of the selected children. Note that the mean of the full distribution is used for covariance matrix generation and not that simply of the selected children; the logic of this is demonstrated in Fig. 5. The combination of these three terms, with proportions controlled by the covariance learning rate (15) is given by

(16) This completes all the necessary information for the next is round of sampling. The principle component analysis of performed here for sampling in (1) and computation of

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Fig. 6. Exploded view of the stacked-patch antenna template used for optimization. The centers of the upper and lower patches share the same (x, y) coordinates. The antenna is backed by a PEC ground plane. Fig. 5. Variation of covariance isocontours when using two different reference means for the estimation of the covariance matrices from the selected children (for only the rank- update portion). The dashed line ellipse represents the new covariance matrix contribution in the last term of (16) when using hxi (new mean) and the solid line represents the use of the old mean hxi . The use of the old mean and solid ellipse best represents the search intentions of the algorithm, that is, movement towards the function minimum and ellipse elongation along the direction of travel.

in (7) and (12). After this, the process repeats until an acceptable cost value is obtained from one of the sampled points, the algorithm converges, or time is expired. In practice, the only parameter the user needs to choose is the sample size, ; everything else is predetermined from the problem properties. This makes a strong case for use of selfadaptive evolutionary strategies such as CMA-ES, as less user time is spent with configuring the algorithm or trial and error of evolutionary parameters (such as crossover and mutation rates of a GA). A suggested minimum sample size is given by [43]

(17) a value chosen to be large enough to make a good estimation of the search space for most cost functions. For very complex cost functions with many local and global optima (multimodal), larger population sizes are typically used to avoid convergence on local optima. Generally, very small populations such as that in (17) yield the fastest optimization times. For simple problems, using a very large population size will usually increase optimization times without any benefit other than some certainty that the best solution has been found. A variation of the CMA-ES that restarts with increasing population sizes has been developed in [52] for very general problems. III. ALGORITHM COMPARISON WITH A STACKED-PATCH ANTENNA OPTIMIZATION PROBLEM The particle swarm optimization technique [1]–[5] is used for comparison to the CMA-ES as it is a popular real-valued

TABLE II OPTIMIZATION PARAMETER RANGES AND DESCRIPTIONS FOR THE STACKED-PATCH ANTENNA ILLUSTRATED IN FIG. 6

optimization strategy and is commonly employed for electromagnetics design problems. The two algorithms were used to optimize a stacked-patch antenna for operation from 1.1 GHz to 1.3 GHz [53]. This was done by coupling the algorithms to the FEKO software package [54], which is based on a full-wave method of moments analysis technique. The geometry of the stacked-patch antenna is shown in Fig. 6. The dielectric substrates were treated by layered media Green’s functions, while the metallic patches were specified as perfect electric conductors. Nine optimization parameters were required to represent the antenna design, with their description and ranges provided in Table II. Aspect ratios were imposed to constrain the dimensions of the metallic patches to practical ranges so that thin strips are excluded from the parameter space. This, however, introduces some parametric inseparability into the problem, making it more challenging for the optimization strategy (as opposed to and directly). specifying values for The broadside gain and S-parameters (VSWR) for each antenna were calculated at five equally spaced frequency points between 1.1 GHz and 1.3 GHz. Design cost to be minimized was computed using (18)

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TABLE III SELECTED PSO ALGORITHM PARAMETERS FOR THE OPTIMIZATION OF THE STACKED-PATCH ANTENNA

Fig. 8. Evolutionary progress for the stacked-patch antenna design problem using PSO with a population of 30 particles. Success is achieved for three seeds, while the other two did not meet the goal in a sufficient number of iterations (100). The black dashed line represents the function value goal of 3.0.

Fig. 7. Evolutionary progress for the stacked-patch antenna design problem using PSO with a population of 20 particles. Success is achieved for two seeds, while the other three did not meet the goal in a sufficient number of iterations (100). The black dashed line represents the function value goal of 3.0.

which contains contributions from the worst input VSWR across the frequency band and the broadside, mid-band (1.2 GHz) gain. A function goal of 3.0 was set for the designs, which typically yielded a 2:1 maximum VSWR and a gain of 5 dB. Approximate time required per function evaluation (5 frequency points) was 90 seconds on a single core of an Intel 2.4 GHz Xeon, quad-core processor running a Linux operating system. Five seeds per algorithm were chosen as a compromise between statistical certainty of performance and total run time. The algorithm parameters in Table III were selected for PSO. An initial population size of 20 particles was chosen, since it performed well on most of the test functions. This small population size, however, resulted in poor success rates. Out of five seeds, three failed to reach the goal after 100 iterations (2000 function evaluations). As shown in Fig. 7, the two that did succeed yielded fast results as expected, achieving the function value goal of 3.0 at 500 and 860 function evaluations. Each curve in Fig. 7 (and additionally, Figs. 8 and 9) represents, for one seed, the best function value achieved from the beginning of the optimization up until the number of function evaluations shown on the x-axis. For many optimization problems, a success rate of 40% would be considered insufficient. It is generally desirable to use a population size that will achieve reasonable success rates (e.g., 80% or higher), therefore the PSO population was gradually increased until an acceptable success rate was achieved. At 30 particles, one additional seed was successful in the allotted 100 iterations, as shown in Fig. 8. Slower performance was exhibited, but at increased algorithm reliability. At a population of 40 particles, PSO gave acceptable success rates, having all

Fig. 9. Evolutionary progress for the stacked-patch antenna optimization problem. A population size of 10 was used for CMA-ES and a population size of 40 was used for PSO. The black dashed line represents the function value goal of 3.0.

five seeds meet the goal with a mean requirement of 1864 function evaluations (simulations). A value of 10 for was chosen for the CMA-ES since it is the minimum suggested population size according to (17). No other algorithm parameters were required for the CMA-ES optimization. A success rate of 100% resulted with five seeds at this small population size; there was no need to increase the population as with PSO. The evolution results for PSO and CMA-ES are shown in Fig. 8, while a statistical comparison is given in Table III. When both algorithms achieved a 100% success rate, the CMA-ES exhibited a 62% reduction in optimization time compared to PSO. At the rarely successful PSO population size of 20, even the seeds that did converge were only comparable in speed to the cases where CMA-ES always yielded a successful result. The resulting antenna performance for PSO (population size of 40) and CMA-ES (population size of 10) are shown in Figs. 10 and 11, respectively. Each algorithm evolved antenna designs with a relatively diverse set of characteristics. More, which leaves room for over, each design has differences between VSWR and gain.

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Fig. 12. Geometry for the optimized ultrawideband antenna arrays. Each d is determined by the evolutionary strategy for the N-element arrays.

TABLE IV OPTIMIZATION STATISTICS FOR THE STACKED-PATCH ANTENNA OPTIMIZATION PROBLEM (CONSIDERING ONLY SUCCESSFUL SEEDS). ASTERISKS INDICATE RUNS WITH LOW SUCCESS RATES; RESULTS CAN ONLY PARTIALLY BE COMPARED TO OTHER RUNS WITH GOOD SUCCESS RATES Fig. 10. VSWR and broadside gain of the five antennas designed using PSO with a population size of 40. Like colors of VSWR and gain correspond to the same antenna.

Fig. 11. VSWR and broadside gain of the five antennas designed using CMA-ES with a population size of 10. Like colors of VSWR and gain correspond to the same antenna.

IV. ULTRAWIDEBAND ARRAY DESIGN The previous stacked-patch antenna clearly represents a design problem that can presently be solved with current optimization strategies, albeit with increased optimization times compared to the new strategy. In the interest of showing the ability of CM-ES to solve an infeasible (or impractical) problem, it will be applied here to the optimization of a challenging class of ultrawideband (UWB) antenna arrays [55]. A great deal of attention has been recently given to the optimization of aperiodic antenna array layouts which do not exhibit the grating lobes associated with periodic arrays when operated over an extended bandwidth. Most design methods incorporate an optimization strategy with a technique for parameter reduction to allow creation of large arrays. For example, the polyfractal arrays in [23]–[25] use fractal tree generator structures to build large arrays from relatively small sets of defining parameters. In this manner, arrays with thousands of elements can be created without overwhelming the optimization tool (in this case, the genetic algorithm). One drawback to these parameter reduction methods is that with much smaller size arrays, the solutions become limited due to the inadequate number of describing parameters. For these smaller arrays, optimizing the location of each element individually can be beneficial if the

optimizer can successfully tackle the large set of parameters (approximately one parameter per antenna element). This direct optimization of antenna element locations within a linear array has been previously attempted with up to 24 elements in [26], yielding fruitful aperiodic low-sidelobe designs for steered arrays (beam steering is analogous to an increase in bandwidth). A similar array design technique will be applied here using many more elements at a much greater bandwidth, allowing the potential creation of better performing UWB arrays. Here, the PSO and CMA-ES will be used to create arrays of 25, 50, 75 and for an 8:1 100 elements with minimum element spacings of frequency bandwidth (when the elements are limited to minimum spacing at the lowest operating frequency). For the arrangement shown in Fig. 12, where the element spacings are directly optimized, this leads to problems of 24, 49, 74 and 99 dimensions, respectively. The element spacings are permitted . Sidelobe values are to vary in the range of calculated from the array factor expression

(19) for an array of linear, uniformly excited elements which is steered to broadside. For each array size, optimizations with various population sizes for both CMA-ES and PSO were performed. Like with the stack-patch design problem, five seeds of each array and population size were run to gather a statistical sample of the performance of both optimization techniques. Table IV shows the permitted optimization time limits in the form of number of iterations and array evaluations. Fig. 18 shows the results of the optimizations, indicating that for any equivalent optimization time (number of array evaluations), CMA-ES always produces a better UWB array design. In these cases, the worst performing

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TABLE V NUMBER OF ITERATIONS AND FUNCTION EVALUATIONS PERMITTED FOR EACH OPTIMIZATION FOR BOTH CMA-ES AND PSO SEEDS. THE OPTIMIZATION TIME IS PREDOMINANTLY DETERMINED BY THE NUMBER OF ARRAY EVALUATIONS

Fig. 15. CMA-ES evolution of the five 55-element arrays. All of the seeds surpass the performance of the RPS array design after the 67th iteration (3,350 array evaluations).

Fig. 16. Array layouts of the five designs shown in Fig. 15 at the 5,000th iteration.

Fig. 13. CMA-ES evolution of the five 46-element arrays. All of the seeds surpass the performance of the polyfractal array design after the 155th iteration (7,750 array evaluations).

Fig. 14. Array layouts of the five designs shown in Fig. 13 at the 1,000th iteration. Easily noticed is how the designs all tend to have dense element positioning in the center and thinning toward the ends of the arrays.

TABLE VI PROPERTIES OF THE FIVE SEEDS FOR EACH OF THE OPTIMIZED DESIGNS

seed of CMA is always better than the best performing seed of PSO. Fig. 18 also highlights the progress of each algorithm though the initial mean best indicators, illustrating how PSO begins to stall, making only marginal improvements over the initial populations with the larger size arrays. CMA-ES, however, produces significant performance improvements over the initial populations for all of the array sizes. The largest optimizations of 100 elements, requiring a significant amount of time (nearly 400 hrs. per seed on a single

Fig. 17. Peak sidelobe level performance of the two best seeds shown in Figs. 13 and 15 over an extended bandwidth. Although the 46 and 55 element arrays are optimized at minimum element spacings of 0:5  and 10 , respectively, they yield good sidelobe suppression well beyond.

array evaluations), were initially only run processor for with 17 and 100 population members at the iteration counts given in Table V. However, in the interest of determining if PSO could surpass CMA-ES in array performance with a larger optimization, a population size of 500 with 5,000 iterations was permitted for PSO. Despite the potential advantage, no benefit over CMA-ES was observed, even compared to the smallest optimization (17 population members). In addition to creating these generic ultrawideband antenna arrays, several designs were optimized based on those found in the literature. Small example designs were selected to best match the capabilities of the method introduced in this section. One example is based on a 46-element polyfractal array found in [23], which has been optimized for minimum peak sidelobe

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Fig. 18. Summary of the CMA-ES and PSO ultrawideband array optimizations for various array and population sizes according to Table V. For seed trials where the population and element counts are the same, CMA-ES and PSO have the same run time (number of array evaluations), yielding a fair comparison between the two algorithms. The initial mean best indicators show the progress of the evolutionary strategies by pointing out where the algorithm begins. Each point is computed by finding the best result of the initial arrays of each seed (the best of the randomly selected initial populations) and taking the average of the five seeds.

level at a minimum element spacing of . In [23], the opti. mized design has a resulting peak sidelobe level of For comparison, the direct optimization technique developed here is applied to design a 46-element aperiodic array subject to . A moderate population size of 50 members was used with a maximum of 1000 iterations. The evolution of the five seeds is shown in Fig. 13, with the array design layouts given in Fig. 14. The final performance of the seeds is given in Table VI. Another example was selected for comparison, the 55-element raised-power series (RPS) array reported in [27]. This , array was optimized at a minimum element spacing of yielding a typical frequency bandwidth of 20:1 at the resulting . Again, five seeds with a popupeak sidelobe level of lation size of 50 members were used, except the element spacing . The evolution ranges were changed to results for the five seeds are shown in Fig. 15 with the corresponding array layouts given in Fig. 16. The final performance of the seeds is provided in Table VI. For all of these examples, the worst seed of each CMA-ES array design yields better performance than their literature counterpart. Hence, for these relatively small array designs, this method proves very useful for obtaining the best performance with a limited set of elements. At their optimized minimum element spacings, the worst of the 46 and 55 element CMA-ES optimized designs produce a respective 4.4 dB and 2.15 dB reduction in peak sidelobe levels versus their polyfractal and RPS counterparts. In Fig. 17, the performance of the two best arrays of the five 46 and 55 element designs are examined over a large bandwidth as in [23] and [27]. V. CONCLUSIONS Through the stacked-patch antenna design problem and optimization of ultrawideband antenna arrays, the CMA-ES has shown itself to be a very powerful and fast optimization strategy. For the stacked-patch design, with comparable success rates, the

CMA-ES reduced the optimization time by approximately 62% on average compared to a conventional PSO. Even with a small PSO population of 20 particles where it yielded fast time-to-success but a reduced success rate, the mean time-to-success (680 NFE), was only comparable to where CMA-ES achieved 100% success rates. It is apparent that for either algorithm, choosing the smallest population size will yield the fastest optimization times (for the successful seeds). This, however, can significantly affect the reliability of the algorithm, therefore a population large enough for reasonable success must be used. In the case of the CMA-ES, small populations yield fast performance in addition to extremely high success rates. In addition to antenna element design, CMA-ES also proves to be a very powerful tool for the optimization of ultrawideband antenna arrays. Because of its ability to handle very large problems, it was demonstrated that arrays whose element spacings are directly optimized can be effectively designed for sizes of at least up to 100 elements with 8:1 frequency bandwidths. Moreover, the method has been shown to be very competitive with other published techniques for ultrawideband array design where the array size domains overlap. For larger arrays, CMA-ES can potentially be used in combination with array parameterization techniques such as the polyfractal or the RPS method. Aside from the high speed of CMA-ES, a great benefit is the self-adaptive nature of internal strategy parameters. This eliminates the need for determining these constants externally, a necessary step when using PSO and most other common optimization techniques, sometimes requiring a great deal of time due to selection through trial and error. Like the stacked-patch antenna and ultrawideband array designs, application of CMA-ES to other electromagnetic design problems is expected to yield great benefit though a reduction in overall optimization time. Alternatively, if time is the limiting factor in an optimization, using CMA-ES will likely yield better designs in the allotted time.

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REFERENCES [1] J. Kennedy and R. Eberhart, “Particle swarm optimization,” in Proc. 9th Int. Conf. on Neural Networks, Nov. 1995, vol. 4, pp. 1942–1948. [2] R. Eberhart and J. Kennedy, “A new optimizer using particle swarm theory,” in IEEE Proc. 6h Int. Symp. on Micro Machine and Human Science, Oct. 1995, pp. 39–43. [3] J. Robinson and Y. Rahmat-Samii, “Particle swarm optimization in electromagnetics,” IEEE Trans. Antennas Propag., vol. 52, pp. 397–407, Feb. 2004. [4] D. Gies, “Particle swarm optimization: applications in electromagnetic design,” M. Eng. thesis, UCLA, Los Angeles, CA, 2004. [5] J. Kennedy and R. Eberhart, Swarm Intelligence. New York: Morgan Kaufmann and Academic Press, 2001. [6] J. H. Holland, “Genetic algorithms and the optimal allocation of trials,” SIAM J. Comput., vol. 2, no. 2, Jun. 1973. [7] , Y. Rahmat-Samii and E. Michielssen, Eds., Electromagnetic Optimization by Genetic Algorithms. New York: Wiley, 1999. [8] R. L. Haupt and D. H. Werner, Genetic Algorithms in Electromagnetics. Hoboken, NJ: Wiley, 2007. [9] D. S. Weile and E. Michielssen, “Genetic algorithm optimization applied to electromagnetics: A review,” IEEE Trans. Antennas Propag., vol. 45, pp. 343–353, Mar. 1997. [10] R. Storn and K. Price, “Differential evolution—A simple and efficient heuristic for global optimization over continuous spaces,” J. Global Optim., vol. 11, no. 4, pp. 341–359, Dec. 1997. [11] K. V. Price, R. M. Storn, and J. A. Lampinen, Differential Evolution: A Practical Approach to Global Optimization, in Natural Computing. Heidelberg: Springer-Verlag, 2005. [12] N. Jin and Y. Rahmat-Samii, “Parallel particle swarm optimization and finite difference time-domain (PSO/FDTD) algorithm for multiband and wide-band patch antenna designs,” IEEE Trans. Antennas Propag., vol. 53, pp. 3459–3468, Nov. 2005. [13] 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, pp. 556–567, Mar. 2007. [14] N. Jin and Y. Rahmat-Samii, “Particle swarm optimization for antenna designs in engineering electromagnetics,” J. Artif. Evol. Applicat., vol. 2008, no. 9, Jan. 2008. [15] F. J. Villegas, T. Cwik, Y. Rahmat-Samii, and M. Manteghi, “A parallel electromagnetic genetic-algorithm optimization (EGO) application for patch antenna design,” IEEE Trans. Antennas Propag., vol. 52, pp. 2424–2435, Sep. 2004. [16] D. H. Werner, P. L. Werner, and K. H. Church, “Genetically engineered multiband fractal antennas,” IEEE Electron. Lett., vol. 37, no. 19, pp. 1150–1151, Sep. 2001. [17] B. Schlobohm, F. Arndt, and J. Kless, “Direct PO optimized dual-offset reflector antennas for small earth stations and for millimeter wave atmospheric sensors,” IEEE Trans. Microwave Theory Tech., vol. 40, pp. 1310–1317, Jun. 1992. [18] A. Hoorfar and Y. Liu, “Antenna optimization using an evolutionary programming algorithm with a hybrid mutation operator,” in Proc. IEEE Int. Symp. on Antennas Propag., 2000, vol. 2, p. 1029. [19] M. John and M. J. Ammann, “Antenna optimization with a computationally efficient multiobjective evolutionary algorithm,” IEEE Trans. Antennas Propag., vol. 57, pp. 260–263, Jan. 2009. [20] D. S. Weile and E. Michielssen, “Integer coded Pareto genetic algorithm design of constrained antenna arrays,” IEEE Electron. Lett., vol. 32, no. 19, pp. 1744–1745, Sep. 1996. [21] 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, pp. 76–86, Jan. 2008. [22] R. L. Haupt, “Thinned arrays using genetic algorithms,” IEEE Trans. Antennas Propag., vol. 42, pp. 993–999, Jul. 1994. [23] J. S. Petko and D. H. Werner, “The evolution of optimal linear polyfractal arrays using genetic algorithms,” IEEE Trans. Antennas Propag., vol. 53, pp. 3604–3615, Nov. 2005. [24] 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, pp. 583–593, Mar. 2007. [25] J. S. Petko and D. H. Werner, “The pareto optimization of ultrawideband polyfractal arrays,” IEEE Trans. Antennas Propag., vol. 56, pp. 97–107, Jan. 2008.

[26] 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, pp. 1732–1742, Dec. 2002. [27] M. D. Gregory and D. H. Werner, “Ultrawideband aperiodic antenna arrays based on optimized raised power series representations,” IEEE Trans. Antennas Propag., vol. 58, Mar. 2010. [28] D. W. Boeringer, D. H. Werner, and D. W. Machuga, “A simultaneous parameter adaptation scheme for genetic algorithms with application to phased array synthesis,” IEEE Trans. Antennas Propag., vol. 53, pp. 356–371, Jan. 2005. [29] D. W. Boeringer and D. H. Werner, “Efficiency-constrained particle swarm optimization of a modified Bernstein polynomial for conformal array excitation amplitude synthesis,” IEEE Trans. Antennas Propag., vol. 53, pp. 2662–2673, Aug. 2005. [30] J. A. Bossard, X. Liang, L. Li, S. Yun, D. H. Werner, B. Weiner, T. S. Mayer, P. F. Cristman, A. Diaz, and I. C. Khoo, “Tunable frequency selective surfaces and negative-zero-positive index metamaterials based on liquid crystals,” IEEE Trans. Antennas Propag., vol. 56, pp. 1308–1320, May 2008. [31] J. A. Bossard, D. H. Werner, T. S. Mayer, J. A. Smith, Y. U. Tang, R. P. Drupp, and L. Li, “The design and fabrication of planar multiband metallodielectric frequency selective surfaces and infrared applications,” IEEE Trans. Antennas Propag., vol. 54, pp. 1265–1276, Apr. 2006. [32] J. A. Bossard, D. H. Werner, T. S. Mayer, and R. P. Drupp, “A novel design methodology for reconfigurable frequency selective surfaces using genetic algorithms,” IEEE Trans. Antennas Propag., vol. 53, pp. 1390–1400, Apr. 2005. [33] M. A. Gingrich and D. H. Werner, “Synthesis of low/zero index of refraction metamaterials from frequency selective surfaces using genetic algorithms,” IEEE Electron. Lett., vol. 41, no. 23, pp. 1266–1267, Nov. 2005. [34] D. J. Kern, D. H. Werner, and M. Lisovich, “Metaferrites: Usign electromagnetic bandgap structures to synthesize metamaterial ferrites,” IEEE Trans. Antennas Propag., vol. 53, pp. 1382–1389, Apr. 2005. [35] D. J. Kern, D. H. Werner, A. Monorchio, L. Lanuzza, and M. J. Wilhelm, “The design synthesis of multiband artificial magnetic conductors using high impedance frequency selective surfaces,” IEEE Trans. Antennas Propag., vol. 53, pp. 8–17, Jan. 2005. [36] S. Genovesi, R. Mittra, A. Monorchio, and G. Manara, “Particle swarm optimization for the design of frequency selective surfaces,” IEEE Antennas Wireless Propag. Lett., vol. 5, pp. 277–279, 2006. [37] M.-I. Lai and S.-K. Jeng, “Compact microstrip dual-band bandpass filter design using genetic algorithm techniques,” IEEE Trans. Microwave Theory Tech., vol. 54, pp. 160–168, Jan. 2006. [38] S. K. Guodos and J. N. Sahalos, “Pareto optimal microwave filter design using multiobjective differential evolution,” IEEE Trans. Antennas Propag., vol. 58, pp. 132–144, Jan. 2010. [39] A. Ostermeier, A. Gawelczyk, and N. Hansen, “A derandomized approach to self adaptation of evolution strategies,” Evol. Comput., vol. 2, no. 4, pp. 369–380, 1994. [40] H.-G. Beyer, “Toward a theory of evolution strategies: Self-adaptation,” IEEE Trans. Evol. Comput., vol. 3, no. 3, 1995. [41] T. Bäck, U. Hammel, and H.-P. Schwefel, “Evolutionary computation: Comments on the history and current state,” IEEE Trans Evol. Comput., vol. 1, pp. 3–17, Apr. 1997. [42] M. Pelikan, D. E. Goldberg, and F. G. Lobo, “A survey of optimization by building and using probabilistic models,” Comput. Opt. Applicat., vol. 21, no. 1, pp. 5–20, 2002. [43] N. Hansen and A. Ostermeier, “Completely derandomized self-adaptation in evolutionary strategies,” Evol. Comput., vol. 9, no. 2, pp. 196–195, 2001. [44] N. Hansen, S. D. Müller, and P. Koumoutsakos, “Reducing the time complexity of the derandomized evolution strategy with covariance matrix adaptation (CMA-ES),” Evol.Comput., vol. 11, no. 1, pp. 1–18, 2003. [45] N. Hansen and S. Kern, “Evaluating the CMA evolution strategy on multimodal test functions,” Paral. Probl. Solv. Nature PPSN VIII, vol. 3242, pp. 282–291, 2004. [46] N. Hansen, “The CMA evolution strategy: A comparing review,” in Towards a New Evolutionary Computation: Studies in Fuzziness and Soft Computing, J. Kacprzyk, Ed. Berlin: Springer-Verlag, 2006, vol. 192, pp. 75–102. [47] D. Whitley, M. Lunacek, and J. Knight, “Ruffled by ridges: How evolutionary algorithms can fail,” GECCO 2004, Lecture Notes in Computer Science, vol. 3103, pp. 294–306, 2004.

GREGORY et al.: FAST OPTIMIZATION OF ELECTROMAGNETIC DESIGN PROBLEMS USING THE CMA-ES

[48] N. Hansen, A. Niederberger, L. Guzzella, and P. Koumoutsakos, “A method for handling uncertainty in evolutionary optimization with an application to feedback control of combustion,” IEEE Trans. Evol. Comput., vol. 13, Feb. 2009. [49] M. Jebalia, A. Auger, M. Schoenauer, F. James, and M. Postel, “Identification of the isotherm function in chromatography using CMA-ES,” Proc. IEEE Congress on Evolutionary Computation, pp. 4289–4296, Sep. 2007. [50] S. Kern, S. D. Müller, N. Hansen, D. Büche, J. Ocenasek, and P. Koumoutsakos, “Learning probability distributions in continuous evolutionary algorithms—A comparative review,” Natural Comput., vol. 3, no. 1, pp. 77–112, Mar. 2004. [51] H.-G. Beyer and H.-P. Schwefel, “Evolution strategies—A comprehensive introduction,” Natural Comput., vol. 1, no. 1, pp. 3–52, Mar. 2002. [52] A. Auger and N. Hansen, “A restart CMA evolution strategy with increasing population size,” Proc. IEEE Congress on Evolutionary Computation, vol. 2, pp. 1769–1776, Sep. 2005. [53] M. D. Gregory, Z. Bayraktar, and D. H. Werner, “Design of high performance compact linear ultra-wideband antenna arrays with the CMA evolutionary strategy,” presented at the IEEE Int. Symp. on Antennas Propag., Jul. 11–17, 2010. [54] FEKO. EM Software and Systems—S.A. Version 5.4 [Online]. Available: www.emssusa.com [55] M. D. Gregory and D. H. Werner, “Fast optimization of electromagnetics design problems through the CMA evolutionary strategy,” presented at the IEEE Int. Symp. on Antennas Propag., Jul. 11–17, 2010. Micah D. Gregory (S’10) was born in Williamsport, PA, in 1984. He received the B.S. degree in electrical engineering from Bucknell University, Lewisburg, PA, in 2006 and the M.S. degree in electrical engineering from the Pennsylvania State University (Penn State), University Park, in 2009, where he is currently working toward the Ph.D. degree. He is currently a Research Assistant for the Computational Electromagnetics and Antennas Research Lab (CEARL), Electrical Engineering Department, 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 is 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.

Zikri Bayraktar (S’10) received the B.S. (honors and distinction) and M.S. degrees in electrical engineering from The Pennsylvania State University (Penn State), University Park, in 2004 and 2006, respectively, where he is currently working toward the Ph.D. degree. Currently, he is a Research Assistant in the Computational Electromagnetics and Antennas Research Laboratory (CEARL), Electrical Engineering Department, Penn State. His research interests are in the field of computational electromagnetics, including design and optimization of artificial magnetic surfaces, electromagnetic bandgap structures, metamaterials and miniaturized antenna elements and arrays. His other research interests include nature based optimization techniques and high performance computing. Mr. Bayraktar has received an IEEE Antennas and Propagation Society Ph.D. Research Award in 2009, the Anthony J. Ferraro Outstanding Doctoral Research Award in Electromagnetics in 2010 and the Melvin P. Bloom Memorial Outstanding Doctoral Research Award in Electrical Engineering at Penn State in 2010.

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Douglas H. Werner (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 Department of Electrical Engineering, Penn State. 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. He has published over 400 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 a book entitled Electromagnetic Optimization by Genetic Algorithms (New York: Wiley Interscience, 1999) as well as for the book entitled Soft Computing in Communications (New York: Springer, 2004). He coauthored the book Genetic Algorithms in Electromagnetics (Hoboken, NJ: Wiley/IEEE, 2007) and completed an invited chapter on “Fractal Antennas” for the popular Antenna Engineering Handbook (New York: McGraw-Hill, 2007). 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 is 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 is a Fellow of the IEEE, the IET and ACES. He 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 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 former Associate Editor of Radio Science and an Editor of the IEEE Antennas and Propagation Magazine.

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Self-Adaptive Differential Evolution Applied to Real-Valued Antenna and Microwave Design Problems Sotirios K. Goudos, Member, IEEE, Katherine Siakavara, Member, IEEE, Theodoros Samaras, Member, IEEE, Elias E. Vafiadis, Member, IEEE, and John N. Sahalos, Life Fellow, IEEE

Abstract—Particle swarm optimization (PSO) is an evolutionary algorithm based on the bird fly. Differential evolution (DE) is a vector population based stochastic optimization method. The fact that both algorithms can handle efficiently arbitrary optimization problems has made them popular for solving problems in electromagnetics. In this paper, we apply a design technique based on a self-adaptive DE (SADE) algorithm to real-valued antenna and microwave design problems. These include linear-array synthesis, patch-antenna design and microstrip filter design. The number of unknowns for the design problems varies from 6 to 60. We compare the self-adaptive DE strategy with popular PSO and DE variants. We evaluate the algorithms’ performance regarding statistical results and convergence speed. The results obtained for different problems show that the DE algorithms outperform the PSO variants in terms of finding best optima. Thus, our results show the advantages of the SADE strategy and the DE in general. However, these results are considered to be indicative and do not generally apply to all optimization problems in electromagnetics. Index Terms—Differential evolution (DE), evolutionary algorithms (EAs), linear array synthesis, microwave filter design, optimization methods, particle swarm optimization (PSO), patch antenna design.

I. INTRODUCTION

S

EVERAL evolutionary algorithms (EAs) have emerged in the past decade that mimic biological entities behavior and evolution In this paper we consider particle swarm optimization (PSO) [1] and Differential evolution (DE) [2], [3]. PSO [1] is an evolutionary algorithm based on the bird fly. It is an easy-to-implement algorithm. PSO has been used successfully in constrained and unconstrained electromagnetic design problems [4]–[24]. Differential evolution (DE) [2], [3] is a population-based stochastic global optimization algorithm. Several DE variants or strategies exist. An overview of both PSO and DE algorithms and the hybridizations of these algorithms with other soft computing tools can be found in [25]. The classical DE strategy has Manuscript received December 13, 2009; revised May 08, 2010; accepted August 28, 2010. Date of publication January 31, 2011; date of current version April 06, 2011. The authors are with the Radiocommunications Laboratory, Department of Physics, Aristotle University of Thessaloniki, GR-54124 Thessaloniki, Greece (e-mail: [email protected]; [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.2011.2109678

been applied to microwave structures [26]–[28], antenna design [29]–[35], signal optimization [36] and microwave imaging applications [37]–[44]. DE produced better results than PSO on numerical benchmark problems with low and medium dimensionality (30 and 100 dimensions) [45]. However, on noisy test problems, DE was outperformed by PSO. In [46] a comparative study between DE and PSO variants is presented for the design of radar absorbing materials (RAM). The number of problem dimensions was 10 and DE outperformed the PSO variants in terms of convergence speed and best values found. The shape reconstruction of a perfectly conducting 2-D scatterer using DE and PSO is presented in [40], [44]. Also both algorithms have been applied to 1-D small-scale inverse scattering problems [43]. In these cases, DE outperformed PSO. In [47] a comparison between DE, PSO and Genetic algorithms (GAs) for circular array design is presented. DE and PSO showed similar performances and both of them had better performance compared to GAs. One of the DE advantages is that very few control parameters have to be adjusted in each algorithm run. However, the control parameters involved in DE are highly dependent on the optimization problem. Therefore, it is not always an easy task to tune these parameters. Recently a novel DE strategy has been applied to numerical benchmark problems that self-adapts the control parameters (SADE) [48]. SADE has been applied successfully to a microwave absorber design problem [49]. In this paper, SADE is compared with other algorithms. The comparison is performed on common real-valued antenna and microwave design problems. These problems include linear-array synthesis with sidelobe level suppression and null control in specified directions. In order to evaluate the algorithms’ performance combined with a numerical solver, we apply the algorithms to the design of a dual-band E-shaped patch-antenna and of a microstrip bandpass filter. As numerical solver we employ FEKO [50], a commercially available EM solver. We compare the SADE with two PSO variants and the classical DE/rand/1/bin strategy. The numerical results show the advantages of the SADE approach and the DE in general. However, these results cannot lead to the general conclusion that DE outperforms PSO in all optimization problems in electromagnetics. This paper is organized as follows: In Section II we describe the PSO and DE algorithms. We present the numerical results in Section III. Finally, we give the conclusion in Section IV.

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GOUDOS et al.: SADE APPLIED TO REAL-VALUED ANTENNA AND MICROWAVE DESIGN PROBLEMS

II. EVOLUTIONARY ALGORITHMS A. Initialization A population (or swarm) in both PSO and DE consists of vectors (or particles) , where is the generation number. The population is initialized randomly from a uniform distribution. Each -dimensional vector represents a possible solution, which is expressed as

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Clerc [51] suggested the use of a different velocity update rule, which introduced a parameter called constriction factor. The role of the constriction factor is to ensure convergence when all the particles have stopped their movement. The velocity update rule is then given by

(4) (5)

(1) The population is initialized as follows: (2) where and are -dimensional vectors of the lower is a uniformly and upper bounds respectively and . The stopping criterion distributed random number within for both PSO and DE is usually the generation number or the number of objective-function evaluations. B. Particle Swarm Optimization In PSO, the particles move in the search space, where each particle position is updated by two optimum values. The first one is the best solution (fitness) that has been achieved so far. This value is called . The other one is the global best value obtained so far by any particle in the swarm. This best value is . called and , the velocity update rule After finding the is an important factor in a PSO algorithm. The most commonly used algorithm defines that the velocity of each particle for every problem dimension is updated with the following:

(3) where sion,

is the particle velocity in the dimendenotes the current iteration and the previous, is the particle position in the nth dimension, , are uniformly distributed random numbers in (0,1), is a parameter known as the inertia weight and and are the learning factors. The parameter (inertia weight) is a constant between 0 and 1. This parameter represents the particle’s fly without any external influence. The higher the value of , or the closer it is to one, the more the particle stays unaffected from pbest and gbest. The parameter represents the influence of the particle memory on its best position, while the parameter represents the influence of the swarm best position. Therefore, in the Inertia Weight PSO (IWPSO) algorithm the parameters to be determined are: the swarm size (or population size), usually 100 or less, the cognitive learning factor and the social learning factor (usually both are set to equal to 2.0), the inertia weight and the maximum number of iterations. It is common practice to linearly decrease the inertia weight starting from 0.9 or 0.95 to 0.4.

where and . This PSO algorithm variant is known as constriction factor PSO (CFPSO). Boundary conditions in PSO play a key role as it is pointed out in [52], [53]. In this paper we have applied the reflective walls boundary conditions. C. Differential Evolution In DE algorithms, the initial population evolves in each generation with the use of three operators: mutation, crossover and selection. Depending on the form of these operators several DE variants or strategies exist in the literature [3], [54]. The choice of the best DE strategy depends on the problem type [55]. The most popular is the one known as DE/rand/1/bin strategy. In this strategy, a mutant vector for each target vector is computed by (6) and are randomly chosen indices from the where , population and is a mutation control parameter. After mutation, the crossover operator is applied to generate a trial vector whose coordinates are given by if if (7) , is a number from a uniform where , a randomly random distribution from the interval chosen index from and the crossover constant . DE uses a greedy selection operator, from the interval which for minimization problems is defined by if otherwise

(8)

, are the fitness values of the trial where and the old vector respectively. Therefore, the newly found trial replaces the old vector only when it produces vector a lower objective-function value than the old one. Otherwise, the old vector remains in the next generation. The stopping criterion for the DE is usually the generation number or the number of objective-function evaluations. D. Self-Adaptive Differential Evolution (SADE) Storn has suggested [3] to choose the differential evolution and from the intervals and control parameters

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, respectively and to set . The correct selection of these control parameter values is, frequently, a problemdependent task. Multiple algorithm runs are often required for fine-tuning the control parameters. In [48] a novel strategy is proposed for the self-adapting of DE control parameters. This strategy is based on DE/rand/1/bin scheme. Each vector is exvalues. The basic idea is based on tended with its own and the evolution of the control parameters. New vectors are found by using the improved values of the control parameters. These vectors are more likely to survive and produce offspring. In turn, the newly found vectors propagate the improved values of the control parameters to the next generation. Therefore, the control parameters are self-adjusted in every generation for each individual according to the following scheme: if otherwise if otherwise

(9)

where

are uniform random numbers , , are the lower and the upper limits of F set to 0.1 and 0.9, respectively and and represent the probabilities to adjust the control parameters. The authors in [48] have set both these probabilities to 0.1 after several trials. They have tested the SADE performance with several low dimension benchmarks. Their conclusion is that the self-adaptive strategy is better or at least comparable to the classical DE DE/rand/1/bin strategy considering the quality of the solutions found. Therefore, by using the self-adaptive strategy the user does not have to adjust the and parameters while the time complexity does not increase. More details about the Self-adaptive DE strategy can be found in [48]. III. NUMERICAL RESULTS It must be pointed out that several PSO and DE variants exist in the literature. In order to select, the best algorithm for every problem one has to consider the problem characteristics. For example, micro-PSO performs very well for microwave image reconstruction [56]. Another key issue is the selection of the algorithm control parameters, which is also in most cases problemdependent. In this paper, we compare SADE with common PSO and DE algorithms. The control parameters for these algorithms are those that commonly perform well regardless of the characteristics of the problem to be solved. We apply all algorithms to real-valued antenna and microwave design problems. These include linear-array synthesis, patch-antenna and microstrip filter design. The linear-array synthesis design cases are executed 50 times for each algorithm. In all other cases, the algorithms are run for 20 independent trials. The best results are compared. All algorithms are compiled using the same compiler (Borland C++ Builder 5.0) on a PC with Intel Core 2 Duo E8500 at 3.16 GHz with 4 GB RAM running Windows XP. The C source code for the DE algorithm was the one given by Storn in [57]. This code was modified to include the self-adaptive DE strategy.

Fig. 1. Geometry of a 2N-element linear array along the x-axis.

The best value, the worst value, the mean and the standard deviation of the last generation computed by each algorithm are to presented here. In the standard DE, is set to 0.5 and 0.9. In the PSO algorithms and are set equal to 2.05. For . The velocity is upCFPSO, these values result in dated asynchronously, which means that the global best position is updated the moment it is found. The stopping criterion for all algorithms is the generation number. For all examples the population size and the number of generations is set equal for all algorithms. The same initial conditions are used for all algorithms. A. Linear-Array Synthesis We consider a 2N-element uniformly excited linear array symmetrically placed along the x-axis (Fig. 1). The array factor in the x-z plane is expressed as (10) where is the wavelength, and are the position and the phase of the th element, respectively, and and are the corresponding vectors. Equation (11) expressed in dB is written as (11) where is the direction of the maximum. The optimization goal is the sidelobe level (SLL) suppression by finding the optimum element positions and phases. This problem is defined by the minimization of the objective function (12) where is the set of theta angles that are outside the angular range of the mainlobe. In order to find the maximum SLL the first zero has to be found first. This is accomplished by using a simple search algorithm that scans the AF values after the mainlobe maximum. A brief description of the search algorithm follows. We start from the mainlobe maximum value and increase . We calculate the difference of the the AF scan in steps of current AF value minus the previous one. If this difference is negative, the angle is increased again and the next AF value is calculated. On the other hand, when the difference becomes positive, which means that the AF scan passed the mainlobe, the first zero is found.

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TABLE I COMPARATIVE RESULTS FOR 32 ELEMENTS POSITION-PHASE SYNTHESIS

= 0:6

d

Fig. 3. Best Array patterns obtained by SADE for the 32-element array case for d : .

= 06

TABLE II COMPARATIVE RESULTS FOR 32 ELEMENTS POSITION-ONLY SYNTHESIS

d

=

Fig. 2. Convergence rate for the 32-element array case (a) position-only syn: . thesis and (b) position-phase synthesis for d

=06

For sidelobe level suppression when null control in specified directions is required, the objective function may be defined as (13) where is the number of the required null directions, are weight factors and is defined as if otherwise.

and

(14)

is the desired null level in dB and the direction of Here, null. the We examine different design options for uniformly excited arrays, which include position-only, phase-only and position-phase synthesis. For position-only and phase-only synthesis the number of unknowns is , while for position-phase synthesis this number is . We assume that for is the minimum distance between two all cases, where adjacent elements. is varied depending on The maximum allowed distance the design case as in [30]. A population of 100 vectors is selected for all algorithms. The total number of iterations is set to 2000. The first design case is that of a 32-element array. We consider first that and we apply all algorithms to position-only and position-phase synthesis. For position-only

synthesis, all algorithms converge to the same best value in all trials. The convergence rate of the average objective function for this design case is given in Fig. 2(a). It is obvious that CFPSO, DE and SADE converge faster than the classical IWPSO algorithm. For position-phase synthesis, the algorithms obtain different results. The convergence-rate plot for this case is shown in Fig. 2(b). The DE strategies and the CFPSO converge faster than the IWPSO. The classical DE strategy converges faster than the SADE strategy but both obtain similar results. Table I presents the results for all algorithms. The self-adaptive DE produces slightly better results than the classical DE and both are better than the PSO ones. The patterns of the best array designs obtained for both position-only and position-phase synthesis are shown in Fig. 3. SADE obtains an SLL of (instead of in [30]) for position-only synthesis. For position-phase synthesis the best SLL value obtained by SADE ( in [30]). is For the second design case we set . Table II shows the results for position-only synthesis. Both DE strategies clearly outperform the PSO algorithms. The DE strategies obtain smaller standard deviation values than the PSO ones. Table III shows the results for position-phase synthesis. SADE

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TABLE III COMPARATIVE RESULTS FOR 32 ELEMENTS POSITION-PHASE SYNTHESIS

d

=

Fig. 5. Best Array patterns obtained by SADE for the 32-element array case . for d

=

TABLE IV ELEMENT POSITIONS AND PHASES FOR THE 32-ELEMENT ARRAY

Fig. 4. Convergence rate for the 32-element array case (a) position-only syn. thesis and (b) position-phase synthesis for d

=

obtains the best objective-function value of all algorithms, while it outperforms the other algorithms in terms of mean and worst values. The convergence-rate plots for these cases are shown in Fig. 4(a) and (b). SADE, DE and CFPSO converge at a similar speed. The patterns of the best designs are shown in Fig. 5. The (instead of maximum SLL values are [30]) and (instead of [30]) for position-only and position-phase synthesis respectively. Table IV shows the position and phase values of the best designs obtained for all the 32-element array design cases. The next design case is that of a 60-element array. We apply the algorithms to phase-only and position-phase synthesis. We as in [30]. Table V presents the statistical reset sults of the objective-function values for both phase-only and position-phase synthesis. It is evident again that both DE strategies produce better results than the PSO ones. For phase-only

synthesis, the classical DE strategy obtains slightly better results than the SADE. However, SADE obtains a lower standard deviation value. For position-phase synthesis, SADE clearly outperforms the other algorithms. CFPSO obtains a better best value than the classical DE, but the classical DE produces better results than the CFPSO algorithm in terms of mean and standard deviation values. Fig. 6(a) and (b) show the convergence-rate plots. CFPSO, DE and SADE converge at similar speeds. Fig. 7 presents the patterns of the best designs obtained for both phase-only and position-phase design cases. The max(instead of imum SLL of the phase-only synthesis is in [30]) while for position-phase synthesis the SLL (instead of in [30]). Table VI value is presents the optimum phases and positions derived by the SADE strategy for phase-only and position-phase synthesis. It must be pointed out that the authors in [30] have not used the classical DE strategy. Instead, they have used a DE variant that generates two trial vectors for every vector of the population using different strategies. The values for the control parameters are not reported for all cases in [30]. For the 32-element

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Fig. 7. Best Array patterns obtained by SADE for the 60-element array case for d = 0:7.

TABLE VI ELEMENT POSITIONS AND PHASES FOR THE 60-ELEMENT ARRAY

Fig. 6. Convergence rate for the 60-element array case (a) phase-only synthesis and (b) position-phase synthesis for d 0:7.

=

TABLE V COMPARATIVE RESULTS FOR 60 ELEMENTS PHASE-ONLY AND POSITION-PHASE SYNTHESIS

case with the population size is set to 320 and the number of iterations is 300. This results in objective-function evaluations. In our results, we have objective-function evaluations. Thus, the number of objective-function evaluations is slightly higher in our case, which could explain the small differences with [30]. We also notice that in the design cases where only positions or phases are optimized the differences with [30] are small, ranging from 0.11 dB to 0.13 dB. This indicates that in such cases where the problem search space is small, it is less probable to obtain a significantly better solution using a number of objective-function evaluations similar to [30]. We obtain slightly

better improvements to the solutions of [30] in the 32-element position-phase synthesis design cases (0.18 dB to 0.25 dB). The better improvement of 2.21 dB is achieved for the 60-element position-phase synthesis. For this case, the search space becomes larger (60 unknowns) and it is therefore more probable to obtain better solutions. The final linear-array synthesis design case is a case of SLL suppression and null control in three directions. The desired null directions are at 30, 32.5 and 35 degrees. The desired null . The array consists of 28 elements as in level is set to

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TABLE VIII ELEMENT POSITIONS FOR THE 28-ELEMENT ARRAY WITH NULL CONTROL

Fig. 8. Convergence rate for the 28-element array case, position-only synthesis 0:7. for d

=

TABLE VII COMPARATIVE RESULTS FOR 28 ELEMENTS POSITION-ONLY SYNTHESIS WITH NULL CONTROL

Fig. 9. Best Array patterns obtained by SADE and DE for the 28-element array = 0:7. case for d

[8]. Table VII shows the results. It is evident that SADE obtains better results than the other algorithms. Up to numerical errors, the same values are obtained by the two DE strategies. The standard deviation values denoted as zero in Table VII are . The DE strategies converge at similar speeds, less than slightly faster than the CFPSO algorithm as shown in Fig. 8. IWPSO converges at slower speed, but the final objective-function value is close to those obtained by the other algorithms. The best patterns obtained by the classical DE and the Selfadaptive DE are shown in Fig. 9. The SLL obtained in [8] using and the null values are below . Both PSO is . The best DE strategies obtain nulls at a level lower than SLL values obtained by the classical DE and the Self-adaptive and respectively. The element DE are positions for these cases are presented in Table VIII.

Fig. 10. Geometry of a E-shaped patch antenna.

B. E-Shaped Patch-Antenna Design The use of microstrip patch antennas in wireless communication systems provides several advantages like low profile, low cost and ease of fabrication. E-shaped patch antennas extend the patch functionality and bandwidth. They are suitable for dual-band or wide-band designs. The design of such antennas requires the determination of the geometrical parameters of the antenna that satisfy the design requirements at the desired frequencies. PSO and DE have been in several occasions [58], [59] applied to E-shaped patch-antenna design. The geometry of an E-shaped patch antenna is given in Fig. 10. Two parallel slots are incorporated into the rectangular patch. A coaxial feed is used, which introduces two possible lengths to generate different resonant frequencies.

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TABLE IX COMPARATIVE RESULTS FOR THE E-SHAPED PATCH-ANTENNA DESIGN

The

E-shaped

patch has six design parameters: . These are the patch width , the patch length , the slot width , the slot length , the slot and the feed position . The ground-plane size position and a 5.5 mm thick air substrate is fixed at is used as in [59]. The design goal is to minimize the magnitude in two frequencies 5 GHz and 5.5 GHz, which implies the minimization of the objective function

Fig. 11. Convergence rate for the E-shaped patch-antenna design.

(15) where is the vector of the antenna geometry. In order to maintain the E-shape, additional restrictions apply to the design parameters. These are [58]

Fig. 12. Simulated S

(16) The computation of the objective function requires the use of a full-wave numerical method. The E-shaped patch antenna is modeled in FEKO. In order to integrate the in-house source code of the evolutionary algorithms with FEKO, a wrapper program is created. The implementation of a separate optimizer that calls external software to do the simulations for the evaluation of the objective function can also be found in [60]. A population of 10 vectors is selected for all algorithms. The total number of generations is set to 200. In this case execution time plays an important role. Therefore, fewer objective-function evaluations are selected than in the linear-array synthesis case. Table IX presents the statistical results of the objective-function values for each algorithm. Fig. 11 shows the convergence rate. SADE seems to perform better than the other algorithms. SADE obtains the best objective-function value, the highest mean value and the smallest standard deviation value. Both DE strategies perform better than the PSO algorithms. IWPSO obtains a better best value than CFPSO but CFPSO produces better mean and standard deviation values than IWPSO. SADE has a faster convergence speed than the other DE algorithm. The DE strategies converge faster than the PSO algorithms. The frequency response of the best designs found by the selfadaptive DE (Des 1) and the classical DE (Des 2) algorithms is given in Fig. 12. A finer mesh is used in FEKO model with 100 frequency points to provide a smooth response. The Des

curves for best designs found by SADE and DE.

TABLE X DESIGN PARAMETERS FOR THE BEST DESIGNS FOUND BY DE AND SADE (mm)

1 antenna has a bandwidth between 4.78 and 5.91 GHz with , while the Des 2 design has a similar bandwidth between 4.8 and 5.91 GHz. For both designs the values are lower than between 4.88 and 5.73 GHz. Table X has the design parameters obtained for both designs. The surface current distributions for designs Des 1 and Des 2 at 5 GHz and 5.5 GHz are shown in Fig. 13. C. Microstrip Bandpass Filter The final example presents the design of a bandpass microstrip filter (Fig. 14). A substrate with dielectric constant equal to 9 and 0.66 mm thickness is considered [61]. Again the filter is modeled in FEKO. In [62], [63] the space-mapping technique is used for filter design. This is accomplished in conjunction with FEKO. Such a filter design problem can be defined by two objectives subject to two constraint functions. in the passband The first objective is to maximize the frequency range. The second objective is to minimize the in the stopband frequency range. Additionally, constraints can

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TABLE XI DESIGN SPECIFICATIONS FOR THE BAND-PASS FILTER

Fig. 13. Simulated surface current distribution of the two best designs (a) Des 1 at 5 GHz, (b) Des 1 at 5.5 GHz, (c) Des 2 at 5 GHz, (d) Des 2 at 5.5 GHz.

Fig. 15. Convergence rate for the microstrip bandpass filter design case. TABLE XII COMPARATIVE RESULTS FOR THE BAND-PASS FILTER DESIGN CASE

Fig. 14. Bandpass microstrip filter geometry.

be set for levels in both the passband and the stopband frequency range. An effective way to combine the above objectives in one objective function is to use an exact penalty method [64]. Therefore, this design problem is defined by the minimization of the objective function

(17) where is the vector of filter geometry, and define the corresponding passband and the stopand define the minimum and band frequency ranges, maximum allowable values in the passband and stopband frequency ranges respectively and is a very large number. In a penalty method the feasible region is expanded, but a large cost or “penalty” is added to the original objective function for solutions that lie outside the original feasible region. Therefore is chosen large enough to ensure that solutions that do not fulfill constrains result in large fitness values. Table XI shows the filter design specifications. For each FEKO run, 17 frequency sweeps are taken in the frequency range 4.3–5.5 GHz. A population of 15 vectors is selected for

TABLE XIII DESIGN PARAMETERS FOR THE BAND-PASS FILTER DESIGN CASE (mm)

all algorithms. The total number of generations is set to 100. The same control parameters that are given in the previous sections are used for all algorithms. In the objective function was set equal to . The results for all algorithms are shown in Table XIII. It is evident that the SADE and the DE outperform the PSO algorithms. The DE obtains the best value but SADE obtains the highest mean and the smallest standard deviation values. The CFPSO performance is also close to that of the DE strategies. Fig. 15 presents the convergence rate. The DE strategies and CFPSO seem to converge at similar speeds. The frequency response of the best design obtained by SADE is depicted in Fig. 16. It is evident that in the frequency bands between 4.3 and 4.6 GHz and between 5.2 and 5.5 GHz the value lies below . Table XIII shows the design parameters for this filter.

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TABLE XIV COMPARATIVE RESULTS FOR 32 ELEMENTS POSITION-ONLY SYNTHESIS  (POPULATION SIZE=30) d

=

Fig. 16. Best filter design case frequency response found by SADE.

Fig. 18. Convergence rate for the 28-element array case, position-only syn0:7 and population size=30. thesis for d

=

TABLE XV COMPARATIVE RESULTS FOR 28 ELEMENTS POSITION-ONLY SYNTHESIS WITH NULL CONTROL (POPULATION SIZE=30)

Fig. 17. Convergence rate for the 32-element array case position-only synthesis  and population size = 30. for d

=

D. Variation of the Population Size We have chosen to run again two examples from the lineararray synthesis cases with a smaller population size. Our goal is to determine if and to what extent convergence rate and statistical results depend on the population size. The population size is set to 30 and all algorithms run for 50 independent trials. The first case is that of position-only synthesis for the 32-ele. The statistical results are shown in ment array with Table XIV while the convergence rate is depicted in Fig. 17. We observe that although the number of objective-function evalua, SADE obtains again the tions is reduced to same best value as with a population size of 100. Also both DE strategies obtain higher best values than those reported in [30]. The convergence-rate plot is quite similar to that of Fig. 4(a). The second case is that of the 28-element array with null control. Table XV shows the statistical results. DE and SADE produce similar results. The standard deviation value denoted as as in Table VII. Fig. 18 zero in Table XV is less than presents the convergence rate. It seems that CFPSO, DE and SADE converge at similar speeds as in Fig. 8. Given the results above, we conclude that a decreasing population size does not significantly modify the convergence rate. However, there is a slight deterioration in all statistical results.

IV. CONCLUSION The SADE strategy has been applied to common design problems in electromagnetics. The number of unknowns for the design problems varies from 6 to 60. The SADE results have been compared with other popular evolutionary algorithms. The DE strategies outperform the PSO algorithms in terms of best values found. However, these results are indicative and do not generally apply to all optimization problems in electromagnetics. Compared with the classical DE strategy the Self-adaptive DE obtains similar convergence rate results and produces in average the same or better results. The main advantage of SADE is the fact that it requires only the adjustment of two parameters: the population size and the number of iterations. CFPSO has produced better results than the IWPSO algorithm and converges at a similar speed with the DE strategies. Both PSO algorithms have presented a larger dispersion of values, which is due to their ability to escape from local optima. All algorithms can be combined with a numerical method. The SADE strategy can be easily applied to other microwave and antenna design problems and it can be used in conjunction with an EM solver software.

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This application of the SADE strategy to other design problems will be part of our future work. In addition, in our future work, we intend to explore the applicability of the SADE strategy to multi-objective DE algorithms. ACKNOWLEDGMENT The authors would like to thank the anonymous reviewers for their detailed and valuable comments. REFERENCES [1] J. Kennedy and R. Eberhart, “Particle swarm optimization,” presented at the IEEE Int. Conf. on Neural Networks, Piscataway, NJ, 1995. [2] R. Storn and K. Price, “Differential evolution-A Simple and efficient adaptive scheme for global optimization over continuous spaces,” 1995, Tech. Rep. TR-95-012. [3] R. Storn and K. Price, “Differential evolution—A simple and efficient heuristic for global optimization over continuous spaces,” J. Global Optimiz., vol. 11, pp. 341–359, 1997. [4] 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, pp. 556–567, 2007. [5] J. Robinson and Y. Rahmat-Samii, “Particle swarm optimization in electromagnetics,” IEEE Trans. Antennas Propag., vol. 52, pp. 397–407, 2004. [6] M. Donelli and A. Massa, “Computational approach based on a particle swarm optimizer for microwave imaging of two-dimensional dielectric scatterers,” IEEE Trans. Microw. Theory Tech., vol. 53, pp. 1761–1776, 2005. [7] S. M. Cui and D. S. Weile, “Application of a parallel particle swarm optimization scheme to the design of electomagnetic absorbers,” IEEE Trans. Antennas Propag., vol. 53, pp. 3616–3624, 2005. [8] M. M. Khodier and C. G. Christodoulou, “Linear array geometry synthesis with minimum sidelobe level and null control using particle swarm optimization,” IEEE Trans. Antennas Propag., vol. 53, pp. 2674–2679, 2005. [9] M. Benedetti, R. Azaro, and A. Massa, “Memory enhanced PSO-based optimization approach for smart antennas control in complex interference scenarios,” IEEE Trans. Antennas Propag., vol. 56, pp. 1939–1947, 2008. [10] S. Selleri, M. Mussetta, P. Pirinoli, R. E. Zich, and L. Matekovits, “Some insight over new variations of the particle swarm optimization method,” IEEE Antennas Wireless Propag. Lett., vol. 5, pp. 235–238, 2006. [11] S. M. Mikki and A. A. Kishk, “Quantum particle swarm optimization for electromagnetics,” IEEE Trans. Antennas Propag., vol. 54, pp. 2764–2775, 2006. [12] Y. Najjar, Y. Moneer, and N. Dib, “Design of optimum gain pyramidal horn with improved formulas using particle swarm optimization,” Int. J. RF Microw. Comput.-Aided Engng., vol. 17, pp. 505–511, 2007. [13] P. J. Bevelacqua and C. A. Balanis, “Minimum sidelobe levels for linear arrays,” IEEE Trans. Antennas Propag., vol. 55, pp. 3442–3449, 2007. [14] J. Nanbo and Y. Rahmat-Samii, “Analysis and particle swarm optimization of correlator antenna arrays for radio astronomy applications,” IEEE Trans. Antennas Propag., vol. 56, pp. 1269–1279, 2008. [15] N. Dib and M. Khodier, “Design and optimization of multi-band wilkinson power divider,” Int. J. RF Microw. Comput.-Aided Engng., vol. 18, pp. 14–20, 2008. [16] S. K. Goudos, I. T. Rekanos, and J. N. Sahalos, “EMI reduction and ICs optimal arrangement inside high-speed networking equipment using particle swarm optimization,” IEEE Trans. Electromagn. Compat., vol. 50, pp. 586–596, 2008. [17] L. Lizzi, F. Viani, R. Azaro, and A. Massa, “A PSO-driven spline-based shaping approach for ultrawideband (UWB) antenna synthesis,” IEEE Trans. Antennas Propag., vol. 56, pp. 2613–2621, 2008. [18] F. Afshinmanesh, A. Marandi, and M. Shahabadi, “Design of a single-feed dual-band dual-polarized printed microstrip antenna using a boolean particle swarm optimization,” IEEE Trans. Antennas Propag., vol. 56, pp. 1845–1852, 2008. [19] K. R. Mahmoud, M. I. Eladawy, R. Bansal, S. H. Zainud-Deen, and S. M. M. Ibrahem, “Analysis of uniform circular arrays for adaptive beamforming applications using particle swarm optimization algorithm,” Int. J. RF Microw. Comput.-Aided Engng., vol. 18, pp. 42–52, 2008.

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[43] A. Semnani, M. Kamyab, and I. T. Rekanos, “Reconstruction of one-dimensional dielectric scatterers using differential evolution and particle swarm optimization,” IEEE Geosci. Remote Sensing Lett., vol. 6, pp. 671–675, Oct. 2009. [44] I. T. Rekanos, “Conducting scatterer reconstruction using differential evolution and particle swarm optimization,” presented at the 23rd Annu. Rev. Progr. Appl. Computat. Electromagn. (ACES), Verona, Italy, 2007. [45] J. Vesterstrom and R. Thomsen, “A comparative study of differential evolution, particle swarm optimization and evolutionary algorithms on numerical benchmark problems,” presented at the Congress on Evolutionary Computation, Portland, OR, 2004. [46] S. K. Goudos, Z. D. Zaharis, K. B. Baltzis, C. S. Hilas, and J. N. Sahalos, “A comparative study of particle swarm optimization and differential evolution on radar absorbing materials design for EMC applications,” presented at the Int. Symp. on Electromagnetic Compatibility-EMC Europe, 2009. [47] M. A. Panduro, C. A. Brizuela, L. I. Balderas, and D. A. Acosta, “A comparison of genetic algorithms, particle swarm optimization and the differential evolution method for the design of scannable circular antenna arrays,” Progr. Electromagn. Res. B, pp. 171–186, 2009. [48] J. Brest, S. Greiner, B. Boskovic, M. Mernik, and V. Zumer, “Selfadapting control parameters in differential evolution: A comparative study on numerical benchmark problems,” IEEE Trans. Evol. Comput., vol. 10, pp. 646–657, 2006. [49] S. K. Goudos, “Design of microwave broadband absorbers using a selfadaptive differential evolution algorithm,” Int. J. RF Microw. Comput.Aided Eng., vol. 19, pp. 364–372, 2009. [50] User’s Manunal, FEKO Suite 5.2 FEKO, 2003 [Online]. Available: www.feko.info [51] M. Clerc, “The swarm and the queen: towards a deterministic and adaptive particle swarm optimization,” presented at the Congress on Evolutionary Computation, Washington, DC, 1999. [52] T. Huang and A. S. Mohan, “A hybrid boundary condition for robust particle swarm optimization,” IEEE Antennas Wireless Propag. Lett., vol. 4, pp. 112–117, 2005. [53] S. Xu and Y. Rahmat-Samii, “Boundary conditions in particle swarm optimization revisited,” IEEE Trans. Antennas Propag., vol. 55, pp. 760–765, Mar. 2007. [54] R. Storn, “Differential evolution research—trends and open questions,” Studies Comput. Intell., vol. 143, pp. 1–31, 2008. [55] E. Mezura-Montes, J. Velazquez-Reyes, and C. A. C. Coello, “A comparative study of differential evolution variants for global optimization,” presented at the Genetic and Evolutionary Computation Conf., Seattle, WA, 2006. [56] T. Huang and A. S. Mohan, “A microparticle swarm optimizer for the reconstruction of microwave images,” IEEE Trans. Antennas Propag., vol. 55, pp. 568–576, Mar. 2007. [57] R. Storn, Differential Evolution Homepage [Online]. Available: http:// www.icsi.berkeley.edu/~storn/code.html [58] N. Jin and Y. Rahmat-Samii, “Parallel particle swarm optimization and finite-difference time-domain (PSO/FDTD) algorithm for multiband and wide-band patch antenna designs,” IEEE Trans. Antennas Propag., vol. 53, pp. 3459–3468, 2005. [59] L. Zhang, Z. Cui, Y. C. Jiao, and F. S. Zhang, “Broadband patch antenna design using differential evolution algorithm,” Microw. Opt. Technol. Lett., vol. 51, pp. 1692–1695, 2009. [60] D. J. Bekers, S. Monni, S. M. van de Berg, A. M. van de Water, B. J. Morsink, C. Alboin, V. Ducros, M. Celikbas, J. Blanche, N. Fiscante, G. Gerini, J. P. Martinaud, M. Rochette, and G. H. C. van Werkhoven, “Optimization of phased arrays integrated with FSS and feeding elements based on parametric models,” presented at the 2nd Eur. Conf. on Antennas and Propagation, 2007. [61] A. Hennings, E. Semouchkina, A. Baker, and G. Semouchkin, “Design optimization and implementation of bandpass filters with normally fed microstrip resonators loaded by high-permittivity dielectric,” IEEE Trans. Microw. Theory Tech., vol. 54, pp. 1253–1261, 2006. [62] S. Koziel and J. W. Bandler, “SMF: A user-friendly software engine for space-mapping-based engineering design optimization,” presented at the Signals, Systems and Electronics Int. Symp., 2007. [63] S. Koziel, J. W. Bandler, and K. Madsen, “A space-mapping framework for engineering Optimization; theory and implementation,” IEEE Trans. Microw. Theory Tech., vol. 54, pp. 3721–3730, 2006. [64] G. Di Pillo, “Exact Penalty Methods,” in Algorithms for Continuous Optimization: The State of the Art, E. Spedicato, Ed. Dordrecht, The Netherlands: Kluwer Academic Publishers, 1994, pp. 1–45.

