Learn about the most recent theoretical and practical advances in radar signal processing using tools and techniques fro

*1,432*
*203*
*16MB*

*English*
*Pages 378
[396]*
*Year 2019*

- Author / Uploaded
- Antonio De Maio (editor)
- Yonina C. Eldar (editor)
- Alexander M. Haimovich (editor)

*Table of contents : CoverFront MatterCompressed Sensing in RadarSignal ProcessingCopyrightDedicationContentsContributorsIntroductionSymbols1 Sub-Nyquist Radar: Principlesand Prototypes2 Clutter Rejection and AdaptiveFiltering in CompressedSensing Radar3 RFI Mitigation Based on CompressiveSensing Methods for UWBRadar Imaging4 Compressed CFAR Techniques5 Sparsity-Based Methods for CFARTarget Detection in STAPRandom Arrays6 Fast and Robust Sparsity-BasedSTAP Methods for NonhomogeneousClutter7 Super-Resolution Radar Imagingvia Convex Optimization8 Adaptive Beamforming viaSparsity-Based Reconstructionof Covariance Matrix9 Spectrum Sensing for CognitiveRadar via Model Sparsity Exploitation10 Cooperative Spectrum Sharingbetween Sparse Sensing-BasedRadar and Communication Systems11 Compressed Sensing Methods forRadar Imaging in the Presence ofPhase Errors and Moving ObjectsIndex*

Compressed Sensing in Radar Signal Processing Learn about the most recent theoretical and practical advances in radar signal processing using tools and techniques from compressive sensing. Providing a broad perspective that fully demonstrates the impact of these tools, the accessible and tutorial-like chapters cover topics such as clutter rejection, CFAR detection, adaptive beamforming, random arrays for radar, space–time adaptive processing, and MIMO radar. Each chapter includes coverage of theoretical principles, a detailed review of current knowledge, and discussion of key applications, and also highlights the potential benefits of using compressed sensing algorithms. A unified notation and numerous cross-references between chapters make it easy to explore different topics side by side. Written by leading experts from both academia and industry, this is the ideal text for researchers, graduate students, and industry professionals working in signal processing and radar. Antonio De Maio is a professor in the Department of Electrical Engineering and Information Technology at the University of Naples Federico II, and a Fellow of the IEEE. Yonina C. Eldar is a professor at the Weizmann Institute of Science. She has authored and edited several books, including Sampling Theory: Beyond Bandlimited Systems and Compressed Sensing: Theory and Applications (Cambridge University Press, 2015; 2012). She is a Fellow of the IEEE and EURASIP, and a member of the Israel National Academy of Science and Humanities. Alexander M. Haimovich is a distinguished professor in the Department of Electrical and Computer Engineering at the New Jersey Institute of Technology, and a Fellow of the IEEE.

Compressed Sensing in Radar Signal Processing Edited by

ANTONIO DE MAIO University of Naples Federico II

YONINA C. ELDAR Weizmann Institute of Science

ALEXANDER M. HAIMOVICH New Jersey Institute of Technology

University Printing House, Cambridge CB2 8BS, United Kingdom One Liberty Plaza, 20th Floor, New York, NY 10006, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia 314–321, 3rd Floor, Plot 3, Splendor Forum, Jasola District Centre, New Delhi – 110025, India 79 Anson Road, #06–04/06, Singapore 079906 Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning, and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781108428293 DOI: 10.1017/9781108552653 © Cambridge University Press 2020 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2020 Printed in the United Kingdom by TJ International Ltd, Padstow Cornwall A catalogue record for this publication is available from the British Library. Library of Congress Cataloging-in-Publication Data Names: De Maio, Antonio, 1974– editor. | Eldar, Yonina C., editor. | Haimovich, Alexander M., 1954– editor. Title: Compressed sensing in radar signal processing / edited by Antonio De Maio, University of Naples Federico II, Yonina C. Eldar, Weizmann Institute of Science, Alexander M. Haimovich, New Jersey Institute of Technology. Description: First edition. | Cambridge, United Kingdom ; New York, NY : Cambridge University Press, [2020] | Includes bibliographical references and index. Identifiers: LCCN 2019014859 | ISBN 9781108428293 (hardback) Subjects: LCSH: Radar. | Compressed sensing (Telecommunication) Classification: LCC TK6580 .C66 2020 | DDC 621.3848/3–dc23 LC record available at https://lccn.loc.gov/2019014859 ISBN 978-1-108-42829-3 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

To my daughter Claudia: my light, my hope, my love – ADM To my husband Shalomi and children Yonatan, Moriah, Tal, Noa, and Roei for their boundless love and for filling my life with endless happiness – YE To my students and collaborators for their contributions to my work on radar – AH

Contents

List of Contributors Introduction List of Symbols 1

Sub-Nyquist Radar: Principles and Prototypes

page xi xiv xx 1

Kumar Vijay Mishra and Yonina C. Eldar

2

1.1 Introduction 1.2 Prior Art and Historical Notes 1.3 Temporal Sub-Nyquist Radar 1.4 Doppler Sub-Nyquist Radar 1.5 Cognitive Sub-Nyquist Radar and Spectral Coexistence 1.6 Spatial Sub-Nyquist: Application to MIMO Radar 1.7 Sub-Nyquist SAR 1.8 Summary References

1 3 5 15 18 29 39 43 44

Clutter Rejection and Adaptive Filtering in Compressed Sensing Radar

49

Peter B. Tuuk

3

2.1 Introduction 2.2 Problem Formulation 2.3 Interference Sources 2.4 Signal Processing Treatment of Clutter 2.5 Measurement Compression 2.6 Estimating Interference Statistics from Compressed Measurements 2.7 Mitigating Clutter in Compressed Sensing Estimation 2.8 Summary References

49 50 53 55 58 59 66 68 69

RFI Mitigation Based on Compressive Sensing Methods for UWB Radar Imaging

72

Tianyi Zhang, Jiaying Ren, Jian Li, David J. Greene, Jeremy A. Johnston, and Lam H. Nguyen

3.1 Introduction 3.2 RPCA for RFI Mitigation 3.3 CLEAN-BIC for RFI Mitigation

72 75 82 vii

viii

4

Contents

3.4 Enhanced Algorithms for RFI Mitigation 3.5 Performance Evaluations 3.6 Conclusions 3.7 Acknowledgment References

91 92 101 102 102

Compressed CFAR Techniques

105

Laura Anitori and Arian Maleki

5

4.1 Introduction 4.2 Radar Signal Model 4.3 Classical Radar Detection 4.4 CS Radar Detection 4.5 Complex Approximate Message Passing (CAMP) Algorithm 4.6 Target Detection Using CAMP 4.7 Adaptive CAMP Algorithm 4.8 Simulation Results 4.9 Experimental Results 4.10 Conclusions References

105 105 106 110 112 115 118 120 127 131 132

Sparsity-Based Methods for CFAR Target Detection in STAP Random Arrays

135

Haley H. Kim and Alexander M. Haimovich

6

5.1 Introduction 5.2 STAP Radar Concepts 5.3 STAP Detection Problem 5.4 Compressive Sensing CFAR Detection 5.5 Numerical Results 5.6 Summary References

135 137 145 148 157 161 162

Fast and Robust Sparsity-Based STAP Methods for Nonhomogeneous Clutter

165

Xiaopeng Yang, Yuze Sun, Xuchen Wu, Teng Long, and Tanpan K. Sarkar

7

6.1 Introduction 6.2 Signal Models 6.3 Sparsity Principle Analysis of STAP 6.4 Fast and Robust Sparsity-Based STAP Methods 6.5 Conclusions References

165 166 168 172 190 190

Super-Resolution Radar Imaging via Convex Optimization

193

Reinhard Heckel

7.1 Introduction

193

8

Contents

ix

7.2 Signal Model and Problem Statement 7.3 Atomic Norm Minimization and Associated Performance Guarantees 7.4 Super-Resolution Radar on a Fine Grid 7.5 Proof Outline 7.6 MIMO Radar 7.7 Discussion and Current and Future Research Directions References

195 199 204 207 211 219 222

Adaptive Beamforming via Sparsity-Based Reconstruction of Covariance Matrix

225

Yujie Gu, Nathan A. Goodman, and Yimin D. Zhang

9

8.1 Introduction 8.2 Adaptive Beamforming Criterion 8.3 Covariance Matrix Reconstruction-Based Adaptive Beamforming 8.4 Simulation Results 8.5 Conclusion References

225 228 234 240 252 252

Spectrum Sensing for Cognitive Radar via Model Sparsity Exploitation

257

Augusto Aubry, Vincenzo Carotenuto, Antonio De Maio, and Mark A. Govoni

10

9.1 Introduction 9.2 System Model and Problem Formulation 9.3 2-D Radio Environmental Map Recovery Strategies 9.4 Performance Analyses 9.5 Conclusions References

257 259 263 270 280 280

Cooperative Spectrum Sharing between Sparse Sensing-Based Radar and Communication Systems

284

Bo Li and Athina P. Petropulu

11

10.1 Introduction 10.2 MIMO Radars Using Sparse Sensing 10.3 Coexistence System Model 10.4 Cooperative Spectrum Sharing 10.5 Numerical Results 10.6 Conclusions References

284 286 293 297 309 315 316

Compressed Sensing Methods for Radar Imaging in the Presence of Phase Errors and Moving Objects

321

Ahmed Shaharyar Khwaja, Naime Ozben Onhon, and Mujdat Cetin

11.1 Introduction and Outline of the Chapter 11.2 Compressed Sensing and Radar Imaging

321 322

x

Contents

11.3 Synthetic Aperture Radar Autofocus and Compressed Sensing 11.4 Synthetic Aperture Radar Moving Target Imaging and Compressed Sensing 11.5 Inverse Synthetic Aperture Radar Imaging and Compressed Sensing 11.6 Conclusions References

328 333 341 349 349

Index

355

Contributors

Laura Anitori Netherlands Organisation for Applied Scientific Research (TNO) Augusto Aubry University of Naples Federico II Vincenzo Carotenuto University of Naples Federico II Mujdat Cetin University of Rochester; Sabanci University Antonio De Maio University of Naples Federico II Yonina C. Eldar Weizmann Institute of Science David J. Greene University of Florida Nathan A. Goodman University of Oklahoma Mark A. Govoni US Army Research Laboratory Yujie Gu Temple University Alexander M. Haimovich New Jersey Institute of Technology

xi

xii

List of Contributors

Reinhard Heckel Rice University Jeremy A. Johnston University of Florida Ahmed Shaharyar Khwaja Sabanci University Haley H. Kim New Jersey Institute of Technology Bo Li Qualcomm Jian Li University of Florida Teng Long Beijing Institute of Technology Arian Maleki Columbia University Kumar Vijay Mishra Technion Israel Institute of Technology Lam H. Nguyen US Army Research Laboratory Naime Ozben Onhon Turkish-German University Athina P. Petropulu Rutgers, State University of New Jersey Jiaying Ren University of Florida Tapan K. Sarkar Syracuse University Yuze Sun Tsinghua University

List of Contributors

Peter B. Tuuk Georgia Tech Research Institute Xuchen Wu Beijing Institute of Technology Xiaopeng Yang Beijing Institute of Technology Tianyi Zhang University of Florida Yimin D. Zhang Temple University

xiii

Introduction

Digital signal processing (DSP) is a revolutionary paradigm shift that enables processing of physical data in the digital domain, where design and implementation are considerably simplified. The success of DSP has driven the development of sensing and processing systems that are more robust, flexible, cheaper, and, consequently, more widely used than their analog counterparts. As a result of this success, the amount of data generated by sensing systems has grown considerably. Furthermore, in modern applications, signals of wider bandwidth are used in order to convey more information and to enable high resolution in the context of imaging. Unfortunately, in many important and emerging applications, the resulting sampling rate is so high that far too many samples need to be transmitted, stored, and processed. In addition, in applications involving very wideband inputs it is often very costly, and sometimes even physically impossible, to build devices capable of acquiring samples at the necessary rate. Thus, despite extraordinary advances in sampling theory and computational power, the acquisition and processing of signals in application areas such as radar, wideband communications, imaging, and medical imaging continue to pose a tremendous challenge. Recent advances in compressed sensing (CS) and sampling theory provide a framework to acquire a wide class of analog signals at rates below the Nyquist rate, and to perform processing at this lower rate as well. Together with the theory, various prototypes have been developed that demonstrate the feasibility of sampling and processing signals at sub-Nyquist rates in a robust and cost-effective fashion. More specifically, CS is a framework that enables acquisition and recovery of sparse vectors from underdetermined linear systems. This research area has seen enormous growth over the past decade and has been explored in many areas of applied mathematics, computer science, statistics, and electrical engineering. At its core, CS enables recovery of sparse high-dimensional vectors from highly incomplete measurements using very efficient optimization algorithms. More specifically, consider a vector x of length n. The vector is said to be k-sparse if it has at most k nonzero components. More generally, CS results apply to signals that are sparse in an appropriate basis or overcomplete representation. The main idea underlying CS is that the vector x can be recovered from measurements y = Ax, where y is of length m n as long as A satisfies certain mathematical properties that render it a suitable CS matrix. The number of measurements m can be chosen on the order of k log n, which in general is much smaller than the length of the vector x. A large body of work has been published on a variety of optimization algorithms that can recover x efficiently and robustly when m ≈ k log n. Loosely xiv

Introduction

xv

speaking, the theory of CS deals with conditions under which the recovery of information has vanishing or small errors. The mathematical framework of CS has inspired new acquisition methods and new signal processing applications in a large variety of areas, including image processing, analog to digital conversion, communication systems, and radar processing. In many of these examples the basic ideas underlying CS need to be extended to include, for example, continuous-time inputs, practical sampling methods, other forms of structure on the input, computational aspects, noise affects, different metrics for recovery performance, nonlinear acquisition methods, and more. Two books devoted to this topic have been published recently, which focus on many of these aspects, as well as on the underlying mathematical results [1,2]. Their main emphasis is on the basic underlying theory and its generalizations, optimization methods, as well as applications primarily to image processing and analog-to-digital conversion. The latter is also covered in depth in [3]. Radar signal processing represents a fertile field for CS applications. By their very nature, radars collect data about surveillance volumes (search radars), targets (tracking radars), terrain and ground targets (imaging radars), or buried objects (radar tomography). From radar’s early days in World War II, through the emergence of digital radar in the 1970s, to today’s advanced systems, the amount of data a radar system has to handle has increased by orders of magnitude. While early digital radars had to contend with 10s and 100s of kbps, today’s radars may be faced with data rates in the Gbps range or more, leading to demanding requirements in cost, hardware, data storage, and processing. The implications of applying CS to radar are potentially enormous: sampling rates could be lowered, the number of antenna elements in large arrays might be reduced and the computers required to handle the data may be downsized. This book aims to present the latest theoretical and practical advances in radar signal processing using tools from CS. In particular, this book offers an up-to-date review of fundamental and practical aspects of sparse reconstruction in radar and remote sensing, demonstrating the potential benefits achievable with the CS paradigm. We take a wider scope than previous edited books on CS-based radars: we do not restrict ourselves to specific disciplines (such as earth observation as in [4]) or applications (such as urban sensing as in [5]), but discuss a variety of diverse application fields, including clutter rejection, constant false alarm rate (CFAR) processing, adaptive beamforming, random arrays for radar, space–time adaptive processing (STAP), multiple input multiple output (MIMO) systems, radar super-resolution, cognitive radar [6] applications involving subNyquist sampling and spectrum sensing, radio frequency interference (RFI) suppression, and synthetic aperture radar (SAR). The book is aimed at postgraduate students, PhD students, researchers, and engineers working on signal processing and its applications to radar systems, as well as researchers in other fields seeking an understanding of the potential applications of CS. To read and fully understand the content it is assumed that the reader has some background in probability theory and random processes, matrix theory, linear algebra, and optimization theory, as well as radar systems. The book is organized into eleven chapters broadly cathegorized into five areas: sub-Nyquist radar (Chapter 1); detection, clutter/interference mitigation, and CFAR techniques (Chapters 2–6); super-resolution

xvi

Introduction

and beamforming (Chapters 7 and 8); radar spectrum sensing/sharing (Chapters 9 and 10); radar imaging (Chapter 11). Each chapter is self-contained and typically covers three main aspects: fundamental theoretical principles, overview of the current state of the art, and emerging/future research directions. Some chapters are also complemented with analyses on real data. Since the chapters are independent, there is flexibility in selecting material both for university courses and short seminars. In Chapter 1, the authors review several sub-Nyquist pulse-Doppler radar systems based on the Xampling framework. Contrary to other CS-based designs, their formulations directly address the reduced-rate analog sampling in space and time, avoid a prohibitive dictionary size, and are robust in the face of noise and clutter. The chapter begins by introducing temporal sub-Nyquist processing for estimating the target locations using less bandwidth than conventional systems. This paves the way to cognitive radars, which share their transmit spectrum with other communication services, thereby providing a robust solution for coexistence in spectrally crowded environments. Next, without impairing Doppler resolution, the authors reduce the dwell time by transmitting interleaved radar pulses in a scarce manner within a coherent processing interval or slow time. Then, they consider MIMO array radars and demonstrate spatial sub-Nyquist processing, which allows the use of few antenna elements without degradation in angular resolution. Finally, they demonstrate application of sub-Nyquist and cognitive radars to imaging systems such as SAR. For each setting, the authors present a stateof-the-art hardware prototype designed to demonstrate the real-time feasibility of sub-Nyquist radars. Chapter 2 discusses the problem of clutter mitigation, which has posed challenges to radar designers and engineers since the early days of radar. Early techniques matured to current approaches like STAP, which use a coherently processed data cube to estimate clutter statistics and to perform adaptive filtering. This chapter examines CS techniques for the mitigation of structured interference, such as clutter. The author first introduces the relevant sensing model and describes results in uncompressed adaptive filtering. This paves the way to the development of models for measurement compression of the coherent data cube and of approaches to estimate and filter clutter from compressed measurements. The chapter includes recent results showing how clutter second-order statistics can be reliably estimated from compressed measurements if the clutter has well-controlled eigenspectrum. Additionally, the covariance of the interference can be incorporated into the CS estimation process to improve performance. RFIs pose serious threats to the proper operations of ultra wideband (UWB) radar systems due to severely degrading their imaging and target detection capabilities. RFI mitigation is a challenging problem, since dynamic RFI sources utilize diverse modulation schemes, hence they are difficult to model precisely. Fortunately, RFI sources possess certain unique properties that can be exploited for their mitigation. In Chapter 3 the authors propose several sparse signal recovery methods for effective RFI mitigation. They first show that the RFI sources possess a low rank property and are sparse in the frequency domain, while in contrast the desired UWB radar echoes are sparse in the time domain. Therefore, robust principal component analysis (RPCA) can be used to simultaneously exploit these properties for effective RFI mitigation. RPCA, however, requires

Introduction

xvii

a fine tuning of a user parameter, which is dependent on the signal-to-interference ratio (SIR). This parameter tuning is not straightforward in practice due to the lack of prior knowledge on the RFI sources and on the desired UWB radar echoes. To avoid the user parameter tuning problem, the authors consider modeling the RFI sources within a pulse repetition interval (PRI) as a sum of sinusoids. The CLEAN algorithm can then be used with the Bayesian information criterion (BIC) to determine the number of sinusoids and to estimate their parameters. They show that CLEAN-BIC is userparameter-free and can be used to remove dominant RFI sources effectively. However, since the sparse property of the UWB radar echoes are not utilized by CLEAN-BIC, the resulting SAR images appear noisy, especially for low SIR values. To take advantage of the merits of both RPCA and CLEAN-BIC algorithms, the authors consider using CLEAN-BIC to estimate SIR, and the estimated SIR value is then used to determine the user parameter for the RPCA algorithm. Finally, the algorithms are applied to both simulated and experimentally measured data for performance evaluation. Chapter 4 is focused on target detection from a set of compressive radar measurements corrupted by additive white Gaussian noise. The complications in the calculation of false alarm and detection probabilities that are caused by the nonlinear nature of target recovery schemes in CS have impeded the application of such systems in practice. In this chapter, the authors aim to show how recent advances in the asymptotic analysis of CS recovery algorithms help to overcome this challenge. Fully adaptive and practical CS target detection schemes are provided together with a detailed analysis of their performance through extensive simulated and experimental data. In Chapter 5, the authors present CFAR detectors for STAP random arrays. The problem is formulated as detection of sparse targets given space–time observations from thinned random arrays. The observations are corrupted by colored Gaussian noise of an unknown covariance matrix, but secondary data are available for estimating the covariance matrix. It is shown that the number of elements required to constrain the peak sidelobe level scales logarithmically with the array aperture, whereas the number of elements of a uniform linear array (ULA) scales linearly with the array aperture. New adaptive detectors are developed that cope with the high sidelobes of random arrays. Performance and complexity analysis demonstrate high performance at a reasonable computation cost with significantly fewer elements than a ULA. In Chapter 6, sparse-based STAP methods are developed by exploiting the intrinsic sparsity of the clutter spatial-temporal power spectrum and of the space–time adaptive weight vectors. First, the signal model of received space–time data for an airborne phased array radar is introduced, and the intrinsic model sparsity for radar STAP is analyzed. Second, leveraging on the sparsity of clutter spatial-temporal power spectrum, a robust and fast iterative sparse recovery method is introduced. It can not only alleviate the effect of noise and dictionary mismatch but can also reduce the computational complexity via recursive inverse matrix calculation. Finally, based on the sparsity of space– time adaptive weight vectors, a fast STAP method based on projection approximation subspace tracking (PAST) with a sparse constraint is discussed. It provides a robust and stable estimation of the clutter subspace when a small set of training samples is available. Based on both the simulated and actual airborne phased array radar data, it is

xviii

Introduction

verified that the developed methods can provide satisfactory performance with a small training sample support in a practical complex nonhomogeneous environment. Chapter 7 considers the use of CS techniques for the resolution of multiple targets. Estimating the relative angles, delays, and Doppler shifts from the received signals allows for the determination of the locations and velocities of objects. However, due to practical constraints, the probing signals have finite bandwidth B, the received signals are observed over a finite time interval of length T only, and in addition, a radar typically has only one or a few transmit and receive antennas. Those constraints fundamentally limit the resolution up to which objects can be localized: the delay and Doppler resolution is proportional to 1/B and 1/T , and a radar with NT transmit and NR receive antennas can only achieve an angular resolution proportional to 1/(NT NR ). The author shows that the continuous angle-delay-Doppler triplets and the corresponding attenuation factors can be resolved at much finer resolution, using ideas from CS. Specifically, provided the angle-delay-Doppler triplets are separated either by factors proportional to 1/(NT NR − 1) in angle, 1/B in delay, or 1/T in Doppler direction, they can be recovered at significantly smaller scale or higher resolution. Traditional adaptive beamformers are very sensitive to model mismatch, especially when the training samples for adaptive beamformer design are contaminated by the desired signal. In Chapter 8, the authors propose a strategy to reconstruct a signalfree interference-plus-noise covariance matrix for adaptive beamformer design. Using the sparsity of sources, the interference covariance matrix can be reconstructed as a weighted sum of the tensor outer products of the interference steering vectors, and the corresponding parameters are estimated from a sparsity-constrained covariance matrix fitting problem. In contrast to classical CS and sparse reconstruction problems, the formulated sparsity-constrained covariance matrix fitting problem can be effectively solved by using the a priori information on array structure rather than using convex relaxation. Simulation results demonstrate that the proposed adaptive beamformer almost always provides near-optimal performance. Chapter 9 deals with two-dimensional (2-D) spectrum sensing in the context of a cognitive radar to gather real-time space–frequency electromagnetic awareness. Assuming a sensor equipped with multiple receive antennas, a formal discrete-time sensing signal model is developed, and two signal processing techniques capable of recovering the space–frequency occupancy map via block sparsity exploitation are presented. The former relies on the iterative adaptive algorithm (IAA) and incorporates a BIC-based stage to foster block-sparsity in the recovery process. The latter resorts to the regularized maximum likelihood (RML) estimation paradigm, which automatically promotes blocksparsity in the 2-D profile evaluation. Some illustrative examples (both on simulated and real data) are provided to compare the different strategies and highlight the effectiveness of the developed approaches. In Chapter 10, a cooperative spectrum-sharing scheme for a MIMO communication system and a sparse sensing-based MIMO radar is presented. Both the radar and the communication systems use transmit precoding. The radar transmit precoder, the radar subsampling scheme, and the communication transmit covariance matrix are jointly designed in order to maximize the radar SIR, while meeting certain communication

Introduction

xix

rate and power constraints. The joint design is implemented at a control center, which is a node with which both systems share physical layer information, and which also performs data fusion for the radar. Efficient algorithms for solving the corresponding optimization problem are presented. The cooperative design significantly improves spectrum sharing performance, and the sparse sensing provides opportunities to control interference. Chapter 11 discusses applications of CS to radar imaging problems with reference to SAR and inverse synthetic aperture radar (ISAR) sensors. The authors first provide the relevant mathematical expressions for CS and SAR necessary to formulate the problem of CS SAR imaging. Thereafter, they consider the case where unknown motion errors are present during the SAR acquisition process. Autofocusing, i.e., the blind compensation of the aforementioned errors, is discussed, and general CS solutions are presented. The chapter ends with a survey of CS methods for ISAR imaging of targets with unknown motion.

References [1] Y. C. Eldar and G. Kutyniok, Compressed Sensing: Theory and Applications. Cambridge University Press, 2012. [2] S. Foucart and H. Rauhut, A Mathematical Introduction to Compressive Sensing. Birkhäuser Basel, 2013, vol. 1, no. 3. [3] Y. C. Eldar, Sampling Theory: Beyond Bandlimited Systems. Cambridge University Press, 2015. [4] C.-H. Chen, Compressive Sensing of Earth Observations. CRC Press, 2017. [5] M. Amin, Compressive Sensing for Urban Radar. CRC Press, 2014. [6] A. Farina, A. De Maio, and S. Haykin, The Impact of Cognition on Radar Technology. Scitech Publishing, Radar, Sonar & Navigation, 2017.

Symbols

A unified notation is used throughout the book. z Z zi Zi,l A = A y x k n m · p (·)T (·)∗ (·)H (·)† tr (·) Rank (·) λmax (·) λmin (·) diag(x) Range (A) I 0 RN CN HN

xx

column vector (lower case) matrix (upper case) ith element of z (i,l)-th entry of Z sensing matrix sparsity matrix product observed measurement vector original signal vector sparsity ambient dimension number of measurements p-norm transpose operator conjugate operator conjugate transpose operator pseudo inverse of the matrix argument trace of the square matrix argument rank of the square matrix argument maximum eigenvalue of the square matrix argument minimum eigenvalue of the square matrix argument N -dimensional diagonal matrix whose ith diagonal element is xi , i = 1,. . .,N, with x ∈ CN range span of the column vectors of the matrix A identity matrix (its size is determined from the context) matrix with zero entries (its size is determined from the context) set of N -dimensional vectors of real numbers set of N -dimensional vectors of complex numbers set of N × N Hermitian matrices for any A ∈ HN , A 0 means that A is a positive semidefinite matrix for any A ∈ HN , A 0 means that A is a positive definite matrix

List of Symbols

j Re(x) Im(x) |x| arg(x) E [·] ⊗ ∂y dy , dx y, ˙ ∂x

standard notation for sets (uppercase letter) cardinality of a set T result of 1 minimization/recovery algorithm support of vector x standard notation for subset of indices length-|T | sub-vector containing the elements of x corresponding to the indices in T m × |T | sub-matrix containing the columns of the m × n matrix A indexed by T imaginary unit real part of the complex number x imaginary part of the complex number x modulus of the complex number x argument of the complex number x statistical expectation Hadamard product Kronecker product first derivative of y with respect to variable x

y, ¨ ∂∂xy2 , ddxy2 P[·] x(t) h(t) xi δk = δk (A)

second derivative of y with respect to variable x probability measure continuous time signal pulse shape measurements of x(t) restricted isometry constant.

T |T | xˆ supp(x) I xT AT

2

2

xxi

Statement of restricted isometry property (RIP): a matrix A satisfies the RIP of order K if (1 − δk )x2 ≤ Ax2 ≤ (1 + δk )x2 for all x with x0 ≤ K.

1

Sub-Nyquist Radar: Principles and Prototypes Kumar Vijay Mishra∗ and Yonina C. Eldar∗∗

1.1

Introduction Radar remote sensing has advanced tremendously since 1950 and is now applied to diverse areas such as military surveillance, meteorology, geology, collision avoidance, and imaging [1]. In monostatic pulse-Doppler radar systems, an antenna transmits a periodic train of known narrowband pulses within a defined coherent processing interval (CPI). When the radiated wave from the radar interacts with moving targets, the amplitude, frequency, and polarization states of the scattered wave change. By monitoring this change, it is possible to infer the targets’ size, location, and radial Doppler velocity. The reflected signal received by the radar antenna is a linear combination of echoes from multiple targets; each of these is an attenuated, time-delayed, and frequency-modulated version of the transmit signal. The delay in the received signal is linearly proportional to the target’s range or its distance from the radar. The frequency modulation encodes the Doppler velocity of the target. The complex amplitude or target’s reflectivity is a function of the target’s size, geometry, propagation, and scattering mechanism. Radar signal processing is aimed at detecting the targets and estimating their parameters from the output of this linear, time-varying system. Traditional radar signal processing employs matched filtering (MF) or pulse compression [2] in the digital domain, wherein the sampled received signal is correlated with a replica of the transmit signal in the delay-Doppler plane. The MF maximizes the signal-to-noise ratio (SNR) in the presence of additive white Gaussian noise. In some specialized systems, this stage is replaced by a mismatched filter with a different optimization metric such as minimization of peak-to-sidelobe ratio of the output. Here, the received signal is correlated with a signal that is close but not identical to the transmit signal [3–5]. While all of these techniques reliably estimate target parameters, their resolution is inversely proportional to the support of the ambiguity function of the transmit pulse, thereby restricting ability to super-resolve targets that are closely spaced. The digital MF operation requires the signal to be sampled at or above the Nyquist sampling rate, which guarantees perfect reconstruction of a bandlimited analog signal [6]. Many modern radar systems use wide bandwidths, typically ranging from hundreds ∗ K.V.M. acknowledges partial support via the Andrew and Erna Finci Viterbi Postdoctoral Fellowship and the Lady Davis Postdoctoral Fellowship. ∗∗ This work is supported by the European Union’s Horizon 2020 research and innovation program under grant agreement no. 646804-ERC-COG-BNYQ.

1

2

Mishra and Eldar

of MHz to GHz, in order to achieve fine radar range resolution. Since the Nyquist sampling rate is twice the baseband bandwidth, the radar receiver requires expensive, highrate analog-to-digital converters (ADCs). The sampled signal is then also processed at high rates, resulting in significant power, cost, storage, and computational overhead. Recently, in order to mitigate this rate bottleneck, new methods have been proposed that sample signals at sub-Nyquist rates and yet are able to estimate the targets’ parameters [6,7]. Analogous trade-offs arise in other aspects of radar system design. For example, the number of transmit pulses governs the resolution in Doppler velocity. The estimation accuracy of target parameters is greatly affected by the radar’s dwell time [1], i.e., the time duration a directional radar beam spends hitting a particular target. Long dwell times imply a large number of transmit pulses and, therefore, high Doppler precision. But, simultaneously, this negatively affects the ability of the radar to look at targets in other directions. Sub-Nyquist sampling approaches have, therefore, been suggested for the pulse dimension or “slow-time” domain in order to break the link between dwell time and Doppler resolution [8–10]. Finally, radars that deploy antenna arrays deal with similar sampling problems in the spatial domain. A phased array radar antenna consists of several radiating elements that form a highly directional radiating beam pattern. Without requiring any mechanical motion, a phased array accomplishes beam-steering electronically by adjusting the relative phase of excitation in the array elements. The operational advantage is the agile scanning of the target scene, ability to track a large number of targets, and efficient search-and-track in the regions of interest [11]. The beam pattern of individual array elements, array geometry, and its size define the overall antenna pattern [12,13], wherein high spatial resolution is achieved by large array apertures. As per the Nyquist Theorem, the array must not admit fewer than two signal samples per spatial period (i.e., radar’s operating wavelength) [14]. Otherwise, it introduces spatial aliasing or multiple beams in the antenna pattern, thereby reducing its directivity. Often an exceedingly large number of radiating elements are required to synthesize a given array aperture in order to enhance the radar’s ability to unambiguously distinguish closely spaced targets; the associated cost, weight, and area may be unacceptable. It is therefore desirable to apply sub-Nyquist techniques to thin a huge array without causing degradation in spatial resolution [15–17]. Sub-Nyquist sampling leads to the development of low-cost, power-efficient, and small-size radar systems that can scan faster and acquire larger volumes than traditional systems. Apart from design benefits, other applications of such systems have been envisioned recently, including imparting hardware-feasible cognitive abilities to the radar [18,19], a role in devising spectrally coexistent systems [20], and extension to imaging [21]. In this chapter, we provide an overview of sub-Nyquist radars, their applications, and hardware realizations. The outline of the chapter is as follows. In the next section, we overview various reduced-rate techniques for radar system design and explain the benefits of our approach to sub-Nyquist radars. In Section 1.3, we describe the principles, algorithms, and hardware realization of temporal sub-Nyquist monostatic pulse-Doppler radar. Section 1.4

Sub-Nyquist Radar: Principles and Prototypes

3

presents an extension of the sub-Nyquist principle to slow time. We then introduce the cognitive radar concept based on sub-Nyquist reception in Section 1.5 and show an application to coexistence in a spectrally crowded environment. Section 1.6 is devoted to spatial sub-Nyquist applications in multiple-input multiple-output (MIMO) array radars. Finally, we consider sub-Nyquist synthetic aperture radar (SAR) imaging in Section 1.7, followed by concluding remarks in Section 1.8.

1.2

Prior Art and Historical Notes There is a large body of literature on reduced-rate sampling techniques for radars. Most of these works employ compressed sensing (CS) methods, which allow recovery of sparse, undersampled signals from random linear measurements [7]. A pre-2010 review of selected applications of CS-based radars can be found in [22]. A qualitative, systemlevel commentary from the point of view of operational radar engineers is available in [23], while CS-based radar imaging studies are summarized in [24]. An excellent overview on sparsity-based SAR imaging methods is provided in [25]. The review in [26] recaps major developments in this area from a nonmathematical perspective. In the following, we review the most significant works relevant to the sub-Nyquist formulations presented in this chapter. On-Grid CS The earliest application of CS toward recovering time delays with subNyquist samples in a noiseless case was formulated in [27]. CS-based parameter estimation for both delay and Doppler shifts was proposed in [28] with samples acquired at the Nyquist rate. These and similar later works [29–31] discretize the delay-Doppler domain, assuming that targets lie on a grid. Subsequently, these ideas were extended to colocated [32,33] and distributed [34] MIMO radars where targets are located on an angle-Doppler-range grid. In practice, target parameters are typically continuous values whose discretization may introduce gridding errors [35]. In particular, [28] constructs a dictionary that exhaustively considers all possible delay-Doppler pairs, thereby rendering the processing computationally expensive. Noise and clutter mitigation are not considered in this literature. Simulations show that such systems typically have poor performance in clutter-contaminated noisy environments. Off-Grid CS A few recent works [36,37] formulate the radar parameter estimation for off-grid targets using atomic norm minimization [38,39]. However, these methods do not address direct analog sampling, the presence of noise, and clutter. Further details on this approach are available in Chapter 7 (Super-resolution radar imaging via convex optimization) of this book. Parametric Recovery A different approach was suggested in [40], which treated radar parameter estimation as the identification of an underlying linear, time-varying system [41]. The proposed two-stage recovery algorithm, largely based on [42], first estimates target delays and then utilizes these recovered delays to estimate Doppler velocities and complex reflectivities. They also provide guarantees for system identification in terms

4

Mishra and Eldar

of the minimum time-bandwidth product of the input signal. However, this method does not handle noise well. Matrix Completion In some radar applications, the received signal samples are processed as data matrices, which, under certain conditions, are low rank. In this context, general works have suggested retrieving the missing entries using matrix completion methods [10,43]. The target parameters are then recovered through classic radar signal processing. These techniques have not been exhaustively evaluated for different signal scenarios and their practical implementations have still not been thoroughly examined. Finite-Rate-of-Innovation (FRI) Sampling The received radar signal from L targets can be modeled with 3L degrees of freedom because three parameters – time delay, Doppler shift, and attenuation coefficient – characterize each target. The classes of signals that have finite degrees of freedom per unit of time are called finite-rate-ofinnovation (FRI) signals [44]. Low-rate sampling and signal recovery strategies for FRI signals have been studied in detail in the past [6, chapter 15]. In [45], a temporal sub-Nyquist radar was proposed to recover delays relying on the FRI model. The Xampling framework [6] was used to obtain Fourier coefficients from low-rate samples with a practical hardware prototype. Similar techniques were later studied for delay channel estimation problems in ultra-wideband communication systems [46,47] and for ultrasound imaging [48]. In [49], Doppler focusing was added to the FRI-Xampling framework to recover both delays and Dopplers. Doppler focusing is a narrowband technique that can be interpreted as low-rate beamforming in the frequency domain, and was applied earlier to ultrasound imaging [50,51]. It considers a chosen center frequency with a band of frequencies around it and coherently processes multiple echoes in this focused region to estimate the delays from low-rate samples. Extensions of Sub-Nyquist Radars The system proposed in [49] reduces samples only in time and not in the Doppler domain. Since the set of frequencies for Doppler focusing is usually fixed a priori, the resultant Doppler resolution is limited by the focusing; it remains inversely proportional to the number of pulses P , as is also the case with conventional radar. In [8], sub-Nyquist processing in slow time was introduced to recover the target range and Doppler by simultaneously transmitting few pulses in the CPI and sampling the received signals at sub-Nyquist rates. Later, [52] proposed a whitening procedure to mitigate the presence of clutter in a sub-Nyquist radar. Spatialdomain compressed sensing (SCS) was examined for a MIMO array radar in [16] and later for phased arrays in [15]. Recently, [17] proposed Xampling in time and space to recover delay, Doppler, and azimuth of the targets by thinning a colocated MIMO array and collecting low-rate samples at each receive element. This sub-Nyquist MIMO radar (SUMMeR) system was also conceptually demonstrated in hardware [18,53]. The formulation in [54] proposes tensor-based 3D sub-Nyquist radar (TenDSuR) that performs thinning in all three domains and recovers the signal via tensor-based recovery. Finally, an extension to SAR was demonstrated in [21]. Table 1.1 summarizes these developments.

Sub-Nyquist Radar: Principles and Prototypes

5

Table 1.1 Sub-Nyquist radars and their corresponding reduction domains.

1.3

Sub-Nyquist system

Temporal

Doppler

Spatial

Monostatic pulsed radar [45] Monostatic pulse-Doppler radar [49] Reduced time-on-target radar [8] MIMO SCS [16] Phased array SCS [15] SUMMeR [15,17] TenDSuR [54] Sub-Nyquist SAR [21]

Yes Yes Yes No No Yes Yes Yes

No No Yes No No No Yes No

No No No Yes Yes Yes Yes No

Temporal Sub-Nyquist Radar Consider a standard pulse-Doppler radar that transmits a pulse train rTX (t) =

P −1

h(t − pτ),

0 ≤ t ≤ P τ,

(1.1)

p=0

consisting of P uniformly spaced known pulses h(t). The interpulse transmit delay τ is the pulse repetition interval (PRI) or fast time; its reciprocal is the pulse repetition frequency (PRF). The entire duration of P pulses in (1.1) is known as the CPI or slow time. The radar operates at carrier frequency fc so that its wavelength is λ = c/fc , where c = 3 × 108 m/s is the speed of light. In a conventional pulse-Doppler radar, the pulse h(t) = hNyq (t) is a time-limited baseband function whose continuous-time Fourier transform (CTFT) is HNyq (f ) = ∞ −j 2πf t dt. It is assumed that most of the signal’s energy lies within the h (t)e Nyq −∞ frequencies ±Bh /2, where Bh denotes the effective signal bandwidth, such that the following approximation holds: B h /2

hNyq (t) ≈

HNyq (f )ej 2πf t df .