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Sotirios K. Goudos (S’00–M’05) was born in Thessaloniki, Greece, in 1968. He received the B.Sc. degree in physics, the M.Sc. degree in electronics, and the Ph.D. degree in physics from the Aristotle University of Thessaloniki, in 1991 1994, and 2001, respectively, and the Master in Information Systems from the University of Macedonia, Greece, in 2005. Since 1996, he has been working in the Telecommunications Center, Aristotle University of Thessaloniki. His research interests include antenna and microwave structures design, electromagnetic compatibility of communication systems, evolutionary computation algorithms and semantic web technologies. Dr. Goudos is a member of the Greek Physics Society and the Greek Computer Society.

Katherine Siakavara (M’04) received the B.Sc. degree in physics, the M.Sc. degree in electronics, and the Ph.D. degree, all from the Aristotle University of Thessaloniki, Greece, in 1977, 1979, and 1982, respectively. She is currently an Assistant Professor in the Department of Physics, University of Thessaloniki, Greece. She has authored or coauthored more than 60 papers in peer reviewed journals and international conferences. Her research interests include applied electromagnetics, antennas and microwaves.

Theodoros Samaras (S’93–A’97–M’02) received the Physics degree from Aristotle University of Thessaloniki, Thessaloniki, Greece, in 1990, the M.Sc. degree in medical physics (with distinction) from the University of Surrey, Surrey, U.K., in 1991, and the Ph.D. degree from the Aristotle University of Thessaloniki, Thessaloniki, Greece, in 1996. In 1998, he was with the Bioelectromagnetics (BIOEM)/Electromagnetic Compatibility (EMC) Group, Swiss Federal Institute of Technology, Zürich, Switzerland, where he was mainly involved with studying the temperature increase due to the absorption of electromagnetic energy in materials and the effect of heat diffusion in electromagnetic dosimetry. He subsequently joined the Hyperthermia Unit, Erasmus Medical Center, Rotterdam, The Netherlands, where he conducted research on treatment quality of superficial hyperthermia. In December 1999, he returned to the Aristotle University of Thessaloniki, where he is currently an Assistant Professor. His research interests include numerical techniques with applications in biomedical technology, EMC and telecommunications. Dr. Samaras was the recipient of a Marie Curie Fellowship awarded by the European Commission.

Elias E. Vafiadis (M’86) was born in Thessaloniki, Greece, in 1952. He received the B.Sc. degree in physics and the M.Sc. degree in electronics from the Aristotle University of Thessaloniki, in 1975 and 1979, respectively, and the Ph.D. degree in electrical engineering from the Democritus University of Thrace, Xanthi, Greece, in 1985. On February 1981, he joined the Electrical Engineering Department, Democritus University of Thrace, as a Research Associate. From 1986 to 1993, he served as a Lecturer and Assistant Professor in the Microwaves Laboratory, Xanthi, Greece. From 1993 to 2003, he was an Assistant Professor and, since 2003, he has been an Associate Professor in the Department of Physics, School of Science, Aristotle University of Thessalonik. His research interests include application of electromagnetic theory to waveguiding and radiating structures and CAD techniques for microwave circuits design. Dr. Vafiadis is a member of the Hellenic Physical Society.

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John N. Sahalos (M’75–SM’84–F’06–LF’10) received the B.Sc. and Ph.D. degrees in physics, in 1967 and 1974, respectively, and the Diploma (BCE+MCE) in Civil Engineering and the Professional Diploma of Postgraduate Studies in Electronic Physics, both in 1975, all from the Aristotle University of Thessaloniki, (AUTH), Greece. From 1971 to 1974, he was a Teaching Assistant and, from 1974 to 1976, he was an Instructor in the Department of Physics, AUTH. In 1976, he worked at the ElectroScience Laboratory, Ohio State University, Columbus, as a Postdoctoral University Fellow. From 1977 to 1986, he was a Professor in the Electrical Engineering Department, University of Thrace, Greece, and Director of the Microwaves Laboratory. Since 1986, he has been a Professor at the School of Science, AUTH, where he is the Director of postgraduate studies in electronic physics and the Director of the Radio-Communications Laboratory (RCL). During 1981-82, he was a Visiting Professor at the Department of Electrical and Computer Engineering, University of Colorado, Boulder. During 1989 to 1990, he was a Visiting Professor at the Technical University of Madrid, Spain. He is the author of three books (in Greek), six book chapters, and more than 350 articles published in the scientific literature. He also is the author of the book The Orthogonal Methods of Array Synthesis, Theory and the ORAMA Computer Tool (Wiley, 2006). His research interests are in the areas of antennas, high frequency techniques, communications, EMC/EMI, microwaves and biomedical engineering.

Dr. Sahalos is a Professional Engineer and a Consultant to industry. In 2002-04, he was on the Board of Directors of the OTE, the largest Telecommunications Company in Southeast Europe. He served as a Technical Advisor on several national and international committees, as well as, in several Mobile Communications Companies. Since 1992, he has been a member of Commissions A and E of URSI. Since 1998, he is the President of the Greek committees of URSI. He is the President of the section on Informatics, Telecommunications and Systems of the National Committee of Research and Technology. He is an honorary member of the Radio-electrology Society, a member of the Greek Physical Society, and a member of the Technical Chamber of Greece. He is the creator and leader of an EMC network with five laboratories (three from the academy and two from the industry). He has been honored with a special investigation fellowship from the Ministry of Education & Science, Spain. He also has been honored by several Institutes and Organizations. He graduated/mentored nearly 25 Ph.D. students and coauthored two best paper awards at international conferences. He received the 2009 IEEE-AP Society award for exceptional performance as a reviewer of the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION. He has been on the editorial board of three scientific journals. He was elected by department representatives of the Aristotle University of Thessaloniki as the Vice-Chairman of the Research Committee of AUTH for the period 2007-2010. Since June 2010, and for the next two years, he will be with the five member International Consulting Committee of the GRNET S.A.

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Improving the Accuracy of the Second-Kind Fredholm Integral Equations by Using the Buffa-Christiansen Functions Su Yan, Student Member, IEEE, Jian-Ming Jin, Fellow, IEEE, and Zaiping Nie, Senior Member, IEEE

Abstract—In computational electromagnetics, the second-kind Fredholm integral equations are known to have very fast iterative convergence but rather poor solution accuracy compared with the first-kind Fredholm integral equations. The error source of the second-kind integral equations can mainly be attributed to the discretization error of the identity operators. In this paper, a scheme is presented to significantly suppress such discretization error by using the Buffa-Christiansen functions as the testing function, leading to much more accurate solutions of the second-kind integral equations, while maintaining their fast convergence properties. Numerical experiments are designed to investigate and demonstrate the accuracy improvement of the second-kind surface integral equations in both perfect electric conductor and dielectric cases by using the presented discretization scheme. Index Terms—Accuracy analysis, Buffa-Christiansen functions, identity operator, magnetic-field integral equation, N-Müller integral equations, Rayleigh-Ritz scheme, second-kind integral equations.

I. INTRODUCTION

S

URFACE integral equations (SIEs) are very widely used in modeling electromagnetic scattering and radiation problems involving perfect electric conductors (PECs) and dielectric objects. Obtained by using dyadic Green’s functions [1] as the integral kernels and integrating over the entire surfaces of the objects under consideration, SIEs can be categorized into the Fredholm integral equations [2] of the first and the second kinds. In computational electromagnetics, the first-kind Fredholm integral equations, or the homogeneous Fredholm integral equations, are known to have very good accuracy, but rather poor

Manuscript received June 22, 2010; revised August 05, 2010; accepted August 19, 2010. Date of publication January 28, 2011; date of current version April 06, 2011. This work was supported in part by the China Scholarship Council (CSC), the National Science Foundation of China (NSFC) under Contract 60728101, the Chinese University 111 Project under Contract B07046, and in part by the Outstanding Ph.D. Research Foundation of UESTC. 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]; [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.2011.2109364

convergence in an iterative solution because of their unbounded integral operators, which produce system matrices with large condition numbers after discretization. In contrast, the secondkind Fredholm integral equations, or the inhomogeneous Fredholm integral equations, usually have a fast convergence rate in an iterative solution since they are in the form of an identity operator plus a compact integral operator. A compact operator, in functional analysis, is a linear operator transforming from a Banach space to another Banach space , such that the image of any bounded subset of under the operator is a relatively compact subset of . Such an operator is a well-bounded operator, and produces a system matrix with eigenvalues clustered around zero. Consequently, the second-kind Fredholm integral equations give rise to the system matrices that have bounded eigenvalues clustered around a non-zero point, which makes the matrices very well-conditioned. However, the second-kind integral equations have a drawback in that their solutions are far less accurate than their first-kind counterparts, and therefore, they are less commonly used for practical applications. In recent years, much effort has been devoted to improving the accuracy of the second-kind Fredholm integral equations, especially the magnetic-field integral equation (MFIE) for the PEC case. Most of these effort focuses on the accuracy loss caused by the integral operation [3]–[12]. Among these studies, some attributed the inaccuracy of the MFIE to the inaccurate evaluation of the impedance elements [3], [5], [10], including the logarithmic singularity in the field integration [6], [8], and the solid angle expression in the MFIE formulation [7]. Some believed that the inappropriate choice of the basis functions caused the problem [4], and hence, proposed the use of the linear-linear basis functions [9], [11] and higher-order vector basis functions [13] to alleviate this problem. Some showed that the improperly chosen solution scheme is another important error source, and proposed the Rayleigh-Ritz scheme for the three-dimensional MFIE to alleviate this error [12]. Some other research effort investigated the error caused by the identity operator in the MFIE. It has been shown that there is actually a large discretization error due to the identity operator [14], which contributes significantly to the total error of the MFIE. In order to alleviate the accuracy loss caused by the identity operator, regularization methods have been proposed for both two-dimensional [15] and three-dimensional [16] cases. The basic idea is to design a “filter” to filter out the high-frequency content in the basis functions and to increase their effective smoothness. Such “filtering,” unfortunately, is not easily applicable to the three-dimensional case. As a result, the regularization method in the three-dimensional case is not as effec-

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tive as its two-dimensional counterpart. More recently, a rotated Buffa-Christiansen (BC) function [17] is adopted as the testing function in the discretization of the MFIE in order to achieve a better accuracy [18]. However, the use of planar patches limits the improvement of the accuracy. Moreover, the limitation of discussion of the second-kind integral equation to the PEC case and the lack of mathematical and numerical explanation of the reason to the accuracy improvement also limit its significance. In sum, with all these methods, the accuracy of the MFIE solution is still worse than that of the electric-field integral equation (EFIE) solution. Therefore, the EFIE is always preferred over the MFIE in the analysis of electromagnetic scattering and radiation by a PEC object, especially when accuracy is important. Although much more complicated, a similar issue has been observed in the dielectric case, where the Poggio-Miller-ChangHarrington-Wu-Tsai (PMCHWT) equations [19]–[22] are the Fredholm integral equations of the first kind and the N-Müller equations [23]–[25] are the Fredholm integral equations of the second kind. Their convergence behavior in an iterative solution and the solution accuracy have been investigated thoroughly [13] and the first-kind equations (PMCHWT equations) have been shown to always have a better accuracy and a worse convergence compared to the second-kind equations (N-Müller equations). Hence, the PMCHWT equations are always preferred for an accurate solution of electromagnetic scattering and radiation by a dielectric object. In this paper, the SIEs for both PEC and dielectric cases are first presented, and the discretization schemes for different integral operators are discussed. The discretization error due to the identity operator is then suppressed by using the rotated BC functions defined on curvilinear triangular patches [26] as the testing function. It is demonstrated through several numerical examples that the accuracy of the second-kind Fredholm integral equations (the MFIE and N-Müller equations), in both PEC and dielectric cases, can be improved significantly using this discretization scheme. Before the conclusion is drawn, the reasons to the accuracy improvement by the proposed scheme are discussed and attributed to the significant suppression of the discretization error of the identity operator and the appropriate adoption of the Rayleigh-Ritz scheme [12]. II. SURFACE INTEGRAL EQUATIONS In this section, the general formulations for electromagnetic problems are first reviewed. The SIEs for both PEC and dielectric cases are then presented. Discussions are made to show how these SIEs can be categorized into the Fredholm integral equations of the first and second kinds. A. General Formulations Consider a scattering problem with an incident plane wave illuminating a homogeneous obstruction having a permittivity and permeability and immersed with a in an infinite homogeneous background medium permittivity and permeability . According to Love’s equivalence principle [27], the solution can be formulated in and terms of an equivalent surface electric current defined on an equivalent surface magnetic current

the surface of the obstruction. Here, stands for the outward pointing unit normal vector. The equivalent surface currents are governed by the EFIEs (1) (2) which can be written in a matrix form as (3) and the MFIEs (4) (5) which can be written in a matrix form as (6) stands for the intrinsic impedance in where ( , 2), stands for the identity operator, and and are the integral operators defined as (7) (8) In the above,

denotes the wavenumber in , denotes the Green’s function in an infinite homogeneous medium with the wavenumber , and denotes or the surface either the surface electric current density magnetic current density . In (8), P.V. stands for the Cauchy instead of is treated principal value integration. In (1)–(6), as the unknown function and is multiplied on (4)–(6) in order to balance the magnitude of each operator in (3) and (6) and make the whole system better conditioned. Equations (1) and (4) are derived from the formulation of the exterior fields in , and (2) and (5) are derived from the formulation of the interior fields in . Applying (1) to (6) to different scatterers with different boundary conditions, we can obtain various SIEs. B. SIEs in the PEC Case If the obstruction is a PEC object, the application of the results in , which, when boundary condition substituted into (1) and (4), yields the EFIE and the MFIE for the PEC case as (9) (10) Although both the EFIE (9) and the MFIE (10) can be solved independently to obtain the equivalent electric current density ,

YAN et al.: IMPROVING THE ACCURACY OF THE SECOND-KIND FREDHOLM IEs BY USING THE BUFFA-CHRISTIANSEN FUNCTIONS

they suffer from the “interior resonance corruption” [28] at some discrete frequencies. As a remedy, their convex combination, the combined-field integral equation (CFIE) [29], can be used

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More explicitly, the PMCHWT equations can be expressed as

(14) and the N-Müller equations can be expressed as (11) is a linear combination factor. where As has been studied extensively [13], [30], the operator is a Fredholm integral operator of the first kind, which has a continuous spectrum distribution (and the corresponding discrete eigenvalue distribution after discretization) clustering at origin and infinity, resulting in an unbounded condition number that approaches infinity when the discretization density is increasingly refined. However, because of its high accuracy and the capability to handle objects with open surfaces, the EFIE is still widely used whenever possible. On the other hand, since the operator is a compact operator, the whole MFIE operator is actually an identity operator plus a compact operator, which makes the MFIE a Fredholm integral equation of the second kind. As mentioned above, the MFIE has a very good iterative convergence but a rather poor solution accuracy, and therefore, is less commonly used compared with the EFIE. As their combination, the CFIE inherits the characteristics from both the EFIE and the MFIE. Hence, it has a better iterative convergence but a worse solution accuracy [31] compared with the EFIE, and its performance depends on the choice of the combination factor . C. SIEs in the Dielectric Case If the obstruction is a dielectric object, by solving either (3) or (6), we can obtain the solution of the scattering problem. However, since both (3) and (6) suffer from the interior resonance corruption, their combinations are usually used. One approach is to combine (1) with (4) to form a CFIE for the exterior region , and combine (2) with (5) to form a CFIE for the interior region . These two equations then form a complete system that [32]. Another approach is to comcan be solved for and bine (1) with (2) and (4) with (5) in a general way as (12) where , , , and are combination factors. The resulting equations can be written as

(13) Several well-known equations can be obtained by choosing different combination factors. For example, 1) by choosing , the PMCHWT equations [19]–[21] can be obtained; and , , , 2) by choosing , the N-Müller equations [23], [24] can be obtained.

(15) Although there are other ways to formulate SIEs for the dielectric case, such as by using a different combination strategy [33], [34], or even applying different forms of the equivalence principle [35], [36], only the PMCHWT and the N-Müller equations are discussed in this paper since they correspond to the Fredholm integral equations of the first and the second kind, respectively, as pointed out in [13], [22], and [25]. As a matter of fact, since in the PMCHWT equations (14), the identity operators are canceled out, leaving only operators, which are compact operators, in the off-diagonal blocks, and operators in the diagonal blocks, the resulting equations are the first-kind Fredholm integral equations. In the N-Müller equations (15), on the other hand, the hyper-singular terms of the operators are canceled, resulting in the compact operators in the off-diagonal blocks [24], [25], and the diagonal blocks are in the forms of the identity operators plus the compact operators. Therefore, the N-Müller equations are the second-kind Fredholm integral equations. Similar to the EFIE and the MFIE in the PEC case, the PMCHWT equations are known to have a better accuracy than the N-Müller equations, while the latter has a faster convergence rate in an iterative solution [13]. III. DISCRETIZATION All the preceding statements on the solution accuracy and iterative convergence rate are based on a certain discretization scheme, which will be described and discussed in this section. The discretization process of an integral equation mainly contains two major steps. The first step is to expand the unknown current density in terms of basis functions, and the second step is to convert the integral equation into a matrix equation through a set of testing functions. Generally speaking, the basis and testing functions can be categorized into two different kinds: the divergence-conforming and the curl-conforming functions. A typical divergence-conforming function is the curvilinear RaoWilton-Glisson (CRWG) function [37], [38], denoted as . By rotating it with respect to the normal direction, a commonly can be obtained. The used curl-conforming function CRWG function has a normal component across the shared edge of two adjacent triangles, whereas the rotated CRWG function has a tangential component along the shared edge, as shown in Fig. 1(a) and (b), respectively. Recently, another divergenceconforming function called the Buffa-Christiansen (BC) function has been proposed [17] and successfully adopted in the implementation of the Calderón preconditioning technique [39], [40]. As a linear combination of the CRWG functions defined on a barycentric refinement of the original triangular mesh, the BC function, denoted as , is strictly divergence-conforming on the

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^ Fig. 1. The sketches of the definition domains of the four functions. (a) The RWG function f . (b) The rotated RWG function n ^ f . (d) The rotated BC function n

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barycentric mesh. At the same time, since its main component function, the BC function is also resembles that of the quasi-curl-conforming on the original mesh. By rotating with respect to the normal direction, the is curl-conforming on the barycentric mesh and quasi-divergence-conforming on the original mesh. Fig. 1(c) and (d) illustrate the definition domains and the main components of a typical BC and rotated BC functions, respectively. Next, we discuss the discretization scheme that can be adopted to discretize the , , and operators, respectively. To this end, the unknown current density is first expanded in terms of a set of basis functions, and the integral operator is then tested by a set of testing functions. It is a physical requirement that the basis function should be divergence-conforming in order to model the current (field) continuity correctly and to term which is related give a good representation of the to the surface (electric/magnetic) charge density. Because of its popularity and simplicity in definition, the CRWG function is employed as the basis function throughout this paper

2f

. (c) The BC function f .

TABLE I COMPARISON OF DIFFERENT TESTING FUNCTIONS IN THE DISCRETIZATION OF OPERATOR (f AS THE BASIS FUNCTION) THE

T

(17) is divergence-conforming, for example, or , If becomes a contour the surface integration of vanishes within each integral, and furthermore, , yielding triangle that supports

(16) (18) where is the number of interior edges in a triangular mesh are the expansion coefficients to be deterof the object, and mined. A. The

Operator

Since the testing procedure is nothing but a mathematical manipulation, in principle, both divergence- and curl-conforming testing functions can be used. However, there are some mathematical issues need to be noted. First, in order to have the operator well-tested, the testing function should be orthogonal to the basis function. From this point of view, the good and . Second, the use of the ditesting functions are vergence-conforming testing function will lead to a contour inoperator, which is not tegration in the discretization of the easy to be evaluated accurately and hence it is undesired. Deas the testing function, the discretization of the opnoting erator yields

where is the outward pointing unit normal vector defined on which comprises the boundaries of the integral boundary . If is curl-conforming, the triangles that support or , the surface integration of vanishes as a result of the Gauss divergence theorem, yielding

(19) Obviously, (19) is easier to implement than (18) since it avoids the evaluation of the contour integral. Shown in Table I is the comparison of different testing functions used in the discretization of the operator. From this table, it is clear that the function is the best candidate for a testing function, since it will result in a well-tested operator and a simple mathematical expression of the integral which can be evaluated accurately. B. The

Operator

At the first glance, the identity operator is usually considered the simplest operator in terms of discretization. In the

YAN et al.: IMPROVING THE ACCURACY OF THE SECOND-KIND FREDHOLM IEs BY USING THE BUFFA-CHRISTIANSEN FUNCTIONS

testing procedure, the identity operator is well-tested as long as the testing function lies in the same direction as that of the basis function. From this point of view, the good testing functions are and , while the first one is commonly used. If the discretization of the operator is considered from a different perspective, it can be found out that, although the discretization of the identity operator may have an analytical expression which allows the evaluation of the integration to be exact, the integral kernel is actually highly singular. Since the discretization of the operator can be expressed as [14]

(20)

is highly singular at . the implied integral kernel It finally turns out that, it is the discretization of the identity operator that contributes significantly to the total error of the second-kind integral equations [14]. Although the regularization methods [15], [16] can be employed to reduce this discretization error, they are not widely used because of the reason mentioned in INTRODUCTION. In this paper, it is shown that the discretization error due to the identity operator can be refunction instead of the duced greatly by choosing the function as the testing function. It is also demonstrated, that by using this discretization scheme, the accuracy of the second-kind Fredholm integral equations can be improved significantly. C. The

Operator

Since in most cases the operator comes along with the operator, the choice of the testing function should be the same as that of the operator, which was discussed earlier. Nevertheless, it should be pointed out that, whatever function is chosen as the testing function, as long as the basis and testing functions lie in the same plane in a specific geometric discretization of the object, the impedance element given by

(21) is always zero, because the magnetic (electric) field (at the point ) produced by the electric (magnetic) current (at the point ) is perpendicular to the plane formed by the current vector and . Clearly, as long as the surface of the object the vector is small, is is smooth, the near-field interaction, where always very weak, because the singularity in the integral kernel has been excluded from (21). This can be regarded as an algebraic interpretation of the concept “compact operator”. Consequently, it is also very clear that, the operator is no longer a

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compact operator if the surface of the object is not smooth, for example, if there are corners or sharp tips on the object. D. SIEs Based on the investigations above, adequate discretization schemes can be adopted to transform the SIEs into matrix equations. Both the conventional scheme and the scheme presented in this paper will be discussed. is the 1) SIEs of the First Kind: According to Table I, most suitable testing function to discretize the EFIE (9). For the PMCHWT equations (14), as long as the diagonal operators are well tested, the whole equations can be solved accurately. function is also a good testing function Therefore, the to discretize the PMCHWT equations. 2) SIEs of the Second Kind: Since the different choices of the testing functions will not change the compactness of the operator, it is sufficient to have a good discretization of the MFIE (10) as long as the operator is well tested. Based on the discusand are the adequate testing funcsion earlier, both tions to discretize the MFIE. However, as will be shown in the testing function, which is following sections, the use of the a conventional way to discretize the MFIE, will produce a much as the testing larger error compared to the choice of the function. Hence, the latter is recommended for the discretization of the MFIE, as proposed in [18]. It is also necessary to point is also a out that, for the discretization of the CFIE (11), good testing function since the operator is rotated by , which corresponds to the conventional way of discretizing the CFIE. testing function is applied directly to However, if the (11), although both the EFIE and the MFIE parts are adequately discretized and well tested, there will be a contour integral in the EFIE part (see Table I). Since the contour of the BC function is very complex, there is no way to achieve both a high accuracy and efficiency in the evaluation of this integral at the same time. In this paper, a mixed discretization scheme is adopted for the testing funcdiscretization of the CFIE, which uses the testing function tion to discretize the EFIE (9) and the to discretize the MFIE (10) before summing them up. Similarly, since the off-diagonal blocks are compact, it is sufficient to have a good discretization of the N-Müller equations (15) as long as the diagonal blocks are well tested. Similar to the and are the adequate MFIE in the PEC case, both testing function has been comtesting functions. In fact, the monly used [24], which resulted in a contour integration in the evaluation of the operators. In contrast, the use of the testing function can not only have a well-tested diagonal blocks, but also avoid the appearance of the undesired contour integral. testing function will More importantly, the use of the significantly suppress the discretization error of the identity operators, and hence improve the accuracy of the N-Müller equations, as will be demonstrated in the next section. IV. ACCURACY IMPROVEMENT OF THE IDENTITY OPERATOR In this section, the accuracy improved by the use of the rotated BC function as the testing function is investigated through a well-defined numerical test. Consider a plane wave traveling in the free space. By defining a closed

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Fig. 2. RMS of the far field as a function of the discretization density of the closed surface. The frequency of the incident plane wave is 50 MHz. Both the CRWG and the rotated BC functions are chosen as the testing functions in the discretization of the identity operators. (a) A sphere with a radius of 1.0 m. (b) A cube with a size of 1:0 1:0 1:0 m .

2

2

Fig. 3. RMS of the far field as a function of the discretization density of the closed surface. The frequency of the incident plane wave is 1.0 GHz. Both the CRWG and the rotated BC functions are chosen as the testing functions in the discretization of the identity operators. (a) A sphere with a radius of 1.0 m. (b) A cube with a size of 1:0 1:0 1:0 m .

2

2

mathematical surface with an arbitrary shape, a set of non-radiating equivalent surface current can be found through the relation [14]

(22) Expanding the equivalent currents with two sets of CRWG basis , and testing the two equations with two sets of functions testing functions , (22) can be transformed into a matrix equation, which can be solved for the expansion coefficients of the should not radiate, their basis functions. Since transverse radiated electric field in the far zone (23) where , can be regarded as the numerical error due to the discretization of the identity operators in (22). Following the definition in [14], we define the far-zone electric field as (24)

and stand for the and components of the where , the sampling points radiated electric field are on the - plane with and for . Then, the root mean square (RMS) of the radiated field, which is also the RMS error due to the discretization of the identity operators, can be calculated as (25) As the numerical test, a 50-MHz plane wave is incident on a closed mathematical surface with a shape of a sphere and a cube, respectively. The radius of the sphere is 1.0 m and the size of . Shown in Fig. 2(a) and (b) are the cube is the RMS error in the calculation of the far field defined by (25) as a function of the discretization density for the sphere and the cube, respectively. The same test is repeated at 1.0 GHz and the results are shown in Fig. 3(a) and (b). In all these figures, both and the rotated BC functions the CRWG functions are chosen as the testing functions for the discretization of the identity operators. It is very evident that by choosing the

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Fig. 4. RMS error of the RCS calculated by the MFIE and the EFIE in the PEC case versus discretization density. Both the CRWG testing scheme and the rotated-BC testing scheme of the MFIE are shown and compared with the EFIE. (a) A 75-MHz, V-polarized plane wave is incident on a PEC sphere with a radius of 1.0 m. (b) A 150-MHz, V-polarized plane wave is incident on a PEC sphere with a radius of 1.0 m.

Fig. 5. RMS error of the RCS calculated by the CFIE with a combination factor of 0.5 and the EFIE in the PEC case versus discretization density. Both the CRWG testing scheme and the mixed testing scheme of the CFIE are shown and compared with the EFIE. (a) A 75-MHz, V-polarized plane wave is incident on a PEC sphere with a radius of 1.0 m. (b) A 150-MHz, V-polarized plane wave is incident on a PEC sphere with a radius of 1.0 m.

rotated BC function as the testing function, the discretization error of the identity operator can be suppressed significantly, which serves as the major reason for the accuracy improvement of the second-kind SIEs, as will be shown in the next section.

V. ACCURACY IMPROVEMENT OF THE SURFACE INTEGRAL EQUATIONS In this section, the accuracy of the first- and the second-kind Fredholm integral equations in both PEC and dielectric cases will be compared, using the discretization schemes discussed in Section III-D. Although it is understood that other factors mentioned in INTRODUCTION, such as the inaccurate evaluation of the operator, also contribute to the error of the second-kind SIEs, only the error due to the discretization of the identity operators is investigated in this paper. Therefore, no special treatments such as those described in [3]–[11] are adopted here. The numerical model used is a sphere with a radius of 1.0 m, which has an analytical Mie-series solution that can be used as the reference data for comparison.

A. The PEC Case The first- and the second-kind Fredholm integral equations in the PEC case are the EFIE (9) and the MFIE (10), respectively. Shown in Fig. 4(a) and (b) are the RMS error in the radar cross section (RCS) of a PEC sphere calculated by the EFIE and the MFIE under the excitation of a 75-MHz and a 150-MHz plane wave, respectively. In these two figures, the accuracy of the EFIE and those of the MFIE with two different testing schemes are compared, with respect to the discretization density. Evidently, by using the CRWG function as the testing function, the MFIE gives a larger error than the EFIE, as has been commonly observed. However, when the rotated BC function is employed as the testing function, the MFIE gives a much smaller error, even smaller than that of the EFIE, thanks to the error suppression in the discretization of the identity operator. By setting the combination factor in the CFIE (11) to be , its accuracy is also investigated by using the CRWG testing scheme and the mixed testing scheme described in the preceding section. From Fig. 5(a) and (b), it is obvious that the accuracy of the CFIE by using the CRWG testing scheme is between those of the EFIE and the MFIE using the same testing scheme, and

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Fig. 6. RMS error of the RCS calculated by the CFIE in the PEC case with different discretization schemes and the iteration counts required in the BiCGstab(1) iterative solution to achieve a relative residual error of 10 , both as a function of the combination factor . A 75-MHz, V-polarized plane wave is incident on a PEC sphere with a radius of 1.0 m. (a) The CFIE with the CRWG testing scheme. (b) The CFIE with the mixed testing scheme.

Fig. 7. RMS error of the RCS calculated by the CFIE in the PEC case with different discretization schemes and the iteration counts required in the BiCGstab(1) iterative solution to achieve a relative residual error of 10 , both as a function of the combination factor . A 131.005-MHz, V-polarized plane wave is incident on a PEC sphere with a radius of 1.0 m. (a) The CFIE with the CRWG testing scheme. (b) The CFIE with the mixed testing scheme.

the accuracy of the CFIE by using the mixed testing scheme is between those of the EFIE and the MFIE using the rotated BC function as the testing function, as expected. Since the MFIE under the mixed discretization scheme has both better accuracy and faster iterative convergence than the EFIE, it is always desired to set the combination factor in the CFIE (11) as small as possible, as long as the existence of the EFIE part is sufficient to eliminate the spurious interior resonance. This is in contrast to the traditional CFIE, in which a compromise has to be made on the choice of the combination factor since a large value yields a slowly convergent but accurate solution, whereas a small value yields a fast convergent but inaccurate solution. This is investigated in Figs. 6 and 7. In Fig. 6(a) and (b), the accuracy and convergence of the CFIE with two different discretization schemes are investigated at 75 MHz, which is far from the interior resonance. The RMS error of the RCS and the iteration counts needed by the BiCGstab(1) [41], [42] iterative solution to achieve a relative residual error (RSS) are shown as functions of the combination factor , of corresponds to the MFIE and corresponds where to the EFIE. From Fig. 6(a), it can be seen that the smallest RMS error of the CFIE with the CRWG testing scheme can be

at the expense of a larger iteration number, achieved at at the while the smallest iteration count can be achieved at cost of a larger error. Therefore, a compromise has to be made between the accuracy and efficiency. From Fig. 6(b), in contrast, both the smallest RMS error and iteration count are achieved at , which corresponds to the MFIE with the rotated-BC testing scheme. To demonstrate the necessity of the EFIE, the same comparison is made at the first resonant frequency of the 1.0-m spherical cavity filled with air. Theoretically, the first resonant frequency of the unit spherical cavity is 131.016 MHz. However, due to the numerical discretization process, there is a small shift on the actual numerical resonant frequency. In this paper, the numerical resonant frequency is found to locate at 131.005 MHz by frequency searching using the MFIE with the rotated-BC testing scheme. Fig. 7(a) and (b) show the comparison of the CFIE with two different discretization schemes at this resonant frequency. It can be seen from both figures that by introducing a small combination factor, the numerical error of the MFIE due to the interior resonance can be effectively suppressed. From Fig. 7(a), it is clear that with CRWG used as the testing function, the optimal combination factor for the , while the optimal choice to achieve the RMS error is

YAN et al.: IMPROVING THE ACCURACY OF THE SECOND-KIND FREDHOLM IEs BY USING THE BUFFA-CHRISTIANSEN FUNCTIONS

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Fig. 8. RMS error of the RCS calculated by different SIEs in the dielectric case versus discretization density. Accuracy of the PMCHWT equations and the N-Müller equations are compared. Both the CRWG testing scheme and the rotated-BC testing scheme of the N-Müller equations are shown and compared with the : and PMCHWT equations. (a) A 75-MHz, V-polarized plane wave is incident on a dielectric sphere with a radius of 1.0 m and the dielectric parameters "  : . (b) A 100-MHz, H-polarized plane wave is incident on a dielectric sphere with a radius of 1.0 m and the dielectric parameters " : and  : .

=10

=40

=26 =10

Fig. 9. Iteration counts required by different SIEs in the dielectric case versus discretization density. The BiCGstab(1) iterative solver is used to solve the SIEs . (a) A 75-MHz, V-polarized plane wave is incident on a dielectric sphere with a radius of 1.0 m and the dielectric parameters to a relative residual error of " : and  : . (b) A 100-MHz, H-polarized plane wave is incident on a dielectric sphere with a radius of 1.0 m and the dielectric parameters " : and  : .

=26 =10 = 10

10

smallest iteration count is around . Hence a compromise is needed. However, from Fig. 7(b), it is seen that with the mixed testing scheme, the optimal combination factor for both . the RMS error and the iteration count is around Following observations can also be made from Figs. 6 and 7. 1) The accuracy of the CFIE with the mixed discretization scheme is better than that with the CRWG discretization scheme in the entire range of . 2) The convergence of the CFIE with the mixed discretization scheme is almost the same as that with the CRWG discretization scheme. B. The Dielectric Case The first- and the second-kind Fredholm integral equations in the dielectric case are the PMCHWT equations (14) and the N-Müller equations (15), respectively. Shown in Fig. 8(a) and (b) are the RMS error in the RCS of a dielectric sphere calculated by (14) and (15), versus the discretization density. In Fig. 8(a), a 75-MHz plane wave is incident on the sphere with a radius of 1.0 m and the dielectric parameter and . In Fig. 8(b), a 100-MHz plane wave is incident on the sphere with the dielectric parameter

=40

and . From these two figures, similar observations to those in the PEC case can be made. By using the CRWG testing scheme, the N-Müller equations give a larger error than the PMCHWT equations, as has been commonly observed. When the proposed rotated-BC testing scheme is employed, the N-Müller equations give a smaller error than the PMCHWT equations. The main reasons of the accuracy improvement of the N-Müller equations are as follows. 1) The use of the rotated BC testing functions suppresses the discretization error of the identity operators significantly. 2) The use of the rotated BC testing functions avoids the contour integral in operators, hence resulting in a more accurate and efficient evaluation of the impedance elements from the discretization of the operators. Fig. 9(a) and (b) show the iteration counts required by the PMCHWT and the N-Müller equations in a BiCGstab(1) itera. Apparently, the tive solution in order to achieve a RSS of convergence rates of the second-kind SIEs using CRWG testing scheme and rotated-BC testing scheme are almost the same, due to the obvious reason that both schemes test the identity operators well and maintain the compactness of the remainder parts, which yield the matrix equations with condition numbers

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Fig. 10. Interior resonance test of the PMCHWT equations, the N-Müller equations with the CRWG testing scheme, and the N-Müller equations with the ro: and  : . The radius of the sphere is 1.0 m and the lowest analytical tated-BC testing scheme for scattering analysis of a dielectric sphere with " resonant frequency of the corresponding spherical cavity filled with air is 131.016 MHz. According to the frequency search using the EFIE for the dielectric case, the numerical resonant frequency of the discretized sphere model is 131.036 MHz. (a) Condition numbers as a function of the frequency in a small band around the analytical resonant frequency. The condition number of the EFIE for the dielectric case is shown as reference. (b) Convergence histories of different equations at the frequency of 131.036 MHz. The convergence history of the EFIE for the dielectric case is shown as reference. to achieve a relative residual error of

= 40

= 10

10

that are not only small, but also invariant with respect to the discretization density. On the other hand, the iteration counts needed by the first-kind SIEs increase exponentially with the increase of the discretization density, similar to that observed in the EFIE for the PEC case. An example is next designed to test the performances of the first- and the second-kind integral equations for the dielectric case at an interior resonant frequency of an object, which is a dielectric sphere with a radius of 1.0 m and the relative and , permittivity and permeability of respectively. The lowest analytical resonant frequency of the corresponding spherical cavity filled with air is 131.016 MHz. Frequency search has been applied using the EFIE (3) for the dielectric case to locate the numerical resonant frequency, which is 131.036 MHz under a specific curvilinear triangular discretization. Fig. 10(a) shows the condition numbers of the impedance matrices obtained by discretizing the EFIE, the PMCHWT, the N-Müller with the CRWG testing scheme, and the N-Müller with the rotated-BC testing scheme, as a function of the frequency in a small band around the analytical resonant frequency. A very small frequency step, which is 1.0 KHz, is used around 131.016 MHz in the frequency search in order to obtain a smooth curve for the condition number and the correct numerical resonant frequency. It can be seen that the rotated-BC testing scheme dose not deteriorate the immunity of the interior resonance corruption of the N-Müller equations. The convergence histories of the four different formulations at the numerical resonant frequency are given in Fig. 10(b). It is clear that the N-Müller equations with both testing schemes much more rapidly can converge to the desired RSS of than their first-kind counterpart which is the PMCHWT equations, while the EFIE for the dielectric case has a very slow convergence due to the interior resonance corruption. VI. DISCUSSION From the theoretical investigation and the numerical demonstration provided in the preceding sections, it is very clear that

by using the proposed discretization scheme, the accuracy of the second-kind integral equations, both in PEC and dielectric cases, can be improved by orders of magnitude. This significant accuracy improvement can be explained from the following aspects. First, as was shown in Section IV, the discretization error due to the identity operator, which is shown in [14] to be the major error source of the solution to a second-kind integral equation, is significantly suppressed by the proposed scheme. Second, the accuracy improvement can also be attributed to the appropriate adoption of the Rayleigh-Ritz scheme [12]. As a result of the variational method, it is well known that the Rayleigh-Ritz scheme can minimize the physical quantities such as the energy or reaction, and therefore, is able to stabilize the numerical solution [43]. In [12], it was shown that when is employed as the basis function, a the CRWG function Rayleigh-Ritz scheme can be constructed for the MFIE by as the testing function. Unfortunately, this choosing the explicit way of constructing the Rayleigh-Ritz scheme results in an ill-conditioned system matrix which is difficult to solve accurately. The proposed discretization scheme in this paper can also be regarded as the Rayleigh-Ritz scheme in an implicit and sense. In fact, if we notice that both the CRWG function the BC function can be expressed as the linear superposition of the CRWG functions defined on the barycentric refinement of the original curvilinear triangular mesh (26)

(27) where the definitions of the weighting factors , , , and , can be found in [26], the MFIE system matrix obtained by using as the basis function and as the testing function can be expressed as (28)

YAN et al.: IMPROVING THE ACCURACY OF THE SECOND-KIND FREDHOLM IEs BY USING THE BUFFA-CHRISTIANSEN FUNCTIONS

where and stand for the transformation matrices mapping and the BC functions defined from the CRWG functions on the original mesh to the CRWG functions defined on the barycentric refinement, while their columns are composed by s in (26) and s in (27), respectively, and is the MFIE system matrix obtained on the barycentric refinement by using as the basis function and as the testing function. Evidentally, is the Rayleigh-Ritz discretization of the MFIE on the barycentric refinement according to [12], while and are nothing but some geometric relations between the original mesh and its barycentric refinement. As a result, is shown to be the linear combination of the rows and columns of with some weighting factors defined in and . Since is obtained from the explicit Rayleigh-Ritz scheme, as its linear combination, can be regarded as the implicit Rayleigh-Ritz discretization of the MFIE, and therefore, is able to minimize the functional and stabilize the solution.

VII. CONCLUSION In this paper, the surface integral equations that are widely used in computational electromagnetics are investigated as the Fredholm integral equations of the first and the second kind. The mathematical characteristics of the operators involved in these integral equations are discussed and the corresponding discretization strategies are studied. The rotated BC function is shown, both theoretically and numerically, to be a better testing function for the discretization of the second-kind integral equations for both the PEC and the dielectric cases. It is demonstrated through some numerical experiments that by using the presented discretization scheme, the discretization error of the identity operator, which is shown to be a major error source of the second-kind integral equations, can be suppressed significantly. It is also shown that the proposed discretization scheme can be regarded as an implicit Rayleigh-Ritz scheme, which is able to minimize the system energy and stabilize the numerical solution. As a result, the overall numerical error of the second-kind surface integral equations in both the PEC and the dielectric cases can be reduced significantly, leading to accurate numerical solutions that are comparable to (or even better than) the existing solutions of their first-kind counterparts. At the same time, the fast convergence of the second-kind integral equations are maintained with the rotated-BC testing scheme. In the PEC case, the CFIE with a mixed discretization scheme is proposed to eliminate the spurious interior resonance corruption, and the optimal choice of the combination factor is shown to . In the dielectric case, the proposed roto be around tated-BC testing scheme maintains the immunity of the spurious interior resonance corruption of the N-Müller equations, leading to an accurate and fast convergent formulation at all frequencies.

REFERENCES [1] C. T. Tai, Dyadic Green Functions in Electromagnetic Theory, 2nd ed. Piscataway, NJ: IEEE Press, 1994. [2] E. I. Fredholm, “Sur une classe d’equations fonctionnelles,” Acta Mathe., vol. 27, pp. 365–390, 1903.

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[3] J. M. Rius, E. Úbeda, and J. Parrón, “On the testing of the magnetic field integral equation with RWG basis functions in method of moments,” IEEE Trans. Antennas Propag., vol. 49, no. 11, pp. 1550–1553, Nov. 2001. [4] Ö. Ergül and L. Gürel, “Improving the accuracy of the MFIE with the choice of basis functions,” in Proc. IEEE AP-S Int. Symp. and URSI Radio Sci. Mtg., Monterey, CA, Jun. 2004, pp. 3389–3392. [5] E. Úbeda and J. M. Rius, “MFIE MoM-formulation with curl-conforming basis functions and accurate kernel-integration in the analysis of perfectly conducting sharp-edged objects,” Microw. Opt. Technol. Lett., vol. 44, no. 4, pp. 354–358, Feb. 2005. [6] L. Gürel and Ö. Ergül, “Singularity of the magnetic-field integral equation and its extraction,” IEEE Antennas Wireless Propag. Lett., vol. 4, pp. 229–232, 2005. [7] Ö. Ergül and L. Gürel, “Solid-angle factor in the magnetic-field integral equation,” Microw. Opt. Technol. Lett., vol. 45, no. 5, pp. 452–456, June 2005. [8] Ö. Ergül and L. Gürel, “Improved testing of the magnetic-field integral equation,” IEEE Antennas Wireless Propag. Lett., vol. 15, pp. 615–617, 2005. [9] Ö. Ergül and L. Gürel, “Improving the accuracy of the magnetic field integral equation with the linear-linear basis functions,” Radio Sci., vol. 41, 2006 [Online]. Available: http://www.cem.bilkent.edu.tr/JP/ JP31.pdf [10] E. Úbeda, A. Heldring, and J. M. Rius, “Accurate computation of the impedance elements of the magnetic-field integral equation with RWG basis functions through field-domain and source-domain integral swapping,” Microw. Opt. Technol. Lett., vol. 49, no. 3, Mar. 2007. [11] Ö. Ergül and L. Gürel, “Linear-linear basis functions for MLFMA solutions of magnetic-field and combined-field integral equations,” IEEE Trans. Antennas Propag., vol. 55, no. 4, pp. 1103–1110, Apr. 2007. [12] S. Yan and Z. Nie, “On the Rayleigh-Ritz scheme of 3D MFIE and its normal solution,” presented at the IEEE Antennas Propag. Symp., San Diego, CA, Jul. 2008. [13] P. Ylä-Oijala, M. Taskinen, and S. Järvenpää, “Analysis of surface integral equations in electromagnetic scattering and radiation problems,” Engineering Analysis with Boundary Elements, vol. 32, no. 3, pp. 196–209, 2008. [14] Ö. Ergül and L. Gürel, “Discretization error due to the identity operator in surface integral equations,” Comput. Phys. Commun., vol. 180, no. 10, pp. 1746–1752, Oct. 2009. [15] C. P. Davis and K. F. Warnick, “High-order convergence with a low-order discretization of the 2-D MFIE,” IEEE Antennas Wireless Propag. Lett., vol. 3, no. 1, pp. 355–358, 2004. [16] K. F. Warnick and A. F. Peterson, “3D MFIE accuracy improvement using regularization,” in Proc. IEEE Antennas and Propagation Society Int. Symp., Honolulu, HI, Jun. 2007, pp. 4857–4860. [17] A. Buffa and S. H. Christiansen, “A dual finite element complex on the barycentric refinement,” Math. Comput., vol. 76, no. 260, pp. 1743–1769, Oct. 2007. [18] K. Cools, F. P. Andriulli, F. Olyslager, and E. Michielssen, “Improving the MFIE’s accuracy by using a mixed discretization,” in Proc. IEEE Antennas Propag. Symp., North Charleston, SC, Jun. 2009. [19] A. J. Poggio and E. K. Miller, “Integral equation solutions of three dimensional scattering problems,” in Computer Techniques for Electromagnetics, R. Mittra, Ed. Elmsford, NY: Permagon, 1973. [20] Y. Chang and R. Harrington, “A surface formulation for characteristic modes of material bodies,” IEEE Trans. Antennas Propag., vol. 25, no. 6, pp. 789–795, June 1977. [21] T.-K. Wu and L. L. Tsai, “Scattering from arbitrarily-shaped lossy dielectric bodies of revolution,” Radio Sci., vol. 12, no. 5, pp. 709–718, 1977. [22] S. Yan, J.-M. Jin, and Z. Nie, “A comparative study of Calderón preconditioners for PMCHWT equations,” IEEE Trans. Antennas Propag., vol. 58, no. 7, pp. 2375–2383, Jul. 2010. [23] C. Müller, Foundations of the Mathematical Theory of Electromagnetic Waves. Berlin, Germany: Springer, 1969. [24] P. Ylä-Oijala and M. Taskinen, “Well-conditioned Müller formulation for electromagnetic scattering by dielectric objects,” IEEE Trans. Antennas Propag., vol. 53, no. 10, pp. 3316–3323, Oct. 2005. [25] S. Yan, J.-M. Jin, and Z. Nie, “Calderón preconditioner: From EFIE and MFIE to N-Müller equations,” IEEE Trans. Antennas Propag., vol. 58, no. 12, pp. 4105–4110, 2010. [26] 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.

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[27] C. A. Balanis, Advanced Engineering Electromagnetics. New York: Wiley, 1989. [28] A. F. Peterson, “The “interior resonance” problem associated with surface integral equations of electromagnetics: Numerical consequences and a survey of remedies,” Electromagnetics, vol. 10, no. 3, pp. 293–312, 1990. [29] J. R. Mautz and R. F. Harrington, “H-field, E-field, and combined-field solutions for conducting bodies of revolution,” Arch. Elektron. Ubertragungstech. (Electron. Commun.), vol. 32, no. 4, pp. 159–164, 1978. [30] 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, no. 3, pp. 316–328, Mar. 1997. [31] L. Gürel and Ö. Ergül, “Contamination of the accuracy of the combined-field integral equation with the discretization error of the magnetic-field integral equation,” IEEE Trans. Antennas Propag., vol. 57, no. 9, pp. 2650–2657, Sep. 2009. [32] P. Ylä-Oijala and M. Taskinen, “Calculation of CFIE impedance matrix ^ RWG functions,” IEEE Trans. Antennas elements with RWG and n Propag., vol. 51, no. 8, pp. 1837–1846, Aug. 2003. [33] X. Q. Sheng, J.-M. Jin, J. Song, W. C. Chew, and C.-C. Lu, “Solution of combined-field integral equation using multilevel fast multipole algorithm for scattering by homogeneous bodies,” IEEE Trans. Antennas Propag., vol. 46, no. 11, pp. 1718–1726, Nov. 1998. [34] P. Ylä-Oijala and M. Taskinen, “A novel combined field integral equation formulation for solving electromagnetic scattering by dielectric and composite objects,” in Proc. IEEE Antennas Propag. Symp., Jul. 2005, vol. 4B, pp. 297–300. [35] M. S. Yeung, “Single integral equation for electromagnetic scattering by three-dimensional homogeneous dielectric objects,” IEEE Trans. Antennas Propag., vol. 47, no. 10, pp. 1615–1622, Oct. 1999. [36] S. Yan and Z. Nie, “A set of novel surface integral equations for electromagnetic scattering from homogeneous penetrable objects,” presented at the Asia-Pacific Microwave Conf., Hong Kong, China, Dec. 2008. [37] S. M. Rao, D. R. Wilton, and A. W. Glissonc, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag., vol. 30, no. 3, pp. 409–418, May 1982. [38] R. D. Graglia, D. R. Wilton, and A. F. Peterson, “Higher order interpolatory vector bases for computational electromagnetics,” IEEE Trans. Antennas Propag., vol. 45, no. 3, pp. 329–342, Mar. 1997. [39] 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, no. 8, pp. 2398–2412, Aug. 2008. [40] S. Yan, J.-M. Jin, and Z. Nie, “EFIE analysis of low-frequency problems with loop-star decomposition and Calderón multiplicative preconditioner,” IEEE Trans. Antennas Propag., vol. 58, no. 3, pp. 857–867, Mar. 2010. [41] G. L. G. Sleijpen and D. R. Fokkema, “BiCGstab(l) for linear equations involving unsymmetric matrices with complex spectrum,” Electronic Trans. Numer. Anal., vol. 1, pp. 11–32, Sep. 1993. [42] 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. [43] W. C. Chew, Waves and Fields in Inhomogeneous Media. New York: IEEE Press, 1995.

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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, China, in 2005, where he is currently working toward the 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, Urbana, under the financial support from the China Scholarship Council. His research interests include numerical methods in computational electromagnetics, especially integral equation based methods, fast algorithms, and preconditioning techniques. Mr. Yan was the recipient of the Best Student Paper Award presented by the IEEE Chengdu Section in 2010.

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 the Y. T. Lo Endowed Chair Professor of Electrical and Computer Engineering and Director of the Electromagnetics Laboratory and Center for Computational Electromagnetics at 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 200 papers in refereed journals and 20 book chapters. He has also authored The Finite Element Method in Electromagnetics (Wiley, 1st ed, 1993, 2nd ed, 2002), Electromagnetic Analysis and Design in Magnetic Resonance Imaging (CRC, 1998), and Theory and Computation of Electromagnetic Fields (Wiley, 2010), 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 was 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 Low-Dispersion Realization of Precise Integration Time-Domain Method Using a Fourth-Order Accurate Finite Difference Scheme Zhong-Ming Bai, Xi-Kui Ma, and Gang Sun

Abstract—A modified precise integration time-domain (PITD) method, called PITD(4) algorithm, is presented in order to mitigate the numerical dispersion errors of a recently proposed PITD method. The PITD(4) method is based on both the fourth-order accurate finite-difference scheme and the precise integration technique. Both the stability condition and the numerical dispersion relations of the PITD(4) method are derived analytically and the effects of spatial and time steps on the numerical dispersion are investigated in detail. It is found that with the precise integration technique, the stability condition of the PITD(4) method is much larger than the Courant-Friedrich-levy (CFL) stability condition of the conventional finite difference time domain; with the fourthorder accurate finite-difference scheme, the numerical dispersion errors of the PITD(4) method are much less than that of the PITD methods. Numerical examples are presented to validate the accuracy and the effectiveness of the PITD(4) method, and to verify our analysis of the numerical dispersion characteristics of the PITD(4) method. Index Terms—Fourth-order accurate difference schemes, precise integration time-domain (PITD) method, numerical dispersion, Maxwell’s equations.

I. INTRODUCTION

D

UE to its simplicity, straightforwardness and easy uses, the finite-difference time-domain (FDTD) method [1], [2] has been accepted extensively as one of the most effective means for solving transient electromagnetic problems. However, the Courant-Friedrich-Levy (CFL) stability condition and the numerical dispersion error have limited the intense utilization of FDTD method. Time step has to be small enough to satisfy the CFL stability condition, and spatial step must be very small to minimize numerical dispersion errors, too. As a result, this leads to a great increase of CPU-time consuming and severely compromises of effectiveness of FDTD method. To improve the computation efficiency, various time-domain techniques have been developed [3]–[5]. Recently, the precise integration time-

Manuscript received September 16, 2009; revised September 03, 2010; accepted September 09, 2010. Date of publication January 31, 2011; date of current version April 06, 2011. This work was supported by the National Nature Science Foundation of China by Grant 50877055. Z.-M. Bai and G. Sun are with the School of Electrical Engineering, Xi’an Jiaotong University, Xi’an 710049, China (e-mail: [email protected]). X.-K. Ma is with the State Key Laboratory of Electrical Insulation and Power Equipment, School of Electrical Engineering, Xi’an Jiaotong University, Xi’an 710049, 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.2011.2109673

domain (PITD) method, hereafter designated as “the conventional PITD method,” has been presented for solving Maxwell’s equations. The precise integration (PI) technique [6] is proposed by Zhong first. It offers a precise algorithm of computing matrix exponential function with an accuracy of machinery precision. So, the ordinary differential equations (ODEs) can be solved exactly with this technique. Zhao and Ma, et al. employ this technique in calculating the transient responses of transmission lines [7], the effects of interconnects in VLSI [8], and the propagation of electromagnetic wave in ferromagnetic sheet [9]. Recently this technique is employed to solve three dimensional Maxwell’s equations [10], [11]. The fundamental idea of the conventional PITD method [10], [11] is to discretize the spatial derivative, so as to reduce Maxwell’s curl equations to a set of ODEs. Despite of the large memory requirement, the conventional PITD scheme has advantages over the conventional Yee’s FDTD method in using a large time step and over the ADI-FDTD method in having high computational accuracy, and numerical dispersion errors of the conventional PITD scheme can be made nearly independent of the time step size [12], [13]. Although the stability condition of conventional PITD method has the superiority of being much larger than CFL stability condition and its numerical dispersion errors can be made nearly independent of the time step size, the numerical dispersion errors of the conventional PITD method have been shown being larger than that of the conventional Yee’s FDTD method while solving wave equations. This behavior is due to the fact that the PITD scheme deals with the spatial operators and the time integration processes separately and that the numerical dispersion error is caused by approximating the spatial derivatives with a second-order central finite difference. Therefore, the approximation of spacial derivatives dominate the performance of the method, i.e., when the time step size increases, the improvement becomes insignificant. Prompted by the above mentioned reasons, we consider using the higher-order finite-difference formula to approximate the spatial derivatives of the field components for constructing a low numerical dispersion algorithm of precise integration time-domain method in this paper. For example, a fourth-order accurate difference scheme is used to obtain the spatial first-derivatives. By means of numerical dispersion analysis, we found that the numerical dispersion error of this modified PITD method is significantly reduced. Practical applications included in this paper indicate that using a fourth-order accurate difference scheme to

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approximate the spatial first-derivative results in a PITD scheme that can achieve significantly accelerated performances compared with the conventional PITD method and the conventional Yee’s FDTD method. Besides, a Gauss quadrature is used to calculate the non-homogeneous term integrations occurring in an actual PITD computation. As a result, the inversion of matrix is avoided and the higher accuracy of numerical results of non-homogeneous term integration is acquired without additional computational cost. The modified PITD method shows better dispersion performance than the FDTD and the conventional PITD methods. This paper is organized as follows. In Section II, a fourthorder accurate difference scheme is used to approximate the spatial derivatives of the electric and magnetic field components so as to reduce Maxwell’s curl equations to a set of ODEs. In Section III, PITD method is reviewed and the Gauss quadrature is used in calculating nonhomogenous term integration. In Section IV, the stability condition of the modified PITD method is derived. In Section V, the numerical dispersion relation of the modified PITD method is derived analytically and its dispersion properties are discussed in detail. In addition, the results are compared with those of the conventional PITD method and the conventional Yee’s FDTD method. In Section VI, two examples are simulated to confirm the theory. Finally, conclusions are drawn.

the PITD’s ODEs) and shown in (4a)–(4f) on the next page. Rewrite (4) as a matrix form

(5) is a one-column vector containing the electric and where magnetic field components defined on the discrete spatial latis a time-invarant coefficient matrix determined by the tices, spatial step sizes and is a column vector introduced by the excitations. In short, later in this paper, we designate the conventional PITD method as PITD(2) with the number in the parenthesis indicating the formal accuracy in space, and the new PITD method proposed here as the PITD(4), respectively. III. PRECISE INTEGRATION TIME-DOMAIN METHOD REVIEW AND GAUSS QUADRATURE According to the theory of ODEs, the solution of the homogenous equation of (5) can be written as (6) Denoting the time step size as ration instants

, we get a series of equal du. So we have (7)

II. DISCRETIZATION OF MAXWELL’S CURL EQUATIONS

where

, which can be computed as

For using the PI technique, Maxwell’s curl equations should be transformed into a set of ODEs. Here, the spatial partial derivatives of Maxwell’s curl equations

(8)

(1a)

where is the subtime step, and is the should preselected arbitrary integer. When selecting be an extremely small time interval. Hence, for the interval of , the truncated Taylor series expansion is of high precision

(1b) need to be discretize into finite difference scheme. The difference scheme used in the conventional PITD method [10] is the traditional Yee central-finite-difference

(9) (2) where which preserves the second-order accuracy. Apparently, the conventional PITD method is only second-order accurate in space and hereby causes significantly nonphysical dispersion and nonphysical anisotropy errors to any electromagnetic wave traveling through a medium. To improve this situation and construct a low numerical dispersion algorithm, we consider using the higher-order finite-difference formula to approximate the spatial derivatives of the field components. Normally, the higher-order-accurate difference schemes for the spatial derivatives can be derived by Taylor’s series expanding method or by fitting polynomial method. Here, the fourth-order accurate difference scheme (3), shown at the bottom of the next page, is used to approximate the spatial partial derivatives so as to reduce Maxwell’s curl equations to a set of ODEs (here called

(10) is very small. So it will almost be rounded-off Note that is added to the unit matrix directly. An effective when rather than the total value of method is to operate on the . As

(11) (12)

BAI et al.: LOW-DISPERSION REALIZATION OF PITD METHOD USING A 4TH ORDER ACCURATE FD SCHEME

and can be computed by staring with (10) and then run the following instruction: do

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Further more, the solution of nonhomogenous equation (5) can be written as

(15) (13) From the above equation, it can be seen that the performance and computational efficiency of the PITD method is also strongly on dependent on the integration of nonhomogenous term the right-hand side (RHS) of (15) in an actual PITD computais assumed to be tion. In [10], the nonhomogenous term linear within the time step

end do Finally

(14) Equations (10), (13), and (14) give the precise-integration technique for computing the exponential matrix . The discussion on the choice of the time step and the choice of number of is included in the Appendix A.