(1.2)

−Bh /2

The total transmit power of the radar is defined as Bh /2 |HNyq (f )|2 df = PT . −Bh /2

1.3.1

(1.3)

Received Signal Model Assume that the radar target scene consists of L non-fluctuating point-targets, according to the Swerling-0 target model [1]. The transmit signal is reflected back by the L targets and these echoes are received by the radar processor. The latter aims at recovering the following information about the L targets from the received signal: time delay τl , which is linearly proportional to the target’s range dl = cτl /2; Doppler frequency νl ,

6

Mishra and Eldar

proportional to the target’s radial velocity vl = cνl /4πfc ; and complex amplitude αl . The target locations are defined with respect to the polar coordinate system of the radar and their range and Doppler are assumed to lie in the unambiguous time-frequency region, i.e., the time delays are no longer than the PRI, and Doppler frequencies are up to the PRF. Typically, the radar assumes the following operating conditions, which leads to a simplified expression for the received signal [49]: A1

A2

A3 A4

A5

“Far targets”: assuming νl 2πfc τl /P τ, target radar distance is large compared to the distance change during the CPI over which attenuation α l is allowed to be constant. “Slow targets”: assuming νl 2πfc /P τBh , target velocity is small enough to allow for constant τl during the CPI. In this case, the following piecewiseconstant approximation holds νl t ≈ νl pτ, for t ∈ [pτ,(p + 1)τ]. “Small acceleration”: assuming dνl /dt c/2fc (P τ)2 , target velocity remains approximately constant during the CPI allowing for constant νl . “No time or Doppler ambiguities”: The delay-Doppler pairs (τl ,νl ) for all l ∈ [1,L] lie in the radar’s unambiguous region of delay-Doppler plane defined by [0,τ] × [−π/τ,π/τ]. The pairs in the set (τl ,νl ) for all l ∈ [1,L] are unique.

Under these assumptions, the received signal can be written as rRX (t) =

P −1 L−1

α l h(t − τl − pτ)e−j νl t + w(t),

(1.4)

p=0 l=0

for 0 ≤ t ≤ P τ, where w(t) is a zero mean wide-sense stationary random signal with autocorrelation rw (s) = σ 2 δ(s). It is convenient to express rRX (t) as a sum of single frames P −1 p rRX (t) + w(t), (1.5) rRX (t) = p=0

where p

rRX (t) =

L−1

α l h(t − τl − pτ)e−j νl pτ,

(1.6)

l=0

for pτ ≤ t ≤ (p + 1)τ is the return signal from the pth pulse. p A classical radar signal processor samples each incoming frame rRX (t) at the Nyquist p rate Bh to yield the digitized samples rRX [n],0 ≤ n ≤ N − 1, where N = τBh . The p signal enhancement process employs an MF for the sampled frames rRX [n]. This is then followed by Doppler processing where a P -point discrete Fourier transform (DFT) is performed on slow-time samples. By stacking all the N DFT vectors together, a delayDoppler map is obtained for the target scene. Finally, the time delays τl and Doppler shifts νl of the targets are located on this map using, e.g., a constant false-alarm rate detector [55].

Sub-Nyquist Radar: Principles and Prototypes

1.3.2

7

Sub-Nyquist Delay-Doppler Recovery Traditional radar systems sample the received signal at the Nyquist rate, determined by p the baseband bandwidth of h(t). Our goal is to recover rRX (t) from its samples taken p far below this rate. We note that over the interval τ, rRX (t) is completely specified by {α l ,τl ,νl }L l=1 , and is an FRI signal with rate of innovation 3L/τ. Hence, in the absence p of noise, one would expect to be able to accurately recover rRX (t) from only a few samples per time τ. Since radar signals tend to be sparse in the time domain, simply acquiring a few data samples at a low rate will not generally yield adequate recovery. Indeed, if the separation between samples is larger than the effective spread in time, then with high probability many of the samples will be close to zero. This implies that presampling analog processing must be performed on the frequency-domain support of the radar signal in order to smear the signal in time before low-rate sampling. The approach we adopt follows the Xampling architecture designed for sampling and processing of analog inputs at rates far below Nyquist, whose underlying structure can be modeled as a union of subspaces (UoS). The input signal belongs to a single subspace, a priori unknown, out of multiple, possibly even infinitely many, candidate subspaces. Xampling consists of two main functions: low-rate analog-to-digital conversion (ADC), in which the input is compressed in the analog domain prior to sampling with commercial devices, and low-rate digital signal processing, in which the input subspace is detected prior to digital signal processing. The resulting sparse recovery is performed using CS techniques adapted to the analog setting. This concept has been applied to both communications [56–59] and radar [49,60], among other applications. Time-varying linear systems, which introduce both time shifts (delays) and frequency shifts (Doppler shifts), such as those arising in surveillance point-target radar systems, fit nicely into the UoS model. Here, a sparse target scene is assumed, allowing the reduction of the sampling rate without sacrificing delay and Doppler resolution. The Xampling-based system is composed of an ADC, which filters the received signal to predetermined frequencies before taking point-wise samples. These compressed samples, or “Xamples,” contain the information needed to recover the desired signal parameters. To demonstrate sub-Nyquist sampling, we begin by deriving an expression for the Fourier coefficients of the received signal and show that the target parameters are embodied in them. Let FR and fNyq be the set of all frequencies in the received signal spectrum and the corresponding Nyquist sampling rate, respectively. Consider the p p Fourier series representation of the aligned frames rRX (t + pτ), with rRX (t) defined in (1.6):

τ

cp [k] = 0

rRX (t + pτ)e−j 2πkt/τ dt = p

1 α l e−j 2πkτl /τ e−j νl pτ, H [k] τ L−1

(1.7)

l=0

f for k ∈ κ, where κ = k = fNyq N f ∈ FR . From (1.7), we see that the unknown parameters {α l ,τl ,νl }L−1 l=0 are embodied in the Fourier coefficients cp [k]. We can estimate these parameters using only a small number of Fourier coefficients, which translates to a low sampling rate.

8

Mishra and Eldar

Figure 1.1 Sub-Nyquist sampling methods: (a) direct sampling; (b) low frequencies only;

(c) multiband sampling.

There are several ways to implement a sub-Nyquist sampler [47,61] in order to obtain a set of Fourier coefficients from low-rate samples of the signal. For simplicity, consider |κ| = K such that q = N/K is an integer defining the sampling reduction factor. In direct sampling (Figure 1.1a), the signal rRx (t) obtained after the anti-aliasing filter is passed through as many analog chains as the number of sub-Nyquist coefficients K. Each branch is modulated by a complex exponential, followed by integration over τ and necessary digital signal processing (DSP). This technique provides the largest flexibility in choosing the Fourier coefficients, but is expensive in terms of hardware. Another approach is to limit the bandwidth of the anti-aliasing filter such that only the lowest K frequencies are free of aliasing (Figure 1.1b). We then sample these lowest K frequencies. Here, the measurements are correlated and a modification in the analog hardware is also required so that the anti-aliasing filter has reduced passband. In the multiband sampling method shown in Figure 1.1c, M disjoint randomly chosen groups of consecutive Fourier coefficients are obtained such that the total number of sampled coefficients is K. This translates to splitting the signal across M branches, passing the downconverted signal through reduced-passband anti-aliasing filters, and then sampling each band with a low-rate ADC. This method can be easily implemented but requires M low-rate ADCs. The sub-Nyquist hardware prototypes developed in [45,49] adopt multiband sampling using four groups of consecutive coefficients. In practice, the specific Fourier coefficients are chosen through extensive software simulations to provide low mutual coherence [6] for CS-based signal recovery. Our goal now is to recover {α l ,τl ,νl }L−1 l=0 from cp [k] for k ∈ κ and 0 ≤ p ≤ P − 1. To that end, [49] adopts the Doppler focusing approach. Consider the DFT of the coefficients cp [k] in the slow-time domain:

˜ ν [k] =

P −1 p=0

cp [k]ej ν pτ =

L−1 P −1 1 α l e−j 2πkτl /τ ej (ν −νl )pτ . H [k] τ l=0

p=0

(1.8)

Sub-Nyquist Radar: Principles and Prototypes

9

Figure 1.2 Sum of exponents |g(ν|νl )| for P = 200, τ = 1 s, and νl = 0 [20,49]. ©2018 IEEE. Reprinted, with permission, from [20].

The key to Doppler focusing follows from the approximation:

P −1 P |ν − νl | < π/P τ j (ν −νl )pτ g(ν|νl ) = e ≈ 0 |ν − νl | ≥ π/P τ,

(1.9)

p=0

as illustrated in Figure 1.2. Denote the normalized focused measurements by τ ˜ ν [k]. ν [k] = P H [k]

(1.10)

As in traditional pulse-Doppler radar, suppose we limit ourselves to the Nyquist grid so that τl /τ = rl /N , where rl is an integer satisfying 0 ≤ rl ≤ N − 1. Then, (1.10) can be approximately written in vector form as

ν = Fκ xν ,

(1.11)

where ν = ν [k0 ] . . . ν [kK−1 ] ,ki ∈ κ for 0 ≤ i ≤ K − 1, Fκ is composed of the K rows of the N × N Fourier matrix indexed by κ, and xν is an L-sparse vector that contains the values α l at the indices rl for the Doppler frequencies νl in the “focus zone,” that is, |ν − νl | < π/P τ. It is convenient to write (1.11) in matrix form, by vertically 1 + P1τ , into concatenating the vectors ν , for ν on the Nyquist grid, namely ν = − 2τ the K × P matrix , as = Fκ X,

(1.12)

where X is formed similarly by vertically concatenating the vectors xν . Note that the matrix Fκ is not square and, as a result, the system of linear equations (1.12) is underdetermined. The system in (1.12) can be solved using any CS algorithm, such as orthogonal matching pursuit (OMP) and 1 minimization [6,7]. A Nyquist receiver needs Bh τ samples to recover the targets. However, as stated shortly in Theorem 1.3.1, the number of samples required by the sub-Nyquist receiver

10

Mishra and Eldar

depends only on the number of targets present and not on Bh . This shows that a subNyquist radar breaks the link between range resolution and transmit bandwidth. In general, only a few targets are present in the radar coverage region leading to a significant reduction in sampling rate. theorem 1.3.1 [49] The minimal number of samples required for perfect recovery of 2 {αl ,τl ,νl }L l=0 in a noiseless environment is 4L . In addition, the number of samples per period is at least 2L, and the number of periods P ≥ 2L. The Doppler focusing operation (1.8) is a continuous operation on the variable ν, and can be performed for any Doppler frequency up to the PRF. With Doppler focusing there are no inherent blind speeds, i.e., target velocities that are undetectable, as occurs with a classic moving target indicator [1]. Since strong amplitudes are indicative of true target existence as opposed to noise, Doppler focusing recovery searches for large magnitude entries and marks them as detections. After detecting each target, its influence is removed from the set of Fourier coefficients in order to reduce masking of weaker targets and to remove spurious targets created by processing sidelobes. It is important to note that our dictionary in (1.12) is indifferent to the Doppler estimation. CS methods, which estimate delay and Doppler simultaneously [28], require a dictionary that grows with the number of pulses. Here, by separating delay and Doppler estimation, the CS dictionary is not a function of P . Moreover, the performance of the sub-Nyquist radar in the presence of noise improves with Doppler focusing. The following theorem states that Doppler focusing increases the per-target SNR by a factor of P . This linear scaling is similar to that obtained using an MF. theorem 1.3.2

[49] Let the prefocusing SNR of the lth target be pl [k] =

|cpl [k]|2 E[|wp [k]|2 ]

where cpl [k] and wp [k] are the signal and white noise Fourier coefficients. Then, the focused SNR for the lth target at center frequency ν is P pl [k]. A continuous-value parameter recovery using Doppler focusing is described in [49]. For practical considerations of computational efficiency, Doppler focusing can be performed on a uniform grid of frequencies so that focused coefficients are computed efficiently using the fast Fourier transform (FFT). Algorithm 1 in this section outlines this approach to solving the P equations (1.12) simultaneously, where, in each iteration, the maximal projection of the observation vectors onto the measurement matrix is retained. The algorithm termination criterion follows from the generalized likelihood ratio test (GLRT) framework presented in [62]. For each iteration, the alternative and null hypotheses in the GLRT problem define the presence or absence of a candidate target, respectively. In Algorithm 1, Qχ22 (ρ) denotes the right-tail probability of the chi-square distribution function with 2 degrees of freedom, C is the complementary set of and PT (1.13) ρ= 2 σ |FR | is the SNR with σ2 the noise variance and PT the total transmit power.

Sub-Nyquist Radar: Principles and Prototypes

11

Algorithm 1 Sub-Nyquist Radar Delay-Doppler Recovery [20,49] Input: Observations cp [k], 0 ≤ p ≤ P − 1 and k ∈ κ, probability of false alarm Pfa , noise variance σ2 , transmitted power PT , total transmitted bandwidth |FR | Output: Estimated target parameters { αˆ l , τˆ l , νˆl }L−1 l=0 1: Create from cp [k] using the FFT (1.8), for k ∈ κ and ν = −1/(2τ) + p/(P τ), 0≤p ≤P −1 2: Compute detection thresholds

PT , γ = Q−12 (1 − N 1 − Pfa ) ρ= 2 (ρ) χ σ |FR | 2 3: 4:

Initialization: residual R0 = , index set 0 = ∅, t = 1 Project residual onto measurement matrix: = FH κ Rt−1

5:

Find the two indices λt = [λt (1) [λ t (1)

6:

λt (2)] such that

λt (2)] = arg maxi,j i,j

Compute the test statistic

(Fκ )λt (1) ((Rt−1 )λt (2) )H ((Fκ )λt (1) )H (Rt−1 )λt (2) σ2 where (M)i denotes the ith column of M 7: If > γ continue; otherwise go to step 12 8: Augment index set t = t {λ t } 9: Find the new signal estimate =

ˆ t|t = (Fκ )† , X t 10:

ˆ C =0 X t| t

Compute new residual ˆ Rt = − (Fκ )t X

Increment t and return to step 4 ˆ = t Estimated support set τ ˆ 1 ˆ ˆˆ ˆ l = N (l,1), νˆl = P τ (l,2), αˆ l = X 13: τ ˆ (l,1), (l,2) 11: 12:

In Section 1.3.4, we introduce a sub-Nyquist prototype implementing the ideas in this section using simple hardware. Before that, we describe how to account for clutter mitigation in sub-Nyquist radar.

1.3.3

Sub-Nyquist Clutter Removal Clutter refers to unwanted echoes from stationary objects such as buildings, trees, chaff, and ground surface as well as moving elements like weather and sea. Since strong clutter echoes hamper detection of desired targets, clutter removal has been investigated

12

Mishra and Eldar

intensively. In the context of CS-based radars, [63] provides a general overview of clutter rejection algorithms. In [64], Capon beamforming is used to reject clutter and then the target is retrieved by exploiting sparse reconstruction methods. On the other hand, a few works such as [65–67] utilize sparsity of the clutter in the mitigation process. Along similar lines, [68] assumes sparse clutter and proposes a GLRT detector. However, they obtain signal samples at the Nyquist rate. Conventionally, clutter is modeled as a random process with Doppler frequency that follows a colored Gaussian noise distribution [69–71]. A standard operation to remove this correlated noise is to use receive filters that maximize the signal-to-clutter-plusnoise (SCNR) ratio. This method is equivalent to first whitening the received signal samples, and then performing matched-filtering with respect to a whitened pulse. Our approach [52] to clutter removal in sub-Nyquist radar is based on this philosophy as it fits well with our Fourier-based analysis. In the presence of clutter and noise, the received signal rq (t) is r(t) = rRX (t) + y(t),

(1.14)

where rRX (t) is the target signal with noise as in (1.5) and y(t) =

C P −1

α c h(t − pτ − τc )ej vc pτ,

(1.15)

p=0 c=1

is the echo from C clutter targets. We assume that the mean clutter amplitude is E[|αc |2 ] = σc2 . Further, the delays τc ∼ U (0,τ) and the clutter Doppler spectrum vc ∼ N (vd ,σd2 ) are independent and identically distributed. Analogous to the target signal in (1.7), the Fourier series representation of the clutter signal is given by 2π 1 α c e−j τ kτc e−j vc pτ . c˜p [k] = H [k] τ C−1

(1.16)

c=0

Let the Fourier series coefficients of the noise be w˜ p [k]. We now form a P × K matrix ˜ p [k], R with kth column given by the Fourier coefficients Rp [k] = cp [k] + c˜p [k] + w k ∈ κ such that R = X + Y + N = FP AFK N H + B, where B = Y + N, FP is the P × P Fourier matrix with (l,k)th element e−j

(1.17) 2π P lk

, FK N is 2π

a submatrix formed by K rows of the N × N Fourier matrix with (l,k)th element ej N lk , H = diag(H [k]) is a K × K diagonal matrix, A is a P × K sparse matrix with complex reflectivity α l at the L indices (rl ,sl ), and Y and N are P × K matrices with (p,k)th elements c˜p [k] and w˜ p [k], respectively. As mentioned previously, noise is white over the indices k (all tones). Our goal is to extract A from the measurements R. For simplicity, we assume that |H [k]|2 is unity for all k. The whitening transformation requires information about the statistics of clutter and noise, which are summarized in the following proposition.

Sub-Nyquist Radar: Principles and Prototypes

13

proposition 1.3.3 [52] The mean of the clutter Fourier coefficients is E[c˜p [k]] = 0, and their correlation is given by 1 2 2 2 Rl1 [k1,k2 ] = E c˜p [k1 ]c˜p+l1 [k2 ] = Cσc2 δk1,k2 e−j vd l1 τ− 2 σd l1 τ . (1.18) The mean and variance of the Fourier coefficients of the noise are, respectively, 1 E Np [k] = 0, E Np [k1 ]Np+l1 [k2 ] = σn2 δk1 k2 δl2 . τ

(1.19)

Our clutter mitigation technique is based on whitening all the tones of the measurements R. It follows from Proposition 1.3.3 that the columns of B are uncorrelated and identically distributed. The covariance matrix M of the columns of B is a Toeplitz matrix with mth diagonal value 1 2

M(m) = Cσc2 e−j vd mτ− 2 σd m

2 τ2

+

1 2 σ δm . τ n

(1.20)

Therefore, the columns of R can be whitened by multiplying on the left by M−1/2 : −1/2 M−1/2 R = M−1/2 FP AFK B, NH + M

(1.21)

where M−1/2 B corresponds to white noise. From here, we proceed with Doppler focusing on M−1/2 R by taking a Hermitian transpose of (1.21) and multiplying on the right by M−1/2 FP : H H H −1 H −1 = H(FK N ) A FP M FP + B M FP = + W,

(1.22)

is white noise for each focused frequency. This equation represents a sparse where W H −1 H matrix recovery problem. For known matrices D1 = H(FK N ) and D2 = FP M FP , we are given measurements = D1 XD2 , and the goal is to retrieve the sparse matrix X = AH . These problems are solved by matrix-sketching algorithms, as described in [72]. It has been shown in [52] that whitened Doppler focusing generally increases the SCNR. Compared to other CS-based radars [27,28], this technique is robust to the presence of clutter, despite sampling at low rates.

1.3.4

Sub-Nyquist Hardware Prototype The first sub-Nyquist radar hardware implementation was presented in [45]. It was then developed further to incorporate Doppler focusing and clutter removal in [49,52], respectively. Since sub-Nyquist techniques manifest themselves mostly in the radar receiver, this prototype emulates receiver processing. The basic prototype (Figure 1.4) consists of an analog front-end (Figure 1.3), fed by a synthetized RF signal using National Instruments (NI) hardware and followed by digital delay-Doppler map recovery. To evaluate the Xampler board, we make use of NI equipment for both system synchronization and RF signal sources. Figure 1.5 shows the entire assembly wrapped in the NI chassis. We transmit 50 pulses with bandwidth 20 MHz. At the receiver, a multiple bandpass sampling approach was chosen, where 4 groups of consecutive Fourier coefficient subsets are selected. Each channel is fed by

14

Mishra and Eldar

Figure 1.3 The four-channel Xampler board [45]. ©2014 IEEE. Reprinted, with permission,

from [45].

Figure 1.4 Sub-Nyquist hardware prototype showing connections between the Xampler board and

NI chassis [8,45,49].

Figure 1.5 NI chassis showing various signal generation and synchronization components.

a local oscillator (LO), which modulates the desired frequency band of the channel to the central frequency of a narrow 80 KHz bandwidth band pass filter (BPF). A fifth LO, common to all 4 channels, modulates the BPF output to a low-frequency band, and sampled with a standard low-rate ADC operating at 250 kHz frequency. The digital samples are acquired by the chassis controller and a MATLAB function is launched that runs Doppler focusing. The digital reconstruction algorithm, performed at a low rate of 250 ksps, allows recovery of the unknown delays and Doppler frequencies of the targets. A block diagram of the system is shown in Figure 1.6.

Sub-Nyquist Radar: Principles and Prototypes

fLO1

28.915MHz Crystal Filter fc = 29MHz

Df3dB = 80KHz

fLO2

LPF fp = 100KHz

14dB 28.915MHz

LPF fp = 100KHz

LPF f3dB = 2.5MHz

14dB

1Æ4 Splitter

Crystal Filter 14dB

fLO3

fc = 29MHz Df3dB = 80KHz

14dB 28.915MHz

14dB

fLO4

Crystal Filter fc = 29MHz

Df3dB = 80KHz

14dB

Df3dB = 80KHz

fs = 125KHz LPF fp = 100KHz

14dB 28.915MHz

Crystal Filter fc = 29MHz

fs = 125KHz

14dB

fs = 125KHz LPF fp = 100KHz

ADC 4 ¥ 250KHz (1MHz Total)

14dB

Input Signal

15

Digital Processing

fs = 125KHz

Figure 1.6 Block diagram of a 4-channel solid-state receiver with 4 up-modulating local

oscillators with respective center frequencies of 28.375, 28.275, 27.65, and 27.391 MHz [45]. ©2014 IEEE. Reprinted, with permission, from [45]

Figure 1.7 Sub-Nyquist prototype experiment [52]. Top left to right: Signal corresponding to

targets, clutter, noise, and all three combined. Bottom left to right: Low rate samples at receiver and delay-Doppler map with true and recovered targets.

Real-time analog experiments show that the system is able to maintain good detection capabilities, while sampling radar signals that require Nyquist rate of about 30 MHz at a total rate of 1 MHz, i.e., 1/30th of the Nyquist rate. We conducted several experiments in order to test the accuracy of our system under various conditions. For example, Figure 1.7 shows results for a hardware experiment where the target scene has seven scatterers with different delays and Doppler frequencies. A few cases of closely spaced targets in the delay-Doppler plane are also included. Clutter is also added to the scene and identified by the system. Our low-rate processing rejects the clutter and successfully detects only targets in the delay-Doppler plane despite sampling at 1/30th of the Nyquist rate. The digital recovery algorithm is efficient as it involves only solving 1D delay recovery problems post FFT-based Doppler focusing and without increasing the size of the dictionary.

1.4

Doppler Sub-Nyquist Radar The temporal sub-Nyquist processing in the fast-time domain described in the previous section breaks the link between signal bandwidth, sampling rate, and range resolution.

16

Mishra and Eldar

The Xampling framework can also be extended in the slow-time or Doppler-frequency domain. The Doppler resolution in classical radar processing is given by 2π/P1 τ, where P1 is the number of pulses transmitted during the CPI. In Doppler domain sub-Nyquist processing, we nonuniformly transmit P2 < P1 pulses and reduce the power consumption and dwell time in a particular direction without loss of Doppler resolution. The advantage is gaining the ability to look at other directions within the same CPI by interleaving transmissions in different directions. A few other CS-based works [9,10] have considered reduced time-on-target (RToT) scenarios without addressing analog sampling. The Doppler sub-Nyquist processing that we review here was introduced in [8], and is based on the prototype and principles presented in the previous section. We consider a nonuniformly transmitting pulse-Doppler radar such that the pth pulse 2 −1 is an ordered set of integers such that mp ≥ p. is sent at time mp τ, where {mp }Pp=0 Then, (1.1) is written as rTX (t) =

P 2 −1

h(t − mp τ),

0 ≤ t ≤ P1 τ.

(1.23)

p=0

The received signal rRX (t) is accordingly expressed as a sum of single frames rRX (t) =

P 2 −1

p

rRX (t),

(1.24)

α l h(t − τl − mp τ)e−j νl mp τ,

(1.25)

p=0

where p

rRX (t) =

L−1 l=0

for 0 ≤ t ≤ P1 τ, is the return signal from the pth pulse. Our goal is to recover the p targets range and Doppler frequency from the received signals rRX (t), with a reduced number of transmit pulses P2 < P1 as well as low-rate samples per pulse.

1.4.1

Xampling in CPI and Delay-Doppler Recovery As before, we consider the Fourier series representation of the aligned frames p rRX (t + mp τ): 1 α l e−j 2πkτl /τ e−j νl mp τ, 0 ≤ k ≤ N − 1, H [k] τ L−1

Xp [k] =

(1.26)

l=0

where N = Bh τ. From (1.26), the Fourier coefficients embody all the information about the unknown parameters {α l ,τl ,νl }L−1 l−0 . The goal is then to recover these parameters from Xp [k], 0 ≤ p ≤ P2 − 1. The low rate sampling technique is as described earlier in Section 1.3.2, but the processing steps to recover the target parameters are different

Sub-Nyquist Radar: Principles and Prototypes

17

to account for sub-Nyquist sampling in Doppler. Let X be the K × P matrix with pth column given by the Fourier coefficients Xp [k], k ∈ κ. Then X can be expressed as P2 T X = HFK N A(FP1 ) ,

(1.27)

P2 where H = τ1 diag(H [k]), FK N is a K × N partial Fourier matrix, FP1 is a P2 × P1 partial Fourier matrix indexed by the values of mp , 1 ≤ p ≤ P2 , and A is an N × P1 sparse matrix with α l values at the L indices {sl ,τl }. We would like to recover A from the measurements X. The system of linear equations (1.27) can be solved by CS techniques. However, this problem is different than the temporal sub-Nyquist formulation of Section 1.3.2, where only the range sensing matrix FK N is a partial DFT. In Doppler sub-Nyquist radar, both P2 range and Doppler sensing matrices (i.e., FK N and FP1 , respectively) are partial DFTs. Analogous to Theorem 1.3.1, we have the following result for the Doppler sub-Nyquist radar:

theorem 1.4.1 [8] The minimal number of samples required for perfect recovery of A for L targets in noiseless settings is 4L2 . In addition, the number of samples per period is at least 2L, and the number of periods P2 ≥ 2L. Note that the number of periods P2 here is for nonuniform transmission while the minimum number of periods in Theorem 1.3.1 pertain to uniformly spaced pulses in the CPI. Theorem 1.4.1 indicates the lower limit of rate reduction in temporal and Doppler domains. To solve for the sparse matrix A in (1.27) one can use the matrix version of OMP or 1 minimization [6]. Alternatively, Doppler focusing is still approximately applicable. The nonuniform discrete Fourier transform of the coefficients Xp [k] is ν [k] =

P 2 −1

Xp [k]e

p=0

j ν mp τ

P L−1 2 −1 1 −j 2πkτl /τ = H [k] αl e ej (ν −νl )mp τ . (1.28) τ l=0

p=0

This can be approximated similar to (1.9) and solved by Algorithm 1, described earlier. However, this is a poor approximation because the P2 points in the sum of exponents P2 −1 j (ν −ν )m τ p are not equally spaced over the unit circle. l p=0 e

1.4.2

RToT Hardware Prototype A hardware implementation of the RToT concept is described in [8]. It uses the subNyquist hardware prototype presented in Section 1.3.4. We evaluated the prototype for a scenario wherein targets are located at two distinct azimuths. Here, P1 = 50 pulses were chosen such that a quarter of them were sent in one direction and the rest in another. The target scenario for both is then simultaneously recovered within the same original CPI (Figure 1.8). In this experiment, the reduction in temporal domain is the same as in Section 1.3.4, i.e., 1/30 of the Nyquist rate. In the Doppler domain, pulses in the two directions are reduced by 75% and 25%, respectively.

18

Mishra and Eldar

Figure 1.8 RToT sub-Nyquist radar prototype [8]. Top left: Targets at two different azimuths.

Bottom left: Echoes from both directions are acquired via nonuniform pulses. Top and bottom right: delay-Doppler maps showing reconstruction for both directions.

1.5

Cognitive Sub-Nyquist Radar and Spectral Coexistence In the previous two sections, we focused on processing the received signal. The receiver design in the sub-Nyquist framework can be exploited to also alter the behavior of the radar transmitter. In this section, we discuss the opportunistic control of the transmitter to impart cognition to the radar and leverage it for spectrum sharing applications. For alternative, non-sub-Nyquist approaches to cognition in radars, we refer the reader to Chapters 9 (“Spectrum sensing for cognitive radar via model sparsity exploitation”) and 10 (“Cooperative spectrum sharing between sparse sensing based radar”) of this book. The unhindered operation of a radar that shares its spectrum with communication systems has captured a great deal of attention within the operational radar community during the last decade [73]. The interest in such spectrum-sharing radars is largely due to the electromagnetic spectrum being a scarce resource and almost all services having a need for greater access to it. Recent research in spectrum sharing radars has focused on S- and C-bands, where the spectrum has seen increasing cohabitation by long-term evolution (LTE) cellular/wireless commercial communication systems. Many synergistic efforts by major agencies are underway for efficient radio spectrum utilization. A significant recent development is the announcement of the Shared Spectrum Access for Radar and Communications (SSPARC) program [74] by the Defense Advanced Research Projects Agency (DARPA). This program is focused on S-band military radars and views spectrum sharing as a cooperative arrangement where the radar and communication services actively exchange information. It defines spectral coexistence as equipping existing radar systems with spectrum sharing capabilities and spectral codesign as developing new systems that utilize opportunistic spectrum access [75]. For a review of spectral interference from different services at IEEE radar bands, see [20]. A variety of system architectures have been proposed for spectrum-sharing radars. Most put emphasis on optimizing the performance of either radar or communications

Sub-Nyquist Radar: Principles and Prototypes

19

while ignoring the performance of the other. The radar-centric architectures [20,76] usually assume fixed interference levels from communication systems and design the system for high probability of detection (Pd ). Similarly, the communications-centric systems attempt to improve performance metrics, like the error vector magnitude and bit/symbol error rate for interference from radar. With the introduction of the SSPARC program, joint radar-communication performance is being investigated [77]. In nearly all cases, real-time exchange of information between radar and communications hardware has not yet been integrated into the system architectures. In a similar vein, our proposed method, described later in this section, incorporates handshaking of spectral information between the two systems. Conventional receiver processing techniques to remove RF interference in radar employ notch filters at hostile frequencies. Typically, spectrum sharing is achieved by notching out the radar waveform’s bandwidth, causing a decrease in range resolution. Our spectrum sharing solution departs from this baseline. The approach we adopt follows the Xampling architecture on which the sub-Nyquist radar prototype described earlier in Section 1.3 is based. We recall that the sub-Nyquist receiver samples and processes only small narrow subbands of the received signal. Hence, we capitalize on the simple observation that if only narrow spectral bands are sampled and processed, then one can restrict the transmit signal to these bands. The concept of transmitting only a few subbands that the receiver processes is one way to formulate a cognitive radar (CRr) [60]. The delay-Doppler recovery is then performed as presented earlier in Section 1.3. The range resolution obtained through this multiband signal spectrum fragmentation can be the same as that of a wideband traditional radar. Furthermore, by concentrating all the available power in the transmitted narrow bands rather than over a wide bandwidth, the CRr increases SNR, as illustrated in Figure 1.9. In the CRr system [60], the support of subbands varies with time to allow for dynamic and flexible adaptation to the environment. Such a system also enables the radar to disguise the transmitted signal as an electronic countermeasure or cope with crowded spectrum by using a smaller interference-free portion. The CRr configuration is key to spectrum sharing since the radar transceiver adapts its transmission to available bands, achieving coexistence with communication signals. To detect vacant bands, a communication receiver is needed that performs spectrum sensing over a large

Figure 1.9 A conventional radar with bandwidth Bh transmits in the band Bh . A cognitive radar Nb transmits only in subbands {Bi }i=1 , but with increased in-band power. The sub-Nyquist receiver samples and processes only these subbands [19].

20

Mishra and Eldar

bandwidth. Such systems have recently received tremendous interest in communications research, which faces a bottleneck in terms of spectrum availability. To increase the efficiency of spectrum managing, dynamic opportunistic exploitation of temporarily vacant spectral bands by secondary users has been considered, under the name of cognitive radio (CRo) [78]. Here, we use a CRo receiver to detect the occupied communication bands, so that our radar transmitter can exploit the spectral holes. One of the main challenges of spectrum sensing in the context of CRo is the sampling rate bottleneck due to the wide signal bandwidth. In this context, we use the Xampling framework to subsample and process the signal [20,56]. Denote the set of all frequencies of the available common spectrum by F. The communication and radar systems occupy subsets FC and FR of F, respectively, such that FC ∩ FR = ∅. Once the CRo receiver has identified FC , it provides the radar with spectral occupancy information. Equipped with this spectral map as well as a known radio environment map (REM) detailing typical interference, the CRr transmitter chooses narrow frequency subbands that minimize interference for its transmission. The radar conveys the frequencies FR to the communication receiver as well, so that it can ignore the radar bands while sensing the spectrum. The combined CRo-CRr system results in spectral coexistence via the Xampling (SpeCX) framework, which optimizes the radar’s performance without interfering with existing communication transmissions. Our hardware prototype for SpeCX, presented in Section 1.5.3, performs real-time recovery of CRo and CRr signals sharing a common spectrum at SNRs as low as −5 dB.

1.5.1

Cognitive Radio We first introduce the signal model, processing, and prototype of CRo in the context of SpeCX. Let xC (t) be a real-valued continuous-time communication signal, supported on F = [−1/2TNyq, +1/2TNyq ] and composed of up to Nsig transmit waveforms, such that xC (t) =

Nsig

si (t),

(1.29)

i=1

where s(t) has unknown carrier frequency fi , and Xc (f ) is the Fourier transform of xC (t). We denote by fNyq = 1/TNyq the Nyquist rate of xC (t). The waveforms, respective carrier frequencies, and bandwidths are unknown. We only assume that the single-sided bandwidth Bci for the ith transmission does not exceed an upper limit B. Such sparse wideband signals belong to the so-called multiband signal model [56,79]. Figure 1.10 illustrates the two-sided spectrum of a multiband signal with K = 2Nsig bands centered around unknown carrier frequencies |fi | ≤ fNyq /2.

− 2

−

−

~

~

−

0

Figure 1.10 Multiband model with K = 6 bands [20].

2

Sub-Nyquist Radar: Principles and Prototypes

Mixing Series Generator

PC–Matlab Based Controller Generates RF Input in Real Time

pi(t)

21

PC - Labview + Matlab Based Controller

yi(t)

i = 1..3 XIUNX VC707 – HighSpeed FPGA

PCIe ×4 to MXI ×4

Signal Generators

NI PXIe-1065 with DC Coupled 4-Channel ADC

x(t) RF Signal

2× NI° USRP-2942R RF Generator

The Cog-Radio Card

Signal ADC *Note: Allows real-time sampling

Figure 1.11 CRo system [80].

Let FC ⊂ F be the unknown support of xC (t). The goal of the CRo communication receiver is to retrieve FC , while sampling and processing xC (t) at low rates in order to reduce system cost and resources. A CRo system was developed earlier [56,80] for blind sensing (see Figure 1.11). Next, we explain the details on combining this system with the sub-Nyquist radar to implement SpeCX. The input signal at the communication receiver of the SpeCX system is x(t) = xC (t) + xR (t),

(1.30)

where xR (t) = rTX (t) + rRX (t) is the radar signal sensed by the communication receiver, composed of the transmitted and received radar signals defined in (1.1) and (1.4), respectively. Since the frequency support of xC (t) is unknown, a classic processor would sample such a signal at its Nyquist rate, which can be prohibitively high. In this work, we instead use the modulated wideband converter (MWC) [56], a sub-Nyquist sampling technique that achieves the lower sampling rate bound for perfect blind recovery of multiband signals, namely twice the Landau rate, and is also practically feasible. The MWC is composed of M parallel channels. In each channel, an analog mixing front end, where xC (t) is multiplied by a mixing function pi (t), aliases the spectrum, such that each band appears in baseband. The mixing functions pi (t) are periodic with period Tp such that fp = 1/Tp ≥ B and have thus the following Fourier expansion: pi (t) =

∞

cil e

j T2π lt p

.

(1.31)

l=−∞

In each channel, the signal next goes through a lowpass filter (LPF) with cut-off frequency is sampled at rate fs ≥ fp , resulting in samples zi [n]. Define fs /2 and fNyq +fs and Fs = [−fs /2,fs /2]. Following the calculations in [56], the N = 2 2fp relation between the known discrete-time Fourier transform of the samples zi [n] and the unknown XC (f ) is given by z(f ) = A(xC (f ) + xR (f )),

f ∈ Fs ,

(1.32)

where z(f ) is a vector of length M with ith element zi (f ) = Zi (ej 2πf Ts ) and the unknown vector xC (f ) is given by xC i (f ) = XC (f + (i − N/2)fp ),

f ∈ Fs ,

(1.33)

22

Mishra and Eldar

Figure 1.12 Schematic implementation of the MWC analog sampling front end and digital signal recovery from low-rate samples [20]. The CRo inputs are the communication signal xC (t) and radar support FR . The communication support output FC is shared with the radar transmitter.

for 1 ≤ i ≤ N . The vector xR i (f ) is defined similarly. The M × N matrix A contains the known coefficients cil such that Ail = ci,−l = cil∗ . The MWC analog mixing front end, shown in Figure 1.12, results in folding the spectrum to baseband with different weights for each frequency interval. The CRo’s goal is now to recover the support of xC (f ) from the low-rate samples z(f ). The recovery of xC (f ) for each f independently is inefficient and not robust to noise. Instead, the support recovery paradigm from [56] exploits the fact that the bands occupy continuous spectral intervals so that xC (f ) are jointly sparse for f ∈ Fp . The continuous to finite block [56] then produces a finite system of equations, called multiple measurement vectors (MMV) from the infinite number of linear systems (1.32). From (1.32), we have Q = ZH , where

Q=

(1.34)

f ∈Fp

z(f )zH (f )df , Z =

f ∈Fp

x(f )xH (f )df ,

(1.35)

are M × M and N × N matrices, respectively. Here, x(f ) = xC (f ) + xR (f ). The matrix Q is then decomposed to a frame V such that Q = VVH . Clearly, there are many ways to select V. One possibility is to construct it by performing an eigendecomposition of Q and choosing V as the matrix of eigenvectors corresponding to the nonzero eigenvalues. The finite dimensional MMV system is then given by V = A(UC + UR ).

(1.36)

The support of the unique sparsest solution of (1.36) is the same as the support of our original set of equations (1.32) [56]. Therefore, the support of UC and UR are disjointed. The frequency support FR of xR (t) is known at the communication receiver. From FR , we derive the support SR of the radar slices xR (f ), which is identical to the support of UR , such that fs + BRi fRi − N/2 < , (1.37) SR = n n − fp 2fp

Sub-Nyquist Radar: Principles and Prototypes

23

for 1 ≤ i ≤ Nb . Our goal can then be stated as recovering the support of UC from V, given the known support SR of UR . This can be formulated as a sparse recovery with partial support knowledge, studied under the framework of modified CS [81]. ModifiedCS has been used to adapt CS recovery algorithms to exploit partial known support. In particular, greedy algorithms, such as OMP, have been modified to OMP with partial known support [82]. Instead of starting with an initial empty support set, one starts with SR as the initial support. In the first iteration, we compute the estimate ˆ SR = A† V, U SR 1

ˆ 1i = 0, U

∀i ∈ / SR ,

(1.38)

and residual ˆ 1. V1 = V − ASR U

(1.39)

The remainder of the algorithm is then identical to OMP. Once the overall support SC SR is known, we have xˆ SC

SR

[n] = A†S

xˆ i [n] = 0,

C

z[n], SR . ∀i ∈ / SC SR

(1.40)

Here, xSC SR (f ) denotes the vector x(f ) reduced to its support, ASC SR is composed of the columns of A indexed by SC SR and † is the Moore–Penrose pseudo-inverse. The occupied communication support is then

fp , for all i ∈ SC . (1.41) FC = f ||f − (i + N/2)fp | ≤ 2

1.5.2

Cognitive Radar After CRo detects the communication signal support, the CRr transmits a pulse h(t) in the unused parts of the spectrum. The transmit signal is supported over Nb disjoint b frequency bands, with bandwidths {Bri }N i=1 centered around the respective frequencies Nb i Nb i {fr }i=1 , such that i=1 Br < Bh . The number of bands Nb is known to the receiver and does not change during operation. The location and extent of the bands Bri and fri are determined by the radar transmitter through an optimization procedure to identify the least contaminated bands (see Section 1.5.2). The resulting transmitted radar signal CTFT is

βi HNyq (f ), f ∈ FRi , for 1 ≤ i ≤ Nb HR (f ) = (1.42) 0, otherwise, where FRi = [fri − Bri /2,fri + Bri /2] is the set of frequencies in the ith band, such that b i FR = N i=1 FR . The parameters β i > 1 are chosen such that the total transmit power PT of the spectrum-sharing radar remains the same as that of the conventional radar:

Bh /2

−Bh /2

|HNyq (f )| df = 2

Nb i=1

Fri

|HR (f )|2 df = PT .

(1.43)

24

Mishra and Eldar

The radar identifies an appropriate transmit frequency set FR ⊂ F \ FC such that the radar’s probability of detection Pd is maximized. For a fixed probability of false alarm Pfa the Pd increases with higher signal to interference and noise ratio [55]. At the spectrum-sharing radar receiver, we employ the sub-Nyquist approach described in Section 1.3.2, where the delay-Doppler map is recovered from the subset of Fourier coefficients defined by FR .

Optimal Radar Transmit Bands We now explain the procedure through which a CRr selects transmit subbands that have minimal spectral interference. The REM is assumed to be known to the radar transmitter in the form of typical interfering energy levels with respect to frequency bands, represented by a vector y ∈ Rq , where q is the number of frequency bands with bandwidth by |F|/q. In addition, the information from the CRo indicates that the radar waveform must avoid all frequencies in the set FC . Therefore, we set y to be equal to ∞ in these bands. Our goal is to select subbands from the set F \ FC with minimal interference. We do that by seeking a block-sparse frequency vector w ∈ Rp with unknown block lengths, where p is the number of discretized frequencies, whose support indicates frequency bands with low interference for the radar. Each entry of w represents a subband of bandwidth bw |F |/p. To this end, we use the structured sparsity framework of [83] based on the onedimensional graph sparsity structure whose nodes denote the p frequency points of w. In order to find the desired block-sparse w, the formulation in [83] replaces the traditional sparse recovery 0 constraint by a more general term c(w), referred to as the coding complexity, such that c(F ) = g log p + |F |, where F ⊂ {1,. . .,p} is a sparse subset of the index set of the coefficients of w and g is the number of connected regions or blocks of F . This coding complexity, which accounts for both the number of discretized frequencies |F | and the number of connected regions g, favors blocks within the graph. In our setting, this reduces to solving the following optimization problem for finding the block-sparse frequency vector w with (yinv )i = 1/yi : minimizew ||yinv − Dw||22 + λc(w),

(1.44)

where λ is a regularization parameter and c(w) is defined by c(w) = minF {c(F )| supp(w) ⊂ F }. The matrix D is q × p matrix and maps each discrete frequency in w to the corresponding band in yinv . That is, the (i,j )th entry of D is equal to 1 if the j th frequency in w belongs to the ith band in y; otherwise, it is equal to 0. Problem (1.44) can be solved using structured OMP [83].

Delay-Doppler Recovery In order to recover the delay-Doppler map from only Nb transmitted narrow bands, CRr employs a sub-Nyquist receiver that we explained earlier in Section 1.3.2. The radar receiver first filters the CRr subbands supported on FR and computes the Fourier coefficients of the received signal. Our resulting spectrum sharing SpeCX framework is summarized in Algorithm 3.

Sub-Nyquist Radar: Principles and Prototypes

25

Algorithm 2 Cognitive Radar Band Selection [20] Input: REM vector y and subbands bandwidth by = |F|/q, shared support F, communication support FC , mapping matrix D, number of discretized frequencies p, number of bands Nb Output: Block sparse vector w, radar support FR 1: Set yi = ∞, for each ith subband not in FC and compute (yinv )i = 1/yi 2: Initialization F0 = ∅, w = 0, t = 1 3: Find the index λ t so that λ t = arg max φ(i), where φ(i) =

4: 5: 6: 7: 8:

ˆ t−1 − yinv )||22 ||Pi (Dw c(i Ft−1 ) − c(Ft−1 )

with Pi = Di (DTi Di )† DTi Augment index set Ft = λt Ft−1 ˆ t|Ft = D†Ft yinv, w ˆ t|F C = 0 Find the new estimate w t If the number of blocks, or connected regions, g(w) > Nb , go to step 7. Otherwise, return to step 3 ˆt = w ˆ t−1 Remove the last index λt so that Ft = Ft−1 and w Compute the radar support FR = j ∈Ft [j bw − |F|/2,(j + 1)bw − |F|/2] with bw = |F|/p

Algorithm 3 Spectral Coexistence via Xampling (SpeCX) [20] Input: Communication signal xC (t) Output: Estimated target parameters { αˆ l , τˆ l , νˆl }L−1 l=0 1: Initialization: perform spectrum sensing at the CRo receiver on xC (t) following the procedure in Section 1.5.1 2: Choose the least noisy subbands for the radar transmit spectrum with respect to detected FC using Algorithm 2 3: Send FR to communication and radar receivers 4: Perform target delay and Doppler estimation using Algorithm 1 5: Perform spectrum sensing at the communication receiver on x(t) = xC (t) + xR (t) following the procedure in Section 1.5.1 6: If FC changes, then the radar transmitter goes back to step 2

For time-delay estimation, [19] compares the performance of conventional and cognitive radars using the extended Ziv–Zakai lower bound (EZB). In a conventional radar, the EZB for a single target delay estimate τˆ0 is 3/2 SN R SN R 4 EZBR ( τˆ0 ) = στ20 · 2Q , (1.45) + 2 2 SN R · F where Q(·) denotes the right tail Gaussian probability function, a (b) is the incomplete gamma function with parameter a and upper limit b, and F is the root-mean-square

26

Mishra and Eldar

(rms) bandwidth of the full-band signal. The bound for CRr is given in the following theorem. theorem 1.5.1 [19] The extended Ziv–Zakai lower bound (EZB) for delay estimation in a cognitive radar is SN R ⎛ ⎞ 3/2 4 SN R ⎠+ EZBCRr ( τˆ 0 ) = στ20 · 2Q ⎝ , (1.46) Nb 2 2 SN Ri · Fi i=1

where SN Ri and Fi are the in-band SNR and rms bandwidth of the ith subband and SN R is the total SNR. b i As noted in [19], since N i=1 Br ⊂ Bh , we have SN R > SN R for given PT . Therefore, the SNR threshold for asymptotic performance of EZBCRr is lower than EZBR . As the noise increases and power remains constant for both radars, the asymptotic performance of EZBCRr is more tolerant to noise than EZBR . The multiband design strategy, besides allowing a dynamic form of the transmitted signal spectrum over only a small portion of the whole bandwidth to enable spectrum sharing, has two additional advantages. First, as we show in hardware experiments (Section 1.5.3), our CS reconstruction achieves the same resolution as traditional Nyquist processing over a significantly smaller bandwidth. Second, the entire transmit power is concentrated in small narrow bands. Therefore, the SNR in the sampled bands is improved, which leads to better parameter estimation, as indicated by Theorem 1.5.1.

1.5.3

SpeCX Prototype Figure 1.13 shows our SpeCX prototype, composed of a CRo receiver and a CRr transceiver. The CRo hardware realizes the system shown in Figure 1.12. At the heart of the system lies our proprietary MWC board [84] that implements the sub-Nyquist Comm Display

Comm Analog Rx Comm Digital Rx

Signal Generator

Radar Analog Rx

Radar Display

Radar Digital Rx

Figure 1.13 Shared spectrum prototype [20]. The system is composed of a signal generator, a CRo receiver based on the MWC, a communication digital receiver, and a CRr analog and digital receiver.

Sub-Nyquist Radar: Principles and Prototypes

27

analog front-end receiver. The card first splits the wideband signal into M = 4 hardware channels, with an expansion factor of q = 5, yielding Mq = 20 virtual channels after digital expansion. In each channel, the signal is then mixed with a periodic sequence pi (t), generated on a dedicated FPGA, with fp = 20 MHz. The sequences are chosen as truncated versions of Gold Codes. These were heuristically found to give good detection results [85], primarily due to small bounded cross-correlations within a set. Next, the modulated signal passes through a Chebyshev LPF of 7th order with a cut off frequency (−3 dB) of 50 MHz. Finally, the low-rate analog signal is sampled by a National Instruments ADC operating at fs = (q + 1)fp = 120 MHz, leading to a total sampling rate of 480 MHz. The digital receiver is implemented on a National Instruments PXIe-1065 computer with DC-coupled ADC. Since the digital processing is performed at the low rate 120 MHz, very low computational load is required in order to achieve real time recovery. MATLAB and LabVIEW platforms are used for digital recovery operations. The prototype is fed with RF signals composed of up to Nsig = 5 real communication transmissions, namely K = 10 spectral bands with total bandwidth occupancy of up to 200 MHz and varying support, with Nyquist rate of 6 GHz. To test the system’s support recovery capabilities, an RF input is generated using vector signal generators, each producing a modulated data channel with individual bandwidth of up to 20 MHz, and carrier frequencies ranging from 250 MHz up to 3.1 GHz. The input transmissions then go through an RF combiner, resulting in a dynamic multiband input signal that enables fast carrier switching for each of the bands. This input is specially designed to allow testing the system’s ability to rapidly sense the input spectrum and adapt to changes, as required by modern CRo and shared spectrum standards, e.g., in the SSPARC program. The system’s effective sampling rate, equal to 480 MHz, is only 8% of the Nyquist rate and 2.4 times the Landau rate. The main advantage of the Xampling framework, demonstrated here, is that sensing is performed in real-time from sub-Nyquist samples for the entire spectral range. Support recovery is digitally performed on the low rate samples. The prototype successfully recovers the support of the CRo transmitted bands, as demonstrated in Figure 1.14. The signal is then reconstructed in real-time. Reconstruction does not a

b

Figure 1.14 SpeCX communication system display [20] showing (a) low rate samples acquired from one MWC channel at rate 120 MHz, and (b) digital reconstruction of the entire spectrum from sub-Nyquist samples.

28

Mishra and Eldar

require interpolation to the Nyquist rate and the active transmissions are recovered at the low rate of 20 MHz, corresponding to the bandwidth of the slices z(f ) defined in (1.32). By combining spectrum sensing and signal reconstruction, the MWC serves as two separate communication devices. The first is a state-of-the-art CRo that performs real time spectrum sensing at sub-Nyquist rates, and the second is a receiver that is able to decode multiple data transmissions simultaneously, regardless of their carrier frequencies, while adapting to real-time spectral changes. The CRr system is based on the sub-Nyquist radar receiver board described in Section 1.3.4. The prototype simulates transmission of P = 50 pulses towards L = 9 targets. The CRr transmits over Nb = 4 bands, selected according to the procedure presented in Section 1.5.2, after the spectrum-sensing process has been completed by the communication receiver. We compare the target detection performance of our CRr with a traditional wideband radar with bandwidth Bh = 20 MHz. The CRr-transmitted bandwidth is thus equal to 3.2% of the wideband. Figure 1.15 shows windows from the graphical user interface (GUI) of our CRr system. Figure 1.15a illustrates the coexistence between the radar transmitted bands

Figure 1.15 SpeCX radar display [20] showing (a) coexisting CRo and CRr (b) CRr spectrum compared with the full-band radar spectrum. The range-Doppler display of detected and true locations of the targets for the case of (a) CRr (four disjoint bands) and (d) all four transmit subbands together forming a contiguous 320 kHz band.

Sub-Nyquist Radar: Principles and Prototypes

29

(thick curve) and the existing communication bands (thin curve). The gain in power is demonstrated in Figure 1.15b, which plots the wideband radar spectrum, CRr, and noise. The true and recovered range-Doppler maps for the CRr (whose transmit signal consists of four disjoint subbands) are shown in Figure 1.15c. All 9 targets are perfectly recovered and clutter is discarded. Figure 1.15d shows the performance when the four subbands are joined together to result in a 320 kHz contiguous band for the radar transmitter. There are many missed detections and false alarms in this case. Let the true and estimated ranges of the ith target be di and dˆi , respectively. Then the rms localization error (RMSLE) of L targets is given by ! " L "1 RMSLE = # (di − dˆi )2 . (1.47) L i=1

In Figure 1.15c–d, the RMSLE is shown as follows: CRr (0.34 km), 320 kHz band or 4 adjacent bands with same bandwidth (8.1 km), and wideband (1.2 km). The poor resolution of the 4 adjacent bands scenario is due to its small aperture. The native range resolution in case of 2 MHz wideband scenario is 75 m. In Figure 1.15c, the CRr is able to detect 9 targets at locations 6.097, 31.764, 35.046, 35.451, 35.479, 81.049, 81.570, 121.442, and 120.922 km. Here, the distance between two closely spaced targets is less than 75 m.

1.6

Spatial Sub-Nyquist: Application to MIMO Radar We now consider extending sub-Nyquist processing to the spatial domain for the particular case of MIMO radar [86]. MIMO radars use an array of several transmit and receive antenna elements, with each transmitter radiating a different, mutually orthogonal waveform. Waveform orthogonality can be in time, frequency or code. Our system is based on the collocated MIMO configuration [87], in which the elements are close to each other so that the radar cross section of a target appears identical in all elements. The MIMO receiver separates and coherently processes the target echoes that correspond to each transmitter. The angular resolution of MIMO using the classic virtual ULA is the same as a phased array with equivalent virtual aperture but many more antenna elements. Conventional MIMO radar’s spatial (angular) and range resolutions are limited by the number of elements and the receiver sampling rate, respectively. Here, we extend the Xampling framework for temporal sub-Nyquist radar in Section 1.3 to both space and time by simultaneously thinning an antenna array and sampling received signals at sub-Nyquist rates. This sub-Nyquist collocated MIMO radar (SUMMeR) recovers the target range, azimuth, and Doppler velocity without loss of any of the aforementioned radar resolutions. In SUMMeR, the radar antenna elements are randomly placed within the aperture, and signal orthogonality is achieved by frequency division multiplexing (FDM). The FDM-based sub-Nyquist MIMO mitigates the range-azimuth coupling by randomizing the element locations in the aperture [88].

30

Mishra and Eldar

Figure 1.16 Location of transmit (diamonds) and receive (triangles) antenna elements within the same physical aperture for (a) conventional MIMO array with T = 5 transmitters and R = 4 receivers, (b) virtual ULA with T R = 20 antenna elements, and (c) randomly thinned MIMO array with M = 4 transmitters and Q = 3 receivers.

1.6.1

Sub-Nyquist Collocated MIMO Radar Model Let the operating wavelength of the radar be λ and the total number of transmit and receive elements be T and R respectively. The classic approach to collocated MIMO adopts a virtual ULA structure, where the receive antennas spaced by λ2 and transmit antennas spaced by R λ2 form two ULAs (or vice versa). Here, the coherent processing of a total of T R channels in the receiver creates a virtual equivalent of a phased array with T R λ2 -spaced receivers and normalized aperture Z = T2R . This standard array structure and the corresponding receiver virtual array are illustrated in Figure 1.16a–b for T = 5 and R = 4. Consider a collocated MIMO radar system that has M < T transmit and Q < R receive antennas. The locations of these antennas are chosen uniformly at random within the aperture of the virtual array mentioned previously, as in Figure 1.16c. The mth transmitting antenna sends P pulses sm (t) =

P −1

hm (t − pτ)ej 2πfc t ,

0 ≤ t ≤ P τ,

(1.48)

p=0

where {hm (t)}M−1 m=0 is a set of narrowband, orthogonal FDM pulses each with CTFT ∞ Hm (ω) =

hm (t)e−j ωt dt.

(1.49)

−∞

For simplicity, we assume that fc τ is an integer. The pulse time support is denoted by Tp . Consider a target scene with L non-fluctuating point targets following the Swerling-0 model [1] whose locations are given by their ranges Rl , Doppler velocity vl , and azimuth angles θl , 1 ≤ l ≤ L. The pulses transmitted by the radar are reflected back by the

Sub-Nyquist Radar: Principles and Prototypes

31

targets and collected at the receive antennas. When the received waveform is downconverted from RF to baseband, we obtain the following signal at the qth antenna, xq (t) =

L P −1 M−1

α l hm (t − pτ − τl ) ej 2πβmq ϑl ej 2πfl

D pτ

,

(1.50)

p=0 m=0 l=1

where α l denotes the complex-valued reflectivity of the lth target, τl = 2Rl /c is the range-time delay the lth target, flD = 2vc l fc is the frequency in the Doppler spectrum, ϑl = sin θl is the azimuth parameter, and βmq is governed by the array structure. We express xq (t) as a sum of single frames xq (t) =

P −1

p

xq (t),

(1.51)

p=0

where p

xq (t) =

L M−1

α l h(t − τl − pτ)ej 2πβmq ϑl ej 2πfl

D pτ

.

(1.52)

m=0 l=1

Our goal is to estimate the time delay τl , azimuth θl , and Doppler shifts flD of each target from low rate samples of xq (t), for 0 ≤ q ≤ Q − 1, and a small number of M channels and Q antennas.

1.6.2

Xampling in Time and Space The application of Xampling in both space and time enables recovery of range, direction, and velocity at sub-Nyquist rates. The sampling technique is the same as in Section 1.3.2, but now the low-rate samples are obtained in both range and azimuth domains. The received signal xq (t) is separated into M channels, aligned, and then normalized. The Fourier coefficients of the received signal corresponding to the channel that processes the mth transmitter echo at the qth receiver are given by p

ym,q [k] =

L

α l ej 2πβmq ϑl e−j

2π τ kτl

e−j 2πfm τl ej 2πfl

D pτ

,

(1.53)

l=1

where − N2 ≤ k ≤ − N2 −1, fm is the (baseband) carrier frequency of the mth transmitter, and N is the number of Fourier coefficients per channel. As in traditional MIMO, assume that the time delays, azimuths, and Doppler frequen1 + cies are aligned to a grid. In particular, τl = TτN sl , ϑl = −1 + T2R rl , and flD = − 2τ 1 u , where s , r , and u are integers satisfying 0 ≤ s ≤ T N − 1, 0 ≤ r ≤ T R − 1, l l l l l l Pτ and 0 ≤ ul ≤ P − 1, respectively. Let Zm be the KQ × P matrix with qth column given p by the vertical concatenation of ym,q [k],k ∈ κ, for 0 ≤ q ≤ Q − 1. We can then write Zm as % $ Zm = Bm ⊗ Am XD FH (1.54) P.

32

Mishra and Eldar

−j 2π fm

2π

n

Bh T Here, Am denotes the K × T N matrix whose (k,n)th element is e−j T N κk n e m with κk the kth element in κ, B is the Q × T R matrix with (q,p)th element 2 e−j 2πβmq (−1+ T R p) , and FP denotes the P × P Fourier matrix. The matrix XD is a T 2 NR × P sparse matrix that contains the values αl at the L indices (rl T N + sl ,ul ). The range and azimuth dictionaries Am and Bm are not square matrices due to lowrate sampling of Fourier coefficients at each receiver and reduction in antenna elements, respectively. Therefore, the system of equations in (1.54) is undetermined in azimuth and range. Our goal is to recover XD from the measurement matrices Zm,0 ≤ m ≤ M − 1. The temporal, spatial, and frequency resolution stipulated by XD are T 1Bh , T2R ,

and

1 Pτ

respectively.

theorem 1.6.1 [17] The minimal number of transmit and receive array elements, i.e., M and Q, respectively, required for perfect recovery of XD with L targets in a noiseless setting are determined by MQ ≥ 2L. In addition, the number of samples per receiver is at least MK ≥ 2L where K is the number of Fourier coefficients sampled per receiver and the number of pulses per transmitter is P ≥ 2L. Theorem 1.6.1 shows that the number of SUMMeR transmit and receive elements as well as samples K depend only on the number of targets present. These design parameters, therefore, can be substantially lesser than the requirements of a Nyquist MIMO array. Similar results for temporal and Doppler sub-Nyquist radars were obtained in Theorems 1.3.1 and 1.6.1.

1.6.3

Range-Azimuth-Doppler Recovery To jointly recover the range, azimuth, and Doppler frequency of the targets, we apply the concept of Doppler focusing from Section 1.3.2 to our MIMO setting. Doppler focusing for a specific frequency ν yields νm,q [k]

=

P −1

ym,q [k]e−j 2πν pτ p

(1.55)

p=0

=

L

α l ej 2πβmq ϑl e−j

2π τ (k+fm τ)τl

l=1

P −1

D −ν )pτ

ej 2π(fl

,

p=0

for − N2 ≤ k ≤ − N2 − 1. Following Section 1.3.2, it holds that P −1 p=0

e

j 2π(flD −ν )pτ

∼ =

P 0

|flD − ν| < otherwise.

1 2P τ ,

(1.56)

Then, for each focused frequency ν, (1.55) reduces to a 2D problem, which can be solved using CS recovery techniques, as summarized in Algorithm 4. Note that step 1

Sub-Nyquist Radar: Principles and Prototypes

33

can be performed using the FFT. In the algorithm description, vec(Z) concatenates the T (l))T where columns of Zm , for 0 ≤ m ≤ M − 1, et (l) = (e0t (l))T · · · (eM−1 t & 'T m m m m T ¯ ¯ et (l) = vec (B ⊗ A )t (l,2)T N+t (l,1) (F )t (l,3) , (1.57) with t (l,i) the (l,i)th element in the index set t at the tth iteration, and Et = [et (1) . . . et (t)]. Once XD is recovered, the delays, azimuths, and Dopplers are estimated as τˆ l =

τL (l,1) ˆ 2L (l,2) ˆD 1 L (l,3) , ϑ l = −1 + , fl = − + . TN TR 2τ Pτ

(1.58)

Since in real scenarios, targets delays, Dopplers, and azimuths are not necessarily aligned to a grid, a finer grid can be used around detection points on the coarse grid to reduce quantization error. This technique adds a step after support detection in each iteration (step 4 in Algorithm 4).

1.6.4

Multi-Carrier and Cognitive Transmission The frequency bands left vacant can be exploited to increase the system’s performance without expanding the total bandwidth of Btot = T Bh . Denote by γ = T /M the compression ratio of the number of transmitters. In multi-carrier SUMMeR, every transmit antenna sends γ pulses, each belonging to a different frequency band, in one PRI. The total number of user bands is M γBh = T Bh . The ith pulse of the pth PRI is transmitted at time i γτ + pτ, for 0 ≤ i < γ and 0 ≤ p ≤ P − 1. The samples are then acquired and processed as described in Sections 1.6.2 and 1.6.3. Besides increasing the detection performance, this method multiplies the Doppler dynamic range by a factor of γ with the same Doppler resolution since the CPI, equal to P τ, is unchanged. Preserving the CPI allows us to maintain the targets’ stationarity. Cognitive transmission described in Section 1.5.2 can also be extended to a SUMMeR system wherein the spectrum of each of the transmitted waveforms is limited to a few nonoverlapping frequency bands while keeping the transmit power per transmitter the same. Cognitive transmission imparts two advantages to the SUMMeR hardware. First, the spatial sub-Nyquist processing of large arrays can be easily designed without replicating the pre-filtering operation for each subband in the hardware. Second, since the total transmit power remains the same, a cognitive signal has more in-band power resulting in an increase in SNR as discussed in Section 1.5.2.

1.6.5

Cognitive SUMMeR Hardware Prototype A cognitive SUMMeR prototype was first presented in [89]. The system realizes a receiver with a maximum of 8 transmit (Tx) and 10 receive (Rx) antenna elements. A scenario includes modeling of pulse transmission, accurate power loss due to wave propagation in a realistic medium, and interaction of a transmit signal with the target. A large variety of scenarios, consisting of different target parameters, i.e., delays,

34

Mishra and Eldar

Algorithm 4 Simultaneous sparse 3D recovery based OMP with focusing [17] Input: Observation matrices Zm , measurement matrices Am , Bm , for all 0 ≤ m ≤ M −1 Output: Index set containing the locations of the non zero indices of X, estimate ˆ for sparse matrix X 1: Perform Doppler focusing for 0 ≤ i ≤ K − 1 and 0 ≤ j ≤ Q − 1: (m, ν )

i,j

=

P −1

j 2π ν pτ Zm . i+j K,p e

p=0 (m, ν )

Initialization: residual R0 = (m, ν ) , index set 0 = ∅, t = 1 3: Project residual onto measurement matrices for 0 ≤ p ≤ P − 1: 2:

ν = AH Rν B, T

T

T

T

T

T

T 0 1 (M−1) ]T , and where A &= [A0 A1 · · · A(M−1) ' ] , B = [B B · · · B (0, ν ) (M−1, ν ) ν R = diag [Rt−1 · · · Rt−1 ] is block diagonal 4: Find the three indices λ t = [λ t (1) λ t (2) λ t (3)] such that [λt (1) λt (2) λt (3)] = arg maxi,j, ν νi,j 5: Augment index set t = t {λ t } 6: Find the new signal estimate

αˆ = [ αˆ 1 . . . αˆ t ]T = (ETt Et )−1 ETt vec(Z) 7:

Compute new residual (m, ν )

Rt

= Zm −

t

αl e

' & & 'T j 2π − 12 + t P(l,3) p m at (l,1) b¯ m t (l,2)

l=1

If t < L, increment t and return to step 2; otherwise stop ˆ = L Estimated support set ˆ 10: Estimated matrix XD : (L (l,2)T N + L (l,1),L (l,3))-th component is given by αˆ l while rest of the elements are zero 8: 9:

Doppler frequencies, and amplitudes, and array configurations, i.e., number of transmitters and receivers and antenna locations, can be examined using the prototype. The waveform generator board produces an analog signal corresponding to the synthesized radar environment, which is amplified and routed to the MIMO radar receiver board. The prototype then samples and processes the signal in real time. The physical array aperture and simulated target response correspond to an X-band (fc = 10 GHz) radar. A conventional 8 × 10 MIMO radar receiver would require simultaneous hardware processing of 80 (or 160 I/Q) data streams. Since a separate sub-Nyquist receiver for each of these 80 channels is expensive, we implement the 8-channel analog processing chain for only 1 receive antenna element, and serialize the received signals of all 10 elements through this chain. This approach allows the prototype to implement a

Sub-Nyquist Radar: Principles and Prototypes

35

Table 1.2 Technical characteristics of the cognitive SUMMeR prototype. Parameters

Mode 1

Mode 2

Mode 3

Mode 4

#Tx, #Rx Element placement Equivalent aperture Angular resolution (sine of DoA)

8,10 Uniform 8 × 10 0.025

8,10 Random 8 × 10 0.025

4,5 Random 8 × 10 0.025

8,10 Random 20 × 20 0.005

Range resolution Signal bandwidth per Tx Pulse width Carrier frequency Unambiguous range Unambiguous DoA PRI Pulses per CPI Unambiguous Doppler

1.25 m 12 MHz (15 MHz including guard-bands) 4.2 μs 10 GHz 15 km 180◦ (from −90◦ to 90◦ ) 100 μs 10 from −75 to 75 m/s

number of receivers greater than 10 as the 8-channel hardware only limits the number of transmitters. If we use the same pre-filtering approach as in Section 1.3.4 for each of the eight channels of our sub-Nyquist MIMO prototype, then the hardware design would need a total of 4 × 8 = 32 BPFs and ADCs excluding the analog filters to separate transmit channels. We sidestep this requirement by adopting cognitive transmission wherein the analog signal of each channel exists only in certain predetermined subbands and consequently, a BPF stage is not required. More importantly, for each channel, a single low-rate ADC subsamples this narrow-band signal as long as the subbands are coset bands so that they do not alias after sampling [46]. This implementation needs only eight low-rate ADCs, one per channel. Another advantage of this approach is flexibility of the prototype in selecting the Xampling slices. Unlike in Section 1.3.4, the number and spectral locations of slices are not permanently fixed, and they can be changed. Table 1.2 lists detailed technical characteristics of the prototype. The system can be configured to operate in various array configurations or modes. Mode 3 and 4 are sub-Nyquist MIMO modes; the hardware switches off the inactive channels and does not sample any data over the corresponding ADCs. Figure 1.18 shows the sub-Nyquist MIMO prototype, user interface and radar display. As shown in Figure 1.19a, the cognitive radar signal occupies only certain subbands in a 15 MHz band. Here, the sliced transmit signal has eight subbands each of width 375 kHz with the frequency range of 1.63–2, 2.16–2.53, 3.05–3.42, 3.88–4.25, 5.66–6.03, 6.51–6.88, 8.64–9.01, and 12.32– 12.69 MHz before subsampling. The total signal bandwidth is 0.375 × 8 = 3 MHz. This signal is subsampled at 7.5 MHz and the subbands locations were chosen so that there is no aliasing between different subbands (Figure 1.19b). A noncognitive signal would have occupied the entire 15 MHz spectrum requiring a Nyquist sampling rate of 30 MHz. Therefore, the use of cognitive transmission enables spectral sampling reduction by a factor of 4 (= 30 MHz/7.5 MHz) for each channel. Depending on whether the guard-bands of the noncognitive transmission are included in the computation or not, the effective signal bandwidth is reduced by a factor of 5 (= 15 MHz/3 MHz) or 4

36

Mishra and Eldar

Figure 1.17 Tx and Rx element locations for the hardware prototype modes over a 6 m antenna aperture. Mode 4’s virtual array equivalent is the 20 × 20 ULA [18].

Figure 1.18 Sub-Nyquist MIMO prototype and user interface. The analog preprocessor module consists of two cards mounted on opposite sides of a common chassis [18].

Figure 1.19 The normalized one-sided spectrum of one channel of a given receiver (a) before and (b) after subsampling with a 7.5 MHz ADC. Each of the subbands spans 375 kHz and is marked with a numeric label. In a noncognitive processing, the signal occupies the entire 15 MHz spectrum before sampling [18].

(= 12 MHz/3 MHz) respectively for each channel. Mode 3 has 50% spatial sampling reduction when compared with Mode 1 or 2. Table 1.3 summarizes the reduction of various resources in Mode 3 when compared with Mode 1. We evaluated the performance of all modes through hardware experiments. We transmitted P = 10 pulses at a PRF of 100 μs and all modes were evaluated against identical target scenarios. In the first experiment, when the angular spacing (in terms of the sine of azimuth) between any two targets was greater than 0.025 and the signal SNR = −8 dB, the recovery performance of the thinned 4 × 5 array in Mode 3 was not worse than Modes 1 and 2. For this experiment, Figures 1.20 and 1.21 show the plan

Sub-Nyquist Radar: Principles and Prototypes

37

Table 1.3 Cognitive SUMMeR Prototype: comparison of resource reduction. Resource

Nyquist Mode 1

Sub-Nyquist Mode 3

Reduction

Bandwidth usage per Tx (including guard-bands) Bandwidth usage per Tx (excluding guard-bands) Temporal sampling rate per channel Spatial sampling rate Tx/Rx hardware channels

15 MHz

3 MHz

80%

12 MHz

3 MHz

75%

30 MHz 8 × 10 80

7.5 MHz 4×5 20

75% 50% 75%

Figure 1.20 Plan position indicator (PPI) display of results for (a) Mode 1 (b) Mode 2 (c) Mode 3 and (d) Mode 4. The origin is the location of the radar. The dark dot indicates the north direction relative to the radar. Positive (negative) distances along the horizontal axis correspond to the east (west) of the radar. Similarly, positive (negative) distances along the vertical axis correspond to the north (south) of the radar. The estimated targets are plotted over the ground truth [18,53].

Figure 1.21 Range-azimuth-Doppler map for the target configurations shown in Figure 1.20 for (a) Mode 1 (b) Mode 2 (c) Mode 3 and (d) Mode 4. The lower axes represent Cartesian coordinates of the polar representation of the PPI plots from Figure 1.20. The vertical axis represents the Doppler spectrum [18,53].

38

Mishra and Eldar

Figure 1.22 PPI plots as in Figure 1.20 for (a) Mode 1 (b) Mode 2 (c) Mode 3 and (d) Mode 4. Only Mode 3 is operating cognitively. All modes have the same overall transmit power per transmitter. The inset plots show the selected region in each PPI display on a magnified scale.

Figure 1.23 Range-azimuth-Doppler maps as in Figure 1.21 for (a) Mode 1 (b) Mode 2 (c) Mode 3

and (d) Mode 4. Only Mode 3 is operating cognitively. All modes have the same overall transmit power per transmitter. The inset plots show the selected region in each map on a magnified scale.

position indicator (PPI) plot and range-azimuth-Doppler maps of all the modes. Here, a successful detection (circle with light fill and no boundary) occurs when the estimated target is within one range cell, one azimuth bin, and one Doppler bin of the ground truth (circle with dark boundary and no fill); otherwise, the estimated target is labeled as a false alarm (circle with dark fill). When a target remains undetected, we label the ground truth location as a missed detection (circle with hatched fill). Finally, we considered a high-noise scenario with SNR = −15 dB. We operated only Mode 3 cognitively and kept all other modes in noncognitive mode. We noticed that the noncognitive Nyquist 8 × 10 Mode 1 array exhibits false alarms while cognitive subNyquist 4 × 5 Mode 3 array is still able to detect all the targets (Figures 1.22 and 1.23), thereby demonstrating robustness to low SNR.

Sub-Nyquist Radar: Principles and Prototypes

1.7

39

Sub-Nyquist SAR Synthetic aperture radar (SAR) and other similar radar techniques were one of the first applications of CS methods (see reviews in [22,25]). SAR imaging data are not naturally sparse in the range-time domain. However, they are often sparse in other domains, such as wavelets. Our motivation to apply sub-Nyquist methods here is to address the following SAR processing challenge. Among the several algorithms that are available to process SAR data, the range-Doppler algorithm (RDA) is most widely used to obtain high-resolution images [90]. Its performance is, however, limited by the range cell migration correction (RCMC) step, which requires oversampled data in order to decouple range and azimuth axes. Recently, [21] proposed a sub-Nyquist SAR that replaces RDA by a Fourier domain method that achieves non-integer nonconstant shifts in the RCMC interpolation via the Fourier series coefficients. This avoids the interpolation step in RCMC and further allows sub-Nyquist sampling that follows the Fourier-domain analysis presented in previous sections. A similar technique was earlier employed in ultrasound imaging [48] to dramatically reduce sampling and processing rates. In this section, we present this Fourier domain RDA processing as a framework for sub-Nyquist sampling of SAR signals. The first part of the sub-Nyquist algorithm exploits the relationship between the signals before and after RCMC in the Fourier domain. We show analytically that a single Fourier coefficient after RCMC can be computed using a small number of Fourier coefficients of the raw data, which translates into low rate sampling as shown in Section 1.3.2. Having the partial Fourier samples after RCMC, the second part of the algorithm is aimed at solving a 2D CS problem in order to reconstruct the image from the low rate samples. Finally, we show that cognitive transmission can also be extended to SAR. We end by demonstrating a prototype that we designed and developed to realize concepts of cognitive SAR (CoSAR) [21].

1.7.1

Traditional SAR Processing via RDA Consider a radar that travels along a path with velocity ν and transmits a time-limited pulse h(t) at PRI T . The pulse has negligible energy at frequencies beyond the bandwidth Bh /2. The transmitted pulses are sent from M different locations, {xm }M−1 m=0 , where x0 is the origin and ||xm − x0 || = m|ν|T . The pulses are transmitted into a scene with reflectivity σ(r). The received signal, after coherent demodulation, is given by (1.59) dm (t) = σ(r)h(t − 2||r − xm ||/c) × wa (xm,r)e−j 4πfc ||r−xm ||/c dr, where ||r − xm || is the distance from the radar to a scatter point and wa (xm,r) is the antenna beam pattern, which varies depending on the SAR operation mode [90]. The main goal of SAR data processing is to construct the scene’s reflectivity map, σ(r), from the raw data. The reflected signal dm (t) at a point m requires sampling at least at the bandwidth Bh , as per the Nyquist sampling theorem. The resulting discrete-time

40

Mishra and Eldar

signal is d[n,m] = dm (nT s), with 0 ≤ n < N = Tfs , where fs = 1/Ts is the sampling rate. RDA processing consists of the following steps. First the sampled raw data is compressed in the range dimension: s[n,m] = d[n,m] ∗ h∗ [−n],

(1.60)

where h[n] is the sampled transmit signal. This data is then transformed to the rangeDoppler domain using DFT along the azimuth: S[n,k] =

M−1

s[n,m]e−j 2πkm/M .

(1.61)

m=0

RCMC is applied assuming a far-field approximation. The purpose of RCMC is to compensate for the effect of range cell migration due to the varied satellite-scatterer distance and to correct the hyperbolic behavior of the target trajectories. The RCMC operator can be written as ˜ C[n,k] = S[n + nak 2,k].

(1.62)

For every Doppler frequency k, the range axis is scaled by 1 + ak 2 . The value of a is predetermined depending on the observation mode. For example, in stripmap SAR, 2 a = 8|ν |Tλ 2 M 2 . This range-variant shift requires values that fall outside the discrete grid. An MF then achieves compression in azimuth via k2

−j π Ka [n] ˜ , Y [n,k] = C[n,k]e

(1.63)

where Ka [n] is the range dependent azimuth chirp rate. Finally, an inverse DFT in the azimuth direction yields the focused image: I [n,m] =

M−1 1 Y [n,k]ej 2πmk/M . M

(1.64)

k=0

There are two ways to implement RCMC: In the first option, RCMC is performed by range interpolation in the range-Doppler domain. However, this interpolation is timeconsuming and computationally demanding. The second approach involves the assumption that the range cell migration is range invariant, at least over a finite range block. In this case, RCMC is implemented using a DFT, linear phase multiply, and inverse DFT per block. However, this implementation’s disadvantage is that samples should overlap in range, and the efficiency gain may not be worth the added complexity.