(16) So we have the following recursive form solution:

(17)

(3)

(4a)

(4b)

(4c)

(4d)

(4e)

(4f)

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Note that in any case, the computation of the inverse matrix is always needed. Unfortunately, in many cases, the matrix obtained from practical problems is noninvertable. Hence, (17) is unavailable directly. In order to solve this difficulty, a scheme is developed in [10]. As indicated in [10], the key to solving the noninvertable matrix is to find the linear relation among the discretized field components. Analyzing a great deal of numerical examples with the PITD method, we found that although the linear relations obtained in [10] could generalize most of the situations, they could not include all, i.e., in different practical problems, new ones often occur. In other words, we should make a concrete analysis of concrete problems with one’s experience in an actual computation. In contrast with [10], in this paper, the Gauss integration quadrature is used to eliminate this deficiency. The nonhomogenous term integration of (15) can be calculated by Gauss integration formula as follows:

Substituting (20) into (4), we get the following PITD’s ODEs (21) for a plane wave propagating through a homogenous lossless , i.e., region of space. Here is a one-column vector containing electric and magnetic field then becomes related to components. The coefficient matrix and , spatial steps, and the spatial frequencies, , and the medium permeability and the permittivity . The is explicit form of

(22)

where (23a) (18) and are the weights and locations of each of the Here, Gauss quadrature points, respectively. The matrix exponential function can also be calculated by the precise integration technique. So the recurrence formula of the PITD(4) method can be written as

(23b) (23c) According to the matrix theory,

can be expressed as (24)

Here is the eigenvector and is the eigenvalues of eigenvalues of the matrix can be obtained as

. The

(19)

(25a)

The computational accuracy of (19) depends on the number of the Gauss quadrature points and the time step size . This approach can reach any arbitrary order of accuracy by using a corresponding and can avoid the computation of matrix inversion.

(25b)

IV. DERIVATION OF STABILITY CONDITION In this section, we will give a derivation of the stability condition for the PITD(4) method by using the von Neumann criterion. In the spatial spectral domain, an electromagnetic component can be expressed in the following discrete form of a plane wave [12]:

(20) , and are the numerical wave numbers along the where , and directions, respectively. is the magnitude of a field component or (here ) that varies with time .

(25c) For the OEDs (21), the following recursive scheme is used to from those at : update the field components at (26) where (27) where, as above, . By substituting (24) into (27) and applying the spectral mapping theorem, the six eigenvalues of can be found as

(28)

BAI et al.: LOW-DISPERSION REALIZATION OF PITD METHOD USING A 4TH ORDER ACCURATE FD SCHEME

Substitution of (25) into (28) results in the six eigenvalues of as

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Substituting (35) into (26), we arrive at (36)

(29a) (29b) (29c) Here

From the determinant operator of the coefficient matrix in the above (36), the dispersion relationship of the PITD(4) method can be derived. So we have

is (37) It can be simplified as the following form: (30) (38)

and

is where (31)

(39)

Based on von Neumann’s criterion, if all the eigenvalues of are less than or equal to unity in magnitude, the numerical solution is stable. The condition of the PITD(4) method to be stable is then

Equation (38) is the numerical dispersion relationship of the PITD(4) method. To study the numerical dispersion properties of the PITD(4) method, the following definitions will be used hereinafter. , which is defined as 1) The grid sampling density . Here ( for a uniform is cubical discretization in 3-D) is the spatial step, and the real wavelength in space. 2) The numerical wavenumber , which can be solved numerically from (38). We have , and in spherical coordinate system. 3) The numerical phase velocity , which is defined as . And the per cell attenuation constant [2]. . 4) The normalized phase velocity . 5) The normalized phase velocity error 6) The Courant number , which is defined as . In the following subsections, several aspects of the numerical dispersion errors of the PITD(4) method are studied in detail and compared with those of the FDTD, FDTD(2,4), and PITD(2) methods.

(32) It is equivalent to (33) Therefore, the stability condition of the PITD(4) method is (34)

. where We can see clearly that the preselected integer and the spatial step size determine the upper limit of the time step size of the PITD(4) method. Meanwhile, the upper limit of the time step size of PITD(4) is much larger than the CFL limit. V. NUMERICAL DISPERSION RELATIONSHIP OF THE PITD(4) METHOD

A. On-Axis Numerical Phase Velocity Versus Spatial Step

Since the phase velocity of numerical wave modes differs from the velocity of real wave, numerical dispersion is an undesired nonphysical effect inherently in the numerical algorithm. The representation of numerical dispersion is that the phase velocity depends on the spatial step size, the time step size, and the wave propagation direction. In this section, the numerical dispersion property of the PITD(4) method will be discussed in detail. Assuming that a plane-wave with an angular frequency propagates in an isotropic, homogeneous, linear, and lossless medium, we have (35)

Suppose that a plane wave propagates along the major axes of space grid, and , then , and . From (38), the variation curves of the normalized phase velocity and the per cell attenuation constant of the PITD(4) can be method as a function of the grid sampling density obtained by utilizing Newton-Raphsion iteration method. Figs. 1, 2, and 3 graph the normalized phase velocity, the normalized phase velocity error, and the per cell attenuation for FDTD, constant versus the grid-sampling density FDTD(2,4), PITD(2), and PITD(4) methods, respectively. and the time step Here, we set Courant number ( is the velocity of real wave) in each method, and

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Fig. 1. Normalized phase velocity versus grid sampling density. Here, S : ;N . , and l

=05

=5

= 32

Fig. 2. Normalized phase velocity versus error (%)grid sampling density. Here, S : ;N , and l .

=05

=5

= 32

the preselected integer , i.e., the number of the subtime in both PITD(2) and PITD(4) methods. step These plots show that for the coarse grid, the numerical wave number can be complex in each method. In this case, the phase velocity of numerical wave can be superluminal or subluminal, but the numerical wave still experiences exponential decay. As the grid points per wavelength increase, the numerical wave number becomes real, and the normalized phase velocity is gradually closes to unity, in each method. Clearly, it can be seen that with the higher-order finite-difference formula to discretize the spatial derivatives of the field components, the normalized phase velocity nears unity more quickly. Obviously, the PITD(4) method can provide a significant improvement over the PITD(2) method. Interestingly, the PITD(4) method shows a significant improvement on the phase velocity of numerical wave compared with the FDTD(2,4) method as grid sampling density is greater than 5.7. It can be observed that the phase velocity of numerical wave may be superluminal in FDTD(2,4) method, and is always subluminal in PITD(4) method.

Fig. 3. Per cell attenuation constant versus grid sampling density. Here, S : ;N , and l .

05

=5

= 32

Fig. 4. Normalized phase velocity versus propagation angle. Here, ;N , and l

N

=5

=5

= 32

=

S = 0:5;

B. Dispersion Anisotropy Anisotropy of the numerical phase velocity is described by the difference between the maximum and minimum normalized phase velocities for different propagation directions at a particular wavelength. Knowledge of the anisotropy of the numerical phase velocity is required for the understanding and evaluation of a PITD scheme. In this subsection, the dispersion anisotropy feature of the PITD(4) method is studied in detail. Suppose that a plane wave propagates on the X-Y plane, , the Courant number is , and the spatial step size . From the dispersion relationship is uniform (38), the variation curve of the normalized phase velocity with wave-propagation angle can be obtained for the PITD(4) method. Figs. 4 to 6 show how the propagation angle affects the numerical phase velocity, with the grid sampling density , and , respectively. Obviously, it can be observed that the curves of the normalized phase velocity of

BAI et al.: LOW-DISPERSION REALIZATION OF PITD METHOD USING A 4TH ORDER ACCURATE FD SCHEME

Fig. 5. Normalized phase velocity versus propagation angle. Here, N ;N . , and l

= 10

=5

= 32

Fig. 6. Normalized phase velocity versus propagation angle. Here, ;N , and l .

N

= 20

=5

= 32

S = 0:5;

S = 0:5;

the PITD(4) and FDTD(2,4) methods are closer to unity and flatter than the PITD(2) and FDTD methods. This means that the anisotropy of numerical wave can be significantly reduced by using the fourth-order finite-difference formula in spatial discretization. In addition, it is interesting to notice that with , the numerical phase the increase of grid sampling density velocity of the PITD(4) method is more quickly closer to the real wave velocity than that of the FDTD(2,4) method. And, the phase velocity of numerical wave is always superluminal in FDTD(2,4) method, and is always subluminal in PITD(4) method, i.e., with the precise integration technique the phase velocity of numerical wave is decreased. C. Effect of Time Step on Phase Velocity In the following, the effect of time step on the numerical dispersion errors of the PITD(4) method is discussed in detail. Suppose that a plane wave propagates along the direction, , . Then , and . Fig. 7 illusand trates the normalized phase velocity versus the Courant number , with , and , respectively. Here, the preselected integer and . The curves of the

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Fig. 7. Normalized phase velocity of the PITD(4) method versus Courant number S . Here, N . ;l

= 20 = 2

Fig. 8. Normalized phase velocity of the PITD(4) method versus propagation ;N . angle. Here, N , and l

= 20

= 20

=2

numerical phase velocity are almost a set of horizontal lines. In other words, the numerical phase velocity can be made nearly independent of the time step size in the PITD(4) method. Compared with the FDTD method, the improvement is wideband and for all Courant number . , Suppose that a plane wave propagates on X-Y plane, . Then and the spatial step size is uniform , and . Fig. 8 shows that how the time step size affects the anisotropy in the PITD(4) and , the method. Here, the preselected integer , and , Courant number respectively. Apparently, we can hardly distinguish the differ, ences between the four curves for . This means that the anisotropy of numerical and wave can be made nearly independent of the time step size in the PITD(4) method. VI. NUMERICAL EXAMPLES In order to validate the accuracy and the effectiveness of the PITD(4) method, two examples are presented in the following subsections. And, the numerical results of the PITD(4) method

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Fig. 9. Normalized electric fields of the scattering by conducting cylinder.

are compared with analytic solutions, the results of the FDTD and the PITD(2) methods.

Fig. 10. The numerical error of the normalized electric fields PITD(2) and PITD(4) method.

E

of FDTD,

A. Scattering Problem in 2-D Case Consider the scattering of a TM wave by a conducting cylinder. The locations of the source, the observing point, and the conducting cylinder are shown in Fig. 9. The diameter of the cylinder is 6 cm. Assuming that the point source to be simulated has the form of (40) where , and GHz. The Courant number is fixed at 0.5. A uniform mesh m is used. with For using the Precise Integration Time domain method directly, the domain decomposition was used in doing related computations [14]. The whole computational domain is decomposed into some subdomains, and then for each time-step, PITD method is used to solve Maxwell’s equations in each subdomain, and the whole domain result is derived from combining the results of all sub-domains. The numerical results of the observing point are obtained with 400 250 Yee cells for FDTD, PITD(2), and PITD(4) methods. In the actual computation, the time step size of FDTD, PITD(2) s. The parameter and PITD(4) is set as of PITD(2) and PITD(4) method is set as . For the case of the theory solution is known [15], the theory solution can be compared with the numerical results of FDTD, PITD(2), and PITD(4) methods. From Fig. 9, as the numerical results of PITD(4) are more accurate than those of FDTD and PITD(2) methods, we can hardly distinguish the difference between the numerical results of the PITD(4) method and the theory results. To see clearly, another kind of error is defined as

where denotes the numerical result of the FDTD, PITD(2), denotes the theory results. From or the PITD(4) method, Fig. 10, we can see clearly that the numerical error of PITD(4) method is smaller than those of FDTD and PITD(2) methods.

Fig. 11. The normalized magnitude of the electric field component E in the frequency domain (air-filled 3D cavity).

B. Air Filled 3-D Cavity The air filled 3-D cavity is shown in Fig. 11. The dimensions , , and m [16]. In actual are computation, a uniform mesh with m is used, leading to a mesh of 6 3 4 cells. The time step was sec. The parameter of PITD(2) and . The value of electric field PITD(4) method is set as is sampled at the time period of . Ficomponent nally, the FFT algorithm is performed to obtain the spectrum of the records. In the magnitude spectrum, the peak values correspond with the eigenfrequencies. As shown in Fig. 11, the computational results of the PITD(2) method are closer to those of the FDTD method, but the computational results of the PITD(4) method are much the same as the theory results. To see clearly, the three basic resonance frequencies are presented in Table I. We can see that the maximum relative errors of the FDTD, PITD(2), and PITD(4) are 3.72%, 3.72%, and 0.41%, respectively.

BAI et al.: LOW-DISPERSION REALIZATION OF PITD METHOD USING A 4TH ORDER ACCURATE FD SCHEME

TABLE I COMPARISON OF ACCURACY OF THREE METHODS

TABLE II INFORMATION ABOUT THE RESONATOR SIMULATION

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APPENDIX A THE DISCUSSION ON THE CHOICE OF THE TIME STEP AND THE CHOICE OF THE NUMBER OF ITERATIONS The conclusion of [17] shows that the relative error of numerical computation is not enlarged in whole recurrent process, i.e., the relative error of exponential matrix of PITD method doesn’t change with the increase of iterations. is the eigenvalues of and is the Assuming that the matrix composed of eigenvectors of . The exponential matrix can be expressed as

..

. (41)

To achieve the PITD(4) method accuracy, we decreased the spatial step size in the FDTD method. An uniform mesh with m is used, leading to a mesh equals of 24 12 16. In this case, the Courant number 0.5. Due to 64 times increase of mesh, for the FDTD method, the computational time increases to 126.56 s and the memory required becomes nearly 2 times. The computational results are shown in Table II and compared with those of the PITD(4) method. The results indicate that for the same accuracy the PITD(4) method is more efficient in terms of the execution time than the FDTD method.

According to the theory of matrix, the error analyze of and can be transformed into the and . Here, is the ones of maximum value of eigenvalues. By using the Taylor series can be expressed as expansion,

(42) when the first terms of Taylor series is chosen for the operation of summing, the truncation error is

VII. CONCLUSION In this paper, a low-dispersion precise integration time-domain method which we called PITD(4) is presented for solving transient electromagnetic problems, which is capable of providing significant improvement over the conventional approach. This method uses a fourth-order accurate finite- difference formula to approximate the spatial derivatives and Gauss quadrature to calculate the nonhomogenous term integration. The stability condition and the numerical dispersion relationship of the PITD(4) method are given analytically, and the superior performance has been shown theoretically as well as confirmed numerically. We have shown that the stability condition of the PITD(4) method is much larger than the CFL stability condition of FDTD, and its numerical dispersion error is nearly independent of time step size. The superiority of this method originates from the successful reduction of the inherent dispersion errors, whose degrading effect becomes significant in large-scale or long-time simulations due to their cumulative nature. With the fourth-order accurate finite-difference formula to approximate the spatial derivatives, the numerical dispersion errors of the PITD(4) method are much less than those of the PITD(2) and the FDTD methods. Therefore, with the same computational accuracy, the spatial sampling rates of the PITD(4) method can be much less than those of the PITD(2) and the FDTD methods.

(43) where . According to the theory of PITD method, after times iterof PITD method is ations, the relative error of (44) Here, we choose the parameter as . Since the maximum eigenvalue of a matrix is fixed; (44) shows the relation, the time step , ship among the relative error of and the parameter . Once the definite accuracy requirement is given, we can get the proper parameter of from (44). Without loss of generality, let

The maximum eigenvalue of is 7.26. Let and . From (44), the relative error of the PITD method is

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The following is the result of the parameter of

of PITD method with .

Compared to the exact result using the function of Matlab

We can see clearly that results of the PITD method have nine significance decimal bits. This does agree with the result of (44). REFERENCES [1] K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwells equations in isotropic media,” IEEE Trans. Antennas Propag., vol. AP-14, no. 3, pp. 302–307, May 1966. [2] A. Taflove and S. C. Hagness, Computational Electrodynamics the Finite-Difference Time-Domain Method, 3rd ed. Reading, MA: Artech House, 2005. [3] F. H. Zheng, Z. Z. Chen, and J. Z. Zhang, “A finite-difference time-domain method without the courant stability conditions,” IEEE Microw. Guided Wave Lett., vol. 9, pp. 441–443, Nov. 1999. [4] Z. Fenghua, C. Zhizhang, and Z. Jiazong, “Toward the development of a three-dimensional unconditionally stable finite-difference time-domain method,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 9, pp. 1550–1558, 2000. [5] T. Namiki, “3-d ADI-FDTD methodłunconditionally stable time-domain algorithm for solving full vector Maxwell’s equations,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 10, pp. 1743–1748, 2000. [6] W. X. Zhong and F. W. Williams, “A precise time-step integration method,” Proc. Inst. Mechan. Eng. Part C-J. Mechan. Eng. Sci., vol. 208, pp. 427–430, 1994. [7] J. Q. Zhao, X. K. Ma, Y. P. Li, and G. Y. Qiu, “Analysis of the electromagnetic transient of multiphase transmission lines by the precise integration method,” (in Chinese) J. High-Voltage Technol., vol. 27, pp. 3–4, 2001. [8] M. Tang and X. K. Ma, “A precise integration algorithm for transient simulation of interconnects in high-speed VLSI,” (in Chinese) J. Electron., vol. 32, pp. 787–790, May 2004. [9] M. Yang and X.-K. Ma, “A semi-integral method for the calculation of electromagnetic pulse propagation in ferromagnetic sheet,” (in Chinese) Trans. China Electrotech. Soc., vol. 1, pp. 89–94, Jan. 2005. [10] X. K. Ma, X. T. Zhao, and Y. Z. Zhao, “A 3-d precise integration timedomain method without the restraints of the Courant-Friedrich-Levy stability condition for the numerical solution of maxwells equations,” IEEE Trans. Microw. Theory Tech., vol. 54, pp. 3026–3037, 2006. [11] X. T. Zhao, Z. G. Wang, and X. K. Ma, “Electromagnetic closed-surface criterion for the 3-d precise integration time-domain method for solving maxwell’s equations,” IEEE Trans. Microw. Theory Tech., vol. 56, pp. 2859–2874, Dec. 2008.

[12] L. Jiang, Z. D. Chen, and J. Mao, “On the numerical stability of the precise integration time-domain (PITD) method,” IEEE Microw. Wireless Compon. Lett., vol. 17, pp. 471–473, Jul. 2007. [13] C. Zhizhang, J. Lele, and M. Junfa, “Numerical dispersion characteristics of the three-dimensional precise integration time-domain method,” in Proc. 2007 Microw. Symp. IEEE/MTT- S Int. (2007), pp. 1971–1974. [14] Z. M. Bai, Y. Z. Zhao, and X. K. Ma, “Analysis and application of subdomain precise integration method for solving Maxwell’s equations,” Trans. China Electrotech. Soc., vol. 25, no. 4, pp. 1–9, Apr. 2010. [15] N. Morita, N. Kumagai, and J. R. Mautz, Integral Equation Methods for Electromagnetics. Boston, MA: Artech House, 1990. [16] Y. Tretiakov, S. Ogurtsov, and G. Pan, “On sampling-biorthogonal time-domain scheme based on Daubechies compactly supported wavelets,” Progr. Electromagn. Res., vol. 47, pp. 213–234, 2004. [17] Y. Xiang, Y. Y. Huang, and W. G. Zeng, “Error analysis and accuracy design for the precise time-integration method,” Chinese J. Computat. Mechan., pp. 276–280, Mar. 2002. Zhong-Ming Bai was born in Shaanxi, China, in 1968. He received the B.Sc. and M.Sc. degrees in electrical engineering from Xi’an Jiaotong University, in 1990 and 2002, respectively, both in electrical engineering. He is currently working toward the Ph.D. degree at Xi’an Jiaotong University. His research interests are in the areas of modeling electromagnetic fields and numerical methods in solving electromagnetic problems.

Xi-Kui Ma was born in Shaanxi, China, in 1958. He received the B.Sc. and M.Sc. degrees in electrical engineering from Xi’an Jiaotong University, in 1982 and 1985, respectively. In 1995, he joined the Faculty of Electrical Engineering, Xi’an Jiaotong University, as a Lecturer, where he became a Professor in 1992. During 1994–1995, he was a Visiting Scientist with the Department of Electrical Engineering and Computer, University of Toronto. His main areas of research interests include electromagnetic field theory and its applications, analytical and numerical methods in solving electromagnetic problems, chaotic dynamics and its applications in power electronics, and the applications of digital control to power electronics. He is the author or coauthor of more than 140 scientific and technical papers on these subjects, and also the author of five books in electromagnetic fields.

Gang Sun was born in Heilongjiang, China, in 1981. He received the M.Sc. degree in electrical engineering from Xi’an Jiaotong University, in 2007, where he is currently working toward the Ph.D. degree. His research interests are in the areas of numerical methods in solving electromagnetic problems.

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High-order Div- and Quasi Curl-Conforming Basis Functions for Calderón Multiplicative Preconditioning of the EFIE Felipe Valdés, Francesco P. Andriulli, Member, IEEE, Kristof Cools, and Eric Michielssen, Fellow, IEEE

Abstract—A new high-order Calderón multiplicative preconditioner (HO-CMP) for the electric field integral equation (EFIE) is presented. In contrast to previous CMPs, the proposed preconditioner allows for high-order surface representations and current expansions by using a novel set of high-order quasi curl-conforming basis functions. Like its predecessors, the HO-CMP can be seamlessly integrated into existing EFIE codes. Numerical results demonstrate that the linear systems of equations obtained using the proposed HO-CMP converge rapidly, regardless of the mesh density and of the order of the current expansion. Index Terms—Electric field integral equation (EFIE), high-order basis functions, integral equations, preconditioning.

I. INTRODUCTION

E

LECTRIC field integral equation (EFIE) solvers find widespread use in the analysis of time-harmonic scattering from perfect electrically conducting (PEC) surfaces [1]. This paper presents a new Calderón multiplicative preconditioner (CMP) for the EFIE which, unlike its predecessors, allows for high-order surface representations and current expansions. The numerical solution of the EFIE requires the discretization of the scatterer’s surface in terms of a mesh of planar or curvilinear triangles or quadrangles, and of its current distribution, by vector basis functions. Discretization of the EFIE means of system of linear equations in the basis leads to a dense functions’ expansion coefficients. The computational cost of it; here eratively solving this system scales as is the complexity of multiplying the system matrix with a trial is the number of iterations required solution vector and for convergence to a prescribed residual. There exist many “fast methods” that reduce the complexity of a matrix-vector multo [2]–[5]. Often tiplication from Manuscript received May 17, 2010; revised September 09, 2010; accepted September 13, 2010. Date of publication January 31, 2011; date of current version April 06, 2011. This work was supported in part by the National Science Foundation under Grant DMS 0713771, by the AFOSR STTR Grant “Multiscale Computational Methods for Study of Electromagnetic Compatibility Phenomena” (FA9550-10-1-0180), by the Sandia Grant “Development of Calderón Multiplicative Preconditioners with Method of Moments Algorithms,” by the KAUST Grant 399813, and in part by an equipment grant from IBM. F. Valdés and E. Michielssen are with the Electrical Engineering and Computer Science Department, University of Michigan, Ann Arbor, MI 48109 USA (e-mail: [email protected]). F. P. Andriulli is with the Politecnio di Torino, Torino 10100, Italy. K. Cools is with the Department of Information Technology (INTEC), Ghent University, B-9000 Ghent, Belgium. 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.2011.2109692

scales with the condition number of the system matrix, with small condition numbers guaranteeing fast convergence. Unfortunately, the condition number of the EFIE system matrix grows rapidly as the mesh discretization density increases [6]. As a result, the cost of solving the EFIE for structures with subwavelength geometric features often is prohibitively high. Techniques for preconditioning the EFIE by leveraging Calderón identities have become quite popular in recent years [7]–[13]. In essence, these techniques exploit the self-regularizing property of the EFIE operator, viz. the fact that the square of the EFIE operator is a compact perturbation of the identity, to produce well-conditioned system matrices even when the mesh includes subwavelength geometric features. Unfortunately, few Calderón preconditioners developed to date are easily integrated into existing codes. The CMP technique proposed in [9] is one of them. The CMP uses two separate discretizations of the EFIE, one in terms of standard Rao-Wilton-Glisson (RWG) basis functions [14], and the other in terms of Buffa-Christiansen (BC) basis functions [15]. The latter are div- and quasi curl-conforming, and geometrically nearly orthogonal to the RWG functions. The effectiveness of the RWG-BC combination in the construction of the CMP stems from the fact that the RWG and BC functions are linked by a well-conditioned Gram matrix and guarantee the annihilation of the square of the discretized hypersingular component of the EFIE operator. We note that Chen and Wilton proposed basis functions similar to the BC ones in the context of analyzing scattering from penetrable objects [16]. Both the BC and Chen-Wilton basis functions are of zeroth-order and designed for use in conjunction with RWG basis functions. In the last decade, EFIE solvers that use high-order representations of the surface and/or the current density have become increasingly popular. A high-fidelity representation of the surface can be achieved using a high-order parametric mapping from a reference cell to the scatterer surface, usually in the form of curvilinear patches (as opposed to flat ones). Among the many high-order basis functions for representing surface current densities, those proposed by Graglia-Wilton-Peterson [17], which comprise of products of scalar polynomials (complete up to order ) and RWG basis functions, are very popular. For a given solution accuracy, high-order EFIE solvers have been shown to be more CPU and memory efficient than their zeroth-order counterparts [18]. That said, they still suffer from ill-conditioning when applied to structures with subwavelength geometric features. To allow for a high-order CMP, a high-order extension of the BC functions is called for. Jan et al. [19] already

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presented an extension of the BC basis functions on curvilinear triangular patches; unfortunately their method does not extend to high-order current representations. This paper presents a true high-order BC extension, viz. a set of high-order div- and quasi curl-conforming functions that, functions, exhibits when used in conjunction with the the aforementioned properties of the BC-RWG pair. The proposed basis functions are constructed as orthogonal projections of the range of the EFIE operator onto div-conforming s defined on a barycentrically refined mesh; preliminary insights into the construction of these basis functions were presented in [20]. Using these basis functions, a high-order CMP (HO-CMP) is implemented and its effectiveness demonstrated via a suite of numerical examples. II. CALDERÓN PRECONDITIONED EFIE AND ITS DISCRETIZATION This section describes the CMP EFIE idea. Section II-A describes the standard EFIE and its classical discretization. Section II-B describes the Calderón-preconditioned EFIE along with its CMP discretization.

is the set of Throughout this paper it is assumed that -order interpolatory Graglia-Wilton-Peterson functions, i.e. [17]. These functions interpolate at and nodes along each of the edges and on each of the patches in , respectively; the total number of functions therefore is ; note that [17]. functions that interpolate at a node internal to a patch or on an edge henceforth will be referred to as patch and edge functions, respectively. For later use we note the Euler identity for a simply connected surface (6) where is the number of vertices in . Substitution of expansion (5) into (1), and testing the resulting equation with curl-conforming functions in yields the linear system of equations (7) where

A. Non-Preconditioned EFIE Solver Consider a closed, simply connected PEC surface residing in a homogeneous medium with permittivity and permeability . The (scaled) current density on induced by the incident satisfies the EFIE [21] time-harmonic electric field

(8) (9) and (10)

(1) where (2) with (3) and (4) Here, and is the outward pointing unit vector normal to at ; is the angular frequency. A time depenis assumed and suppressed. The subdence scripts “ ” and “ ” stand for “singular” (vector potential) and “hyper-singular” (scalar potential), respectively. To numerically of planar or curvisolve(1), is approximated by a mesh linear triangles with minimum edge size , and is expressed as

Here denotes the inner product between to vector functions and on . When analyzing electromagnetic phenomena involving elecis large, trically large and/or complex structures, i.e., when (7) cannot be solved directly and iterative solvers are called for. The computational cost of solving (7) iteratively is proportional by a trial to the cost of multiplying the impedance matrix required to solution vector and the number of iterations typically is proportional reach a desired residual error; ’s condition number, viz. the ratio of ’s largest and to smallest singular values. Unfortunately, the singular values of comprise two branches, one accumulating at the operator zero, and the other at infinity [6]. Thus the condition number grows without bound as is increasingly well-approxof imated, i.e. as and/or . When this happens the number of iterations required for convergence often is prohibitively high. B. Calderón Preconditioned EFIE Solver A well-conditioned EFIE can be obtained by leveraging ’s self-regularizing property expressed by the Calderón identity [6], [9], [10], (11)

(5) with are expansion coefficients of where , of a set of the div-conforming basis functions .

in terms (12)

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The operator is compact on smooth surfaces: its singular [6], values accumulate at zero and the same holds true for has a bounded spec[22]. It follows that the operator trum with singular values accumulating at 1/4. Equation (11) implies that the Calderón-preconditioned EFIE (13) may be amenable to stable discretization regardless of the mesh density or basis function order. is by Unfortunately, the discretization of no means trivial. The literature abounds with techniques for discretizing (14) that separately handle the first three terms in the above expan[7], [8]. Howsion, explicitly leaving out the fourth as ever, the implementation of these techniques into existing codes is quite intrusive. The CMP proposed in [9] does not suffer from as the product of this drawback. The CMP approximates and with two impedance matrices , separated by a Gram matrix that accounts for the possible lack of (bi-)orthogonality between the functions in and . In other words, the CMP matrix equation reads (15) where (16) (17) and (18) is the matrix of overlap integrals of functions in and . Equation (15) does not require the decomposition of matrix eland into their singular (vector potential) and ements in hypersingular (scalar potential) components, simplifying its implementation. That said, (15) only will be well-conditioned if C1. the functions in and are div-conforming; is well-conditioned; this ensures the C2. the matrix rapid iterative solution of for trial solution vectors while solving(15); this requirement precludes the choice as such leads to a singular Gram matrix; upon C3. the sets and ensure the cancellation of discretization, i.e.

Fig. 1. RWG and BC functions defined for edge n in S . Functions are plotted ^ on top of S . (a) Div-conforming RWG, f . (b) Curl-conforming RWG, n f . (c) Div-conforming BC, f~ . (d) Curl-conforming BC, n^ f~ .

2

2

forming Buffa-Christiansen basis functions, used by all CMP implementations reported to date [9], [10], [12], [13], [19].

(19) III. ZEROTH-ORDER QUASI CURL-CONFORMING BASIS FUNCTIONS

where (20) If (19) is not satisfied, the desirable spectral properties of will not be inherited by . and The above criteria are satisfied by the sets , the set of (zeroth-order) div- and quasi curl-con-

This section reviews the construction of the BC basis functions and their main properties [9], [10], [15]. , the set contains Just as basis functions. Contrary to the current of the RWG function , which crosses edge (Fig. 1(a)), that of the BC function flows along edge (Fig. 1(c)). Consider the barycentri-

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cally refined mesh , obtained by adding the three medians to each triangle of the original mesh . Each BC basis function is a linear combination of div-conforming RWGs defined [9], [10], [15]. Even though BC functions are strictly on div-conforming, they also are quasi curl-conforming in that they (Fig. 1(b)). This renresemble curl-conforming RWGs in and ) ders the Gram matrix in (18) (with well-conditioned. That is, the sets and fulfill conditions C1 and C2 above. To show that these sets also satisfy condition C3, consider the space spanned by “div-conforming solenoidal RWG” functions (21) with ; the are charge-free and could, for example, be “loop” functions describing current flowing around all (Fig. 2(a)) [23], [24]. The set but one of the vertices in can be complemented by a set such that . The set contains “divconforming non-solenoidal RWG” functions (22) ; the with all produce charge and could, for example, be “star” functions describing current flowing out of all but one patch (Fig. 2(b)) [23], [24]. Similarly, consider the space in spanned by “div-conforming solenoidal BC” functions (23) equals that of dimensionality of ; indeed, it can be verified that an appropriate linear combination of the BC functions associated with the describes a divergence-free three edges of a patch in current circulating the patch (Fig. 2(c)) [10]. The set can be complemented by a set such that . The set contains “div-conforming non-solenoidal BC” functions The

(24) equals that of Again, the dimensionality of (Fig. 2(d)) [10]. , , and , are Next, assume that the matrices not constructed using the sets and , but instead from and with functions and in the left and right subset labeled 1 through through , respectively; note the reverse order of the “ ” and ” superscripts for functions in and “ . It is clear from (20) and (4) that the entries and vanish when the source function is solenoidal or the test function is irrotational, which implies

Fig. 2. Div-conforming RWG and BC solenoidal and non-solenoidal functions defined in S . Note that functions are plotted on top of S . (a) Div-conforming RWG solenoidal function f , describing current flowing around vertex n in S . (b) Div-conforming RWG non-solenoidal function f , describing current flowing out of patch n in S . (c) Div-conforming BC solenoidal function f~ , describing current flowing around patch n in S . (d) Div-conforming BC nonsolenoidal function f~ , describing current flowing out of vertex n in S .

The blocks in these matrices have dimensions (25)

(26)

VALDÉS et al.: HIGH-ORDER DIV- AND QUASI CURL-CONFORMING BASIS FUNCTIONS FOR CMP OF THE EFIE

Since an irrotational function can be written as the surface gradient of a scalar function , and a solenoidal function can be written as the surface curl of a scalar function , the inner product of two such functions can be expressed as (27) which can be transformed by partial integration into

and (36) judiciously chosen such that system (15) is well-conditioned. Throughout this section, notation introduced previously for spaces and sets applicable to will be reused and extended for all spaces and functions derived from the barycentrically refined by adding bars on top of symbols. That is, mesh

(28) Therefore, the Gram matrix

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(37) where the sets

has the form

(38)

(29) and and so does its inverse

(39) (30) From (25) and (30), it is clear that . The fact that the dimension of the solenoidal subspace of the RWG basis functions equals that of the non-solenoidal subspace of the BC basis functions (and vice-versa), is essential for the CMP upon technique to work, as it ensures the cancellation of discretization. IV. HIGH-ORDER QUASI CURL-CONFORMING BASIS FUNCTIONS In this section, a set of div- and quasi curl-conforming highorder extensions of the BC functions is proposed. In the con, the new set is meant to struction of (15)–(18), for any be used alongside (31) Here

complements the set of RWG functions such that . The sets (32)

and (33) span the solenoidal and non-solenoidal subspaces of , respectively. Likewise, will be constructed as (34) with sets (35)

span the solenoidal and non-solenoidal subspaces of , respectively; and denote and RWG basis functions defined on , etc. To guarantee that system (15) has a low condition number, sets and must satisfy the above conditions C1 through C3. To ensure functions in are div-conforming, they will be constructed as linear combinations of the div-conforming functions in . Bases for the high-order solenoidal and non-solenoidal and will be built sepsubspaces , arately. To arrive at a well-conditioned Gram matrix functions in will be constructed so as to “resemble” those in . To ensure the cancellation , the cardinality of will be matched to that of , i.e., and . Section IV-A details the Helmholtz decomposition of the , , and . Section IV-B explains how to spaces basis functions such that the Gram matrix construct the is well-conditioned. A. Helmholtz Decomposition As described in [25], bases for and can be constructed by separating into edge and (internal-to-) patch subspaces. One way of constructing these subspaces is as follows. 1) For each patch in a. Define as the matrix that maps all patch functions (columns of onto their charges (divergence) at points in the patch (rows of ) b. Perform a singular value decomposition (SVD) on : . c. The last columns of are associated with zero singular values, and they describe patch solenoidal functions (Fig. 3(a)). describe patch nond. All other columns of solenoidal functions (Fig. 3(b)).

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that the patch solenoidal functions are now associated with singular values equal to . . c. Perform a SVD on : d. The last columns of are associated with zero singular values, and they describe edge solenoidal functions (Fig. 3(c)). contains: To summarize, the set (i) high-order patch functions; high-order edge functions. (ii) and contains highLikewise, and are order patch functions. The cardinalities of therefore (40) and (41)

Fig. 3. Div-conforming F solenoidal and non-solenoidal functions defined in S . Note that functions are plotted on top of S . (a) Div-conforming F patch solenoidal function f , its support (shaded area) is limited to a patch in S . (b) Div-conforming F patch non-solenoidal function f , its support (shaded area) is limited to a patch in S . (c) Div-conforming F edge , its support (shaded area) include the two patches solenoidal function f sharing the edge in S .

2) For each edge in a. Define as the matrix that maps edge and the overlapping patch functions (columns of onto their charges at points in the patches (rows of ). b. Define , where is a non-zero and are the singular vectors assoconstant, and ciated with the patch solenoidal functions identified in step 1 for the two patches that share the edge. Note

Of course Once the high-order solenoidal and non-solenoidal functions have been obtained as described above, they can be (separately) linearly combined to form a more convenient basis for and , respectively. A partial local orthogonalization can be performed as follows: 1) For each edge in , orthogonalize the solenoidal functions associated with it. , separately orthogonalize the 2) For each patch in solenoidal and non-solenoidal functions. After this partial orthogonalization has been performed, all are orthogonal to one another, but not necfunctions in essarily orthogonal to any or all functions in . Furthermore, , only those which are patch based among the functions in are orthogonal to one another, but not necessarily orthogonal to any or all of the edge solenoidal functions. A full local orthogonalization can also be performed. The difference with respect to the previous one being that now patchbased solenoidal and non-solenoidal functions are orthogonalare orthogonal ized altogether. Hence all functions in to one another, and also orthogonal to all patch based functions, . but not necessarily to any or all edge based functions in ) For future use, we define the matrix (of size as linear combinations of functhat expresses functions in tions in , i.e. its th column contains the coefficients after the orthogonalization process. Simiobtained for larly, the matrix (of size ) expresses functions in as linear combinations of functions in . of . As is Next, consider the barycentric refinement has simply connected, so is ; hence edges, vertices, and patches. The total number of high-order solenoidal and nonsolenoidal functions is

(42)

(43)

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2

^ Fig. 4. Div- and quasi curl-conforming functions in F~ , approximating those in nF . Note that functions are plotted on top of S . (a) n f , i.e. curlconforming counterpart of the patch solenoidal function f depicted in Fig. 3(a). (b) Div-conforming patch non-solenoidal function f~ approximating depicted in Fig. 3(c). (d) Div-conforming edge non-solenoidal n^ f . (c) n^ f , i.e. curl-conforming counterpart of the edge solenoidal function f ^ f ^ f function f~ approximating n . (e) n , i.e. curl-conforming counterpart of the patch non-solenoidal function f depicted in Fig. 3(b). ^ (f) Div-conforming patch solenoidal function f~ approximating n f .

2

2

2

2

2

Matrices and can be obtained just as described before, operating on the functions in . is built as a linear combination of funcEach function in tions in , i.e. (44)

Likewise, each function in , i.e. tion of functions in

is built as a linear combina-

(45)

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Note that the cardinality of matches that , thereby ensuring the cancellation of .

. Similarly, enforcing (48) yields the foland : lowing equation for the expansion coefficients

(54)

B. Well-Conditioning of the Gram Matrix Due to the way the sets and are defined, the Gram matrix can be decomposed into four blocks

with (55)

(46) where , , , and are matrices of size , , , respectively. The block and is nothing but the Gram matrix encountered in the zeroth-order case, and it is of course well-conditioned [9], [10]. That said, to be well-conditioned the exin order for the block in (44), and in (45) need to be pansion coefficients chosen appropriately. Clearly, if we insist that (47) and (48) then the entries of will be approximately those of . This suggests that the condition number of the former matrix should be similar to that of the latter. is chosen to be the To achieve the resemblance in (47), orthogonal projection of

onto

, i.e.,

(49) Substituting (44) into (49) yields a system of linear equations : for the expansion coefficients

(50) Equation (50) can be expressed in matrix form as

(51) where the Gram matrices

and

are (52) (53)

(56) and . As an example on how these two orthogonal projections perform, consider the div-conforming patch (edge) solenoidal funcdepicted in Fig. 3(a) (Fig. 3(c)). Its curl-conforming tion is shown in Fig. 4(a) (Fig. 4(c)). The counterpart orthogonal projection of the latter is the div-conforming patch , depicted in Fig. 4(b) (edge) non-solenoidal function (Fig. 4(d)). Similarly, consider the patch non-solenoidal funcdepicted in Fig. 3(b). Its curl-conforming countertion is shown in Fig. 4(e). The orthogonal projecpart tion of the latter is the div-conforming patch solenoidal funcdepicted in Fig. 4(f). Note that the support of tion is a couple of “barycentric patches bigger” than the support of . This “extra space” is required as a return path for to provide a charge-free approxthe current described by . imation of are built to resemble Since the functions in , the condition numbers of those in and are expected to depend on the way the functions and are obtained. Indeed, if functions in in and are not orthogonalized in any way described at the and end of Section IV-A, the condition numbers of grow without bound with . As it will be shown later in Section VI, partial and full local orthogonalization of the funcand reduce the aforementioned growth on tions in the condition numbers to a minimum. An ideal scenario would and are built as be one in which the functions in one orthogonal set of functions, such that equals the would be as close as it identity. Hence the matrix can be to the identity matrix. Of course, such orthogonalization cannot be performed “locally” therefore it is far from being practical due to its computational cost.

V. IMPLEMENTATION OF THE HO-CMP This section provides details on the construction of the basis functions in and their use in the HO-CMP. First, explicit expressions for the matrices and are given in terms of Gram matrices and basis transformations. With these matrices, , , and are given. Finally, issues expressions for relating to computational cost are discussed.

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Fig. 5. Bistatic RCS of a PEC sphere of radius 1 m. illuminated by a 30 MHz

x^-polarized plane wave traveling in the z^ direction. The surface of the sphere is

modeled with 32 curvilinear patches. The current density is modeled with basis functions of orders p = 0, 1, 2, 3. The number of unknowns ranges from 48 (p = 0) to 576 (p = 3): (a) Bistatic RCS in the x-z plane. (b) Relative error in the RCS with respect to Mie series solution.

The evaluation of Gram matrices:

in (51) requires the computation of two and . Since each func-

tion in is a linear combination of functions in , the Gram matrix in (52) can be obtained as the product (57) Similarly, the Gram matrix puted as

The evaluation of and of tained as the product

in (54) involves the computation . The former can be ob-

in (53) can be com(59) (58)

where in

Fig. 6. Bistatic RCS of a PEC star-shaped object illuminated by a 30 MHz x^-polarized plane wave traveling in the z^ direction. The surface of the object is modeled with 102 curvilinear patches. The current density is modeled with basis functions of orders p = 0, 1, 2, 3. The number of unknowns ranges from 153 (p = 0) to 1836 (p = 3). (a) Bistatic RCS in the x-z plane. (b) Relative error in the RCS with respect to the solution obtained using basis functions of order p = 4.

is the matrix (of size ) that expresses functions as linear combinations of functions in .

Finally,

can be computed as

(60)

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Substitution of the above expressions into (51) and (54) yields

TABLE I CONDITION NUMBERS OF , , AND FOR THREE DIFFERENT MESH DISCRETIZATIONS OF A PEC SPHERE

G

T

T

G

T

T

G

T

T

(61) (62) All matrices on the right hand side of (61) and (62) are sparse operations. Note that and can be multiplied by a vector in the inversion of matrix need not be performed explicitly, instead its operation on any vector can be obtained by solving the linear system iteratively. as deOf course, orthogonalization of the functions in scribed in the previous section makes the matrix well-conditioned. Thus, the evaluation of (61) has an overall computa. Similar considerations apply tional cost that scales as to the evaluation of (62), with the exception that if the functions are orthogonalized, then is nothing but in the identity, therefore no system need to be solved. The implementation of the HO-CMP follows the same structure of the zeroth-order CMP (see [9]), which makes use of maand , that express functions in BC and RWG as trices , respectively. The malinear combinations of functions in encountered in the zeroth-order CMP is extended here trix to defined as

TABLE II , , AND FOR THREE DIFFERENT CONDITION NUMBERS OF MESH DISCRETIZATIONS OF A PEC STAR-SHAPED OBJECT

(63)

where and are given in (61) and (62) respectively. can be found in Explicit expressions for the entries of encountered in the zeroth-order CMP is [9]. The matrix replaced here by the matrix , defined earlier in this section. in (15) can be discretized as follows: Using , the matrix

TABLE III , , AND FOR THREE DIFFERENT CONDITION NUMBERS OF MESH DISCRETIZATIONS OF A PEC CUBE

(64) where is the matrix that expresses functions in . Similarly, combinations of functions in can be discretized as

as linear in (15)

(65) is the matrix that expresses functions in where . combinations of functions in Finally, with being discretized as

as linear

(66) the evaluation of (16) and (17) can now be carried out. The computational cost of solving (15) is that of multiplying the matrix times the number of iterations required to reach a prescribed residual error. Evaluation of a vector times involves multiplying first by as in (65), then by

the inverse of as in (66), and finally by as in (64). As mentioned previously, the cost of multiplying , , and by a vector scales as . Thus, the cost of multiplying (and therefore ) by a vector also scales as . Provided that is well-conditioned, and it is, then its inverse can be multiplied by a vector using just a few (i.e.,

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CONDITION NUMBERS OF

G

, AND

T

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TABLE IV THREE DIFFERENT HELMHOLTZ DECOMPOSITION STRATEGIES

FOR

) iterations of an iterative solver like the generalized minimal residual (GMRES) [26] or the transpose-free quasiminimal residual (TFQMR) [27]. Using the multilevel fast multiby a vector scales pole method [3], the cost of multiplying where is the cost of multiplying by a as is greater than vector. Indeed, even though the dimension of by a factor of 6, the additional degrees of freedom inthat of troduced by the barycentric mesh do not change the number of multipoles required for field expansion compared to that used . Therefore, the cost of multiplying when multiplying by increases only by an additive linear term. The fact that the number of iterations required for the HO-CMP to converge is much smaller than that of the standard EFIE justifies the use of the former scheme. VI. NUMERICAL RESULTS This section presents several examples that demonstrate the effectiveness of the basis functions presented in this paper and its performance in the HO-CMP. The results emphasize the main advantage of using a HO-CMP: high-order accuracy in the solutions, without compromising the number of iterations needed for convergence. The results presented here are obtained using a parallel EFIE MoM solver, which uses the proposed HO-CMP or a standard diagonal preconditioner. This solver uses a TFQMR-based iterative method [27] to solve the EFIE MoM systems. A. High-Order Accuracy The first two examples demonstrate the convergence of the radar cross section (RCS) as the order of the basis functions in the HO-CMP is increased. Each example comprises a smooth PEC object: a sphere of radius 1 m, and a star-shaped object whose surface is parameterized as , both illuminated by a 30 MHz, -polarized plane wave traveling in the direction. Fig. 5(a) (Fig. 6(a)) shows the bistatic RCS of the PEC sphere (star-shaped object) when computed with basis functions of orders , 1, 2, 3. Fig. 5(b) (Fig. 6(b)) shows the relative error of the computed RCS of the PEC sphere (star-shaped object) with respect to Mie series (4th-order) solution. In these examples, the geometric models consist of 32 patches for the sphere and 102 patches for the star-shaped object. Each patch is obtained by means of an exact mapping from a reference patch onto the surface of the object. The evaluation of basis functions on curvilinear patches requires the computation of a

Jacobian function, which requires additional computation time when compared to flat patches [17]. The overhead introduced by the evaluation of the Jacobian is more than compensated however by the reduction in the number of patches required to accurately describe the sphere surface. B. Condition Number The following three examples illustrate the behavior of the condition numbers of the non-preconditioned EFIE and HO-CMP system matrices as the surface current expansion is and/or . increasingly well-approximated, i.e., as Table I shows the condition numbers of , , and , obtained with several mesh discretizations of the PEC sphere of Fig. 5(a) using basis functions of orders , 2, 3, 4. Similarly, Tables II and III show the same data for the star-shaped object of Fig. 6(a) and a PEC cube with sidelength of 1 m, respectively. These results show that for a fixed order , the condition numbers of and remain bounded as the mesh density is increased, whereas the condition number of does not. is By virtue of the Calderón identity in (11), the operator spectrally equivalent to the identity operator. Hence the condidepends on how well the sets and can tion number of discretize the identity operator, i.e. the Gram matrix . As mentioned in Section IV-B, the growth in the condition number (and therefore of ) with is related to the of and are obtained. way in which the functions in and Table IV shows the condition numbers of for three different ways of obtaining these sets, and for orders , 2, 3, 4. As expected, full local orthogonalization of the and result in lower condition numbers functions in and that are more stable with for the matrices respect to when compared to partial local orthogonalization. Also, as conjectured at the end of Section IV-B, a global orthogand yields and onalization of the functions in matrices with condition numbers that are almost independent of . C. Speed of Convergence The three examples in this section compare the speed of convergence of the diagonally-preconditioned EFIE and HO-CMP when solved iteratively. Fig. 7(a)–(e) show the residual error versus iteration count achieved by a TFQMR solver during the iterative solution of the matrix systems obtained by discretizing the diagonally-preconditioned EFIE and HO-CMP with basis functions of orders

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Fig. 7. Residual history of diagonally-preconditioned EFIE (dashed lines) and HO-CMP (solid lines) for the case of a PEC sphere of radius 1 m, illuminated ^-polarized plane wave traveling in the z^ direction. Four different discretizations are used, ranging from 32 to 810 curvilinear elements. by a 30 MHz, x Results are shown for several orders of the basis functions: (a) order 1; (b) order 2; (c) order 3; (d) order 4; (e) order 5.

, 2, 3, 4, 5. The geometry is a PEC sphere of radius 1 m. Similarly, Fig. 8(a)–(e) show the same data for a PEC cube with sidelength of 1 m. In both examples, the excitation is a 30

MHz, -polarized plane wave traveling in the direction, and the prescribed accuracy (relative residual error) for the TFQMR solver is . As dictated by the condition number of ,

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Fig. 8. Residual history of diagonally-preconditioned EFIE (dashed lines) and HO-CMP (solid lines) for the case of a PEC cube of side 1 m, illuminated by ^-polarized plane wave traveling in the z^ direction. Four different discretizations are used, ranging from 24 to 918 elements. Results are shown a 30 MHz, x for several orders of the basis functions: (a) order 1; (b) order 2; (c) order 3; (d) order 4; (e) order 5.

the number of iterations required for the HO-CMP to reach the prescribed accuracy does not grow as the discretization density is increased. In contrast, the diagonally-preconditioned EFIE requires an increasing number of iterations as

the mesh becomes denser. Moreover, this behavior worsens as the order of the basis functions is increased, severely penalizing the efficiency and accuracy of high-order basis functions.

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Fig. 9. Monopole antenna excited with a voltage delta-gap. (a) Mesh and dimensions of the antenna. (b) Divergence of the current density induced on the antenna, for a frequency of 3.55 GHz. (c) Radiation pattern in the x-y plane for two different frequencies. (d) Residual history of diagonally-preconditioned EFIE (dashed , 1. lines) and HO-CMP (solid lines), for a frequency of 5.5 GHz for orders p

=0

Next, the diagonally-preconditioned EFIE and HO-CMP are used to analyze scattering from a printed monopole antenna similar to the one presented in [28]. The antenna geometry and mesh are shown in Fig. 9(a). Note that the dielectric substrate has not been considered here. The antenna is fed with a voltage deltagap. The divergence of the electric current, i.e., the (scaled) charge distribution on the surface of the antenna is plotted in Fig. 9(b). The current distribution in this example was obtained using the HO-CMP, with basis functions of order and a frequency of 3.55 GHz. The radiation pattern of the antenna is plotted in Fig. 9(c) for two different frequencies: 3.55 and 5.5 GHz. Finally, Fig. 9(d) shows the residual error versus iteration count achieved by a TFQMR solver during the iterative solution of the matrix systems stemming from the diagonally-pre-

conditioned EFIE and HO-CMP with basis functions of orders , 1. The last example involves a model of the Airbus A380 shown in Fig. 10(a). The surface of the aircraft is discretized using second-order curvilinear patches, allowing the use of (relatively) large patches on smooth surfaces (wings and main body), and small patches near fine geometric features (engines and wing tips). The airplane is illuminated by a -polarized plane wave traveling in the direction. Fig. 10(b) shows the bistatic RCS obtained for four different frequencies, ranging from 1.5 to 30 MHz. Fig. 10(c) and 10(d) show the divergence of the current density induced on the surface of the aircraft, at frequencies of 6 MHz and 30 MHz, respectively. Note that at 30 MHz the high-order basis functions allow for the use

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^ direction. (a) Mesh and dimensions of the aircraft; second order curvilinear Fig. 10. Airbus A380 model illuminated by y^-polarized plane wave traveling in the x patches are used to discretize the surface. (b) Bistatic RCS in the x-y plane for four different frequencies. (c) Divergence of the current density induced on the aircraft, for a frequency of 6 MHz. (d) Divergence of the current density induced on the aircraft, for a frequency of 30 MHz. (e) Residual history of diagonallypreconditioned EFIE (dashed lines) and HO-CMP (solid lines), for a frequency of 6 MHz for orders p , 2, 3. (f) Residual history of diagonally-preconditioned EFIE (dashed lines) and HO-CMP (solid lines), for a frequency of 30 MHz for orders p , 1, 2.

=0

of less than 5 patches per wavelength on the wings and main body of the aircraft. Finally, Fig. 10(e) shows the residual error versus iteration count achieved by a TFQMR solver during the iterative solution of the matrix systems obtained by discretizing the diagonally-preconditioned EFIE and HO-CMP with basis , 2, 3. In this case, the excitation functions of orders frequency is 6 MHz. Similarly, Fig. 10(f) shows the residual error versus iteration count achieved by a TFQMR solver for an excitation frequency of 30 MHz. Using basis functions of order , it took 30 minutes and 16852 iterations for the diagonally preconditioned EFIE to converge to a prescribed . For the HO-CMP it took 11 relative residual error of minutes and 485 iterations. Using basis functions of order

=1

, the diagonally preconditioned EFIE could only reach a after 8.6 hours and 100000 relative residual error of iterations. For the HO-CMP it took 1.2 hours and 383 iterations . to reach the prescribed relative residual error of VII. CONCLUSION In this paper, the CMP technique is extended to high-order by building a set of high-order div- and quasi curl-conforming basis extensions of the BC basis functions used by all CMP implementations reported to date. Numerical results demonstrate fast convergence rates of the HO-CMP, regardless of the mesh density and the order of the basis functions used. The HO-CMP presented here can be used in the presence of open

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surfaces with minor modifications. In addition, the basis functions presented here can also be used in high-order Calderón preconditioned formulations for analyzing scattering from penetrable objects.

REFERENCES [1] A. J. Poggio and E. K. Miller, “Integral equation solutions of three-dimensional scattering problems,” in Computer Techniques for Electromagnetics, R. Mittra, Ed. New York: Pergamon, 1973. [2] W. C. Chew, J. M. Jin, C. C. Lu, E. Michielssen, and J. M. Song, “Fast solution methods in electromagnetics,” IEEE Trans. Antennas Propag., vol. 45, no. 3, pp. 533–543, Mar. 1997. [3] J. M. Song, C. C. Lu, and W. C. Chew, “Multilevel fast-multipole algorithm for solving electromagnetic scattering by large complex objects,” IEEE Trans. Antennas Propag., vol. 45, no. 10, pp. 1488–1493, Oct. 1997. [4] E. Bleszynski, M. Bleszynski, and T. Jaroszewic, “AIM: Adaptive integral method for solving large-scale electromagnetic scattering and radiation problems,” Radio Sci., vol. 31, pp. 1225–1151, 1996. [5] A. Dutt and V. Rokhlin, “Fast Fourier transforms for nonequispaced data,” SIAM J. Sci. Comput., vol. 14, pp. 1368–1393, 1993. [6] J.-C. Nedéléc, Acoustic and Electromagnetic Equations. New York: Springer-Verlag, 2000. [7] H. Contopanagos, B. Dembart, M. Epton, J. J. Ottusch, V. Rokhlin, J. L. Visher, and S. M. Wandzura, “Well-conditioned boundary integral equations for three-dimensional electromagnetic scattering,” IEEE Trans. Antennas Propag., vol. 50, no. 12, pp. 1824–1830, Dec. 2002. [8] 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, no. 9, pp. 2262–2266, Sep. 2004. [9] F. P. Andriulli, K. Cools, H. Bagci, 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, no. 8, pp. 2398–2412, Aug. 2008. [10] K. Cools, F. P. Andriulli, F. Olyslager, and E. Michielssen, “Time-domain Calderón identities and their application to the integral equations analysis of scattering by PEC objects part I: Preconditioning,” IEEE Trans. Antennas Propag., vol. 57, pp. 2352–2364, Aug. 2009. [11] F. P. Andriulli and E. Michielssen, “A regularized combined field integral equation for scattering from 2-D perfect electrically conducting objects,” IEEE Trans. Antennas Propag., vol. 55, pp. 2522–2529, Sep. 2007. [12] F. Valdés,, F. P. Andriulli, H. Bagci, and E. Michielssen, “On the discretization of single source integral equations for analyzing scattering from homogeneous penetrable objects,” in Proc. IEEE Antennas and Propagation Society Int. Symp., Jul. 5–11, 2008, pp. 1–4. [13] 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. [14] S. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag., vol. AP-30, pp. 409–418, May 1982. [15] A. Buffa and S. Christiansen, “A dual finite element complex on the barycentric refinement,” Math. Comp., vol. 76, pp. 1743–1769, 2007. [16] Q. Chen and D. R. Wilton, “Electromagnetic scattering by three-dimensional arbitrary complex material/conducting bodies,” in Proc. IEEE AP-S Symp., 1990, vol. 2, pp. 590–593. [17] R. D. Graglia, D. R. Wilton, and A. F. Peterson, “Higher order interpolatory vector bases for computational electromagnetics,” IEEE Trans. Antennas Propag., vol. 45, no. 3, pp. 329–342, Mar. 1997. [18] K. C. Donepudi, J. M. Jin, S. Velamoarambil, J. M. Song, and W. C. Chew, “A higher-order parallelized multilevel fast multipole algorithm for 3D scattering,” IEEE Trans. Antennas Propag., vol. 49, pp. 1078–1078, Jul. 2001. [19] S. Yan, J.-M. Jin, and Z. Nie, “Implementation of the Calderón multiplicative preconditioner for the EFIE solution with curvilinear triangular patches,” presented at the IEEE Antennas and Propagation Society Int. Symp., Jun. 2009.

[20] F. Valdés, F. P. Andriulli, K. Cools, and E. Michielssen, “High-order quasi-curl conforming functions for multiplicative Calderón preconditioning of the EFIE,” presented at the IEEE Antennas and Propagation Society Int. Symp., Jun. 2009. [21] 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. 53, pp. 3316–3323, Oct. 2005. [22] R. Kress, Linear Integral Equations, 2nd ed. New York: Springer, 1999. [23] 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, 2000. [24] W. Wu, A. W. Glisson, and D. Kajfez, “A study of two numerical solutions procedures for the electric field integral equation at low frequency,” Appl. Comput. Electromagn. Soc. J., vol. 10, pp. 69–80, 1995. [25] R. A. Wildman and D. S. Weile, “An accurate broadband method of moments using higher order basis functions and tree-loop decomposition,” IEEE Trans. Antennas Propag., vol. 52, no. 11, pp. 3005–3011, Nov. 2004. [26] Y. Saad and M. H. Schultz, “GMRES: A generalized minimal residual algorithm for solving non-symmetric linear systems,” SIAM J. Sci. Stat. Comput. 3, vol. 7, no. 3, pp. 856–869, 1986. [27] R. W. Freund, “A transpose-free quiasi-minimal residual algorithm for non-Hermitian linear systems,” SIAM J. Sci. Comput., vol. 14, no. 2, pp. 470–482, 1993. [28] M. A. Antoniades and G. V. Eleftheriades, “A broadband dual-mode monopole antenna using NRI-TL metamaterial loading,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 258–261, May 2009.

Felipe Valdés received the B.S. degree in electrical engineering from the Pontificia Universidad Católica de Chile, Santiago, Chile, in 2004. He is currently working toward the Ph.D. degree at the University of Michigan at Ann Arbor. Since August 2006, he has been a Research Assistant at the Radiation Laboratory, University of Michigan at Ann Arbor. His main research interest is in computational electromagnetics, with focus on preconditioning and single source integral equations. Mr. Valdés was the recipient of a Fulbright Doctoral Fellowship in 2006–2010.

Francesco P. Andriulli (S’05–M’09) received the Laurea degree in electrical engineering from the Politecnico di Torino, Italy, in 2004, the M.S. degree in electrical engineering and computer science from the University of Illinois at Chicago, in 2004, and the Ph.D. degree in electrical engineering from the University of Michigan at Ann Arbor in 2008. Since 2008 he has been a Research Associate with the Politecnico di Torino. His research interests are in computational electromagnetics with focus on preconditioning and fast solution of frequency and time domain integral equations, integral equation theory, hierarchical techniques, and single source integral equations. Dr. Andriulli was awarded the University of Michigan International Student Fellowship and the University of Michigan Horace H. Rackham Predoctoral Fellowship. He was the recipient of the best student paper award at the 2007 URSI North American Radio Science Meeting. He received the first place prize of the student paper context of the 2008 IEEE Antennas and Propagation Society International Symposium, where he authored and coauthored other two finalist papers. He was the recipient of the 2009 RMTG Award for junior researchers and was awarded a URSI Young Scientist Award at the 2010 International Symposium on Electromagnetic Theory.

VALDÉS et al.: HIGH-ORDER DIV- AND QUASI CURL-CONFORMING BASIS FUNCTIONS FOR CMP OF THE EFIE

Kristof Cools was born in Merksplas, Belgium, in 1981. He received the M.S. degree in physical engineering from Ghent University, Belgium, in 2004. His master’s dissertation dealt with the full wave simulation of metamaterials using the low frequency multilevel fast multipole method. He received the Ph.D. degree from the University of Michigan at Ann Arbor, in 2008, under the advisership of Prof. Femke Olyslager and Prof. Eric Michielssen. In August 2004, he joined the Electromagnetics Group, Department of Information Technology (INTEC), Ghent University. His research focuses on the spectral properties of the boundary integral operators of electromagnetics.