1.7.2

Fourier Domain RDA and Sub-Nyquist SAR In this section, we introduce a new RDA processing technique implemented in frequency using the Fourier series coefficients of the raw data. This paves the way for substantial reduction in the number of samples in the time-domain interpolation needed

Sub-Nyquist Radar: Principles and Prototypes

41

to obtain the same image quality and without any assumptions on the signal structure or the invariance of range blocks. We begin with the continuous version of (1.62): Ck (t) = Sk (t(1 + ak 2 )),

(1.65)

where Sk (nTs ) = S[n,k]. The Fourier series coefficients of Ck (t) over the interval [0,T ) can be expressed as [21] 1 Ck [l] = T

T Sk (t)qk,l (t),

(1.66)

0

where qk,l (t) approximates the scaling operation in (1.62). The Fourier coefficients of the continuous-time signals Sk (t) and qk,l (t) are, respectively, Sk [n] and 1 l −j π(n+ 1 2 ) 1+ak sinc n + Qk,l [n] = e . (1.67) 1 + ak 2 1 + ak 2 It can be shown that most of the energy of the set Qk,l [n] is concentrated around a specific component nk,l . Thus, for every Doppler frequency k, the Fourier series coefficients of the scaled signal, Ck (t), can be calculated as a linear combination of a local choice of Fourier series coefficients of Sk (t) Sk [n]Qk,l [−n], (1.68) Ck [l] = n∈ν (k,l)

where ν(k,l) is the set of indices that dictate the decay property of Qk,l [n]. Assuming the Fourier series coefficients Dm [l] of the raw data dm (t) can be acquired directly, the range compression is achieved in the Fourier domain as D˜ m [l] = T Dm [l]H ∗ [l],

(1.69)

where H [l] are the Fourier series coefficients of the transmitted pulse h(t). Applying azimuth DFT gives Sk [l] that can be used in (1.68) to perform Fourier domain RCMC. The inverse DFT on the coefficients Ck [l] provides the corrected sampled signal after RCMC. One could then proceed with the remaining steps of RDA, i.e., (1.63) and (1.64), to complete the processing. The number of Fourier coefficients required can be further reduced if a basis (e.g., wavelet) is found in which the desired image is sparse. Then, the relationship between Ck [l] and the raw data samples Dm [l] can be exploited to solve for the coefficients in the sparse basis using fewer Fourier coefficients. In [21], it was suggested that we modify a fast iterative shrinkage-thresholding algorithm (FISTA) to solve this problem and achieve full data reconstruction from the partial measurements with reasonable computational load. Similar to cognitive pulse-Doppler radar (Section 1.5.2) and cognitive SUMMeR (Section 1.6.5), sub-Nyquist SAR systems can also be modified to fit cognitive radar requirements and allow for dynamic transmission and reception of several narrow frequency bands. We present the hardware prototype of such a system in the next subsection.

42

Mishra and Eldar

Figure 1.24 Cognitive SAR (CoSAR) prototype and (inset) analog preprocessor.

Figure 1.25 CoSAR GUI showing the cognitive and noncognitive chirp waveforms along with the sampled subbands at top right.

1.7.3

Hardware Prototype We designed and developed a hardware prototype of a CoSAR system and evaluated Fourier domain RDA processing in real time. Figure 1.24 shows the entire setup. The PRI is 51.2 μs and carrier frequency of the signal is 90 MHz. A control interface (Figure 1.25) activates the prototype that generates the desired I/Q signal and feeds it to the analog preprocessor (inset). The analog preprocessor filters have 30 dB stopband attenuation in order to filter out interference from neighboring channels. The digital receiver obtains and processes samples at low rates. The processed image is then shown on the radar display. We used a 5 MHz cognitive chirp signal whose only 4 narrow subbands of 625 kHz bandwidth were sampled and processed by the digital receiver. The Xampling and RCMC are performed at 1/4th and 1/8th of the Nyquist rate, respectively. Similar to the cognitive SUMMeR system, our CoSAR prototype can operate in both cognitive and noncognitive modes. Figure 1.26 shows results of these modes at Nyquist and sub-Nyquist sampling rates at SNR = 2dB. The range and cross-range (azimuth) resolutions are 30 and 10 m, respectively. When compared with the Nyquist rate of 10 MHz, the combined sampling rate of the 4 slices is 2.5 MHz leading to reduction of rate by 75%. We note that CoSAR reconstruction exhibits smaller error

Sub-Nyquist Radar: Principles and Prototypes

43

Figure 1.26 Comparison of prototype outputs for an image of a ship.

than the noncognitive Nyquist processing in low SNR scenarios, despite sampling at a significantly reduced sampling rate. Further, the prototype demonstrates operation of SAR using narrow subbands that can be adaptively changed. This opens up the possibility of spectral coexistence of SAR with other satellite-borne services.

1.8

Summary In this chapter, we reviewed sub-Nyquist radar principles, algorithms, prototypes, and applications. Our focus was on pulse-Doppler systems for which sub-Nyquist processing can be individually applied to temporal, Doppler, and spatial domains. Our approach has distinct advantages over several past CS-based designs. The proposed sub-Nyquist radar receivers perform low-rate sampling and processing, which can be implemented with simple hardware, impose no restrictions on the transmitter, use a CS dictionary that does not scale up with the problem size, and exhibit robustness to clutter and noise. We presented colocated MIMO radar as an application where joint spatiotemporal sub-Nyquist processing leads to reduction in antenna elements and savings in signal bandwidth. In SAR imaging, sub-Nyquist processing in the Fourier domain leads to sampling rate reduction without compromising high-quality and high-resolution imaging. We demonstrated that sub-Nyquist receivers lead to the feasibility of cognitive radar, which transmits thinned spectrum signals. This development was significant in making the spectral coexistence of radar with a communication service possible. We also extended cognition ideas based on sub-Nyquist processing to MIMO and SAR systems. Most importantly, we emphasized that sub-Nyquist radars are realizable in hardware for each of the systems described in this chapter. The hardware prototypes were inhouse and custom-made using many off-the-shelf components. The systems operate in real-time and their performance is robust to high noise and clutter. We believe that such practical implementations pave the way to delivering the promise of reduced-rate processing in radar remote sensing.

44

Mishra and Eldar

References [1] M. I. Skolnik, Radar Handbook, 3rd edn. McGraw-Hill, 2008. [2] P. Z. Peebles, Radar Principles. Wiley-Interscience, 1998. [3] J. E. Cilliers and J. C. Smit, “Pulse compression sidelobe reduction by minimization of lp -norms,” IEEE Transactions on Aerospace and Electronic Systems, vol. 43, no. 3, pp. 1238–1247, 2007. [4] J. George, K. V. Mishra, C. M. Nguyen, and V. Chandrasekar, “Implementation of blind zone and range-velocity ambiguity mitigation for solid-state weather radar,” in IEEE Radar Conference, 2010, pp. 1434–1438. [5] K. V. Mishra, V. Chandrasekar, C. Nguyen, and M. Vega, “The signal processor system for the NASA dual-frequency dual-polarized Doppler radar,” in IEEE International Geoscience and Remote Sensing Symposium, 2012, pp. 4774–4777. [6] Y. C. Eldar, Sampling Theory: Beyond Bandlimited Systems. Cambridge University Press, 2015. [7] Y. C. Eldar and G. Kutyniok, Compressed Sensing: Theory and Applications. Cambridge University Press, 2012. [8] D. Cohen and Y. C. Eldar, “Reduced time-on-target in pulse Doppler radar: Slow time domain compressed sensing,” in IEEE Radar Conference, 2016, pp. 1–4. [9] J. Akhtar, B. Torvik, and K. E. Olsen, “Compressed sensing with interleaving slow-time pulses and hybrid sparse image reconstruction,” in IEEE Radar Conference, 2017, pp. 0006– 0010. [10] K. V. Mishra, A. Kruger, and W. F. Krajewski, “Compressed sensing applied to weather radar,” in IEEE International Geoscience and Remote Sensing Symposium, 2014, pp. 1832– 1835. [11] R. P. Shenoy, “Phased array antennas,” in Advanced Radar Techniques and Systems, G. Galati, Ed. Peter Peregrinus, 1993. [12] E. T. Bayliss, “Design of monopulse antenna difference patterns with low sidelobes,” Bell System Technical Journal, vol. 47, no. 5, pp. 623–650, 1968. [13] D. K. Cheng, “Optimization techniques for antenna arrays,” Proceedings of the IEEE, vol. 59, no. 12, pp. 1664–1674, 1971. [14] R. L. Haupt, Timed Arrays: Wideband and Time Varying Antenna Arrays. John Wiley & Sons, 2015. [15] K. V. Mishra, I. Kahane, A. Kaufmann, and Y. C. Eldar, “High spatial resolution radar using thinned arrays,” in IEEE Radar Conference, 2017, pp. 1119–1124. [16] M. Rossi, A. M. Haimovich, and Y. C. Eldar, “Spatial compressive sensing for MIMO radar,” IEEE Transactions on Signal Processing, vol. 62, no. 2, pp. 419–430, 2014. [17] D. Cohen, D. Cohen, Y. C. Eldar, and A. M. Haimovich, “SUMMeR: Sub-Nyquist MIMO radar,” IEEE Transactions on Signal Processing, vol. 66, no. 16, pp. 4315–4330, 2018. [18] K. V. Mishra, E. Shoshan, M. Namer et al., “Cognitive sub-Nyquist hardware prototype of a collocated MIMO radar,” in International Workshop on Compressed Sensing Theory and its Applications to Radar, Sonar and Remote Sensing, 2016, pp. 56–60. [19] K. V. Mishra and Y. C. Eldar, “Performance of time delay estimation in a cognitive radar,” in IEEE International Conference on Acoustics, Speech and Signal Processing, 2017, pp. 3141–3145.

Sub-Nyquist Radar: Principles and Prototypes

45

[20] D. Cohen, K. V. Mishra, and Y. C. Eldar, “Spectrum sharing radar: Coexistence via Xampling,” IEEE Transactions on Aerospace and Electronic Systems, vol. 29, no. 3, pp. 1279–1296, 2018. [21] K. Aberman and Y. C. Eldar, “Sub-Nyquist SAR via Fourier domain range-Doppler processing,” IEEE Transactions on Geoscience and Remote Sensing, vol. 55, no. 11, pp. 6228–6244, 2017. [22] J. H. Ender, “On compressive sensing applied to radar,” Signal Processing, vol. 90, no. 5, pp. 1402–1414, 2010. [23] N. A. Goodman and L. C. Potter, “Pitfalls and possibilities of radar compressive sensing,” Applied Optics, vol. 54, no. 8, pp. C1–C13, 2015. [24] L. Zhao, L. Wang, L. Yang, A. M. Zoubir, and G. Bi, “The race to improve radar imagery: An overview of recent progress in statistical sparsity-based techniques,” IEEE Signal Processing Magazine, vol. 33, no. 6, pp. 85–102, 2016. [25] M. Cetin, I. Stojanovic, O. Onhon, et al., “Sparsity-driven synthetic aperture radar imaging: Reconstruction, autofocusing, moving targets, and compressed sensing,” IEEE Signal Processing Magazine, vol. 31, no. 4, pp. 27–40, 2014. [26] M. A. Hadi, S. Alshebeili, K. Jamil, and F. E. A. El-Samie, “Compressive sensing applied to radar systems: An overview,” Signal, Image and Video Processing, vol. 9, no. 1, pp. 25–39, 2015. [27] R. Baraniuk and P. Steeghs, “Compressive radar imaging,” in IEEE Radar Conference, 2007, pp. 128–133. [28] M. A. Herman and T. Strohmer, “High-resolution radar via compressed sensing,” IEEE Transactions on Signal Processing, vol. 57, no. 6, pp. 2275–2284, 2009. [29] Y.-S. Yoon and M. G. Amin, “Compressed sensing technique for high-resolution radar imaging,” in Signal Processing, Sensor Fusion, and Target Recognition XVII, vol. 6968, 2008, p. 69681A. [30] X. Tan, W. Roberts, J. Li, and P. Stoica, “Range-Doppler imaging via a train of probing pulses,” IEEE Transactions on Signal Processing, vol. 57, no. 3, pp. 1084–1097, 2009. [31] J. Zhang, D. Zhu, and G. Zhang, “Adaptive compressed sensing radar oriented toward cognitive detection in dynamic sparse target scene,” IEEE Transactions on Signal Processing, vol. 60, no. 4, pp. 1718–1729, 2012. [32] C.-Y. Chen, “Signal processing algorithms for MIMO radar,” PhD dissertation, California Institute of Technology, 2009. [33] Y. Yu, A. P. Petropulu, and H. V. Poor, “Measurement matrix design for compressive sensing–based MIMO radar,” IEEE Transactions on Signal Processing, vol. 59, no. 11, pp. 5338–5352, 2011. [34] Y. Yu, A. P. Petropulu, and H. V. Poor, “MIMO radar using compressive sampling,” IEEE Journal on Selected Topics in Signal Processing, vol. 4, no. 1, pp. 146–163, 2010. [35] Y. Chi, L. L. Scharf, A. Pezeshki, and A. R. Calderbank, “Sensitivity to basis mismatch in compressed sensing,” IEEE Transactions on Signal Processing, vol. 59, no. 5, pp. 2182– 2195, 2011. [36] R. Heckel, V. I. Morgenshtern, and M. Soltanolkotabi, “Super-resolution radar,” Information and Inference: A Journal of the IMA, vol. 5, no. 1, pp. 22–75, 2016. [37] R. Heckel, “Super-resolution MIMO radar,” in IEEE International Symposium on Information Theory, 2016, pp. 1416–1420. [38] G. Tang, B. N. Bhaskar, P. Shah, and B. Recht, “Compressed sensing off-the-grid,” IEEE Transactions on Information Theory, vol. 59, no. 11, pp. 7465–7490, 2013.

46

Mishra and Eldar

[39] K. V. Mishra, M. Cho, A. Kruger, and W. Xu, “Super-resolution line spectrum estimation with block priors,” in Asilomar Conference on Signals, Systems and Computers, 2014, pp. 1211–1215. [40] W. U. Bajwa, K. Gedalyahu, and Y. C. Eldar, “Identification of parametric underspread linear systems and super-resolution radar,” IEEE Transactions on Signal Processing, vol. 59, no. 6, pp. 2548–2561, 2011. [41] W. Kozek and G. E. Pfander, “Identification of operators with bandlimited symbols,” SIAM Journal on Mathematical Analysis, vol. 37, no. 3, pp. 867–888, 2005. [42] K. Gedalyahu and Y. C. Eldar, “Time-delay estimation from low-rate samples: A union of subspaces approach,” IEEE Transactions on Signal Processing, vol. 58, no. 6, pp. 3017– 3031, 2010. [43] S. Sun, W. U. Bajwa, and A. P. Petropulu, “MIMO-MC radar: A MIMO radar approach based on matrix completion,” IEEE Transactions on Aerospace and Electronic Systems, vol. 51, no. 3, pp. 1839–1852, 2015. [44] M. Vetterli, P. Marziliano, and T. Blu, “Sampling signals with finite rate of innovation,” IEEE Transactions on Signal Processing, vol. 50, no. 6, pp. 1417–1428, 2002. [45] E. Baransky, G. Itzhak, I. Shmuel et al., “A sub-Nyquist radar prototype: Hardware and algorithms,” IEEE Transactions on Aerospace and Electronic Systems, vol. 50, pp. 809–822, 2014. [46] K. M. Cohen, C. Attias, B. Farbman, I. Tselniker, and Y. C. Eldar, “Channel estimation in UWB channels using compressed sensing,” in IEEE International Conference on Acoustics, Speech and Signal Processing, 2014, pp. 1966–1970. [47] K. V. Mishra and Y. C. Eldar, “Sub-Nyquist channel estimation over IEEE 802.11ad link,” in IEEE International Conference on Sampling Theory and Applications, 2017, pp. 355–359. [48] T. Chernyakova and Y. C. Eldar, “Fourier-domain beamforming: The path to compressed ultrasound imaging,” IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, vol. 61, no. 8, pp. 1252–1267, 2014. [49] O. Bar-Ilan and Y. C. Eldar, “Sub-Nyquist radar via Doppler focusing,” IEEE Transactions on Signal Processing, vol. 62, pp. 1796–1811, 2014. [50] R. Tur, Y. C. Eldar, and Z. Friedman, “Innovation rate sampling of pulse streams with application to ultrasound imaging,” IEEE Transactions on Signal Processing, vol. 59, no. 4, pp. 1827–1842, 2011. [51] N. Wagner, Y. C. Eldar, and Z. Friedman, “Compressed beamforming in ultrasound imaging,” IEEE Transactions on Signal Processing, vol. 60, no. 9, pp. 4643–4657, 2012. [52] Y. C. Eldar, R. Levi, and A. Cohen, “Clutter removal in sub-Nyquist radar,” IEEE Signal Processing Letters, vol. 22, no. 2, pp. 177–181, 2015. [53] D. Cohen, K. V. Mishra, D. Cohen et al., “Sub-Nyquist MIMO radar prototype with Doppler processing,” in IEEE Radar Conference, 2017, pp. 1179–1184. [54] S. Na, K. V. Mishra, Y. Liu, Y. C. Eldar, and X. Wang, “TenDSuR: Tensor-based 3D subNyquist radar,” IEEE Signal Processing Letters, 2018, in press. [55] S. M. Kay, Fundamentals of Statistical Signal Processing, Volume 2: Detection Theory. Prentice Hall, 1998. [56] M. Mishali and Y. C. Eldar, “From theory to practice: Sub-Nyquist sampling of sparse wideband analog signals,” IEEE Journal on Selected Topics in Signal Processing, vol. 4, no. 2, pp. 375–391, 2010. [57] M. Mishali and Y. C. Eldar, “Sub-Nyquist sampling: Bridging theory and practice,” IEEE Signal Processing Magazine, vol. 28, no. 6, pp. 98–124, 2011.

Sub-Nyquist Radar: Principles and Prototypes

47

[58] D. Cohen and Y. C. Eldar, “Sub-Nyquist sampling for power spectrum sensing in cognitive radios: A unified approach,” IEEE Transactions on Signal Processing, vol. 62, no. 15, pp. 3897–3910, 2014. [59] D. Cohen and Y. C. Eldar, “Sub-Nyquist cyclostationary detection for cognitive radio,” IEEE Transactions on Signal Processing, vol. 65, no. 11, pp. 3004–3019, 2017. [60] D. Cohen, A. Dikopoltsev, R. Ifraimov, and Y. C. Eldar, “Towards sub-Nyquist cognitive radar,” in IEEE Radar Conference, 2016, pp. 1–4. [61] K. Gedalyahu, R. Tur, and Y. C. Eldar, “Multichannel sampling of pulse streams at the rate of innovation,” IEEE Transactions on Signal Processing, vol. 59, no. 4, pp. 1491–1504, 2011. [62] L. L. Scharf and B. Friedlander, “Matched subspace detectors,” IEEE Transactions on Signal Processing, vol. 42, no. 8, pp. 2146–2157, 1994. [63] P. B. Tuuk and S. L. Marple, “Compressed sensing radar amid noise and clutter using interference covariance information,” IEEE Transactions on Aerospace and Electronic Systems, vol. 50, no. 2, pp. 887–897, 2014. [64] Y. Yu, S. Sun, and A. P. Petropulu, “A Capon beamforming method for clutter suppression in colocated compressive sensing based MIMO radars,” in SPIE Defense, Security, and Sensing, 2013, pp. 87 170J. [65] K. Sun, H. Zhang, G. Li, H. Meng, and X. Wang, “A novel STAP algorithm using sparse recovery technique,” in IEEE International Geoscience and Remote Sensing Symposium, vol. 5, 2009, pp. V–336. [66] X. Yang, Y. Sun, T. Zeng, and T. Long, “Iterative roubust sparse recoery method based on focuss for space-time adaptive processing,” in IET International Radar Conference 2015, 2015, pp. 1–6. [67] Z. Ma, Y. Liu, H. Meng, and X. Wang, “Jointly sparse recovery of multiple snapshots in STAP,” in IEEE Radar Conference, 2013, pp. 1–4. [68] Z. Wang, H. Li, and B. Himed, “A sparsity based GLRT for moving target detection in distributed MIMO radar on moving platforms,” in Asilomar Conference on Signals, Systems and Computers, 2015, pp. 90–94. [69] S. Kay, “Optimal signal design for detection of Gaussian point targets in stationary Gaussian clutter/reverberation,” IEEE Journal of Selected Topics in Signal Processing, vol. 1, no. 1, pp. 31–41, 2007. [70] L. E. Brennan and L. Reed, “Theory of adaptive radar,” IEEE Transactions on Aerospace and Electronic Systems, no. 2, pp. 237–252, 1973. [71] L. E. Brennan and I. S. Reed, “Optimum processing of unequally spaced radar pulse trains for clutter rejection,” IEEE Transactions on Aerospace and Electronic Systems, no. 3, pp. 474–477, 1968. [72] T. Wimalajeewa, Y. C. Eldar, and P. K. Varshney, “Recovery of sparse matrices via matrix sketching,” arXiv preprint arXiv:1311.2448, 2013. [73] H. Griffiths, L. Cohen, S. Watts, E. Mokole, C. Baker, M. Wicks, and S. Blunt, “Radar spectrum engineering and management: Technical and regulatory issues,” Proceedings of the IEEE, vol. 103, no. 1, pp. 85–102, 2015. [74] G. M. Jacyna, B. Fell, and D. McLemore, “A high-level overview of fundamental limits studies for the DARPA SSPARC program,” in IEEE Radar Conference, 2016, pp. 1–6. [75] J. R. Guerci, R. M. Guerci, A. Lackpour, and D. Moskowitz, “Joint design and operation of shared spectrum access for radar and communications,” in IEEE Radar Conference, 2015, pp. 0761–0766.

48

Mishra and Eldar

[76] K. V. Mishra, A. Zhitnikov, and Y. C. Eldar, “Spectrum sharing solution for automotive radar,” in IEEE 85th Vehicular Technology Conference, 2017, pp. 1–5. [77] A. R. Chiriyath, B. Paul, G. M. Jacyna, and D. W. Bliss, “Inner bounds on performance of radar and communications co-existence,” IEEE Transactions on Signal Processing, vol. 64, no. 2, pp. 464–474, 2016. [78] D. Cohen, S. Tsiper, and Y. C. Eldar, “Analog to digital cognitive radio: Sampling, detection and hardware,” IEEE Signal Processing Magazine, vol. 35, no. 1, pp. 137–166, 2018. [79] M. Mishali and Y. C. Eldar, “Blind multi-band signal reconstruction: Compressed sensing for analog signals,” IEEE Transactions on Signal Processing, vol. 57, no. 3, pp. 993–1009, 2009. [80] D. Cohen, S. Tsiper, and Y. C. Eldar, “Analog to digital cognitive radio,” in Handbook of Cognitive Radio, W. Zhang, Ed. Springer Singapore, 2017. [81] N. Vaswani and W. Lu, “Modified-cs: Modifying compressive sensing for problems with partially known support,” IEEE Transactions on Signal Processing, vol. 58, no. 9, pp. 4595– 4607, 2010. [82] V. Stankovi, L. Stankovi, and S. Cheng, “Compressive image sampling with side information,” in IEEE International Conference Image Processing, 2009, pp. 3037–3040. [83] J. Huang, T. Zhang, and D. Metaxas, “Learning with structured sparsity,” Journal of Machine Learning Research, vol. 12, no. 11, pp. 3371–3412, 2011. [84] M. Mishali, Y. C. Eldar, O. Dounaevsky, and E. Shoshan, “Xampling: Analog to digital at sub-Nyquist rates,” IET Circuits, Devices & Systems, vol. 5, pp. 8–20, 2011. [85] M. Mishali and Y. C. Eldar, “Expected RIP: Conditioning of the modulated wideband converter,” in IEEE Information Theory Workshop, 2009, pp. 343–347. [86] E. Fishler, A. Haimovich, R. Blum, D. Chizhik, L. Cimini, and R. Valenzuela, “MIMO radar: An idea whose time has come,” in IEEE Radar Conference, 2004, pp. 71–78. [87] J. Li and P. Stoica, “MIMO radar with colocated antennas,” IEEE Signal Processing Magazine, vol. 24, no. 5, pp. 106–114, 2007. [88] D. Cohen, D. Cohen, and Y. C. Eldar, “High resolution FDMA MIMO radar,” arXiv preprint arXiv:1711.06560, 2017. [89] K. V. Mishra, Y. C. Eldar, E. Shoshan, M. Namer, and M. Meltsin, “A cognitive sub-Nyquist MIMO radar prototype,” arXiv preprint arXiv:1807.09126, 2018. [90] C. F. Barnes, Synthetic Aperture Radar: Wave Theory Foundations: Analysis and Algorithms. Barnes (self-published), 2014.

2

Clutter Rejection and Adaptive Filtering in Compressed Sensing Radar Peter B. Tuuk

2.1

Introduction Clutter returns have posed challenges to radar designers and engineers since the early days of technology development and use. Indeed, as early as the Second World War, attention was paid to mitigating unwanted detections from terrain [1]. Generally, clutter returns were mitigated by constructing the observation geometry so that targets were above the radar and sensed against the background of sky. Early techniques included a simple notch filter at the transmitted frequency that removed returns with zero Doppler shift. This was effective in some cases for stationary radar systems, but did not compensate for the effects of platform motion that shifted the frequency of clutter returns. A subsequent development, pioneered in the 1950s, was the displaced phase center technique that phase-shifted the returns from a series of pulses to align them, allowing for more effective cancellation [2]. The next major development in airborne radar was a revolution: pulse-Doppler radar, enabled by high-accuracy timing circuitry and early digital memory, which coherently processed a set of pulses for clutter rejection and other purposes. These techniques allowed effective airborne early warning development and look-down, shoot-down modes for fighter aircraft [3]. One such early pulse-Doppler radar, the AWG-10, was employed on the McDonnell-Douglas F-4 Phantom. In the late 1970s and 1980s, space–time adaptive processing (STAP) was introduced [4]. STAP takes advantage of improvements in digital signal processing to extend the clutter cancellation to the two-dimensional domain. It does so by introducing a spatial or array channel dimension. These additional degrees of freedom allow improved cancellation in the joint domain and extend work on clutter cancellation to that of other structured interference sources. In the years since, STAP theory and practice have improved with the introduction of more array channels, prior-knowledge-aided processing, and the introduction of computationally expensive matrix decompositions. As STAP becomes a more mature technology, it is migrating to smaller platforms. But cost, size, weight, power, and other considerations make large, multi-channel, highbandwidth array antennas infeasible in these settings. Compressed sensing (CS) offers the hope that lower sampling requirements and data volumes could simplify data acquisition requirements and allow advanced techniques on lower-end platforms. In some contexts the computational costs of CS reconstruction are prohibitive today. But in others, the signal acquisition problem is intractable under traditional Nyquist-rate sampling. 49

50

Tuuk

As CS techniques, approaches, and technologies mature, the need to consider additional sources of interference beyond noise becomes more pressing. It is at this point that this chapter picks up the thread, examining the topic of CS in radar with a focus on mitigating structured interference, such as clutter, in the CS context. To do so, we introduce work from adaptive filtering and low-rank matrix approximation. Recent results in this area show that if the interference has low rank statistics of the covariance can be reliably estimated from highly compressed measurements. In addition, the covariance of the interference can be incorporated into the CS estimation process to improve performance.

2.2

Problem Formulation The radar identifies objects within its field of regard by transmitting radio frequency electromagnetic energy into the surrounding medium. This energy propagates through the medium impinging on objects in that environment. The objects reflect some portion of the energy back to the radar where it is processed to estimate characteristics of the environment.

2.2.1

Data Cube The most basic use of a radar system is to calculate range to a target by measuring the time between transmission of a pulse and the time the reflection from the target is received. Another fundamental measurement that may be made with a radar receiver is to calculate target velocity by measuring the Doppler shift of the reflected pulse. For multi-pulse radar the Doppler shift is calculated over multiple pulses to increase the observation time, thereby improving the Doppler resolution. Multichannel digital receivers may estimate the angle of arrival of reflected energy using the differential time or phase delay between measurements at the sampled channels. These three sampling dimensions correspond to three dimensions in the target space: 1.

2.

3.

Receiver Channel: Elements of the antenna array are separated into some number of channels. The received energy collected by the elements of a channel is coherently combined and sampled. This sampling dimension is used to determine the direction of the arrival of signals from targets. Slow Time: The radar transmits a series of pulses, samples the returns from each, and processes these samples coherently. This set of pulses constitutes a coherent processing interval (CPI), and the pulses are transmitted with some frequency, the pulse repetition frequency (PRF). This sampling dimension is used to determine the range rate of targets. Fast Time: The analog-to-digital converter samples the incoming radio-frequency signal at a rate determined by the bandwidth of the transmitted waveform. This sampling dimension is used to determine the range of targets.

This three-dimensional conception of the received signal is known as the data cube [5]. Let the dimensionality of this cube be Nc × Ns × Nf , for the channel, slow time, and

Clutter Rejection and Adaptive Filtering

51

fast time sampling dimensions. Further define n = Nc Ns Nf . This data cube can be vectorized as y ∈ Cn and approximated by a linear combination of basis elements in a sensing operator: y = Sx,

(2.1)

where y contains the samples in time and space. The vector x is the unknown vector that describes the target scene; it is a vector that gives radar a cross section of scatterers at each location in the observation extent. Generally this vector is sparse because many gridpoints contain no target.

2.2.2

Linear Sensing Model The sensing operator S is a linear transform from the discretized target space (angle, radial velocity, and range of dimension Na × Nv × Nr ) to the sampled data cube. The full sensing matrix can be constructed from the bases that describe the response along the range, angle, and Doppler dimensions: S = Sr ⊗ Sa ⊗ Sd,

(2.2)

where ⊗ is the Kronecker product. These models describes simple propagation phenomena. Let there be a target at range ri from the antenna and angle θi from the array boresight with range rate vi . d ri .) This target is illuminated (Range rate being the time derivative of range, vi = dt by a series of ns identical waveforms with carrier frequency f0 , i.e., each waveform w(t) = e2πjf0 t e2πj φ(t) with pulse repetition interval Ts . The waveform w(t) has phase φ(t) and bandwidth β, whether by swept frequency chirp, phase code sequence, or some other modulation function. The illumination experienced at the ith target is then ei (t) = αi

n s −1

w(t − qTs − (ri + vi qTs )/c)

(2.3)

q=0

for some scalar αi , where c is the speed of light. A moving target imparts a Doppler frequency shift on the waveform proportional to its radial velocity (positive shift for decreasing range), and some of this energy is reflected back to the antenna array to be received. The array consists of ne individual array elements uniformly separated by a distance d. We neglect the element pattern of any array element, and instead model them as isotropic receivers. Each of these array elements makes nf uniformlyspaced fast-time samples on in-phase and quadrature channels, i.e., these samples are points in the complex plane. The signal reflected from target i received at element k is fi,k (t) = γi

n s −1 q=0

v 2(ri + qvi Ts ) − kd sin θi 2πj i w t − qTs − e f0 c c

(2.4)

for some scalar γi . In a digital receiver fi,k (t) is sampled in fast time every Ts = 1/β seconds.

52

Tuuk

To express this in the range basis, S r , each column is a shifted copy of the transmitted waveform, w(t), with leading and trailing zeros. The first column is the response from a target at the minimum range in the range window and in each subsequent column the waveform is shifted by one entry. ⎡ ⎢ ⎢ ⎢ Sr = ⎢ ⎢ ⎣

w(Ts ) w(2 Ts ) w(3 Ts )

0 w(Ts ) w(2 Ts )

0

0

0 ··· 0 ··· w(Ts ) · · · .. . 0

···

0 0 0

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(2.5)

w(nr Ts )

The angle and Doppler bases, S a and S d , are both frequency bases. The Doppler frequency is evaluated across pulses in the coherent processing interval. The angle basis is the relative phase delay introduced at the elements of the antenna array as an incoming planar wavefront reaches each element sequentially. The time delay between receivers becomes a simple phase shift for narrowband signals. For wideband signals, the spatial phase pattern becomes frequency-dependent. To coherently process signals in a wideband setting, other approaches can be used to minimize the losses due to phase mismatch. These include true time delay units or corrections applied in the digital domain to counteract the known phase errors introduced. For radar systems made up of subarrays, true phase combining can be used at the subarray level if the subarray is small enough or the bandwidth is small enough to adequately control phase migration. This linear model can be represented explicitly as a matrix, with each column of the matrix S being the return from a target at the corresponding range-angle-Doppler position in space. But the size of the matrix grows rapidly as the dimensions of the sample space and search space increase. For instance, for a system with 8 channels, 128 pulses, and 512 range samples, the matrix S has 2.8 × 1011 elements. Therefore, the most efficient way to implement this model is not by explicitly storing the matrix but by performing discrete Fourier transforms along the angle and velocity dimensions and a convolution in the range dimension. For computational gains, these can be implemented using the fast Fourier transform (FFT). By this means the storage requirements and processing time can be reduced considerably. This linear model suffices for analysis and estimation. Of course targets move through continuous space, and so any discretization will necessarily be only an approximation. Finer discretization can reduce the associated errors, but at the cost of increasing the correlation between columns of the matrix. With any processing there are diminishing returns as the discretization becomes finer than the fundamental resolution of the sensor. This chapter expresses many of these concepts in a single dimension assuming a uniform linear array, but all these results are generalizable to planar and nonuniform arrays with somewhat more complicated notation. In this book we neglect that fourth dimension in the interest of clarity and computational tractability. Furthermore, any polarization effects are neglected in this model. For systems that record dual polarization an added dimension could be used to represent that variable.

ϕ ϕ A

...

D

ϕ ϕ

53

Signal and Data Processor

Antenna Subarray Elements

Clutter Rejection and Adaptive Filtering

Figure 2.1 A simplified block diagram of a single subarray of a large airborne phased array radar.

The signal path includes a low-noise amplifier, analog filtering, phase shifting, RF combining, mixing to IF, and sampling. The digital samples from this and all other subarrays feed into the digital signal and data processor for pulse compression, beamforming, detection, association, tracking, prediction, and other functions.

2.2.3

Matched Filtering The received-signal processing chain is built of a number subsystems: antenna, analog signal processing, digital signal processing, and data processing. These subsystems sequentially refine this input signal. This includes using a set of matched filters to generate an estimate of the true range profile. These matched filters can be expressed as the conjugate transpose of the sensing model: xˆ mf = S H y.

(2.6)

The matched filters in the spatial and Doppler dimensions amount to Fourier transforms, and the matched filter in the fast-time dimension is a convolution, which can be performed in the Fourier domain as well. The upstream portion of this processing is shown in Figure 2.1 for a notional radar subarray.

2.3

Interference Sources

2.3.1

Measurement Noise White noise is the simplest and most commonly treated interference type. In some applications it is the dominant interference source, particularly for targets at long ranges and for radars that detect targets against an empty background. Thermal noise that accumulates along the analog processing chain and at the analog-to-digital converter (ADC). This noise can be represented as a circular Gaussian vector n ∈ Cn with a variance set by the noise level of the receivers and other elements in the processing chain. Noise can be introduced into the sensing model as y = Sx + n.

(2.7)

Quantization noise, though deterministically related to the sampled signal, can also be treated as Gaussian for ADC converters with typical resolution.

54

Tuuk

2.3.2

Correlated Interference Other sources of interference exist, some of which exhibit structure according to the mechanism by which they are caused. Mitigating these sources of interference requires a different approach than using longer waveforms or more pulses; such techniques increase the target energy but also increase the correlated interference energy. This interference is said to be correlated because its structure can be described statistically by its expected autocorrelation. Correlated interference takes on a number of forms, including distributed ground clutter, in-band radio frequency interference (RFI), and differences between the true and the modeled sensing system. Clutter returns come from energy that is reflected back to the radar by objects other than the intended targets. Thus, the definition of clutter is application-specific. For a system designed to detect motor vehicles in forested terrain, the trees and ground are the clutter. However, for the purpose of performing geographic land use surveys, the ground and trees are the desired scene. Clutter and other structured interference degrade performance in this and other applications. This work considers the case where the targets are moving objects, especially motor vehicles, and the clutter consists of terrain, foliage, and buildings. To include this in the sensing model, a clutter vector c is introduced, which contains the geometry- and terrain-dependent clutter reflectivity at the grid locations. y = S(x + c) + n.

(2.8)

The clutter is illuminated by the same waveform as the targets themselves and thus signal processing that is used to improve the signal-to-noise ratio of the targets will cause the clutter to be accentuated as well. So more energetic waveforms or more integration will not be useful in improving the signal-to-clutter ratio (SCR). By contrast RFI exibits strong spatial correlation but little or no correlation between pulses, so for this interference source the signal-to-interference ratio can be improved by longer integration times or higher power. Clutter can result from land or sea surface reflections, and the statistics of the interference depend strongly on the particularities of the terrain being surveilled. Much research has focused on describing the expected returns from clutter as a function of terrain, radar band, radar resolution, and other parameters. The simplest model is to assume a Gaussian distribution; for low-bandwidth radars this assumption is often sufficient. A common distribution for clutter amplitude is gamma function [6,7]. But detailed analysis of several collected datasets shows significant skewness and kurtosis that do not match either the normal or gamma distributions. Having examined K, log-normal, Weibull, and Rayleigh distributions, work in [8] showed that the Weibull distribution best described the data collected in a flight test over open farmland in Saskatchewan. In [9] research led to a compound distribution composed of two separate gamma distributions that describe the modulation and speckle observed in high-resolution radar.

Probability of Occurance

Clutter Rejection and Adaptive Filtering

55

1 One Realization Underlying Distribution

0.8 0.6 0.4 0.2 0 0

0.5

1 1.5 2 2.5 3 Amplitude Value (Scaled to Mean)

3.5

4

Figure 2.2 The sampled distribution of the observed clutter in one realization, along with the

underlying gamma distribution, which is defined by a shape parameter of 10/3.

So with all this in mind, we have elected to use a constant gamma-distributed random model variable as in [6], with a shape factor of 10/3 in our examples. This model approximates a terrain with relatively open land and the absence of man-made scatterers. Figure 2.2 shows the amplitude of clutter observed in a realization of this clutter model. Other models may be more appropriate at very low grazing angles and for other types of terrain [8,9].

2.4

Signal Processing Treatment of Clutter Processing using only the matched filtering applies a nonadaptive technique to a target scene. However the inter-bin correlation structure can be used to improve target detection performance. Several approaches to treating the correlated clutter problem are described in this section.

2.4.1

Early Techniques The two-pulse canceler is a filtering technique in which the returns from two consecutive pulses are subtracted. If the clutter is stationary from pulse to pulse, it will be nulled by this filtering. If targets exhibit radial motion, their return will not be subject to as much nulling. The null produced by this technique can be quite broad, and might hide slow-moving targets or targets moving nearly perpendicular to the radar’s radial vector. In addition, this filter introduces nulls at Doppler frequencies equal to integer multiples of the PRF, which correspond to blind velocities [10]. The canceler technique can be employed noncoherently with reduced performance if radar system phase tolerances are not sufficient to support pulse-to-pulse coherent processing. A number of improvements can be made to this basic filtering concept. Differencing filters can be cascaded to improve filter null depth. Additionally, the PRF can be staggered within a CPI to resolve blind velocities and null the clutter at zero frequency while passing moving targets.

56

Tuuk

2.4.2

Space–Time Adaptive Processing The principal technique for reducing the effects of structured interference in a multichannel radar system is adaptive filtering. This technique estimates the interference statistics to produce a filter, which is applied to the received signals. In the multichannel pulse-Doppler radar case under examination, a two-dimensional filter, STAP, may be employed. STAP uses the three-dimensional data cube to estimate the interference structure over the joint angle-velocity space using the range samples as training data [4,11–13]. Especially in downward-looking airborne radar, the clutter may interfere with the target in either the Doppler or angle dimension but by processing them jointly the target often falls outside the clutter support. The STAP filter defines the optimal filter for maximizing signal-to-noise ratio in Gaussian interference. The measurements defined in (2.8) with noise, clutter, and other interference y = S(x + c) + n can be broken into signal and interference portions: y = (Sx) + (Sc + n). Let the measurement data y be reorganized as Y = [y 1 | . . . |y Nf ] ∈ CNcs ×Nf , where Ncs = Nc Ns . Putting aside any target energy, each of these vectors can be approximated by a complex normal distribution y i ∼ CN (0Ncs ,R). This interference distribution has both clutter and noise components: the noise contributes a diagonal component to the covariance matrix, while the clutter contributes a component that is low-rank (or approximately so) as a result of the sensing geometry and the timeline. Thus R = R n + R c = ν 2 I Ncs + V DV H ,

(2.9)

where ν 2 is the measurement noise variance, V ∈ CNcs ×k with structure determined by the characteristics of the observed clutter, and D is diagonal with entries σ12,. . .,σk2 . Here we model the clutter as having rank k ≤ Ncs , though the degree to which this low-rank model holds true will be examined in greater detail in this chapter. Clutter is stronger than noise (necessitating adaptive filtering), though this may be quite variable. The sample covariance matrix used in STAP is then defined as: ˆ = N −1 Y Y H . R f

(2.10)

As more samples from this distribution are collected, Nf /Ncs → ∞, and this estimate will converge to the true covariance R. This estimate of the interference statistics can be inverted to form a signal-to-interference-plus-noise ratio (SINR)-maximizing linear filter, [12] which leads to an estimate of the target scene for range bin i as ˆ −1 y i , xˆ i = κ S H R

(2.11)

where the scaling factor κ = (S H R −1 S)−1 . The quantity of training data available to estimate the covariance, as in (2.10), is often limited. The statistics of the interference must be estimated from data in adjacent or nearby range bins to avoid using nonrepresentative statistics in nonhomogeneous environments. The Reed–Mallet–Brennan rule states that a stable estimate can be made from a set of training data with twice as many snapshots as degrees of freedom in the covariance [14]. Various approaches that have been applied in attempts to maximize performance with limited data are discussed next.

Clutter Rejection and Adaptive Filtering

57

Diagonal loading is a commonly used technique to improve sample covariance accuracy and stability [15]. It adds a diagonal (usually a scaled identity) component to regularize the covariance matrix estimate before inversion. After introducing the diagonal loading term δ, the sample covariance matrix estimate becomes: ˆ f ull = N −1 Y Y H + δI Ncs , R f

(2.12)

which is identified as the “full” estimate because it uses the full (Nyquist-rate) set of data, and to distinguish it from other estimators to be introduced in this chapter. Diagonal loading has the effect of reducing the matrix condition number if the matrix is badly conditioned. It also places a limit on the filter null depth. The optimal loading level depends on the type of covariance estimate, as well as the high sensitivity to the clutter-to-noise ratio and even the temporal sampling rate. Optimal selection is largely based on heuristic approaches and empirical results. For this work we use a diagonal loading factor of 10−6 , which will appropriately regularize the inverse without degrading performance by overwhelming the observed sample structure. Figures 2.3 and 2.4 illustrate the ability of the STAP filter to reduce the contribution of clutter in the estimate while maintaining target detectability in uncluttered regions.

Figure 2.3 The matched filter estimate reshaped into the range-angle-Doppler cube and projected along each of the three dimensions. The true target location is r = 290 m, θ = −30◦ , and

R

A

V

A

v = 1 m/s. The marker indicates the true target location in each view.

R

V

Figure 2.4 The STAP estimate reshaped into the range-angle-Doppler cube and projected along each of the three dimensions. The true target location is r = 290 m, θ = −30◦ , and v = 1 m/s.

The marker indicates the true target location in each view.

58

Tuuk

The line of clutter appears as a slice in the third frame of the figures due to the interaction of the sensor platform motion with the stationary ground clutter. In the matched-filteronly estimate that clutter line is such a strong ridge in the angle-Doppler plane that it renders the target invisible. In the STAP estimate, the clutter has been nulled by the filter and the target can be easily located in all three projections. This is a high SCR example to enhance the visibility of the target relative to the clutter. This same framework will be used to generate experimental results in the latter portion of this chapter, although with different parameters for signal, clutter, and noise power. The accuracy of the estimate of the interference covariance decreases with the number of degrees of freedom in the estimate (for a fixed training data set). To improve the stability of the estimates at the expense of resolution, reduced-rank STAP is utilized. These techniques work well because the interference information can be described using relatively few basis vectors. In [16], it is shown that signal dependent rank reduction can improve performance. The cross-spectral metric for designing the rank reducer incorporates information on the desired signal steering vectors, not just the interference statistics to improve performance in cases of limited training data. In [17], it is shown that reduced-rank STAP techniques offer better performance than full-rank estimation under constraints on training time and data. In [18], it is shown how one may select a dimensionality reduction basis efficiently. In addition to a performance improvement given limited training data, these reduced-rank techniques also offer a reduced computational burden relative to a full-rank algorithm. The optimal rank may be set by processing timeline constraints, but even in the unconstrained case, learning the optimal number of dimensions to use for estimating the covariance is not straightforward.

2.5

Measurement Compression If y or Y is the set of Nyquist-sampled measurements, then it represents all the (bandlimited) electromagnetic information passing over the measurement aperture during the period of observation (a single coherent processing interval). The compressed measurements, which undersamples the incident signals by a factor of u, are modeled as another vector z = Cy = C(Y ),

(2.13)

where z ∈ Cm , C ∈ Cm×n , m = Nc Ns Nf /u = n/u, and linear operator C(·) : CNcs ×Nf → Cm

(2.14)

with adjoint C ∗ (·). To be effective in a CS estimation framework, these compressive measurements must be incoherent with the sparsifying basis S. A simple estimate of the target vector from the compressed measurements z can be computed by performing a compressed matched filter, which applies the matched filter to the available measured data

Clutter Rejection and Adaptive Filtering

xˆ cmf = S H C H z.

59

(2.15)

This method has low computational cost but does not necessarily yield a sparse solution. A CS estimate of the target vector can be computed by solving a convex linear optimization problem, such as an 1 -regularized least-squares xˆ cs = arg min ||z − CSx||22 + λ ||x||1 , x

(2.16)

where the first term of the minimization objective is the Euclidean norm of the residual that enforces fidelity to the measured data, and the second term is the 1 norm of the estimate that promotes sparsity in the solution. The parameter λ enables a trade off between these competing priorities. Much attention has been paid to how the random sampling at the core of CS can be realized in hardware. The exact content of the compression matrix C will depend on the measurement process it describes. In pulse-Doppler radar, this process may introduce incoherence in the following ways: • • •

2.6

in fast time by mixing incoming signals with pseudo-random modulation sequences before low-pass filtering and sampling slowly [19] in slow time by staggering the pulse repetition interval [20] in the spatial domain using a random measurement array [21,22] or coprime thinned array [23].

Estimating Interference Statistics from Compressed Measurements We have seen that interference covariance can be estimated from samples via the STAP approach. The central question of this work is how well samples produced from a compression operation C, as in (2.13), can be used to form an estimate of the covariance matrix. The simplest approach is compressed sample matrix inversion (SMI), in which the adjoint of the compression operator is used to bring the samples back to the full ambient signal space before forming the covariance estimate: ˆ comp = N −1 C ∗ (z)C ∗ (z)H + δI Ncs . R f

(2.17)

As in (2.12) a diagonal loading term is introduced to regularize the matrix inverse.

2.6.1

Matrix Completion Related to CS recovery of sparse vectors is the recovery of low-rank matrices from fewer samples than matrix elements. This area of work is known as matrix completion. In the same way that the 1 norm offers a convex relaxation of a direct sparsity (0 “norm”) objective, the nuclear norm (denoted as ||·||∗ ), which is the sum of the matrix singular values, offers a convex relaxation of the rank(·) objective. Let some unknown matrix X ∈ Cn×n exist with rank r. Some number m observations, M i,j , of this matrix are available only at the support set (i,j ) ∈ . In [24] it is shown

60

Tuuk

that low-rank matrix recovery from few measurements is not ill-posed and is convex. The convex optimization problem min ||X||∗ s. t. X i,j = M i,j , (i,j ) ∈

(2.18)

will recover the matrix X exactly with high probability if m ≥ Cn1.2 r log n, for a specified constant C. In [25], the topic of matrix completion is surveyed and it is shown that n × n matrices of rank r can be recovered from m noise-corrupted direct samples via nuclear norm minimization with high probability if m ≥ Cnr log2 n with an error on the order of the noise level. In [26], this idea is expanded to show that matrices can be recovered from expansion coefficients with respect to a known matrix basis as long as that basis is not coherent with the matrix being recovered. This is directly analogous to the required incoherence between the spasifying basis and the sensing basis in CS theory. The work in [27] provides information theoretic lower bounds on the number of samples needed to recover certain types of low-rank matrices. One result is that m = Cnr log n is the lower limit on the number of samples needed to recover a random n × n matrix with rank r. The work in [28] treats the topic of estimating simultaneously sparse and low-rank matrices from rank-one measurements. This measurement model consists of a series of sketches of the underlying matrix. In [29] this same “sketching” measurement model is considered and shows specifically that the number of measurements required for stable estimation scales linearly with the rank of the matrix and the sparsity of the matrix and with the logarithm of the number of rows. Though this model differs from that which we will develop in this chapter, the results provide a basis for confidence that by exploiting the low-rank nature of the covariance, an improvement in performance can be expected. In [30], it is shown that the low-rank assumption can be used to improve the estimation accuracy of the covariance matrix using a standard STAP benchmark dataset in the case of limited training data. Their proposed algorithm out-performs other lower computational complexity algorithms by using a dictionary learning approach. In [31,32], a CS–STAP technique is developed in which a small amount of training data, in some cases one snapshot, can be used to estimate the covariance statistics and build the whitening filter. This technique assumes direct sparsity in the interference covariance matrix. The validity of this assumption depends on the type of interference being described. For certain types of electromagnetic interference, or for certain types of man-made clutter, a concentration in this domain can be very pronounced. For other types of natural ground cover, the spectrum can be more distributed. This distribution is especially pronounced for foliage being blown in the wind. In [33] an approach is proposed to estimate the sample covariance matrix (that would be obtained from the uncompressed data) from a set of compressed measurements. The goal of estimating the sample covariance matrix is slightly different than that of estimating the underlying interference statistics, but does isolate the two stages of sampling limits: first, a sample covariance matrix deviates from the true covariance matrix because it is based on a limited number of realizations, and second, the compressed sample covariance matrix differs from the full sample covariance matrix because of the

Clutter Rejection and Adaptive Filtering

61

dimensionality reduction. The nature of the second limit depends on the manner in which the samples are compressed and the estimator used to generate the sample covariance matrix. Also, we note that approaches from the low-rank matrix approximation body of theory have been applied to the moving-target detection problem, for example in [34].

2.6.2

Iterative Singular Value Thresholding In [35], an algorithm for efficient matrix completion is provided; this approach uses iterative singular value thresholding (SVT) and projection back onto the observation set. This algorithm is able to recover large matrices in low run-time relative to interiorpoint methods. Starting from M, the observations, and P (·), the projection onto the observation domain, the iteration involves repeated application of X = shrink(Y,τ) Y = Y + δP (M − X)

(2.19)

returning X. In this iteration, shrinkage to the threshold τ imposes low rank (spectral sparsity) by reducing the magnitude of the singular values, as specified in (2.20). A larger value of τ results in a lower-rank representation. The shrink(·) operation is soft-thresholding of the singular values of its argument. When the singular value decomposition (SVD) of matrix U SV H , where S = diag(s) = diag([s1 ,. . ., sn ]T ), shrink(A;τ) := U diag(soft(s;τ))V H , soft(s;τ) := [soft(s i ;τ), i ∈ 1 . . . n]T , si max(0,|si | − τ). soft(s i ;τ) := |si |

(2.20)

The iteration in (2.19) solves 1 ||X||2F 2 s. t. P (X) = P (M). min τ ||X||∗ +

(2.21)

This iteration approaches the direct nuclear norm objective for large values of τ, and smaller τ improves the stability of the solution. This technique solves for large 1,000 × 1,000 matrices in several seconds on a personal computer. This technique is the basis for the low-rank covariance estimation technique developed in this chapter.

2.6.3

Performance Evaluation To review, the identified covariance estimates are: • • • • •

R true : true second-order statistics of the interference in (2.9) ˆ f ull = N −1 Y Y H + δI Ncs in (2.12) R f ˆ svt : following iteration in (2.19) R ˆ comp = N −1 C ∗ (z)C ∗ (z)H + δI Ncs in (2.17) R f

ˆ none = I Ncs R

Tuuk

10

Eigenvalue Mag.

62

0

10–1

Structure 1: Simple Plateau Struecture 2: Triple Plateau Structure 3: Exponential Decay Structure 4: Plateau and Decay

–2

10

–3

10

0

5

10

15

20

25

30

Eigenvalue Index Figure 2.5 Synthetic structured interference eigenvalue decay functions. These functions are

shown with width parameter of 10. For other values, of this parameter, the spectral structure is stretched or compressed proportionally.

We test four different spectrally structured synthetic data matrices to better understand the performance of the SVT algorithm under various conditions. For an input rank width r: 1. 2. 3.

4.

The first decay function has r equal eigenvalues. The second decay function has r eigenvalues with unity magnitude, r with magnitude 1/4, and r with magnitude 1/16. The third decay function has eigenvalues decaying, with the ith eigenvalue having magnitude % $ exp −(35i/r)7/10 %. $ (2.22) σi = exp −(35/r)7/10 The fourth decay funtion has eigenvalues decaying, with the ith eigenvalue having magnitude σi =

1,

% $ exp −(35(i−r)/r)7/10 , exp(−(35/r)7/10 )

i≤r otherwise.

(2.23)

These four synthetic interference structures are illustrated in Figure 2.5. This set of spectral structures includes two with exponential decay in the true eigenvalues of the interference. This condition is included in an attempt to represent true sensing problems in which the interference does not vanish. Selected parameters used for these simulations are provided in Table 2.1. To build up to the performance summary statistics we first include details for a particular example of generating these estimates and using them to filter interference. In this case, the true interference structure is structure 4, shown in Figure 2.5. The number of snapshots is 500, the number of spatial channels is 8, the number of pulses is 128,

Clutter Rejection and Adaptive Filtering

63

Table 2.1 Parameters used for synthetic structured interference experiments. Value

Range bins Pulses Spatial channels Clutter spectral decay width Clutter-to-noise ratio Signal-to-interference ratio

100–500 128 8 5–20 0–40 dB 0 dB

Eigenvalue Magnitude

Parameter

10 –2 10 –3 10

True Cov. Full SMI Compr. SMI Compr. SVT

–4

10

0

10

1

10

2

10

3

Eigenvalue Index Figure 2.6 The spectral decay structures of the various covariance estimates of the synthetic

structured interference. The SVT estimate more closely matches the truth than the compressed SMI does, which uses the same data as input.

the CNR is 15 dB, and the rank width parameter is 10. The various covariance estimates exhibit varying spectral structure and are shown in Figure 2.6. To illustrate performance over a broader problem space we conducted a parametric sweep over several relevant problem input variables: under-sampling factor, clutter-tonoise ratio (CNR), number of range samples (snapshots), and clutter eigenvalue decay function. These results are all generated using Na = 8 spatial channels and Ns = 128 pulses. The results of this evaluation are shown the following figures. Figure 2.7 shows how performance varies as the under-sampling factor varies. It is evident that for low and moderate under-sampling factors the SVT estimator is able to perform as well as the full-data SMI, but under these same conditions the performance of compressed SMI falls off sharply. If the experiment is limited to favorable interference conditions (low rank, simple structure), the SVT estimate performs very well, out to an under-sampling factor of 20, as shown in Figure 2.8. Figure 2.9 shows how performance varies as the number of pre-compression range bins varies. It shows that the SVT estimate is best able to take advantage of a longer observation interval to improve the estimation accuracy, even if those range bins will

Tuuk

Probability of Detection

1 True Cov. Full SMI Compress SVT Compress SMI No Cov.

0.8

0.6

0.4

0.2

0 1

2

5

10

20

Under-Sampling Factor Figure 2.7 Average probability of detection as a function of the data under-sampling factor for

cases of synthetic structured interference. As the under-sampling factor increases and less data is available for the two compressed estimates, the accuracy of those estimates degrades. Notably, the SVT estimate maintains much better performance than compressed SMI as the USF increases. 1

Probability of Detection

64

0.8

0.6

0.4

True Cov. Full SMI Compress SVT Compress SMI No Cov.

0.2

0 1

2

5

10

20

Under-Sampling Factor Figure 2.8 Average probability of detection as a function of the data under-sampling factor

for cases with favorable interference structure. The SVT estimate based on the compressed data performs better than the diagonally loaded estimate based on the full data and nearly as well as the true covariance matrix. The interference rank is 5 and the structure is simple (structure ID = 1).

be compressed to reduce the number samples, a larger ambient dimension improves the estimation performance. This somewhat challenges at least one motivation for reducing the number of samples: non-stationarity interference forcing a reduced range sample space. But for cases in which acquiring the samples themselves is a greater challenge

Clutter Rejection and Adaptive Filtering

65

Probability of Detection

1

0.8 True Cov. Full SMI Compress SVT Compress SMI No Cov.

0.6

0.4

0.2

0 100

200

300

400

500

Number of Range Samples Figure 2.9 Average probability of detection as a function of the number of range samples for

cases of synthetic structured interference. As the number of range samples (snapshots) increases, the two compressed estimates, which use compressed data (compressed snapshots), both improve in estimation accuracy.

Probability of Detection

1

0.8 True Cov. Full SMI Compress SVT Compress SMI No Cov.

0.6

0.4

0.2

0

Clutter Structure ID Figure 2.10 Average probability of detection as a function of interference structure for cases of synthetic structured interference. The performance of various estimators is tested with four different clutter structures, having the four eigenspectra illustrated in Figure 2.5. The performance of the estimators varies as a function of the spectra, with lower rank clutter being easier to filter out and higher rank clutter being more difficult.

than inherent interference constraints, this result shows that the SVT estimator could provide improved performance. Figure 2.10 shows how performance varies over the different clutter eigenspectra shown in Figure 2.5.

66

Tuuk

2.7

Mitigating Clutter in Compressed Sensing Estimation Estimating the statistics of clutter from compressed measurements is necessary, but not sufficient for the detection of targets embedded therein. A variety of techniques exist for estimation of the scene, including an approaches that uses an estimated interference covariance matrix and other approaches that do not. One of the first approaches to this problem, from [36], uses a mask over the presumed clutter ridge of the angle-Doppler domain. This allows traditional CS-based solution techniques to be applied outside that mask where targets are presumed sparse. The extent of the clutter region may be reasonably estimated from platform motion and sensing geometry. However, this limits minimum detectable velocity and does not readily extend to electromagnetic interference (EMI) and other structured interference sources. In [37], the clutter covariance is assumed known and a set of Capon beamforming weights is built into the sparsity matrix to favor detection of targets and suppress detection of clutter. The objective vector is assumed to be block-sparse, with high response in all snapshots. This is combined with an arbitrary random sensing matrix and tools from convex optimization. The approach in [38] uses the clutter covariance to modify the optimization norm or, equivalently, to modify the system model. For a combined sensing and sampling model, matrix A and interference covariance matrix R with inverse R −1 , which have ¯ = P A, and modified a Cholesky decomposition P H P , a modified model matrix A measurement vector y¯ = P y, are introduced. These can be used in a standard sparse recovery formulation: ¯ − y¯ 2 + λ ||x||1 . xˆ = arg min = Ax 2 x

This is solved with the FISTA algorithm in the published work, but any desired optimization routine could be applied. Using the statistics of the clutter to develop a whitening filter is also the approach taken in [39]. In this case, the whitening filter is derived in the frequency domain without any spatial degrees of freedom. The resultant technique is applied to an experimental CS-based radar system and provides improvement up to a factor of the number of pulses. None of the mentioned CS solution methods, as described, are equipped to handle ˆ is some estimate of the space–time covariance matrix of the structured interference. If R ˜ =R ˆ ⊗ I Nf is its expansion into the full measurement fully sampled interference, and R domain, then the covariance matrix of the interference in the compressed domain can be ˜ H . This use of the dimensionally reduced covariance matrix ˆ c = C RC expressed as R is a non-data-adaptive reduced-dimension STAP formulation. The literature of reduceddimension STAP provides a natural way to obtain an estimate of the scene from the compressed measurements while including the covariance information. Define ˆ −1 xˆ cstap = S H C H R c z.

(2.24)

This is analogous to the xˆ stap solution in that the matched filter is post-multiplied by the covariance matrix inverse to whiten the interference. It is identical to the STAP

Clutter Rejection and Adaptive Filtering

67

solution in the case that C is the identity matrix (i.e., the fully sampled case). To compute this estimate, one incurs the cost of inverting the covariance matrix (or parametrically estimating the inverse), but gains a good deal of clutter suppression as we will show in our results. However, this technique does not necessarily favor sparse solutions. Work in [40,41] proposes a covariance-aware CS (CA CS) that accounts for structured interference in a CS framework. As in STAP methods, this is accomplished via the interference covariance matrix inverse. The original 1 regularized least squares problem of (2.16) can be modified as follows: ˆ −1 xˆ cacs = arg min (z − CSx)H R c (z − CSx) + γ ||x||1 . x

(2.25)

This generalization of (2.16) is identical (for some value of γ) in the clutter-free case. ˆ −1 ˆ c is a scaled identity matrix. Here γ is set relative to the entries in R In that case, R c . . $ −1 % ˆ Specifically γ = λ R c , using the Frobenius norm where ||Q||F = Tr QQH F and Tr(·) gives the trace of a matrix. This relationship is used so as to provide similar performance as in (2.16) for a selected value of λ. The first term in the objective function penalizes deviations from the measurements in interference-free regions, but less so in interference regions. This term is akin to a Mahalanobis distance. The second term penalizes large entries in the solution. Thus 1

Probability of Detection

0.8

0.6 CA CS, usf = 20 Full STAP Full Adjoint CS, usf = 20 CSTAP, usf = 20 Comp. MF, usf = 20

0.4

0.2

0 –50

–40

–30 –20 Input SCR (dB)

–10

0

Figure 2.11 The CA CS method subsamples the data just as the standard CS method does, however it takes into account the covariance matrix that describes the interference structure. By doing so, it improves the probability of detection over the CS case as well as beyond the fully sampled, matched filter case that does not use the covariance information. These results are shown with a probability of false alarm of 0.005.

68

Tuuk

the entries in the interfered region are unconstrained by the first term and they are allowed to be driven to zero by the second term. The first term maintains fidelity to the measurements in the clear areas while the second term promotes sparsity. Here the γ parameter serves the same purpose as λ in (2.16): balancing the weight of the sparsitypromoting 1 norm against the fidelity-preserving 2 norm. Figure 2.11 compares the performance of various estimation techniques on problems with clutter as the dominant interference. Here CA CS shows success beyond that of the CS estimates. By using the covariance information, CA CS can even surpass the performance of the fully sampled (but covariance-ignorant) matched filter. Still, the fully sampled adaptive STAP filter remains the gold standard. Of particular interest is the 20× under-sampled CA CS solution that achieves nearly the same detection performance as the STAP solution. Also notable is the fact that the performance of the classically formulated estimation methods (compressed STAP and compressed adjoint) is equal to that of the CS-formulated ones (CA CS and CS). When compared to the compressed STAP (CSTAP) methods, there is no advantage to using CA CS. It is also evident that the compressed adjoint performs as well as the CS solution in the interference limited case. This result holds for a range of under-sampling factors, u, as well as for both the covariance-aware (CA CS vs. CSTAP) and covarianceunaware (CS vs. compressed adjoint) cases. Furthermore, this performance equivalence holds over a range of tested probability of false alarm settings.

2.8

Summary As has been shown, clutter can be of great detriment to terrestrial and airborne radar systems. And mitigation of clutter can be a significant driver of radar design requirements, in terms of dynamic range and space–time degrees of freedom. These statements hold true whether in a classical or CS radar system. A rich body of literature that describs techniques for mitigating clutter has led to the STAP filter that estimates interference statistics for a range bin of interest from samples of nearby bins. This estimate is used to whiten the bin under test prior to subsequent processing. This approach, leavened with ideas from sparse estimation and low-rank matrix approximation, can be extended to the compressed domain. For the estimation of interference covariance information, the mechanisms of lowrank matrix approximation have given new perspective to prior work in rank-reduced and dimensionality-reduced STAP estimation. And recent results show that even in the presence of sampling compression the interference covariance matrix can be accurately estimated if it has a compact eigen-structure. This structure can be used with singular value thresholding or other low-rank approximation techniques to recover the full and the compressed covariance matrices to high accuracy. And for the covariance information can be used in the recovery of a sparse scene from compressed measurements. The whitening approach taken in STAP can be extended into the sparsity-favoring convex objective function by a reshaping of the distance norm. This allows the full machinery of CS to be applied to the problem.

Clutter Rejection and Adaptive Filtering

69

However, there is significant computational cost associated with estimation of a large spatiotemporal covariance matrix through low-rank matrix approximation and application of that covariance matrix to estimate the targets in an iterative optimization algorithm. This cost is, at present, too high to see widespread adoption of compressed measurements in the airborne radar domain. So work remains in these areas. A number of promising avenues exist, including the application of advances in online and streaming CS to clutter estimation and mitigation problems, further incorporation of fundamental results in random matrix theory to help set algorithm thresholds, and the improvement of random sampling for large arrays in both the time and space domains.

References [1] L. Brown, A Radar History of World War II: Technical and Military Imperatives. Institute of Physics Publishing, 1999. [2] F. R. Dickey, M. Labitt, and F. M. Staudaher, “Development of airborne moving target radar for long range surveillance,” Aerospace and Electronic Systems, IEEE Transactions on, vol. 27, no. 6, pp. 959–972, Nov. 1991. [3] L. P. Goetz and J. D. Albright, “Airborne pulse-doppler radar,” IRE Transactions on Military Electronics, vol. MIL-5, no. 2, pp. 116–126, Apr. 1961. [4] L. Brennan and L. Reed, “Theory of adaptive radar,” Aerospace and Electronic Systems, IEEE Transactions on, vol. 9, no. 2, pp. 237–252, Mar. 1973. [5] M. A. Richards, Fundamentals of Radar Signal Processing. McGraw-Hill, 2005. [6] D. Shnidman, “Radar detection in clutter,” Aerospace and Electronic Systems, IEEE Transactions on, vol. 41, no. 3, pp. 1056–1067, July 2005. [7] D. Shnidman, “Generalized radar clutter model,” Aerospace and Electronic Systems, IEEE Transactions on, vol. 35, no. 3, pp. 857–865, July 1999. [8] J. Billingsley, A. Farina, F. Gini, M. Greco, and L. Verrazzani, “Statistical analyses of measured radar ground clutter data,” Aerospace and Electronic Systems, IEEE Transactions on, vol. 35, no. 2, pp. 579–593, Apr. 1999. [9] V. Anastassopoulos, G. Lampropoulos, A. Drosopoulos, and N. Rey, “High resolution radar clutter statistics,” Aerospace and Electronic Systems, IEEE Transactions on, vol. 35, no. 1, pp. 43–60, Jan. 1999. [10] R. McAulay, “A theory for optimal MTI digital signal processing part i. receiver synthesis,” Massachusetts Institute of Technology Lincoln Laboratory, Lexington, MA, Tech. Rep. 1972-14, Feb. 1972. [Online]. Available: www.ll.mit.edu/mission/aviation/ publications/publication-files/technical_notes/McAulay_1972_TN-1972-14i_WW-18358 .pdf. [11] J. Ward, “Space-time adaptive processing for airborne radar,” MIT Lincoln Laboratory, Lexington, MA, Tech. Rep. 1015, 1994. [12] W. Melvin, “A STAP overview,” Aerospace and Electronic Systems Magazine, IEEE, vol. 19, no. 1, pp. 19–35, Jan. 2004. [13] J. Guerci, Space-Time Adaptive Processing for Radar. Artech House, 2003. [14] I. Reed, J. Mallett, and L. Brennan, “Rapid convergence rate in adaptive arrays,” Aerospace and Electronic Systems, IEEE Transactions on, vol. 10, no. 6, pp. 853–863, Nov. 1974.

70

Tuuk

[15] B. D. Carlson, “Covariance matrix estimation errors and diagonal loading in adaptive arrays,” Aerospace and Electronic Systems, IEEE Transactions on, vol. 24, no. 4, pp. 397– 401, July 1988. [16] J. Guerci, J. Goldstein, and I. Reed, “Optimal and adaptive reduced-rank STAP,” Aerospace and Electronic Systems, IEEE Transactions on, vol. 36, no. 2, pp. 647–663, Apr. 2000. [17] C. Peckham, A. Haimovich, T. Ayoub, J. Goldstein, and I. Reid, “Reduced-rank STAP performance analysis,” Aerospace and Electronic Systems, IEEE Transactions on, vol. 36, no. 2, pp. 664–676, Apr. 2000. [18] R. Fa, R. de Lamare, and L. Wang, “Reduced-rank STAP schemes for airborne radar based on switched joint interpolation, decimation and filtering algorithm,” Signal Processing, IEEE Transactions on, vol. 58, no. 8, pp. 4182–4194, Aug. 2010. [19] J. Tropp, J. Laska, M. Duarte, J. Romberg, and R. Baraniuk, “Beyond Nyquist: Efficient sampling of sparse bandlimited signals,” Information Theory, IEEE Transactions on, vol. 56, no. 1, pp. 520–544, Jan. 2010. [20] L. Zhen, W. Xizhang, and L. Xiang, “CS-based moving target detection in random PRI radar,” in Geoscience and Remote Sensing Symposium (IGARSS), 2012 IEEE International, July 2012, pp. 7476–7479. [21] L. Carin, “On the relationship between compressive sensing and random sensor arrays,” Antennas and Propagation Magazine, IEEE, vol. 51, no. 5, pp. 72–81, Oct. 2009. [22] L. Carin, D. Liu, and B. Guo, “Coherence, compressive sensing, and random sensor arrays,” IEEE Antennas and Propagation Magazine, vol. 53, no. 4, pp. 28–39, Aug. 2011. [23] P. P. Vaidyanathan and P. Pal, “Sparse sensing with co-prime samplers and arrays,” Signal Processing, IEEE Transactions on, vol. 59, no. 2, pp. 573–586, Feb. 2011. [24] E. J. Candès and B. Recht, “Exact matrix completion via convex optimization,” Foundations of Computational Mathematics, vol. 9, no. 6, pp. 717–772, Dec. 2009. [25] E. J. Candès and Y. Plan, “Matrix completion with noise,” Proceedings of the IEEE, vol. 98, no. 6, pp. 925–936, Apr. 2010. [26] D. Gross, “Recovering low-rank matrices from few coefficients in any basis,” Information Theory, IEEE Transactions on, vol. 57, no. 3, pp. 1548–1566, Mar. 2011. [27] E. J. Candès and T. Tao, “The power of convex relaxation: Near-optimal matrix completion,” Information Theory, IEEE Transactions on, vol. 56, no. 5, pp. 2053–2080, May 2010. [28] Y. Chen, Y. Chi, and A. J. Goldsmith, “Exact and stable covariance estimation from quadratic sampling via convex programming,” Information Theory, IEEE Transactions on, vol. 61, no. 7, pp. 4034–4059, July 2015. [29] S. Bahmani and J. Romberg, “Near-optimal estimation of simultaneously sparse and lowrank matrices from nested linear measurements,” Information and Inference: A Journal of the IMA, vol. 5, no. 3, p. 331, Sept. 2016. [30] L. Bai, S. Roy, and M. Rangaswamy, “Compressive radar clutter subspace estimation using dictionary learning,” in Radar Conference (RADAR), 2013 IEEE, Apr. 2013, pp. 1–6. [31] K. Sun, H. Zhang, G. Li, H. Meng, and X. Wang, “A novel STAP algorithm using sparse recovery technique,” in Geoscience and Remote Sensing Symposium, 2009 IEEE International, vol. 5, July 2009, pp. V-336-V-339. [32] K. Sun, H. Meng, Y. Wang, and X. Wang, “Direct data domain STAP using sparse representation of clutter spectrum,” Signal Processing, vol. 91, no. 9, pp. 2222–2236, Sept. 2011. [33] F. Pourkamali-Anaraki, “Estimation of the sample covariance matrix from compressive measurements,” IET Signal Processing, vol. 10, no. 9, pp. 1089–1095, Dec. 2016.

Clutter Rejection and Adaptive Filtering

71

[34] S. Sen, “Low-rank matrix decomposition and spatio-temporal sparse recovery for stap radar,” Selected Topics in Signal Processing, IEEE Journal of, vol. 9, no. 8, pp. 1510–1523, Dec. 2015. [35] J.-F. Cai, E. J. Candès, and Z. Shen, “A singular value thresholding algorithm for matrix completion,” SIAM Journal on Optimization, vol. 20, no. 4, pp. 1956–1982, Mar. 2010. [36] I. Selesnick, S. Pillai, K. Y. Li, and B. Himed, “Angle-doppler processing using sparse regularization,” in Acoustics Speech and Signal Processing (ICASSP), 2010 IEEE International Conference on, Mar. 2010, pp. 2750–2753. [37] Y. Yu, S. Sun, and A. P. Petropulu, “A capon beamforming method for clutter suppression in colocated compressive sensing based MIMO radars,” in SPIE Conference Proceedings, vol. 8717, May 2013, p. 87170J. [Online]. Available: https://doi.org/10.1117/12.2015635. [38] J. T. Parker and L. C. Potter, “A Bayesian perspective on sparse regularization for STAP post-processing,” in Radar Conference, 2010 IEEE, May 2010, pp. 1471–1475. [39] Y. C. Eldar, R. Levi, and A. Cohen, “Clutter removal in sub-nyquist radar,” IEEE Signal Processing Letters, vol. 22, no. 2, pp. 177–181, Feb. 2015. [40] P. B. Tuuk and S. L. Marple, “Compressed sensing radar amid noise and clutter using interference covariance information,” Aerospace and Electronic Systems, IEEE Transactions on, vol. 50, no. 2, pp. 887–897, Apr. 2014. [41] P. B. Tuuk and S. L. Marple, “Compressed sensing radar amid noise and clutter,” in Signals, Systems and Computers, 2012 Conference Record of the Forty-Sixth Asilomar Conference on, Nov. 2012, pp. 446–450.

3

RFI Mitigation Based on Compressive Sensing Methods for UWB Radar Imaging Tianyi Zhang, Jiaying Ren, Jian Li, David J. Greene, Jeremy A. Johnston, and Lam H. Nguyen

3.1

Introduction Ultra-wideband (UWB) radar operating at low frequencies (for example, from under 100 MHz to several GHz) has been used in a wide range of applications, including landmine and unexploded ordinance (UXO) detection using ground-penetrating radar (GPR), imaging with foliage-penetrating (FOPEN) radar, as well as detecting hidden humans or objects via through-wall imaging [1]. Examples of such radar systems, including the US Army Research Laboratory (ARL)’s synchronous impulse reconstruction (SIRE) system, which is a forward-looking GPR (FLGPR) system, and the ARL’s BoomSAR system, are shown in [1]. These UWB systems are important to both military and civilian applications. One significant challenge for the proper operations of an UWB radar system is dealing with the severe radio frequency interference (RFI) they encounter, since there are many competing users within the UWB frequency range in which they operate. Typical RFI sources include FM radio transmitters, TV broadcast transmitters, cellular phones; their operating frequency bands tend to overlap with that of UWB radar systems [2]. Figure 3.1 shows an example of the spectrum of a radar echo signal from a single pulse repetition interval (PRI), the spectrum of an RFI source (containing strong narrowband interferes), and the spectrum of the radar echo signal merged with the RFI. Note that the presence of RFI causes significant distortions to the spectrum of the received signal. Therefore, the RFI signal poses a significant hindrance to the proper operation of UWB radar in terms of reduced signal-to-noise ratio and degraded radar image quality. Therefore, effective RFI mitigation is critically important for the proper functioning of a UWB radar system. RFI mitigation is a notoriously challenging problem because RFI signals are difficult to predict and model accurately due to their dynamic range and diverse modulation schemes. Many methods have been developed for RFI mitigation, including RFI suppression via filtering techniques and RFI extraction based on RFI estimation. The former suppression approaches, including notch-filtering, subband filtering and adaptive filtering, though popular due to their simplicity, usually introduce sidelobes in the time-domain and suffer from filter transients and reduced data length [2–6]. The latter RFI extraction methods are composed of techniques based on, for example, parametric modeling [7], spectral decomposition [8], eigensubspace decomposition [9,10],

72

RFI Mitigation Based on Compressive Sensing Methods for UWB Radar Imaging

73

160 Original SAR Data RFI data SAR+RFI

Amplitude(dB)

140

120

100

80

60 0

500

1000

1500

Frequency (MHz)

Figure 3.1 Comparison of spectrum obtained from experimental data collected by ARL’s

BoomSAR.

and independent component analysis (ICA) [10,11]. Most of these methods can only provide satisfying performance under certain assumptions or constraints; most assumptions and constraints are often invalid in practice due to the difficulties of modeling complex RFI sources. For instance, principal-component-based techniques, such as ICA and eigendecomposition, heavily depend on orthogonal subspaces and have difficulty in distinguishing between RFI sources and UWB radar echoes when they have similar power levels within the same subspace [12]. The recent development of compressive sensing (CS) theory has stimulated numerous investigations on exploiting sparsity and low-rank properties for RFI mitigation. Early sparsity-based recovery methods solve the RFI mitigation problem by modeling both the desired UWB radar echo signal and the RFI sources as sparse with respect to welldesigned dictionaries [1,12]. These methods work well but suffer from a significant drawback: they require an additional dictionary-learning step based on the measured or estimated data. In [13,14], an improved method based on a joint sparse and low-rank model is proposed. By simply exploiting the low-rank property of the RFI sources, the method does not require any specific prior knowledge on the interferences. The robust principal component analysis (RPCA) approach can be used to extract RFI sources from the observed data [15]. More specifically, this approach models the RFI contamination across multiple PRIs within a coherent processing interval (CPI) using a general, lowrank structure while treating the UWB radar echo signals as sparse impulse outliers. Unlike the previous CS-based approaches, this technique is completely adaptive to highly time-varying operating environments and does not require any prior knowledge on the dictionary for the desired UWB radar signals and the unwanted RFI sources. Additionally, the RPCA-based RFI mitigation method can be easily incorporated into diverse UWB radar systems as a preprocessing stage before further signal processing

74

Zhang, Ren, Li, Greene, Johnston, and Nguyen

and image formation [15]. However, RPCA requires the fine-tuning of a user parameter, which is nontrivial in practical applications due to the lack of prior information on the RFI sources and the desired radar echo signals. Since the desired UWB radar echo signals are relatively weak with a flat spectrum and RFI sources are typically strong, with sparse, narrow spectral peaks, the RFI sources can be approximately modeled as sparse spectral lines. The parameters of RFI sources can be estimated using the CLEAN algorithm, which is a conceptually and computationally simple approach widely used in diverse applications [16–19]. After parameterization, the RFI sources can be extracted out from the observed RFI-contaminated data. Moreover, the CLEAN algorithm can be used with the Bayesian information criterion (BIC) [20–22] to estimate the number of spectral lines needed to model the RFI sources. Unlike the aforementioned RPCA method, the CLEAN-BIC approach does not require the selection of any user parameter and hence can be easily used in practical applications. However, the recovered UWB radar echoes that were obtained by the CLEANBIC approach are not sparsified, and the resulting radar images appear noisy in the presence of severe RFI and noise contaminations. To take advantage of the merits of both RPCA and CLEAN-BIC, we consider a hybrid method, referred to as HM. We first utilize the CLEAN-BIC algorithm to estimate the signal-to-interference ratio (SIR) of the received RFI-contaminated data. If the estimated SIR is above a certain threshold, we use CLEAN-BIC to recover the desired UWB radar echo signals. Otherwise, we use RPCA with the user parameter recommended in [23] for RFI mitigation. Furthermore, we introduce a framework of choosing the proper user parameter for the RPCA method for RFI mitigation via first using CLEAN-BIC to estimate the SIR. The RPCA algorithm, with its user parameter determined by the estimated SIR, is then used for RFI mitigation. We refer to this approach as RPCA-CB. Finally, both simulated and experimental results are presented to evaluate the RFI mitigation performance of the aforementioned algorithms in the presence of various levels of RFI contaminations. Specifically, our experiments are conducted using the measured RFI-free radar echo data set with two different RFI data sets: a simulated RFI-only data set and measured RFI-only data set. The measured UWB data set was collected by the ARL using their impulse-based, low-frequency UWB BoomSAR system covering a frequency band from approximate 50 MHz to 1150 MHz. The UWB BoomSAR was mounted on a platform that emulated an airborne geometry. Two transmitters and two receivers were used to collect data in different polarizations. The measured data used in these experiments was configured in a horizontal transmit, horizontal receive (HH) polarization. The simulated RFI-only data set has the RFI sources modeled as a sum of sinusoids, whereas the measured RFI-only data set is collected by the ARL radar receiver with the antenna pointing toward Washington, DC [1,12]. We show that RPCACB outperforms the aforementioned RPCA, CLEAN-BIC, and HM algorithms. Notation: Most of the notation used in this chapter is listed in the list of Symbols in the Preface. For this chapter, we need to add some definitions. det (·) denotes the determinant of a matrix. x k· and x ·k refer to the k-th row and k-th column of matrix X, respectively. R ∈ CN×M denotes the complex-valued N × M matrix. For a matrix,

RFI Mitigation Based on Compressive Sensing Methods for UWB Radar Imaging

75

·p means the p element-wise norm of this matrix, ·F is the Frobenius norm of a matrix. · ∞ means the infinite norm of a matrix and · ∗ denotes the nuclear norm of a matrix, that is, the sum of the singular values of the matrix. The subscript of I denotes / to denote the estimate the size of the identity matrix. To avoid confusion, we also use (·) T of a parameter. X,Y = tr(X Y ) denotes the inner product of two matrices X and Y .