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Eric Michielssen (M’95–SM’99–F’02) received the M.S. degree in electrical engineering (summa cum laude) from the Katholieke Universiteit Leuven (KUL, Belgium) in 1987 and the Ph.D. degree in electrical engineering from the University of Illinois at Urbana-Champaign (UIUC), in 1992. He joined the faculty of the UIUC Department of Electrical and Computer Engineering in 1993, reaching the rank of Full Professor in 2002. In 2005, he joined the University of Michigan as a Professor of electrical engineering and computer science where, since 2009, he has been the Director of the University of Michigan Computational Science Certificate Program. He authored or coauthored over one 160 journal papers and book chapters and over 280 papers in conference proceedings. His research interests include all aspects of theoretical and applied computational electromagnetics. His research focuses on the development of fast frequency and time domain integral-equation-based techniques for analyzing electromagnetic phenomena, and the development of robust optimizers for the synthesis of electromagnetic/optical devices. Dr. Michielssen is a Fellow of the IEEE and a member of URSI Commission B. He received a Belgian American Educational Foundation Fellowship in 1988 and a Schlumberger Fellowship in 1990. He was the recipient of a 1994 International Union of Radio Scientists (URSI) Young Scientist Fellowship, a 1995 National Science Foundation CAREER Award, and the 1998 Applied Computational Electromagnetics Society (ACES) Valued Service Award. He was named the 1999 URSI United States National Committee Henry G. Booker Fellow and selected as the recipient of the 1999 URSI Koga Gold Medal. He was also awarded the UIUC’s 2001 Xerox Award for Faculty Research, appointed 2002 Beckman Fellow in the UIUC Center for Advanced Studies, named 2003 Scholar in the Tel Aviv University Sackler Center for Advanced Studies, and selected as UIUC 2003 University and Sony Scholar. He served as the Technical Chairman of the 1997 Applied Computational Electromagnetics Society (ACES) Symposium (Review of Progress in Applied Computational Electromagnetics, March 1997, Monterrey, CA), and served on the ACES Board of Directors (1998–2001 and 2002–2003) and as ACES Vice-President (1998–2001). From 1997 to 1999, he was as an Associate Editor for Radio Science, and from 1998 to 2008 he served as Associate Editor for the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION.

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On the FDTD Formulations for Modeling Wideband Lorentzian Media Zhili Lin, Member, IEEE, Yuntuan Fang, Jiandong Hu, and Chunxi Zhang

Abstract—Several popular numerical approaches for modeling the dispersion characteristics of complex media are applied to simulate the wave propagation in the wideband Lorentzian (WBL) media under the second-order or fourth-order accurate finite-difference time-domain (FDTD) formulation. Their implementing algorithms and work performance for this type of dispersive materials are studied and compared to each other. The coefficients of a general second-order discrete numerical model pertaining to each of these approaches are derived respectively and a unified set of FDTD updating equations for implementing them is presented. In light of the analytical results calculated from the given numerical dispersion equations that determines the effective permittivity and permeability of a corresponding artificial WBL medium and the simulation results from the practical FDTD simulations for an exemplified WBL medium, the modeling accuracy and stability limits of these approaches are presented by evaluating the absolute errors of both the propagation and transmission functions of a plane electromagnetic pulse wave propagating in the WBL material and transmitting through an interface between the vacuum and the WBL medium under the second-order or fourth-order FDTD formulation combined with each of the discussed approaches. Index Terms—Finite-difference time-domain (FDTD) method, numerical dispersion, wideband Lorentzian media.

I. INTRODUCTION VER the past several decades, the finite-difference timedomain (FDTD) method has been proven to be one of the most popular and powerful numerical tools for modeling various complex media [1] and quite a variety of FDTD approaches have been developed to simulate the dispersion characteristics of the well-known narrowband Lorentzian (NBL) media, the representative ones among them are such as those reported in [2]–[7] and with their modeling accuracy and numerical stability studied in [8], [9]. However, the FDTD formulations for another type of Lorentzian media, the so-called wideband Lorentzian (WBL) media, have rarely been investigated in literature except for the two proposed in [10] and [11], which are based on the recursive convolution (RC) approach [2] and the auxiliary difference

O

Manuscript received November 28, 2009; revised September 12, 2010; accepted September 13, 2010. Date of publication January 31, 2011; date of current version April 06, 2011. This work was supported by the Fundamental Research Funds for the Central Universities of China, under Research Project YWF-10-02-025. Z. Lin and C. Zhang are with the School of Instrumentation Science and Optoelectronics Engineering, Beihang University, Beijing 100191, China (e-mail: [email protected]). Y. Fang is with the Department of Physics, Zhenjiang Watercraft College, Zhenjiang 212003, China. J. Hu is with the Department of Electrical Engineering, College of Mechanical and Electrical Engineering, Henan Agricultural University, Zhengzhou 450002, 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.2011.2109677

equation (ADE) approach [5], [6], respectively. This may be due to the fact that a WBL medium has a Lorentzian resonance with its half linewidth, , larger than its angular resonance frequency, . It makes some popular FDTD approaches initially proposed for modeling various types of dispersive media unable to model this type of Lorentzian media in a direct way because involved is no longer the auxiliary parameter with a real value that could be applied in the time-stepped updating equations. The approaches that can not be applied for modeling the WBL media directly include the recursive convolution (RC) approach [2], the piecewise linear recursive convolution (PLRC) approach [3], and the Z-Transform (ZT) approach [4]. As a matter of fact, in order to model a WBL medium by these three approaches, the WBL dispersion should first be represented with the subtraction of two Debye-like dispersions and then each of the Debye-like dispersions can be modeled solely [10]. However, this issue doesn’t happen to the category of FDTD approaches based on frequency-domain approximations, such as the auxiliary difference equation (ADE) approach [5], [6] and the bilinear-transform (BT) approach [7], where the parameter is not involved in their time-stepped updating equations. As so many approaches are existing, our first drive behind this work is to derive a unified FDTD model that could represent all the previously-mentioned FDTD approaches in modeling the constitutive relation for this type of Lorentzian media and further to evaluate their work performance in practical FDTD simulations after they are combined with the time-stepped updating equations under the second-order or fourth-order accurate FDTD formulation. In view of practical applications, it is of importance and interest to investigate and compare the modeling accuracy and numerical stability of these approaches so as to find a more accurate and suitable one among them for simulating a specific FDTD problem. The subsequent parts of this work are organized as follows. In Section II, we first give the relevant definition of the permittivity of a WBL medium. The several widely-used approaches for implementing the dispersion characteristics of WBL media are examined and expressed in the form of a general second-order discrete numerical model with different sets of model coefficients, based on which a unified set of updating equations is possibly given. In Section III, the numerical dispersion equation for the general second-order model under the second-order or fourth-order FDTD formulation is given. The concepts on the numerical and effective permittivity and permeability of WBL media modeled by these approaches are discussed to evaluate their work performance and the von Neumann method is utilized to give an approximation of the stability limit on time step for each of the involved approaches. In Section IV, numerical examples with the wave propagation in an exemplified medium and

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the wave transmission through the interface between the WBL medium and vacuum simulated by the discussed approaches are presented respectively. Some explanations on the reason why some approaches outperform others for the specific exemplified modeling problem are given and verified by the practical FDTD simulation results as compared to the analytical solutions. In Section V, the conclusions are made about the work.

, , , under the notations , , and , respectively. in (1) can be easily The ohmic term implemented in the FDTD algorithm [1] with satisfactorily high accuracy based on the bilinear transform

II. FORMULATION

with , . Because its truncation error is at the order of magnitude of , quite lower than the numerical dispersion error of second-order accurate FDTD algorithms when is with a small value as it should be in many practical FDTD simulations, we focus in this paper mainly on the FDTD approaches for formulating the WBL dispersion given by (5) or (6) in the following discussions. To implement (5), one potential approach is the recursive convolution (RC) approach [2] initially designed for modeling the Debye or NBL dispersions and lately utilized to simulate the electromagnetic problems involving the WBL media [10]. Following the full formulation given therein, the Z-domain discrete model pertinent to this approach for WBL dispersion can be derived that (5) is in fact approximated based on the following transformation

The dispersion characteristics of a WBL medium, such as the FR-4 fiber glass epoxy often used in printed circuit boards (PCBs) in the microwave frequency region, can be described by its frequency-domain relative permittivity with a form (1) where the susceptibility term is given by (2) and and are the infinite-frequency and static relative dielectric constants, is the angular resonance frequency, is half of the Lorentzian resonance linewidth, and is the conductivity representing an ohmic loss. As mentioned in Section I, the Lorentzian media can be divided into two categories, the WBL media and the NBL media, depending on their intrinsic material parameters. For a WBL medium, is larger than , while for [10]. To model the Lorentzian a NBL medium is less than media by the RC, RLRC, and ZT approaches, an auxiliary paneeds to be utilized in some coefficients rameter of time-stepped updating equations [2]–[4]. However, is with an imaginary value for the case of a WBL medium, indicating that these approaches can not be used to implement (2) directly by a similar means as that for a NBL medium [8]. Therefore for these approaches, we should first decompose the susceptibility term (2) in (1) into the subtraction of two Debye-like dispersions [10] as

(7)

(8) where , for , 2. If the enhanced version of RC approach, the so-called PLRC approach [3], is applied instead, (5) would be approximated with a higher accuracy by

(9) where for , 2. Another popular FDTD approach, widely used for modeling various dispersive media based on the Z transform (ZT) [4], is to approximate (5) via another transformation

(3)

(10)

(4)

As to implement (6) in the FDTD algorithm, noting that and the values of is much smaller than unit one in practical simulations, we have

where

and , . That is, the RC, RLRC, and ZT approaches are formulated to implement (3) rather than (2). On the other hand, the updating equations for the ADE and BT approaches don’t involve the parameter , so that they can be used directly to model (2) like that for a NBL media without any modification. For the sake of compactness of the following expressions and given that the time step chosen is , we can rewrite (3) and (2) into

(11) and (12) By substituting the above two approximations into (6) and after some mathematical operations, we obtain the discrete model pertaining to the ADE approach [5], [6], [11]

(5) (6)

(13)

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TABLE I THE COEFFICIENTS OF THE GENERAL SECOND-ORDER DISCRETE MODEL FOR THE INDICATED APPROACHES AND THEIR APPROXIMATE STABILITY LIMITS ON TIME STEP IN MODELING AN EXEMPLIFIED WBL MEDIUM UNDER THE SPECIFIC SIMULATING PARAMETERS

To compact the expressions, the following notations are utilized: g

=e

or, alternatively, by using another set of approximations that (14)

,h

= 0

,p

=A

=

.

Then the unified set of updating formulas for , , and from their values at previous time steps and step is given by step

at time at time

and (15) we have the discrete model based on the BT approach [7]

(21) (16)

(22)

where , and . Interestingly enough, all the discussed approaches for modeling the WBL dispersion can be generalized into the following second-order discrete model in the Z domain

(23)

(17) where the sets of six model coefficients are different for various FDTD approaches and for the convenience of application and comparison, they are summarized in Table I. Based on (7)–(17), the complete updating equations for the whole WBL model (1) can be formulated, which could be further combined with the updating equations for the second-order or fourth-order FDTD formulation of Maxwell’s curl equations. We first define the following quantities:

and the spatial indices have where stands for time instant been suppressed for clarity. To complete a single time iteration of a practical FDTD simulation, the updating equations for the Faraday’s law and Ampere-Maxwell law should be performed before and after (21)–(23), respectively. Under the classical procedure for a standard second-order FDTD method, they are [1] (24) (25) while for the fourth-order FDTD formulation [12], they are

(18) (19)

(26)

(20)

(27)

and

LIN et al.: ON THE FDTD FORMULATIONS FOR MODELING WIDEBAND LORENTZIAN MEDIA

where represents the impressed current density at time instant and here it is served as the wave source. ’ denotes a second-order discrete spatial curl operator, ‘ and field components at the nodes such as for the and , respectively, and in (24) and (25) should be calculated as

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velocity of propagation of the numerical solution for a wave in free space, , different from the theoretical speed of light, . As a matter of fact, the numerical dispersion relation of the second-order accurate central-difference FDTD algorithm [1] is given for modeling the free space or vacuum by (32)

(28)

in the Z domain or, equivalently (33)

(29) respectively. denotes at a fourth-order discrete spatial curl and field components operator, such as for the nodes and , respecand in (26) and (27) are fortively, mulated as

in the frequency domain, where , is the complex wave number of the discrete mode in -direction for is the size of the discretization vacuum and cell along the axis. Under the fourth-order FDTD formulation for modeling the vacuum [12], (32) becomes (34) with in Z domain and its counterpart in frequency domain is obtained as

(30) and

(35) . Effectively, (33) or (35) also under the replacement represents the dispersion relation of an artificial vacuum with a set of effective relative permittivity and permeability implicitly governed by (36) is a function of where and can be numerically determined by (33) or (35). When and , we have since when is with a small value. Furthermore, from (18)–(20), the relative numerical permittivity (37)

(31) respectively. Update equations for other field components can be derived similarly. III. MODELING ACCURACY AND NUMERICAL STABILITY The updating equations (21)–(23) are only for implementing the constitutive relation of a dispersive WBL medium. To give a close inspection of the work performance of the discussed approaches in practical FDTD simulations, we should also consider the spurious influence of the inherent numerical dispersion error arising from the second-order or fourth-order difference approximation of derivatives of continuous field functions in the Maxwell’s equations (24), (25) or (26), (27), which makes the

from each of the previously discussed approaches can be easily calculated in the Z-domain by substituting (19) and (20) into (18). Then the numerical dispersion equation for the second-order FDTD formulation combined with the discussed approaches for modeling a WBL medium is approximately given by (38) in the Z domain [12], [13] or, equivalently (39)

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in the frequency domain, where and is the complex wave number of the discrete mode for WBL media in the -direction. As to model WBL media under the fourth-order accurate FDTD formulation, the numerical dispersion equation is approximately given by

and (40) for modeling dispersive and lossy media by the FDTD method are just good approximations to the exact numerical dispersion relations as we can see lately from the data shown in Fig. 4, Fig. 5, and Fig. 6. In fact, the exact numerical dispersion varies temporally as a function of time and also depends on the values of transient electric field when it is updated by (21)–(23). This is due to the ratio between the quantity at time step

(40) in the Z domain, or

(44) and the quantity at previous time step (45)

(41) in the frequency domain. Similarly, (39) or (41) represents an artificial WBL medium with the effective relative permittivity and permeability (42) (43) where could be numerically calculated from (39) or (41). That is, a WBL given medium with a frequency-dependent permittivity is modeled as an by (1) and a constant permeability artificial WBL medium with the effective permittivity and permeability given by (42) and (43) under the second-order or fourth-order FDTD formulation combined with the discussed approaches for implementing the WBL dispersion. When and , we have and . Thence the working performance of any of the discussed approach is governed by the difference between the permittivity and permeability of the real and artificial and , respectively. From (39), media, it should be noted that due to the relatively large numerical dispersion error from the second-order FDTD formulation, an approach with higher accuracy in implementing the WBL dispersion as discussed in Section II doesn’t always result in higher final modeling accuracy and better work performance. Thus under the second-order formulation, a preferred approach should not only be able to model the dispersion characteristics of the simulated dispersive medium accurately enough, but also could compensate the numerical dispersion error resulting from the second-order difference approximation of continuous field functions, where the latter is depending on the size of spatial discretization relative to the shortest wavelength and the time step applied. On the contrary, since the numerical dispersion error of the fourth-order FDTD formulation is much smaller than that of the second-order one, an approach more competent to implement the WBL dispersion will also result in a higher final modeling accuracy. However, it should also be noted that the numerical dispersion (32) or (33) for the FDTD modeling of vacuum or nondispersive media are exactly accurate, but the numerical dispersion (38)

. However, the difference is not exactly equal to approaches zero between them is reasonably small when and if the amplitudes of electric field don’t change dramatically at adjacent time steps. The von Neumann method [14] is useful and convenient for estimating the numerical stability limits of the several discussed for a plane wave approaches, which can be written as is any of the real or complex roots of (38) [13], [14], where or (40) for the case of WBL media under the second-order or fourth-order FDTD formulation. By the method of numerical with the stability condisearching for the range of values of tion being fulfilled, we can find the maximum one among them, which is called the stability limit. Because the function in (30) is often assigned with unit one in the process of derivation of stability limit, these conditions also give a sufficient stability limit for the case of non-plane waves with a generous tolerance. However, since the numerical equations (39) and (40) are just good approximation formulae to the exact numerical dispersion relations that only can be exactly determined from practical simulation results, it should be emphasized that the derived stability limits are also just with approximate values and potential instability may occur when is not small enough for the high frequency components of the modeled electromagnetic wave propagating in dispersive or lossy media even though the stability limit on time step derived by the von Neumann method is fulfilled. Meanwhile, due to the finite machine precision problem, late-time instability would also possible happen because the roots may move slightly outside the unit circle of complex plane although the von Neumann analysis shows they are within or on the unit circle. IV. NUMERICAL EXAMPLES AND DISCUSSIONS Consider a practical problem of simulating the propagation of an electromagnetic wave in a WBL medium with the ma, , terial parameters , , and in the frequency range from 100 MHz to 5 GHz [10]. The grid size is chosen as 2 mm, less than one twentieth of the shortest wavelength of the source spectrum, so that the Courant stability limit [1] for this one-dimensional (1-D) FDTD problem is in modeling a non-dispersive dielectric with a constant under the second-order FDTD under the formulation [1] and fourth-order FDTD formulation [12]. Under these modeling parameters and from (38) and (40), we can calculate the approx-

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Fig. 1. The frequency distribution of the absolute difference between the nu) of each of the indicated approaches in immerical permittivity "^ (e plementing the constitutive relation of the exemplified WBL medium and its theoretical counterpart "^ (2f ), where the influence of the numerical dispersion error from the FDTD method has not yet been considered.

Fig. 2. The analytical results on the frequency distributions of the absolute difand permeability  of ~ ferences between the effective permittivity "~ each of the indicated approaches in modeling the exemplified WBL medium under the second-order FDTD formulation and their theoretical counterparts " and  = 1, respectively.

imate stability limits on time step of all the RC, PLRC, ZT, ADE, and BT approaches for the specific WBL medium and and modeling parameters, as shown with respective to in the two bottom rows of Table I for the second-order and fourth-order FDTD formulations, respectively. For the convenience of comparison, a sufficiently stable time step is applied for the specific spatial discretization in the following calculations and practical FDTD simulations. In Fig. 1, , the magnitude of the difwe plot the error ference between the numerical relative permittivity of each approach and its theoretical counterpart , as a function of frequency, to evaluate their abilities in approximating the constitutive relation of the exemplified WBL medium, respectively. It is evident that the BT approach performs better than all the other approaches in implementing the WBL dispersion in the whole frequency range of interest and ZT works worst due to a real-valued discrepancy [8] in numerical permittivity as compared to its theoretical permittivity. However, when they are combined with the second-order FDTD updating equations (24), (25) for the two Maxwell’s curl equations, the work performance of all the previously discussed approaches in practical FDTD simulations changes accordingly due to the relatively large numerical dispersion error of the FDTD method resulting from the second-order accurate difference approximation of partial derivatives in the Maxwell’s curl equations. In Fig. 2, we show the frequency distribution of , the magnitude of the difference the error between the effective permittivity pertaining to the second-order FDTD formulation combined with each of the . discussed approaches and its theoretical counterpart Also shown in the same figure is the frequency distribution of , the magnitude the logarithmic error of the difference between the effective permeability pertaining to the second-order FDTD formulation and its theo. We can see from Fig. 2 that under retical counterpart the second-order FDTD formulation, the work performance of

the PLRC, ADE, and BT approaches is nearly identical due to the relatively large dispersion error from the second-order accurate FDTD formulation, but all of them are better than that of the RC approach as a whole. As for the ZT approach, it is better than all the other approaches in the high frequency range above 2.5 GHz, but worse in the low frequency range. By comparing Fig. 1 and Fig. 2, we find that the reason is that the error of ZT approach in implementing the WBL dispersion is fortunately and partially compensated by the numerical dispersion error of the second-order FDTD formulation in the high frequency region, especially near the frequency 3.5 GHz. At low frequencies, the numerical dispersion error decrease dramatically, but the error of ZT approach in implementing the WBL dispersion remains nearly unchanged as we can see from Fig. 1. Thus the latter error can not sufficiently counteract the influence of the former error, which results in a big redundant error from the ZT approach in the low frequency range as shown in Fig. 2. This reasoning is also evidenced when these approaches are incorporated into the fourth-order FDTD formulation. The fre, the magnitude of quency distribution of the frequency-dependent difference between the effective perof the fourth-order FDTD formulation committivity bined with each of the discussed approaches and its theoretis shown in Fig. 3, which is nearly the ical counterpart except the PLRC apsame as the error proach. This is because the numerical dispersion error of the fourth-order FDTD formulation is substantially suppressed as compared to the second-order FDTD formulation if we compare the two dashed lines shown in Fig. 3 and Fig. 2, respectively. Thus the modeling accuracy is mainly determined by the ability of each approach in mimicking the constitutive relation . of WBL media, that is, the error To further verify this hypothesis, two numerical examples are presented with practical 1-D FDTD simulations involving the FDTD modeling of the exemplified WBL medium. In the

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Fig. 3. The analytical results on the frequency distributions of the absolute difand permeability  of ~ ferences between the effective permittivity "~ each of the indicated approaches in modeling the exemplified WBL medium under the fourth-order FDTD formulation and their theoretical counterparts " and  = 1, respectively.

Fig. 4. The absolute difference between the propagation function P~ (2f ) calculated from the FDTD simulation results under the second-order FDTD formulation for each of the indicated approaches and the theoretical propagation function P (2f ). The dashed line corresponds to the absolute difference between P~ (2f ) and the analytical propagation function P~ (2f ) based on the concept of effective permittivity and permeability given by (42) and (43) under the second-order FDTD formulation.

first numerical example, the propagation of a differentiated Blackman-Harris pulse in the exemplified WBL medium in the positive direction is simulated under the second-order and fourth-order FDTD formulations combined with each of the discussed approaches for implementing the constitutive relation of WBL dispersion. The spectrum of applied source is spanning across the frequency range from dc to 5 GHz. The whole cells is filled with computational space consisting of the exemplified WBL medium and truncated at both ends by two 16-cell-thick perfectly matched layers (PMLs). The current source that generates the pulse is located at the cell, and two detectors are placed at the and cells to record the reference wave and the propagating wave , respectively. Thus, the practical modeling ability of the second-order or fourth-order FDTD formulation combined with each of the discussed approaches for simulating a WBL medium can be evaluated by the difference between the propagation function (46) denoting a fast calculated from the simulation results with Fourier transform and its theoretical counterpart given by (47) with an time convention and calculated from the fact that propagates the reference wave cells, or 1.6 meters. To compare the two quantities, we plot the in a logarithmic scale pertaining to each error of the discussed approaches under the second-order formulation in Fig. 4 and under the fourth-order formulation in Fig. 5, respectively. We can find that the lines denoting the errors obtained from the simulation results agree with those indicated in Fig. 2 and Fig. 3. This fact proves that the artificial WBL medium does have an effective permittivity and permeability given by (42)

Fig. 5. The absolute difference between the propagation function P~ (2f ) calculated from the FDTD simulation results under the fourth-order FDTD formulation for each of the indicated approaches and the theoretical propagation function P (2f ). The dashed line corresponds to the absolute difference between P~ (2f ) and the analytical propagation function P~ (2f ) based on the concept of effective permittivity and permeability given by (42) and (43) under the fourth-order FDTD formulation.

and (43) in stead of and 1, respectively, and therefore should be numerically equal to its analytical solution given by (48) This is clearly verified by the much smaller absolute erunder the second-order and rors fourth-order FDTD formulations, indicated by the two dashed lines we plot in Fig. 4 and Fig. 5, respectively, which are nearly

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frequency range but worse than them in the low frequency range. Also shown in Fig. 6 with a dashed line is the frequency distriand its bution of the magnitude of the difference between analytical counterpart [15] of given by

(51)

Fig. 6. The absolute difference between the transmission function T~ (2f ) calculated from the FDTD simulation results under the second-order FDTD formulation for each of the indicated approaches and the theoretical transmission function T (2f ). The dashed line corresponds to the absolute difference between T~ (2f ) and the analytical transmission function T~ (2f ) based on the concept of effective permittivity and permeability given by (42) and (43) under the second-order FDTD formulation.

overlapped for all the discussed approaches and therefore only one of the five lines is plotted in each figure. The errors are at and should be attributed to the the order of magnitude of approximation nature of the numerical dispersion (38) and (40). The second numerical example goes to the simulation of the transmission process of the same electromagnetic pulse through an interface between the exemplified WBL medium and the vacuum. The space size and modeling parameters are the same as those applied in the first example except that the half part of the computational space from the 501st cell to its right-hand end is filled with the free space instead of the WBL medium and cell is now utithence the second detector at the lized to record the transmitting wave rather than the . By performing the FDTD simupropagating wave lations using the discussed approaches for modeling the WBL constitutive relation under the second-order FDTD formulation, we obtain the transmission function (49) calculated from the simulation results and its theoretical counterpart given by (50) We illustrate in Fig. 6 the frequency distribution of the absolute difference between and in logarithmic scale for each of the indicated approaches under the second-order FDTD formulation. Once again, the results shown in Fig. 6 verifies the previous conclusions that under the second-order FDTD formulation, the PLRC, ADE and BT approaches work better than the RC approach in modeling the wave propagation and transmission in and at the interface of the exemplified WBL medium, while the ZT approach work better than all the others in the high

which is nearly the same for all the discussed approaches and therefore only one line is plotted. From Fig. 6, we can find that the order of magnitude of the difference between and is much smaller than that between and for each of the discussed approaches, again verifying that the modeled WBL medium has an effective permittivity and permeability given by (42) and (43) and the modeled vacuum has an effective permittivity and permeability given by (36). V. CONCLUSION The basic principles of several popular FDTD approaches for modeling the so-called WBL media are investigated and compared with each other in a detailed and systematical way. The model coefficients of a general second-order discrete numerical model pertaining to each of these approaches are derived respectively and a unified set of FDTD updating equation for implementing them is given. Their work performance under the second-order or fourth-order FDTD formulation is studied and demonstrated by evaluating the wave propagation in an exemplified WBL medium and transmission functions and at the interface of the exemplified WBL medium and the vacuum. The implicit expressions for both the numerical and effective permittivity and permeability are also presented and verified by the simulation results. We find that under the second-order FDTD formulation, the working behaviors of the PLRC, ADE, and BT models are nearly identical and better than the RC model as a whole for the given FDTD problem, while the ZT works better than all the other models in the high frequency region, but worse in the low frequency region. However, under the fourth-order FDTD formulation with a lower numerical dispersion error, the performance of the ZT approach become worse then the others due to its weakness in implementing the constitutive relation of a WBL medium in the frequency of interest. It should be mentioned here that the conclusions made about the modeling accuracy and numerical stability of these approaches are only demonstrated for the exemplified WBL medium and under the specific modeling parameters. The performance of the various approaches for other WBL materials may vary since there are quite a variety of WBL media with different material parameters that can not be completely covered in this work, as well as the modeling parameters on spatial or temporal discretization. However, we have presented in this work the general formulae for the implementation of the various approaches for WBL media and the procedures that guide how to evaluate their performance for modeling this type of media with different material parameters can be followed in a similar way as those

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for the examples examined in this work. The results of this paper may find potential applications in the design and electromagnetic compatibility analysis of any electronic devices and structures made of WBL materials like the PBC boards. REFERENCES [1] A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 2nd ed. Norwood, MA: Artech House, 2000. [2] R. J. Luebbers, F. P. Hunsberger, K. S. Kunz, R. B. Standler, and M. Schneider, “A frequency-dependent finite-difference time-domain formulation for dispersive materials,” IEEE Trans. Electromagn. Compat., vol. 32, no. 3, pp. 222–227, Aug. 1990. [3] D. F. Kelley and R. J. Luebbers, “Piecewise linear recursive convolution for dispersive media using FDTD,” IEEE Trans. Antennas Propag., vol. 44, no. 6, pp. 792–797, Jun. 1996. [4] D. M. Sullivan, “Frequency-dependent FDTD methods using Z transforms,” IEEE Trans. Antenna Propag., vol. 40, no. 10, pp. 1223–1230, Oct. 1996. [5] M. Okoniewski, M. Mrozowski, and M. A. Stuchly, “Simple treatment of multi-term dispersion in FDTD,” IEEE Microw. Guided Wave Lett., vol. 7, no. 5, pp. 121–123, May 1997. [6] Y. Takayama and W. Klaus, “Reinterpretation of the auxiliary difference equation method for FDTD,” IEEE Microw. Wireless Compon. Lett., vol. 12, no. 3, pp. 102–104, Mar. 2002. [7] C. Hulse and A. Knoesen, “Dispersive models for the finite-difference time-domain method: Design, analysis, and implementation,” J. Opt. Soc. Amer. A, vol. 11, no. 6, pp. 1802–1811, Jun. 1994. [8] Z. Lin and L. Thylén, “On the accuracy and stability of several widely used FDTD approaches for modeling Lorentz dielectrics,” IEEE Trans. Antennas Propag., vol. 57, no. 10, pp. 3378–3381, Oct. 2009. [9] Z. Lin, P. Ou, Y. Jia, and C. Zhang, “A highly accurate FDTD model for simulating Lorentz dielectric dispersion,” IEEE Photon. Technol. Lett., vol. 21, no. 22, pp. 1692–1694, Nov. 2009. [10] M. Y. Koledinsteva, J. L. Drewniak, D. J. Pommerenke, G. Antonini, A. Orlandi, and K. N. Rozanov, “Wide-band Lorentzian media in the FDTD algorithm,” IEEE Trans. Electromagn. Compat., vol. 47, no. 2, pp. 392–399, May 2005. [11] S. Aksoy, “An alternative algorithm for both narrowband and wideband Lorentzian dispersive materials modeling in the finite-difference timedomain method,” IEEE Trans. Microw. Theory Tech., vol. 55, no. 4, pp. 703–708, Apr. 2007. [12] K. P. Hwang and J. Y. Ihm, “A stable fourth-order FDTD method for modeling electrically long dielectric waveguides,” J. Lightw. Technol., vol. 24, no. 2, pp. 1048–1056, Feb. 2006. [13] W. H. Weedon and C. M. Rappaport, “A general method for FDTD modeling of wave propagation in arbitrary frequency-dependent media,” IEEE Trans. Antennas Propag., vol. 45, no. 3, pp. 401–410, Mar. 1997. [14] J. A. Pereda, L. A. Vielva, Á. Vegas, and A. Prieto, “Analyzing the stability of the FDTD technique by combining the von Neumann method with the Routh-Hurwitz criterion,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 2, pp. 377–381, Feb. 2001. [15] A. Christ and N. Kuster, “Correction of the numerical reflection coefficient of the finite-difference time-domain method for efficient simulation of vertical-cavity surface-emitting lasers,” J. Opt. Soc. Amer. B, vol. 20, no. 7, pp. 1401–1408, Jul. 2003.

Zhili Lin (M’10) received the B.S. and Ph.D. degrees from Zhejiang University, Hangzhou, China, in 2002 and 2007, respectively. From November 2007 to November 2008, he was a Postdoctoral Research Fellow with the Department of Microelectronics and Applied Physics, School of Information and Communication Technology, Royal Institute of Technology, Sweden, before he joined the Department of Electro-Optical Engineering, School of Instrumentation Science and Opto-electronics Engineering, Beihang University, Beijing, China, as a faculty member. His research interests include computational electromagnetics, microwave photonics, optical waveguides, and fiber-optic sensors. He has authored or coauthored more than 20 papers in refereed journals and conference proceedings. Dr. Lin is a member of the American Physical Society (APS), the Optical Society of America (OSA), and a Senior Member of the Chinese Optical Society (COS).

Yuntuan Fang received the Master degrees from the Nanjing Normal University, Nanjing, China, in 2000. From August 2007 to August 2008, he was a Visiting Scholar with the Department of Microelectronics and Applied Physics, School of Information and Communication Technology, Royal Institute of Technology, Sweden. He is currently with the Department of Physics, Zhenjiang Watercraft College, Zhenjiang, China. He has authored or coauthored more than 40 papers in refereed journals. His research interests include computational electromagnetics, microwave photonics, photonic crystals, and optical communications.

Jiandong Hu received the Ph.D. degree in optical engineering from the College of Information Science and Engineering, Zhejiang University, Hangzhou, China, in 2005. He is currently a Professor with the Department of Electrical Engineering, Henan Agricultural University, Zhengzhou, China. He has authored or coauthored more than 60 papers in refereed journals and conference proceedings. His research interests include electromagnetic modeling and simulation, optical surface plasma resonance, and photoelectric signals acquisition and processing.

Chunxi Zhang received the Ph.D. degree from Zhejiang University, Hangzhou, China, in 1996. He is currently a Professor and Vice Dean of the School of Instrumentation Science and Opto-electronics Engineering, Beihang University, Beijing, China. He has authored or coauthored more than 100 papers in refereed journals and holds over 30 patents. His research interests focus on signal processing, optoelectronic detection, optical waveguides, and integrated optics.

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An Analytical Expression for 3-D Dyadic FDTD-Compatible Green’s Function in Infinite Free Space via z-Transform and Partial Difference Operators Shyh-Kang Jeng, Senior Member, IEEE

Abstract—FDTD solutions have their own properties distinct from the discrete samples of corresponding continuous wave solutions. Thus, the discrete equivalent to the Green’s function is needed for applications like the one using a hybrid absorbing boundary condition which couples the FDTD algorithm with integral operators for nonconvex scatterers. In this paper we propose a new closed-form expression for the 3-D dyadic FDTD-compatible Green’s function in infinite free space via a novel approach with the ordinary z-transform along with the spatial partial difference operators. The final expression involves a summation of standing wave modes with time-varying coefficients. The propagation of waves in the Yee’s grid can be interpreted by the selective property of the time-varying coefficients, which is very different from the conventional concept of a traveling wave. The traditional dispersion analysis using plane waves for the FDTD algorithm in a source-free region may not be applicable to explain the wave propagation phenomenon through our analytic expression, because the corresponding z-transform diverges for z on the unit circle. Index Terms—FDTD methods, Green functions, operators (mathematics), Z transforms.

I. INTRODUCTION

T

HE need for an Yee’s FDTD compatible Green’s function (also known as the discrete Green’s function) has been identified by Kastner [1] and Vazquez and Parini [2]. In [1] Kastner mentioned that Boag et al. proposed a hybrid absorbing boundary condition for nonconvex scatterers [3], which couples FDTD and integral operators. Kastner argued that since FDTD solutions have their own dispersion, anisotropy, and stability properties, applications coupling FDTD and integral operators create the need to produce discrete equivalents to the integrator and the corresponding Green’s function. In fact, to link the sources and the fields, a discrete dyadic Green’s function is required. In [2], Vazquez and Parini pointed out that with a 3-D discrete dyadic Green’s function convolving with currents, radiation and scattering problems can be numerically solved on the Yee’s grid avoiding absorbing boundary conditions, and sup-

pressing free space nodes to save computer memory storage requirements. In the meantime, it seems that conventional analyses on the essence of Yee’s FDTD algorithm are confined to plane wave propagation in a source-free region [4]. Very few mathematical analyses on the FDTD algorithm for problems with sources have been done before [1], [5]–[9]. Thus an analytic expression for the discrete dyadic Green’s function will provide some new insights for the FDTD solutions to problems with sources. In [1] Kastner derived 1-D and 3-D scalar discrete Green’s functions using the multidimensional z-transform, but in the 3-D case he can only get the temporal response for field points along the axis. In [2] Vazquez and Parini obtained complicated expressions for 2-D and 3-D discrete Green’s functions by the multidimensional z-transform in spatial domain and a technique introduced by Boole. They later applied those expressions with convolution to model antenna radiation in free space [8]. In this paper we will propose a new closed-form expression for the 3-D discrete dyadic Green’s function in infinite free space via a novel approach with the ordinary z-transform along with the spatial partial difference operators. The partial difference operators have been widely used to deal with partial difference equations by mathematicians [10], but it is believed new to combine the partial difference operators with the z-transform. Similar approach has been applied by the author to obtain an analytical expression for the 2-D discrete Green’s function in a recent conference paper [9]. Finally, in this paper some numerical results together with discussions on interesting phenomena are also included. II. PROBLEM Consider an infinite free space excited by electric and magnetic current sources. The corresponding Yee’s FDTD equations are

Manuscript received May 17, 2010; revised August 10, 2010; accepted November 07, 2010. Date of publication January 31, 2011; date of current version April 06, 2011. This work was supported by the National Science Council, Taiwan, under Grant NSC97-2221-E-002-143-MY3. The author is with the Department of Electrical Engineering and Graduate Institute of Communication Engineering, National Taiwan University, Taipei, Taiwan (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieee.ieeexplore.org. Digital Object Identifier 10.1109/TAP.2011.2109363 0018-926X/$26.00 © 2011 IEEE

(1)

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To get the discrete dyadic Green’s functions, we need to let one vector component of the electric current or the magnetic current source to be a delta function in space and time in turn, and find the corresponding field components. A complete definition will be given in Section IV. III. PARTIAL DIFFERENCE OPERATORS

(2)

Define the forward partial difference operator for a funcas tion , etc. Similarly, the backward partial difference operator is defined by , and so on. These operators can easily be generalized to operate on vector functions. Let (7)

(8)

(3)

(9)

(10) Then the Yee’s FDTD equations (1)–(6) can be written as (4) (11) (12) where (13) (5)

(14) (15)

(16)

(6) Here the Yee cell size is , the sampling in, and terval in time is , where and are the light speed and the intrinsic impedance of free space, respectively.

IV. DISCRETE DYADIC GREEN’S FUNCTION The electric discrete dyadic Green’s functions are defined as

(17)

JENG: ANALYTICAL EXPRESSION FOR 3-D DYADIC FDTD-COMPATIBLE GREEN’S FUNCTION IN INFINITE FREE SPACE

and

(18) , and where tric fields (7) due to the electric current sources

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Here we adopt the one-sided, instead of the two-sided z-transform, because any FDTD program has to start from a certain time index, and the initial conditions can be taken into considerations in later discussions. Taking the z-transform of (11)–(12), we achieve the Yee’s FDTD equations in z-domain

are the elec(28) (19) (20)

and (21) , and respectively, and are the electric fields (7) due to the magnetic current sources (10) in the form of (19)–(21), respectively. Notation for , and for . Similarly, for , and otherwise. The magnetic discrete dyadic Green’s function can be defined similarly, with suitable change of subscripts. They are related to magnetic field (8) instead of electric field (7). Since the electric field and the magnetic field are dual, we will deal with only the electric discrete dyadic Green’s function in this paper. V. FORMULATION

(29) For brevity, the dependence on the spatial indices has been suppressed from herein. The initial fields can be absorbed into equivalent sources in the z-domain (30) (31) Using (30)–(31), we can express (28)–(29) as (32) (33)

B. z-Domain Discrete Dyadic Green’s Function Substituting (33) into (32) leads to an equation involving only the electric field in the z-domain

A. Yee’s FDTD Equations in z-Domain Define the z-transform of (7)–(10) as

(34) (22)

Here

is the identity matrix and (35)

(23)

The electric field can be obtained symbolically as (36)

(24)

Taking the inverse matrix symbolically, we found with the help of Mathematica

(25) (37) From these we can prove

(38) (39) (26)

(27)

(40)

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The symbolic inverse of (39) can be obtained by infinite series expansion as

The three components of

are (45) (46)

(47)

(41) where

is a multinomial coefficient, and

.

We can prove by mathematical induction that (48) (42)

where

is a binomial co-

efficient; and

is the partial advancing operator . The effect of the partial advancing operator on a function of integer indices can be expressed as, for example, . Substituting (37)–(42) back to (36), we get

(49)

(50) (51) (52) (53)

(43) Here are the three components of the vector (35). The results for and can be obtained and corresponding from (43) by rotating the subscripts summation indices and . Considering only the electric discrete dyadic Green’s function (17) due to (21), we have (44)

have been eliminated, leaving Here the summations over only a single term, because of the delta function in (44), which also results in the nonzero lower limit of the summation over . The well-known recurrence identity in combinatorics (54) has also been applied to simplify the results. The components of and can be derived from (45)–(53) by suitable rotation of subscripts and corresponding summation indices .

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C. Time-Domain Discrete Dyadic Green’s Function Taking the inverse z-transform of (45)–(47) and applying the residue theorem with the contour in the region of convergence of (45)–(47), which is outside of the unit circle and encloses

(55)

leads to the final analytical expressions for the three components of (56) (57)

(58) for , and for . Here, The other components of the electric discrete dyadic Green’s function (17) can be derived from (56)–(58) by suitable rotation of subscripts and corresponding summation indices . When is reduced to the 2-D discrete-time Green’s function and approach infinity, derived in [9]. As both can be reduced to a form identical with the 1-D special case, (34) given in [1], with suitable changes of coordinate axes. The final results (56)–(58) are also consistent with the analytical expressions given in [2] by rearranging the polynomial terms and summations (using Jacobi polynomials). However, our results seem to be in a more compact form and may need less computation than those in [2]. It is also found that mutual cancellation of adjacent terms in (48)–(50) to speed up calculation can not be carried out, because the powers of the three Courant numbers are different in different terms. VI. NUMERICAL RESULTS AND DISCUSSIONS Fig. 1 shows the temporal responses for the discrete dyadic Green’s function by (56)–(58) and by the direct implementation of Yee’s FDTD algorithm. The parameters for the computation . The reare sults by (56)–(58) overlap with those by direct implementation of FDTD, confirming the validity of the analytical expressions. Note that here (56)–(58) are computed by Mathematica, with which the binomial and multinomial coefficients can be calculated accurately. The discrete dyadic Green’s functions shown in Fig. 1 is obviously different from the corresponding continuous dyadic

Fig. 1. Temporal responses of the 3-D discrete dyadic Green’s function computed by direct implementation of FDTD and analytic expressions (overlapped).

Green’s functions sampled at the Yee nodes and the time after dividing by

,

(59)

(60)

(61) is the unit step function, is the Dirac’s where is the derivative of . Also delta function, and , and are in these equations the azimuthal and polar angles for the spherical coordinates corresponding to the Cartesian coordinates , and , respectively, with respect to . Notations and stand for the normalized distances (62) (63)

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(64) For the same parameters used for computing Fig. 1, , and equal 64.3545, 64.5872, and 64.8074, respectively, close to the onset time of the main time responses in Fig. 1. Numerical results for (59)–(64) are not shown in Fig. 1, because the delta function is difficult to display, and the unit step function as well as the derivative of the delta function parts of all three field components involve , which remains there after normalization. Although (56)–(58) are very different from (59)–(61), we will see that the continuous temporal responses due to a point source with smooth temporal variation are close to the discrete responses by convolving the discrete Green’s dyadic functions with the sampled source signal. Consider a point electric cur. The generated rent source continuous electric fields sampled at the Yee cells and the time , after dividing by , will be Fig. 2. Electric fields due to a point source, evaluated by direct implementation of Yee’s algorithm, by convolution with the discrete dyadic Green’s function, and by sampling the continuous wave solution. Note that the curves by the direct FDTD implementation and those by convolution overlap.

(65) (70)

(66)

Fig. 2 shows the results evaluated by (65)–(67), (68)–(70), and direct implementation of Yee’s FDTD algorithm, with (71), shown at the bottom of the page, and . In the figure the curves obtained via direct implementation of the Yee’s algorithm overlap with those computed by (68)–(70), verifying our FDTD-compatible Green’s dyadic function again. At the same time, the FDTD results just deviate from the continuous results slightly. This provides a support for using FDTD to approximately solve general continuous wave problems. Next, let’s get back to the discrete dyadic Green’s function. Equations (56)–(58) can be interpreted as summations of , standing wave modes respectively, with temporal coefficients

(67) For the same problem, the FDTD solution can be expressed as (68) (69)

over the

index . This is very different from the traveling wave expressions for the continuous dyadic Green’s function. Although (56)–(58) are expressions with standing wave modes, numerical results obtained from them still show a wave propagating outward. Since (45)–(47) diverge for z on the unit circle, results of the conventional dispersion analysis of the FDTD algorithm by assuming the field to be a uniform plane wave in the form of , with , may not be applicable to explain the propagation phenomenon here.

(71)

JENG: ANALYTICAL EXPRESSION FOR 3-D DYADIC FDTD-COMPATIBLE GREEN’S FUNCTION IN INFINITE FREE SPACE

Fig. 3. Spatial distribution of the magnitude of g on the plane k = 0.

Fig. 4. Spatial distribution of the magnitude of g on the plane k = 0.

[

[

i; j; k ] in dB for m = 50

Fig. 5. Spatial distribution of the magnitude of g on the plane k = 0.

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[

i; j; k ] in dB for m = 50

i; j; k ] in dB for m = 50 Fig. 6. Modal coefficients (

m (vertical axis).

Figs. 3 to 5 exhibit the spatial distribution of the magnitudes of the standing wave mode in dB for with by (48)–(50) on the plane . Note that the numerical results have a very large dynamic range, and are better presented by the magnitude in dB, ignoring the fast are set to changing signs. Besides, magnitudes less than dB. We may see that the boundaries of the modes are in be a rhombus-like shape. This can be explained by that (48)–(50) , and , respectively. are zero when These conditions are corresponding to regions with the Manhattan-like distances from the source to the field point being smaller than a threshold. Obviously, modes with larger index will occupy larger areas. Note also that the Manhattan-like distance is a more natural measure of distance for the Yee’s grid than the Euclidean distance.

n+m ) in dB for various n (horizontal axis) and m+2

2

Though (56)–(58) are in terms of the standing wave modes, the disturbance can still propagate outward. The reason may be obtained by observing the temporal modal coefficients in dB plotted in Fig. 6. From the figure, we see that for a given (the horizontal axis), the coefficients

will pick out the

modes whose modal index (the vertical axis) approximately satisfies . Thus when increases, the selected mode will be also with an increasing , which corresponds to a standing wave mode distributed in a larger area, and the disturbance spreads out. The wave propagation phenomenon may be observed further by calculating the time responses (56)–(58), for a larger

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Fig. 7. Spatial distribution of the magnitude of = 45 on the plane = 0.

n

k

G

[

i

= ; j; k

+1 2

]

in dB at

Fig. 9. Spatial distribution of the magnitude of = 45 on the plane = 0.

n

k

G

G

[

i; j;k

=

+ 1 2]

in dB at

; ; =

Fig. 10. [0 0 1 2] computed by direct implementation of FDTD and analytic expressions (overlapped). Fig. 8. Spatial distribution of the magnitude of = 45 on the plane = 0.

n

k

G

[

i; j

= ;k

+1 2

]

in dB at

time index, and observe the spatial distribution of the magnitude. For example, Figs. 7 to 9 displays the spatial distribution on the plane. The set of magnitude in dB for of outer peaks form a shape like a circle, very close to our expectation according to the experience in dealing with the continuous wave case. The field between the circle-like contour and the rhombus-like wavefront becomes much smaller in the meanwhile. Another difference of the discrete dyadic Green’s functions from the continuous dyadic Green’s functions is that the continuous dyadic Green’s functions become singular when the field point is identical with the source point, while the discrete dyadic Green’s functions are always finite. For example, (61) diverges , but (58) keeps bounded, even when when , as shown in Fig. 10, with .

Finally, we would like to share some experiences in evaluating the numerical values of (68)–(70). These equations involve several binomial and multinomial coefficients. It has been known by many researchers that the binomial and multinomial coefficients become very large for large , and this will cause numerical problems, if they are calculated by programs written , in which the inby common languages like FORTRAN or C tegers are represented by few computer words of fixed number (32, 64, or 128) of bits. To avoid such an issue, in this study we compute (68)–(70) by the symbolic mathematics software Mathematica using its built-in functions Binomial and Multinomial. In Mathematica large integers are represented by arrays of computer words. Thus accurate results can be obtained through arbitrary precision arithmetic. The evaluation of binomial coefficients is limited by $MaxNumber, the maximum arbitrary-precision number that can be represented in the computer using Mathematica, and the available memory. For a 32-bit notebook computer with 2.6 GHz duo CPU and 3 GB RAM, $Maxnumber

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developing a hybrid absorbing boundary condition. The z-transform and operator approach is possible to be utilized to find the discrete dyadic Green’s function in domains like an infinite rectangular waveguide, but taking the inverse z-transform will be more involved. ACKNOWLEDGMENT

Fig. 11. CPU time required for generating the discrete Green’s function (56) and for doing the discrete convolution in (68) from n = 0 to the upper limit shown on the horizontal axis. Here (i; j; k ) = (10; 20; 30).

The author wishes to thank Prof. S. S. Cheng at the National Tsing Hua University, Taiwan, for the suggestion of using the operator theory to solve partial difference equations. In addition, very helpful and inspiring comments for improving the manuscript from the anonymous reviewers are much appreciated. Thanks are also given to Dr. N. Lariviere and Dr. P. Mokhasi, Technical Support, Wolfram Research, Inc., for providing information on the representation of large integers and the limit of arbitrary-precision arithmetic in Mathematica. REFERENCES

is about . Using Mathematica 7 on that computer, binomial coefficients such as one million choose half million can be calculated almost instantly, ten million choose five million is computed within two seconds, and one hundred million choose fifty million have been obtained in about five minutes; however, one trillion choose five hundred million cannot be evaluated because of insufficient memory. Fig. 11 displays the CPU time required for generating a component of the discrete dyadic Green’s function (56) and for direct implementation of the discrete convolution. This log-log diagram shows that the slopes for the CPU time to generate (56) and to compute convolution are around 2.43 and 1.76, respectively, when , the upper limit of for computation, is large. Thus the complexities are about and for generating the discrete dyadic Green’s functions and for doing the discrete convolution, respectively. We thus claim that accurate numerical results for (68)–(70) as is around several thousand can be obtained, though we stopped at , because of the long computing time expected. A possible way to speed up the computation might be using the approximate approach of applying a window introduced in [8].

[1] R. Kastner, “A multidimensional z-transform evaluation of the discrete finite difference time domain Green’s function,” IEEE Trans. Antennas Propag., vol. 54, pp. 1215–1222, Apr. 2006. [2] J. Vazquez and C. G. Parini, “Discrete Green’s function formulation of FDTD method for electromagnetic modelling,” Electron. Lett., vol. 35, pp. 554–555, 1999. [3] A. Boag, U. Shemer, and R. Kastner, “Hybrid absorbing boundary conditions based on fast nonuniform grid integration for nonconvex scatterers,” Microw. Opt. Technol., vol. 43, pp. 102–106, Oct. 2004. [4] A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 2nd ed. Boston, MA: Artech House, 2000, ch. 4. [5] R. Holtzman and R. Kastner, “The time-domain discrete Green’s function method (GFM) characterizing the FDTD grid boundary,” IEEE Trans. Antennas Propag., vol. 49, pp. 1079–1093, 2001. [6] R. Holtzman, R. Kasrner, E. Heyman, and R. Ziolkowski, “Stability analysis of the Green’s function method (GFM) used as an ABC for arbitrarily shaped boundaries,” IEEE Trans. Antennas Propag., vol. 50, pp. 1017–1029, 2002. [7] N. Rospsha and R. Kastner, “Closed form FDTD-compatible Green’s function based on combinatorics,” J. Computat. Phys., vol. 226, pp. 798–817, 2007. [8] J. Vazquez and C. G. Parini, “Antenna modeling using discrete Green’s function formulation of FDTD method,” Electron. Lett., vol. 35, pp. 1033–1034, 1999. [9] S. K. Jeng, “An analytical expression for 2-D FDTD-compatible Green’s function in infinite free space via z-transform and partial difference operators,” presented at the IEEE Antennas Propag. Soc. Int. Symp., Toronto, Ontario, Canada, Jul. 11–17, 2010. [10] S. S. Cheng, Partial Difference Equations. London, U.K.: Taylor and Francis, 2003.

VII. CONCLUSION A new analytical expression for the 3-D FDTD-compatible dyadic Green’s function in infinite free space has been derived using z-transform and spatial partial difference operators. The final expression involves a summation of standing wave modes with time-varying coefficients. The propagation of waves in the Yee’s grid can be explained by the selective property of the time-varying coefficients, which is very different from the conventional concept of a traveling wave. Results of the traditional dispersion analysis using plane waves for the FDTD algorithm in a source-free region may not be applicable to interpret our analytic expression, because the z-transform corresponding to our expression diverges for z on the unit circle. A discrete equivalent of the Stratton-Chu integral formula may be derived next for

Shyh-Kang Jeng (M’86–SM’98) received the B.S.E.E. and the Ph.D. degrees from National Taiwan University, Taipei, Taiwan, in 1979 and 1983, respectively. In 1981, he joined the faculty of the Department of Electrical Engineering, National Taiwan University, where he is now a Professor. Between 1985 and 1993, he was a Visiting Research Associate Professor and a Visiting Research Professor with the University of Illinois, Urbana-Champaign. In 1999, he visited the Center for Computer Research in Music and Acoustics, Stanford University, Stanford, CA, for six months. His research interest includes time-domain electromagnetic field computation techniques, antenna design, multimedia signal processing, computational neuroscience, and computational cognitive neuroscience. Dr. Jeng is a recipient of the 1998 Outstanding Research Award of National Science Council and the 2004 Outstanding Teaching Award of National Taiwan University.

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Pulsed Beams Expansion Algorithms for Time-Dependent Point-Source Radiation. A Basic Algorithm and a Standard-Pulsed-Beams Algorithm Yael Gluk and Ehud Heyman, Fellow, IEEE

Abstract—We introduce two pragmatic pulsed beam (PB) expansion algorithms for radiation from a time-dependant point source, whereby the field is expanded in terms of PB propagators emerging from the source in all directions. The algorithms are based on a rigorous expansion identity involving a continuous angular spectrum of complex source pulsed beams (CSPB). The present algorithms, however, are structured upon a discrete lattice of beam directions, and utilize the iso-diffracting PB propagators (ID-PB) that may readily be tracked in non-uniform media. In the basic algorithm, the PB propagators are determined by the analytic-signal extension of the source and hence have to be re-calculated for any given source. To circumvent this difficulty we introduce a more pragmatic algorithm that utilizes time-samples of the source signal and expresses the radiated field using a set of standard-PB propagators defined by a given filter ( ). The optimal expansion parameters, namely the beam collimation, the angular spectrum discretization, the overall number of PB needed to reconstruct the field, as well as the choice of the filter ( ), are determined analytically as functions of the observation range and the excitation pulse. Guidelines for choosing these parameters are provided and are verified numerically. Index Terms—Beam summation method, complex source pulsed beams, iso-diffracting pulsed beams, transient fields.

I. INTRODUCTION AND PROBLEM SCOPE

B

EAM SUMMATION (BS) formulations are an important tool in wave theory since they provide a framework for ray-based construction of spectrally uniform solutions in complex configurations. In these formulations, the source-field is expanded into a spectrum of beams that emanate at a given set of points and directions in the source domain, and thereafter are tracked locally in the medium along ray trajectories. The field is then obtained by summing up the beam contributions at the observation point. We used here the generic term “beams” for either “Gaussian beams” (GB) or “pulsed beams” (PB) which are our propagators for time-harmonic or time-dependent fields, respectively. Manuscript received January 03, 2010; revised May 29, 2010; accepted July 01, 2010. Date of publication November 01, 2010; date of current version April 06, 2011. This work was supported in part by the Israeli Science Foundation, under Grant 674/07 and in part by NATO’s Public Diplomacy Division in the framework of “Science for Peace” program under Grant SfP982376. Y. Gluk was with the School of Electrical Engineering, Tel Aviv University, Tel Aviv 69978, Israel. She is now with the telecommunication industry in Israel (e-mail: [email protected]). E. Heyman is with the School of Electrical Engineering, Tel Aviv University, Tel Aviv 69978, Israel (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.2010.2090475

The main advantages of the beam formulations are spectral localization (the summation involves only the beams that pass near the observation point) and spectral uniformity (the beam propagators are insensitive to ray catastrophes, like caustics and foci). Thus, the BS approach, when applied judiciously, combines the uniform features of spectral representations (like plane waves or Maslov integrals) with the algorithmic ease of ray representations. Furthermore, unlike the “ray search” approach, the “beam shooting” approach circumvent the need to search for the rays that lead from the source to the observation point. The advantages of the BS approach in dealing with large and rather complex propagation environments have been recognized and utilized in various applications in electromagnetics, computational geophysics and underwater acoustics (see recent reviews in [1], [2]). There are several classes of BS expansion schemes for localized sources or for extended source distributions in the frequency or in the time domains. This approach has been introduced first in the context of quasi-monochromatic point source configurations [3]–[10]. The present work is concerned with a time-dependent point-source configuration. In the above we referred only to point source configurations: the interested reader may refer to the review articles in [1], [2] for a broader perspective, or to recent publications that have dealt with radiation from extended aperture-source distribution in the ultra wide band frequency domain [11], [12] or in the time domain [13]. Specifically, we are concerned with the PB expansion of the radiation by a time-dependent point source with an arbitrary ex. In free space, the field is given by citation pulse (1) define the space-time coordinates, where is the source coordinates and is the wave speed. The BS algorithm presented here is based on an exact identity whereby the field in (1) is described as a spectral integral of complex source pulsed beams (CSPB) emerging from the source in all directions [14] (see also [2], [15] and Section III). This representation may be considered an exact localized alternative to the time dependent Sommerfeld integral. The CSPBs, which serve as the “basis functions” (or propagators) in the exact spectral expansion, are exact causal solutions of the time dependent wave equation that are modeled by radiation from a time-dependent source with complex spacetime coordinates. Actually, the complex source model is just a mathematical trick to obtain compact solutions: physically,

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GLUK AND HEYMAN: PBs EXPANSION ALGORITHMS FOR TIME-DEPENDENT POINT-SOURCE RADIATION

these waves are generated by causal radiation from a time dependent source distribution with finite support (see [16] and also in [1], [2]). The set of CSPBs that radiate from the source in all directions provides a complete basis in the spectral expansion identity. This identity is valid regardless of the degree of collimation of the CSPBs, yet it is desirable to use well collimated CSPBs since they are localized near their propagation trajectories and may be approximated by the so called iso-diffracting pulsed beams (ID-PB). We prefer to use the ID-PB not only because they are expressed by simpler and more explicit expression, but also because they define a class of approximate PB solutions in general inhomogeneous media that propagate along (generally curved) ray trajectories (see [17] and also [1], [2]). Their dynamics is fully described by complex Hamilton-Jacobi equations along these trajectories. The “iso-diffracting” designation is implied by the frequency-independence nature of this dynamics. Following the discussion above, our goal in this work is to derive a pragmatic PB expansion algorithm for radiation from a time-dependent point source, utilizing collimated ID-PBs that can be tracked in complex configurations, involving inhomogeneous media and curved interfaces [1], [2], [17], [18], edge diffractions [19], or even rough surface scattering [20], [21]. Our goal is to calibrate the algorithm and the parameters for an efficient numerical implementation. Toward this goal we determine here the conditions for replacing the CSPBs in the exact expansion identity of [2], [14] by the ID-PBs. We also determine the discretization of the angular spectrum of PB’s directions. The motivation is to use a sparse lattice of directions in order to minimize the number of PBs to be tracked in the medium. The angular spectrum of PBs can be further truncated since the PBs that pass far from the observer do not contribute there. We derive analytic expressions for the discretization and truncation conditions as a function of the excitation pulse-length and the propagators’ properties. We present, in fact, two expansion algorithms: In the basic algorithm of Section IV, the waveforms of the PB propagators are determined by the analytic-signal extension of the source in (1), and hence they have to be re-calculated for any given source and for all complex times.1 To circumvent this difficulty we introduce in Section V a more pragmatic algorithm that utiand also expresses the radiated field lizes time samples of using a set of standard-PB propagators whose waveform is de. termined by a known filter The layout of the presentation is as follows: Section II deals with the properties of the CSPBs and of the ID-PBs. The spacetime shape of these wavepackets are described by the analytic that, in general, may be chosen quite extension of a pulse arbitrarily. The wavepacket is thus parameterized by referring to the general properties of analytic signals in the complex time domain which are summarized in the Appendix. Specifically,

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we introduce a new set of parameters that quantifies the analytic signal decay in the complex time domain. These parameters are essential in quantifying the expansion parameters, as discussed below. The exact PB expansion of [2], [14] utilizing a continuous angular spectrum of CSPBs is briefly reviewed in Section III. that defines the PBs is given now by source The pulse excitation pulse in (1). The basic and the standard-PB expansion algorithms are presented next in Sections IV and V, respectively. These sections also explore, both theoretically and numerically, the discrete implementations of these algorithms and the mechanisms that control the expansion error. We also provide specific guidelines for choosing the expansion parameters, namely, the beam collimation, the discrete lattice of beam directions, the spectral truncation, and the expansion filters . The presentation ends with a summary and conclusions Section VI. II. THE PB PROPAGATORS This section reviews the basic properties of the PB propagators. We start in Section II-A with the globally exact complex source pulsed beams (CSPB), which also provide the “basis function” in the exact expansion identity (24), and then continue with the approximate ID-PB solutions in Section II-B. The readers are referred to [1], [16] for further details. A. The Complex Source Pulsed Beams (CSPB) [1], [2] 1) Field Solutions: The CSPB is modeled by extending the space-time source coordinate to the complex domain. The most general case may be expressed as (2) As will be shown, the real point is the center of the beam waist, the real vector defines the beam direction and its magis nitude is the beam collimation length. The condition on explained later on (e.g., in the discussions following (6) and (9)). The field due to the complex source in (2) is obtained as an analytic extension of the real point-source solution and is given by (cf. (1)) (3) where and is the complex distance from to as defined in (6). The multiplication by the parameter is a normalization whose motivation is clarified in (10). The wave field

in (3) depends on a rather arbitrary pulse

. Actually, it is expressed in terms of , the analytic , defined via the one sided inverse signal counterpart of Fourier transform (4)

1In this paper we define “analytic signals” as the complex signal extensions of

real (physical) signals that admit complex time-argument. This is an extension of the conventional definition [22] where analytic signals are complex signals of a real time-argument whose real part equal the real (physical) signals. More details on this distinction are discussed in connections with (4) and in the Appendix.

where is the frequency spectrum of . The integral in , the lower half (4) converges for real and hence for all of the complex plane, thus defining an analytic function there

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(see further discussion in the Appendix). As follows from (A3), . Hencethe physical signal for real is given by forth, analytic signals are denoted by an over . We also use to denote the temporal pulse that defines the subscript in wavefield. It should be noted that the observation time in (3) is real, but in view of the complex space-time source coordinates in (2), the time-argument of in (3) is complex with negative imaginary part (see discussion following (9)). in (3) is an exact complex One may readily verify that space. As shown solution of the wave equation in the real below, its PB properties are mainly determined by the properties . The of the pulse and by the complex distance function physical field solution is given by (5) Actually,

defines two real solutions:

and

. In view of the general properties of analytic signals, it follows that the latter is a temporal Hilbert transform of the

(6a)

Similarly defines a family of one-sided hyperboloids that emerge from the source disk (Fig. 1). The values of corresponding to of (6) are in the range . The is covered by the set such that half space for , the hyperboloids shrink to the positive -axis and , they tend to the complement of the source disk in the as plane. Likewise, the half-space is covered by the with along the negative -axis. set It follows from the discussion above that for in (2), the

(6b)

time-argument on

(6c)

all as required by the analyticity of . The argument is real only for along the positive -axis. 3) Paraxial Approximation of : The paraxial zone near the -axis deserves special attention. From (6b) with (6c) one , where finds for

former (see (A3)). Henceforth we concentrate only on since

or any linear combination of

and

Fig. 1. The oblate spheroidal coordinate system. All units are normalized with respect to the collimation length b. The system is continuous everywhere except across the source-disk branch cut (wiggly lines), where the s hyperbolae are discontinuous.