3.2

RPCA for RFI Mitigation

3.2.1

Problem Formulation Consider an RFI-contaminated UWB radar system, with M PRIs within a CPI and N samples, referred to as fast-time samples, per PRI. Then, the observed data, denoted by matrix Y ∈ RN×M , can be modeled as follows: Y = X + R,

(3.1)

where each column y ·m of Y denotes a data vector within a PRI with N fast-time samples, composed of the desired UWB radar echo signal x ·m and the RFI-only signal r ·m , m = 1,2,. . .,M. Here, m is the PRI index, i.e., the slow-time index. Our goal is to extract the desired UWB radar signal X and the RFI signal R from the observed RFI-contaminated data Y . We exploit two main assumptions to accomplish the RFI mitigation task: (1) the desired UWB radar echo matrix X is sparse and (2) the RFI matrix R is low rank [15]. The sparse nature of the desired UWB radar echo matrix is confirmed and illustrated in Figure 3.2, where a typical example of the UWB impulse radar echoes from multiple PRIs, i.e., across multiple slow-time indices, within a CPI, as measured by an ARL radar receiver in the absence of RFI, is depicted. These measured UWB radar echo signals contain occasional narrow backscattered pulses, which indicate the distances between the targets and the radar and the reflection coefficients of the targets. Due to the sparsity of strong targets, X is in general quite sparse in the fast-time domain. It is worth mentioning that this sparse property of the radar echo signal is also valid for stepped-frequency or chirp UWB radar systems, since the echoes received by these radar systems can be easily converted into sparse narrow pulses through pulse compression. The low-rank property of the RFI sources has been observed and utilized in [13,14]. Figure 3.3 gives the fast-time RFI-only spectra from multiple PRIs within a CPI using data measured by the ARL radar. RFI sources, such as FM radios, digital TV and cellular phones, tend to have strong sparse peaks in the fast-time frequency domain, whereas, in contrast, the entire frequency band of the UWB radar system is occupied by the radar echoes. Figure 3.4 shows that the singular values of the measured RFI-only data matrix R decreases rapidly and the bottom 90% of these singular values are zero or close to zero, confirming again that R possesses the low-rank property. Within a small time window, such as within a CPI, the low-rank property of the RFI sources appears to be due to the sinusoidal carrier approximations resulting from various modulation schemes popular in today’s wireless communications systems [15].

Zhang, Ren, Li, Greene, Johnston, and Nguyen

(dB) 0 200 –5 400 –10

Fast-Time Samples

600 800

–15 1000 –20

1200 1400

–25

1600 –30

1800 2000

–35 200

400

600

800

1000

1200

1400

1600

1800

Slow-Time Index

Figure 3.2 An example of a measured ARL UWB impulse radar echo signal in the fast-time vs.

slow-time domain in the absence of RFI. (dB)

10 8

0

15

–5 –10 –15 10 Frequency (Hz)

76

–20 –25 –30 5

–35 –40 –45

0

–50 200

400

600

800

1000

1200

1400

1600

1800

Slow-Time Index

Figure 3.3 An example of fast-time RFI spectrum vs. slow-time index of the RFI-only data

measured by the ARL radar receiver.

RFI Mitigation Based on Compressive Sensing Methods for UWB Radar Imaging

77

12

10

Singular Value

8

6

4

2

0 0

200

400

600

800

1000

1200

1400

1600

1800

Figure 3.4 The singular values of the RFI-only matrix R measured by the ARL radar receiver.

3.2.2

RPCA RPCA has been widely used to recover the low-rank and sparse components from their mixtures [23]. A natural choice of extracting X and R from the collected data Y is to use the following RPCA optimization metric [15]: min

||R||∗ + ζ||X||1

s.t. Y = X + R,

(3.2)

where · ∗ denotes the nuclear norm that promotes the low rank property, · 1 represents the 1 element-wise norm, which is a sparsity-enforcing metric, and ζ is a user parameter used to balance the trade-off between the two components. The user 1 in [24]. parameter ζ is recommended to be ζ 0 = √max{N,M} Following from [24], we employ the augmented Lagrange multiplier (ALM) method to solve the RPCA problem efficiently. First, we rewrite the RPCA problem (3.2) as follows: min

||R||∗ + ζ||X||1

s.t. Y − X − R = 0.

(3.3)

Then, the Lagrangian function is given by: μ L(X,R,Z,μ) = ||R||∗ + ζ||X||1 + < Z,Y − X − R > + ||Y − X − R||2F , (3.4) 2 where Z is the Lagrange multiplier and μ > 0 is the penalty parameter.

78

Zhang, Ren, Li, Greene, Johnston, and Nguyen

Algorithm 1 (RPCA via the Inexact ALM method) Input: Observation matrix Y ∈ RN×M , and tuning parameter ζ 1: Z ∗0 = Y /J (Y ), where J (Y ) = max(||Y ||2,ζ−1 ||Y ||∞ ); 2: X0 = 0;μ0 > 0; k = 0. 3: while not converged do 4: // Lines 5-6 solve R ∗k+1 = arg minR L(Xk ,R,Z k ,μk ). 5: (U,,V ) = svd(Y − X k + μ−1 k Z k ); 6: R k+1 = U Sμ−1 []V T , where Sε []ij = max{1 − ε/| ij |,0} ij ; k

7: // Lines 8 solves Xk+1 = arg minX L(X,R k+1,Z k ,μk ). 8: X k+1 = Sζ/μk [Y − R k+1 + μ−1 k Zk ] 9: Z k+1 = Z k + μk (Y − R k+1 − Xk+1 ). 10: Update μk to μk+1 11: k ←k+1 12: end while Output: (Xk ,R k ).

There are two types of ALM algorithms that can be used to solve the RPCA problem: the exact ALM algorithm and the inexact ALM algorithm [24]. Compared with the inexact ALM algorithm, which updates X∗k and R ∗k once when solving the sub-problem: (X∗k+1,R ∗k+1 ) = arg min L(X,R,Z ∗k ,μk ), X,R

(3.5)

the exact ALM method, though performing sightly better due to using the iterative thresholding approach to solve this sub-problem, requires a much longer computation time even for moderate N and M. Therefore, we focus herein on using the inexact ALM method. The detailed steps of the inexact ALM algorithm are summarized in Algorithm 1 [24]. More implementation details can be found in [24]. In this section we provide initial evaluations of the RFI mitigation performance of the RPCA algorithm using the measured data collected by ARL’s BoomSAR system. (Further performance evaluations are given in Section 3.5.) The measured data consists of the measured RFI-free radar echo data (see Figure 3.2) and the measured RFI-only data (see Figure 3.3). We scale the RFI-only data based on the desired SIR value, i.e., ||X||2F ||R||2F

, and add the scaled version of the RFI-only data to the RFI-free data X to obtain

the RFI-contaminated data Y . By using all the data within the CPI, we can form synthetic aperture radar (SAR) images. Figure 3.5 shows the original RFI-free SAR image that was obtained from the data in Figure 3.2. Figure 3.6 shows the RFI-contaminated SAR image that was obtained from Y for SIR = −10 dB. Figures 3.7 and 3.8 compare the recovered SAR images after RFI mitigation using the RPCA method with different ζ values for SIR = −10 dB. Visually, for SIR at −10 dB, RPCA with ζ = 0.4ζ0 significantly outperforms RPCA with the recommended ζ0 , since the recovered SAR image that was obtained by the latter is much sparser than the original RFI-free SAR image, with many small targets missing. Consider now the case of more severe RFI; Figure 3.9 shows the RFI-contaminated SAR image for SIR = −30 dB. Figures 3.10 and 3.11

RFI Mitigation Based on Compressive Sensing Methods for UWB Radar Imaging

79

Down Range (meters)

(dB) 40

0

50

–5

60

–10

70

–15

80

–20

90

–25

100

–30

110

–35 –740

–720

–700

–680

–660

–640

–620

–600

–580

–560

–540

Cross Range (meters)

Figure 3.5 Original RFI-free SAR image obtained by using the back projection algorithm on the

measured RFI-free ARL radar data.

Down Range (meters)

(dB) 40

0

50

–5

60

–10

70

–15

80

–20

90

–25

100

–30

110

–35 –740

–720

–700

–680

–660

–640

–620

–600

–580

–560

–540

Cross Range (meters)

Figure 3.6 SAR image of the measured SAR data set contaminated by measured RFI signals with

a SIR of −10 dB.

Zhang, Ren, Li, Greene, Johnston, and Nguyen

Down Range (meters)

(dB) 40

0

50

–5

60

–10

70

–15

80

–20

90

–25

100

–30

–35

110 –740

–720

–700

–680

–660

–640

–620

–600

–580

–560

–540

Cross Range (meters)

Figure 3.7 Recovered SAR image from the measured SAR data set contaminated by measured

RFI signals with a SIR of −10 dB obtained by using RPCA with ζ = 0.4ζ0 . (dB)

Down Range (meters)

80

40

0

50

–5

60

–10

70

–15

80

–20

90

–25

100

–30

110

–35 –740

–720

–700

–680

–660

–640

–620

–600

–580

–560

–540

Cross Range (meters)

Figure 3.8 Recovered SAR image from the measured SAR data set contaminated by measured

RFI signals with a SIR of −10 dB, obtained by using RPCA with ζ = ζ0 .

RFI Mitigation Based on Compressive Sensing Methods for UWB Radar Imaging

81

(dB) 40

0

50

–5

60

–10

70

–15

80

–20

90

–25

100

–30

–35

110 –740

–720

–700

–680

–660

–640

–620 –600

–580

–560

–540

Cross Range (meters)

Figure 3.9 SAR image from the measured SAR data set contaminated by measured RFI signals

with a SIR of −30 dB.

Down Range (meters)

(dB) 40

0

50

–5

60

–10

70

–15

80

–20

90

–25

100

–30

110

–35 –740

–720

–700

–680

–660

–640

–620

–600

–580

–560

–540

Cross Range (meters)

Figure 3.10 Recovered SAR image from the measured SAR data set contaminated by measured RFI signals with a SIR of −30 dB obtained by using RPCA with ζ = 0.4ζ0 .

82

Zhang, Ren, Li, Greene, Johnston, and Nguyen

Down Range (meters)

(dB) 40

0

50

–5

60

–10

70

–15

80

–20

90

–25

100

–30

110

–35 –740

–720

–700

–680

–660

–640

–620

–600

–580

–560

–540

Cross Range (meters)

Figure 3.11 Recovered SAR image from the measured SAR data set contaminated by measured RFI signals with a SIR of −30 dB obtained by using RPCA with ζ = ζ 0 .

compare the recovered SAR images after RFI mitigation using the RPCA method with different ζ values for SIR = −30 dB. Note that most of the target features are still retained in the resulting SAR image that was obtained by RPCA with ζ = ζ 0 , whereas the SAR image obtained by RPCA with ζ = 0.4ζ 0 has a fairly high noise level, with only a few strong targets discernible. Therefore, it appears that the choice of the user parameter ζ has a significant impact on the RFI mitigation performance of the RPCA algorithm and the resulting SAR image quality. Thus, a fine-tuning of ζ based on the SIR value is warranted. However, this parameter selection is not straightforward in practical applications, since the SIR value is usually not known a priori.

3.3

CLEAN-BIC for RFI Mitigation

3.3.1

RFI Sinusoidal Model Using the fact that the RFI sources tend to have strong, narrow peaks, we now consider approximately modeling the narrowband RFI sources within each PRI as sparse spectral lines in the frequency domain (see Figure 3.1). Since the UWB radar target echo signals are relatively weak and have flat spectrum in the fast-time frequency domain, they, along

RFI Mitigation Based on Compressive Sensing Methods for UWB Radar Imaging

83

with other uncertainties and approximations, are modeled as noise. Within a PRI, the observed data, denoted by vector y ∈ RN , can be modeled as follows: y = x + r,

(3.6)

where r ∈ RN denotes the RFI signal, which can be modeled as a sum of sinusoids: r(n) =

Kc

ak cos(ωk n + φk )

k=1

=

Kc 0 ak k=1

=

2Kc

2

ej (ωk n+φk ) +

ak −j (ωk n+φk ) 1 e 2

(3.7)

˜

a˜ k ej (ω˜ k n+φk ), n = 0,1,. . .,N − 1

k=1

with r(n) denoting the n-th element of r, ak ,ωk ,φk denoting the amplitude, frequency and phase of the k-th RFI source, and Kc denoting the number of the RFI sources. Since r(n) is real-valued, we state that a˜ k = a˜ Kc +k = a2k , ω˜ k = −ω˜ Kc +k = ωk , φ˜ k = −φ˜ Kc +k = φk , k = 1,. . .,Kc . In (3.6), x ∈ RN denotes the desired UWB radar echo signal within one PRI plus other uncertainties and approximations, and x will be treated as additive noise when estimating the sinusoidal parameters in r. Our goal is to recover x by estimating the sinusoidal parameters in r and removing the estimated rˆ of r from the RFI-contaminated data y. To reduce approximation errors, the sinusoidal parameters are allowed to change from one PRI to another. Hence the CLEAN-BIC algorithm presented in the next subsection is applied to one PRI at a time.

3.3.2

CLEAN-BIC To remove the RFI sources, modeled as a sum of sinusoids, from the observed RFIcontaminated data, we first estimate the sinusoidal parameters of these RFI sources. CLEAN, with its name meaning “cleaning the undesired component from the data,” is a simple and robust sinusoidal parameter estimation algorithm [18,19], first introduced in [16]. From as early as 1988, CLEAN has been used for the extraction of a single interference signal [17]. Since then, several works have considered using extensions of CLEAN for RFI mitigation [25–28]. The basic idea of CLEAN is to first estimate the parameters of the strongest sinusoid by minimizing the following nonlinear least-squares (NLS) criterion: ˆ˜ 1, αˆ˜ 1 } = arg min ||y − w(ω˜ 1 ) α˜ 1 ||2, {ω 2 {ω˜ 1, α˜ 1 }

(3.8)

84

Zhang, Ren, Li, Greene, Johnston, and Nguyen

˜

˜ ˜ = [1 ej ω˜ · · · ej ω(N−1) where α˜ k = a˜ k ej φk ,k = 1,2,. . .,2Kc , w(ω) ]T . The minimization of the cost function above with respect to ω˜ 1 , α˜ 1 yields:

ˆ˜ 1 = arg max |wH ( ω˜ 1 )y|2, ω

(3.9)

ω˜ 1

wH ( ω˜ 1 )y . αˆ˜ 1 = ˆ˜ 1 ω˜ 1 =ω N

(3.10)

ˆ˜ 1 is obtained as the location of the dominant peak of the scaled periodogram, Hence, ω H |w ( ω˜ 1 )y|2 , which can be efficiently computed by using the FFT with the data sequence y padded with zeros. Note that we consider fitting y to a complex-valued sinusoid in (3.8) because we can implement (3.9) using computationally efficient FFT operations. Note also that the accuracy of the frequency estimate of CLEAN can be improved by using Newton’s method following the FFT so that the accuracy of the frequency estimate of CLEAN is not limited to the grid size determined by the zero-padding that ˆ˜ 1 is determined, calculating αˆ˜ 1 using (3.10) is is used for the FFT operations. Once ω ˆ˜ ˆ˜ 1 easily, too. Next, since we have straightforward. We can obtain aˆ˜ 1 and φ 1 from α ak mentioned earlier that a˜ k = a˜ K +k = , ω˜ k = −ω˜ K +k = ωk , φ˜ k = −φ˜ K +k = 2

c

c

c

φk , k = 1,. . .,Kc , due to the real-valued data vector r, the dominant RFI source can be estimated as ˆ

ˆ

ˆ ˆ ˜ ˜ rˆ1 (n) = aˆ˜ 1 ej (ω˜ 1 n+φ1 ) + aˆ˜ 1 e−j (ω˜ 1 n+φ1 )

= aˆ 1 cos(ωˆ 1 n + φˆ 1 ), n = 0,1,. . .,N − 1.

(3.11)

From here, it is subtracted out from the RFI-contaminated data to get the residue data as follows: y 2 = y − rˆ 1,

(3.12)

where rˆ 1 is a vector formed from {ˆr1 (n)}N−1 n=0 . Then, the same process is repeated on the residue data y 2 to estimate the second-strongest sinusoid. For k = 3,. . .,Kc , let y k = y k−1 − rˆ k−1 . This process is repeated, to obtain rˆ k from y k , until the desired number of sinusoids Kc is reached. The CLEAN algorithm can be used with the BIC [20–22] to estimate the number of sinusoids needed to model the RFI sources. The BIC rule, when assuming K sinusoids in zero-mean white Gaussian noise with unknown variance σ 2 , has the following form [20,29,30]: BIC(K) = −2 ln p(y, θˆ K ) + ln[det(Jˆ K )],

(3.13)

ˆ denotes the likelihood function of y, θK = [γT ,σ2 ]T denotes the where p(y, θ) K K unknown parameter vector of y, γ K = [a1,. . .,aK ,ω1,. . .,ωK , φ1,. . .,φK ]T denotes the unknown sinusoidal parameter vector, and J K is the following Fisher information matrix (FIM): ∂ 2 ln p(y,θK ) J K = −E . (3.14) ∂ θK ∂ θTK

RFI Mitigation Based on Compressive Sensing Methods for UWB Radar Imaging

85

The negative log-likelihood function of y has the form: 2 + − 2 ln p(y,θK ) = N ln 2π + N ln σK

||y − r˘ K ||22 2 σK

,

(3.15)

where r˘ K is the same as r, except that it corresponds to the assumed K (instead of Kc ) 2 can be obtained by sinusoids. The maximum likelihood (ML) estimates of γ K and σK minimizing the negative log-likelihood function in (3.15): γˆ K = arg min ||y − r˘ K ||22, γK

2 σˆ K =

1 . ||y − r˘ K ||22 γ K =γˆ K N

(3.16) (3.17)

Then, the corresponding value of the likelihood function is given by [20]: 2 − 2 ln p(y, θˆ K ) = constant + N ln σˆ K ,

where 2 σˆ K

2 N K 1 ˆ = aˆ k cos(ωˆ k n + φk ) . y(n) − N n=1

(3.18)

(3.19)

k=1

Note that in the case of sinusoidal signals, we have the following approximation for sufficiently large values of N [20,31,32]: K N Jˆ K K N ≈ K N J K K N = O(1), where

2 KN =

1 I N 3/2 K

0

3 0 , 1 I N 1/2 2K+1

(3.20)

(3.21)

with I K denoting the K × K identity matrix. Then we obtain by a simple calculation: ˆ ln[det(Jˆ K )] = ln[det(K −2 N )] + ln[det(K N J K K N )] = (2K + 1) ln N + 3K ln N + O{1}

(3.22)

= (5K + 1) ln N + O{1}. Hence, the BIC metric takes on the following form [20] for our problem of interest: 2 + 5K ln N, BIC(K) = N ln σˆ K

(3.23)

where we have kept only the terms that depend on K. Because obtaining the ML estimate of γ is computationally expensive, we use CLEAN instead. In practical applications, where the sinusoidal data model is in itself an approximation, our empirical experience suggests that using CLEAN in lieu of ML yields similar RFI extraction performance. Combined with the BIC rule, the steps of the CLEAN-BIC algorithm for RFI removal can be summarized as follows:

Zhang, Ren, Li, Greene, Johnston, and Nguyen

Algorithm 2 (CLEAN-BIC) Input: Observation vector y ∈ RN , maximum number of sinusoids Kmax . 1: Assume K = 1. Let y 1 = y. Estimate the sinusoidal parameters {aˆ 1, ωˆ 1, φˆ 1 } from the original data vector y 1 using FFT followed by using the Newton’s method for fine search. Determine rˆ 1 . 2: Calculate BIC(1) by (3.23). 3: for K = 2 : Kmax 4: Compute y K = y K−1 − rˆ K−1 . 5: Estimate the sinusoidal parameters {aˆ K , ωˆ K , φˆ K } from y K . 6: Calculate BIC(K). 7: end 8: Determine Kˆ c that minimizes BIC(K), K = 1,. . .,Kmax . 9: Obtain γˆ Kˆ , the estimate of γ, and the corresponding rˆ Kˆ , the estimate of r. c c 10: xˆ = y − rˆ Kˆ . c ˆ rˆ Kˆ ). Output: (x, c

104

4.9

4.8

4.7

BIC(K)

86

4.6

4.5

4.4

4.3

4.2 0

50

100

150

200

250

300

K Figure 3.12 BIC curve obtained via CLEAN-BIC for one PRI using the RFI-contaminated data measured by the ARL radar for SIR = −30 dB.

Note that we need to iterate the algorithm until the iteration number Kmax is reached, since the BIC curve is usually not a smooth convex function of K. Figure 3.12 shows BIC(K) versus K for one PRI via CLEAN-BIC on the RFI-contaminated data that was measured by the ARL radar when SIR = −30 dB.

RFI Mitigation Based on Compressive Sensing Methods for UWB Radar Imaging

87

(dB)

10 8

0

15

–5 –10 –15 Frequency (Hz)

10 –20 –25 –30 5

–35 –40 –45

0

–50 200

400

600

800

1000

1200

1400

1600

1800

Slow-Time Index

Figure 3.13 Fast-time RFI spectrum vs. slow-time index of the noise-free, simulated RFI data.

We now evaluate the performance of the CLEAN-BIC algorithm using the same measured data set as used in Section 3.2. First, we simulate a set of noise-free RFI data using 10 sinusoids. Figure 3.13 shows the fast-time spectrum versus slow-time index image of the simulated RFI sources. The RFI-contaminated data is generated by combining the measured radar echo data with the scaled simulated RFI data based on the desired SIR. Figure 3.14 shows the contaminated SAR image at SIR = −10 dB. Figure 3.15 shows the recovered SAR image at SIR = −10 dB. Figure 3.16 shows the contaminated SAR image at SIR = −30 dB. Figure 3.17 shows the recovered SAR image at SIR = −30 dB. Comparing Figures 3.16 and 3.17 with the original RFI-free SAR image in Figure 3.5, we note that for the noise-free simulated RFI data, most targets in the recovered SAR images are discernible and the noise levels of the recovered SAR images are low, meaning that CLEAN-BIC does an excellent job of eliminating the simple simulated and noise-free RFI sources even at very low SIR levels. Next, we use CLEAN-BIC on the UWB radar echo data contaminated by the measured RFI signal (see Figure 3.3). Figures 3.18 and 3.19 show the recovered SAR images after RFI mitigation using CLEAN-BIC for SIR = −10 dB and −30 dB, respectively. Visually, for SIR = −10 dB, CLEAN-BIC performs well because it removes most of the RFI signals and retains most targets. However, with a SIR of −30 dB, the recovered SAR image that was obtained by CLEAN-BIC suffers from a high noise level, as compared to its RPCA counterpart in Figure 3.11. The high noise level is due to the fact that the measured RFI data inevitably contains noise (including thermal noise, model approximation errors, and other uncertainties), which is amplified when the SIR

Zhang, Ren, Li, Greene, Johnston, and Nguyen

Down Range (meters)

(dB) 40

0

50

–5

60

–10

70

–15

80

–20

90

–25

100

–30

110

–35 740

–720

–700

–680

–660

–640

–620

–600

–580

–560

–540

Cross Range (meters)

Figure 3.14 SAR image of the measured SAR data set contaminated by simulated RFI signals with a SIR of −10 dB. (dB)

Down Range (meters)

88

40

0

50

–5

60

–10

70

–15

80

–20

90

–25

100

–30

110

–35 –740

–720

–700

–680

–660

–640

–620

–600

–580

–560

–540

Cross Range (meters)

Figure 3.15 Recovered SAR image of the measured SAR data set contaminated by simulated RFI signals with a SIR of −10 dB, obtained with CLEAN-BIC.

RFI Mitigation Based on Compressive Sensing Methods for UWB Radar Imaging

89

Down Range (meters)

(dB) 40

0

50

–5

60

–10

70

–15

80

–20

90

–25

100

–30

110

–35

–740

–720

–700

–680

–660

–640

–620

–600

–580

–560

–540

Cross Range (meters)

Figure 3.16 SAR image of the measured SAR data set contaminated by simulated RFI signals with a SIR of −30 dB.

Down Range (meters)

(dB) 40

0

50

–5

60

–10

70

–15

80

–20

90

–25

100

–30

110

–35 –740

–720

–700

–680

–660

–640

–620

–600

–580

–560

–540

Cross Range (meters)

Figure 3.17 Recovered SAR image of the measured SAR data set contaminated by simulated RFI signals with a SIR of −30 dB obtained by using CLEAN-BIC.

Zhang, Ren, Li, Greene, Johnston, and Nguyen

Down Range (meters)

(dB) 40

0

50

–5

60

–10

70

–15

80

–20

90

–25

100

–30

110

–35 –740

–720

–700

–680

–660

–640

–620

–600

–580

–560

–540

Cross Range (meters)

Figure 3.18 Recovered SAR image of the measured SAR data set contaminated by measured RFI signals with a SIR of −10 dB obtained by using CLEAN-BIC. (dB)

Down Range (meters)

90

40

0

50

–5

60

–10

70

–15

80

–20

90

–25

100

–30

110

–35 –740

–720

–700

–680

–660

–640

–620

–600

–580

–560

–540

Cross Range (meters)

Figure 3.19 Recovered SAR image of the measured SAR data set contaminated by measured RFI signals with a SIR of −30 dB obtained by using CLEAN-BIC.

RFI Mitigation Based on Compressive Sensing Methods for UWB Radar Imaging

91

is decreased. Moreover, unlike RPCA, CLEAN-BIC does not make any attempt to sparsify the radar echo signal, and returns a recovered SAR image with a high noise level, especially when the SIR is low.

3.4

Enhanced Algorithms for RFI Mitigation

3.4.1

The Hybrid Method To combine the merits of both RPCA and CLEAN-BIC, we consider a hybrid method / via CLEAN-BIC. Then, the (HM). We first obtain the estimate of the RFI sources, R, 2 2 / / SIR is estimated as Y − RF /RF . In our previous analysis on RPCA, we have shown that RPCA with the recommended ζ0 works well when the SIR is low. We have also shown that CLEAN-BIC does a satisfying job at high SIR. Therefore, if the estimated / that was obtained via CLEAN-BIC SIR is high, such as −10 dB or higher, we use the R / = Y − R. / If the estimated SIR is to recover the desired UWB radar echo signals via X / lower than −10 dB, we then use RPCA with the recommended ζ0 to obtain X.

3.4.2

RPCA-CB We further consider using CLEAN-BIC to determine the user parameter ζ for RPCA, since it appears that ζ should be tuned based on the SIR. This method is referred to as RPCA-CB (RPCA with its user parameter determined via CLEAN-BIC). Similar to / Then, the user parameter ζ the HM method, we first use CLEAN-BIC to obtain R. for RPCA is selected based on the estimated SIR. The process of RPCA-CB can be illustrated by Figure 3.20. More details on the selection of ζ based on the estimated SIR will be depicted in Section 3.5.

Figure 3.20 A flow graph depicting the connections between RPCA and CLEAN-BIC in RPCA-CB.

92

Zhang, Ren, Li, Greene, Johnston, and Nguyen

3.5

Performance Evaluations We evaluate the RFI mitigation performance of the aforementioned four methods and compare them to each other in this section, using both simulated and experimentally measured data.

Data Sets Our performance evaluations are conducted using the measured RFI-free UWB radar echo data set with two different RFI-only data sets: a realistically simulated RFI-only data set and a measured noisy RFI-only data set. The measured RFI-free UWB radar echo data set (see Figures 3.2 and 3.5) is collected by using the ARL BoomSAR system, with a frequency band spanning from approximately 50 MHz to 1150 MHz. The UWB BoomSAR is mounted on a platform that emulates an airborne geometry. Two transmitters and two receivers are used to collect data in different polarizations. The measured data used herein is configured in a horizontal transmit, horizontal receive (HH) polarizations. The measured RFI-only data set (see Figure 3.3) is collected by the ARL radar receiver with the antenna pointing toward Washington, DC [1,12]. Since the ARL experimental hardware system is used to measure the RFI-only data (or more precisely, the data that is free of the UWB radar echoes), the measured RFI-only data is inevitably contaminated only by the hardware system noise. The realistically simulated RFI-only data set (see Figure 3.21) is noise-free and composed of sinusoids estimated by using (dB)

10 8

0

15

–5 –10 –15 10 Frequency (Hz)

3.5.1

–20 –25 –30 5

–35 –40 –45

0

–50 200

400

600

800

1000

1200

1400

1600

1800

Slow-Time Index

Figure 3.21 Fast-time RFI spectrum vs. slow-time index for the noise-free realistically simulated

RFI data.

RFI Mitigation Based on Compressive Sensing Methods for UWB Radar Imaging

93

the CLEAN-BIC algorithm on the measured ARL RFI-only data. Note that the realistically simulated RFI-only data utilized herein is much more complex than that used in Section 3.3 and consists of about 80 sinusoids per PRI. We note that Figure 3.21 resembles its counterpart, obtained from measured data. All data sets have N = 2048 fast-time samples and M = 1892 slow-time samples. The received RFI-contaminated radar data is obtained by adjusting the power of the RFI-only data based on the desired SIR and adding it to the RFI-free UWB radar echo signal. Since the measured RFI-only data set contains noise, as we lower the SIR, we are increasing the noise level of the received RFI-contaminated radar data as well. Note that we have no prior information on the noise in the measured data.

3.5.2

Evaluation Metric The SAR image is obtained using the back projection algorithm on the recovered radar data after RFI mitigation. We utilize the signal-to-noise ratio (SNR) metric of the recovered SAR image to benchmark the RFI mitigation performance: ZF 4 , SNR Metric = 20log10 4 4Z − / Z4

(3.24)

F

where Z is the normalized, original, and RFI-free SAR image, and / Z is the normalized, recovered SAR image.

3.5.3

Analysis of RPCA First, we investigate the performance of RPCA by varying the user parameter ζ for various SIR levels. Tables 3.1 and 3.2 compare the SNR metric of the recovered SAR images that were obtained by applying RPCA to RFI-contaminated data over a wide range of SIR levels. Apparently, for both the simulated and measured RFI-only data, the choice of the user parameter ζ has a significant impact on the RFI mitigation performance. For the given data sets, the proper selection of ζ (boldfaced in Tables 3.1 and 3.2) is related to the SIR levels.

Table 3.1 The SNR metric of the recovered SAR images obtained by using RPCA with data obtained from the ARL RFI-free radar data and the scaled simulated RFI-only data. SNR Metric (dB) ζ 0.4ζ0 0.6ζ0 0.8ζ0 1.0ζ0 1.2ζ0

SIR (dB)

0

−10

−20

9.22552 10.76778 7.47976 4.94639 3.49384

3.42188 7.92373 6.84925 4.66251 3.24972

−3.42708 4.78473 5.93832 4.38511 2.99793

−30 −12.12427 0.56811 4.57629 4.11642 2.93142

Zhang, Ren, Li, Greene, Johnston, and Nguyen

Table 3.2 The SNR metric of the recovered SAR images obtained by using RPCA with data obtained from the ARL RFI-free radar data and the scaled measured ARL RFI-only data. SIR (dB)

SNR Metric (dB) ζ 0.4ζ0 0.6ζ0 0.8ζ 0 1.0ζ0 1.2ζ0

0

−10

−20

9.16536 10.7068 7.49281 4.98311 3.53173

3.09982 7.40891 6.72988 4.76060 3.39350

−4.20045 2.49460 4.22138 3.92003 3.11635

−30 −13.2707 −5.52660 −3.01897 −1.74045 −0.95133

1.2

1.1

Parameter Coefficient β

94

1

0.9

0.8

0.7

0.6

0.5 –30

–25

–20

–15

–10

–5

0

SIR

Figure 3.22 Exponential curve to fit the relationship between the SIR of the observed data and the corresponding proper user parameter ζ. Data was obtained from combining the ARL RFI-free radar data with the scaled measured ARL RFI-only data (ζ = βζ0 , where β is the parameter coefficient).

Note that the relationship between the proper ζ and the SIR should be nonlinear because the appropriate ζ should approach infinity as the SIR approaches negative infinity, and zero as the SIR approaches infinity. A natural choice is to use the following exponential equation to express the relationship between the proper ζ denoted as ζ = βζ0 and the SIR: β = ηexp(−γSIR),

(3.25)

where η and γ are the parameters that can be obtained by using, for example, the MATLAB Curve Fitting Toolbox. Figure 3.22 gives the curve fitting for the RFI-contaminated

RFI Mitigation Based on Compressive Sensing Methods for UWB Radar Imaging

95

Table 3.3 The SNR metric of recovered SAR images obtained by using the CLEAN-BIC algorithm. Data was obtained from combining the ARL RFI-free radar data with the scaled, realistically simulated RFI-only data. 0

−10

−20

−30

9.51874

6.54657

5.21395

1.40243

SIR (dB) SNR Metric (dB)

Table 3.4 The SNR metric of recovered SAR images obtained by using the CLEAN-BIC algorithm. Data was obtained from combining the ARL RFI-free radar data with the scaled, measured ARL RFI-only data. 0

−10

−20

−30

9.44348

5.90590

1.41839

−6.64253

SIR (dB) SNR Metric (dB)

data that was obtained using the measured RFI-only data with the measured RFI-free radar echo data, based on (3.25) (η = 0.5027, γ = 0.02761). It can be demonstrated that the curve in Figure 3.22 also works well for the realistically simulated RFI-only data. For the RPCA-CB algorithm, we use this curve to determine the RPCA user parameter based on the estimated SIR that was obtained via CLEAN-BIC.

3.5.4

Analysis of CLEAN-BIC Next, Tables 3.3 and 3.4 compare the RFI mitigation performance using the SNR metric for the CLEAN-BIC algorithm over a wide range of SIR levels. Compared with the simple simulated RFI-only data used in Section 3.3, the realistically simulated RFI-only data used here is composed of many sinusoids and the frequencies of some of the sinusoids are very closely spaced. Due to the limited resolution of CLEAN-BIC, it cannot remove the sinusoids completely, even though the sinusoids are obtained by CLEANBIC from the measured RFI-only data. As a result, the SNR metric of the recovered SAR images of CLEAN-BIC decreases with the increasing SIR level. Moreover, in the case of measured RFI-only data, the SNR metric of the recovered SAR images is more severely affected by the SIR levels due to the presence of noise in the measured RFI-only data and the sinusoidal model error.

3.5.5

Analysis of Enhanced Methods and Comparisons Finally, we evaluate the RFI mitigation performance of the enhanced algorithms, including HM and RPCA-CB, and compare their performance with their RPCA and CLEAN-BIC counterparts. Herein, the user parameter ζ in RPCA is set as ζ0 , as recommended in [24]. Since both of the enhanced algorithms are based on the estimated SIR values that were obtained via CLEAN-BIC, we first show the SIR estimation results of CLEAN-BIC in Table 3.5. Apparently, the CLEAN-BIC algorithm can provide sufficiently accurate SIR estimates for the enhanced algorithms.

Zhang, Ren, Li, Greene, Johnston, and Nguyen

Table 3.5 The estimated SIR obtained via CLEAN-BIC for data obtained from combining the ARL RFI-free radar data with the scaled realistically simulated and measured ARL RFI-only data. True SIR (dB) Estimated SIR (dB) Simulated RFI Measured RFI

0

−10

−20

−30

1.6975 1.8753

−8.7983 −8.3260

−18.8360 −16.1593

−26.5150 −19.6084

(dB)

Down Range (meters)

96

40

0

50

–5

60

–10

70

–15

80

–20

90

–25

100

–30

110

–35 –740

–720

–700

–680

–660

–640

–620

–600

–580

–560

–540

Cross Range (meters)

Figure 3.23 Recovered SAR image of the measured SAR data set contaminated by realistically simulated RFI-only signals with a SIR of −10 dB obtained by using HM.

Figures 3.23–3.26 show the recovered SAR images using these two enhanced algorithms with data obtained from combining the measured ARL radar data with realistically simulated RFI-only data for SIR = −10 dB and SIR = −30 dB, respectively. Compared with the RFI-free case in Figure 3.5, we see that the enhanced algorithms work well for both strong and weak realistically simulated RFI cases. Figure 3.27 compares the SNR metric of the recovered SAR images for various SIR levels for the realistically simulated RFI case. Clearly, RPCA achieves a relatively low SNR value with a SIR value higher than −10 dB, and CLEAN-BIC has a low SNR value in the case of severe RFI. However, the enhanced algorithms, which combine the merits of both RPCA and CLEAN-BIC, has a satisfying performance over a wide range of SIR levels. Moreover, the RPCA-CB algorithm has the best performance among all four algorithms. Figures 3.28–3.31 present the recovered SAR images using these two enhanced algorithms with the RFI-contaminated data that was obtained by combining the measured

RFI Mitigation Based on Compressive Sensing Methods for UWB Radar Imaging

97

Down Range (meters)

(dB) 40

0

50

–5

60

–10

70

–15

80

–20

90

–25

100

–30

110

–35 –740

–720

–700

–680

–660

–640

–620

–600

–580

–560

–540

Cross Range (meters)

Figure 3.24 Recovered SAR image of the measured SAR data set contaminated by realistically simulated RFI-only signals with a SIR of −10 dB obtained by using RPCA-CB.

Down Range (meters)

(dB) 40

0

50

–5

60

–10

70

–15

80

–20

90

–25

100

–30

110

–35 –740

–720

–700

–680

–660

–640

–620

–600

–580

–560

–540

Cross Range (meters)

Figure 3.25 Recovered SAR image of the measured SAR data set contaminated by realistically simulated RFI-only signals with a SIR of −30 dB obtained by using HM.

Zhang, Ren, Li, Greene, Johnston, and Nguyen

Down Range (meters)

(dB) 40

0

50

–5

60

–10

70

–15

80

–20

90

–25

100

–30

110

–35 –740

–720

–700

–680

–660

–640

–620

–600

–580

–560

–540

Cross Range (meters)

Figure 3.26 Recovered SAR image of the measured SAR data set contaminated by realistically simulated RFI-only signals with a SIR of −30 dB obtained by using RPCA-CB. 12 10 8 6 4

SNR(dB)

98

2 RPCA

0

CLEAN–BIC –2 HM –4

RPCA–CB

–6 –8 –30

–25

–20

–15 SIR(dB)

–10

–5

Figure 3.27 Comparison of RFI mitigation performances for various SIR values using the simulated RFI-only data and the measured ARL RFI-free UWB radar data.

0

RFI Mitigation Based on Compressive Sensing Methods for UWB Radar Imaging

99

Down Range (meters)

(dB) 40

0

50

–5

60

–10

70

–15

80

–20

90

–25

100

–30

110

–35 –740

–720

–700

–680

–660

–640

–620

–600

–580

–560

–540

Cross Range (meters)

Figure 3.28 Recovered SAR image of the measured SAR data set contaminated by measured-only RFI signals with a SIR of −10 dB obtained by using HM.

Down Range (meters)

(dB) 40

0

50

–5

60

–10

70

–15

80

–20

90

–25

100

–30

–35

110 –740

–720

–700

–680

–660

–640

–620

–600

–580

–560

–540

Cross Range (meters)

Figure 3.29 Recovered SAR image of the measured SAR data set contaminated by measured RFI-only signals with a SIR of −10 dB obtained by using RPCA-CB.

Zhang, Ren, Li, Greene, Johnston, and Nguyen

Down Range (meters)

(dB) 40

0

50

–5

60

–10

70

–15

80

–20

90

–25

100

–30

110

–35 –740

–720

–700

–680

–660

–640

–620

–600

–580

–560

–540

Cross Range (meters)

Figure 3.30 Recovered SAR image of the measured SAR data set contaminated by measured RFI-only signals with a SIR of −30 dB obtained by using HM. (dB)

Down Range (meters)

100

40

0

50

–5

60

–10

70

–15

80

–20

90

–25

100

–30

–35

110 –740

–720

–700

–680

–660

–640

–620

–600

–580

–560

–540

Cross Range (meters)

Figure 3.31 Recovered SAR image of the measured SAR data set contaminated by measured RFI-only signals with a SIR of −30 dB obtained by using RPCA-CB.

RFI Mitigation Based on Compressive Sensing Methods for UWB Radar Imaging

101

12 10 8 6

SNR(dB)

4 2 RPCA 0

CLEAN

–2

HM RPCA-CB

–4 –6 –8 –30

–25

–20

–15 SIR(dB)

–10

–5

0

Figure 3.32 Comparison of RFI mitigation performances for various SIR values using the measured ARL RFI-only data and the measured ARL RFI-free UWB radar data.

RFI-free SAR data with the scaled measured RFI-only data for SIR = −10 dB and SIR = −30 dB, respectively. Similar to the realistically simulated RFI case discussed in the previous paragraph, the enhanced algorithms perform better than their RPCA counterparts (see Figures 3.8 and 3.11) and their CLEAN-BIC counterparts (see Figures 3.18 and 3.19). Figure 3.32 compares the RFI mitigation performance using the SNR metric of the recovered SAR images using all four methods for various SIR values for the case of measured RFI data. Again, the enhanced algorithms outperform their RPCA and CLEAN-BIC counterparts. Note also that RPCA-CB outperforms HM slightly at high SIR values.

3.6

Conclusions In this chapter, we proposed several sparse signal recovery methods for effective RFI mitigation. We first demonstrated that the RFI sources are low-rank and sparse in the frequency domain; in contrast, the desired UWB radar echoes are sparse in the time domain. RPCA can be used to exploit these properties for effective RFI mitigation; however, RPCA requires fine tuning of a user parameter that is nontrivial in practical applications due to the lack of prior knowledge about the RFI sources and UWB radar echoes. To avoid the user parameter tuning problem, we introduced the CLEAN-BIC algorithm for RFI mitigation, via modeling the RFI sources within a PRI as the sum

102

Zhang, Ren, Li, Greene, Johnston, and Nguyen

of sinusoids. We have shown that CLEAN-BIC can be used to remove dominant RFI sources effectively; however, since the sparse property of the desired UWB radar echoes is not utilized by CLEAN-BIC, the resulting SAR images contain high noise levels, especially at low SIR levels. To combine the merits of both RPCA and CLEAN-BIC, we proposed a hybrid method, or HM, wherein the use of RPCA or CLEAN-BIC is based on the estimated SIR value obtained via CLEAN-BIC. Furthermore, we have considered determining the user parameter for the RPCA algorithm based on using the estimated SIR value that was obtained via CLEAN-BIC; the resulting algorithm is referred to as RPCA-CB. These enhanced algorithms were applied to RFI-contaminated data that was obtained from combining the measured RFI-free radar echo data with either realistically simulated or experimentally measured RFI-only data for performance evaluations. We have shown that the estimated methods outperformed both RPCA and CLEAN-BIC. Finally, RPCA-CB has been shown to outperform HM slightly, especially at high SIR levels.

3.7

Acknowledgment This material is based upon work supported in part by the US Army Research Laboratory and the US Army Research Office under grant number W911NF-16-2-0223.

References [1] L. H. Nguyen, T. Tran, and T. Do, “Sparse models and sparse recovery for ultra-wideband SAR applications,” IEEE Transactions on Aerospace and Electronic Systems, vol. 50, no. 2, pp. 940–958, 2014. [2] T. Koutsoudis and L. A. Lovas, “RF interference suppression in ultrawideband radar receivers,” in Algorithms for Synthetic Aperture Radar Imagery II. International Society for Optics and Photonics, 1995, pp. 107–119. [3] D. O. Carhoun, “Adaptive nulling and spatial spectral estimation using an iterated principal components decomposition,” in 1991 International Conference on Acoustics, Speech, and Signal Processing, 1991, pp. 3309–3312. [4] H. Subbaram and K. Abend, “Interference suppression via orthogonal projections: a performance analysis,” IEEE Transactions on Antennas and Propagation, vol. 41, no. 9, pp. 1187–1194, 1993. [5] V. T. Vu, T. K. Sjögren, M. I. Pettersson, and L. Håkasson, “An approach to suppress RFI in ultrawideband low frequency SAR,” in Radar Conference, 2010 IEEE. IEEE, 2010, pp. 1381–1385. [6] X. Luo, L. M. H. Ulander, J. Askne, G. Smith, and P. O. Frolind, “RFI suppression in ultrawideband SAR systems using LMS filters in frequency domain,” Electronics Letters, vol. 37, no. 4, pp. 241–243, 2001. [7] T. Miller, L. Potter, and J. McCorkle, “RFI suppression for ultra wideband radar,” IEEE Transactions on Aerospace and Electronic Systems, vol. 33, no. 4, pp. 1142–1156, 1997. [8] X. Wang, W. Yu, X. Qi, and Y. Liu, “RFI suppression in SAR based on approximated spectral decomposition algorithm,” Electronics Letters, vol. 48, no. 10, pp. 594–596, 2012.

RFI Mitigation Based on Compressive Sensing Methods for UWB Radar Imaging

103

[9] F. Zhou, R. Wu, M. Xing, and Z. Bao, “Eigensubspace-based filtering with application in narrow-band interference suppression for SAR,” IEEE Geoscience and Remote Sensing Letters, vol. 4, no. 1, pp. 75–79, 2007. [10] F. Zhou and M. Tao, “Research on methods for narrow-band interference suppression in synthetic aperture radar data,” IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing, vol. 8, no. 7, pp. 3476–3485, 2015. [11] F. Zhou, M. Tao, X. Bai, and J. Liu, “Narrow-band interference suppression for SAR based on independent component analysis,” IEEE Transactions on Geoscience and Remote Sensing, vol. 51, no. 10, pp. 4952–4960, 2013. [12] L. H. Nguyen and T. D. Tran, “Efficient and robust RFI extraction via sparse recovery,” IEEE Journal of Selected Topics in Applied Earth Observations and Remote Sensing, vol. 9, no. 6, pp. 2104–2117, 2016. [13] L. H. Nguyen, M. D. Dao, and T. D. Tran, “Estimating unknown sparsity in compressed sensing,” in 2014 IEEE International Conference on Image Processing (ICIP), 2014, pp. 116–120. [14] L. H. Nguyen, M. D. Dao, and T. D. Tran, “Joint sparse and low-rank model for radiofrequency interference suppression in ultra-wideband radar applications,” in 2014 48th Asilomar Conference on Signals, Systems and Computers, 2014, pp. 864–868. [15] L. H. Nguyen and T. D. Tran, “RFI-radar signal separation via simultaneous low-rank and sparse recovery,” in 2016 IEEE Radar Conference (RadarConf), 2016, pp. 1–5. [16] J. A. Högbom, “Aperture synthesis with a non-regular distribution of interferometer baselines,” Astronomy and Astrophysics Supplement Series, vol. 15, p. 417, 1974. [17] J. Tsao and B. D. Steinberg, “Reduction of sidelobe and speckle artifacts in microwave imaging: The CLEAN technique,” IEEE Transactions on Antennas and Propagation, vol. 36, no. 4, pp. 543–556, 1988. [18] P. T. Gough, “A fast spectral estimation algorithm based on the FFT,” IEEE transactions on Signal Processing, vol. 42, no. 6, pp. 1317–1322, 1994. [19] J. Li and P. Stoica, “Efficient mixed-spectrum estimation with applications to target feature extraction,” IEEE transactions on signal processing, vol. 44, no. 2, pp. 281–295, 1996. [20] P. Stoica and Y. Selen, “Model-order selection: a review of information criterion rules,” IEEE Signal Processing Magazine, vol. 21, no. 4, pp. 36–47, 2004. [21] T. Yardibi, J. Li, P. Stoica, M. Xue, and A. B. Baggeroer, “Source localization and sensing: A nonparametric iterative adaptive approach based on weighted least squares,” IEEE Transactions on Aerospace and Electronic Systems, vol. 46, no. 1, pp. 425–443, 2010. [22] P. Stoica and P. Babu, “On the proper forms of BIC for model order selection,” IEEE Transactions on Signal Processing, vol. 60, no. 9, pp. 4956–4961, 2012. [23] E. J. Candès, X. Li, Y. Ma, and J. Wright, “Robust principal component analysis?” Journal of the ACM, vol. 58, p. 11, 2011. [24] Z. Lin, M. Chen, and Y. Ma, “The augmented lagrange multiplier method for exact recovery of corrupted low-rank matrices,” arXiv preprint arXiv:1009.5055, 2010. [25] P. A. Fridman and W. A. Baan, “RFI mitigation methods in radio astronomy,” Astronomy & Astrophysics, vol. 378, no. 1, pp. 327–344, 2001. [26] A. Camps, J. Gourrion, J. Miguel Tarongí et al., “RFI analysis in smos imagery,” in Geoscience and Remote Sensing Symposium (IGARSS), 2010 IEEE International. IEEE, 2010, pp. 2007–2010.

104

Zhang, Ren, Li, Greene, Johnston, and Nguyen

[27] X. Peng, F. Hu, F. He, D. Zhu, Y. Cheng, H. Hu, and T. Zheng, “An improved clean algorithm for RFI mitigation in aperture synthesis radiometers,” in 2017 IEEE International Geoscience and Remote Sensing Symposium (IGARSS). IEEE, 2017, pp. 3448–3451. [28] F. Hu, X. Peng, F. He, et al., “RFI mitigation in aperture synthesis radiometers using a modified clean algorithm,” IEEE Geoscience and Remote Sensing Letters, vol. 14, no. 1, pp. 13–17, 2017. [29] G. Schwarz, “Estimating the dimension of a model,” The Annals of Statistics, vol. 6, no. 2, pp. 461–464, 1978. [30] J. Rissanen, “Modeling by shortest data description,” Automatica, vol. 14, no. 5, pp. 465–471, 1978. [31] P. Stoica and R. L. Moses, Introduction to Spectral Analysis. Prentice Hall, 1997, vol. 1. [32] P. M. Djuric, “Asymptotic MAP criteria for model selection,” IEEE Transactions on Signal Processing, vol. 46, no. 10, pp. 2726–2735, 1998.

4

Compressed CFAR Techniques Laura Anitori and Arian Maleki

4.1

Introduction In this chapter we study the problem of target detection from a set of compressive radar measurements that are corrupted by additive white Gaussian noise. The complications in the calculation of false alarm and detection probabilities that are caused by the nonlinear nature of target recovery schemes in compressed sensing represent a major limitation in the application of such techniques in real radar systems. In this chapter, we show how recent advances in the asymptotic analysis of the recovery algorithms help us overcome this challenge. We first review some standard tools that are used in classical radar detection, such as the matched filter and the constant false alarm rate (CFAR) processor. Then, we clarify the main challenges one would face in implementing such schemes for target detection from compressed measurements. We show how the recently developed asymptotic analyses of recovery algorithms enable us to overcome these challenges, and develop fully adaptive target detection schemes. Finally, we evaluate the performance of these schemes through theoretical analyses and extensive simulations on synthetic and experimental data.1

4.2

Radar Signal Model Consider a one-dimensional radar, which measures the target echoes as a function of range (or time delay) x(t) over an observation interval t ∈ (Tmin,Tmax ).2 The radar transmits a radio frequency (RF) signal modulated by a waveform a(t), and, in the absence of noise and interference, the received and demodulated signal can be mathematically represented by the convolution y(t) = C a(t − τ)x(τ)δτ, (4.1) where C is a complex constant including the target radar cross section (RCS), phase terms and propagation effects. The time delay variable t can be mapped to the range via 1 This chapter is based on chapters 2, 3, and 4 of [1] and ©[2013] IEEE. Reprinted, with permission,

from [2]. 2 The extension to azimuth and Doppler domains can be easily obtained in a similar way; see for example

[3,4].

105

106

Anitori and Maleki

the equation t = 2rc , where c is the speed of light and r is range (or distance). If there are k point targets located at ranges ri ,i = 1,. . .,k, corresponding to time delays τi = 2rc i , the target reflectivity distribution can be expressed as x(t) = ki=1 xi δ(t − τi ), where xi is the ith target RCS [5–7]. Hence, the (complex) baseband, received signal can be rewritten as k y(t) = xi a(t − τi ), (4.2) i=1

where xi ,i = 1,. . .,k is a (complex) amplitude proportional to, among others, the target RCS, target distance and transmitted power [8]. In the remainder of this chapter we consider |xi |2 as the power received from a target at position i. It is often convenient to discretize the range and represent signals in vector forms. To this end, let the vector x represent the target response (or scene) at discretized range = r1 + n R, where R is bins,3 i.e., r = [r1,. . .,rn ], with r1 = cT2min , rn = cTmax 2 the range bin size. Furthermore, assume that targets can only be present at locations corresponding exactly to discrete grid points. Using the Nyquist sampling theorem, the received signal y(t) is sampled at a rate fs ≥ B, where B is the bandwidth of the transmitted signal. Then, the sampled received signal y(tl ),l = 1,. . .,L in (4.2) can be rewritten in vector form as k y = Ax = xi a i , (4.3) i=1

where each column a i ,i = 1,. . .,n of the matrix A is a time delayed version of the sampled transmitted waveform corresponding to the received signal from a target at range bin i (the ith model), and x is a length n vector with amplitude xi at indices i corresponding to target located at ranges ri and zero elsewhere. Taking the noise into account, we obtain y = Ax + n,

(4.4)

where we consider n ∼ CN (0,σ2 I ).4

4.3

Classical Radar Detection

4.3.1

The Matched Filter It is well known that the matched filter (MF) is the filter that optimizes the signal-tonoise ratio (SNR) of a known signal in white Gaussian noise; see, e.g., [9]. The impulse response of the MF is a time-reversed and conjugated copy of the (known) transmitted signal, and the range response of a point target after MF is given by the autocorrelation function of the transmitted waveform. For a radar transmitting an unmodulated pulse, 3 Because of the relation between time delay and range, we will use the two interchangably. 4 In case of clutter or correlated noise, prewhitening filters are often applied to obtain the formulation in

(4.4). In that case, the matrix A includes also the pre-whitening.

Compressed CFAR Techniques

107

the range resolution is given by δR = cT2 . Hence, improving the range resolution requires shortening the pulse duration, and results in reduced transmitted energy (for a given fixed peak power). A common way to improve resolution without reducing the pulse length is to use frequency or phase modulated pulses such as linear frequency modulation (LFM) or chirp waveform, Barker codes, and pseudorandom noise (PN) sequences. In this case, the output of the MF is a compressed pulse with resolution c , where B is the bandwidth of the transmitted pulse. Because of this property, δR = 2B the operation of matched filtering is referred to as pulse compression. The SNR gain of the MF after pulse compression is given by the time bandwidth product BT . Although pulse compression by matched filtering results in significant range resolution improvement compared to unmodulated pulses, one of the issues that needs to be addressed is the sidelobe level (SLL). For example, the autocorrelation function of an LFM pulse exhibits large sidelobes (about −13 dB) with respect to the mainlobe [10]. Large sidelobes of strong targets may result in masking of weaker targets in a multiple targets scenario and in a severe increase of the false alarm rate.

4.3.2

Target Detection For the detection of a single target (with known parameters) embedded in white Gaussian noise with known variance, the use of statistical decision theory shows that the optimum (Neyman–Pearson) receiver consists of an MF followed by a fixed-threshold detector [11–13]. Using the Neyman–Pearson theorem, it is possible to design the detector threshold to achieve a false alarm probability not exceeding a predetermined value of, say, α. Note that the optimum detector, based on MF, is derived for the case of one known signal in white Gaussian noise. This is hardly the case in practical operations, where the target amplitude, phase, time delay (range), and Doppler frequency are unknown. When the target parameters are unknown, a common approach is to set up a generalized likelihood ratio test (GLRT) for each discrete time delay τi on a predefined grid. If the target’s phase has a uniform distribution, then we obtain the test statistic [13] T H1 ∗ a (t − τi )y(t)dt ≷ γ, (4.5) 0

H0

where T is the received signal length. In other words, GLRT computes the envelope (or power) of the MF output at all discrete time delays and compares it with a threshold to determine the presence (declare hypothesis H1 ) or absence (declare hypothesis H0 ) of a target at time delay τi . Using the discrete linear model introduced in (4.4), we can rewrite the test statistic in (4.5) as |xˆi | = |a i H y|,

i = 1,. . .,n

(4.6)

where superscript H indicates the conjugate transpose of a vector. In words, the MF computes the cross-correlation of the received signal with time-delayed versions of the transmitted waveform. Combining the MF outputs for all time delays, we can write the MF discrete output signal in vector form as xˆ MF = [xˆ1,. . ., xˆn ]T , where

108

Anitori and Maleki

xˆ MF = AH y.

(4.7)

The envelope (or power) of each component of the vector xˆ MF is compared to the threshold γ to decide upon the presence of a target. For the envelope detector, the threshold γ should be set to

γ = −σ 2 ln α (4.8) to achieve a false alarm probability (FAP) equal to α [9].

4.3.3

Detection of Multiple Targets Often in real radar applications, there is more than one target in the received window. The task of a radar is to determine how many targets are present and to estimate their parameters. If an MF is used at the receiver, expanding (4.7) we obtain xˆ MF = AH y = AH (Ax + n) = x + (AH A − I )x + AH n.

(4.9)

It can be seen in (4.9) that each entry xˆ i of the MF output is the sum of the true target response at location i, xi , plus the interference caused by the presence of other possible targets at locations j != i ( nj=1,j !=i xj a H i a j ), plus Gaussian noise. Note that the interference is proportional to the cross-correlation between the time-delayed version of the transmitted waveform [14]; in the ideal case that the time-delayed versions of the transmitted waveforms are orthogonal and have unit norm, i.e., AH A = I , we obtain xˆ MF = x + z,

(4.10)

where z ∼ CN (0,σ2 I ). Therefore, if each target is exactly on a grid point and the matrix A is orthogonal, the components of the vector xˆ MF are independent of one another and each range bin can be treated independently. One can then apply hypothesis tests to each time delay to estimate simultaneously the number of targets and their time delays. For practical frequency or phase-modulated waveforms, the orthogonality condition is never met. Hence, each target interferes, through its sidelobe, with the detection of the others. This phenomenon is known as target masking. A reduction of the SLL can be accomplished by applying a weighting function during matched filtering [9]. However, the weighted MF output is no longer matched to the transmitted signal, and therefore, in addition to reducing the SLL, the output SNR is also reduced. Alternatively, one can design waveforms with low SLLs [15–17] or use different pulse compression filters, such as adaptive pulse compression (APC) [5,6] or mismatched filters [18–22]. A review of mismatched filters can be found in [14]. Note that the design of mismatched filters is based on iterative algorithms and the optimum filter weights have to be estimated separately for each range bin. Therefore, such schemes are computationally more demanding than the MF. Performing an independent binary GLRT at each range bin does not take into account the interaction between targets. Hence when the sidelobes are not sufficiently

Compressed CFAR Techniques

109

suppressed, a better detection/estimation strategy would be the multiple hypothesis test. In this approach one considers all possible combinations of the number of targets and their locations, thus taking into account the interaction among all possible targets. Note that this approach should perform 2n different tests, which is computationally prohibitive for n > 30. In Section 4.4, we introduce the 1 -norm minimization as an alternative for estimating the location and magnitudes of one or multiple targets.

4.3.4

Constant False Alarm Rate Detectors In (4.8) and all the discussion so far we have assumed that the noise power is known, and can be used to set the threshold γ in (4.5). This assumption is not always met in practice, since the noise plus interference power is varying and not known in advance. Hence, in classical radar detectors a constant false alarm rate (CFAR) processor is employed. In CFAR schemes, the cell under test (CUT) xˆi (which corresponds to the output of the receive filter at time delay τi ), is tested for the presence of a target against a threshold that is derived from an estimated clutter plus noise power. The 2Nw cells (CFAR window) surrounding the CUT are used to derive an estimate of the local background and they are assumed to be target-free. Commonly, 2NG guard cells immediately adjacent to the CUT are excluded from the CFAR window to deal with extended targets and large sidelobes. The advantage of CFAR schemes is that they are able to maintain a CFAR via adaptation of the threshold to a changing environment. The general form of a CFAR test is H1

X ≷ βY,

(4.11)

H0

where the random variable X represents some function (generally envelope or power) of the CUT xˆi , β is a threshold multiplier that controls the false alarm rate, and Y is also a random variable that is obtained from the cells in the CFAR window [xˆi−Nw −NG ,. . ., xˆi−NG −1, xˆi+NG +1,. . ., xˆi+Nw +NG ]. A cell-averaging CFAR (CA-CFAR) detector is a well-known CFAR scheme in which Y is the average of the power (or envelope) of the cells in the CFAR window [23–28]. The CA-CFAR detector is optimal for the detection of a target in the presence of homogeneous i.i.d. Gaussian noise. However, when clutter changes rapidly, interfering targets are present in the CFAR window, or the clutter and noise distributions are not Gaussian, the CA-CFAR detector performance degrades severely. For this reason many alternative CFAR schemes have been devised, such as greatest of (GO), smallest of (SO), trimmed mean (TM), logarithmic (LOG), and order statistic (OS) CFAR processors [26,29–36]. Each CFAR scheme is suitable for a specific clutter and interference scenario. In most CFAR detectors it is assumed the distribution of the noise follows a parametric model, such as Gaussian, with unknown mean and variance, and that the parameters of the model can be estimated from the data. Depending on the characteristics of the expected noise and interference scenario, the most appropriate CFAR scheme can be designed. Clearly, one has to know the relation between the CFAR threshold multiplier and the probability of false alarm, so that β can be adjusted to maintain a constant false alarm probability during the observation time.

110

Anitori and Maleki

4.4

CS Radar Detection

4.4.1

1 −Norm Minimization Compressive sensing (CS) is a novel technique for data acquisition and processing that allows reconstruction of sparse signals from a number of measurements m much smaller than the one dictated by the Shannon–Nyquist sampling theorem. Thus, CS uses the same model as in (4.4), except that now the number of measurements is reduced from n to m, and the sensing matrix A is of size m × n, with m < n. Since we want to estimate the target response x of size n from a set of measurements m, the problem is ill-posed. However, exploiting the prior information about sparsity of the scene, it has been proved [37–39] that a good estimate of x can be obtained from the noisy, subsampled measurements y by solving the 1 -regularized least squares, also known as the LASSO or basis pursuit denoising (BPDN) [40,41], given by xˆ = arg min x

1 y − Ax22 + λ x1, 2

(4.12)

where λ is called the regularization parameter, and controls the trade-off between the sparsity of the solution and the 2 -norm of the residual. Alternatively, one can solve the constrained problem [39,42] min x1 x

s.t. y − Ax2 ≤ ε,

(4.13)

where ε is a threshold proportional to the noise variance. The relation between λ and ε that makes the two problems equivalent is data-dependent, and does not have an explicit form. Standard techniques, such as interior point method, can be used for solving (4.12). However, the computational costs of such methods have encouraged researchers to consider iterative algorithms with inexpensive per-iteration computations. See for example [43,44] and references therein. These iterative algorithms exploit the fact that the optimization problem arg minx 21 u − x22 + λx1 has an explicit solution x = η(u;λ) (|u|− λ)ej u 1(|u| > λ), where η(·;λ) is called the complex soft thresholding function and 1 is the indicator function. The complex soft thresholding function acts (componentwise) on the amplitudes of the input vector u and produces a sparse signal by shrinking to zero all the elements of u whose amplitude is below the threshold, or regularization parameter, λ, thus enforcing sparsity on the solution. The components of u that are above the threshold will be shrinked toward zero by an amount equal to the threshold λ, and their phase is unchanged. The complex soft thresholding function is shown in Figure 4.1. Iterative soft thresholding (IST) uses the following iterations to solve (4.12). Starting with xˆ 0 = 0, at each iteration t the estimate xˆ of the vector x is updated using & ' (4.14) xˆ t+1 = η xˆ t + AH (y − Axˆ t );λ . Therefore, at each iteration, the current residual is projected along the waveforms, and added to the previous solution. In other words, at each iteration of (4.14), the algorithm

Compressed CFAR Techniques

(a) Amplitude

111

(b) Phase

Figure 4.1 Complex soft thresholding function.

moves in the negative gradient direction (of the objective function) −AH (y − Axˆ t ), and then applies the soft thresholding function to enforce sparsity on the solution. At convergence, the estimated sparse signal xˆ will contain many zero components, and a few non–zeros. As λ increases, the solution to (4.14) will become more sparse and, if λ > AH y∞ , the only feasible solution is xˆ = 0. It is clear that 1 -minimization is implicitly performing target detection; we can interpret the nonzero elements of xˆ as detected targets. However, note that if the recovered amplitude at a target location, say xˆ i , is zero, then there is no way to recover the target, not even in any subsequent processing stage. In other words, if the estimated amplitude resulting from 1 -norm minimization is zero, then the target is simply lost, i.e., we have a missed detection. Furthermore, at locations not containing the targets, we expect most of the estimates resulting from 1 -norm minimization to have zero amplitude. However, depending on the SNR and the ratio m/n, some of the noise samples are reconstructed with non–zero amplitude, and therefore, if no further processing is applied, they will produce false targets in the recovered range profile. Increasing the threshold λ will reduce the number of reconstructed noise samples, but it may also result in suppressing the targets. In this sense, the sparse estimate produced by 1 -norm minimization is comparable to the output of a classical detector, with the parameters λ or ε used in the recovery controlling both the detection and false alarm probabilities. Despite this similarity with classical radar detection, we should emphasize that the relation between λ and false alarm probability in each cell is extremely complicated. In particular, especially for m < n, the interference among targets’ sidelobes is not negligible and is not easy to model either due to the nonlinear nature of the 1 -minimization. These complications make the design of CFAR schemes very challenging for CS radar systems. In the rest of this chapter, we will focus on this problem and describe how the asymptotic theory of compressed sensing can help us address this challenge.

4.4.2

Main Challenge: The Design of CS Radar Detectors In this chapter, we focus on the design and analysis of CS radar detectors. Particularly, we are interested in (1) determining a strategy for the optimal detection of targets from CS measurements, and (2) designing an adaptive CFAR detector to achieve a desired pair of detection probability Pd , and false alarm probability Pf a . To design an adaptive

112

Anitori and Maleki

detector in the CS framework, one has to deal with a number of issues that are related mostly to the non-linearity of the 1 -recovery. As explained in the previous sections, classical radar architectures (without CS) use well-established signal processing and detection schemes, such as MF and CFAR processors. Unlike classical radars, in which the relation between the threshold parameter and the false alarm is straightforward, in CS the relation between λ (or ε) and the false alarm rate is data-dependent and complicated. For the design of possible CS radar detectors, it is essential to determine the (statistical) properties of the signal obtained from 1 -norm minimization algorithms. To solve the CS detection problem, we employ some of the recent advances in the asymptotic analysis of compressed sensing algorithms. We will show that a debiasing step can make the problem of CS radar detection similar to that of classical radar detection. We will then combine this debiasing step with classical tools such as CFAR to build practical CS radar detection schemes. A key ingredient for obtaining the debiasing step is the complex approximate message passing (CAMP) algorithm [44], which will be introduced in the next section.

4.5

Complex Approximate Message Passing (CAMP) Algorithm The complex approximate message passing [44] is an iterative algorithm for solving (4.12). CAMP is an extension of the original approximate message passing (AMP) algorithm, first proposed in [45] for real-valued signals, to the case of signals and measurements in the complex domain. The AMP algorithm and its properties have been thoroughly investigated in [46–52]. However, in radar it is most common to work with signals in the complex domain. Therefore, in the remainder of this chapter we will concentrate exclusively on CAMP. As we will see in the next sections, properties of CAMP enable us to achieve the following objectives: • • • •

characterize the distribution of the noise after 1 -norm minimization; establish the relation between the regularization parameter λ of LASSO and the quality, in terms of SNR, of the recovered solution; adaptively set the regularization parameter λ in a way that optimizes the recovery SNR; design a fully adaptive CS radar detector that can be combined with classical CFAR processing.

We first review the iterations of the CAMP algorithm, that is given in Table 4.1. In the algorithm, · denotes the average of a vector, ηI and ηR are the imaginary and real parts of the complex soft thresholding function, ∂ ηI

∂ ηR ∂xR

is the partial derivative of ηR with

respect to the real part of the input, ∂xI is the partial derivative of ηI with respect to the imaginary part of the input, δ = m/n is the compression factor, and maxiter is the (user specified) maximum number of iterations. Furthermore, note that the soft thresholding function is applied with threshold parameter τσt , where σt is the standard deviation of the noise at iteration t and τ is a (fixed) user specified threshold. We will see in the

Compressed CFAR Techniques

113

Table 4.1 Pseudocode for the ideal CAMP algorithm. ©[2013] IEEE. Reprinted, with permission, from [2]. Ideal CAMP Algorithm Input: y, A, τ, x, δ = m/n Initialization xˆ 0 = 0, z0 = y for t = 1 : maxiter x˜ t = A† zt−1 + xˆ t−1 σt = std(x˜ t − x) R I 1 $ ∂ η (x˜ t ;τσ ) + ∂ η (x˜ t ;τσ )% zt = y − Ax˜ t−1 + zt−1 2δ t t ∂xR ∂xI xˆ t = η(x˜ t ;τσt ) end ˜ x,σ ˆ ∗ Output: x,

next subsection how the parameter τ of CAMP relates to the regularization parameter λ in (4.12). We now explain each variable in the CAMP algorithm: (i) (ii)

(iii)

ˆ xˆ t : the estimate of x at iteration t. Under a proper tuning of τ, xˆ t → x(λ) as t → ∞. The relation between τ and λ is given later in (4.17). x˜ t : the non-sparse and “noisy” estimate of x. Define the vector wt = x˜ t − x, which represents the “noise vector” at iteration t of CAMP. The distribution of wt is “close” to a zero-mean Gaussian probability density function. This property of wt will be clarified and proved in Section 4.5.1. σt : the standard deviation of wt . Furthermore, we define σ∗ limt→∞ σt .

Using this terminology, the operations executed in CAMP can be explained as follows. First, CAMP calculates a noisy, non-sparse estimate of the signal x, which is given by x˜ t = A† zt−1 + xˆ t−1 . Then, to make this estimate sparse, the soft thresholding function is applied to x˜ t to obtain the sparse vector xˆ t = η(x˜ t ;τσt ). This algorithm is referred to as ideal CAMP. The word ideal in the algorithm’s name refers to the fact that the sought x is used inside the iterations of CAMP for estimating the noise standard deviation σt . Therefore although this is not a practical algorithm, since it uses in its iterations the vector that is trying to estimate, this assumption is only used here for the clarity of presentation. A practical scheme for estimating σt is described in Section 4.7.

4.5.1

State Evolution: A Framework for the Analysis of CAMP Before we proceed to the design of practical CS radar detectors, we need to answer the following three questions: (i) Under what conditions is the Gaussianity of wt accurate? (ii) Can the performance of CAMP be predicted theoretically? (iii) What is the connection between CAMP and LASSO that was defined in (4.12)? The first question has been carefully studied in [44,48,50,53]. It is proved that under the asymptotic setting n → ∞, while δ = m/n and ρ = k/m are fixed, the Gaussianity heuristic is correct. To clarify this claim consider the following definition from [50].

114

Anitori and Maleki

definition 4.1 For a given (δ,ρ) ∈ [0,1]2 , a sequence of instances {x(n),A(n),n(n)} is called a converging sequence if the following conditions hold: • • •

The empirical distribution of x(n) ∈ Rn converges weakly to a probability measure pX with bounded second moment as n → ∞. The empirical distribution of n ∈ Rm (m = δn) converges weakly to a probability measure pn with bounded second moment as n → ∞. The elements of A(n) ∈ Rm×n are i.i.d. drawn from a Gaussian distribution.

theorem 4.2 [50] Let {x(n),A(n),n(n)} be a converging sequence, and let x˜ t (n) be the estimate provided by the CAMP algorithm. The empirical law of wt (n) = x˜ t (n) − x t (n) converges to a zero-mean Gaussian distribution almost surely as n → ∞. Theorem 4.2 has been proved for Gaussian measurement matrices. However, empirical studies have already confirmed that this theoretical prediction holds for other sensing matrices with i.i.d. elements other than Gaussian [45]. Also, in Section 4.8.1 we study the validity of Theorem 4.2 for partial Fourier matrices, which are of particular interest in radar applications. Using the Gaussianity of the noise vector wt , we can answer the second question raised at the beginning of this subsection, regarding the theoretical performance of the algorithm. In fact, given that the noise has a zero-mean Gaussian distribution, to predict the performance of CAMP we only need to track the standard deviation of the noise σt across the iterations of the algorithm. This is performed through what is called the state evolution (SE). Under the asymptotic setting, the value of the standard deviation at time t + 1 is calculated from σt according to the following equation: 2 σt+1 = (σt ),

where (σt ) = σ2 +

2 ' 1 & E η(X + σt Z;τσt ) − X , δ

(4.15)

(4.16)

Z ∼ CN (0,1), σ2 is the input noise variance and the expectation is with respect to the two independent random variables X ∼ pX and Z, where pX denotes the marginal distribution of x. It has been proved in [44] that the function is a concave function of σt2 , and therefore the iteration (4.15) has at most one stable fixed point, which we refer to as σ∗2 . Also, CAMP converges to this fixed point exponentially fast (linear convergence according to optimization literature). An example of how the function can be calculated in closed form for a given distribution pX of X is provided in appendix A of [2]. Finally, let us answer the third question we raised at the beginning of this subsection regarding the connection between LASSO and CAMP. In [44] it is proved that if τ satisfies R ∂η 1 ∂ ηI λ τσ∗ 1− E (4.17) (X + σ∗ Z;τσ∗ )+ (X + σ∗ Z;τσ∗ ) , 2δ ∂xR ∂xI then CAMP with threshold τ solves the LASSO in (4.12) with parameter λ.

Compressed CFAR Techniques

4.6

115

Target Detection Using CAMP Using the properties of the CAMP algorithm described earlier, in this section we propose two CS target detection schemes and analyze their performance through their receiver operating characteristic (ROC) curves. Let k be the number of targets, i.e., the number of nonzero coefficients in x, and define G as the distribution of the nonzero elements of x. In this section k, G, and σ are assumed to be known. A more realistic case in which none of these parameters are known will be studied in Section 4.7. The two architectures we consider are displayed in Figure 4.2. In Architecture 1, the measurements y are given to a recovery algorithm (say CAMP or LASSO). This algorithm returns a sparse vector xˆ that has a few nonzero values. In this case the nonzero elements of xˆ can be considered as detected targets, where the soft thresholding operation inside the recovery algorithm performs the detection function. Since the estimated xˆ (support and amplitude of the nonzero entries) depends on threshold parameter τ in CAMP or the regularization parameter λ in LASSO, this parameter will also automatically control both the false alarm α (or Pf a ) and detection (Pd ) probabilities. √ proposition 1 Consider the CAMP iteration with threshold τα = − ln α. If A(n),x(n),w(n) is a converging sequence, then n 1 1{xˆ t !=0,x i =0} = α i t→∞ n→∞ n − k

lim lim

(4.18)

i=1

almost surely, where 1{.,.} denotes the indicator function. Also, τα is the only value of τ for which (4.18) holds. Proof

Define z ∼ CN (0,1). According to [53] we have n 1 2 1{xˆ i (n)!=0,xi (n)=0} = Prob(|σ∗ z| > τα σ∗ ) = e−τα . t→∞ n→∞ n − k

lim lim

i=1

Note that this proposition does not provide any information on the relation between τ and the detection probability. This issue will be discussed later. In contrast to Architecture 1, Architecture 2 is inspired by classical radar detection schemes. In fact, classical radar detection usually comprises two sequential stages: in the first stage, called the estimation stage here, a noisy estimate of the signal is computed often through a matched filtering with the goal of maximizing the output SNR (and therefore Pd ).5 In the second stage, the noisy estimate obtained from the first stage is fed to a detection block, whose threshold parameter is set to achieve a predefined Pf a = α. Using a similar approach, in Architecture 2 we propose to first use CAMP to obtain a noisy, non-sparse estimate of the signal x˜ = x + w. Similarly to the matched filter in 5 As explained in Section 4.3.2, the MF is optimal (in terms of SNR) only for the case of a single target in

white Gaussian noise. For the case of multiple targets, the optimality is only satisfied if the matrix A is orthogonal.