,

by a complex phase , may be obtained by multiplying , and then taking the real part. 2) The Complex Distance Function : The complex disis defined by tance from to

Equation (6a) is the general definition of in the coordinates, while (6b) is expressed in terms of the so called , defined such that the “beam coordinates” axis extends from along the direction and are the transversal coordinates normal to the axis. In these coordinates, and , i.e., , thus leading to the expression in (6b). Here and henceforth, a “hat” over a vector defines a unit vector. Finally, the condition in (6c) defines as a single valued function. The reason for this particular choice of the square-root sign is explained in (9) (see also [1]). This single valued square root introduces a branch cut in . From (6b), the set of branch points is the circle in the plane. The branch cut implied by (6c) is the disk in that plane, henceforth termed the “source disk” (see Fig. 1). serve as the The real and imaginary parts of natural coordinates for the CSPB and have distinct physical roles. These coordinates define an oblate spheroidal system (see defines a family of spheroids such that the Fig. 1). shrinks to the source disk while as increases spheroid they tend to spheres the spheroids become larger, and for with radius . From (3), is identified as the real propagation delay of the wavefield, hence, the spheroids are . identified as the “wave-fronts” of

in (3) has a non-positive imaginary part for

(7) is due to the condition where the sign for across the source in (6c), giving rise to a discontinuity of disk. As will be shown, the resulting PB is strong only near the positive axis (upper sign in (7)). The second order term of (7) is used in (10) for the ID-PB approximation of the CSPB, while the next order term will be used in (22) to determine the region of validity of this approximation. : Another important case 4) Far-Zone Approximation of where from (6b) is the far zone region (8) where measures the polar angle from the positive -axis. 5) Properties of the CSPB: The discussion above implies that is continuous everywhere in except across the source is an exact homogeneous solution of the disk. It follows that time dependent wave equation everywhere except across this disk. This disk therefore represents the location of the phys[16]. Furthermore, since , ical sources that give rise to

GLUK AND HEYMAN: PBs EXPANSION ALGORITHMS FOR TIME-DEPENDENT POINT-SOURCE RADIATION

the real propagation delay in , increases monotonically away is strictly outgoing (or causal) with respect to from this disk, the source disk, and satisfies the radiation condition as (see also (8)). in (3) has the characteristics of a PB, a The wave function and collimated space-time wave packet that emerges from propagates along the positive -axis. Confinement along this

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where (11b) Substituting in (10), we obtain the following explicit expression for the complex time delay (see also (9))

axis is effected by the pulse shape of while transverse confinement is due to two factors: (i) the general property of an-

(12)

which decay in the lower half of the comalytic signals plex -plane as the imaginary part of becomes negative (see Appendix), and (ii) the imaginary part of the propagation delay in (3). This propagation delay is given by

Recalling the discussion in (9), the real and imaginary , namely and parts of , define the paraxial delay and the paraxial is identified as the wavefront radius decay of the PB. Thus controls the rate of decay as grows of curvature, while away from the beam axis. The parameter is identified as the collimation length: For , is essentially independent of and the PB propagates essentially without spreading, while , hence the wavepacket diverges for . along the cones

(9) we find from the Noting that following (6) that vanishes on the posdiscussion on itive -axis and increases away from it along the ellipsoids along the negative axis. Conup to its maximum is strongest on the positive -axis and weakest sequently, along the negative axis. The beamwidth depends therefore on in the lower half of two factors: (i) the rate of decay of away from the the -plane, and (ii) the rate of increase of -axis. The former is determined only on the pulse-length of , while the latter depends on the collimation parameter . We defer any further discussion to the next section which is devoted to the well collimated case. For other properties of the general CSPB, the reader is referred to [1], [2].

The observations above followed from the waveform term in (10). Referring to the amplitude term there we note that it is essentially constant for , while for it decays like . The amplitude term also affect the real PB field . In order to obtain an explicit expression for the real field we express the analytic signal for a given as a sum of two real signals (see (A3))

and

(13) B. The Paraxial Iso-Diffracting PB (ID-PB) 1) Field Solutions: Under well collimation conditions the CSPB is localized near the propagation axis , hence we may use the second order paraxial approximation of in (7). Using also from (2), the exact CSPB solution in (3) becomes

and is the Hilbert transform (A4) of . Note that and decay as increases. Substituting (13) in (10) and taking the real part we obtain where

(14) (10) We denote this solution by in order to make a clear distinc. We also added explicitly tion from the exact CSPB solution which define, respectively, the waist the parameters location, the direction and the collimation of the ID-PB. The wave field

in (10) consists of an amplitude term and

has been added a waveform term . We note that the factor into (3) only in order normalize the amplitude term in (10) to . unity at The space-time shape of the wavefield is mainly controlled by the waveform term. In order to parameterize this shape we separate the argument of and define

in (10) into real and imaginary parts

(11a)

are now functions of and are given in (12). Thus, in with is approximately , but as increases the near zone , it is it is gradually Hilbert transformed and finally, for . On the axis, this expression dominated by simplifies using , and . 2) The Pulse Length and the Beam Width: The wavepacket is parameterized by the pulselength and beamwidth , both are functions of the PB position . The pulselength is defined on the beam axis . Assuming that the pulse is centered about , the pulselength is defined by (see (A7)) (15) where is the propagation delay of the center of mass, and is the temporal norm in (A5). If the pulse has no d.c. component, it follows from the discussion after (A12) that in the right hand side in (15) may be replaced by its analytic

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counterpart . Using (10) and referring again to the discussion in (A12), we find that

(16)

is the pulselength of as defined in (A7). where at a given range is defined as the The beam width transversal coordinate where the pulse energy reduces to some of its value on the axis, i.e., value (17) of (10) and noting that the transversal decay of substituting the PB is generated by , the imaginary part of the complex time delay in (12), we find that (17) is equivalent to

(18)

. Referring to (A8), (A9) where from (12) we find that that satisfies (18) is given by the condition (19) where and are descriptors of the pulse : is the pulse length (A7) while the dimensionless parameter defined in (A8), (A9) quantifies the value of where the norm of

decays to a given value of the norm for . Substituting from (12) into (19) we obtain

(20) is the waist width at . Note that in the near zone , but in the far zone the PB opens up along a constant diffraction angle (21) are obtained if . Thus collimated PBs with As an example, consider the twice differentiated analytic pulse in (A15) where is a parameter. (see (A18)) and that is given in We find that and obtain from (A19). For a 10% beamwidth we use . (A19) Fig. 2 depicts the propagating CSPB of (3) for using the parameters , (i.e., the well collimation condition in (21) is satisfied). The figure depicts snapshots in the near, intermediate, and far zones with respect to . The pulse-length and the 10% beamwidth are and , respectively, both are in agreement with the analysis above. Fig. 3 depicts the normalized error (in %) of

9 = 10 = (horizontal vertical) plane

( ) = Re = 2

(0 )

Fig. 2. The exact CSPB of (3) for f t  t iT , with the param, cT satisfying the well collimation condition cT =b . eters b The figure depicts snapshots at (a) t b= c when the wavepacket is in the b=c in the intermediate zone; and (c) at t b=c near zone; (b) at t in the far zone. The field is depicted in the rotational symmetric ;  . For clarity we used different scales for the  = and  axes.

=1

1 =2 ( )=

the ID-PB relative to the globally exact CSPB , using the same format as in Fig. 2. One readily observes that the error increases with , but it is still very small for the observation ranges depicted, in agreement with the analysis below. 3) The Error of the Paraxial PB Approximation: The paraxial of (14) is based on the second order approximaPB solution in (7). It is valid as long as the next (fourth) order tion of term in (7) is smaller than the pulse length . In order to explore this error, we rewrite the error term as (22a) (22b) (22c)

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Fig. 4. The error of the ID-PB in the far zone. (a) Snapshot of the exact CSPB 9 of Fig. 2 at ct = 10 . (b) Snapshot of the normalized error (in %) of the ID-PB 8 relative to 9 . The field is depicted in the spherical coordinates (r; ) = (horizontal=vertical) where, in addition,  is normalized with respect to  .

we depict the field in the spherical coordinate frame , and we also normalize with respect to . One observes that the error here is 1% and it is much larger than in Fig. 3, in accord with the analysis in (23). Fig. 3. Snapshot of error of the paraxial PB solution 8 of (14) relative to the exact CSPB 9 of (3). The error is normalized by the maximum value of the field. The figure format and the pulse are the same as in Fig. 2. Note the error increase with the distance. (a) t = b=2c; (b) t = b=c; (c) t = 2b=c.

where in (22b) we kept only the leading order real and imaginary , while in (22c) we considered parts in the far-zone range only values of within the beamwidth region and substituted . One readily observes that the far zone error is dominated by the real time delay in (22c) which increases linearly with . The paraxial solution is . Substituting from therefore valid only if (20) we find that the solution is valid for (23) Fig. 4 depicts a snapshot of the CSPB of Fig. 2 at a such that the wavepacket is located very late time . The figure also shows the normalized error around relative to . In order to accommodate the of the ID-PB far zone transversal expansion of the wavepacket (see Fig. 2(c))

III. THE PB EXPANSION IDENTITY We now return to the problem outlined in (1) and consider the PB expansion scheme for a general point source in free space . The exact PB expansion for the timewith excitation pulse dependent Green’s function in (1) corresponding to a source with excitation pulse is given by (see proof in at [14] and a more refined version of this proof in [2]) (24) where and the tion”

is the spherical angle direction about integration spans all directions. The “basis func(henceforth referred to as the “propagators”)

are the CSPBs of (5) and (3), but with the pulse replaced by , the analytic signal extension of the excitation pulse and the prime denotes a temporal derivative. For the sake of simplicity we expressed the CSPB propagators in (24) in an abbreviated form. Actually, referring to the notations in (3), they are given by (25)

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where is a on the right hand unit vector in the direction. Comparing side of (25) to (5) and (3) we note that they are CSPBs due to the -dependent complex source coordinates (26) Here

A. Angular Spectrum Truncation If the propagators in (24) are collimated then the propagators passing far from the observation point have a negligible contribution there. The integration may be truncated to a cone of angle about the observation direction, i.e., (31)

is a complex parameter such that (27)

The unit vector and the parameter define, respectively, in the direction and the magnitude of the imaginary part of (26), hence, recalling (2), they define the direction and the collimation of the CSPBs. Thus, the integration in (24) expresses as an angular spectrum of the physical field radiating from CSPBs emerging in all directions . The source disks of these CSPBs are centered at the real part in (26), namely at . Thus if then of and the CSPBs emerge all the source disks are centered at then are located radially from this point. If, however, on a sphere with radius centered at and are swept as a , the CSPBs emerge function of the beams directions . If on this sphere, but if , they radially outward from toward the center and then continue emerge inward from outward in the direction. We therefore prefer to use or . In the former case, the CSPB’s waists are centered at and the beam start diverging as they emerge from , while from in the latter, the waists are located at a distance and the CSPBs radiate radially outward, so that they are more collimated as they reach he observation point . From this discussion it follows that the expansion in (24) is valid only outside the domain covered by the union of all the around source disks, i.e., outside a sphere of radius

To establish the truncation error in (31) we assume, without loss and on the -axis. of generality, that The integral in (31) becomes independent, yielding (for convenience we omit the Re and consider the analytic signal field) (32) where the prime denotes a derivative with respect to time, and from (6) (33) Changing the integration variable to via (33) we obtain

(34) and using

we obtain

(35) (28) In practice, however, this restriction may be relaxed as discussed after (44). For the reader’s convenience, we rewrite the propagators in (25) explicitly as (see (3))

(29) where is the complex distance (6) from in (26) to . In (24) we used a somewhat simplified form of the expansion formula. Actually, in the exact expansion, the propagators should be the sum

is the physical field in (1) hence is recognized as the trunis given by (33) with . cation error. In this term, depends on the The magnitude of the truncation error imaginary part of the argument of . To get an explicit error parametrization we assume that is sufficiently small so that we may utilize the paraxial ID-PB propagators as in (10). In this case (36) where , with , are the longitudinal/transversal coordinates of , expressed with respect to the beam that emerges in the direction (see (7)), and since is on the -axis,

(30) We note that the terms simply have an additional negative imaginary time shift relative to the term. The series (30) is rapidly converging and it is dominated by the term, which is the one used in (25). Finally, the proof of the expansion identity follows by substituting the series (29) into (25) [2], [14].

(37) The truncation error becomes (38) is controlled by the imaginary part of the time arguwhere where ment in , given by

GLUK AND HEYMAN: PBs EXPANSION ALGORITHMS FOR TIME-DEPENDENT POINT-SOURCE RADIATION

, as follows from (11) by replacing with the distance from the beam waist. The relative error becomes

(39)

The last term in (39) can be quantified in terms of the parameter defined in (A8), (A9) for a given pulse and . It where follows (see also (19)) that is the prescribed error-level in (39) and is the pulselength of , giving (40) The truncation condition may be further parameterized in terms of the beamwidth of the propagators in (20) defined for a given . From (40) and (20) one finds that beamwidth level (41) Considering the example discussed in (19), we used there for a 10% beamwidth level, obtaining . For error level we find from (A19) , and . finally from (41)

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along the beam-axis and are the transversal coordinates . Referring to the notations in (10), we wrote with which in (44a) explicitly the PB parameters define, respectively, the waist location, the direction and the collimation of the ID-PB. Thus, (43) describes the field as a sum of in the ID-PBs with collimation length , emerging from direction, having waist at . Note that the condition in (28) which is due , to the “source-disk singularities” of the CSPB propagators do has been removed in (43) since the ID-PB propagators not have such singularities. The validity of the expansion in (43) is explored analytically and numerically for both in Section IV-C-3. It is shown there that although the expansion it is more efficient for . can be used for , which implies Finally, we added in (43) the condition that the summation involves only outgoing beams (this condition is included implicitly in the CSPB expansion (24) in view discussed in Section II-A-5). of the causal properties of A. The Beam Lattice The lattice of beam directions is formed by dividing the unit sphere into a mesh of triangles with common nodes. The corners while the spherical angle associdefine the beam directions ated with the th beam is (45)

IV. DISCRETE IMPLEMENTATION: THE BASIC PB EXPANSION ALGORITHM The PB summation representation in (31) is applied numerand the ically by using a discrete lattice of beam directions , giving associated spherical angle differentials (42) (43) In (43) we also replaced the CSPB’s used in (31) by the of (10) (see (44)). This replacement is justified ID-PB’s at is described since is small so that the contribution of there. The error introduced by rewell by contribution of placing by is explored in Fig. 9, where it is shown that this replacement introduces an error only at very large ranges due to the ID-PB error discussed in (23) (see also (52)). Yet, as discussed in the Introduction, it is desirable to use ID-PBs not only since they have a much simpler form but mainly since they can be tracked locally in inhomogeneous media. in (43) are given by (10) with and The ID-PB . For the reader’s convenience, we rewrite them explicitly using the notations of (10) (44a) (44b) where are the coordinates of in the beam coordinates system, defined such that the axis extends from

are the areas of the triangles associated with the th where is node. Since on the average there are 6 triangles per node, . roughly twice the average triangle size There are several techniques to create a relatively uniform mesh. The one used here is based on a successive bisections of an Icosahedron, which is a Platonic solid with 20 identical equilateral triangle faces and 12 polyhedron vertices that are located on a unit sphere (Fig. 5(a)). To create a denser lattice, the triangle edges are bisected and then the radius vectors leading to the medians are extended up to the unit sphere to determine the intermediate beam directions. Finally each triangle is divided into 4 triangles by connecting the three points where these radius vectors intersect the unit sphere (Fig. 5(b)). This process is repeated so that the number of triangles after bisections is , and the average discretized spherical angle is (46) where the factor 2 has been as added in view of (45). This process is continued until the desired discretization level determined in (48) is achieved. B. Choosing the Expansion Parameters The expansion depends on four parameters: the collimation length , the beam waist location , the discretization , and the spectral truncation . We would like to optimized the choice of these parameters for a given source waveform and an observation range in order to obtain an efficient representation using a small number of PBs, while keeping the error below a given level.

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of (37) are the values of for the last beam where included in the summation (i.e., in the beam passing furthest , (50) reduces to from ). For , where the last expression follows from (20), (21). It fol, and thereby the number of beams lows that for needed to describe the field, are essentially independent of . tends to The number of beams for (51) where in the last expression we used (48), (49). The equality is chosen as large as possible acin the last term applies if cording to (48). , on the other hand, (50) reduces to For so that and the number of beams increase as deceases and the efficiency of the BS representation deteriorates in that sense. This phenomenon may readily be explained from simple geometrical considerations and is demonstrated by the numerical examples below (see Fig. 7). We conclude that should be chosen such that where is the smallest observation range of interest. Alternatively speaking, for a chosen , the algorithm is valid for Fig. 5. Icosahedron and meshed icosahedron. (a) The basic Icosahedron; (b) icosahedron divided twice into 320 triangles.

To simplify the analysis we assume here that , i.e., the waists of all the expansion beams are located at . The spectral truncation and the discretization step is quanand . Foltified now by two proportionality parameters lowing the discussion in (41), the truncation condition can be expressed in the form (47) where is the value of for the last beam included in the summation. In other words, condition (47) implies that the summation involves all beams passing a distance smaller from the observation point. In Section IV-C-1 than and compare it with the theoretical we explore the value of . value determined in (41). A typical value is is proportional to the Likewise, the discretization step “beam spot-size” , i.e., (48) where

is related to the diffraction angle

of (21) via (49)

The value of the proportionality constant needed is explored in Section IV-C-2. A typical value is . It is desirable to possible, subject to the condition in (48). Note use the largest is controlled by , where though that the choice of is the pulselength of while defines the PB’s collimation. The choice of will be discuss below. The number of beams needed in the summation depends on and on the truncation angle . In order to quantified as a function of , we rewrite the condition in (47) as (50)

(52) The lower limitation on is due to the increasing number of as noted above. The upper limitation beams required for is due to the far zone error of the ID-PB propagators noted in (23), which leads to a large expansion error in that range (see Fig. 9(b)). This error does not exist when using the exact CSPB propagators (see Fig. 9(a)), although, as noted previously, using ID-PBs is preferable since they can be tracked in complex configurations. Equation (52) is used to choose the collimation parameter for a give application (i.e., given range and excitation pulse). Another consideration in choosing is the beamwidth, which should be sufficiently narrow to allow local tracking in complicated configurations. Thus, if the range of observation distances as noted above, then is large and is chosen such that . the beams spread and might become too wide for It is desirable, therefore, to use a different for each range of observation distances, where we also note that larger requires a finer beam lattice (smaller in (48), (49)). An efficient algorithm is obtained if the beam discretization is performed in a self-consistent fashion so that the course lattice (for shorter ranges and smaller ) is a decimated version of the finer one. In that way only one set of beams for the finest lattice needs to be tracked in the medium, whereas for the coarser lattices, one may use properly decimated sub sets. The self consistent lattice refinement proceeds as follows. into two-ocWe divide the propagation range , , etc. For the taves range-bins and then calculate the shortest range-bin, we choose coarsest lattice using the bisection procedure outlined in (46) until the condition in (48) is satisfied. For the next range-bin we and create a finer lattice by bisecting the choose in the finer coarsest lattice as outlined in (46). The area of lattice is now 4 times smaller than the area for the shorter range

GLUK AND HEYMAN: PBs EXPANSION ALGORITHMS FOR TIME-DEPENDENT POINT-SOURCE RADIATION

Fig. 6. The relative error (in %) of the ID-PB summation algorithm in (43) as a function of the spectral truncation parameter  in (47). Parameters: q (t) = Re



(i.e., n

t 0 iT T

( ), = = 6 in (46)).

p3 (i.e., T

= 1),

c = 1, b = 10 , d = 3 2 10

,

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Fig. 7. The number of beams included in the summation as a function of

c

b



qt

t 0 iT T

R

( ), = 1, = 1000, 2000, 4000. Parameters: ( ) = Re = 1, = 3 and = 3 5. The number of beams is related to 10

for

d

2



:

the truncation angle. For large truncation angle more beams are included in the summation.

bin, and hence complies again with (48) for the relevant . This process continues until we cover the entire observation range. C. Numerical Tests The PB summation algorithm (43) is applied numerically for (see (A15)). The the excitation pulse and the pulse length is taken time unit is chosen such that (i.e., , see (A18)). The field is calculated to be between the source as a function of the distance and the observation point. The relative error is defined as (53) where is the exact Green’s function solution and is the PB expanded solution in (43). 1) Truncation Effects: Fig. 6 depicts the error as a function for several observation ranges of the truncation parameter . We used (so that as needed for collimated . Note from (49) that PBs), and (we used here a 10% beamwidth as in the example ), so that . after (21), for which From Fig. 6 we conclude that parameterizing the truncation as in (47) indeed provides the in terms of the ratio correct physical description for the truncation error since the and not on . The error reduces as error-curves depend on increases, but it remains constant beyond where it is dominated by the discretization step . Specifically, choosing yields a 1% error (in accord with the analysis in (41)). Fig. 7 depicts the overall number of beams in the summation as a function of and for several values of , but for the and spectral truncasame spectral discretization . One readily observes that the number of beams tion increases for while for it tends to an -independent value (see (51)). Note also that in this example for , as follows from (51) where while is fixed (recall from (48), (49) that if one chooses then the number of beams for is independent of ). 2) Discretization Effects: In general it is desired to have the largest discretization step possible in order to minimize the number of beams that need to be tracked in the medium. Fig. 8 as a function of and . explores the criteria for choosing

Fig. 8. The relative discretization error (in %) for the ID-PB summation algorithm (43) versus T . Parameters: q (t) = Re d = 3 10 , and  = 3:5.

2



t 0 iT ), c

(

= 1,

R

=

b,

Since the lattice generation procedure of Section IV-A produces (see (46)), we keep fixed and exonly discrete values for plore the error as a function of . The source is for which (see (A18)). The results in is controlled by , Fig. 8 verify that the error for a fixed in agreement with the analysis in (48), (49). The error decreases increases until is stabilizes at (for all as three values of in the figure), beyond which it is dominated by the spectral truncation effect. A 1% error is obtained for . From (48), (49) using for a 10% beamwidth (see the example after (21)), we obtain . 3) The Overall Error: Finally we explore the overall error of the algorithm as a function of for a given implementation and i.e., a given excitation pulse and a given choice of , . We also compare the error obtain by using the exact CSPB of (3) versus the ID-PB propagators of (10). propagators The results are depicted in Fig. 9(a) and (b), respectively. In the the error is below 1% near and intermediate zones by . and is essentially unaffected by the replacement of , on the other hand, one observes an exponentially For increasing error for which is absent for . This error is due to the growing error in the paraxial delay of the ID-PB as discussed (22c), which becomes effective beyond the range defined in (23).

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of their samples and a properly chosen interpolating kernel (in, viz terpolant) (54) where the sampling interval and the sampling frequency satisfy the Nyquist condition (55) The interpolant should have the following spectrum (for simplicity we consider only positive frequencies since is symmetrical) (56) . If (see (55)), the signal is over-sampled. The spectral range can then be used to taper smoothly from at to 0 at , obtaining and propagators smoother and more localized interpolants (see examples in Section V-C below). Fig. 9. The relative expansion error in % versus the observation range R using (a) the CSPB summation algorithm in (42), and (b) the ID-PB summation algo-

0

The analytic signal extension with is given therefore by

for complex argument

rithm in (43). Pulse parameters: q(t) = Re  (t iT ) with T = 1 (i.e., = 3). Expansion parameters:  = 3:5,  = 0:2, d = 3 10 and b = 1000, 2000, or 4000. Note the exponential growth of the error in case (b) for large R as noted in (52).

T

p

2

V. THE STANDARD-PB EXPANSION ALGORITHM The PB propagators in the basic PB summation algorithm (43) are described in terms of the analytic source signal . Since the analytic signals are known analytically only for a limfor a general source ited class of functions, calculating requires the calculation of for all complex in the lower half plane. In this section we introduce a modified algorithm that circumvent this difficulty. It involves a set of standard-PB propagators that are governed by standard waveforms , and can acwhich is specified only by its time-samples. commodate any The algorithm is based on the analytic-signal sampling theorem which is presented in Section V-A. This theorem enables for complex from the calculation of the analytic signal samples of the physical signal , using a set of standard an. In Section V-B, this representation is alytic interpolants substituted into the basic expansion scheme of (43), thus expressing the radiated field as a sum of standard-PB propagators that are governed now by the interpolants

(57)

is the analytic signal extension of the where lated via (4) (or via (A2)).

, calcu-

B. The Standard-PB Expansion Scheme Substituting (57) into (43) we obtain (58) are the same as in (43) The standard ID-PB propagators except that the source function there is replaced here by the . They emerge from the source at times and interpolant . They are expressed explicitly (44b) with in all directions . The parameters of the expansion, namely the collimation length of the propagators, the beam waist location , the discretization and the spectral truncation are determined by the source bandwidth and the observation range, following the same considerations discussed in Section IV-B. C. The Analytic Interpolants

, instead of

For the sake of completeness, we consider a class of analytic

We consider the class of band limited functions with an upper frequency . Such functions can be expanded in terms

. It is assumed that is band-limited in the interpolants range ; the case is a special case. Within , the spectrum of must satisfy as in (56), it should taper smoothly to zero. As mentioned while outside after (56), smoother taper yields smooth and localized spatio. However, extending the filters temporal PB propagators

in (44b). A. The Analytic-Signal Sampling Theorem

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Finally we recall that the standard ID-PB propagators in (58) are given by (43) with the source function there replaced by one of the interpolants above. Plots of these propagators (in a somewhat different context) are given in [23, Fig. 4]. Fig. 10. Frequency characterization of a generic filter.

D. Algorithm Implementation and a Numerical Test beyond implies that the sampling rate must be larger than the Nyquist rate, as discussed below in (65). of the form deWe therefore consider a standard filter and . Referscribed in Fig. 10, wherein ring to (56), for while outside this band it tapers to zero in the lower and upper tapering zones whose and , respectively. We consider polynowidths are vanishing derivatives at mials tapers of order that have the end points. Specifically we consider linear and cubic tapers such that

(59) otherwise where we use with the center of the lower taper zone, and

otherwise.

being with

(60)

The corresponding analytic filters are (61)

(62) , the lower transition zone For the special case where shrinks to zero and the low frequency contribution becomes , i.e., (63) (64) The low frequency transition zone affects mainly the large convergence. For in the lower half plane , decays like , where is the lowest frequency in the spectrum of . The high frequency transition affect mainly the width of the main lob of for small as well as the level of the side lobes. The magnitude of the peak at is .

imAs noted above, using filters that extend beyond plies that the sampling rate must be larger than the Nyquist rate . Specifically, since the spectrum of the interpolants vanishes only for , it follows from (56) that the sampling rate should satisfy (65) , the lowest possible rate is obtained if For a given taper is taken to be equal to , giving . It should be noted though that the oversampling does not necessarily increases the computational complexity. As follows form from (58), the propagators in each spectral direction a train of identical standard-PB’s , all having the same propagation trajectory. Thus, the PB propagators need to be calculated only once for each . A related issue is the sparseness of the beam lattice. In the basic ID-PB summation formulation (43), the propagators depend on the source whereas in the standard-PB formudepend on whose frelation of (58), the propagators is wider than . Consequently quency spectrum are narrower than and may require a denser beam lattice than required by the basic formulation (recall from (48), is roughly proportional to the (49) that the discretization as small pulselength). It is therefore suggested to choose as possible. This issue is explored here by comparing expansions with two and , different lattices: A fine lattice that supports both but not . and a sparser lattice that supports In the demonstration we choose the basic interpolant, whose spectrum is given by (cf. (59), (60)) for and 0 otherwise, with time-domain counterpart and the corresponding analytic interpolant We use this

. interpolant to expand the field due to the source

with so that (see may be truncated at (A18). Since the spectrum of with no noticeable error, we choose in , which is twice the bandwidth needed. The corresponding sampling rate . is We now compare the algorithm performances for two lattices: and a sparser one with a dense lattice with . From Fig. 8, both lattices are dense enough to pro. They vide an expansion error of less than 0.1% for this were chosen since the former is sufficiently dense to support , whereas the latter is not. the -propagators with This observation is substantiated in Fig. 11 which explores the expansion error when is used as a source in the basic expansion scheme of (43), so that the PB propaga-

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for the sparse lattice (dashed-dotted line); (c) The standard-PB interpolant with expansion scheme (58) using the and the dense lattice (solid line); and (d) Same as (c) but for the sparse lattice (dotted line). The interesting observation is that the errors in all four cases are of the same order. Thus, although the with are not supported standard-PB propagators by the sparse lattice as discussed above, they nevertheless yield a small expansion error even when they are used in conjunction do complies with this lattice, since the low frequency source with this lattice. The final conclusion is that for the present source signal, the lattice should support the source function but not necessarily the standard-PB propagators. VI. SUMMARY AND CONCLUSIONS Fig. 11. The error using the basic ID-PB expansion algorithm (43) for the source function q (t) = sinc(! t) as a function of ! . Expansion parameters: b = R = 1000,  = 3:5, and we use two different beam lattices:  = 3 10 and  = 12 10 .

Fig. 12. The relative expansion errors as a function of the observation range R using the basic expansion algorithm with the PB propagators 8 and the standard-PB algorithm with the propagators 8 . For each scheme we also compare two beam lattices: a dense lattice  = 3 10 that supports both 8 and 8 , and a sparse lattice  = 12 10 that supports 8 but not 8 . Problem pa-

0

p

rameters: q (t) = Re  (t iT ) with T = 2 3 (i.e., T = 2) and c = 1. Expansion parameters: b = 1000,  = 3:5. The, standard-PB are defined by the interpolant h (t) with ! = 10.

tors are . The figure depicts the error as a function of for the two lattices. One observes that for , the denser lattice supports the propagator with an expansion error of less than 0.3%, whereas the latter is too sparse and yields an error of 15%. Note that in both cases the error starts growing . at Having done these preliminary studies we now compare in Fig. 12 the relative expansion error versus the observation range for the four different expansions, all of them with , and : (a) The basic expansion scheme (43) using the dense lattice (dashed line); (b) Same as case (a) but

We presented two pulsed beam (PB) expansion algorithms (43) and (58) for calculating radiation from a time-dependant point source. They are based upon the exact expansion identity (24) that involves an angular spectrum of complex source emerging from the source in all dipulsed beams (CSPB) rections. This identity is valid regardless of the degree of collimation of the CSPBs, yet it is desirable to use collimated propagators that are localized near their propagation trajectories. In this well collimated regime, the CSPB are approximated by the iso-diffracting pulsed beams (ID-PB) that have a much simpler form and may be tracked locally in general inhomogeneous media. The final algorithms in (43) and (58) are expressed therefore in terms of a discrete spectrum of well collimated ID-PBs. In the basic expansion algorithm of (43), the PB propagators depend on the source in (1), and hence they need to be re-calculated for any given . Specifically, it is required to calfor all complex . The culate the associated analytic signal modified algorithm (58) circumvents this difficulty by utilizing a set of standard ID-PB propagators defined by a set of known . This algorithm may accommodate, analytic signals (filters) in fact, any specified by its time-samples . A particular set of analytic filters is presented in Section V-C. Both algorithms are structured upon a discrete lattice of beam directions, described in Section IV-A and Fig. 5. In general, the discretization step should be as large as possible in order to use the smallest number of beam propagators to be tracked in the medium. In Sections IV and V it has been shown that the spectral discretization and the spectral truncation needed for a given error-level are determined by the shape-parameters of the PB propagators, namely the pulselength , the beamwidth and the diffraction angle . These parameters are calculated in Section II-B. It is shown that they are governed by the space-dependence of the complex time decay of PB, and also by the rate of decay of the analytic signals in the complex time domain. To this end we define in the Appendix a new set of parameters that quantifies this decay. For the readers convenience the key results are summarized below. 1) The CSPB and the ID-PB are given in (3) and (10), respectively. Their space-time shape is governed of an arbitrary signal by the analytic signal extension . Their characteristics, namely the pulse length , the

GLUK AND HEYMAN: PBs EXPANSION ALGORITHMS FOR TIME-DEPENDENT POINT-SOURCE RADIATION

beam width and the diffraction angle are quantified in (15), (20) and (21), respectively. These values depend on the definitions of the beamwidth level in (17). These characteristics depend on the imaginary part of the delay function (see (9) and (12) for the CSPB and the ID-PB, respectively) and on the decay rate of

in the

in the complex complex -plane. The properties of were quantified in the Appendix. In particular, we introthat quantifies duced in (A8), (A9) a new parameter the decay rate in the complex -plane (see also (A19)). and truncation needed for 2) The spectral discretization a given error-level are determined by the pulselength , the beamwidth and the diffraction angle , see (47) and (48), respectively. The proportionality values were found in Section IV-C (Figs. 8 and 6) to be and (see also (41)). 3) The numerical algorithms have been explored in Sections IV-C and V-D. The algorithms are applicable in the range discussed in (52). The lower limitation on in (52) is due to the increasing number of beams required for (see Fig. 7). The upper limitation in (52) is due to the far zone error of the ID-PB propagators (see (23)) which leads to a large expansion error in that range (see Fig. 9(b)). This error does not exist if one uses the exact CSPB propagators (cf. Fig. 9(a)), but, as discussed above and in the Introduction, it is desirable to use ID-PBs since they can be tracked in complex configurations. Eq. (52) is used to choose the collimation parameter for a give application (i.e., given range and excitation pulse). APPENDIX ANALYTIC SIGNALS Analytic signals are extensions of real time signals that can accommodate complex time variables. Therefore, they are a useful tool in time-domain wave theory which involves complex time delays, for example when dealing with time-dependent plane-wave representations and/or with evanescent spectra [24], [25]. In the present context of the CSPB, the analytic signals are used to accommodate the complex propagation delays implied by the complex source. This space-dependent complex time-shift gives rise to the particular space-time shape of the wavepacket. The wavepacket shape is also governed by the complex time decay of the analytic signal that describes the CSPB (see Section II). In this Appendix we summarize the properties of analytic signals and define a set of parameters to quantify their behavior in the complex time domain. These parameters are used in the text in order to parameterize the space-time shape of the wavepacket, which, in turn define the expansion parameters (the spectral discretization and truncation).

The corresponding analytic signal is defined via the one sided inverse Fourier transform in (4). As noted there, this integral defines an analytic function complex -plane. real signal

,

, with frequency spec-

(A1)

in

, the lower half of the

may also be defined directly from the via (A2)

The limit of via signal

on the real axis is therefore related to the real

(A3) where (A4) is the Hilbert transform with denoting Cauchy’s principal value. It should be noted that the conventional definition of analytic signals [22] involves real time-argument, and it is the real limit of the complex definition in (4) (or its time-domain counterpart in (A2), (A3)). The complex-time extension is an important tool in wave theory that has been utilized extensively in various publications dealing with pulsed beams solutions or spectral representations in the time domain (see reviews in [1], [2]). An important corollary of (A2)–(A3) is the real signal rep, , where resentation of the analytic signal is a parameter. The real and imaginary parts of this signal for form a Hilbert transform pair, i.e., defining then property has been utilized in (13), (14). B.

. This

Parameterization of Analytic Signals

Real signals are typically parameterized by the following definitions: (A5)

(A6) (A7) A new important parameter which is relevant for the PB theory for which the norm of the real signal is the complex shift

A. Definition Consider a given real signal trum

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the norm for

, , i.e.,

, decays to some value

of

(A8)

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where the first equality follows from (A11). The parameter quantifies the rate of decay of the analytic signals as a function of the imaginary time shift . Typically, this parameter depends on , hence we introduce a dimensionless parameter

provides sufficiently localized PB’s soluThe case tions and is therefore used as a representative of this class of functions. The analytic and real signals are (A15a)

(A9) (A15b) which is invariant under stretching of the axis. It is convenient to perform the norm calculations using the analytic signal counterparts of the real signals. In these calculations, we utilize the following unitarity, orthogonality and Parseval’s theorems for the Hilbert transform, namely

(A15c) and are symmetric and antisymmetric about and tively, and for large they decay like norm of

is calculated conveniently from

, respec. The

via (A11)

(A10) (A11) (A16)

where is the Heaviside step function. A useful relation is

(A12)

For the calculations in (A7) we note that hence we need to . Referring to the discussion in (A12) we note calculate that

is also an analytic signal so that from (A11)

To prove it, we rewrite thus leading to the result on the right hand side of (A12). is an analytic signal iff An important corollary is that (i.e., has no d.c. component). To prove it we note that if

is an analytic signal then , but this is possible only if the last term in

(A12) vanishes. is symmetric (or antisymmetric) then If, in addition, posses the opposite symmetry. In this case , and finally C.

.

(A17) The pulselength is obtained now by substituting (A16) and (A17) in (A7) (A18) Finally we calculate the parameter

. Substituting in (A8) together with from which we obtain and finally, using (A9) and (A18)

We use the analytic pulse as an example for the manipulation above. The -times differentiated analytic pulse is defined as (A13)

where the analytic

is a parameter. pulse. Its frequency spectrum for

is is (A14)

The th derivative suppresses the low frequency components thus localizing the time domain signal. Specifically, for in .

in (A15).

Using (A16) we obtain (A16) we obtain

Pulse

of (A8) for

(A19)

REFERENCES [1] E. Heyman and L. B. Felsen, “Gaussian beam and pulsed beam dynamics: Complex source and spectrum formulations within and beyond paraxial asymptotics,” J. Opt. Soc. Am. A, vol. 18, no. 7, pp. 1588–1610, 2001. [2] E. Heyman, “Pulsed beam solutions for propagation and scattering problems,” in Scattering: Scattering and Inverse Scattering in Pure and Applied Science, R. Pike and P. Sabatier, Eds. New York: Academic Press, 2002, vol. 1, ch. 1.5.4, pp. 295–315. [3] M. M. Popov, “A new method of computation of wave fields using Gaussian beams,” Wave Motion, vol. 4, pp. 85–97, 1982. ˇ [4] V. Cervený, M. M. Popov, and I. Pˇsenˇcik, “Computation of wave fields in inhomogeneous media—Gaussian beam approach,” Geophys. J. R. Astr. Soc., vol. 70, pp. 109–128, 1982.

GLUK AND HEYMAN: PBs EXPANSION ALGORITHMS FOR TIME-DEPENDENT POINT-SOURCE RADIATION

ˇ [5] V. Cervený, “Gaussian beam synthetic seismogram,” J. Geophys., vol. 58, pp. 44–72, 1985. [6] V. M. Babich and M. M. Popov, “Gaussian summation method (review),” Radiophys. Quantum Electron., vol. 39, pp. 1063–1081, 1989. [7] B. S. White, A. Norris, A. Bayliss, and R. Burridge, “Some remarks on the Gaussian beam summation method,” Geophys. J. R. Astr. Soc., vol. 89, pp. 579–636, 1987. [8] A. N. Norris, “Complex point-source representation of real sources and the Gaussian beam summation method,” J. Opt. Soc. Am. A, vol. 3, pp. 2005–2010, 1986. [9] I. T. Lu, L. B. Felsen, and Y. Z. Ruan, “Spectral aspects of the Gaussian beam method: Reflection from homogeneous half space,” Geophys. J.R. Astr. Soc., vol. 89, pp. 915–922, 1987. ˇ [10] V. Cervený, Seismic Ray Theory. Cambridge, U.K.: Cambridge Univ. Press, 2000. [11] A. Shlivinski, E. Heyman, A. Boag, and C. Letrou, “A phase-space beam summation formulation for wideband radiation,” IEEE Trans. Antennas Propag., vol. AP-52, pp. 2042–2056, 2004. [12] A. Shlivinski, E. Heyman, and A. Boag, “A phase-space beam summation formulation for ultra wideband radiation—Part II: A multi-band scheme,” IEEE Trans. Antennas Propag., vol. AP-53, pp. 948–957, 2005. [13] A. Shlivinski, E. Heyman, and A. Boag, “A pulsed beam summation formulation for short pulse radiation based on windowed Radon transform (WRT) frames,” IEEE Trans. Antennas Propag., vol. AP-53, pp. 3030–3048, 2005. [14] E. Heyman, “Complex source pulsed beam representation of transient radiation,” Wave Motion, vol. 11, pp. 337–349, 1989. [15] T. B. Hansen and A. N. Norris, “Exact complex source representations of transient radiation,” Wave Motion, vol. 26, pp. 101–115, 1997. [16] E. Heyman, V. Lomakin, and G. Kaiser, “Physical source realization of complex source pulsed beams,” J. Acoust. Soc. Am, vol. 107, pp. 1880–1891, 2000. [17] E. Heyman, “Pulsed beam propagation in an inhomogeneous medium,” IEEE Trans. Antennas Propag., vol. 42, pp. 311–319, 1994. [18] T. Heilpern, E. Heyman, and V. Timchenko, “A beam summation algorithm for wave radiation and guidance in stratified media,” J. Acoust. Soc. Am., vol. 121, no. 4, pp. 1856–1864, 2007. [19] M. Katsav and E. Heyman, “Gaussian beam summation representation of half plane diffraction: A full 3D formulation,” IEEE Trans. Antennas Propag., vol. 57, pp. 1081–1094, 2009. [20] G. Gordon, E. Heyman, and R. Mazar, “A phase-space Gaussian beam summation representation of rough surface scattering,” J. Acoustical Soc. Am., vol. 117, pp. 1911–1921, 2005. [21] G. Gordon, E. Heyman, and R. Mazar, “Phase space beam summation analysis of rough surface waveguide,” J. Acoustical Soc. Am., vol. 117, pp. 1922–1932, 2005. [22] R. Bracewell, The Fourier Transform and Its Applications, 3rd ed. New York: McGraw-Hill, 1999. [23] A. Shlivinski and E. Heyman, “Windowed Radon transform frames,” Appl. Comput. Harmon. Analy. (ACHA), vol. 26, pp. 322–343, 2009, 10.1016/j.acha.2008.07.003.

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[24] E. Heyman and L. B. Felsen, “Weakly dispersive spectral theory of transients, part I: Formulation and interpretation,” IEEE Trans. Antennas Propag., vol. AP-35, pp. 80–86, 1987. [25] E. Heyman, “Transient wave spectrum representation for radiation from volume source distribution,” J. Math. Phys., vol. 37, pp. 658–681, 1996.

Yael Gluk was born in Kfar Sava, Israel, in August 1980. She received dual B.Sc. degrees, one in electrical engineering and the other in physics, and the M.Sc. degree in electrical engineering all from Tel Aviv University, Tel Aviv, Israel, in 2004 and 2008, respectively. From 2001 until 2006, she worked as a Research Engineer in various hi-tech companies. Currently she works in a startup company in the telecommunication industry.

Ehud Heyman (S’80–M’82–SM’88–F’01) was born in Tel Aviv, Israel, in 1952. He received the B.Sc. degree (summa cum laude) in electrical engineering from Tel Aviv University, the M.Sc. degree in electrical engineering (with distinction) from The Technion-Israel Institute of Technology, Haifa, and the Ph.D. degree in electro-physics from the Polytechnic Institute of New York, (now Polytechnic Institute of NYU), Brooklyn, in 1977, 1979, and 1982, respectively. While at the Polytechnic he was a Research Fellow and later a Postdoctoral Fellow, as well as a Rothschild, a Fullbright, and a Hebrew Technical Institute Fellow. In 1983, he joined the Department of Physical Electronics at Tel Aviv University, where he is now a Professor of electromagnetic theory. He served as the Department Head, the Head of the School of Electrical Engineering, and since 2006 he is serving as Dean of Engineering. From 1991 to 1992, he was on sabbatical at Northeastern University, Boston, the Massachusetts Institute of Technology, Cambridge, and A. J. Devaney Association, Boston. He spent several summers as a Visiting Professor at various universities. He has published over 100 journal articles and has been an invited speaker at many international conferences. His research interests involve analytic methods in wave theory, including asymptotic and time-domain techniques for propagation and scattering, beam and pulsed beam fields, short-pulse antennas, inverse scattering, target identification, imaging and synthetic aperture radar, propagation in random medium. Prof. Heyman is a Member of Sigma Xi and the Chairman of the Israeli National Committee for Radio Sciences (URSI). He is an Associate Editor of the IEEE Press Series on Electromagnetic Waves and was an Associate Editor of the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION.

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Communications Compact UWB Antenna With Multiple Band-Notches for WiMAX and WLAN Ming-Chun Tang, Shaoqiu Xiao, Tianwei Deng, Duo Wang, Jian Guan, Bingzhong Wang, and Guang-Ding Ge

Abstract—In order to prevent interference problem due to existing nearby communication systems within the UWB operating frequency, a compact triple band-notch UWB antenna is presented in this communication. This antenna, designed for the rejection of interference with Worldwide Interoperability for Microwave Access (WiMAX), lower and upper wireless local area networks (WLANs) covering 3.3–3.6 GHz, 5.15–5.35 GHz and 5.725–5.825 GHz, provides three notched bands by only one structure with simple design. Based on simulation and measurement, it shows that the proposed antenna can guarantee a wide bandwidth from 3.03 to 11.4 GHz with triple unwanted band-notches successfully. Index Terms—Complementary co-directional SRR, triple notched bands, UWB antenna.

I. INTRODUCTION UWB technology has gained a lot of popularity among researchers and wireless industries after the FCC permitted its civil application within the frequency band from 3.1 to 10.6 GHz [1]. With its rapid growth, the ever developing UWB systems raise their demand for compact and low-cost antennas with omnidirectional radiation patterns [2]–[6]. Given the challenges encountered in the UWB antenna design, such as the system interferences, it necessitates the rejection of interference with some narrow bands for UWB applications in other communication systems, for example, the existing WiMAX and WLAN covering the 3.3–3.6 GHz, 5.15–5.35 GHz and 5.725–5.825 GHz [7]. Most examples of compact UWB antennas with band notched characteristic to minimize potential interference have been reported recently. Generally, the existing techniques in extensive use can be classified into the following two categories: One method focuses on loading diverse parasitic elements on the antennas, such as strip near patch [8], split ring resonators (SRRs) or stepped impedance resonators (SIRs) near feed line [9], [10] and ring-shaped patch near ground [11]. The other effective method is embedding various slots, such as arc-shaped slot [12], U-shaped slot [13], square-shaped slot [14], pi-shaped slot [15], H-shaped slot [16], fractal slot [17] and complementary edge-coupled SRR- shaped slot [18]. However, these methods unavoidably exhibit some inherent defects in practical applications. The designs for frequency band rejection function bring Manuscript received March 15, 2010; revised July 04, 2010; accepted October 09, 2010. Date of publication February 10, 2011; date of current version April 06, 2011. This work was supported in part by the Hi-Tech Research and Development Program of China (Grant No. 2009AA01Z231), in part by the National Natural Science Foundation of China (Grant No. 60872034), and in part by the National Defense Pre-Research Foundation of China (Grant No. 08DZ0229, 09DZ0204). The authors are with the Institute of Applied Physics, University of Electronic Science and Technology of China, Chengdu 610054, China (e-mail: [email protected]; [email protected]; [email protected]; [email protected]; [email protected]; [email protected]; [email protected]). Digital Object Identifier 10.1109/TAP.2011.2109684

Fig. 1. Topology of different SRRs. (a) Single SRR, (b) Edge-coupled SRR, (c) Co-directional SRR.

occupation of too large space on the antenna as well as strong coupling between band-notch structures. Consequently, most reported antennas can only perform one notched band and fail to meet the requirement of avoiding multiple interferences caused by the coexisting systems. Indeed, scarce dual or triple band-notch antennas can be realized only by loading the rejection function designs of different types, different numbers at different spaces. Therefore, to implement those approaches will require too much space and accordingly result in complicated design. Moreover, the strength and width of each notched band, not being suitable for the rejection of bands for other communication systems, exhibit poor performance. As a result, the design of compact antennas with multiple suitable band-notches has become an imperative research area for both the academia and the industry. Featuring of the multi-frequency signal rejection characteristic, complementary co-directional SRR is promising for UWB antennas to ensure multiple notched bands. With its help in this communication, single, dual and triple notched bands can be easily achieved at such a compact antenna respectively. Especially, the proposed triple band-notch design could provide three suitable notched bands in small enough size. Based on the analysis of radiation, waveform distortion and transmission performance, the proposed antennas demonstrate to be suitable for UWB applications with good rejection of other communication systems interference. II. COMPACT UWB ANTENNAS WITH SINGLE DUAL BAND-NOTCHES

AND

A. Different Types of SRR When time-harmonic external or local magnetic field is applied along the x- axis, it is not difficult to see how the induced current distribution on the electrically small SRR in Fig. 1(a) [19], [20]. The ring, behaving as a circuit driven by an external electromotive force, shows corresponding magnetic resonance and exhibits a band gap above magnetic resonance. According to Babinet’s principle [21], when the time-harmonic external or local electric field is applied along the x- axis, complementary SRR could show resonant behavior of the effective permittivity and restrain signal propagation in a narrow band in the vicinity of the resonant frequency. Two types of double-SRR structure are further demonstrated in Figs. 1(b) and 1(c). Similarly, when magnetic field is applied, the SRRs could exhibit corresponding magnetic resonance [19]. In Fig. 1(b) of the traditional edge-coupled SRR, its only one distinct fundamental resonance frequency can be determined by capacitance between the rings due to their different induced charge distributions [20]. In contrast, when the inner ring

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Fig. 2. Proposed compact UWB antenna. (a) Its sketch with the dimensions: L = 30 mm, L = 25:65 mm, L = 26:95 mm, L = 10:65 mm, W = 25 mm, W = 2:46 mm, R = 15:06 mm, R = 28 mm, R = 16 mm. (b) Voltage standing wave ratio (VSWR) characteristics.

Fig. 4. Dual band-notch UWB antenna. (a) Its sketch with the dimensions: L = 14:15 mm, L = 14 mm, W = 0:3 mm, W = 0:3 mm, R = 3:42 mm, R = 2:68 mm, g = 0:5 mm, g = 3:35 mm. (b) VSWR characteristics. (c) Effect of the complementary co-directional SRR relative positions on the band-notches. Fig. 3. Single band-notch UWB antenna. (a) Its sketch with the dimensions: L = 13:9 mm, W = 0:8 mm, R = 3:39 mm, g = 1:5 mm. (b) VSWR characteristics.

is placed along the same direction as the outer ring in Fig. 1(c), the capacitance coupling between the rings decreases drastically for their similar induced charge distributions. Thus, the co-directional SRR can exhibit dual distinct fundamental magnetic resonance frequencies for each ring. Analogously, these two corresponding complementary double-SRR structures of Figs. 1(b) and 1(c), out of Babinet’s principle [21], could gain electric resonant behaviors and restrain signal propagation. Verified by our simulation study, the complementary edge-coupled SRR could provide only one distinct band gap, while the complementary co-directional SRR, with dual distinct fundamental electric resonance frequencies for dual frequency rejection bands, is of our interest. B. Compact Planar Antenna Design Based on the UWB antenna techniques in literatures [2], [8] and [22], a compact planar UWB antenna is presented in Fig. 2(a). In this design, a substrate (FR4) with the permittivity constant "r = 4:4 of the height 1.5 mm is selected and a 50 - SMA is connected to the end of the feeding strip and the edge of the ground plane. One measurement of this antenna is carried out with AV3618 Vector Network Analyzer. As shown in Fig. 2(b), the measured results indicate a wide bandwidth 3.05–11.03 GHz with VSWR < 2. This result is in agreement with the simulation (carried out by HFSS [23]) except for a slight deviation at higher frequencies, which can be ignored [2], [20] since the antenna is for use in the 3.1–10.6 GHz band. C. Compact Antenna With Single Band-Notch According to the small size of complementary single SRR and its strong signal rejection performance (aforementioned in Section II-A), it is proposed to etch at the patch of the compact UWB antenna to

achieve a strong notched band in the operation frequency region in Fig. 3(a). The complementary SRR is arranged in the middle of the patch close to the feeding strip with the gap opposite to the z -axis. As our simulation finds, when the complementary SRR gets closer to the feeding strip, its peak rejection goes higher and rejection band goes wider. Hence, by adjusting the dimensions of complementary SRR, the desired single notched band can be easily achieved.Fig. 3(b) shows the simulated and measured VSWRs versus frequency of the antenna. This UWB antenna could provide sufficiently wide impedance bandwidth (VSWR < 2) of 3.02–12 GHz and even more at higher frequency and show effective rejection frequency band with high peak rejection centered around 5.25 GHz. Based on above analysis of the resonance mechanism and our parametric study on the complementary single SRR, the notched frequency can be empirically approximated by fnotch

c = (Linner + Louter )

1

p

"eff

(1)

where c is the speed of light and "eff  ("r )=2 the approximated effective dielectric constant; Linner and Louter are the lengths of inner edge and outer edge of complementary SRR, respectively. D. Compact Antenna With Dual Band-Notches As aforementioned in Section II-A, the complementary co-directional SRR can show distinct double band gaps due to the weaker mutual coupling between inner and outer rings even the two band gaps are adjacent. Thus, complementary co-directional SRR is selected to obtain adjacent dual notched bands for lower WLAN and upper WLAN here. Parameters of two rings are all designed based on (1). Fig. 4(a) presents the sketch of the dual band-notch UWB antenna. It is noted that the centers of the inner ring and outer ring are not at the same position (the reason will be analyzed in next paragraph). Fig. 4(b) shows its measured and simulated VSWRs versus frequency. The antenna could

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Fig. 5. Triple band-notch UWB antenna. (a) Its sketch with the dimensions: L = 14:2 mm, L = 14:3 mm, L = 15:2 mm, R = 5:2 mm, R = 3:48 mm, R = 2:75 mm, g = 0:3 mm, g = 3:8 mm, g = 6 mm. (b) VSWR characteristics.

provide sufficiently wide impedance bandwidth (VSWR < 2) covering 3.08–12 GHz or more with the dual notched bands. Measured dual notched bands are 4.85-5.35 GHz and 5.65–6.08 GHz (where VSWR > 2) respectively, covering lower WLAN and upper WLAN successfully. Therefore, by loading complementary co-directional SRR with different centers, the SRR can provide good dual band-notch performance. Owning to its dual band-notch structure, the complementary co-directional SRR can reduce the design space to achieve dual notched bands in comparison with the complementary edge-coupled SRR [24], [25]. Furthermore, changing the relative position of complementary SRRs is an effective method to adjust each peak rejection and the rejection bandwidth. In Fig. 4(c), the relative positions of the inner and outer complementary rings are discussed. When the inner ring gets close to the feeding strip, its peak rejection goes higher and rejection band goes wider. Meanwhile, for lower notched band, the peak rejection goes lower and rejection band goes narrower. It can be seen from this phenomenon that, when their gaps get closer, the influence on reciprocal resonance strengths becomes more notable.

Fig. 6. The current distribution at (a) 3.4 GHz, (b) 5.25 GHz, (c) 5.78 GHz.

III. COMPACT UWB ANTENNA WITH TRIPLE BAND-NOTCHES As aforementioned in Section I, the UWB antennas with triple band notches are already presented in [9], [10], [16] and [25]. However, each band-notch structure placed at different positions, takes too much space. Here, a novel structure with three complementary co-directional SRRs is proposed, placing at the superposition to greatly save the space. Considering the weak coupling between the rings in complementary co-directional SRR and the strong notched bands, we further add another complementary ring and investigate a triple band-notch UWB antenna, shown in Fig. 5(a). The triple band-notch UWB antenna is fabricated and measured. Fig. 5(b) exhibits the measured and simulated VSWRs versus frequency of the triple band-notch antenna. The proposed antenna could provide sufficiently wide impedance bandwidth covering 3.02–11.1 GHz with the triple notched bands, which operate centered around 3.4 GHz, 5.25 GHz, and 5.78 GHz, respectively. Fig. 6 shows the current distributions at three center notched bands. The dimensions of three complementary SRRs [determined by (1)] are corresponding to three notched bands. When the antenna is working at the center of lower notched band near 3.4 GHz, the outer complementary SRR behaves as a separator in Fig. 6(a), which almost has no relation to the other band-notches. Similarly, the middle complementary SRR operates as a second separator for the center of middle notched band near 5.25 GHz in Fig. 6(b). From Fig. 6(c), the upper notched band near 5.78 GHz is ensured by the inner complementary SRR. Additionally, as a certain current crowded on the ground plane

Fig. 7. (a) Measured gains of the antennas with and without triple notched bands over the entire UWB frequency band and measured radiation patterns of triple band-notch antenna along (b) xz and (c) xy -cut planes.

near the feed line would affect the antenna performance, we take simulation and find that the dimension of ground plane, especially L3 , has a significant effect on the triple band-notches performance, as well as impedance bandwidth. The boresight gains at +x direction versus frequency of the reference antenna (without band-notch) and triple band-notch UWB antenna are measured and displayed in Fig. 7(a). It is observed that the average gain of the reference antenna is about 2.9–5.5 dBi over the entire operating band, exhibiting general flat gain performance [26]. When the triple band notches are loaded, the sharp frequency notched characteristic is obtained due to the frequency notched function. The gain reduction is about 13.6 dB at 3.4 GHz, 7.9 dB at 5.25 GHz and 7.6 dB at 5.78 GHz, respectively. The measured far-field radiation patterns for the triple band-notch UWB antenna at 3 GHz, 6.5 GHz, and 9 GHz are plotted in Figs. 7(b)

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Fig. 8. Phase responses for different identical antenna pairs. (a) Reference antenna in Fig. 2(a). (b) Triple band-notch antenna in Fig. 5(a).

and 7(c). It is seen that the radiation patterns in xy -cut plane (H - plane) are almost omnidirectional and the radiation patterns in xy -cut plane (E -plane) are monopole alike. Clearly, the triple notch UWB antenna has a good radiation performance. When it operates at 3.4 GHz, 5.25 GHz and 5.78 GHz, the gains in the two planes drop significantly, in accordance with the results in Fig. 7(a). It is noted that the radiation pattern at each notched band in H -plane is not omnidirectional because of the dramatically increasing current distribution near three band notches [27], shown in Fig. 6. To examine the antenna performance in time domain, the phases of transmission response and waveform distortion in the operating frequency region are discussed. Here, a time-domain finite integration technique (CST Microwave Studio) [28] is used to carry out the following simulations. A pair of identical antennas is placed face to face with a distance 0.5 m [9], [26]. It can be seen that the phase response from the original antenna is generally linear in Fig. 8(a). By contrast, the phase response for triple band-notch antenna provided in Fig. 8(b), though witnessing few ripples at notched bands, is also linear. And then, the normalized source spectrum and the pulses are portrayed in Fig. 9(a) and (b) respectively. The frequency spectrum of modulated Gaussian pulse is chosen corresponding to 3–11 GHz. For original antenna pair, the received signal experiences slight distortion shown in Fig. 9(c), which can be explained by the linear phase response shown in Fig. 8(a) [26]. The received pulse [in Fig. 9(c)] is then transformed into frequency spectrum, as shown in Fig. 9(d). Clearly, we can observe that the operating frequency spectrum also covers UWB region well. Likewise, the similar work is taken to evaluate the triple bandnotch antenna performance. Compared with original antenna pair, the triple band-notch antenna pair experiences a little spread in the waveform, indicating that part of the source pulse can not be received as shown in Fig. 9(e) [26]. And in Fig. 9(f), three narrow frequency spectrum bands [corresponding to the results in Figs. 7(a) and 8(b)] takes sharply a great drop in the signal amplitude, reflecting to be hardly transmitted and received by the band-notch antennas. This result in Fig. 9(f) validates that the source pulse, locating within the notched bandwidth, is accurately filtered by the band-notch structure of antenna. Therefore, Fig. 9 illustrates that, the proposed band-notch antenna can restrain the signal transmission within narrow band-notch spectrums effectively and maintain the integrity of signal transmission outside. Furthermore, the antenna transfer functions (jS21 j) between pairs of identical antennas are also measured and discussed. In the measurement, the antenna pair, also aligned face to face with a distance 0.5 m, is connected to the two ports of the vector network analyzer indoor [10], [27]. In Fig. 10, the original antenna pair exhibits approximately flat magnitude (040 to 033 dB) of transfer gain in UWB frequency region. For the triple band-notch antenna pair, the transfer gain remains flat on the whole, except in the notched bands centered around 3.4 GHz, 5.25 GHz, 5.78 GHz, where the magnitudes reduce dramatically by

Fig. 9. Transmitted and received waveforms for different identical antenna pairs. (a) Source spectrum. (b) Transmitted waveform. (c) Received waveforms by reference antenna. (d) Received spectrum by reference antenna. (e) Received waveforms by triple band-notch antenna. (f) Received spectrum by triple band-notch antenna.

Fig. 10. Experimental results on transfer function of identical antenna pairs.

18–25 dB. Accordingly, the experiment shows agreement with the results in Figs. 7(a) and 9, which further validates its excellent multiple band-notches in UWB communications. IV. CONCLUSION We proposed a compact antenna for the rejection of interference with WiMAX and WLAN. Analysis results show that the proposed antenna guarantees a bandwidth wider than the region 3.1–10.6 GHz with unwanted band-notches and keep omnidirectional radiation performance

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successfully. The performances of the proposed antenna prove it is suitable for UWB applications.

REFERENCES [1] Federal Communications Commission Revision of Part 15 of the Commission’s Rules Regarding Ultra-Wideband Transmission System From 3.1 to 10.6 GHz. Washington, DC: Federal Communications Commission, 2002, pp. 98–153. [2] Z. N. Chen, T. S. P. See, and X. Qing, “Small printed ultra wideband antenna with reduced ground plane effect,” IEEE Trans. Antennas Propag., vol. 55, no. 2, pp. 383–388, Feb. 2007. [3] M. Gopikrishna, D. D. Krishna, C. K. Anandan, P. Mohanan, and K. Vasudevan, “Design of a compact semi-elliptic monopole slot antenna for UWB systems,” IEEE trans. Antennas propag., vol. 57, no. 6, pp. 1834–1837, Jun. 2009. [4] A. M. Abbosh, “Miniaturization of planar ultra-wideband antenna via corrugation,” IEEE Antennas Wireless Propag. Lett., vol. 7, pp. 685–688, 2008. [5] M. Ojaroudi, G. Kohneshahri, and J. Noory, “Small modified monopole antenna for UWB application,” IET Microw. Antennas Propag., vol. 3, no. 5, pp. 863–869, 2009. [6] M. Ojaroudi, C. Ghobadi, and J. Nourinia, “Small square monopole antenna with inverted T-shaped notch in the ground plane for UWB application,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 728–731, 2009. [7] Y. D. Dong, W. Hong, Z. Q. Kuai, C. Yu, Y. Zhang, J. Y. Zhou, and J. Chen, “Development of ultrawideband antenna with multiple bandnotched characteristics using half mode substrate integrated waveguide cavity technology,” IEEE Trans. Antennas Propag., vol. 57, no. 12, pp. 2894–2902, Dec. 2009. [8] K. S. Ryu and A. A. Kishk, “UWB antenna with single or dual bandnotches for lower WLAN band and upper WLAN band,” IEEE Trans. Antennas Propag., vol. 57, no. 12, pp. 3942–3950, Dec. 2009. [9] Y. Zhang, W. Hong, C. Yu, Z. Kuai, Y. Don, and J. Zhou, “Planar ultra wideband antennas with multiple notched bands based on etched slots on the patch and/or split ring resonators on the feed line,” IEEE Trans. Antennas Propag., vol. 56, no. 9, pp. 3063–3068, Sep. 2008. [10] Y. Zhang, W. Hong, C. Yu, J. Zhou, and Z. Kuai, “Design and implementation of planar ultra-wideband antennas with multiple notched bands based on stepped impedance resonators,” IET Microw. Antennas Propag., vol. 3, no. 7, pp. 1051–1059, 2009. [11] K.-H. Kim and S.-O. Park, “Design of the band-rejected UWB antenna with the ring-shaped parasitic patch,” Microw. Opt. Technol. Lett., vol. 48, no. 7, pp. 1310–1313, 2006. [12] S. J. Hong, J. W. Shin, H. Park, and J. H. Choi, “Analysis of the band-stop techniques for ultrawideband antenna,” Microw. Opt. Technol. Lett., vol. 49, pp. 1058–1062, 2007. [13] W. S. Lee, D. Z. Kim, K. J. Kim, and J. W. Yu, “Wideband planar monopole antennas with dual band-notched characteristics,” IEEE Trans. Microw. Theory Tech., vol. 54, pp. 2800–2806, 2006. [14] S. Hu, H. Chen, C. L. Law, Z. Shen, L. Zui, W. Zhang, and W. Dou, “Backscattering cross section of ultrawideband antennas,” IEEE Antennas Wireless Propag. Lett., vol. 6, pp. 70–73, 2007. [15] Y. L. Zhao, Y. C. Jiao, G. Zhao, L. Zhang, Y. Song, and Z. B. Wong, “Compact planar monopole UWB antenna with band-notched characteristic,” Microw. Opt. Technol. Lett., vol. 50, pp. 2656–2658, 2008. [16] J.-Y. Deng, Y.-Z. Yin, S.-G. Zhou, and Q.-Z. Liu, “Compact ultrawideband antenna with tri-band notched characteristic,” Electron. Lett., vol. 44, no. 21, Oct. 2008. [17] W. J. Lui, C. H. Cheng, and H. B. Zhu, “Compact frequency notched ultra-wideband fractal printed slot antenna,” IEEE Microw. Wireless Compon. Lett., vol. 16, no. 4, pp. 224–226, Apr. 2006. [18] J. Kim, C. S. Cho, and J. W. Lee, “5.2 GHz notched ultra-wideband antenna using slot-type SRR,” Electron. Lett., vol. 42, no. 6, Mar. 2006. [19] R. Marqués, F. Mesa, J. Martel, and F. Medina, “Comparative analysis of edge- and broadside coupled split ring resonators for metamaterial design-theory and experiments,” IEEE Trans. Antennas Propag., vol. 51, no. 10, pp. 2572–2581, Oct. 2003. [20] T. Koschny, P. Markoˇs, D. R. Smith, and C. M. Soukoulis, “Resonant and antiresonant frequency dependence of the effective parameters of metamaterials,” Phys. Rev. E, vol. 68, p. 065602, 2003.

[21] F. Falcone, T. Lopetegi, M. A. G. Laso, J. D. Baena, J. Bonache, M. Beruete, R. Marqués, F. Martín, and M. Sorolla, “Babinet principle applied to the design of metasurfaces and metamaterials,” Phys. Rev. Lett., vol. 93, p. 197401, 2004. [22] Z. N. Chen, N. Yang, Y. X. Guo, and M. Y. W. Chia, “An investigation into measurement of handset antennas,” IEEE Trans. Instrum. Meas., vol. 54, pp. 1100–1110, Jun. 2005. [23] Ansoft High Frequency Structure Simulation (HFSS). ver. 10 Ansoft Corp, 2005. [24] L. Liu, Y. Z. Yin, C. Jie, J. P. Xiong, and Z. Cui, “A compact printed antenna using slot-type CSRR for 5.2 GHz/5.8 GHz band-notched UWB application,” Microw. Opt. Technol. Lett., vol. 50, no. 12, pp. 3239–3242, Dec. 2008. [25] J. Ding, Z. Lin, Z. Ying, and S. He, “A compact ultra-wideband slot antenna with multiple notch frequency bands,” Microw. Opt. Technol. Lett, vol. 49, no. 12, pp. 3056–3060, Dec. 2007. [26] X. N. Low, Z. N. Chen, and T. S. P. See, “A UWB dipole antenna with enhanced impedance and gain performance,” IEEE Trans. Antennas Propag., vol. 54, no. 10, pp. 2959–2966, Oct. 2009. [27] W.-J. Lui, C. Cheng, and H. Zhu, “Improved frequency notched ultrawideband slot antenna using square ring resonator,” IEEE Trans. Antennas Propag., vol. 55, no. 9, pp. 2445–2450, Sep. 2007. [28] CST Microwave Studio, Computer Simulation Technology (CST), Ver. 4.2 [Online]. Available: www.cst.com

A Wideband Stacked Offset Microstrip Antenna With Improved Gain and Low Cross Polarization V. P. Sarin, M. S. Nishamol, D. Tony, C. K. Aanandan, P. Mohanan, and K. Vasudevan

Abstract—A

broadband

cross polarization level

printed

microstrip

antenna

having

15 dB with improved gain in the entire

frequency band is presented. Principle of stacking is implemented on a strip loaded slotted broadband patch antenna for enhancing the gain without affecting the broadband impedance matching characteristics and offsetting the position of the upper patch excites a lower resonance which enhances the bandwidth further. The antenna has a dimension of 42 55 4.8 mm when printed on a substrate of dielectric constant 4.2 and has a 2:1 VSWR bandwidth of 34.9%. The antenna exhibits a peak gain of 8.07 dBi and a good front to back ratio better than 12 dB is observed throughout the entire operating band. Simulated and experimental reflection characteristics of the antenna with and without stacking along with offset variation studies, radiation patterns and gain of the final antenna are presented. Index Terms—Broadband microstrip antenna, gain, offset, stacking.

I. INTRODUCTION The United States Federal Communication Commission (FCC) has allocated new frequency bands in the 5–6 GHz range under the unlicensed National Information Infrastructure (U-NII) for high speed WLAN. Also European Telecommunications Standards Institute (ETSI) has dedicated a 150 MHz band from 5.15–5.35 GHz for WLAN applications. Although most WLAN applications incorporate Manuscript received May 06, 2010; revised July 13, 2010; accepted August 11, 2010. Date of publication January 31, 2011; date of current version April 06, 2011. This work was supported in part by the University Grants Commission (UGC), Government of India and Department of Science and Technology (DST), Government of India. The authors are with Centre for Research in Electromagnetics and Antennas (CREMA), Department of Electronics, Cochin University of Science and Technology, Cochin-22, Kerala, India (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2011.2109362

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Fig. 2. Reflection coefficient without stacking. Fig. 1. Geometry of the proposed antenna. (a) Top view. (b) Cross sectional view. (c) Patch dimensions.

omnidirectional antennas, directional antennas like patch antennas, can be used to suppress unwanted radio frequency emissions as well as unwanted interferences in other directions. The major disadvantage of patch antennas is their inherent narrow bandwidth which limits their applications in wireless communications. A lot of techniques are used to enhance the bandwidth of patch antennas. A simple technique to increase the bandwidth of patch antenna is to use low dielectric substrates [1]. Increasing substrate thickness enhances antenna bandwidth. But this technique has a disadvantage of high cross polarization level which leads to distortion in the co-polarization patterns. The method of phase cancellation [2], [3] can effectively suppress the cross polar power, but it requires complex matching networks to provide 180 phase difference between the feeds. Stacking is an effective technique for increasing the bandwidth of a standard microstrip antenna. Gain enhancement for a stacked standard microstrip configuration can be brought about by loading the parasitic element at a height between 0:3–0:5 which gives the antenna a larger volume [4]. The electromagnetically coupled E shaped patch antenna [5] exhibits a peak gain of only 6 dBi and has a dimension of 60 2 60 2 1.6 mm3 . Waterhouse et al. has proposed an aperture-stacked patch antenna [6] which utilizes more than three layers and impedance matching techniques are used at the feed end to attain a very high bandwidth. The WLAN antenna [7] uses a Yagi array configuration including two reflectors and four director elements to enhance the gain and it occupies a larger surface area. The important design issue of broadband antennas is attaining good cross polarization characteristics while maintaining compactness. Usually increase in the cross polar power reduces the gain of the antenna. In this communication stacking is successfully implemented on the strip loaded broadband patch antenna [8] without affecting its broadband impedance matching characteristics. Offsetting the position of the upper patch as in [9] introduces an additional lower resonance to improve the bandwidth. The antenna has a peak gain of 8.07 dBi with good cross polarization characteristics and a good F=B ratio, greater than 12 dB, which makes it suitable for integrating with wireless communication devices. The antenna has a 2:1 VSWR bandwidth of 34.9% from 3.73 GHz to 5.73 GHz and it is a good choice for Broadband mobile access (4.4–5 GHz), 5.2 GHz centered WLAN (5.15–5.35 GHz), HIPERLAN2 (5.45–5.725 GHz) and HiSWANa (5.15–5.25 GHz) communication bands. II. GEOMETRY OF THE ANTENNA The geometry of the proposed antenna is shown in Fig. 1. Initially a square microstrip patch antenna of dimension L1 2 L1 mm2 is

Fig. 3. Reflection characteristics with stacking.

fabricated on a substrate of dielectric constant 4.2 with a loss tangent 0.02 and height 1.6 mm. A 45 tilted square slot of dimension L2 2 L2 mm2 is etched on the center of the square patch. For achieving good impedance matching characteristics a rectangular strip of dimension L3 2 W mm2 is incorporated symmetrically at the top corner of the slot. The same antenna fabricated on the same substrate is stacked over the initial antenna and an offset in its position along the +Y direction yields the final antenna configuration. The offset between the two patches is denoted by L0 . The antenna is electromagnetically coupled using a 50 microstrip transmission line fabricated on the same substrate. The total dimension of the antenna is found to be 42 2 55 2 4.8 mm3 . III. RESULTS AND DISCUSSIONS The simulation and the experimental studies of the antenna are done using Ansoft HFSS and HP8510C Network analyzer respectively. Fig. 2 shows the simulated and experimental reflection coefficient of the antenna without stacking. The three major resonances are found to be at 4.5 GHz, 5.6 GHz and 6.14 GHz. The first and second resonances are contributed by the patch itself and the third resonance is caused by the surface current variation on both the patch and the impedance matching strip as described in [8]. The strip added on the top portion of the slot acts like a series LC circuit which suppresses the high capacitive reactance for the first two resonances and it introduces the third resonance. These three resonances merge together to ensure broadband operation. The symmetrical opposing currents on the two sides of the strips cancel the fields along the on axis giving reduced gain for the higher resonance. Stacking the same antenna over the initial antenna without any offset in the position of the parasitic patch reduces the input impedance of the resonances. Fig. 3 shows the reflection coefficient of the stacked antenna without offset. It is noted that the resonances are shifted towards the lower side as compared to the antenna without the parasitic patch. This lower shift is due to the change in the effective dielectric constant

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Fig. 4. Effect of offset parameter L .

Fig. 5. Reflection characteristics of the final antenna. TABLE I MEASURED CROSS-POLAR LEVEL AND 3 dB BEAMWIDTH

Fig. 6. Measured radiation patterns of the antenna at (a) 3.8 GHz, (b) 4.27 GHz, (c) 4.76 GHz, (d) 5.1 GHz and (e) at 5.49 GHz.

of the substrate because the placement of the upper patch affects the fringing fields of the bottom patch. It can be seen that the minor resonance around 5 GHz for the initial antenna is found to be predominant for the stacked configuration and is found to be around 4.8 GHz. The other resonances are found to be at 4.3 GHz, 5.1 GHz and at 5.4 GHz. The final antenna is obtained by offsetting the top patch along the +Y direction. Fig. 4 shows the simulated reflection characteristics of the antenna with the offset parameter L0 . It can be seen that offsetting the upper patch introduces an additional lower resonance and as L0 is increased this resonance detaches away from the 4.3 GHz centered resonance and shifts towards the lower side while all the other resonances remains the same. An optimum offset of L0 = 4 mm is selected to obtain the maximum bandwidth of the antenna. The offset structure itself acts as the origin of the new lower resonance [9]. Fig. 5 depicts the simulated and experimental reflection coefficient of the antenna at the optimum design. A good agreement is observed between the simulated and experimental results. The antenna offers five resonances around 3.8 GHz, 4.27 GHz, 4.76 GHz, 5.1 GHz and 5.49 GHz having a 2:1 VSWR

Fig. 7. Measured gain of the antenna.

bandwidth of 34.9% from 3.73 GHz to 5.73 GHz. The antenna parameters are found to be L1 = 35 mm, L2 = 17:5 mm, L3 = 7:8 mm, W = 2 mm, Lf = 10:5 mm, L0 = 4 mm at the optimum design. The experimental radiation patterns of the antenna at the resonances are shown in Fig. 6. It is observed that for all the resonances, symmetrical stable XZ copolar pattern is observed for the entire operating band and the YZ copolar patterns show fluctuations. For the first four resonances the YZ plane cross polarization is lower as compared to that of the XZ plane. The maximum cross polar isolation is found to be 33 dB at 5.1 GHz in the XZ plane and 36.7 dB at 5.49 GHz in the YZ plane patterns. The variation in cross polar level and 3 dB beam width over the entire frequency range is studied and illustrated in Table I. The gain of the antenna is shown in Fig. 7. The antenna shows a maximum

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gain of 8.07 dBi at 3.9 GHz which is higher than that of a conventional patch antenna.

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Dual-Band Circularly-Polarized CPW-Fed Slot Antenna With a Small Frequency Ratio and Wide Bandwidths Can-Hui Chen and E. K. N. Yung

IV. CONCLUSION A simple offset stacked broadband patch antenna with improved gain and low cross polarization is presented. The proposed design simultaneously exhibits improvement in gain, good cross polar isolation > 15 dB and good F=B ratio better than 12 dB throughout the entire frequency range. The antenna has a 2:1 VSWR bandwidth of 34.9% from 3.73 GHz to 5.73 GHz which is suitable for Broadband Wireless Access, 5.2 GHz WLAN and HiSWANa communication bands.

REFERENCES [1] Y.-X. Guo, C.-L. Mak, K.-M. Luk, and K.-F. Lee, “Analysis and design of L-probe proximity fed patch antennas,” IEEE Trans. Antennas Propag., vol. 49, no. 2, pp. 145–149, 2001. [2] X.-Y. Zhang, Q. Xue, B.-J. Hu, and S.-L. Xie, “A wideband antenna with dual printed L-probes for cross-polarization supprression,” IEEE Antennas Wireless Propag. Lett., vol. 5, pp. 388–390, 2006. [3] C.-L. Mak, H. Wong, and K.-M. Luk, “High-gain and wide-band single-layer patch antenna for wireless communications,” IEEE Trans. Vehucular Tech., vol. 54, no. 1, pp. 33–40, 2005. [4] E. Nishiyama, M. Aikawa, and S. Egashira, “Stacked microstrip antenna for wideband and high gain,” IEE Proc. Microw. Antennas Propag., vol. 151, pp. 143–148, 2004. [5] Yuehe, K. P. Esselle, and T. S. Bird, “E-shaped patch antennas for high-speed wireless networks,” IEEE Trans. Antennas Propag., vol. 52, no. 12, pp. 3213–3219, 2004. [6] S. D. Targonski, R. B. Waterhouse, and D. M. Pozar, “Design of wideband aperture stacked patch microstrip antennas,” IEEE Trans. Antennas Propag., vol. 46, no. 9, pp. 1245–1251, 1998. [7] G. R. DeJean and M. M. Tentzeris, “A new high-gain microstrip Yagi array antenna with a high front-to-back ratio for WLAN and millimeter-wave applications,” IEEE Trans. Antennas Propag., vol. 55, no. 2, pp. 298–304, 2007. [8] V. P. Sarin, N. Nassar, V. Deepu, C. K. Aanandan, P. Mohanan, and K. Vasudevan, “Wideband printed microstrip antenna for wireless communications,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 779–781, 2009. [9] E. Rajo-Iglesias, G. Villaseca-Sanchez, and C. Martin-Pascual, “Input impedance behaviour in offset stacked patches,” IEEE Antennas Wireless Propag. Lett., vol. 1, pp. 28–30, 2002.

Abstract—A novel dual-band circularly-polarized (CP) CPW-fed slot antenna is proposed in this communication. This antenna characterizes a small frequency ratio and wide CP bandwidths. The dual-band operations are realized by using two parallel monopoles, one curved monopole and one fork-shaped monopole. In addition, a crane-shaped strip is placed in the ground plane to achieve circular polarization. The experimental results show that the antenna has the axial ratio bandwidths of 9% for the lower band and 11% for the upper band. Both bands are left handed. The frequency ratio of the upper band to the lower band is 1.286 (1.98 GHz/1.54 GHz). Other characteristics such as impedance bandwidths, radiation patterns and gains will also be presented. Index Terms—Circular polarization, coplanar waveguide (CPW)-fed, dual-band, slot antenna.

I. INTRODUCTION The rapid development of wireless communication services has stimulated the increasing need for dual-band antennas. In the last decade, various types of dual-band antennas have been proposed. CPW-fed slot antenna has become one of the most promising candidates for dual-band applications because of its favorable characteristics, such as wide bandwidths, light weight, low profile, and easy integration with monolithic microwave integrated circuits. The CPW-fed dual-band antennas can be divided into three types based on their polarization status. The first type is the dual-band linearly-polarized (LP) antenna. Such type includes the band-notch antenna [1], the annular ring slot antenna [2] and the dual-monopole antennas [3]–[5]. The second type is the dual-band CP antenna with dual senses of circular polarization. The dual CP bands operations can be realized by introducing an inverted-L slit [6] or two spiral slots [7] in the ground plane. Their upper-to-lower band frequency ratios are around 1.38. However, the antenna proposed in [6] is a dual-band antenna with respect to its axial ratio characteristic. It has only one band as far as its impedance characteristic is concerned. There is no isolation between the dual CP bands. Poor isolation may introduce unwanted interference from the stopband. The third type is the dual-band CP antenna with single sense of circular polarization in both bands. An example of this type is the CPW-fed circular fractal slot antenna [8]. By using the self-similar iterative configuration, the dual-band and wideband operations can be obtained. And its both bands are right-hand circularly polarized. However, the minimum axial ratio is not obtained in the boresight direction and its CP frequency ratio is larger than 1.9. In many applications, a small frequency ratio is required. Recently, a probe-fed annular-ring patch antenna with an unequal cross-slot loaded in the ground has been proposed in [9] for dual-band CP operations. The frequency ratio of this antenna can reach as small as 1.21. But its axial ratio bandwidths are about only 1%. Manuscript received June 08, 2009; revised December 04, 2009; accepted October 26, 2010. Date of publication January 28, 2011; date of current version April 06, 2011. The authors are with Department of Electronic Engineering, City University of Hong Kong, Kowloon, Hong Kong SAR, China (e-mail: cchen0@student. cityu.edu.hk). 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.2011.2109347

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Fig. 2. Simulated reflection coefficient and axial ratio of different values of Ld.

Fig. 1. Geometry of the proposed antenna (unit in mm).

In this communication, a novel dual-band circularly polarized CPW-fed slot antenna is proposed to achieve a small frequency ratio and wide CP bandwidths simultaneously. Based on the idea of using two simple monopoles [3] to generate dual-band operations, we also employ two deformed parallel monopoles, i.e., one curved monopole and one fork-shaped monopole, in the feeding line to achieve the same purpose. Meanwhile, the proposed antenna is circularly polarized. A crane-shaped strip is used as a perturbed element to excite the circularly polarized radiation, whose function is the same as that of the T-strip [10]. Therefore the configuration of this antenna is a combination of the dual-monopole antenna [3] and the square slot CP antenna [10]. As a result, the proposed antenna inherits the characteristics of a small adjustable frequency ratio and the wide CP bandwidth. In addition, a composite vector current method will be used to explain mechanism of generation the dual left-hand CP bands. Both the experimental and simulated results demonstrate that the proposed antenna can achieve wide CP bandwidths and a small frequency ratio. II. DESIGN

OF

CPW-FED SLOT ANTENNA FOR DUAL-BAND CIRCULAR POLARIZATION

The proposed antenna geometry and its dimension are shown in Fig. 1. There are two deformed parallel monopoles in the feeding line. The longer monopole is deformed into a rectangular shape because of the limited space of the square slot. The shorter monopole is deformed into a fork shape for parameter tuning. The longer one surrounds the shorter one. The gap between these two monopoles is 0.5 mm except that the lower left gap is 1 mm. Unlike the T-shaped strips used in the conventional CPW-fed CP antennas in [7] and [10], a crane-shaped strip is adopted instead. We will show that the bottom horizontal part of the crane strip is important for the dual-band operations in later section. The substrate used is FR4 of thickness h = 1:6 mm, relative permittivity "r = 4:4. The antenna is a square with size of 70 2 70 mm2 . The gap between the feeding strip and the ground plane is 0.5 mm and the gap Dc is 0.5 mm. Other parameters of the antenna can be found in Fig. 1. The size of the ground plane will affect the antenna performance,

Fig. 3. Simulated reflection coefficient and axial ratio of different values of Li.

especially the axial ratio characteristic. Therefore we should choose a ground plane with a suitable size. The antenna designed in this communication is left-hand circularly polarized for both bands. Right-hand CP antenna can be achieved by mirror the current design along the y-axis. III. NUMERICAL ANALYSIS OF THE DUAL-BAND CIRCULARLY POLARIZED CPW-FED SLOT ANTENNA In order to investigate the effects of various parameters on the antenna performance, parametric studies have been carried out by the IE3D simulator [11]. Through out the studies presented in this section, all other parameters that have not been mentioned are fixed to the values shown in Fig. 1. The simulated reflection coefficient and axial ratio for different lengths of the outer curved monopole are shown in Fig. 2. It can be seen that increasing of the length of curved monopole will lower the center frequencies of both impedance bands. But the change of the lower band is more obvious than that of the upper band. The center frequency of the lower CP band will shift to lower frequency as the length increases. However, the increase has little influence on the upper band. Fig. 3 shows the simulated reflection coefficient and axial ratio for different lengths of the left arm of the inner fork-like monopole. The reflection coefficient of the lower band becomes much

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Wg

Fig. 4. Simulated reflection coefficient and axial ratio of different values of .

Fig. 5. Simulated reflection coefficient and axial ratio of different values of and different status of bottom strip.

Wg = 0 5mm

Dc = 2 5 mm

Lc = 2 mm

Dc = 0 5 mm

Fig. 6. Simulated reflection coefficients of the proposed antenna with different monopole status.

Dc

smaller when the length increases. In the meantime, the reflection coefficient of the upper band becomes much bigger. For the axial ratio characteristic, the changing rule of the inner monopole is similar to that of outer monopole. By taking advantage of these characteristics, changing of the lengths of the inner and outer monopoles can be a complementary measure in the tuning process. Fig. 4 illustrates the ) between the two monopoles on the antenna influence of the gap ( performance. Note that moving downward the bottom edge of the curved monopole won’t change its length. It can be observed that as the gap is widened, the upper CP band moves upward and the lower CP band moves downward, making the frequency ratio become larger. For is adopted in this communication. : the optimum design, The effect of crane-shaped strip on the antenna performance is shown in Fig. 5. It is found that when the whole crane strip is placed upward : ), another band centered at 1.1 GHz is by 2 mm (i.e., introduced but not CP. Meanwhile, the axial ratio characteristic of : is selected the lower band also deteriorates. Therefore as the optimum value for the current design. Fig. 5 shows another case that the horizontal bottom strip is removed from the crane (i.e., ). That is to say, its shape will change from a crane to an inverted-L. It is seen that the upper CP band shifts upward greatly. Moreover, the upper impedance band and the lower CP band disappear. This shows that the crane-shaped strip is important for the

Wg

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Fig. 7. Effect of ground plane’s size on the axial ratio characteristic.

dual-band operations. In addition, the additional band merges with the lower band, which results in a much wider lower band. It means that increasing the gap or reducing the length will easily introduce the additional band. The circular polarization generation mechanism of the proposed antenna is the same as that of the antenna in [10]. The crane-shape strip is orthogonal to the feeding signal strip and acts as a perturbed element to generate circularly polarized radiation. The parallel feeding monopoles split the impedance band into dual bands. Therefore some changes should be made to the perturbed element to adapt CP bands with the impedance bands. Fig. 6 shows the reflection coefficients for different status of the feeding monopoles of the proposed antenna. The length of the longer monopole is 91.9 mm. And the length of the left arm (shown in Fig. 6) of shorter monopole is 33.2 mm. Due to the deformation, the first resonance of longer monopole occurs at 1.15 GHz and its second resonance occurs at 2.25 GHz. However, the reflection coefficient of the first resonance is very large. The two arms of shorter monopole create two close resonances, which result in a wider bandwidth centered at 1.48 GHz. If the right arm is cut, only one resonance located at 1.65 GHz is observed. When using both monopoles in the antenna, we can see that

Dc

Lc

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Fig. 8. Simulated vector current distributions at 1.58 GHz: (a) 0 degree; (b) 90 degree; (c) composite vector current.

Fig. 9. Simulated vector current distributions at 1.76 GHz: (a) 0 degree; (b) 90 degree; (c) composite vector current.

the longer monopole creates the upper band at its second resonance while the shorter one creates the lower band. By contrast, the longer monopole creates the lower band while the shorter one creates the upper band for the conventional dual-monopole antenna in [3]. Fig. 7 illustrates the effect of the ground plane’s size on the axial ratio characteristic. Since the impedance characteristic is not so sensitive to the change of size, it is not plotted in the graph. There are two ways of changing the size. One way is to change the ground plane’s size without changing the slot’s size. The other way is to change the sizes of the ground plane and the slot at the same time. From Fig. 7, we can see that increase of the size will make the dual frequencies come closer in the former case. However, in the latter case, it can be found that the upper band shifts to higher frequency while the lower band changes a little when the both sizes decrease. Similar to the analysis method of the vector current [13], we employ a composite vector current method in this communication. Fig. 8 shows the vector current distributions of the proposed antenna at 1.58 GHz viewed from the +z direction. If we only look at the Fig. 8(a) and (b), it is very difficult to draw a conclusion that the antenna is left-hand circularly polarized. However, when taking the sum of all the vector currents both in the feeding line and in the ground plane, we will have a much clearer view about generation mechanism of left-hand circular

polarization. The composite vector current is 6966 77:3 A=m at phase 0 degree and 5576 0 12:9 A=m at phase 90 degree. In other words, after one-quarter period, the composite vector current rotates in left-hand direction by 90.2 degrees in the +z direction, which satisfies the requirement of the spatial and temporal quadrature for circular polarization [12], even though there is a slight drop in magnitude. Fig. 9 shows the vector current distributions at 1.76 GHz. We can find that, after one-quarter period, the composite vector current becomes from 5076 45 A=m to 3596 65:5 A=m. It rotates in right-hand direction by 20.5 degrees, which is far from satisfying the quadrature requirement. Fig. 10 shows the vector current distributions at 1.9 GHz. After one-quarter period, the composite vector current changes from 3906 111:9 A=m to 4576 20:6 A=m, rotating in left-hand direction by 91.3 degrees, which satisfies the quadrature requirement for circular polarization again. The above analysis can explain the why the dual-band left-hand circular polarization can be achieved in the proposed antenna.

IV. MEASURED RESULTS In order to verify the numerical results, a CPW-fed slot antenna based on the parameters shown in Fig. 1 is finally fabricated and mea-

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Fig. 10. Simulated vector current distributions at 1.9 GHz: (a) 0 degree; (b) 90 degree; (c) composite vector current.

Fig. 11. Measured and simulated reflection coefficients for the proposed antenna.

spectively. The measured results are in reasonable agreement with the simulated results. The discrepancies between them may be attributed to the low-cost lossy FR4 substrate used in the experiment. The measured impedance bandwidths are 17% (from 1.45 to 1.72 GHz) for the lower band and 21% (from 1.86 to 2.29 GHz) for the upper band. And the simulated impedance bandwidths are 16% (from 1.45 to 1.70 GHz) and 28% (from 1.86 to 2.47 GHz) respectively. The measured CP bandwidths are 9% (from 1.47 to 1.61 GHz) and 11% (from 1.87 to 2.09 GHz) for the lower and upper bands against the simulated bandwidths of 11% (from 1.46 to 1.63 GHz) and 11% (from 1.86 to 2.08 GHz). It is observed that both CP bands are completely inside their respective impedance bands. The CP frequency ratio is 1.286 (1.98 GHz/1.54 GHz). As seen in Fig. 12, the measured and simulated lowest axial ratios occur around 1.58 GHz and 1.92 GHz. Moreover, the variation trends of the measured result are consistent with those of the simulated result. Note that the co-polarizations of the lower and the upper band are the same, i.e., LHCP. Fig. 13 shows the measured and simulated radiation patterns in the XOZ plane at four discrete frequencies within the dual operational bands, i.e., 1.52 GHz, 1.60 GHz, 1.92 GHz and 2.04 GHz. It can be found that cross-polarizations can keep 15 dB lower than the co-polarizations in the boresight direction. Meanwhile, these four patterns can remain stable and their 3-dB beamwidths are around 80 degrees. Note that the printed slot antenna is a bidirectional radiator and the radiation patterns in both sides are almost the same. But the polarization of the lower half plane is the opposite of that of the upper half plane. Fig. 14 shows the measured gains of the proposed antenna. The gains of the lower band and the upper band are defined to left-hand isotropic circularly polarized antenna. It can be seen that within the dual CP bands, the gains are between 2.5 to 4 dBi. V. CONCLUSION

Fig. 12. Measured and simulated axial ratios for the proposed antenna.

sured. Figs. 11 and 12 illustrate the simulated and measured results of reflection coefficients and axial ratios of the proposed antenna re-

A novel dual-band circularly polarized CPW-fed slot antenna with a small frequency ratio and wide CP bandwidths has been proposed and discussed. By employing two parallel monopoles in the feeding line and a crane-shaped strip in the ground plane, a small CP frequency ratio and wide CP bandwidths can be realized simultaneously. The measured impedance and axial ratio bandwidths are 17%, 9% for the lower band and 21%, 11% for the upper band. And the frequency ratio of the upper band to the lower band is 1.286.

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Fig. 13. Measured and simulated radiation patterns for the proposed antenna: (a) 1.52 GHz; (b) 1.6 GHz; (c) 1.92 GHz; (d) 2.04 GHz.

Fig. 14. Measured gains versus frequency.

REFERENCES [1] Y. C. Lin and K. J. Hung, “Compact ultrawideband rectangular aperture antenna and band-notched designs,” IEEE Trans. Antennas Propag., vol. 54, pp. 3075–3081, Nov. 2006. [2] J.-S. Chen, “Dual-frequency annular-ring slot antennas fed by CPW feed and microstrip line feed,” IEEE Trans. Antennas Propag., vol. 53, no. 1, pp. 569–571, Jan. 2005. [3] H.-D. Chen and H.-T. Chen, “A CPW-Fed dual-frequency monopole antenna,” IEEE Trans. Antennas Propag., vol. 52, no. 4, pp. 978–982, Apr. 2004. [4] T. H. Kim and D. C. Park, “CPW-fed compact monopole antenna for dual-band WLAN applications,” Elect. Lett., vol. 41, no. 6, pp. 291–293, Mar. 2005.

[5] W. -C. Liu and C. -F. Hsu, “Dual-band CPW-fed Y-shaped monopole antenna for PCS/WLAN application,” Elect. Lett., vol. 41, no. 7, pp. 390–391, Mar. 2005. [6] C. F. Jou, J. W. Wu, and C. J. Wang, “Novel broadband monopole antennas with dual-band circular polarization,” IEEE Trans. Antennas Propag., vol. 57, no. 4, pp. 1027–1034, Apr. 2009. [7] C. Chen and E. K. N. Yung, “Dual-band dual-sense circularly-polarized CPW-fed slot antenna with two spiral slots loaded,” IEEE Trans. Antennas Propag., vol. 57, no. 6, pp. 1829–1833, Jun. 2009. [8] D.-C. Chang, B. H. Zeng, and J.-C. Liu, “CPW-fed circular fractal slot antenna design for dual-band applications,” IEEE Trans. Antennas Propag., vol. 56, no. 12, pp. 3630–3636, Dec. 2008. [9] X. L. Bao and M. J. Ammann, “Dual-frequency circularly-polarized patch antenna with compact size and small frequency ratio,” IEEE Trans. Antennas Propag., vol. 55, no. 7, pp. 2104–2107, Jul. 2007. [10] 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, no. 8, pp. 2141–2144, Aug. 2003. [11] IE3D Manual9.04 ed. Zeland software Inc. [12] W. L. Stutzman, Polarization in Electromagnetic Systems. Norwood, MA: Artech House, 1993. [13] X. L. Bao, M. J. Ammann, and P. McEvoy, “Microstrip-fed wideband circularly polarized printed antenna,” IEEE Trans. Antennas Propag., vol. 58, no. 10, pp. 3150–3156, Oct. 2010.

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Rectangular Dielectric Resonator Antennas With Enhanced Gain Aldo Petosa and Soulideth Thirakoune

Abstract—Rectangular dielectric resonator antennas were designed to operate at high-order modes to achieve enhanced gain. A simple model was developed to predict the radiation patterns of the higher-order modes. Prototypes designed at 11 GHz showed gains of up to 5 dB above those obtained by the fundamental mode. Index Terms—Dielectric resonator antennas (DRAs).

Fig. 1. Geometry of a rectangular DRA.

I. INTRODUCTION The design flexibility offered by dielectric resonator antennas (DRAs) makes them attractive alternatives to other low-gain antennas such as microstrip antennas or dipoles [1], [2]. They also maintain a high radiation efficiency, even at millimeter-waves frequencies due to the lack of surface wave losses and minimal conductor losses. DRAs can be designed from various shapes of dielectric material. Rectangular shapes are often selected due to their fabrication simplicity and improved degrees of freedom compared to other basic shapes such as cylindrical or hemispherical. Rectangular DRAs have typically been excited in the lowest order mode TEx 11 (or TEy11 ), 0 <   1, which radiate like an x-directed (or y-directed) short magnetic dipole, with typical gains of about 5 dBi, when placed on a large ground plane. Several methods have been proposed to enhance the gain of DRAs without resorting to arraying elements together using a feed network. Stacking DRAs on top of each other has been primarily used as a method to increase impedance bandwidth [3]–[7]. In some of these cases, however, stacking can also improve the directivity of the DRA with enhancements of up to about 3 dB above that of a single DRA. The use of a shallow pyramidal horn with a height of 0.15 times the free space wavelength () and a square aperture of dimensions 1:22 1:2 has been shown to increase the gain of a DRA to nearly 10 dBi [8]. Placing a circularly polarized DRA within a circular cavity was shown to increase the gain to over 13 dBic for a cavity diameter of approximately 2 and height of 0:5 [9], [10]. Finally, the use of a thin (0:25g , where g is the guided wavelength) dielectric superstrate placed about 0:5 above the DRA has been shown to enhance the gain up to 16 dBi for a square superstrate with dimensions of 3:2 2 3:2 [11]. Most of these techniques require a significant increase in surface area, which may not be available for applications such as portable wireless communications devices. The higher-order modes of rectangular DRAs have already been used in multi-mode operations to enhance the impedance bandwidth [12]–[14]. However, no thorough study has yet been carried out to determine the enhancement in directivity that can be achieved by exciting higher-order modes. Recently, the use of higher-order modes was proposed for enhancing the gain of rectangular DRAs [15] where finite difference time domain (FDTD) simulations showed that gains of up Manuscript received May 11, 2010; revised June 29, 2010; accepted August 30, 2010. Date of publication February 04, 2011; date of current version April 06, 2011. The authors are with the Advanced Antenna Technology Lab, Communications Research Centre Canada, Ottawa, ON K2H 8S2, Canada (e-mail: aldo. [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.2011.2109690

to 10 dBi could be achieved. In [15], the dielectric waveguide model was also used to analyze these higher modes and a simple model was proposed for predicting the radiation patterns. However, an error in the model was made in [15] which is corrected in Section II of this article. Section III presents a recommended procedure for designing the higher-order mode DRAs. The experimental results of three prototype designs are then shown in Section IV. A summary and brief discussion conclude the communication. II. DIELECTRIC WAVEGUIDE MODEL Rectangular DRAs have been modeled as truncated dielectric waveguides in order to predict their resonant frequencies and radiation Q-factors [16]. For a rectangular DRA having a dielectric constant of "r and dimensions of w, h(= b=2) and d and mounted on a perfect infinite ground plane, as shown in Fig. 1, the resonant frequency fmn of x mode can be predicted using the following transcendental the TEmn equation: 2 0 k2 kx tan k2x d = ("r 0 1) kmn x

(1)

where

kmn = 2fcmn ; ky = m w ; 2 kz = n b and kx2 + ky2 + kz2 = "r kmn

and c is the speed of light. The E- and H-fields for the various modes within the DRA can be approximated using the equations in Appendix A. When the DRA is mounted on a ground plane, the even modes in the z-direction (i.e., n = 2N; N = 1; 2; 3; . . .) will be short-circuited and only the odd modes (n = 2N + 1) can exist. Fig. 2(a) depicts the normalized Hx -fields, using (A4), for the fundamental and two higher-order modes in the rectangular DRA. This field configuration within the DRA can be approximated by a set of short magnetic dipoles, separated by a distance s, as shown in Fig. 2(b), which can then be used to predict the far-field radiation patterns. (Note that the modes with m > 1 and n = 1 are not of interest, since they produce a broadside null in the radiation pattern.) The spacing, s, between the magnetic dipoles of the higher-order modes will affect the radiation pattern of the DRA and can, to a certain extent, be controlled by the aspect ratio of the DRA. Fig. 2(c) shows the final model used to predict the radiation patterns of the DRAs, where image theory is used to remove the ground plane. The double arrow for the short magnetic dipole located at z = 0 indicates that it has twice the amplitude of the other dipoles. (This was neglected in the model developed in [15]). Fig. 3 shows the normalized far-field patterns of the DRA operating in the fundamental

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Fig. 2. H -field configuration and radiation model for the DRA (a) H-Fields. (b) Radiation model. (c) Radiation model (using image theory).

TEx11

mode (the “x” superscript is suppressed henceforth) and the higher-order modes TE13 and TE15 , based on the radiation models in Fig. 2(c) where simple array theory is used and the effects of mutual coupling between dipoles are ignored. The higher-order modes show a significant narrowing of the beam width, indicating an increase in directivity over the DRA operating in the fundamental mode. A spacing of s = 0:4 offers a good compromise between the beamwidth and sidelobe levels. For a DRA operating in the TE15 mode, the required height to achieve this spacing is h  .

Fig. 3. Normalized patterns based on the dielectric waveguide model with s

0:4.

=

III. DESIGN PROCEDURE To verify the predictions of the model, DRAs were designed to resonate in the TE11 , TE13 and TE15 modes, henceforth referred to as (m; n) = (1; 1), (1,3) and (1,5), respectively, all at approximately 11 GHz. Since rectangular DRAs have two degrees of freedom, there is no unique set of dimensions for a given resonant frequency and dielectric constant. This is one of the advantages of rectangular DRAs since the designer has significant flexibility in choosing the aspect ratio to suit the intended application. For this study, the following design approach was taken. The dielectric constant was first selected to be "r = 10. This was chosen partially based on material availability and also to maintain a reasonable impedance bandwidth. (Higher dielectric constants would result in more compact designs but with narrower bandwidth.) Next the height of the DRA was chosen to be approximately =3, =2 and  for the (1,1), (1,3) and (1,5) modes, respectively. This was done so that the spacing between the higher-order modes would be approximately s = 0:4. Since there is still one degree of freedom, a further constraint was placed such that w = d. (This choice was somewhat arbitrary and other w=d ratios could just as easily have been selected. Having a square cross-section does, however, somewhat simplify the fabrication process.) The values for w and d were determined by substituting h and "r into (1) and trying different values for w and d in an iterative fashion until the desired resonant frequency was arrived at. Models using these initial dimensions were then analyzed using a commercial finite difference time-domain (FDTD) software package and were adjusted somewhat to obtain the desired number of modes at

Fig. 4. DRA designs (dimensions in mm).

approximately 11 GHz. The resulting dimensions for the three DRAs are shown in Fig. 4. Fig. 5 shows the simulated normalized jHx j-field of the DRA operating in the (1,5) mode at 11 GHz. The modes are approximately spaced s = 11:9 mm apart which corresponds to 0:44 at 11 GHz. The separation between modes can be somewhat controlled by adjusting the aspect ratio of the DRA. IV. MEASURED RESULTS Three DRA prototypes were fabricated from a low-loss microwave material having a dielectric constant of "r = 10 and a loss tangent of tan  < 0:002. Each DRA was mounted on a 150 mm 2 100 mm ground plane and excited using a microstrip line through an aperturecoupled slot, as shown in Fig. 6. The value of the stub length s in Fig. 6 was adjusted to match each individual DRA. The value of s was 3 mm for DRA1 and DRA2 and 6 mm for DRA3. The measured magnitude

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Fig. 5. Simulated j

H

j

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fields of DRA 3 at 11 GHz.

m = 1; n = 1).

Fig. 8. Normalized patterns of DRA 1 (

Fig. 6. Geometry of the microstrip-fed aperture.

The normalized patterns for the three DRAs are shown in Figs. 8–10. For all three DRAs, the measured cross-polarization levels were 20 dB or more below the peak co-polarization levels. In each figure, the radiation pattern obtained from the simple dielectric waveguide model is overlaid. The agreement between the simple radiation model and the measured H-plane patterns is very close. There is some discrepancy in the E-plane patterns due to scattering from the finite size ground plane edges, which is not accounted for in the simple radiation model. Nevertheless, this simple model is a good predictor of the radiation patterns resulting from exciting higher-order modes of the rectangular DRA. The gain (the sum of the measured realized gain plus the mismatch loss) for the three DRAs is plotted as a function of frequency in Fig. 11. The gains for all three DRAs peak at 11 GHz, with values of 5.5, 8.2 and 10.2 dBi, for the (1,1), (1,3) and (1,5) modes, respectively. V. SUMMARY AND CONCLUSIONS

Fig. 7. Measured j

S

j

of the DRA prototypes.

of the reflection coefficient (jS11 j) for each of the three DRAs is shown in Fig. 7.

The higher-order modes of a rectangular DRA were used to produce radiation patterns with enhanced gain. Based on the field configurations within the DRA obtained from the dielectric waveguide model, the radiation patterns were predicted using an array of short magnetic dipoles, whose number depends on the mode being excited within the DRA. This model predicts that a rectangular DRA operating in a higher order mode will radiate a more directive pattern. In addition, by adjusting the aspect ratio of the rectangular DRA, some degree of pattern control is possible for the case of the higher order modes. (This is not true for the fundamental mode, which radiates like a single short magnetic dipole, independent of the aspect ratio of the DRA.) Comparisons of the predicted patterns to the normalized measured patterns showed good agreement, especially in the H-plane, where the effects of the finite ground plane (not considered in the model) are negligible. Measured patterns from fabricated prototypes showed that gains of up to 10.2 dBi were achieved for a DRA operating in the (m = 1; n = 5). The advantage of this approach for enhancing gain compared to some of the other cited techniques lies in the smaller

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Fig. 11. Gain of the DRA prototypes.

m = 1;n = 3).

Fig. 9. Normalized patterns of DRA 2 (

Finally, it should be noted that although this investigation focused on rectangular DRAs, the excitation of the appropriate higher-order modes in DRAs of other shapes such as cylindrical, should also result in enhanced gain performance. APPENDIX Approximation of the E- and H-fields in a rectangular DRA based on the dielectric waveguide model

Ex = 0 (ky y) sin (kz z) Ey = Akz cos (kx x) cos sin (ky y) cos (kz z) sin (ky y) cos (kz z) Ez = 0 Aky cos (kx x) cos (ky y) sin (kz z) 2 2 + kz cos (k x) sin (ky y) cos (kz z) Hx = A jk2yf x cos (ky y) sin (kz z) mn sin (kz z) k x ky Hy = A j 2f sin (kx x) cos ((kky yy)) cos sin (kz z) y mn (ky y) sin (kz z) x kz Hz = A j 2kf sin (kx x) cos sin (ky y) cos (kz z) mn

(A1) (A2) (A3) (A4) (A5) (A6)

where the upper functions are chosen when the corresponding values of m or n are odd and the lower when m or n are even. A is an arbitrary constant of proportionality.

REFERENCES

m = 1;n = 5).

Fig. 10. Normalized patterns of DRA 3 (

area requirements, which is an important consideration for many applications such as mobile wireless communications, where space limitations are of major concern. The maximum achievable gain attainable by exciting these higher-order modes will be determined primarily by physical constraints and practical limitations. For example, simulations show that exciting the (1,7) mode of a rectangular DRA increases the directivity to 13.7 dBi. However, such a DRA designed at 11 GHz would require a height of 90 mm, which would probably find very limited practical applications.

[1] Dielectric Resonator Antennas, K. W. Luk and K. W. Leung, Eds. Hertfordshire, England: Research Studies Press, 2003. [2] A. Petosa, Dielectric Resonator Antenna Handbook. Norwood, MA: Artech House, 2007. [3] Y. Hwang, Y. P. Zhang, K. M. Luk, and E. K. N. Yung, “Gain-enhanced miniaturised rectangular dielectric resonator antenna,” IEE Electron. Lett., vol. 33, no. 5, pp. 350–352, Feb. 1997. [4] K. M. Luk, K. W. Leung, and K. Y. Chow, “Bandwidth and gain enhancement of a dielectric resonator antenna with the use of a stacking element,” Microw. Opt. Technol. Lett., vol. 14, no. 4, pp. 215–217, Mar. 1997. [5] C. Nannini, J. M. Ribero, J. Y. Dauvignac, and C. Pichot, “Bifrequency behaviour and bandwidth enhancement of a dielectric resonator antenna,” Microw. Opt. Technol. Lett., vol. 42, no. 5, pp. 432–434, Sep. 2004. [6] A. A. Kishk, “Directive Yagi-Uda dielectric resonator antennas,” Microw. Opt. Technol. Lett., vol. 44, no. 5, pp. 451–453, Mar. 2005.

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[7] K. W. Leung, K. Y. Chow, K. M. Luk, and E. K. N. Yung, “Offset dual-disk dielectric resonator antenna of very high permittivity,” IEE Electron. Lett., vol. 32, no. 22, pp. 2038–2039, Oct. 1996. [8] Nasmuddin and K. Esselle, “Antennas with dielectric resonators and surface mounted short horns for high gain and large bandwidth,” IET Proc. Microw., Antennas Propag., vol. 1, no. 3, pp. 723–729, Jun. 2007. [9] J. Carrie et al., “A. K-Band circularly polarized cavity backed dielectric resonator,” in IEEE Antennas and Propagation Symp. Digest AP-S, Baltimore, MD, Jul. 1996, pp. 734–737. [10] J. Carrie et al., “A Ka-band circularly polarized dielectric resonator modelled using the transmission-line matrix method,” in Proc. Antenna Technology and Applied Electromagnetics Symp. ANTEM 96, Montreal, Canada, Aug. 1996, pp. 709–712. [11] Y. M. M. Antar, “Antennas for wireless communications: Recent advances using dielectric resonators,” IET Proc. Circuits, Devices Syst., vol. 2, no. 1, pp. 133–138, Feb. 2008. [12] G. Bit-Babik, C. DiNallo, and A. Faraone, “Multimode dielectric resonator antenna of very high permittivity,” in Proc. IEEE Antennas and Propagation Symp., Monterey, CA, Jun. 2004, vol. 2, pp. 1383–1386. [13] B. Li and K. W. Leung, “A wideband strip-fed rectangular dielectric resonator antenna,” in Proc. IEEE Antennas and Propagation Symp., DC, Washington, Jul. 2005, vol. 2, pp. 172–175. [14] C. S. de Young and S. A. Long, “Wideband cylindrical and rectangular dielectric resonator antennas,” IEEE Antennas Wireless Propag. Lett., vol. 5, pp. 426–429, 2006. [15] A. Petosa, S. Thirakoune, and A. Ittipiboon, “Higher-order modes in rectangular DRAs for gain enhancement,” presented at the ANTEM 2009, Banff, Canada, Feb. 2009. [16] R. K. Mongia and A. Ittipiboon, “Theoretical and experimental investigation on rectangular dielectric resonator antennas,” IEEE Trans. Antennas Propag., vol. 45, pp. 1348–1356, Sep. 1997.

Design of a Microstrip Monopole Slot Antenna With Unidirectional Radiation Characteristics Chien-Jen Wang and Te-Liang Sun

Abstract—A design procedure for a wideband planar unidirectional antenna is presented based on the monopole slot antenna. Owing to the additional capacitance contributed by a stub-protruded feedline and two symmetrical stubs at the slot edge, the impedance-matching condition is improved; meanwhile, the resonance frequency of the monopole slot antenna is significantly shifted down, and an improvement in impedance bandwidth is achieved. By modifying the width of the ground plane, connecting one metallic finger at the upper right corner, and etching two asymmetrical slits in the ground plane as reflectors, the radiation directivity of the monopole slot antenna can be enhanced. The proposed antenna achieves a measured 10 dB at the center frequency of 2.31 GHz, 97.4% bandwidth for S better than 10 dB front-to-back ratio. Index Terms—Corrugated grating structure, front-to-back ratio, monopole slot antenna, protruded stub.

I. INTRODUCTION Owing to planar geometry, wide impedance bandwidth and simple structure, the monopole slot antenna receives much attention in the field of wireless communication systems. A quarter-wavelength open Manuscript received March 03, 2010; revised September 01, 2010; accepted September 01, 2010. Date of publication January 31, 2011; date of current version April 06, 2011. This work was supported in part by the National University of Tainan and National Science Council, Taiwan, under Grants NSC 96-2221-E024-001 and 99-2221-E024-001. The authors are with the Department of Electrical Engineering, National University of Tainan, Tainan, Taiwan. (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.2011.2109682

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slot cut in the finite ground plane and fed by a microstrip transmission line, a wide impedance bandwidth and bi-directional radiation are provided [1]. Several different slot shapes were introduced to generate additional resonances so that bandwidth enhancement was achieved [2]. Furthermore, the occupied area of the antenna was decreased. In [3], a pentagon-shape slot was etched as a radiator on the ground plane, and a feeding line with a pentagon stub was used. This antenna demonstrates a superior ultrawideband (UWB) impedance bandwidth. A multiband monopole slot antenna utilized in wireless wide area network (WWAN) application was studied in [4]. By using a step-shaped microstrip feedline to electromagnetically couple the power into three monopole slots, three resonant modes were excited. The proposed antenna is promising as an internal antenna in the thin-profile laptop computer. Due to the monopole topology [5], co-polarized radiation patterns of the monopole slot antenna at different frequencies look omni-directional. By placing a back metal plate as a power reflector below a printed open-slot antenna, a back-lobe level was reduced and unidirectional radiation characteristics in free space were achieved [6]. In this communication, we etched two slits of different lengths in the ground plane of the monopole slot antenna in order to decrease the back-lobe. Techniques such as adding one finger stub at the upper right corner and changing the ground geometry are also utilized to enhance the front-to-back (F/B) ratio of the co-polarized radiation in the operated band. A stub-protruded feeding structure and two identical rectangular stubs inserted into the radiated region are used in order to improve the impedance matching condition. II. ANTENNA DESIGN Fig. 1 shows the schematic configuration of the unidirectional monopole slot antenna with the F/B ratio of up to 10 dB over the operated bands. The design parameters of the proposed antenna after the optimization process are listed in Table I. FR-4 is used as a substrate with the dielectric constant of 4.4 and the thickness of 0.8 mm, respectively. The proposed antenna is composed of a quarter-wavelength open rectangular slot etched in the ground plane, a microstrip stub-protruded feedline, two slits of different lengths etched in the ground plane and a finger stub connecting at the upper right corner of the ground plane. A protruded stub of width Fw = 4:5 mm and length FL1 = 11 mm is connected to a 50- feedline printed on the other side of the monopole slot. At the slot edge, two rectangular stubs are applied to form an internal capacitance for good impedance-matching condition. The geometry of the ground plane is varied in order to modify the radiation patterns. In our experiment, Gx2 is tuned from 35 mm to 65 mm when Gx1 is fixed at 35 mm. Similarly to utilize the corrugated grating structure near the slot antenna [7], in order to achieve good reduction of the back-lobe, two slits of different lengths are etched in the left part of the ground plane. The lower slit is uniform, and the upper slit is rippled. III. RESULTS AND DISCUSSION In our study, the finite element method (FEM) commercial software used is the high frequency structure simulator (HFSS) from Ansoft Corporation. Fig. 2 shows the comparison of simulated reflection coefficients (S11 ) of the three cases of the monopole rectangular-slot antenna, which are the conventional monopole slot antenna with a uniform feedline one with a stub-protruded feedline and the other with a protruded feedline and two stubs at the slot edge. Due to the protruding element at the end of the feeding line, the coupling between the feedline and the slot edge increases, resulting in frequency reduction of the lower resonance so that the impedance bandwidth effectively increases. Similar to the technique of adding a parasitic capacitance at the feedline

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Fig. 2. Comparison of simulated S of the three feeding cases, which are the uniform feedline, one with the protruded feedline, and the protruded one with the tuning stubs in the slot. (a) Uniform feeding, (b) protruded feeding, (c) with tuning stubs.

stubs at the slot edge, an internal capacitance is formed due to a voltage difference between the two identical stubs. The impedance-matching condition at the middle frequency band (about at 3.0 GHz) is improved. By utilizing these two techniques, the impedance bandwidth increases from 52% (2.40–4.09 GHz) to 81% (1.68–3.97 GHz) and the lower frequency band is shifted down. In other words, the antenna size can be reduced. The distributions of the simulated surface current density on the mm and ground plane for two cases at 1.8 GHz, including x2 mm, are shown in Fig. 3(a) and (b). The injected power x2 is coupled into the open-slot and radiates into space in the form of space wave and surface wave. The space wave is a standing-wave mode and radiates out in the broadside direction, which locates at the z-axis (Fig. 1). The surface wave is a traveling-wave mode. For the monopole slot antenna, the power leaks out mainly through the surface wave. The surface wave flows along the edges of the open-slot and radiates out   and  . At in the -y-axis direction, positioned at the open end of the slot, some of the residual power flows out along the x-axial edges of the ground plane. It is observed that, for the case of x2 mm, the current densities along the two horizontal (the y-axis) edges of the ground plane are high. The edge currents flow left, and propagate out like the traveling wave. The phenomenon may be a major cause of back radiation. Shown in Fig. 3(b), when the area of the upper part of the ground plane increases, the densities of the surface currents flowing along the edges decreases, and the back-lobe can be – mm, suppressed. With an increase in the width from X2 there is no apparent variation in the impedance matching condition of the antenna. The comparison of the simulated radiation patterns at 1.8 and 3.6 GHz for different X2 is shown in Fig. 4. The simulated results of the F/B ratio and the main-beam angle are listed in Table II. It is noted that the main beam (labeling front : F) locates at angles of   and  , and the back-lobe (labeling back: B) is at  and   . The yz-plane radiation pattern consists of two    to 180 at and . When X2 increases, parts:  resulting in a decrease in current densities, the back-lobe reduces and of the main beam bethe F/B ratio increases. However, the angle comes tilted. Hence, X2 is set at 60 mm in order to maintain the power  and  . radiation in the end-fire direction, which is at

G = 35

G = 60

8 = 270

Fig. 1. Schematic configuration of the unidirectional monopole slot antenna. (a) 3-dimensional view, (b) top view, (c) bottom view. TABLE I GEOMETRICAL PARAMETERS OF THE PROPOSED MONOPOLE SLOT ANTENNA AFTER THE OPTIMIZED DESIGN PROCEDURE

= 90

G = 35

G = 35 65

G

8 = 270 8 = 90 =0

[8], this capacitance is a capacitive loading to the slot for canceling the inductive reactance of the slot at lower frequency. Hence, the lower resonance is shifted down and the bandwidth increases. After adding the

= 90 = 90 8 = 90 G

8 = 270 G (8) 8 = 270

= 90

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Fig. 4. Comparison of the simulated radiation patterns for different

G

.

TABLE II COMPARISON OF THE SIMULATED RESULTS OF THE F/B RATIO AND MAIN-BEAM ANGLE OF X-Y PLANE FOR DIFFERENT

G

Fig. 3. Distribution of the simulated surface current density on the ground plane for the symmetrical- and asymmetrical-ground-plane cases at 1.8 GHz. (a) mm, (b) mm, mm, (c) Slits embedded.

G = G = 35

G = 60

G = 35

G = 60

For X2 mm, the F/B ratios at four operated frequencies are inmm. creased by at least 1 dB compared to the case where X2 It is noted that, in our experiment, the impedance matching condition of the open slot antenna is slightly affected by adding the finger stub at the corner of the ground plane. Fig. 5 presents the comparison of the F/B ratio of the monopole slot antennas without and with the finger stub operated from 1.7 GHz to 3.8 GHz. After adding the finger stub at

G = 35

Fig. 5. Comparison of the simulated F/B ratio of the monopole slot antennas without and with the finger stub.

the ground plane, the F/B ratios apparently improve, especially at the middle and high frequency bands.

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Fig. 8. Comparison of the simulated radiation patterns of the monopole slot antennas without and with the reflectors.

Fig. 6. Comparison of the simulated reflection coefficients of the monopole slot antennas without and with the reflectors. (a) No slit (b) uniform slit (c) rippled slit.

Fig. 9. Reflection coefficients of the conventional and proposed monopole slot antennas.

Fig. 7. Comparison of the simulated radiation patterns of the antennas without and with the slits at 3.0 GHz.

Figs. 6 and 7 provide the comparisons of the reflection coefficients and the 3.0-GHz radiation patterns of three monopole slot antennas without and with two types of slits. The effect of embedding the two slits in the ground plane is apparent on its impedance bandwidth. It is observed that, when the lower slit is rippled, the bottom-sided power is effectively suppressed and the back-lobes at 3.0 GHz are reduced. Fig. 3(b) and (c) show the distributions of the surface current density on the ground plane without and with the embedded slits. Part of the surface currents at the y-axial edges of the ground plane leads to the x-axial slits, and then the direction of the original back radiation is changed from the y-axis to the x-axis; hence, the back-lobe at 8 = 90 and  = 90 is suppressed. The results of the simulated F/B ratios from 1.7 GHz to 3.8 GHz are shown in Fig. 8. When the slits are embedded, the F/B ratios around 1.7 GHz and 3.3 GHz increase significantly.

Fig. 9 exhibits the comparison of the reflection coefficients of the conventional and proposed monopole slot antennas. In accordance with the measured results, the frequency bands of the conventional and proposed antennas range from 2.93 to 4.00 GHz and from 1.68 to 3.93 GHz. As can be observed from the figure, an improvement in impedance matching condition has been achieved, obtaining resonant bandwidths of 97.4% (2.25 GHz for the center frequency at 2.31 GHz). Compared to the conventional monopole slot antenna, it is also seen that the lower resonant frequency of the proposed antenna decreases from 2.93 GHz to 1.68 GHz, a reduction of 1.25 GHz. The proposed monopole slot antenna achieves the twin advantages of size reduction and bandwidth enhancement. Fig. 10 shows the comparison of the measured radiation patterns for the conventional rectangular-slot and the proposed antennas. It is noted that the xz-plane radiation pattern consists of two parts:  = 0 to 180 at 8 = 0 and 8 = 180 . Due to the surface currents distributing along the slits, the cross-polarized radiation in the yz-plane is larger than that in the xy-plane. The radiation patterns in the xz-plane are eight-like due to the radiation phenomenon of the slot antenna. Table III lists the measured results of the F/B ratio for the four types of monopole slot antenna, including the conventional rectangular slot [1], L-slot [2], pentagon slot [3], and the proposed antennas. The positions, labeling front (F) and back (B), are hereby located at the angles

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10 dB at 1.8, 2.4, 3.0 and 3.6 GHz. Hence, compared to other antennas, the proposed antenna achieves unidirectional radiation, similar to the traditional end-fire radiating antennas, such as the Yagi-Uda antenna. Fig. 11 presents the simulated maximum gain and radiation efficiency proposed monopole slot antennas. For most of the operated of the conventional and frequencies, the maximum gain of the proposed antenna is higher than that of the conventional antenna. The lowering of radiation efficiency of the proposed antenna at high operated frequencies may be attributed to the coax loss and the substrate loss. The measures taken to prevent coax feed radiation are not treated. Unbalanced antennas with an unbalanced feedline (coax) can lead to radiation from the outside of the coax. In addition, because of the end-fire (in the y-axis) radiation, most of the power, which is in the form of the surface wave propagation, leaks out along the y-axis, and is attenuated by the substrate material. IV. CONCLUSION

Fig. 10. Comparisons of the measured radiation patterns of the conventional and proposed antennas.

A wideband unidirectional monopole slot antenna has been demonstrated in this communication. Simulated and measured results show that the techniques of using a stub-protruded feeding line, inserting two symmetrical stubs into the radiating slot effectively reduces the resonant frequency, miniaturizes the antenna size, and enhances the impedance bandwidth. Furthermore, by adding a finger stub at the corner of the ground plane, changing the geometry of the ground plane, and embedding slits in the ground plane, the front/back ratio of the monopole slot antenna increases such that the unidirectional radiation characteristics can be achieved. ACKNOWLEDGMENT The authors are grateful to the National Center for High-performance Computing (NCHC) and the Chip Implementation Center (CIC) of the National Applied Research Laboratories, Taiwan, for their simulation software support and facilities.

REFERENCES

Fig. 11. Comparison of the simulated maximum gain and radiation efficiency of the conventional and proposed antennas.

TABLE III RESULTS OF THE MEASURED F/B RATIO FOR THE CONVENTIONAL AND PROPOSED ANTENNAS

of ( = 90 ; 8 = 270 ) and ( = 90 ; 8 = 90 ). It is seen that the F/B ratios of the proposed monopole slot antenna are larger than

[1] S. K. Sharma, N. Jacob, and L. Shafai, “Low profile wide band slot antenna for wireless communications,” in IEEE Antennas Propagat. Soc. Int. Symp. Digest, San Antonio, TX, Jun. 2002, vol. 1, pp. 390–393. [2] S. I. Latif, L. Shafai, and S. K. Sharma, “Bandwidth enhancement and size reduction of microstrip slot antenna,” IEEE Trans. Antennas Propag., vol. 53, pp. 994–1003, Mar. 2005. [3] S. K. Rajgopal and S. K. Sharma, “Investigations on ultra-wideband pentagon shape microstrip slot antenna for wireless communications,” IEEE Trans. Antennas Propag., vol. 57, pp. 1353–1359, May 2009. [4] K.-L. Wong and L.-C. Lee, “Multiband printed monopole slot antenna for WWAN operation in the laptop computer,” IEEE Trans. Antennas Propag., vol. 57, pp. 324–330, Feb. 2009. [5] A. P. Zhao and J. Rahola, “Quarter-wavelength wideband slot antenna for 3–5 GHz mobile applications,” IEEE Antennas Wireless Propag. Lett., vol. 4, pp. 421–424, 2005. [6] Y.-W. Jang, “Experimental characteristics of a printed open-slot antenna with reflector for PCS, DCS, and IMT-2000,” Microw. Opt. Technol. Lett., vol. 41, no. 5, pp. 348–350, Jun. 2004. [7] S. Sugawara, Y. Maita, K. Adachi, and K. Mizuno, “Characteristics of an mm-wave tapered slot antenna with corrugated edges,” IEEE MTT-S IMS Digest, vol. 2, pp. 533–536, Jun. 1998. [8] Y. S. Wang and S. J. Chung, “A short open-end slot antenna with equivalent circuit analysis,” IEEE Trans. Antennas Propag., vol. 58, pp. 1771–1775, May 2010.

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Compact and Tunable Slot-Loop Antenna Pei-Ling Chi, Rod Waterhouse, and Tatsuo Itoh

Abstract—A compact and tunable slot-loop antenna is studied in this communication. By periodically loading the slot line with varactor diodes, the slow-wave loop is capable of operating at frequencies from 2.34 GHz to 4 GHz with the measured input reflection coefficient better than 7 5 dB 0 072 (8 1 mm while occupying a small footprint of 0 072 8 1 mm), where is the guided wavelength of a slot line at 2.34 GHz. The slot-loop in the ground plane is excited by a simple microstrip line, which 2. enables the impedance matching from 3.2 GHz to 4 GHz for VSWR A LC low-pass circuit is configured into the bias network to isolate the RF signal from the DC power path and consistent radiation patterns are experimentally obtained across the entire tunable frequencies of interest. The slot loop combined with the feed line and the bias circuitry is only 40 mm 38 97 mm. Experimental results validate the feasibility of the presented antenna and agree well with simulation data. Index Terms—Compact antenna, slot-loop, tunable antenna.

I. INTRODUCTION Compact and broadband antennas are highly desired nowadays for communication terminals as they occupy limited space and can interface with various communication standards. Miniaturization techniques that increase the effective permittivity or/and permeability of the structure and therefore decrease the guided wavelength are well-known [1], [2]. The development of these slow-wave antennas, however, is usually at the expense of considerably reduced bandwidth, which makes them difficult to be used for multiple systems. For most wireless communication systems, the instantaneous bandwidth is relatively narrow and therefore if we can tune the frequency of the antenna over the entire operational bandwidth (defined as the effective bandwidth), we may be able to meet the somewhat conflicting requirements of a small radiator and a large bandwidth of operation. Thus, an electrically tunable antenna may provide an alternative solution. Over the past few decades, there have been extensive studies on tunable antennas. One class of the tunable antennas takes advantage of PIN diodes to create multiple current paths and therefore resonant frequencies depending on the on or off diode state [3]–[5]. In order to configure multiple paths, tunable antennas using PIN diodes are typically layout-complicated and unable to present continuous frequency tunability. Recently, RF MEMS varactors and switches [6]–[9] find many applications in tunable antennas as they demonstrate great tuning potential in terms of the low power consumption, better linearity and a large capacitance ratio. In spite of these superior features, the fabrication technology is relatively costly and the MEMS devices are therefore not as accessible as the commercial tuning elements. Alternatively, loading varactor diodes to resonant antennas to demonstrate frequency or polarization agility has been widely employed [10]–[15]. In [11], the independent control of orthogonal polarizations using four varactor Manuscript received April 30, 2010; revised July 09, 2010; accepted September 11, 2010. Date of publication January 31, 2011; date of current version April 06, 2011. This work was supported by the ONR STTR Phase II Program, Topic#: N06-T032. P.-L. Chi and T. Itoh are with the Department of Electrical Engineering, University of California at Los Angeles, Los Angeles, CA 90095 USA (e-mail: [email protected]; [email protected]). R. Waterhouse is with Pharad LLC, Glen Burnie, MD 21061 USA (e-mail: rwaterhouse@ pharad.com). 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.2011.2109687

Fig. 1. Schematic of the proposed tunable slot-loop antenna. The microstrip line (dashed line) is on the opposite side of the substrate with respect to the slot loop (bold line). Eight varactor diodes are periodically loaded across the slot in the positions where the gap width is reduced as indicated. All dimensions are in mm.

diodes was presented. By properly choosing the varactor locations, the frequency tuning ratio, the second to the first resonant frequencies, can be significantly increased in dual-band slot antennas [13]. As far as the miniaturization is concerned, embedding one or two varactor diodes to reduce the antenna operating frequency, as is generally observed in the literature [14], [15], is less effective. Based on our previous study [16], a slot loop periodically loaded by capacitors shows ability to attempt considerable size reduction and moreover, can be easily analyzed by applying the periodic boundary condition. As aforementioned, the instantaneous bandwidth is, however, adversely reduced for this capacitor-loaded slow-wave antenna. Therefore, in this communication, a slot loop evenly loaded with eight varactor diodes is applied to create a miniaturized antenna with a considerably increased effective bandwidth. With regard to the impedance matching across the tunable frequencies, the electromagnetically coupled feed using the microstrip line beneath the slot loop is considered to provide more design parameters as compared to the direct CPW feed employed in [16]. Experimental results show the maximum input reflection coefficient of 07:5 dB, ranging from 2.34 GHz to 4 GHz, was achieved. Meanwhile, the effective impedance bandwidth from 3.2 GHz to 4 GHz was obtained for VSWR  2. In order to isolate the RF signal from the bias line, influencing the radiation characteristics, a LC low-pass circuit is included in the bias network in this communication. Furthermore, consistent radiation patterns in the tunable frequency range were observed. The entire structure, including the feed line and bias circuit, occupies an area of 40 mm 2 38:97 mm. Experimental data agree well to the simulation results. II. SLOW-WAVE ANTENNA SCHEMATIC USING VARACTOR DIODES A. Slow-Wave Antenna Fig. 1 shows the proposed varactor-loaded slot-loop antenna. The square slot loop (8:1 mm 2 8:1 mm) is excited by a microstrip line, which is located on the other side of the substrate. In order to enable the antenna to operate over the tunable frequency range, a feed line composed of two sections of different widths was used for more degrees of freedom in optimization. It was found that compared to the CPW-fed slot-loop antenna [16], the microstrip-coupled excitation configuration makes it easier to achieve wideband impedance matching. The slot is mounted with varactors at a 45 load period where the perimeter of the loop is one wavelength at 9 GHz. At the load positions, reduced gap widths are especially arranged in layout for soldering varactors, which are smaller than the slot width itself. The bias circuitry is arranged as follows. To bias the diodes across the slot, the DC power applied to the center patch is injected from the other side through a via-hole whereas

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Fig. 2. Comparison between the slow wave enhancement factor = and miniaturization factor f =f for the proposed tunable slot-loop antenna.

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Fig. 3. The extracted junction capacitance C and series resistance R in the equivalent lumped-element model versus the bias voltage. The parasitic capacitance C and parasitic inductance L are assumed as 0.06 pF and 0.2 nH, respectively.

the surrounding metal plane is grounded. Furthermore, a LC low-pass circuit is used to isolate the RF current flowing on the center patch from the pulled bias line to avoid affecting antenna radiation characteristics from the latter (see Fig. 4). By periodically loading the varactor diodes across the slot, the guided wave is slowed down as a result of an increased effective propagation constant. The propagation constant of a loaded structure in terms of the unloaded propagation constant 0 is as follows [17]:

=

1 cos01 (cos( 0d) 0 fCZ0 sin( 0 d))

d

(1)

where f , Z0 , d and C are the frequency, characteristic impedance of the host structure, load period and load capacitance, respectively. The ratio of the propagation constant to 0 , or the slow wave enhancement factor = 0 , was shown in good agreement with the miniaturization factor in [2]. In this communication, given the fixed load period, the degree of the miniaturization is determined by the capacitance tuning ratio of varactors. Fig. 2 plots the slow wave enhancement factor versus the available varactor capacitance Ct compared with the respective miniaturization factors of the slot loop loaded with equivalent varactor capacitances. Please note miniaturization is defined as the ratio of the unloaded to the loaded loop resonant frequencies, fr0 =fr and is calculated in the full-wave simulation. The varactor lumped-element model was embedded into the simulation environment and carried out using the Ansoft Designer 4.0. As observed from Fig. 2, good agreement is achieved and moreover, approximately twice the resonant frequency can be reduced from Ct  0:2 pF to Ct  1:4 pF, which is the case for the proposed slot-loop antenna with frequency tunable range from 2.34 GHz to 4 GHz. B. Diode Modeling In order to improve the simulation accuracy, the characterization technique provided in [18] is applied to extract the equivalent lumpedelement parameters of a varactor diode. The Microsemi MPV 2100 surface mount varactor diode is used [19]. From the manufacturer’s datasheet, the parasitic inductance Lp and parasitic capacitance Cp are assumed as 0.2 nH (the worst case) and 0.06 pF in the model. Instead of mounting the diode at the open-end of a known transmission line, which is ordinarily used for diode characterization in the past, the proposed configuration in [18] especially incorporates the impedance matching circuit to improve the extraction accuracy. Because of the close proximity to the edge of the Smith chart, the real part of the impedance of the varactor is very sensitive to the calibration plane where the diode is mounted, which increases the possibility of extraction error based on the former implementation. Furthermore, while the DC voltage is applied to the cathode of the diode through the high

Fig. 4. Photographs of the fabricated tunable slot-loop antenna. (a) Top view of the structure showing the microstrip feed line and the LC low-pass bias network (the detail is shown in the inset), and (b) back view of the structure showing the slot loop periodically loaded by the 8 varactor diodes.

impedance line in the bias network, the impedance matching circuit in [18] provides the ground voltage to the anode from the shorted stub. Thus, a bias tee is not required in this bias scheme. Fig. 3 shows the extracted junction capacitance Cj and series resistance Rs biased at several voltage points. The equivalent lumped-element model is included as an inset. As observed, the junction capacitance and series resistance are decreased with the bias voltage as expected for the reverse-biased varactor diodes. III. SIMULATION, EXPERIMENTS AND DISCUSSIONS The proposed varactor-loaded slot-loop antenna was built on a Duroid 5880 substrate of r = 2:2 and thickness 1.57 mm. Fig. 4 shows photographs of the fabricated antenna. Eight MPV 2100 varactor diodes are periodically mounted across the slot. Two wires are pulled from the center patch (with the L = 22 nH and C = 8 pF low-pass circuit in-between) and the ground plane, respectively, to bias the diodes. The entire footprint is 40 mm238:97 mm. The return loss measurement was taken with an Agilent 8515A network analyzer and is illustrated in Fig. 5. From 4 volts to 22 volts (close to the breakdown

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TABLE I CALCULATED RADIATION EFFICIENCY FOR THE PROPOSED TUNABLE ANTENNA UNDER DIFFERENT BIAS VOLTAGES

Fig. 5. The measured jS

j

for the proposed tunable slot-loop antenna.

Fig. 6. Comparison between measured and simulated reflection coefficients j for the proposed tunable slot-loop antenna biased at (a) 22 volts, (b) 20 jS volts, (c) 15 volts, and (d) 4 volts.

Fig. 7. Measured radiation patterns for the proposed tunable slot-loop antenna biased at (a) 20 volts, (b) 15 volts, (c) 10 volts, and (d) 4 volts.

voltage VB ), the resonant frequency is tuned from 2.34 GHz to 4 GHz with better than 07:5 dB reflection coefficient. The measured impedance bandwidth for VSWR  2 is about 22.2%, ranging from 3.2 GHz to 4 GHz. Furthermore, simulated and measured results are compared in Fig. 6. The antenna reflection coefficients for varactor diodes biased at 4 volts, 15 volts, 20 volts and 22 volts are presented in this figure. Simulation results show that the impedance matching better than 014 dB was obtained across the tunable frequency range from 1.87 GHz to 4.14 GHz, which manifests the advantage of the chosen excitation configuration. Compared to the simulation results, good agreement is observed for the Vdc  15 volts. The discrepancy at 4 volts may be ascribed to two reasons. First, at the resonant frequency (2.34 GHz) when 4-volts reverse bias is applied, the slot-loop antenna is electrically small and therefore the measurement of an electrically small antenna, as is known, is susceptible to nearby objects. Please note that, an electrically small antenna (as well as the radiation capability) is determined by its physical dimension with respect to the free space wavelength at frequency of interest and thus an antenna given a fixed physical size will become electrically smaller as the operating or resonant frequency decreases. On the other hand, the resonance occurs as long as the resonant condition is satisfied for the antenna resonator at the frequency, which can be decreased significantly by artificially loading the radiator itself such as the substrate loading and reactive element loading. Our proposed tunable antenna (or any other

slow-wave antenna in the literature) is such an example. By simultaneously applying both loading approaches, the resonant frequency is decreased considerably at the expense of degraded radiation efficiency (see Table I). Second, the error tolerance of the parameter extraction at lower bias voltage is increased. Please note that in Fig. 3 the slopes of Cj and Rs curves are steeper at lower bias voltages. Normalized radiation patterns were measured and are shown in Fig. 7. The radiation patterns were measured in an anechoic chamber. As observed, the typical radiation patterns (solid curve for the H-plane and dashed curve for the E-plane) for the slot-loop antenna are maintained across the frequency tunable range. The E-plane and H-plane of the slot loop is parallel to the xz and yz planes in Fig. 1, respectively. Furthermore, the pattern ripples in the measurement may be due to the bias wires. The measured maximum gain versus the bias voltage is plotted in Fig. 8. The radiation gain is lowered with the decreased bias voltage when the varactor capacitance is increased. Three reasons may contribute to this outcome. First, the antenna itself becomes electrically smaller and the antenna radiation capability is deteriorated as a result [20]. Second, the resistance loss from the varactor diode increases at lower bias voltages (see Fig. 3). Third, the increased return loss when biased at a lower voltage degrades the antenna gain. The measured maximum gain at 20 volts is 1.71 dBi while the value drops to 08:25 dBi at 4 volts. Compared to numerical calculation, the measured gains are less by approximately 1.6 dBi. The fabrication

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Fig. 8. Measured and simulated radiation gains for the proposed tunable slotloop antenna.

inaccuracy and measurement error might contribute to the discrepancy. The antenna radiation efficiency was calculated and tabulated in Table I, where the decreasing radiation efficiency at the lower resonant frequencies indicates the electrically smaller size of the loop antenna. Since the measured antenna gain is less than its simulation counterpart, it should be noted that, the measured radiation efficiency is expected to be lower than the simulated result. IV. CONCLUSION A compact and tunable slot-loop antenna is studied in this communication. By loading varactor diodes periodically along the loop perimeter and utilizing a simple microstrip feed line on the back side of the substrate, the proposed slot-loop antenna presents the frequency tunable range from 2.34 GHz to 4 GHz with the maximum input reflection coefficient of 07:5 dB while occupying a small loop area of 0:072 g 2 0:072 g (8:1 mm 2 8:1 mm), where g is the guided wavelength of a slot line at 2.34 GHz. The measured impedance bandwidth for VSWR  2 is about 22.2%, ranging from 3.2 GHz to 4 GHz. Meanwhile, consistent radiation patterns are obtained in the entire tunable frequencies. Experimental results verify the feasibility of the proposed tunable antenna. ACKNOWLEDGMENT The authors gratefully acknowledge the support of and technical discussions with, D. Arceo, J. Rockway and J. Allen from SPAWAR, San Diego.

REFERENCES [1] C. R. Rowell and R. D. Murch, “A capacitively loaded PIFA for compact mobile telephone handsets,” IEEE Trans. Antennas Propag., vol. 45, no. 5, pp. 837–842, May 1997. [2] P.-L. Chi, R. Waterhouse, and T. Itoh, “Antenna miniaturization using slow wave enhancement factor from loaded transmission line models,” IEEE Trans. Antennas Propag., vol. 59, no. 1, pp. 48–57, Jan. 2011. [3] A.-F. Sheta and S. F. Mahmoud, “A widely tunable compact patch antenna,” IEEE Antennas Wireless Propag. Lett., vol. 7, pp. 40–42, 2008. [4] M. Komulainen, M. Berg, H. Jantunen, E. T. Salonen, and C. Free, “A frequency tuning method for a planar inverted-F antenna,” IEEE Trans. Antennas Propag., vol. 56, pp. 944–950, Apr. 2008.

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[5] S. Nikolaou, R. Bairavasubramanian, C. Lugo, Jr, J. I. Carrasquillo, D. C. Thompson, G. E. Ponchak, J. Papapolymerou, and M. M. Tentzeris, “Pattern and frequency reconfigurable annular slot antenna using PIN diodes,” IEEE Trans. Antennas Propag., vol. 54, no. 2, pp. 439–448, Feb. 2006. [6] K. Topalli, E. Erdil, O. A. Civi, S. Demir, S. Koc, and T. Akin, “Tunable dual-frequency RF MEMS rectangular slot ring antenna,” Sens. Actuators A: Phys., vol. 156, no. 2, pp. 373–380, Dec. 2009. [7] N. Kingsley, D. E. Anagnostou, M. Tentzeris, and J. Papapolymerou, “RF MEMS sequentially reconfigurable Sierpinski antenna on a flexible organic substrate with novel DC-biasing technique,” J. Microelectromech. Syst., vol. 16, no. 5, pp. 1185–1192, Oct. 2007. [8] J. H. Schaffner, R. Y. Loo, D. F. Sievenpiper, F. A. Dolezal, G. L. Tangonan, J. S. Colburn, J. J. Lynch, J. J. Lee, S. W. Livingston, R. J. Broas, and M. Wu, “Reconfigurable aperture antennas using RF MEMS switches for multi-octave tunability and beam steering,” in Proc. IEEE AP-S Int. Symp., Salt Lake City, UT, 2000, pp. 321–324. [9] M. Nishigaki, T. Nagano, T. Miyazaki, T. Kawakubo, K. Itaya, M. Nishio, and S. Sekine, “Piezoelectric MEMS variable capacitor for a UHF band tunable built-in antenna,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2007, pp. 2079–2082. [10] L. Huang and P. Russer, “Electrically tunable antenna design procedure for mobile applications,” IEEE Trans. Microw. Theory Tech., vol. 56, no. 12, pp. 2789–2797, Dec. 2008. [11] 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. [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] 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. [14] K. L. Virga and Y. Rahmat-Samii, “Low-profile enhanced-bandwidth PIFA antennas for wireless communications packaging,” IEEE Trans. Microw. Theory Tech., vol. 45, no. 10, pp. 1879–1888, Oct. 1997. [15] B. R. Holland, R. Ramadoss, S. Pandey, and P. Agrawal, “Tunable coplanar patch antenna using varactor,” Electron. Lett., vol. 42, no. 6, pp. 319–321, Mar. 2006. [16] P.-L. Chi, K. Leong, R. Waterhouse, and T. Itoh, “A miniaturized CPW-fed capacitor-loaded slot-loop antenna,” in Proc. IEEE ISSSE Int. Symp., Aug. 2, 2007, pp. 595–598. [17] D. M. Pozar, Microwave Engineering, 2nd ed. New York: Wiley, 1998. [18] G. H. Stauffer and R. Collins, “Finding the lumped element varactor diode model,” High Frequency Electronics, pp. 22–28, Nov. 2003. [19] Monolithic Microwave Surface Mount Varactor Diodes (MPV1965, MPV 2100). Lowell, MA, Microsemi Microwave Products Division [Online]. Available: http://www.microsemi.com/datasheets/ msc1598.pdf [20] W. L. Stutzman and G. A. Thiele, Antenna Theory and Design, 2nd ed. New York: Wiley, 1998.

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Design of SIW Cavity-Backed Circular-Polarized Antennas Using Two Different Feeding Transitions Dong-Yeon Kim, Jae W. Lee, Taek K. Lee, and Choon Sik Cho

Abstract—Two circular-polarized circular patch antennas which have novel feeding structures such as a substrate integrated waveguide (SIW), a cavity-backed resonator and two different feeding transitions, are proposed and experimentally investigated in terms of electrical performances, including reflection coefficients, optimized parameter values, circular polarized antenna gain, axial ratios and radiation patterns. By inserting asymmetrical inductive via arrays into the interface region between the circular patch and SIW feeding structure, it is found that an enhancement of input impedance bandwidth has been achieved. In addition, in order to check the effects of feeding transition types on the electrical performances of the main radiator, two different feeding transitions, namely microstrip-to-SIW and coax-to-SIW, have been studied by considering reflection coefficients, gain, axial ratios and radiation patterns. As a result, it is experimentally proved that a broadband impedance bandwidth of 17.32% and 14.42% under the criteria of less than VSWR 2:1 and 1.5:1, respectively, have been obtained and an RHCP axial ratio of 2.34% with a maximum gain of 7.79 dBic has been accomplished by using the proposed antenna with coax-to-SIW transition operating at the X-band of 10 GHz center frequency. Index Terms—Asymmetric inductive diaphragm, cavity-backed resonator, circular polarization, sequential feeding, substrate integrated waveguide (SIW).

I. INTRODUCTION Until now, a variety of satellite antennas have been designed and suggested because of their importance in satellite communication. The general requirements for satellite antennas are light weight, stabilized gain, good return loss characteristics and a compatible radiation pattern to narrow/wideband electromagnetic interferences caused by unexpected aerospace environmental phenomena. First and foremost, circularly polarized characteristics in antennas are the most important in many wireless communication systems, including in satellite communication. In order to generate circular polarization (CP) in an antenna, a perturbed circular patch and microstrip [1], [2]/aperture coupled feeding [3] has been conventionally used at the expense of transmission line loss. Moreover, array antennas employing sequentially rotated feeding networks have been implemented to improve total antenna gain, the purity of axial ratio and axial ratio bandwidth [4]. In [5] considering the structure of 2 2 16 phased array antennas suitable for the generation of circular polarization at Ku-band, it is conjectured that the axial ratio and feeding losses of each element have been improved by applying the notch and stub and using a hollow metallic rectangular waveguide, Manuscript received November 03, 2009; revised June 13, 2010; accepted September 14, 2010. Date of publication January 31, 2011; date of current version April 06, 2011. This work was supported by the NSL (National Space Lab) program through the Korea Science and Engineering Foundation funded by the Ministry of Education, Science and Technology (no. S10801000159-08A010015910). D.-Y. Kim is with the Institute of New Media Communication, School of Electrical Engineering and Computer Science, Seoul National University, Seoul 151-742, Korea (e-mail: [email protected]). J. W. Lee, T. K. Lee and C. S. Cho are with the School of Electronics, Telecommunication and Computer Engineering, Korea Aerospace University, Goyang, Gyeonggi-do, Korea (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.2011.2109675

respectively. However, as a drawback, the structure proposed in [5] is a little bulky. Hence, in order to complement this shortcoming in terms of the bulky feeding structure, a feeding network adopting partially substrate integrated waveguide (SIW) technology has been introduced to replace the conventional metallic waveguide [6]. Nevertheless, the microstrip line, which may lead to distortion and degradation in radiation characteristics, still exists for sequentially rotated feeding. Consequently, single or array antennas fed by well-established SIW transmission lines or sequentially rotated feeding networks are required to minimize the distortion of radiation characteristics. In general, SIW is well known as a representative transmission line with low loss replacing the conventional microstrip line which causes performance degradation due to the co-existence in the same plane with the antenna radiator in terms of reflection coefficient and axial ratio. In addition, it has some advantages in terms of its high power signal transfer and miniaturization with the help of the rectangular waveguide type and easy fabrication on PCB, respectively [7]. As a good candidate for low loss antennas, various types of LP- and CP-generating antennas using SIW technology have been developed and studied [8], [9] and [12]. The authors in [8], [9] have suggested an X-band LP-generating planar slot antenna and RHCP antenna generating two orthogonal cavity modes as a function of lengths of two crossed slot arms. Even though the previously suggested structures in [8], [9] have adopted GCPW feeding networks and obtained good electrical performances, they have had some drawbacks such as a narrow impedance bandwidth of less than 3%. On the other hand, the authors of this work have already published work regarding a CP-generating antenna using an SIW-based circular ring-slot as a single element [12]. In the proposed structure in [12], the RHCP generation has been obtained as a result from the separation of the single resonant mode to dual resonant modes with two orthogonal zero-potential planes having a phase difference of 90 . At the same time, a wide impedance bandwidth of 18.74% (VSWR 2:1) has been achieved through the use of an asymmetric inductive diaphragm embedded into the SIW feeding part. As another method for increasing the impedance bandwidth, reflection-canceling posts and slots have been introduced in [10], [11]. In this communication, an RHCP-generating SIW single antenna suitable for an element of sequentially rotated phased array antennas is proposed for high data transfer rate in satellite communication. Particularly, the proposed SIW antenna with a cavity-backed resonator is unified into the SIW feeding network with two different transitions by differentiating from the conventional single antenna using a microstrip feeding line. II. PROPOSED ANTENNA GEOMETRY AND DESIGN PROCEDURES On the contrary to a single antenna element employing a conventional microstrip feeding line, an SIW-fed structure, which leads to an enhancement of the shielding effect protecting from the external noise and induces a stabilization of radiation patterns, is proposed by using metallic via arrays acting as a perfect electric wall (PEW). The proposed antenna geometry shown in Fig. 1 is mainly composed of three parts. The first one is a combination of a conventional perturbed circular patch with a shorted-end via (or probe) and an SIW cavity where the former is essential for the generation of circular polarization and the latter is for a higher gain and a lower backlobe radiation. The second one is an SIW-fed impedance matching network integrated with an asymmetric inductive diaphragm using via arrays. The role of this impedance matching network has been already clarified in [12]. The last one is a microstrip-to-SIW transition in Fig. 1(a) or coax-to-SIW transition in Fig. 1(b) essential for measurements. The total occupying

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TABLE I PROPOSED ANTENNA PARAMETERS AND OPTIMIZED VALUES ACCORDING TO THE TRANSITION TYPES

Fig. 1. Proposed antenna geometry using (a) microstrip-to-SIW transition and (b) coax-to-SIW transition.

area including transition region is 35 mm 2 45:2 mm. The entire antenna structure has been designed by employing an RT/Duroid 5880 substrate as a single layer having a relative permittivity of 2.2, loss tangent of 0.0009 at 10 GHz and a thickness of 1.57 mm. The final optimized parameter values according to the transition types are listed in Table I. A. Radiating Element for Generating Circular Polarization The used circular patch operating at 10 GHz with the RHCP characteristic is proposed and designed by using a conventional equation [13], resulting in 5.2 mm as a radius (r ). The roughly estimated initial value of the main radius has been optimized to 4.65 mm with the help of commercially available software, CST MWS based on a finite-difference time-domain algorithm [14]. The finally optimized value includes the coupling effects between the upper metal plate of the SIW feeding structure and the circular patch mounted on the upper patch. From the reflection characteristics of the CP-generating circular patch proposed in this communication, it seems that the dual resonance is due to the separation into the dual resonant TM11 -mode at the nearest resonant frequencies of the single resonant TM11 -mode generated by the conventional perturbed ratio (1s=S ) of the circular patch. This phenomenon can be explained through the electric field distributions inside the cavity as shown in Fig. 2. As a detailed description, at first, Fig. 2(a) depicts the electric field distributions at

Fig. 2. Electric field distributions inside the cavity at each resonant frequency (a) the first resonant TM11-mode at 9.89 GHz and (b) the second resonant TM11-mode at 10.58 GHz.

9.89 GHz as a lower resonant frequency and shows the zero-potential plane formed at azimuthal angles () of 45 and 225 . On the other hand, it is conjectured that the upper resonant frequency occurs at 10.58 GHz as shown in Fig. 2(b) due to the reduced size. At this time, the zero-potential plane is located at 135 and 315 in an azimuthal angle. Hence, the two zero-potential planes are orthogonal and form a phase difference of 90 each other. This phase difference leads to an RHCP rotation of the electric field inside the cavity and a maximum purity of axial ratio around 10 GHz as a center frequency. The perturbed design parameters, w1 and w2 , determining the generation of dual TM11 -modes occupy the total area 1s = 2 (w1 1 w2 ) = 2(w1 )2 =x = 2x 1 (w2 )2 and govern the ratio of the perturbed and original circular patch area (k = 1s=S ). The parameter indexes x and k defined as x = w1 =w2 and k = 1s=S are essential for accomplishing the electrical performances such as the lower reflection coefficients, an enhancement of the optimized axial ratio bandwidth and purity improvement and could be optimized to 1.357 when w1 = 1:92 mm and w2 = 1:415 mm and 0.08, respectively. As another critical parameter for generating RHCP, a shorted-end via connected between the circular patch on the upper plate and ground plane on the lower plate is located at a position 0.5 mm away from the center of the circular patch to the negative y direction with a via diameter of 0.3 mm such that zero-reference planes indicating zero-field points in a SIW cavity constitutes 90 in a rotational angle. In addition, the circular array vias (j ) replacing a metalized circular-cylindrical

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cavity has been adopted for performance improvement in radiation patterns and gain. Finally, the distance, s, between the circular patch and the upper metal plate of SIW cavity has been controlled for obtaining the minimum effect on the self-resonance of the circular patch and optimized by an EM full-wave simulator. B. SIW Feeding Structure With Embedded Impedance Matching Circuitry The width, a1 , of the SIW feeding structure within the operating frequency regime from 8.2 to 12.4 GHz has been determined as 16 mm by setting the cutoff frequency of the fundamental mode (TE10 -mode) to approximately 6.3 GHz. Moreover, in order to minimize the leakage property, including radiation loss through side walls, conduction loss and dielectric loss due to the imperfect manufacturing process of metalized via arrays, the diameter (v1 ) of vias and the distance (d1 ) between vias have been optimized to 1 mm and 1.5 mm, respectively, by considering the conditions of d1 =v1 < 2:5 and v1 =a1 < 1:8 [15]. Consider the antenna configuration with the traveling wave-type SIW feeding structure as shown in Fig. 1. The important parameters for this configuration are the pattern parameters, a4 and h3 , etched on the upper plane and the distance, h1 , of the SIW feeding line from each transition. Especially, since the fundamental TE10 -mode of SIW and the resonant TM11 -mode of the perturbed circular patch are different from each other, the optimized parameters considering the smooth energy transition are required to satisfy the matched impedance characteristics. Figs. 3(a) and 3(c) show the reflection coefficients or impedance matching characteristics according to the variations of h1 and h3 , respectively. As a result, it is obtained that the optimized parameters of the proposed antenna using microstrip-to-SIW transition h1 and h3 are 9.6 mm and 3.292 mm, respectively, for achieving the maximum impedance bandwidth under the condition of less than VSWR 1.5:1 and an improvement of more than 10 dB with respect to the reflection coefficients. As a function of additional impedance matching to overcome the narrow bandwidth characteristic due to the microstrip patch antenna, metallic via arrays with parameters d2 and v2 acting as an asymmetric inductive diaphragm with zero thickness have been introduced, inserted and unified into the SIW feeding structure. Fig. 3(b) shows the impedance matching characteristics according to the parameter h2 which is the distance between the asymmetrical inductive via arrays and the feeding line as shown in Fig. 1(a). In a similar way, the parameter h2 in the case of coax-to-SIW transition as shown in Fig. 1(b), showing the distance between asymmetrical inductive via arrays and the back-shorted via arrays, can be optimized. The asymmetric inductive diaphragm in Fig. 4(b) can be modeled by a shunt inductor in a two-port network and the magnitude of susceptance (Z0 =X ) is controlled by the width of the diaphragm (d0 in Fig. 4(a)) [16]. In addition, the width of the diaphragm is dependent on the number of via arrays and the separate distance (d2 ). Here, Z0 means the wave impedance of the rectangular waveguide. As shown in Fig. 3(d) and (e), the matching condition at the antenna input port can be obtained from the variation of the width of the SIW-based asymmetric inductive diaphragm. In practical situations, to realize the via arrays with radii as small as possible to take into account the theoretical via effect of zero thickness as suggested in [16], v2 , the diameter of via has been set up to be 0.3 mm. On the other hand, Fig. 5 delineates the input impedance variation at the feeding point according to the number of via arrays. From the impedance loci within the frequency range (8.79 to 10.84 GHz), it is seen that the inductive characteristic is dominant and the loci move to the center point. As an optimized via array, a case of 7 EA vias shown in Fig. 5 has been selected and makes the input impedance locus entirely

Fig. 3. Impedance matching characteristics according to the variation of (a) h , (b) h , (c) h , (d) d , (e) the number of via arrays consisting of asymmetric inductive diaphragm in Fig. 1 (the other parameter values are fixed as in Table I).

move to the inside of a circle satisfying VSWR = 2. The physical behavior means that the total width of via arrays plays a role in the

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Fig. 4. Embedded asymmetric inductive diaphragm (a) SIW impedance matching circuitry and its equivalent metallic rectangular waveguide model. (b) Equivalent circuit.

Fig. 6. The reflection coefficients and simulated total efficiency of two proposed antennas using (a) microstrip-to-SIW transition and (b) coax-to-SIW transition, respectively.

Fig. 5. Input impedance variation of proposed antenna according to the number of asymmetric inductive diaphragm via.

generation of additional susceptance to the equivalent input circuit and making the imaginary part of input impedance zero. As a result, it is ensured that the optimum solution for input impedance matching could be obtained from the variations of the number of via arrays and the control of distance between the nearest vias as shown in Figs. 3(d) and 3(e). Furthermore, the optimum position of SIW-based matching circuitry comprising via array is mainly dependent on the design parameter h2 as shown in Fig. 1 and has been determined from the parametric studies using an FDTD-based commercially available simulator. At this time, the simulated results in Fig. 3 have been computed according to the variation of interested parameters in each plot by fixing the other parameters as the same as the values listed in Table I. C. Two Types of Transition for Measurements In order to extract the electrical characteristics of only the proposed antenna, it is necessary to design the transition having a lower return loss and higher transfer characteristics without the distortion in radiation properties. Two types of transitions, microstrip-to-SIW and

coax-to-SIW, have been proposed and implemented for measurement [17], [18]. In the case of microstrip-to-SIW transition transforming quasi TEM-mode into SIW fundamental TE10 -mode as shown in Fig. 1(a), the antenna parameters of a2 and h4 for transition have been optimized for the proper transfer characteristics and summarized in Table I. On the other hand, in order to obtain the optimized impedance matching and an enhancement of impedance bandwidth, the capacitive ring slot shown in Fig. 1(b) has been inserted into a lower conductor plate of the PCB substrate with diameter r1 , equal to 3.2 mm in the case of coax-to-SIW transition [18]. As a critical element for easy power transition, the feeding point h6 of the SMA connector shown in Fig. 1(b) should be located at g =4 away from the back-shorting via arrays preventing the reflected power from being cancelled, where g means the guided wavelength. III. RESULTS AND DISCUSSION The reflection coefficients of the proposed antennas with two different feeding structures have been simulated and measured by using an EM full-wave simulator and Agilent vector network analyzer (N5230A) in a laboratory environment. Fig. 6 shows a good agreement between the simulated and measured data in both of the cases using the two different feeding structures. In addition, the simulated total efficiency is suggested in Fig. 6 with the measured reflection coefficients. In both cases, the total radiating efficiency is more than 90% by taking the finite conductivity of the copper and the loss tangent of the considered dielectric material during the simulation process.

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Fig. 8. Radiation patterns of the proposed antenna using microstrip-to-SIW transition at 10.3 GHz (a) yz plane (b) zx plane.

Fig. 7. Measured right-handed circular polarized (RHCP) gain and axial ratio results of two proposed antennas, respectively.

In the case of using microstrip-to-SIW transition, as in Fig. 1(a), it is seen from Fig. 6(a) that the simulated impedance bandwidths are 2.06 GHz ranging from 8.87 to 10.93 GHz and 1.25 GHz raging from 9.06 to 10.31 GHz, respectively, under the criteria of VSWR 2:1 and 1.5:1, whereas the measured results are 2.33 GHz (from 8.82 to 11.15 GHz) and 1.36 GHz (from 9.02 to 10.38 GHz), respectively, under the conditions of less than VSWR 2:1 and 1.5:1. On the other hand, Fig. 6(b) shows that in the case of using coax-to-SIW transition, as in Fig. 1(b), the simulated (measured) impedance bandwidths are 1.815 GHz (1.82 GHz) and 1.6 GHz (1.52 GHz), respectively, under the same conditions listed above. As additional aspects to antenna performances, the axial ratios of RHCP and gain characteristics have been treated in Fig. 7. As a reference in CP purity, the simulated data using only a waveguide port without transition, which is impractical in real situations and results in 03 dB axial ratio bandwidth ranging from 9.94 to 10.18 GHz, have been included for comparison in Fig. 7. As expected from Figs. 7 and 8, the maximum axial ratio of the SIW antenna using microstrip-to-SIW transition amounts to approximately 03:4 dB at 10.3 GHz and leads to the degradation of purity in RHCP generation. It is conjectured that one of the reasons is due to the generation of higher-order modes from the wide line width caused by employing the substrate of lower permittivity and relatively higher thickness. As another reasonable effect on CP purity, the co-existence of a soldered SMA connector on the input port and circular patch radiator positioned in the same direction on the upper plane may have a possibility of affecting the radiation pattern and axial ratio characteristics. As a result, the enhanced purity of axial ratio and stable high gain more than 6 dBic over the operating bandwidth with maximum RHCP gain, 7.79 dBic at 9.85 GHz, can be obtained by using coax-to-SIW transition as shown in Fig. 7. From Figs. 7 and 9, which show the measured axial ratio and radiation patterns, it is seen that the maximum axial ratio and 03 dB bandwidth in axial ratio of the proposed antenna using coax-to-SIW transition are 01:3 dB at 10.2 GHz and 240 MHz ranging from 10.13 to 10.37 GHz, respectively, which are almost similar to the simulated data. In addition, Figs. 8 and 9, which show the radiation patterns according to the transition types, represent that the proposed antenna employing coax-to-SIW transition has more stable RHCP radiation patterns and a lower level in cross-polarization than that using microstrip-to-SIW transition. However, as a disadvantage of the proposed antenna radiator without transition structure, it is noticed that the beam-tilting phenomenon in yz plane should occur because of the mismatch at the boundary between the SIW transmission line and probe in the circular patch.

Fig. 9. Radiation patterns of the proposed antenna using coax-to-SIW transition at 10.2 GHz (a) yz plane and (b) zx plane.

IV. CONCLUSION A circularly polarized antenna with a unified integration of matching feeding structure and SIW-typed cavity in a single layer has been suggested and implemented in this communication. Particularly it is shown that the low loss characteristic of the feeding line and broadband impedance bandwidth of the proposed antenna have been accomplished by using an SIW structure compared to the conventional microstrip transmission line. In addition, in order to improve the reflection coefficients remarkably, the asymmetric inductive diaphragm consisting of via arrays has been adopted as a control of input susceptance. In terms of input impedance bandwidth, it is found that the optimized antenna results in 17.32% and 14.42% as a percentage bandwidth under the criteria of less than VSWR 2:1 and 1.5:1, respectively. From the results of comparison of the radiation patterns in yz- and zx-plane, it is seen that the cross-polarization level of the coax-to-SIW transition case is lower than that of the microstrip-to-SIW transition case. Furthermore, it is guaranteed that the single element and the low-loss feeding network using SIW can be easily extended to array systems and low loss multi-port feeding networks in satellite communication.

REFERENCES [1] M. Haneishi, S. Yoshida, and N. Goto, “A broadband microstrip array composed of single-feed type circularly polarized microstrip antennas,” in Proc. IEEE Antennas and Propag. Symp., May 1982, vol. 20, pp. 160–163. [2] W.-K. Lo, C.-H. Chan, and K.-M. Luk, “Bandwidth enhancement of circularly polarized microstrip patch antenna using multiple L-shaped probe feeds,” Microw. Opt. Technol. Lett., vol. 42, no. 4, pp. 263–265, Aug. 2004. [3] A. Ghiotto, M. Bourry, and K. Wu, “Cross-slot coupled elliptical patch antenna circularly polarized for localization,” Microw. Opt. Technol. Lett., vol. 49, no. 2, pp. 336–339, Feb. 2007.

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[4] P. S. Hall, “Application of sequential feeding to wide bandwidth, circularly polarized microstrip patch arrays,” in Inst. Elect. Eng. Proc. H, Oct. 1989, vol. 136, pp. 390–398. [5] M. Shahabadi, D. Busuioc, A. Borji, and S. Safavi-Naeini, “Low-cost, high-efficiency quasi-planar array of waveguide-fed circularly polarized microstrip antennas,” IEEE Trans. Antennas Propag., vol. 53, no. 6, pp. 2036–2043, Jun. 2005. [6] A. Borji, D. Busuioc, and S. Safavi-Naeini, “Efficient, low-cost integrated waveguide-fed planar antenna array for Ku-band applications,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 336–339, 2008. [7] Y. J. Cheng, K. Wu, and W. Hong, “Power handling capability of substrate integrated waveguide interconnects and related transmission line systems,” IEEE Trans. Adv. Packag., vol. 31, pp. 900–909, Nov. 2008. [8] G. Q. Luo, Z. F. Hu, L. X. Dong, and L. L. Sun, “Planar slot antenna backed by substrate integrated waveguide cavity,” IEEE Antennas Wireless Propag. Lett., vol. 7, pp. 236–239, 2008. [9] G. Q. Luo, Z. F. Hu, Y. Liang, L. Y. Yu, and L. L. Sun, “Development of low profile cavity backed crossed slot antennas for planar integration,” IEEE Trans. Antennas Propag., vol. 57, no. 10, pp. 2972–2979, Oct. 2009. [10] S. Park, Y. Okajima, J. Hirokawa, and M. Ando, “A slotted post-wall waveguide array with interdigital structure for 45 linear and dual polarization,” IEEE Antennas Propag., vol. 53, no. 9, pp. 2865–2871, Sep. 2005. [11] J. Hirokawa and M. Ando, “45 linearly polarized post-wall waveguide- fed parallel-plate slot arrays,” Proc. Inst. Elect. Eng. Microw. Antennas Propag., vol. 147, pp. 515–519, Dec. 2000. [12] D. Kim, J. W. Lee, C. S. Cho, and T. K. Lee, “X-band circular ringslot antenna embedded in single-layered SIW for circular polarization,” Electron. Lett., vol. 45, no. 13, pp. 668–669, Jun. 2009. [13] C. A. Balanis, Antenna Theory: Analysis and Design. Hoboken, NJ: Wiley, 2005, ch. 14. [14] CST Microwave Studio (MWS) CST Corporation, 2008 [Online]. Available: http://www.cst.com [15] M. Bozzi, M. Pasian, L. Perregrini, and K. Wu, “On the losses in substrate integrated waveguides,” in Proc. 37th Eur. Microwave Conf., Munich, Germany, Oct. 2007, pp. 384–387. [16] N. Marcuvitz, Waveguide Handbook. New York: McGraw-Hill, 1951. [17] D. Deslandes and K. Wu, “Integrated microstrip and rectangular waveguide in planar form,” IEEE Microwave Wireless Compon. Lett., vol. 11, no. 2, pp. 68–70, Feb. 2001. [18] K. Song, Y. Fan, and Y. Zhang, “Eight-way substrate integrated waveguide power divider with low insertion loss,” IEEE Trans. Microwave Theory Tech., vol. 56, pp. 1473–1477, Jun. 2008.

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UWB Dielectric Resonator Antenna Having Consistent Omnidirectional Pattern and Low Cross-Polarization Characteristics Kenny Seungwoo Ryu and Ahmed A. Kishk

Abstract—A new simple compact monopole type dielectric resonator antenna (DRA) for ultrawideband (UWB, 3.1-10.6 GHz) applications is presented. The design combines the advantages of small size DRA and thin planar monopole antennas. The design provides high-radiation efficiency, consistent omnidirectional characteristics, and low cross polarization within the entire band. The antenna size is 15 33 mm with 5.08 mm thickness. The dielectric resonator (DR) is shaped to house the excitation feed and the dielectric substrate is cut to house the DR. The coplanar waveguide is used to feed the antenna.

2

Index Terms—Dielectric resonator (DR) antenna, low cross-polarization level, omnidirectional pattern, ultrawideband (UWB) antenna.

I. INTRODUCTION

Recently, many UWB antennas have been proposed [1]–[6], since the Federal Communication Commission (FCC) allowed 3.1–10.6 GHz unlicensed band for UWB communication. Challenges of the feasible UWB antenna design include the UWB performance issues of the sufficient impedance matching bandwidth, the compact antenna size, high radiation efficiency, avoiding the interference problem of the nearby communication band, constant gain, constant group delay or linear phase, and getting a consistent uniform radiation pattern to avoid undesirable distortions of the radiated and received pulse [1]. Lots of efforts have been performed to improve the performance of the UWB antenna especially to avoid the interference problem [2]–[4]. However, among them, the radiation pattern problems of the UWB antenna still need to be solved, because omnidirectional or near-omnidirectional radiation patterns (defined for [7] to be gain variations less than 3 dB in the H plane) usually cannot be achieved for higher operating frequencies. Some techniques such as the step shaped element [7], orthogonal element [8], and S-shaped element [9] for the broadband monopole antenna were proposed to improve the omnidirectional radiation pattern characteristics within the whole operating frequency band. In addition, planar broadband monopole antennas are known to suffer from high cross-polarization radiation levels [1]. However, no efforts were reported to reduce the cross-polarization levels with consistent omnidirectional patterns for the planar broadband monopole antennas. The dielectric resonator antenna (DRA) is one of the attractive candidate antennas for UWB application due to several striking characteristics such as high radiation efficiency, low dissipation loss, light weight, and small size. Significant efforts for the DRA have been reported to achieve wide bandwidth enhancements over the past two decades [10]–[17]. Manuscript received October 13, 2009; revised August 16, 2010; accepted August 25, 2010. Date of publication January 31, 2011; date of current version April 06, 2011. The authors are with the Department of Electrical Engineering, University of Mississippi, University, MI 38677 USA (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.2011.2109676

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Fig. 1. Geometry of the proposed inserted UWB DRA excited by CPW feeding. (a) Top view. (b) Isometric view. (c) Side view. Fig. 3. Isometric view of: (a) inserted DRA excited by microstrip feeding; (b) planar DRA excited by CPW feeding; (c) planar DRA excited by microstrip feeding.

Fig. 2. Reflection coefficients of the UWB DRA with/without dielectric resonator when strip width (F ) is (a) 2.16 mm; (b) 4.0 mm; (c) 6.0 mm.

Here, we present inserted DRA type with omnidirectional pattern. The combination mechanism of the inserted structure and dielectric resonator characteristic provide a consistent excellent omnidirectional radiation pattern and low cross polarization level within the operating frequency band. Four primary types of UWB DRAs are compared to show the performance of the cross polarization levels and omnidirectional patterns. The analyses in this communication are performed using the commercial code HFSS [18]. II. ANTENNA CONFIGURATION The proposed UWB DRA called inserted DRA excited by CPW is shown in Fig. 1. The size of the DRA has 12 mm width, 13 mm length, and 5.08 mm thickness with a dielectric constant of 10.2, and it is supported by a 15 2 33 mm2 RT6002 substrate with a dielectric constant of 2.94 and a substrate thickness of 0.762 mm. The antenna parameters : , 2 : , 3 , 1 , are shown in Fig. 1 with 1 , 1 : , 1 : , : , and 2 : mm. 2 1

G = 0 15 G = 0 45 G = 2 F = 6 F = 7 W = 2 16 L = 8 5 T = 2 54 T = 1 778 III. NUMERICAL INVESTIGATION

Fig. 2 shows the reflection coefficients of the antenna with different strip widths 1 with the DR and without the DR. Our proposed UWB DRA with omnidirectional pattern uses two modes so that mainly the lower band is contributed by dielectric resonator itself and the upper band is affected by the rectangular wide strip. It can be observed that the reflection coefficients of the metallic element alone do not have significant difference, while the reflection

(F )

coefficients with dielectric resonator have significant difference around 8 GHz. When the strip width is narrow, the strip acts as a resonator providing a band-notch, so that a high quality factor (Q) of around 20 is obtained. However, the wide strip width provides lower quality factor around 2, which implies that the wide strip does not act as a resonator. This technique is the vital to improve the impedance matching bandwidth compared with previous planar type broadband monopole DRAs [10], [11]. The geometries of several UWB DRAs are shown in Fig. 3 to compare the performance with our proposed inserted DRA excited by CPW. Two excitation types, namely CPW and microstrip line feed, are used with two different DRA locations with respect to the dielectric substrate over the ground plane edge. For a DR placed on top of the dielectric substrate we refer to it as planar DRA and the other is inserted DR inside the substrate we refer to it as inserted DRA. Fig. 4 shows the matching frequency band of the four types of the UWB DRAs for 0 dB reflection coefficients with a bandwidth of 115% for the CPW feeding and 109% for the microstrip feeding. It can be illustrated that the CPW-fed technique will give a slightly wider impedance matching bandwidth depending on the matching transformer. Fig. 5 shows the comparisons of the H-plane of the four types of the UWB DRAs at 3.1, 5, 8, and 10 GHz. It is well known that electric currents exist on the ground plane, so the radiation from the ground plane is inevitable. Therefore, the performance of the printed UWB antenna is significantly affected by the shape and size of the ground plane in terms of the impedance bandwidth and radiation patterns. With the truncated ground plane and extending the microstrip line to be the excitation probe, a discontinuity is created and the fringing electric field lines at the end of the microstrip line joined to the printed probe causes some direct radiation that contributes to the cross polarization. Also, in the planar DRA type, the asymmetry of the structure with the probe feed disturbs the symmetry and the omnidirectionality of the radiation patterns with the high cross polarization levels. The CPW feeding has less effect in that regard as its fringing fields between the ground plane and printed probe is symmetry and only the asymmetry of the structure causes the asymmetry of the omnidirectional patterns. In the case of the inserted DRA, the structure symmetry is much better than the planar type. The cross-polarization is caused by the microstrip line discontinuity, which is removed by using the CPW feeding. Also, having the main contribution of the radiation patterns due to polarization current induced in the DR that has no sharp edge current component, rather

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Fig. 4. Reflection coefficients for the inserted DRA and planar DRA excited by CPW and microstrip feeding technique.

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Fig. 6. Maximum cross-polarization levels of the H-plane within the operating frequency band.

Fig. 7. The difference between the maximum and minimum antenna gain simulated in the H plane.

Fig. 5. Radiation patterns for four different antennas: (a) Proposed inserted DRA excited by CPW feeding. (b) Inserted DRA excited by microstrip feeding. (c) Planar DRA excited by CPW feeding. (d) Planar DRA excited by microstrip feeding; 3.1 GHz; . . . . . . 5 GHz; - - - - - 8 GHz; : : : . 10 GHz.

00 00

than conduction current induced on the planar printed UWB antennas and concentrated at the edges that contribute to the high cross-polarization. Fig. 6 illustrates the maximum cross-polarization level of the H-plane within operating frequency. The differences of the maximum cross polarization levels between the inserted DRA type and the planar DRA type are around 20 dB. In addition, we can notice that CPW feeding provides smaller cross-polarization level than the microstrip feeding. Getting consistent omnidirectional radiation patterns in the H-plane within the operating UWB frequency band from printed planar antenna is a very challenging goal. The reason is that a wide monopole element is needed to achieve the UWB bandwidth. The width of the antenna

is usually comparable to or larger than a quarter-wavelength of the highest operating frequency. Therefore, radiation patterns are affected by the difference of the path lengths caused by the wide monopole element. To overcome this problem, the orthogonal radiating element, and stair-shaped radiating element were used for broadband monopole antenna. They reduced the maximum path length in a certain direction and introduced the other path length in an orthogonal direction. That is; they provided three dimensional path lengths, which is the key mechanism of the improving omnidirectional pattern over the whole frequency band. Ideally, the H-plane gain variation of the monopole antenna should be 0 dB, because the shape of the radiation pattern is perfect circle. However, planar UWB antennas have difficulty to achieve less than 3 dB gain variation [7]. Fig. 7 shows the difference between the maximum and minimum gain variations in the H-plane within the operating frequency band. It is clearly seen that the inserted DRA does not have distorted circle shape of H-plane but almost perfect circle shape of H-plane, while the planar DRA does not have nearly perfect circle shape of H-plane at the upper end of the frequency band due to the asymmetry of the structure. The gain variations in the H-plane for the inserted DRA excited by CPW feeding are less than 3 dB within operating frequency band. The symmetry of the antenna around the excitation provides good omnidirectional pattern over the entire frequency band. DR naturally has a volume structure, which does not have the

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Fig. 9. Maximum cross-polarization levels of the E-plane within the operating frequency band.

TABLE I SIZE COMPARISONS OF UWB ANTENNA

Fig. 8. E-plane radiation patterns, yz plane (left) and xz plane (right). (a) Proposed inserted DRA excited by CPW feeding. (b) Inserted DRA excited by microstrip feeding. (c) Planar DRA excited by CPW feeding. (d) Planar DRA excited by microstrip feeding; Co-polar, : : : : . X-polar at 3.1 GHz; - - - - - - - Co-polar, . . . . . . X-polar at 8 GHz.

00 0 00

strong edge currents that exist in the x and y direction of the planar antennas. Therefore, the DRA provides better cross-polarization as well as better omnidirectional patterns. Fig. 8 shows the E-plane radiation patterns (yz-cut and xz-cut) of the four types of the UWB DRAs at 3.1 and 8 GHz. It shows more clearly the effect of the unbalanced current distribution of xz direction and the symmetric structure effect. The proposed inserted DRA structure excited by CPW feeding provides not only the lowest cross-polarization levels but also good eight-shape omnidirectional patterns. It is also

clear from the null shifted of the xz plane as the frequency increases when microstrip line feeding or planar DRA structure is used due to unbalanced current distribution of xz plane and asymmetric structure, and consequently high cross polarization levels of the yz plane. Moreover, planar DRA types have noticeable asymmetric structure due to the volume of the DR, so that the cross-polarization levels of the yz plane are really high and the eight-shape patterns of the xz plane are distorted at high frequency. Fig. 9 illustrates the maximum cross polarization level of the E-plane within the operating frequency band. The cross-polarization levels of the xz-cut are small for all the cases. Therefore, we only plot cross polarization levels at the xz-cut of the proposed inserted DRA excited by CPW feeding. The cross polarization level of the yz-cut indicates the cross polarization level due to the unbalanced current distribution of the microstrip feeding and asymmetric structure. Our proposed inserted DRA excited by CPW feeding provides less than 030 dB cross-polarization levels within the operating frequency band. Table I shows the size of the several printed planar UWB antennas with proposed UWB DRA. Three dimensional dielectric resonator radiating element can reduce the width of radiating element compared with the common UWB antennas due to the volume characteristics. Therefore, approximately 50% width reduction is achieved compared with the common UWB antennas. IV. RESULTS Fig. 10 shows the photo of the inserted DRA excited by CPW feeding. The measurements are performed using the HP8510c network analyzer. Fig. 11 shows the simulated and measured reflection coefficients of the proposed antenna. It is from 3.07 GHz to 11.5 GHz in simulation and from 3.25 to 10.82 GHz in measurement. Generally, the RF cable from the Vector Network Analyzer significantly affects in the measurement of the small antenna [19]. In addition, slightly

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Fig. 10. The photo of inserted DRA excited by CPW feeding.

Fig. 11. Measured and simulated reflection coefficients.

Fig. 13. Measured radiation patterns, H plane (left) E plane (right) at: (a) 3.1 GHz;(b) 5 GHz; (c) 8 GHz; (d) 10 GHz. Fig. 12. Radiation efficiency and gain versus frequency.

V. CONCLUSION deep milling between the ground plane and feeding line affect the measurement result. Fig. 12 shows the simulated radiation efficiency with the gain of the antenna versus frequency for the proposed DRA. The antenna achieves higher than 96.5% radiation efficiency within the operating frequency band. The far field radiation patterns of the proposed inserted DRA excited by CPW feeding are also measured. Fig. 13 plots the measured radiation patterns at four different frequencies (3.1, 5, 8, and 10 GHz). We can see the good omnidirectional radiation patterns with low cross polarization level.

A novel UWB DRA for achieving improved omnidirectional patterns with an extremely low cross polarization level was achieved. The proposed antenna used dielectric resonator instead of metallic element and better symmetrical structure was achieved using the CPW feeding compared with planar type DRAs. Therefore, the inserted DR achieved the lowest cross polarization levels, consistent omnidirectional radiation patterns, and high radiation efficiency with compact size.

REFERENCES [1] Z. N. Chen, Antennas for Portable Devices.

New York: Wiley, 2007.

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[2] K. S. Ryu and A. A. Kishk, “UWB antenna with single or dual bandnotches for lower WLAN band and upper WLAN band,” IEEE Trans. Antennas Propag., vol. 57, pp. 3942–3950, 2009. [3] W. C. Liu and S. M. Chen, “Ultra-wideband printed fork-shaped monopole antenna with a band-rejection characteristic,” Microw. Opt. Technol Lett., vol. 49, pp. 1536–1538, 2007. [4] T. Dissanayake and K. P. Esselle, “Prediction of the notch frequency of slot loaded printed UWB antennas,” IEEE Trans. Antennas Propag., vol. 55, pp. 3320–3325, 2007. [5] M. Zhang, Y. Z. Yin, J. Ma, Y. Wang, and W. C. Xiao, “A racket-shaped slot UWB antenna coupled with parasitic strips for band-notched application,” Progr. Electromagn. Res. Lett., vol. 16, pp. 35–44, 2010. [6] J. N. Cheng, S. M. Deng, C. L. Tsai, and S. S. Bor, “Impedance improvement of miniature UWB antenna,” Microw. Opt. Technol. Lett., vol. 49, pp. 2982–2983, 2007. [7] K. L. Wong, S. W. Su, and C. L. Tang, “Broadband omnidirectional metal-plate monopole antenna,” IEEE Trans. Antennas Propag., vol. 53, pp. 581–583, 2005. [8] P. V. Anob, K. P. Ray, and G. Kumar, “Wideband orthogonal square monopole antenna with semi-circular base,” in Proc. IEEE Antennas Propag. Soc. Int. Symp., Jul. 2001, vol. 3, pp. 294–297. [9] X. H. Wu and A. A. Kishk, “Study of an ultrawideband omnidirectional rolled monopole antenna with trapezoidal cuts,” IEEE Trans. Antennas Propag., vol. 56, pp. 259–263, Jan. 2008. [10] M. N. Suma, P. V. Bijumon, M. T. Sebastian, and P. Mohanan, “A compact hybrid CPW fed planar monopole/dielectric resonator antenna,” J. Eur. Ceram. Soc., vol. 27, pp. 3001–3004, 2007. [11] T. Chang and J. Kiang, “Broadband DR-loaded planar monopole,” in Proc. IEEE Antennas Propag. Soc. Int. Symp., Honolulu, HI, Jun. 2007, pp. 553–556. [12] K. S. Ryu and A. A. Kishk, “UWB dielectric resonator antenna mounted on a vertical ground plane edge,” in Proc. IEEE Antennas Propag. Soc. Int. Symp., North Charleston, SC, Jun. 2009, pp. 1–4. [13] G. P. Junker, A. A. Kishk, A. W. Glisson, and D. Kajfez, “Effect of an air gap on a cylindrical dielectric resonator antennas operating in the TM01 mode,” Electron. Lett., vol. 30, pp. 97–98, 1994. [14] A. A. Kishk, “Wideband dielectric resonator antenna in truncated tetrahedron form excited by a coaxial probe,” IEEE Trans. Antennas Propag., vol. 51, pp. 2907–2912, 2003. [15] A. A. Kishk, B. Ahn, and D. Kajfez, “Broadband stacked dielectric resonator antennas,” Electron. Lett., vol. 25, pp. 1232–1233, 1989. [16] M. Lapierre, Y. M. M. Antar, A. Ittipiboon, and A. Petosa, “Ultra-wideband monopole/dielectric resonator antenna,” IEEE Microw. Wireless Compon. Lett., vol. 15, pp. 7–9, 2005. [17] K. S. Ryu and A. A. Kishk, “Ultra-wideband dielectric resonator antenna with broadside patterns mounted on a vertical ground plane edge,” IEEE Trans. Antennas Propag., vol. 58, pp. 1047–1053, 2010. [18] HFSS: High Frequency Structure Simulator Based on Finite Element Method 2007, v. 11.0.2, Ansoft Corp.. [19] Z. N. Chen, N. Yang, Y. Guo, and M. Y. W. Chia, “An investigation into measurement of handset antennas,” IEEE Trans. Instrum. Meas., vol. 54, pp. 1100–1110, 2005.

Analog Direction of Arrival Estimation Using an Electronically-Scanned CRLH Leaky-Wave Antenna S. Abielmona, H. V. Nguyen, and C. Caloz

Abstract—A novel direction of arrival (DoA) estimation system based on a leaky-wave antenna (LWA) is presented. Compared to traditional array-based DoA techniques, which require multiple radiating elements, a large feeding network, and expensive phase shifters consuming power, the proposed scheme requires only a simple, compact, and low-cost radiating element consuming no power. The system estimates the DoA by measuring the received power at both ports of the LWA using detectors as the beam is scanned. It is implemented here using a composite right/left-handed (CRLH) leaky-wave antenna (LWA), selected for its efficient electronic full-space scanning capability. A detailed theoretical explanation is presented. Three methods of operation are described: forward scanning, backward scanning, and full-space scanning. Furthermore, full-wave simulation and experimental results both validate the theoretical analysis. To the best of the authors’ knowledge, this is the first time a LWA is employed in a DoA system. Index Terms—Composite right/left-handed (CRLH), direction of arrival (DoA), leaky-wave antenna (LWA).

I. INTRODUCTION Direction-of-arrival (DoA) estimation is becoming increasingly prevalent today for applications such as local positioning for user-tracking, location-based services in WiFi networks, as well as radar and MIMO-based systems. DoA systems must therefore employ an efficient and cost-effective antennas to become commercially viable. [1]. Most DoA systems utilize an antenna array with each antenna connected to an RF down-converter, followed by an A/D converter and subsequently fed into a processor, where various algorithms can be implemented for DoA estimation [2]. Since this architecture is an all-DSP solution, it is fast, efficient and flexible with the DoA’s estimation accuracy being independent of the array’s effective aperture. However, increasing the number of antennas directly leads to an increase in cost, power-consumption, and space because of the RF and A/D converters [3]. Another kind of array employed in DoA systems is the electronically steerable parasitic array radiator [4], or ESPAR, which utilizes a single central radiator with surrounding parasitic elements. In this approach, only one RF down-converter and A/D converter are employed, connected to the main radiator, which reduces the cost and power-consumption [5]. However, since the radiator and parasitics are mainly monopoles, the antenna is non-planar with a non-negligible form-factor. The composite right/left-handed (CRLH) electronically-scanned leaky-wave antenna (ES-LWA) [6]–[8], referred to here simply as ES-LWA, possesses some unique properties, such as fixed frequency, fundamental-mode full-space scanning (including broadside) while being planar, compact, low-loss, and low cost due to the absence of a feeding network. Due to these aforementioned properties, it is suitable to be used in cost-effective DoA systems. Manuscript received September 02, 2009; revised May 05, 2010; accepted September 12, 2010. Date of publication January 31, 2011; date of current version April 06, 2011. The authors are with the Poly-GRAMES Research Center, École Polytechnique de Montréal, Montréal, and Centre de Recherche En Électronique Radiofréquence, QC H3T 1J4, Canada (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.2011.2109672

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Fig. 1. Proposed DoA estimation system based on received power measurements showing the ES-LWA employing voltage-controlled reverse-biased varactors and two power detectors. The two symmetrical beams at  are due to port symmetry for an imposed voltage (V1,V2).

We present in this work a LWA-based DoA estimation system firstly introduced in [9]. This is an analog system consisting of a fixed frequency ES-LWA terminated with two power detectors. The DoA is provided by measuring the differential power from these two detectors. This system could also be implemented with other types of LWAs. However, the CRLH LWA has been preferred here for its efficient electronic full-space scanning capability. The communication is organized as follows. Section II describes the principle of operation for the DoA estimation technique employed here, Section III presents the full-wave proof-of-principle results, Section IV presents the measured experimental results for the ES-LWA prototype and the DoA estimation, and Section V outlines some conclusions. II. ANALOG DOA ESTIMATION BASED ON RECEIVED POWER The principle of operation for the analog DoA estimation is illustrated in Fig. 1 as a system composed of a ES-LWA and two power detectors (P D). As an incident signal impinges on the ES-LWA at an angle of i , the pointing antenna beam with an angle of p is scanned across the visible space while constantly recording the received power at P D1 and P D2. When the incident signal is aligned with the pointing beam, i.e., i = p , the received power at one of the detectors reaches a maximum. As will be explained, the differential received power between of the PDs reveals the DoA. This approach is analogous to the DoA estimation using a conventional array where appropriate weights are applied to scan the beam in the “look” direction [2]. In order to scan the beam in a particular direction, an electronicallyscanning mechanism is employed in the CRLH LWA with the following scanning relation [7], [10]

p (V )  sin01 (!0 ; V 1; V 2) k0

(1)

where k0 = !0 =c is the free-space wavenumber, and is the CRLH’s dispersion relation controlled by the varactors Cvar1 (V 1 0 V 2) and Cvar2 (V 2), which consume essentially no power due to their reversebiasing state. An illustration of the scanning law is shown in the left inset of Fig. 1. With two independent voltage controls for the varactors, the ES-LWA scans more efficiently the full-space with good matching [11]. Employing the ES-LWA’s as a receiver with two power detectors connected to its output ports, DoA estimation can be accomplished using two distinct methods of operation: 1) Forward (0  p     +90 ), 2) Backward (0  p  090 ). In this manner, it will be

Fig. 2. DoA estimation scenarios (forward or backward scanning) for signals incoming from the right and/or left quadrants.

shown that full-space DoA estimation is feasible with either forward or backward half-space scanning only. Fig. 2 illustrates the two methods of operation along with the analog DoA estimation algorithm. Since only one method is required to do full-space DoA estimation, only the forward method is described in detail, with the backward method being similar in operation. In the forward method of operation, shown in the first row of Fig. 2, the ES-LWA’s pointing beam is scanned in the forward region only   (0  p  +90 ). This corresponds to the CRLH’s right-handed region inferring parallel phase and group velocities [10]. Now, scenarios #1 and #2 show an example of an incident signal from either the right or left quadrant, respectively, with an angle i . Due to phase matching between the ES-LWA and the incident signal, the ES-LWA’s phase velocity must be parallel to the longitudinal component of the incoming signal. This is illustrated in Fig. 2 where phase ( ) is flowing to the left and to the right for scenarios #1 and #2, respectively. Furthermore and most importantly, the ES-LWA’s group velocity must flow in the same direction as as they must be parallel because the CRLH is operating in the right-handed region. Therefore, the Poynting vector (S ) is shown as flowing towards P D1 and P D2 for scenarios #1 and #2, respectively, indicating that P D1 receives higher power than P D2 in scenario #1 and vice versa for scenario #2. The DoA estimation algorithm is now described, based on the above, for scenario #1 showing an incident signal with angle i in the right quadrant. As the pointing beam is scanned from 0  p  +90 , the received power at P D1 continuously increases, reaches a maximum at i = p , then decreases, while the power at P D2 remains constant. By measuring the differential power of the power detectors as 1P =

P D1 0 P D2

(2)

with P D1 > P D2, the peak of the positive differential resultant shown in the last row of Fig. 2 indicates the DoA angle iDoA . Similarly, the DoA estimation algorithm reveals a negative differential resultant for scenario #2, since in this case P D1 < P D2, where once again the peak indicates the DoA angle iDoA . Consequently, full-space DoA

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Fig. 3. Frequency-scanned CRLH LWA full-wave simulation results (HFSS) verifying the direction of power flow. The 13-cm, 12-unit cell CRLH LWA is simulated on a RO3003 substrate with an  and : . (a) Incident wave with  as in scenario #2 (forward method) in Fig. 2 with with maximum maximum Poynting vector and substrate current towards P D . The CRLH LWA’s f is 4.25 GHz. (b) Incident wave from broadside  as in scenario #3 Poynting vector and substrate currents equally out of P D and P D . The CRLH LWA’s f is 3.9 GHz (c) Incident wave with  (backward mode) in Fig. 2 with Poynting vector and substrate current towards P D . The CRLH LWA’s f is 3.45 GHz.

=3

height = 1 524 mm 2 1 2

estimation is achieved with only half-space forward scanning, where the quadrant of the incident wave is determined by the sign of P (positive and negative for the right and left quadrants, respectively), while the DoA angle iDoA is obtained from its peak value. In a complimentary but opposite manner, the backward method can also achieve full-space DoA estimation. Half-space scanning is now undertaken in the backward range   p  0  where the CRLH operates in the left-handed region, making the phase and group velocities anti-parallel. This is illustrated in the second row of Fig. 2. Of note is that the leaky mode responsible for radiation in both distinct methods is always the fundamental mode—or, more precisely, the fundamental space harmonic—of the CRLH LWA. The proposed analog DoA system can also operate in a third mode, not shown here. This third mode scans the full-space 0   p   while measuring the received power at only one PD with the other port terminated by a matched load. It can be considered as a combination of scenario #1 and #3 or #2 and #4. In this mode, the DoA is determined by simply noting the angle for which peak power at the PD is achieved. Compared to the forward and backward methods, the full-space method employs only a single power detector, but requires varactors having a larger capacitance range. As with every antenna, the antenna’s gain varies with the scanning angle. For LWAs, the gain variation is also further dependent on the leakage factor being non-constant [12], although design techniques may be used to mitigate this issue [13]. Since this analog DoA estimation scheme is based on power measurements, it’s important that the antenna exhibit equal-gain patterns versus scan angle. For a non-specific antenna design, such as the one adopted in this work, the antenna’s gain is manually equalized in order to achieve a constant gain antenna. This equalization can be easily accomplished from the known gain versus angle characteristic of the antenna. It consists de facto in normalizing the radiation patterns of all angles to the most achievable maximum by adding a correction factor so as to obtain a constant gain versus scan angle. In this manner, the received power is no longer a function of the antenna’s gain, but only of the scan angle. This method is applied in this work for both the full-wave and the experimental results.

= 28

1

( =0 ) = 031

1

(0

90 )

( 90

+90 )

III. FULL-WAVE RESULTS Fig. 3 shows the full-wave simulation results (HFSS) of a frequencyscanned CRLH LWA in receive mode. Without any loss of generality, a frequency-scanned LWA is used here due to the difficulty of integrating varactors in HFSS. As expected, Fig. 3(a) and (c) verify the direction of power flow for scenarios #1 and #2 of Fig. 2, respectively, while

Fig. 4. DoA estimation results (crosses represent recorded results and solid line represents the fitting function) based on full-wave (HFSS) simulations for the CRLH LWA of Fig. 3. Scenarios #1–#4 correspond to the same scenarios in Fig. 2.

Fig. 3(b) shows the results for the broadside case showing equal power out of both detector ports. Fig. 4 shows the differential power results from the full-wave simulations for all four scenarios in Fig. 2. The power detectors, P D and P D in Fig. 3, are represented by a two-dimensional planar surface perpendicular to the direction of propagation at the two ports, over which the Poynting vector is integrated to obtain the total out-flowing power. As expected, the full-wave results validate the DoA estimation algorithm outlined in the previous section. For all scenarios, the DoA estimation is fairly accurate due to the application of the gain equalization method discussed previously. Furthermore, the effect of the LWA’s beam-width can also be noticed when examining the peaks of scenarios #1 and #3 in Fig. 4, where the latter is wider than the former due to its larger beam-width, rendering DoA estimation in the forward mode more accurate. As mentioned in [13], the beam-widths and the gains can be equalized by an appropriate design approach.

1

2

IV. EXPERIMENTAL RESULTS

( = 2 445 GHz) (1 )

: ES-LWA prototype, comThe fixed-frequency f0 : g = : , posed of 14 unit cells 0 each with a size of p is shown in Fig. 5. The varactors are placed in series with the inter-digital capacitors and stub inductors [7], [11]. The measured reflection for all pointing angles. coefficient is below 0 Fig. 6 shows the measured radiation patterns for the forward space   p   . The antenna is always positioned so that port 1 is on the left side while port 2 is on the right side of the measurement

10 dB

(0

+40 )

= 8 5 mm( 9 3)

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Fig. 5. The 14-unit cell ES-LWA prototype with series (Aeroflex MSV34, 075) and shunt (Aeroflex MSV34, 064) varactors controlled independently by voltages V 1 and V 2, respectively.





Fig. 6. Forward (0  +90 ) measured and normalized radiation patterns for the ES-LWA of Fig. 5 operating at f = 2:445 GHz in receive mode (V 1 = 2:2–18 V , V 2 = 0:2–6 V ). Legend: solid patterns measured at port 1 utilized in scenario #1; dashed patterns measured at port 2 utilized in scenario #2. Radiation patterns with stars, crosses, circles, and squares are for 5 , 15 , 25 , and 35 , respectively.

apparatus, as per Fig. 2. Radiation patterns marked with “Port 1,” utilized in DoA estimations for scenario #1, are measured by connecting the ES-LWA’s port 1 to the measurement apparatus while terminating port 2 with a 50 load. The opposite is done for the ones marked with “Port 2,” utilized in DoA estimations for scenario #2, with the termination now at port 1. The gain variation for the patterns is about 2 dB. Note that not all the measured patterns are shown in Fig. 6. The radiation patterns for the backward scanning (0  p  040 ) are also measured and behave similarly but are not shown here. Fig. 7 shows the experimental DoA estimation results. Signals at f0 = 2:445 GHz with various incident angles are tested while electronically scanning the beam and recording the received power at both ports 1 and 2 using a dual-channel power meter (Anritsu ML2437A with power sensitivity of 070 dBm). Fig. 7(a) corresponds to scenario #1 of Fig. 2 in which an incident signal from the right quadrant at i = 15 illuminates the ES-LWA. As seen, while the beam is electronically-scanned in the forward region (0  p  +40 ), the received power at P D1 continuously increases to reach a maximum at p = 15 , while the received power at P D2 continuously decreases. Subsequently, the differential power 1P of (2) reaches a positive maximum at iDoA = 15 indicating the angle of arrival from the right quadrant. Similarly, Fig. 7(b) corresponds to scenario #2 of Fig. 2 with an incident signal in the left quadrant at i = 25 . As seen, the differential power 1P now shows a negative maximum for this scenario with

Fig. 7. Experimental DoA estimation results for the forward method of operation showing power received at P D 1 and P D 2 along with the differential power (1P = P D 1 P D 2). (a) Scenario #1 in Fig. 2. (b) Scenario #2 in Fig. 2.

0

iDoA = 25 indicating the angle of arrival from the left quadrant. This experimentally validates the theory presented in Section II where full-space DoA estimation is achieved with only forward half-space scanning. Some notes now follow regarding this analog DoA estimation method where several limitations are inherent. Firstly, since this approach is based on received power measurements, the antenna must be well matched, as it is here, since a poorly matched antenna causes reflections at the power detectors, thus minimizing the received signal levels. Secondly, in order to have high resolution between at least two incident signals, and therefore able to detect the largest number of signals, the antenna must possess a narrow beam by having a larger aperture. Also, a narrower antenna beam alleviates the stringent constraint on the power meter sensitivity by increasing the power variation as the beams are scanned. Nonetheless, this limitation is not specific to the LWA, but also applies to all antenna arrays. Thirdly, DoA estimation is also limited in the endfire or backfire direction due to the nulls caused by the ground plane, where this limitation is also not specific to the LWA only. Finally, in near-broadside situations, the DoA estimation algorithm fails due to the overlap of the radiation patterns from ports 1 and 2. This results in little power discrimination between P D1 and P D2, where 1P results in a minimum instead of a maximum, falsely indicating that the incident signal is from broadside instead of near broadside.

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V. CONCLUSIONS A DoA estimation system based on a leaky-wave antenna has been presented for the first time. The proposed scheme requires only a simple, compact, and low-cost radiating element consuming no power, compared to complex, bulky, and expensive array-based DoA techniques. The system operates at a fixed-frequency employing an electronically-scanned composite right/left-handed leaky-wave antenna due to its advantages such as fundamental-mode full-space scanning (including broadside) while being planar, compact, low-loss, and low cost due to the absence of a feeding network. An alternative digital DoA solution utilizing the CRLH ES-LWA along with the MUSIC algorithm, providing super-resolution and alleviating the limitation mentioned in Section IV, is currently under development in the authors’ group.

On the Mathematical Link Between the MUSIC Algorithm and Interferometric Imaging Greg Hislop and Christophe Craeye

Abstract—Radio astronomy and direction finding are closely related fields. However, there is almost no overlap between the two sets of literature. Interferometric imaging and the MUSIC algorithm are perhaps the most popular algorithms in the respective domains. This paper presents the exact mathematical link between them. Index Terms—Direction of arrival estimation, interferometry, microwave imaging, MUSIC, radio astronomy.

I. INTRODUCTION ACKNOWLEDGMENT The authors would like to thank ANSYS Corp. for their generous donation of the HFSS software. They would also like to thank Metelics Aeroflex and DLI Laboratories for their respective donations of varactors and capacitors.

REFERENCES [1] R. Ramanathan and J. Redi, “A brief overview of ad hoc networks: challenges and directions,” IEEE Commun. Mag., vol. 40, no. 5, pp. 20–22, 2002. [2] L. C. Godara, “Application of antenna arrays to mobile communications, part II: Beam-forming and direction-of-Arrival considerations,” Proc. IEEE, vol. 85, no. 8, pp. 1195–1245, 1997. [3] 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. [4] T. Ohira and K. Gyoda, “Hand-held microwave direction of arrival finder based on varactor-tuned analog aerial beamforming,” in Proc. Asia-Pacific Microwave Conf., Taipei, Taiwan, Dec. 2001, vol. 2, pp. 585–588. [5] C. Sun and N. Karmakar, “Direction of arrival estimation with a novel single-port smart antenna,” EURASIP J. Applied Signal Processing, vol. 9, pp. 1364–1375, 2004. [6] S. Lim, C. Caloz, and T. Itoh, “Electronically-scanned composite right/ left-handed microstrip leaky-wave antenna,” IEEE Microw. Wireless Compon. Lett., vol. 14, no. 6, pp. 277–279, May 2004. [7] S. Lim, C. Caloz, and T. Itoh, “Metamaterial-based electronically-controlled transmission line structure as a novel leaky-wave antenna with tunable radiation angle and beamwidth,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 1, pp. 161–173, Nov. 2005. [8] C. Caloz, T. Itoh, and A. Rennings, “CRLH traveling-wave and resonant metamaterial antennas,” Antennas Propag. Mag., vol. 50, no. 5, pp. 25–39, Oct. 2008. [9] S. Abielmona, H. V. Nguyen, and C. Caloz, “Direction of arrival estimation using an electronically-scanned CRLH leaky-wave antenna,” presented at the CNC/USNC URSI National Radio Science Meeting, Charleston, SC, Jun. 2009. [10] C. Caloz and T. Itoh, Electromagnetic Matematerials, Transmission Line Theory and Microwave Applications. Hoboken/Piscataway, NJ: Wiley/IEEE Press, 2005. [11] R. Siragusa, H. V. Nguyen, C. Caloz, and S. Tedjini, “Efficient electronically scanned CRLH leaky-wave antenna using independent double tuning for impedance equalization,” presented at the CNC/USNC URSI National Radio Science Meeting, San Diego, CA, Jul. 2008. [12] A. Oliner and D. Jackson, Antenna Engineering Handbook, J. Volakis, Ed., 4th ed. New York: McGraw-Hill, 2007, ch. 11. [13] S. Paulotto, P. Baccarelli, F. Frezza, and D. R. Jackson, “Full-wave modal dispersion analysis and broadside optimization for a class of microstrip CRLH leaky-wave antennas,” IEEE Trans. Microw. Theory Tech., vol. 56, no. 12, pp. 2826–2837, Dec. 2008.

The problems of radio direction finding and interferometric imaging for radio astronomy are closely linked. Both problems involve the localization or imaging of radio sources which are in the far field of a comparatively compact array of receiving antennas. Both techniques use the cross correlations of the signals received across the array elements and attempt to find the power coming from different incident directions. However, despite their apparent similarities there is almost no overlap between the literature of radio interferometric imaging [1]–[3] and direction finding [4]–[6]. Indeed these two groups of citations, barely mention the existence of each other. There are differences between the areas. For example, direction finding is often a one-dimensional problem [5], [6]. That is only the azimuth of a source needs to be found (assuming it is Earth bound or guaranteed to be at a low elevation). Thus much of the research focuses on linear arrays and algorithms specific to them [5]–[7]. For radio astronomy, imaging of the whole celestial plane is required, hence two dimensional arrays are needed. While radio astronomy finds the intensity across all angles, only the directions of arrival are of concern to direction finding problems, thus there exists a branch of techniques which avoid scanning the search space and directly find the incident directions [8], [9]. Another difference is that in radio astronomy the sources are always uncorrelated, while in direction finding correlated sources are sometimes present [7], [9], [10]. Despite these differences, if one limits their interest to two dimensional arrays, scanning systems and uncorrelated signals, a very large and significant part of the direction finding literature remains. This includes perhaps the most popular of direction finding algorithms, MUSIC [11]. This paper considers an algorithm from each of the domains and gives the exact mathematical link between them. From radio astronomy, interferometric imaging is considered. To enable a concise and mathematically exact comparison, the paper will focus on the fundamental result of the Van Cittert-Zernike Theorem [1], [3]. That is the construction of the “dirty image” via the inverse Fourier transformation of the visibility data. All of the competing radio astronomy techniques and the pre/post processing are well understood and compared in the literature [1], [3]. Manuscript received May 12, 2010; revised August 23, 2010; accepted August 23, 2010. Date of publication January 31, 2011; date of current version April 06, 2011. This paper was supported by the Région Wallone RFTAG Project. The authors are with the Université catholique de Louvain, ICTEAM-ELEN, 1348 Louvain la Neuve, Belgium (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2011.2109691

0018-926X/$26.00 © 2011 IEEE

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 59, NO. 4, APRIL 2011

From the direction finding community, the MUSIC algorithm will be addressed. There are many other direction finding algorithms, such as minimum variance, ESPRIT, neural networks, etc. [5]. But the direction finding literature already contains numerous excellent comparisons of these techniques [4]–[6]. Thus this paper focuses on providing the mathematically exact link between MUSIC and the formation of radio astronomy’s “dirty image.” Of the algorithms in the two fields, these were chosen due to their popularity and the presence of an exact and unpublished link. Throughout the paper a cartesian co-ordinate system (with unit vec^; y^; z^) will be used with L array elements located in the x-y plane. tors x All sources will be considered incident from the positive z direction. 1 is assumed and suppressed A ej!t time dependence, where j = and  is the wavelength of a narrowband source. As the nomenclature differs between the two domains, this paper will start with a brief description of some commonly used terms in the direction finding (Section II) and radio astronomy communities (Section III). Section IV will then start with the definition of the MUSIC algorithm and will alter its form so as to enable comparison with interferometry. Section V will take the interferometry definition and alter it. The exact mathematical relationship between the two is stated in Section VI, a discussion of the results follows in Section VII and finally Section VIII concludes the paper.

p0

II. GLOSSARY OF DIRECTION FINDING TERMS • The minimum variance algorithm is a direction finding method obtained by applying complex weights to each channel of an array and optimizing the weights so as to minimize the total output power in all directions while preserving a constant sensitivity in the direction of interest. • The MUSIC pseudo spectrum is the power-like spectrum produced by the MUSIC algorithm (it does not satisfy the definition of a power spectrum).  is a matrix containing the cross correla• The correlation matrix R tions between signals at all possible antenna pairs.  is the vector wavenumber for an arbitrary incident plane wave • k and is used to define the search direction. ) is a vector representing the phase factors • A steering vector e(k across an array due to a source coming from a given direction.  S subspaces are obtained by separating • The noise U N and signal U the eigenvectors of the correlation matrix into two groups based on the magnitude of their eigenvalues. • • • • •

•

III. GLOSSARY OF RADIO ASTRONOMY TERMS The celestial sphere is a spherical surface representing the sky. (l; m) are the x and y components of a unit vector used to represent a direction of arrival. I (l; m) is the intensity of the celestial sphere (W=sr=Hz ). Baselines (u; v ) are the distances between pairs of antennas (in wavelengths), u and v are the x and y components respectively. The visibility function V (u; v ) is the correlation between signals received by antenna pairs. Thus the elements of the correlation  are the same as the terms of the visibility V (u; v ). matrix R The normalized equivalent antenna area An (l; m) is the collecting surface of an antenna normalized by the area on boresight and by 1 l2 m2 (the z cosine of the search direction).

p 0 0

IV. MUSIC MUSIC is a variant of the minimum variance algorithm for direction finding. The minimum variance algorithm produces a direction dependent image [6]

P (k) =

1 01  H  e(k) R e(k)

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) is the algorithm’s estimate of the power received from where P (k the given search direction (the distribution’s maximums are the source locations) and H denotes the conjugate transpose. The ith element of e(k) is the phase factor for the ith antenna and is defined as

ei = ej k1r where ri is the position of the ith antenna in the array. 

MUSIC differs from the minimum variance method by using the prior knowledge that there are M (M < L where L is the number of antennas) uncorrelated sources. It performs an eigen-decomposition of the correlation matrix and assumes that the largest M eigenvalues and their corresponding vectors (the signal space) represent the incoming M signals (and noise correlated with the signals) while the other eigenvalues and vectors (noise space) represent the noise. The method reconstructs the correlation matrix using only the noise space with unity eigenvalues. The MUSIC equivalent to (1) is [6], [11] 1 (3) P (k) = H  H  e(k) UN U N e(k)  N are the eigenvectors of the noise subspace. where the columns of U  S ) are related as follows: Note that the noise and signal subspaces (U

I = US US + UN UN

H

H

(4)

where I is the identity matrix. It is obvious that one will obtain the same results if one inverts (3) and searches for minimums rather than maximums. By doing this and substituting (4) into (3) one obtains (5) P 01 (k) = e(k)HIe(k) 0 e(k)HUS US e(k): Note the first term of this equation is equal to L and by ignoring it and H

instead searching for the maximums of the second term (without the negative sign) the same source directions will be located. This results in a new simpler distribution within which one searches for the maximums, that approximate the directions of arrival (6) P 0 (k) = e(k)HUS US e(k): S U H is a filtered version of R  , where the noise space eigenIn (6), U S H

values are set to zero and those of the signal space to one. Representing H 0 US US as a filtered matrix R with elements ai i (i1 and i2 represent the rows and columns), one obtains

P 0 (k) =

ai i

ei e3i :

i

(7)

i

V. INTERFEROMETRY The fundamental equation for radio interferometric imaging may be developed using the Van Cittert-Zernike theorem. Assuming identical isotropic antennas, it may be written [3, p. 73] as

1

I (l; m) =

1

01 01

V (u; v)ej 2(ul+vm) dudv:

(8)

The propagation vector may be written in terms of l and m as

k =

2



(lx^ + my^ +

p

1

0 l2 0 m2z^)

(9)

and the baselines u and v are related to the antenna positions via

ri

0 r

= (ux^ + v y^ + 0^ z ): (10) Using (2), (8)–(10) and discretising the integrand in (8) one finds i

I (l; m) =

bi i

(1)

(2)

where bi

i

i

ei ei3

i

. are the elements of the original correlation matrix R

(11)

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In (11) the intensity across the celestial plane is a two dimensional discrete Fourier transform of the correlation matrix as a function of the antenna baselines. Equation (7) differs only by performing an eigenspace filtering operation on the correlation matrix prior to Fourier transforming. Thus both (7) and (11) may be viewed as a separation of the power across the celestial sphere into a Fourier series where each element of the series projects a different pattern onto the celestial sphere and the correlations between antenna measurements provide the series coefficients. VI. RELATING THE TWO METHODS Comparing (7) and (11), the primary relationship between interferometric imaging and MUSIC based direction finding is obvious. MUSIC or more exactly the authors’ redefined version of MUSIC (which gives the same source directions as the original) is equivalent to interferometry with the correlation matrix replaced by a filtered version of itself. This filtered correlation matrix is obtained by: 1) performing an eigen-decomposition; to divide 2) using the prior knowledge on the number of sources the eigen-space into signal and noise components; 3) and reconstructing the correlation matrix with the signal eigenvalues set to 1 and the noise eigenvalues set to 0. The above stated difference is similar to the difference between minimum variance and MUSIC. It should, however, be noted that interferometric imaging and minimum variance are not equivalent and this will be discussed in the following section. Two other differences exist between MUSIC and interferometric imaging, they are: • MUSIC is used for direction finding, and thus only the maximums of the distribution are of interest. While interferometry is an imaging method, and as such uses the whole distribution; • Interferometry uses CLEAN [1], [3] which is a powerful postprocessing method. MUSIC does not use post processing.

M

VII. DISCUSSION This paper has presented an exact mathematical link between MUSIC and interferometric imaging. The two have been shown to be nearly identical, with a filtering operation on the correlation matrix providing the primary difference. This subtle difference is a result of the slightly differing applications of the two techniques. MUSIC is concerned with finding the direction of arrival of a finite number of sources while interferometry is an imaging method. As such MUSIC often relies on the prior knowledge of how many sources are present, while interferometry does not. Thus, MUSIC can separate the noise and signal spaces. MUSIC aims to find all sources irrelevant of their power, it changes eigenvalues in the signal space to 1, thus improving the probability of separating weak signals from stronger ones. Reviewing the above mathematics, it may appear that interferometric imaging (8) is equivalent to the minimum variance algorithm (1). The separation of the noise and signal eigen-spaces of the correlation matrix is the difference between minimum variance (1) and MUSIC (3). While, interferometric imaging is related via the eigen-space separation to an altered form of the MUSIC algorithm (7). However, to obtain this altered form, one uses (4) which is only valid once the eigenvalues are set to unity. Thus, to link the interferometry and minimum variance methods one must first establish the link with MUSIC. There are several reasons why the alternative MUSIC definition (6) may be preferred to the original (3). Firstly, as it is defined using (3) [5], [6], MUSIC’s advantage over other algorithms is often described as being due to it searching for where there is no noise as opposed to searching for maximums of the signal power. However, the alternative definition (6) portrays a more logical reason. It shows that the performance is obtained from the knowledge of the number of sources present, enabling one to perform noise filtering of the correlation matrix via eigen-decomposition. Secondly, (3) is a pseudo spectrum, as it has no physical interpretation. While (6) is interferometry with a filtered correlation matrix and has the same units as interferometry, W/sr/Hz. Lastly, (6) is simpler and more compact, while the inversion operation

in (3) artificially steepens the distribution without actually improving accuracy. One situation where the redefined MUSIC algorithm applies is in [12]. Here, two techniques are compared, for the problem of locating malfunctioning elements in large arrays from field measurements. The MUSIC method is compared with elementary Fourier propagation for imaging fields across an array. MUSIC gives higher resolution while the Fourier method gives the actual field values rather than the unit less MUSIC pseudo spectrum. If (7) were used, units of power would be obtained, while benefiting from MUSIC’s eigen-space separation and avoiding MUSIC’s false sense of precision. The authors have assumed ideal isotropic antennas for simplicity. However, practical antennas may be accurately accounted for. For nearly identical embedded element patterns, the left hand side of (7) ) and (11) may be multiplied by the antenna equivalent area n ( [3]. For non identical embedded element patterns n ( ) may be considered a function of ( 1 2 ) and divided into the elements of the sums on the right hand side of (7) and (11) [3]. The above mathematics raise an interesting question. Can Radio Astronomy use eigen-decomposition and noise signal space separation to filter noise? This would assume that the specific application knows that there are less uncorrelated sources than antenna positions and can obtain an upper estimate of this number. When direction finding is taught or presented in text books [5], [6], it is stated that it is related to radio astronomy, but mathematical links are not established. When teaching interferometric imaging for radio astronomy the links are not mentioned [2], [3]. The link in this paper will be useful as a teaching aid enabling an understanding of the relationship between the two domains.

i ;i

A l;m A l;m

VIII. CONCLUSION This paper has presented the exact mathematical link between radio interferometric imaging and the MUSIC direction finding algorithm. The primary difference between MUSIC and interferometric imaging is the use of the eigen decomposition for noise filtering. An alternative definition of the MUSIC algorithm was proposed which gives the same results as the original but portrays the eigen decomposition as a noise filtering operation and produces a true power spectrum rather than the traditional and unit less MUSIC pseudo spectrum.

REFERENCES [1] G. Taylor, C. Carilli, and R. Perley, “Synthesis imaging in radio astronomy two,” presented at the Astronomical Society of the Pacific Conf. Series, 1999. [2] B. Burke and F. Graham-Smith, An Introduction to Radio Astronomy, 3rd ed. Cambridge, U.K.: Cambridge Univ. Press, 2010. [3] A. Thomson, J. Moran, and G. Swenson, Interferometry and Synthesis in Radio Astronomy. New York: Wiley Interscience, 2004. [4] E. Tuncer and B. Friedlander, Classical and Modern Direction-of-Arrival Estimation. The Netherlands: Elsevier, 2009. [5] H. V. Trees, Optimum Array Processing: Part IV of Detection, Estimation, and Modulation Theory. New York: Wiley Interscience, 2002. [6] D. Johnson and D. Dudgeon, Array Signal Processing: Concepts and Techniques. Englewood Cliffs, NJ: Prentice Hall, 1993. [7] U. Sarac, F. Harmanci, and T. Akgul, “Experimental analysis of detection and localization of multiple emitters in multipath environments,” IEEE Antennas Propag. Mag., vol. 50, pp. 61–70, Oct. 2008. [8] J. Liang, “Joint azimuth and elevation direction finding using cumulant,” IEEE Sensors J., vol. 9, no. 4, pp. 390–398, Apr. 2009. [9] J. Gu, P. Wei, and H. Tai, “Fast direction-of-arrival estimation with known waveforms and linear operators,” IET Signal Processing, vol. 2, no. 1, pp. 27–36, Mar. 2008. [10] G. Hislop and C. Craeye, “Spatial smoothing for 2D direction finding with passive RFID tags,” presented at the Loughborough Antennas and Propagation Conference (LAP2009), Loughborough, U.K., Nov. 16–17, 2009. [11] R. Schmidt, “A Signal Subspace Approach to Multiple Emitter Location and Signal Parameter Estimation,” Ph.D. dissertation, Stanford Univ., Stanford, CT, 1981. [12] A. Buonanno and M. D’Urso, “On the diagnosis of arbitrary geometry fully active arrays,” presented at the EUCAP, Eur. Conf. on Antennas and Propagation, Barcelona, Spain, Apr. 2010.

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On the Nature of Oscillations in Discretizations of the Extended Integral Equation G. Fikioris, N. L. Tsitsas, and I. Psarros

Abstract—Using analytical methods, previous studies have shown that it is possible for oscillations to occur in the auxiliary surface current determined by applying the method of auxiliary sources (MAS) to problems of scattering by perfect conductors of a very simple shape. Such oscillations are inherent to MAS and would occur even in a hypothetical computer with ideal hardware and software. Because the integral equation relevant to MAS very much resembles the “extended integral equation” (in which the unknown is the actual surface current on the conductor), one might surmise that similar oscillations also occur in discretizations of the latter equation. In this communication, we use analytical means to show that this is not the case. Therefore, any oscillations that do occur in discretizations of the extended integral equation are—at least for “sufficiently simple” problems—likely to be due to matrix ill-conditioning, which magnifies errors that would otherwise be unimportant. Index Terms—Convergence of numerical methods, Fredholm integral equations, scattering.

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order to satisfy the EIE for the total electric field. The surface currents are then approximated by the complete systems of the DS’s fields. The method of auxiliary sources (MAS) [14]–[16] stems from an integral equation—to be abbreviated in this communication by MASIE—which is very similar to the EIE: Whereas the EIE seeks the actual surface current (on the PEC) that cancels the incident field on a closed auxiliary surface inside the PEC, the MASIE seeks an auxiliary surface current inside the PEC that cancels the incident field on it. Thus, both equations employ an auxiliary surface inside the PEC surface, but the roles of the two surfaces are interchanged. The aforementioned resemblance makes it natural to ask whether the findings of [17] and [18]—which concern the MAS and the MASIE—apply to the EIE and its discretizations; this is the question addressed herein. [17] considers the two-dimensional (2-D) problem in which the PEC is an infinitely long circular cylinder illuminated by an infinitely long, constant-current line source (“cylindrical problem”). The auxiliary surface is a similar cylinder, located inside the PEC. If is the number of auxiliary sources, both the MASIE and the usual N 2 N MAS system of algebraic equations are solved explicitly. It is then shown analytically that the MASIE is nonsolvable only if the radius of the auxiliary surface is smaller than a certain critical radius. In the nonsolvable case, the MAS currents present unphysical oscillations, when is sufficiently large.1 In [18], asymptotic methods are used to show that the oscillations are exponentially large in the parameter . [18] also performs a similar investigation for a PEC planar surface illuminated by a 2-D line source (“planar problem”). The findings are analogous, indicating that the aforementioned behavior is representative: indeed the MAS solutions of more complicated problems can also exhibit rapid and intense oscillations of the above type, which would occur even with ideal hardware and software, (see Section V of [18]). Using analytical methods, parallel to those in [17], the present communication demonstrates a very different behavior for the EIE, despite its close resemblance to the MASIE: For the cylindrical problem, we show irrespective of the position of the auxiliary surface that (i) the EIE is solvable; (ii) the solution of the discrete version of the EIE, when properly normalized, converges to the true solution in the limit ! 1 (now is the number of auxiliary sources on the PEC surface). As in [17], we proceed from first principles and only invoke fundamental concepts of electromagnetics. Similar conclusions hold for the planar problem, not discussed here for brevity. References [17] and [18] demonstrate analytically that nonsolvability and associated oscillations do occur when MAS/MASIE is applied to the cylindrical and planar problems. The present work considers the EIE and shows that such phenomena do not occur in the two simple problems. The absence of nonsolvability/oscillations in two simple problems cannot of course guarantee their absence in any scattering problem. Nonetheless, our results are not irrelevant to more complicated problems. For instance, our results lead one to suspect that oscillations observed in a problem with a sufficiently smooth boundary may be of different nature than those of [17] and [18]. In Section V, we show that oscillations can occur in the problem of scattering by a 2-D ellipse, but then demonstrate conclusively that they must be blamed on matrix ill-conditioning, which magnifies errors that would otherwise be unimportant.

N

N

I. INTRODUCTION This communication is relevant to problems of scattering by closed, smooth perfect electrical conductors (PEC), illuminated externally. The usual boundary condition requires cancellation of the scattered and incident tangential electric fields on the PEC surface. The extended boundary condition (EBC) requires a cancellation in the interior of the PEC surface. The extended integral equation (EIE) [1]–[3] (also known as the null-field integral equation) can be derived by satisfying the EBC in a properly selected closed interior auxiliary surface. The unknown quantity of the EIE is the surface current on the PEC. The EIE forms the first step of many numerical methods, such as (i) the null-field method (NFM) [1]–[5], (also known as the T-matrix method [6] or the extended boundary condition method [7], [8]; see also the reviews [9] and [10]), (ii) the dual-surface integral equations method (DSIEM) [11], (iii) the combined Helmholtz integral equation formulation (CHIEF) [12], and (iv) the null-field method with discrete sources (NFM-DS) [13], [14, Ch. 10]. The NFM first utilizes the EIE to determine the unknown surface currents and then substitutes into the exterior electric field integral representation to obtain the unknown scattered field. The DSIEM imposes an appropriate boundary condition on the magnetic or electric field on a fictitious surface inside the PEC surface. The resulting integral equation, multiplied by a constant, is then added to the corresponding original magnetic or electric field integral equation to obtain the dual-surface electricand magnetic-field integral equations. The CHIEF combines two sets of integral equations, one of which stems from the EIE, to obtain an overdetermined system solved by a least squares procedure. The key step in the NFM-DS is to use DS’s located inside the PEC surface in Manuscript received January 07, 2010; revised September 02, 2010; accepted September 09, 2010. Date of publication January 31, 2011; date of current version April 06, 2011. The work of G. Fikioris was supported by the PEBE 2009 Basic Research Program. G. Fikioris and I. Psarros are with the School of Electrical and Computer Engineering, National Technical University, GR 157-73 Zografou, Athens, Greece (e-mail: [email protected]; [email protected]). N. L. Tsitsas is with the School of Applied Mathematical and Physical Sciences, National Technical University, GR 157-73 Zografou, Athens, 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.2011.2109679

N

N

N

II. SOLVABILITY OF THE EIE As in [17] and [18], denote the polar coordinates of a point on the PEC cylinder and the auxiliary surface by ( cyl cyl ) and ( aux aux ). The cylinder is illuminated externally by a line source 0i!t time at ( ) = ( l 0), with aux cyl l . Assume a

 ; ; 

 ;



obs ): (18)

We can verify these remarks by explicitly calculating A and B (whose definitions appear very similar) and showing that they are indeed dif(1) ferent: To find B , substitute H0 (kRcyl;obs ) in (17) using

1 Jn (kcyl ) n=01 2 Hn(1) (kobs )ein( 0 +

H0(1) (kRcyl;obs ) =

)

(obs

> cyl ) (19)

As with the series in (13), the one in (19) behaves like Taylor series for

(N

! 1)

(14)

=

k2 I 4!"0

1 Hn(1) (k l ) in Jn (kcyl )Hn(1) (kobs ) (1) e Hn (kcyl ) n=01 +

: (20)

As discussed in [17], (i) the series in (20) is convergent, not only when obs > cyl , but in the extended region obs > cri ; (ii) the series for the derivative @B=@obs also converges in obs > cri ; (iii) as a result, the series in (20) is analytic and can be used to define the desired quantity (i.e., the analytic continuation of EzS to the interior), at least when cri < obs < cyl . The quantity A follows from the proper addition theorem

H0(1) (kRcyl;obs ) =

(15)

1 Jn (kobs ) n=01 2 Hn(1) (kcyl )ein( 0 +

)

(cyl

> obs ) (21)

in place of (19). The same procedure gives

which is independent of aux . By (11) and (15), the limit 2` N I` Nlim !1 2cyl cyl = N

tion. We then recognize the resulting integral as being proportional to the Fourier coefficient of Jsz (cyl ), which can be found from (6). For obs > cyl , we thus obtain

B

The terms for n = 0 are independent of N and have been written separately. Using the large- n Bessel and Hankel functions asymptotic approximations, we see that the nth term in the second series in (13) is of the order of (aux =cyl )nN =(nN ) for large n. Since aux < cyl , this behaves like the nth term of the uniformly convergent Taylor series for ln(1 + x)(jxj < 1) [21, Eq. 4.1.24]. We can thus find the limit of this series as N ! 1 term-by-term, and the limit is zero. Similarly, the limit of the other three series in (13) and (14) is also zero. Thus, in the limit, we keep only the n = 0 (separate) terms in (13) and (14), so that (12) reduces to

m(1) (k l )  0 N1 H(1)

IV. ANALYTIC CONTINUATION OF THE SCATTERED FIELD

ln(1 + x), so we can interchange the orders of integration and summa-

n=01 n=1 0Jm (kaux )Hm(1)(k l ) 01 1 (1) + JnN +m (kaux )HnN 0 +m (k l ): n=01 n=1

I (m)

1417

A= (16)

exists irrespective of the position of the auxiliary surface and equals the Jsz (cyl ) in (6). Our discretization method thus converges. This implies that—as opposed to MAS—there are no oscillations.

k2 I 4!"0

1 Jn (kobs )Hn(1)(k l ) exp(inobs ): n=01 +

(22)

Clearly, A differs from B . Furthermore, use of the addition theorem and comparison with the first term in the RHS of (2) shows that A is minus the incident field, just as expected. Thus, the singularities of the analytic continuation have no relevance to the EIE, something that must still hold for more complicated scattering problems.

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V. FURTHER RELEVANCE TO MORE COMPLICATED PROBLEMS In Section III, for the cylindrical problem, we found no oscillations of the type encountered in MAS. As a consequence, oscillations encountered in an EIE solution of a “sufficiently simple” scattering problem with a smooth boundary are likely to be due to matrix ill-conditioning, which can greatly magnify hardware and software errors that normally (in problems involving smaller condition numbers) would have negligible effects on the final results. For EIE solutions, such errors arise because of finite computer word length and the imperfect computation of the Bessel functions.2 The fact that effects such as roundoff can be the cause of oscillations that resemble those of [17] and [18] has been noted elsewhere, in the context of thin-wire integral equations; see [23] and, especially, [20]. In that context, matrix condition numbers increase exponentially with matrix size [23, Fig. 3]. For further examples of solvable Fredholm integral equations of the first kind whose discretizations oscillate, see [24]–[28]. For Hermitian kernels, the solution can be written in terms of eigenvalues and corresponding eigenfunctions, and this gives rise to the following explanation [28]: “Changes in the eigenvalues due to rounding and truncation errors can radically alter the coefficients of the higher eigenfunctions and cause characteristic spurious oscillations in numerical results.” With the aid of singular values, singular functions, and Picard’s theorem [19], the above explanation can be extended to other compact operators: see [20, Eq. (11)] and note from [20, Sec. IV.C] that for the present cylindrical problem, the high-order singular functions are highly oscillatory. For the case of the EIE, we illustrate the above points by showing representative results for an elliptical 2-D PEC scatterer illuminated by a z -polarized plane wave incident from inc = =6. The auxiliary surface is the maximum inscribed circle, and the eccentricity is large enough for the ellipse’s two foci—which are the singularities of the analytic continuation of the scattered field [14, Ch. 5]—to lie outside the circle. The system solved was the obvious generalization of (10), with the discrete currents and collocation points equispaced in  and with the first point placed along the ellipse’s major axis at  = 0; a singular matrix can be avoided if N is not a multiple of 4. Fig. 2(a) and (b) show the real and imaginary parts of the computed normalized currents Ip =I . The dashed curves, corresponding to the standard Matlab routine that explicitly computes the matrix inverse, are seen to present oscillations, which are greatest near the two directions along the ellipse’s major axis. These oscillations very much resemble the ones in [17] and [18]. Nonetheless, the present oscillations are of a very different nature. We verified this conclusively simply by using a more accurate system solver: the solid curves in Fig. 2—which were obtained using the Matlab routine that produces the solution using Gaussian elimination (GE)—barely present oscillations (at the scale of the figure). In the case of Fig. 2, at least, the oscillations must be blamed on the original solver, which magnifies effects due to finite computer word length and special function computations. Needless to say, when N is sufficiently small, both solvers yielded non-oscillatory, coincident solutions. With the exception of the regions near the oscillations of the GE results (as mentioned, those oscillations are only barely visible in Fig. 2) the GE results—when normalized according to (16)—coincided with normalized GE results obtained with smaller values of N and associated with smaller condition numbers. In other words, our discretization method in conjunction with GE satisfactorily converges, and gives correct results, before serious oscillations occur. On the other hand, extensive tests with larger N revealed larger condition numbers and more intense oscillations; such oscillations eventually appear even in the case of GE. 2An iterative method (rather than a direct one) of solving the linear system would introduce additional errors, as would any further special function (apart from Bessel function) computations, as well as numerical integrations.

Fig. 2. Real (a) and imaginary (b) parts of the computed normalized currents I =I as function of element number p for N ; = , and for an ellipse with major and minor semi-axes ka : and kb : respectively (this yields an eccentricity of 0.784). Dashed curves correspond to standard Matlab matrix inversion, and the solid ones to Matlab’s Gaussian elimination.

= 99 =29

= 6 =18

We also compared to results obtained by another Matlab solver, the Moore-Penrose generalized inverse. For our problem, this was the best solver we tried: It yielded nonoscillatory results even for N larger than 99. And those results, when normalized according to (16), coincided with the normalized GE results of Fig. 2, with the exception of the regions near the oscillations of the GE results. Also, we checked that Bessel-function computations are important by adding random errors to the computed values of the Bessel functions. We found much more intense oscillations. In more difficult cases, it may not be easy to correct the problematic results. But understanding their origin is important; such understanding may eventually help one correct, or make one certain that correction using better hardware/software is impossible (as with the oscillations in [17] and [18]). Besides computing the condition number, it is useful to check whether observed oscillations are hardware/software dependent. For example, if one observes the same results by changing the solver and/or the numerical integration routines, the observed oscillations are probably similar in nature to those in [17] and [18]. But if the results change (without necessarily improving), then effects such as those responsible in Fig. 2 are surely present. For the case of Fig. 2, we found that condition is number is nearly 1012 and observed that condition numbers increase exponentially with matrix size: the log of the condition number appears to increase linearly after N = 40 and up to a certain value where computation of the condition number itself becomes problematic. For that case, we observed changes in the dashed

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curve if we changed the specific computer implementation. (The results in Fig. 2 were implemented in Matlab 7.2 with double precision arithmetic and executed in an Intel Pentium (R) 4, CPU 3.00 GHz with 1 GB of RAM; we obtained different results with an Intel Pentium M, CPU 1.60 GHz with 504 MB of RAM). Such numerical experiments would lead one to blame ill-conditioning even if one were not able to remove the oscillations with improved solvers.

VI. CONCLUSION For problems of scattering by closed, perfectly conducting surfaces, both the MAS and the EIE use a closed, interior auxiliary surface, but the roles of the two surfaces are interchanged. For two simple problems, and for the case of MAS, previous works have shown that the underlying integral equation is nonsolvable in certain cases, and that associated oscillations must necessarily occur in its discretization. Because of the resemblance to MAS, the present work conducts a similar investigation for the case of the EIE. For the same two simple problems, it is shown that similar difficulties do not occur: Solvability is shown by two different analytical methods, while the lack of “MAS-type” oscillations is a consequence of the demonstrated convergence of the discretized solution to the true solution. For a 2-D problem with an elliptical scatterer, oscillations resembling those of MAS are observed. Careful numerical experiments, however, reveal that these oscillations are of an entirely different nature.

REFERENCES [1] P. C. Waterman, “Matrix formulation for electromagnetic scattering,” Proc. IEEE, vol. 53, pp. 805–812, 1965. [2] P. C. Waterman, “New formulation of acoustic scattering,” J. Acoust. Society Amer., vol. 45, pp. 1417–1429, 1968. [3] P. A. Martin, “On the null-field equations for the exterior problems of acoustics,” Quart. J. Mechan. Appl. Math., vol. 33, pp. 385–396, 1980. [4] D. Colton and R. Kress, “The unique solvability of the null field equations of acoustics,” Quart. J. Mechan. Appl. Math., vol. 36, pp. 87–95, 1983. [5] A. G. Dallas, “On the Convergence and Numerical Stability of the Second Waterman Scheme for Approximation of the Acoustic Field Scattered by a Hard Object,” Dept. Math. Sciences, Univ. Delaware, Tech. Rep. 2000-7:1-35, 2000. [6] B. Peterson and S. Ström, “T matrix formulation of electromagnetic scattering from multilayered scatterers,” Phy. Rev. D, vol. 10, pp. 2670–2684, Oct. 1974. [7] T. Wriedt and A. Doicu, “Comparison between various formulations of the extended boundary condition method,” Opt. Commun., vol. 142, pp. 91–98, 1997. [8] W. C. Chew, Waves and Fields in Inhomogeneous Media. New York: IEEE Press, 1995, pp. 453–460. [9] F. M. Kahnert, “Numerical methods in electromagnetic scattering theory,” J. Quant. Spectr. Radiative Transf., vol. 79–80, pp. 775–824, 2003. [10] M. I. Mishchenko, G. Videen, V. A. Babenko, N. I. Khlebtsov, and T. Wriedt, “T-matrix theory of electromagnetic scattering by particles and its applications: A comprehensive reference database,” J. Quant. Spectr. Radiative Transf., vol. 88, pp. 357–406, 2004. [11] R. A. Shore and A. D. Yaghjian, “Dual-surface integral equations in electromagnetic scattering,” IEEE Trans. Antennas Propag., vol. 53, pp. 1706–1709, May 2005. [12] H. A. Schenck, “Improved integral formulation for acoustic radiation problems,” J. Acoust. Society Amer., vol. 44, pp. 41–58, 1967. [13] T. Wriedt, “Review of the null-field method with discrete sources,” J. Quant. Spectr. Radiative Transf., vol. 106, pp. 535–545, 2007. [14] Generalized Multipole Techniques for Electromagnetic and Light Scattering (Vol. 4 in Computational Methods in Mechanics), T. Wriedt, Ed. Amsterdam, The Netherlands: Elsevier, 1999.

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[15] D. I. Kaklamani and H. T. Anastassiu, “Aspects of the method of auxiliary sources (MAS) in computational electromagnetics,” IEEE Antennas Propag. Mag., vol. 44, no. 3, pp. 48–64, Jun. 2002. [16] A. Doicu, Y. Eremin, and T. Wriedt, Acoustic and Electromagnetic Scattering Using Discrete Sources. London, U.K.: Academic Press, 2000. [17] G. Fikioris, “On two types of convergence in the method of auxiliary sources,” IEEE Trans. Antennas Propag., vol. 54, no. 7, pp. 2022–2033, Jul. 2006. [18] G. Fikioris and I. Psarros, “On the phenomenon of oscillations in the method of auxiliary sources,” IEEE Trans. Antennas Propag., vol. 55, no. 5, pp. 1293–1304, May 2007. [19] R. Kress, Linear Integral Equations, 2nd ed. New York: Springer, 1999. [20] P. J. Papakanellos, G. Fikioris, and A. Michalopoulou, “On the oscillations appearing in numerical solutions of solvable and nonsolvable integral equations for thin-wire antennas,” IEEE Trans. Antennas Propag., vol. 58, no. 5, pp. 1635–1644, May 2010. [21] Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables, ser. National Bureau of Standards Applied Mathematics, I. Abramowitz and I. A. Stegun, Eds. Washington, DC: U.S. Government Printing Office, 1972, vol. 55. [22] E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, 4th ed. New York: Cambridge Univ. Press, 1927, sec. sec. 5.31, (reprinted, 1992). [23] 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, no. 8, pp. 1847–1854, Aug. 2003. [24] A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems. New York: John Wily, 1977, p. 82, (translated from the Russian by F. John). [25] V. B. Glasko, Inverse Problems of Mathematical Physics. New York: American Institute of Physics, 1988, p. 9. [26] C. W. Groetsch, Inverse Problems in the Mathematical Sciences. Braunschweig: Vieweg, 1993, p. 92. [27] A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems. New York: Springer, 1996, p. 13. [28] B. Noble, “The numerical solution of integral equations,” in The State of the Art in Numerical Analysis, D. Jacobs, Ed. New York: Academic Press, 1976, ch. VIII.3, p. 939.

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Time Domain UTD-PO Solution for the Multiple Diffraction of Spherical Waves for UWB Signals Peng Liu, Jundong Tan, and Yunliang Long

Abstract—The time domain (TD) solution for the multiple diffraction of spherical waves after an arbitrary number of obstacles with different shapes such as knife-edges and wedges, is presented. The proposed TD solution is based on the representation of the inverse Laplace transform of the corresponding frequency domain solution in closed form, as it is given by a hybrid of the uniform theory of diffraction (UTD)-physic optics (PO) solution. The proposed formulation, validated with the results from technical literature, does not need to incorporate the TD version of the higher-order diffraction coefficients due to a recursive relation, thus significantly reducing both the mathematical complexity and computation time of the formulation when compared with other TD solutions. Index Terms—Radio propagation, time-frequency analysis, ultrawideband (UWB), uniform theory of diffraction (UTD).

I. INTRODUCTION The ultrawideband (UWB) communication system has received considerable attention in recent years [1]. Such system employs the pulses with very short duration, usually a fraction of 1 ns, which cover extremely large bandwidth in the frequency domain (FD). The large bandwidth of the UWB signals offers high bit rates and precise positioning and ranging, however, it also cause the pulse distortion in the time domain (TD) due to the frequency selectivity of attenuation in the channel. It is more convenient and efficient to directly study the UWB pulse propagation in the TD when the pulse is very narrow. Moreover, with the aid of the TD version of the uniform theory of diffraction (UTD), the impulse response of the channel model can be calculated and it is convolved with the transmitted pulse to predict the received signals. By doing this, all the detailed features which are concerned in the UWB system, such as the time delay, the power and the pulse shape/distortion can be easily obtained. The TD result can give more insight, e.g., in baseband system analysis. It is also helpful in determining time delay parameters of the channel in areas such as synchronization, positioning and detection. In radio propagation, the main physical mechanisms that need to be considered are reflections, transmissions and diffractions. In [2], the TD reflection coefficient for half space was presented. The TD reflection and transmission coefficient through a slab were presented in [3]. For the diffraction phenomena, the TD-UTD solutions which can be very accurate for early-to-intermediate time are commonly used. In [4]–[10], TD single or double diffraction solutions for the canonical objects were obtained. However, in real environment, the transmitted signals usually undergo multiple interactions, and the obstacles along the path usually have quite different shapes. Therefore, in order to predict the received signals accurately, it is necessary to extend the single or double diffraction solutions to the multiple diffraction solution. For Manuscript received May 24, 2009; revised March 30, 2010; accepted November 17, 2010. Date of publication January 28, 2011; date of current version April 06, 2011. This work was supported in part by the NSFC-Guangdong (U0635003, U0935002) and in part by the Research Program of Guangzhou (2010Y1-C401). The authors are with the Department of Electronics and Communication Engineering, Sun Yat-Sen University, 510006 Guangzhou, China (e-mail: [email protected]). Digital Object Identifier 10.1109/TAP.2011.2109346

Fig. 1. Geometry of spherical wave incidence over an array of different obstacles with a constant spacing w .

such multiple diffraction phenomena, R. Qiu first derived the channel impulse response of Wolfish-Bertoni’s urban propagation configuration in a closed form and quantified the impacts of pulse distortion on a UWB system performance [11]. Karousos and Tzaras have presented a TD-UTD solution in [12] for the multiple-diffraction caused by a cascade of multi-modeled obstacles such as knife-edges and wedges. Such solution is obtained by incorporating the TD representation of the higher-order diffraction coefficients which provided accurate results while at the same time increasing both the mathematical complexity and computation time of the formulation. In this communication, a new TD solution in terms of TD-UTD coefficients for modeling the multiple-diffraction of spherical waves by an array of multi-modeled obstacles such as knife-edges and metallic or nonperfectly conducting wedges is presented. Such solution does not need to incorporate the TD representation of the higher-order diffraction coefficients and only single diffraction is involved in the calculation, which provide an easier and faster method without any compromise on the accuracy of the results. The TD solution is valid for obstacles with the same height, which is relative to the base station antenna height, and separated by a constant distance. II. THEORETICAL MODELS The propagation environment considered in this work is shown in Fig. 1, where an array of multi-modeled obstacles (absorbing knifeedges, metallic or nonperfectly conducting wedges with the same interior angles) has been taken into account. The obstacles are assumed to have the same height which is relative to the base station antenna height and separated by a constant distance w . The transmitter is at an arbitrary height and is located at a certain distance d from the array of structure. The incident spherical wave impinges on the first obstacle with an angle . We will first borrow some well-known FD results from radio propagation literature and extend it to a more general solution. Then, the inverse Laplace transform is applied to obtain the TD solution. A. Frequency Domain Model For the scenario in Fig. 1, a hybrid of UTD-PO formulation is presented by extending the solution in [13] for the analysis of multiplediffraction caused by an array of obstacles consisting of only knifeedges or only wedges. Therefore, just as the formulations given in [13], the new solution does not need to incorporate the slope or higher-order diffraction coefficients due to a recursive relation in which only single diffractions are involved in the calculation even for the overlapping transition case; that is, if we are dealing with the shadow boundary when the source, the obstacles and the field point are on a straight line. Regarding the above, for N  1, the electric field at the reference point in Fig. 1 can be expressed as:

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(i) Base station antenna above or level with the average height of the obstacles (  0)

EN (!) = 1 N

2

N01

Em (!) m=0 s0 e0jk(s 0s sN0m +

(N

)

B. Time Domain Model In this section, our main task is to derive the TD version of (1) and (8) to predict the received signals directly in the TD. The TD expression will give us more insight into UWB applications. Therefore, for base station antenna above or level with the average height of the obstacles (  0), (1) is translated into the TD as

eN (t) = 1 N

s0

0 m)w [s0 + (N 0 m)w]

2D(!; Lm )e0jk(N0m)w

1421

3

(1)

where Em (! ) is the electric field at the m + 1th obstacle and

i 0jks E0 (!) = E e s0

(2)

is the electric field at the first obstacle and E i the relative amplitude of a spherical source in Fig. 1, k is the wave number, sN0m is the distance from the source to the N 0 m + 1th obstacle in Fig. 1, w is the constant spacing between obstacles and D(!; Lm ) is the UTD coefficient which can be calculated by different formulations according to the modeling of the obstacles. For the absorbing knife-edge, the UTD coefficient is given in [14]

D(!; Lm ) =

0 exp( p0j=4) 2

2k

1

cos( =2)

F 2kLm cos2 ( =2)

(3)

with  = 3=2 and 0 = =2+  are the diffraction and incident angle, respectively. =  0 0 . For nonperfectly conducting wedges, the diffraction coefficient is given in [15]

D(!; Lm ) = D1 + R0s;h Rns;h D2 + R0s;h D3 + Rns;h D4

(4)

where R0;n is the Fresnel reflection coefficient for the zero- and n-face. If the wedge is metallic, then R0;n is 01 for soft and +1 for hard polarization. Also, there is

Di (!) =

0 exp(p0j=4) cot(ai )F 2kLm n2 sin2 (ai ) 2n

2k

(5)

where

a1 =  0 ( 0 0 ) =2n; a2 =  + ( 0 0 ) =2n a3 =  0 ( + 0 ) =2n; a4 =  + ( + 0 ) =2n

(6)

with  = 3=2 0 =2 and 0 = =2 0 =2 + , n is the exterior angle of the wedge. The distance parameters Lm that are used in the above formulas can be obtained as

Lm = s0 (N 0 m)w m = 0; 1; . . . . . . ; N 0 1: s0 + (N 0 m)w