Anitori and Maleki

(a) Architecture 1

(b) Architecture 2 Figure 4.2 Block diagrams of the proposed architectures for radar detection in the CS framework.

©[2013] IEEE. Reprinted, with permission, from [2]. 0.6 δ = 0.6

0.55

δ = 0.2

0.5 0.45

σ*

116

0.4 0.35 0.3 0.25 0.5

1

1.5

τ

2

2.5

3

Figure 4.3 Fixed point σ∗ versus threshold τ for ideal CAMP with σ = 0.23, δ = 0.2 (dashed

line), and δ = 0.6 (solid line), ρ = 0.1. The nonzero entries in x all have amplitudes equal to 1 and phase uniformly distributed between −π and π. The sensing matrix A has i.i.d. Gaussian entries. These curves are obtained using the analytical equation derived in appendix A of [2]. ©[2013] IEEE. Reprinted, with permission, from [2].

classical radar, in the first stage we aim to maximize the output SNR. This can be done by setting the CAMP threshold τ to a value that minimizes σ∗ . Figure 4.3 exhibits the dependence of σ∗ on τ for two distinct values of δ. As it is clear from the figure, there is a value of τ, say τo , for which σ∗ is minimized. In Section 4.7, we will propose a simple method for calculating τo . √ In the second stage, a detection bock with fixed threshold κ = σ∗ − ln(α) is applied to the noisy estimate x˜ to control the false alarm rate. From the Gaussianity of w, in the asymptotic setting this choice of κ results in the false alarm probability α as derived in Proposition 1. With respect to Architecture 1, Architecture 2 has two properties that are very useful for practical radar applications. Namely: 1.

All the parameters can be optimized and estimated adaptively and efficiently, even without prior knowledge of k, G, and σ. This will be clarified later.

Compressed CFAR Techniques

117

1 1

0.8

0.9998

0.6 0.4

0.6

P

d

0.2

0.4 Arch. 2, Theoretical Arch. 1, Theoretical MC

0.2 0 −6 10

−4

−2

10

10

0

10

Pfa

Figure 4.4 ROC curves for Architectures 1 and 2 with δ = 0.6, ρ = 0.1 and σ 2 = 0.05. All the

nonzero coefficients of x have amplitude equal to 1 and phase uniformly distributed between −π and π. The solid (Architecture 2) and dashed (Architecture 1) lines are obtained from the theoretical predictions using the SE equation. The dots are obtained by Monte Carlo (MC) simulations using Ideal CAMP. The sensing matrix for MC simulations is i.i.d. Gaussian. ©[2013] IEEE. Reprinted, with permission, from [2].

2.

From a detection perspective, Architecture 2 outperforms Architecture 1. Suppose that the Gaussianity of wt holds. Let τo be the optimal value of τ that leads to the minimum σ∗ . Then, the following theorem proves that the detection performance of Architecture 2 is better than that of Architecture 1. theorem 4.3 Set the probability of false alarm to α for both Architecture 1 that uses τα and Architecture 2 that uses τo in CAMP. If Pd,1 and Pd,2 are the detection probabilities of the two schemes, then Pd,1 ≤ Pd,2 . Furthermore, the equality is satisfied at only one specific value α = e−τo . 2

For a more general version of this result the reader may refer to [54]. Theorem 4.3 is proved in [2]. Figure 4.4 exhibits ROC curves for Architectures 1 and 2. The solid and dashed lines are obtained from the analytical equations derived in appendix A of [2]. The theoretical ROC curves are confirmed by Monte Carlo simulations (dots). In the simulations, we run the ideal CAMP algorithm given in Table 4.1 for several values of Pf a ranging from 10−1 to 10−5 . One interesting phenomenon that is observed in Figure 4.4 is that for Architecture 1, in the region around Pf a = 0.4 (the zoomed area), the probability of detection, Pd,1 , decreases as Pf a increases. This is a counterintuitive behavior, but can be explained in the following way. Recall that in Architecture 1 the CAMP threshold τα varies with Pf a . Therefore, σ∗ (and hence the CAMP reconstruction SNR) also changes with Pf a , and is not constant along the ROC curve. This explains why Pd,1 reaches its maximum at around Pf a = 0.4 and then decreases again as the Pf a goes to 1 (the SNR is maximized at τ = τo and then it descreases again). Instead, in Architecture 2, σ∗ is fixed to its minimum along the ROC curve (SNR is

118

Anitori and Maleki

constant), and therefore Pd,2 increases with increasing FAP. From this figure it can also be observed that, as predicted by Theorem 4.3, Pd,1 = Pd,2 happens at only one value of FAP (α = 0.22).

4.7

Adaptive CAMP Algorithm In the ideal CAMP algorithm, we made a few assumptions that are not met in real applications. (i) (ii)

x is assumed to be known. This is required for the calculation of the standard deviation of the noise. ρ, σ, and G are assumed to be known for calculating the theoretical fixed point solution.

In this section, we show how these assumptions can be removed. The main objective here is to propose a practical, fully adaptive scheme that does not require any prior information about the signal and can adapt to the changes in the signal and the noise level. More specifically, the three main issues that we would like to address are: (i) how to obtain a good estimate of σt without knowing x; (ii) how to obtain a good estimate of τo for Architecture 2, efficiently and accurately without using the SE (which depends on the unknown parameters ρ, σ, and G); (iii) how to replace κ in Architecture 2 with an adaptive threshold that is able to maintain CFAR property in the multiple targets scenario. Several different schemes can be used to answer the first question. For the moment, suppose that x = 0. If this assumption is true, then according to Theorem 4.2, x˜ t is a Gaussian random vector with mean zero and variance σt . Hence, the following expression gives a good estimate of σt . 1 / (4.19) median(|x˜ t |). σt = ln 2 However, in the presence of targets, i.e., x != 0, this is a biased estimator. More specifi.i.d.

ically, for large values of n, we have x˜ t = x + wt , where xi ∼ G(x) + (1 − )δ(x) with = δρ 1 and δ(x) denotes a point mass at zero, and the elements of wt look like a complex Gaussian random variable with mean zero and standard deviation σt . ˜ The goal is to estimate the median u∗ of |wt |. However, since we only have access to x, ˆ = 12 . Note that we estimate the median of |wt | as the uˆ that satisfies Prob(|x˜it | > u) since the median is robust to outliers (i.e., multiple targets), we still expect (4.19) to offer a good estimate of σt . The following proposition confirms this claim. proposition 2 [2] The error of the estimated median is bounded from above by | ln(1 − )| |uˆ − u∗ | ≤ . √ σ∗ 2 ln 2

(4.20)

Note that when the scene is very sparse, i.e., is small, | ln(1 − )| ≈ and, hence, the error is proportional to the sparsity level. In such cases, the proposed estimator

Compressed CFAR Techniques

119

Table 4.2 CAMP-based algorithms. Algorithm

Inputs

Outputs

Ideal CAMP Median CAMP Adaptive CAMP

A,y,x,σ,τ A,y,τ A,y

ˆ x,σ ˜ ∗ x, ˆ x, ˜ σˆ ∗ x, ˆ x, ˜ σˆ ∗, τˆ o x,

provides a good estimate of σt . We should also emphasize that, when the number of targets is large and the median estimator is not reliable, we can use more recent and better estimates of σt that are proposed in [52]. The algorithm that uses the estimate in (4.19) instead of σt will be referred to as CAMP or median CAMP in the rest of this chapeter. The second question we raised at the start of this section was concerned with estimating the optimal threshold τo in Architecture 2. Suppose that we know or can estimate τmax such that τo < τmax . We will discuss how this parameter can be set later. Define a sequence of thresholds τ = {τ }L =1 such that τ1 = τmax and τ = τ−1 − δ τ , and δ τ is a user-defined parameter that controls the step size. Starting from τmax , at each new iteration , CAMP is initialized with xˆ 0 = xˆ −1 and z0 = z−1 . Using the solution of CAMP at the previous iteration − 1 as an initial value for the current iteration, CAMP needs only a few iterations to converge to the solution,6 and therefore the entire process ˆ = [xˆ 1, xˆ 2,. . ., xˆ L ] of is very fast. After L iterations, we have a matrix of solutions X size n × L, where each column contains the CAMP solution for a given τ . Also, we ˆ o is chosen as the one that have L estimates { σˆ ∗ }L =1 . The optimum estimated threshold τ minimizes the estimated CAMP output noise variance σˆ ∗2 . At the first iteration ( = 1 and t = 1) τmax can be set as τmax = AH y∞ / σˆ 0 . In fact, if the CAMP algorithm is initialized with xˆ 0 = 0 and z0 = y, then x˜ = AH y, where σˆ 0 is an estimate of the standard deviation of the noise. In this case, any value of τ larger than τmax will lead to the same estimate xˆ 1 = 0. So far we have resolved two of the issues we raised at the beginning of this section, i.e., estimating σˆ t and τˆ o . We will refer to the resulting algorithm as adaptive CAMP, since both the noise variance σˆ t and the threshold τˆ o are adaptively estimated. To clarify the differences between ideal, median, and adaptive CAMP, Table 4.2 shows the input and output variables for each of the three algorithms. Please recall that ideal CAMP is not an algorithm that can be used in practice, as it requires the knowledge of the true vector x.

4.7.1

Adaptive CAMP CFAR Radar Detector As discussed previously, if the noise variance σ2 is known, we can use the asymptotic results to find the fixed threshold κ that achieves the desired FAP; see e.g., appendix B in [1]. In practice, however, the noise statistics are not known in advance. In classical 6 Typically, 10 iterations are sufficent to converge to a solution.

120

Anitori and Maleki

Figure 4.5 Block diagram of the adaptive CAMP CFAR detector. ©[2013] IEEE. Reprinted, with

permission, from [2].

radar detectors, often a CFAR processor estimates the unknown background plus interference level. In Architecture 2, since the signal x˜ is modeled as the sum of targets plus Gaussian noise (just as after a classical MF), this estimate can be directly input to a conventional CFAR processor. A block diagram of the Adaptive CAMP CFAR detector based on Architecture 2 is shown in Figure 4.5. Replacing the fixed threshold detector with a CFAR detector in Architecture 2 provides similar results as in classical CFAR without CS. All we need to determine is the input/output SNR relations of CAMP, so that we can use the output SNR in the detector equation for the prediction of Pd . A method for estimating the output SNR of CAMP for a given problem is provided in [2]. Please note that, since the sparse estimate xˆ contains many zeros, this vector could not be used in a CFAR processor.

4.8

Simulation Results In this section we investigate the performance of the proposed CS architectures using Monte Carlo (MC) simulations and compare it with the theoretical results derived using the SE. We compute ROC curves for the cases of fixed threshold and CA-CFAR detectors. The proposed CS detection schemes are analyzed using both Gaussian sensing matrices, for which SE applies, and partial Fourier matrices,7 which are of particular interest in radar applications [55–57]. In the remainder of this chapter we will use as reference SNR the output SNR of an (ideal) MF,8 so that the SNR given in the results will be independent of n or m. We define the SNR at the input (SNRin ) and output (SNR) of the MF and CAMP Architecture 2 respectively as SNRin,MF =

|xi |2 , nσ2

SNRin,CS =

|xi |2 , mσ2

SNRMF =

|xi |2 , σ2

SNRCS =

|xi |2 . σ∗2

(4.21)

where |xi |2 is the received power from a target at bin i. Given the output MF SNR, the input SNR of both MF and CAMP, which depends through n and m on the specific problem being investigated, can be easily derived using (4.21). 7 Recall that an m × n partial Fourier matrix can be obtained from an n × n discrete Fourier transform

matrix by preserving only a random subset m of the original n matrix rows. 8 Recall that, in a multiple target scenario, the MF SNR is optimum and independent of the number of

targets as long as each target is exactly on a grid point and the matrix A is orthogonal. We assume these (ideal) conditions are satisfied when computing the MF SNR.

Compressed CFAR Techniques

δ = 0.7, ρ = 0.2

δ = 0.2, ρ = 0.1 6

Density

4

δ = 0.9, ρ = 0.5

121

δ = 0.2, ρ = 0.7

2

1

1

0.5

4 2 2 0 −0.4 −0.2

0 0.2 Data

0.4

0

−0.2

0 Data

0.2

0

−0.5

0 Data

0.5

0 −2

0 Data

2

(a) Histogram of the real part of w. δ = 0.2, ρ = 0.1

δ = 0.7, ρ = 0.2 6

Density

4

δ = 0.9, ρ = 0.5

δ = 0.2, ρ = 0.7

2

1

1

0.5

4 2 2 0

−0.2

0 Data

0.2

0

−0.2

0 Data

0.2

0

−0.5

0 Data

0.5

0

−1

0 Data

1

(b) Histogram of the imaginary part of w. Figure 4.6 Histograms (bars) of (a) the real and (b) imaginary parts of the noise signal w for different combinations (δ,ρ) using CAMP with threshold τ = 1.8, σ = 0.1 and n = 4000. The solid line shows a Gaussian distribution fitted to the histograms. The sensing matrix is partial Fourier. ©[2013] IEEE. Reprinted, with permission, from [2].

4.8.1

Gaussianity of w Using Partial Fourier Matrices The Gaussianity of the reconstructed noise vector wt for a partial Fourier sensing matrix was not demonstrated theoretically in the previous sections. Therefore, we resorted to Monte Carlo simulations to investigate it using two methodologies. First we studied the empirical distribution of w at convergence for different combinations of δ and ρ. The histograms were obtained using CAMP with a fixed threshold τ (not necessarily optimal) and for a fixed value of σ = 0.1. A few examples of such histograms are shown in Figure 4.6. We further investigated the Gaussianity of w using both quantile plots and the Kolmogorov–Smirnov (KS) test [58], for different values and combinations of n, δ, and ρ. The results of these simulations are reported in [1]. All our simulations confirmed that the Gaussianity of the noise vector is preserved for partial Fourier matrices as well.

4.8.2

Accuracy of State Evolution The accuracy of the SE was investigated by comparing the theoretical results obtained from (4.16) with simulation results obtained using the ideal CAMP algorithm. Additionally, we are interested in investigating how the behavior of σ∗ changes for the case of a partial Fourier sensing matrix as compared to the case of Gaussian sensing matrix, for which the theoretical results from SE apply. Figure 4.7 compares σ∗ obtained from ideal CAMP for the case of complex Gaussian and partial Fourier sensing matrices with the theoretical one from the SE. From this figure we observe that:

122

Anitori and Maleki

δ = 0.05

0.3

0.29

0.28

0.28 *

σ

σ

*

0.29

0.27 0.26

0.27 0.26

0.25

0.25

2

τ

2.5

3

1.5

δ = 0.2

0.3

2

0.29

0.28

0.28

2.5

3

*

0.29

τ δ = 0.5

0.3

σ

*

σ

δ = 0.1 Ideal CAMP, Gaussian Ideal CAMP, Fourier Theoretical SE

0.3

0.27

0.27

0.26

0.26

0.25

0.25 1.5

2

τ

2.5

3

1

1.5

2

τ

2.5

3

Figure 4.7 σ∗ versus τ using ideal CAMP for both complex Gaussian and partial Fourier sensing

matrices. The empirical curves are obtained for several values of δ by averaging over 100 MC samples. σ 2 = 0.05,ρ = 0.05, n = 4000. The theoretical SE curve shows the analytical σ∗ . ©[2013] IEEE. Reprinted, with permission, from [2].

(i) (ii) (iii)

(iv)

4.8.3

SE correctly predicts the performance of CAMP for the Gaussian sensing matrix. SE does not predict the performance of CAMP for partial Fourier matrices. However, for τ = τo , the value of σ∗ for Fourier and Gaussian matrices is very similar. As δ → 0 the predictions of SE become more accurate for the partial Fourier matrix. However, as δ → 1, i.e., as the number of measurements increases, the columns of the partial Fourier matrix become deterministic and orthogonal, and hence the true behavior deviates from the SE, which is derived assuming matrices with i.i.d. entries. For the partial Fourier matrix, the optimal threshold τo seems to be almost the same for different values of δ. Interestingly, although the curves of σ∗ are different for different δ’s, for a fixed δ and for τ > τo the variation of the output variance is much smaller in the partial Fourier case than in the Gaussian case. This behavior will have an impact on the difference in performance between Architecture 1 and Architecture 2 for the case of partial Fourier and Gaussian matrices, as the SNR of Architecture 1 varies less along the ROC curves and it is closer to the optimal SNR for the partial Fourier case.

Effects of the Median Estimator in CAMP In this section, we investigate the performance of the proposed algorithms and architectures when the true σt is replaced with the estimated σˆ t from (4.19) for the case x != 0. Figure 4.8 shows the estimated output noise standard deviation for both ideal

Compressed CFAR Techniques

δ = 0.05

0.29

0.29

0.28

0.28

0.27

0.27 0.26

0.26

0.25

0.25 2

2.5

τ

1.5

3

δ = 0.2

0.3

2

0.29

0.29

0.28

0.28

2.5

τ

3

δ = 0.5

0.3

σ

*

*

σ

δ = 0.1 Median CAMP, Gaussian Median CAMP, Fourier Ideal CAMP, Gaussian Ideal CAMP, Fourier

0.3

σ*

σ*

0.3

123

0.27 0.26

0.27 0.26

0.25

0.25 1.5

2

2.5

τ

3

1

1.5

2

2.5

τ

3

Figure 4.8 Output noise standard deviation versus τ for both ideal (σ∗ ) and median (/ σ∗ ) CAMP

for complex Gaussian and partial Fourier sensing matrices. The curves are obtained for several values of δ by averaging over 100 MC realizations. σ2 = 0.05,ρ = 0.05, n = 4000. ©[2013] IEEE. Reprinted, with permission, from [2]. 1

1

0.9

0.95

0.8 0.9 Pd

P

d

0.7 0.6 0.5

0.85 0.8

0.4 Median CAMP Ideal CAMP

0.3 10

−4

10

−3

Pfa

10

−2

(a) Gaussian sensing matrix

10

−1

0.75

10

Median CAMP Ideal CAMP −4

−3

10

10

−2

10−1

Pfa

(b) Partial Fourier sensing matrix

Figure 4.9 ROC curves for Architecture 1 using both ideal and median CAMP. n = 1000, δ = 0.6, ρ = 0.1, a 2 = 1, and σ 2 = 0.05 (corresponding to a MF SNR = 13 dB).

©[2013] IEEE. Reprinted, with permission, from [2].

and median CAMP. As expected, we observe that σˆ ∗ deviates from the ideal CAMP case because of the bias introduced by the estimator. Furthermore, the deviation diminishes as = δρ decreases, as predicted by the upper bound provided in (4.20). Overestimating / σ∗ in Architecture 1 results in a loss of detection performance. This is shown in Figure 4.9, where we plot the ROC for Architecture 1 using both ideal and median CAMP. The loss of detection performance is explained by the fact that the soft thresholding function in CAMP uses the parameter τα σˆ ∗ , which, for a fixed

124

Anitori and Maleki

τα , increases with σˆ ∗ . If the overall threshold increases, then the detection probability will decrease. In Figure 4.9, we also observe that Architecture 1 performs better when the sensing matrix is the partial Fourier. This has to do with the behavior of the noise variance σ∗2 , and therefore the SNR, versus the threshold, that changes along the ROC curve. As can be seen from Figure 4.8, in the partial Fourier case the variance curve becomes flatter than the curve obtained in the Gaussian case as δ increases. As a result, when the false-alarm probability decreases, in the partial Fourier case the SNR along the ROC curves for Architecture 1 deviates much less from the optimum SNR that is achieved for τ = τo .

4.8.4

Performance of a Fully Adaptive CAMP CFAR Detector In this section, we investigate the performance of the fully adaptive CAMP CFAR detector using ROC curves, and compare its performance to the compressive matched filter (CMF) [56,59], which is the filter matched to the subsampled waveform. In Architecture 2, the CA-CFAR processor is preceded by a square law (SL) detector (see Figure 4.5) and has a CFAR window of length 20 with 4 guard cells. In the Monte Carlo simulations, we consider the case of a signal consisting of multiple targets, and the detection probability is estimated for each target separately. Since all targets are generated having equal amplitude, the plots show the results for only one of the targets. Please note that, although most commonly in the radar literature the ROC plots are shown for the case of a single target, in the CS case, having a single target represents an extremely sparse and favorable scenario. Instead, in the case of multiple targets we observe both the effects of reconstruction and of the CFAR processor. However, even if there are multiple targets, the results of the CFAR processor will be independent on the number of actual targets but will depend exclusively on the CAMP output SNR as long as the targets are not in the CFAR window of one another. Therefore, in the MC simulations we never generate targets at locations that fall within the CFAR window of another target. It is well known that if there are interfering targets in the CFAR window of the CUT, the estimated noise variance will raise, leading to an increased threshold that can potentially result in missing the target yet to be detected. This is a classical problem in CFAR processing, and in literature several alternative CFAR schemes have been proposed to deal with this scenario, such as the ordered statistic (OS)-CFAR [32]. This and other types of CFAR detectors can be similarly used in Architeture 2 instead of the CA-CFAR processor. In Figure 4.10, we show the ROC curves for Architecture 2 for the cases of: (a) ideal CAMP with an ideal (fixed threshold) detector, (b) ideal CAMP followed by the CA-CFAR detector, (c) adaptive CAMP with an ideal (fixed threshold) detector, and (d) fully adaptive scheme consisting of adaptive CAMP followed by a CA-CFAR processor. In the same figure the theoretical curve of a CA-CFAR processor with the same window length is also shown. The SNR used in the analytical CA-CFAR Pd equation is set to 11.55 dB for the Gaussian sensing matrix and 11.9 dB for the partial Fourier sensing matrix9 . For the Gaussian sensing matrix the optimal threshold for Architecture 2 (using ideal CAMP) is computed analytically using the SE. For the partial Fourier sensing 9 The SNR has been estimated during simulations.

Compressed CFAR Techniques

125

1 0.95

0.8

0.9

0.7

0.85

0.6 Pd

Pd

0.9

0.5 0.4

Ideal CAMP CA-CFAR Adaptive CAMP FT Adaptive CAMP CA-CFAR Ideal CAMP FT Theoretical CA-CFAR CMF FT CMF CA-CFAR

0.3 0.2 0.1 –4 10

–3

10

–2

10 Pfa

(a) Gaussian sensing matrix

–1

10

0.8 0.75 0.7

Ideal CAMP CA-CFAR Adaptive CAMP FT Ideal CAMP FT Adaptive CAMP CA-CFAR Theoretical CA-CFAR CMF FT CMF CA-CFAR

0.65 0.6 10 –3

Pfa

10 –2

10 –1

(b) Partial Fourier sensing matrix

Figure 4.10 ROC curves for Architecture 2 with different combinations of CAMP algorithms and detection schemes. Here n = 1000, δ = 0.6, ρ = 0.1, a 2 = 1, and σ 2 = 0.05 (corresponding to

a MF SNR = 13 dB). FT denotes the use of an (ideal) fixed threshold detector. ©[2013] IEEE. Reprinted, with permission, from [2].

matrix, the threshold in Ideal CAMP is derived from a plot like the ones shown in Figure 4.7 for the case δ = 0.6, and is equal to τo = 1.85. As expected, adaptivity imposes extra losses in performance. One loss is due to the use of adaptive instead of ideal CAMP, which is caused by the error in the estimated τo . Another loss is caused by the CFAR processor and its estimate of the noise standard deviation, and this is the well-known CFAR loss. From Figure 4.10 the following observations can be made. First, adaptive CAMP introduces almost no loss in the detection performance of Architecture 2. This can be seen by observing that the perfomance of adaptive and ideal CAMP combined with the same detector (either fixed threshold or CA-CFAR) are almost indistinguishable. The reason for this is that, although σˆ ∗ estimated in adaptive CAMP is biased, the value of τˆ o at which the minimum σˆ ∗ occurs is very close to the true optimal τo (see Figure 4.8) computed in ideal CAMP, resulting in an almost optimal SNR even in the adaptive case. The main loss instead is introduced by the adaptive CFAR detector, as can be seen by comparing the curves obtained using the fixed threshold detector against the ones obtained using the CA-CFAR processor, both with ideal or adaptive CAMP. This loss is however not introduced by CAMP, but it is the well-known CFAR loss [26]. Furthermore, if one compares the curve of adaptive CAMP plus CFAR with the theoretical curve of a CA-CFAR processor (without CS) computed analytically using the same parameters (SNR, CFAR window length, and number of guard cells), it can be noted that the performance of the CA-CFAR detector appears to be independent of the fact that the input to the detector is obtained by running CAMP instead of a conventional MF. Second, we observe that Architecture 2 significantly outperforms the CMF. This should be expected since the output of the MF using the subsampled waveform will result in severe target sidelobes (interference), which in turn leads to both an increase of the false alarm rate and possibly the masking of weaker targets, there by reducing their detection probability.

Anitori and Maleki

4

4

3.5

3.5

3

3 P

2

10

2.5

−log

fa

fa

P

10

−log

126

1.5

2.5 2 1.5

Arch.1 Arch.2 + CA−CFAR

1 1

1.5

2 2.5 −log α 10

(a)

3

3.5

Arch.1 Arch.2 + CA−CFAR

1 4

1

1.5

2 2.5 −log10 α

3

3.5

4

(b)

Figure 4.11 Estimated FAP versus design FAP α for Architectures 1 and 2. n = 1000, and

δ = 0.6. (a) Complex Gaussian sensing matrix; (b) Partial Fourier sensing matrix.

By comparing Figures 4.9 and 4.10 it can be seen that, in the fixed threshold case, Architecture 2 always outperforms Architecture 1, as predicted by Theorem 4.3. Also in the adaptive case, Architecture 2 followed by a CA-CFAR processor outperforms Architecture 1 using median-based CAMP. However, the difference between the two schemes can vary significantly with the system parameters (δ,ρ,σ), sensing matrix type, and CFAR window length. For instance, for the value of δ used in these figures, we observe that Architecture 1 performs much better in the Fourier case than in the Gaussian sensing matrix case. Also, the loss in detection performance is significantly reduced compared to the adaptive detector. This again depends on the behavior of σ∗2 versus τ. In general, to predict how the two architectures will perform one should observe the behavior of the output noise variance as a function of the threshold τ. If the variation of σ∗ versus τ is small, in the ideal detector case the ROC curves of the two architectures will be almost identical, with Architecture 2 always slightly better. In Figure 4.11 the estimated FAP is shown for both Architecture 1 (which is nonadaptive, and uses median-based CAMP) and Architecture 2, which uses adaptive CAMP in combination with a CA-CFAR processor. The desired FAP α, on the x-axis, is used to obtain the threshold multiplier β for the CFAR processor in Architecture 2 and to derive the value of the fixed CAMP threshold τα in Architecture 1. As expected, Figure 4.11 shows that, in homogeneous Gaussian noise, the proposed architectures posses the CFAR property. In simulating the FAP for Figure 4.11, according to hypothesis H0 (target absent) we generated a measurement vector y with standard Gaussian distribution and x = 0. However, in practical scenarios, where the noise level may change across range, or in the presence of one or multiple targets located anywhere in the signal x, Architecture 1 can not achieve CFAR. This is because the noise estimate computed by median CAMP is not performed locally, as in a CFAR processor, but is based on the whole received signal using the median, which is a biased estimator if x != 0. In Architecture 2 instead, separating the reconstruction from the detection stage gives more flexibility, and, e.g., in a multiple interfering targets scenario, the CA processor can be replaced with a more robust CFAR scheme, such as OS-CFAR.

Compressed CFAR Techniques

4.9

127

Experimental Results To further validate our theorethical findings, we performed a set of experimental measurements using the software defined LabRadOr experimental radar system at Fraunhofer FHR, Germany. A set of two CS digital stepped frequency (SF) waveforms were designed to perform the measurements, and five stationary corner reflectors were used as targets. A description of the radar systems and the transmitted waveform is provided in the next two subsections, together with the results.

4.9.1

Radar System LabRadOr is a software-defined pulsed radar, with maximum transmit power of 32 dBm (and an attenuator of 1 dB step size) using separate transmit (TX) and receive (RX) reflector antennas, with gain of 31.6 dB each. The digital waveform designed by the user is transferred from the control computer to an FPGA, where a digital-to-analog converter (DAC) converts the digital data to an analog signal. The analog waveform is then transferred to an RF front-end for up conversion to the carrier frequency fc = 8.9 GHz. At the receiver, after down conversion, the signal is returned to the FPGA, where an analog-to-digital converter (ADC) samples the received analog signal at 2 GHz sampling rate. The samples are then transferred to the control unit where they are stored for further processing. Because of internal FPGA limitations, the maximum number of samples per sweep that can be recorded is 1,024, thus limiting the receiver record window length to 512ns. The start time of the record window can be set by the user within the pulse repetition interval (PRI), which is fixed and equal to 10ms. Since our objective is to perform SF measurements, but the maximum TX pulse length is limited to 512ns, we transmit one frequency per pulse, and later combine all the m frequencies (thus m pulses) to obtain a single SF measurement. Hence, we assume that the scene is stationary at least to within m PRI seconds. Also, since both the corners and the radar are fixed, the target amplitudes can be modeled as Swerling Case 0 [60].10

4.9.2

Transmitted Waveform A set of stepped frequency waveforms is used in the experiments. The TX signal consists of a number of discrete frequencies fm , covering the band from 100 to 900 MHz, with a range resolution of δR = 18.75 cm. In the Nyquist case (that represents unambiguous mapping of ranges to phases over the whole bandwidth) we used n = 200 frequencies over a bandwidth of 800 MHz, and each frequency is transmitted for 0.512 μs, thus implying a bandwidth of Bf = 1.95 MHz. Sequential frequencies are separated by f = 4 MHz, resulting in an unambiguous range of Run = 37.5 m. To obtain CS waveforms, a subset of m frequencies is chosen uniformly at random from the Nyquist waveform (m < n). We considered the cases of m = 50 and 100, 10 Since we are interested in the detection problem from a single range measurements, we kept the targets

fixed and did not perform any Doppler measurements.

Anitori and Maleki

Frequency [MHz]

128

1000

1000

800

800

600

600

400

400

200

200

0 0

20

40

60

80

100

0 0

5

10

15

Time [us]

Time [us]

(a)

(b)

20

25

Figure 4.12 Spectrogram of TX waveform. (a) Nyquist waveform with n = 200. (b) CS waveform

with m = 50 and δ = 0.25.

that correspond, for n = 200 and k = 5, to δ = 0.5,0.25 and ρ = 0.05,0.1. The spectrograms of the Nyquist waveform and of one of the CS TX waveforms for the case m = 50 (δ = 0.25) are shown in Figure 4.12. As shown in [55,61], for stepped frequency waveforms the sensing matrix A is a Fourier matrix (partial, in the CS case). Please note that during the measurements, the same total power is transmitted in each burst, irrespective of the number of transmitted frequencies. This was achieved by adjusting accordingly the per frequency transmitted power such that when the number of measurements is reduced by a factor δ, the power per transmitted frequency PT is 1/δ times higher than in the Nyquist waveform case, so that the total transmitted energy (PT × m/Bf ) stays the same for all waveforms. This was done to ensure that the detection performance behavior for different undersampling factors was not caused by a reduction of transmitted power, but is solely due to the quality of the recovery for different subsampling regimes.

4.9.3

Performance of Adaptive CAMP with CFAR Detector In this section, we investigate the performance of the proposed CAMP based detection schemes using the measured data. Figure 4.13 shows the signals reconstructed using the proposed CAMP-based architectures in addition to MF. For Architecture 1 with median CAMP, τα was set using Pf a = 10−4 . For Architecture 2, the CAMP threshold τˆ o is adaptively estimated at each measurement. The 5 corner reflectors are indicated as T1, T2, T3, T4, and T5 in the figure. Note that since targets are not exactly on grid points, there is a leakage of target power into several range bins, both for MF as well as for CS.11 This is the so-called straddling loss, which is always present in real measurements. In Figure 4.14 we plot the estimated output noise standard deviation (σˆ ∗ ) for the experimental data, where the range profiles were reconstructed using median CAMP for different values of the threshold τ. The curve is obtained by averaging over all 300 11 In -norm minimization, as in classical MF, a clustering and interpolation step could be included after 1

detection to improve range estimation and reduce straddeling losses, see, e.g., [62].

Compressed CFAR Techniques

Amplitude [au]

104

T5

T1

CAMP Arch. 1 CAMP Arch. 2 MF

129

T2 T3 T4

103

102

101 0

5

10

15

20 Range [m]

25

30

35

Figure 4.13 Estimated range profile using: CAMP Architectures 1 and 2, and MF. For the MF n = 200; for CS m = 100 and δ = 0.5. The y-axis is in log scale. ©[2013] IEEE. Reprinted, with permission, from [2].

Estimated σ

*

1600

δ = 0.5 δ = 0.25

1400

1200

1000

800 1.5

2

τ

2.5

Figure 4.14 Estimated σ∗ versus τ using median CAMP.

measurements. Note that, for the same input noise variance, for δ = 0.25 the output noise power is always higher than for δ = 0.5, also implying that for the same target received power the SNR decreases with δ, as predicted by the SE. We also see that the behavior of the estimated output noise standard deviation resembles the one shown is Section 4.8.3 for the simulated data. For obtaining the ROC and FAP curves, since the SNR is very high for all targets (in all cases above 20 dB), to evaluate the performance of the detectors at medium SNR values, we added white Gaussian noise to the raw frequency data samples, resulting in an equivalent MF output SNR of 17.2, 16.6, 14, 10.2 and 26 dB respectively, from the closest corner to the farthest one. For estimating the ROC cuves, we adopted the following procedure. The FAP is estimated by averaging the detections on all other cells, excluding the true targets’ cells

Anitori and Maleki

1

1

0.9

0.9 T1

0.8

T1

0.8 T3

T3

0.7

0.7

0.6

T4

d

0.6

P

d

P

130

0.5 0.4

0.5 0.4

0.3

0.3

T4

0.2

0.2

Arch. 2 CA-CFAR Arch. 2 FT Arch. 1

0.1 0 10

-3

10

-2

10

Pfa

(a) δ = 0.25

-1

10

0.1 0

10-3

10-2

10-1

100

Pf

(b) δ = 0.5

Figure 4.15 ROC curves for Architecture 1 (a) and Architecture 2 (b) using both fixed threshold

(FT) and CA-CFAR detectors. The curves correspond to targets T1, T3, and T4 in Figure 4.13. Adapted from [2], ©[2013] IEEE.

plus four guard cells (because of straddling). The detections instead are obtained for each target separatly by counting the detections at the location of the corresponding target highest peak. Figure 4.15 shows the ROC curves for three of the five targets (T1, T3, and T4), having different SNRs, for δ = 0.5 and 0.25. The ROC curves are estimated for each target using both Architecture 1 and Architecture 2. In Architecture 2, CAMP recovery is followed by either a fixed threshold (FT) detector or a CA-CFAR processor.12 Please note that, since in Architecture 1 the estimated sparse signal xˆ can never contain more than m out of n nonzero coefficients, FAPs higher than δ cannot be estimated. Again, we emphasize that in CAMP the reconstruction SNR is the ratio of the target power to the system plus reconstruction noise power (σ∗2 ). Therefore, while the total transmitted power remains fixed for δ = 0.5 and 0.25, the reconstruction SNR for each target depends on the architecture used in addition to the compression factor δ. As observed in the simulated results, when the number of measurements is reduced the reconstruction noise variance σ∗ increases, and CAMP SNR decreases. Since a loss in SNR translates directly into a loss in detection probability, for a given FAP, CAMP will perform better for larger δ. In agreement with our theoretical findings, from Figure 4.15 we observe that the detection probability of Architecture 2 with the fixed threshold detector is always higher than the one of Architecture 1. Additionally, Architecture 2 followed by a CA-CFAR processor exhibits a detection performance loss compared to the fixed threshold case. This is again the CA-CFAR loss. For the same figure we observe that the two proposed architectures perform very similarly at very low or very high SNRs. However, at low FAPs and high Pd , which is the most relevant case in practical situations, Architecture 2 always outperforms Architecture 1. Also note that Architecture 1 is designed in a way that it is very similar to an OS-CFAR detector, in which the CFAR window consists of the whole signal, i.e., 12 For the CA-CFAR processor we used the same parameters as in the simulations, i.e., four guard cells and

a CFAR window of length 20.

3

3

2.5

2.5

2

2

−log10Pfa

−log10Pfa

Compressed CFAR Techniques

1.5 1

131

1.5 1

δ = 0.5 δ = 0.25

0.5 0 0

1

−log10α

2

(a) Fixed threshold detector

δ = 0.5 δ = 0.25

0.5 3

0 0

1

−log10α

2

3

(b) CA-CFAR detector

Figure 4.16 Estimated FAP versus design FAP α for CAMP Architecture 2 using fixed threshold detector (a), and CA-CFAR detector (b). δ = 0.5 (dashed line) and δ = 0.25 (solid line).

2Nw = m, and it also includes the CUT. Because in these measurements is very small, the bias introduced by the median estimate used in Architecture 1 is also very small and therefore the overall noise estimate of median CAMP is better than the noise estimate of the CA-CFAR detector that is using only 20 bins. However, a serious disadvantage of Architecture 1 is that, since the entire signal is used in the noise estimation and the threshold τα is fixed, this scheme can inherently not adapt to local variation of noise level. This makes Architecture 1 unsuitable for many radar applications. In Architecture 2 instead, one has the flexibility to choose both the most appropriate CFAR processor and CFAR window length, depending on the specific scenario. This of course will also depend on the target distribution in range. In fact, if for example multiple targets are present in the same CFAR window, then Architecture 2 will have a significant loss in performance, unless more clever CFAR schemes are used. Instead Architecture 1 is insensitive to the targets’ locations, and therefore would have the same performance. As we only performed measurements with targets, we are unable to evaluate the CFAR property of Architecture 1. However, for CAMP Architecture 2, we can demonstrate that our model x˜ = x + σ∗ w is correct by estimating the FAP from the reconstructed noisy signal x˜ by excluding the range bins corresponding to the target locations plus four guard cells. If our model is correct, and the noise in the signal x˜ is Gaussian, then the estimated FAP should correspond to the design FAP used to set the detector threshold. This should be true for both the fixed threshold and the CFAR detector. This is demonstrated in Figure 4.16, where the estimated Pf a is plotted versus the design FAP α for CAMP Architecture 2 using both the CA-CFAR and the fixed threshold detector. From this figure we observe that, as expected, the estimated FAP matches the design one, confirming that our model is correct.

4.10

Conclusions In this chapter, we studied the problem of CS radar detection. The nonlinearity of target reconstruction algorithms in CS makes the calculation of false alarm and detection

132

Anitori and Maleki

probabilities data-dependent and complicated. We showed how the recent advances in the asymptotic analysis of CS recovery algorithms enable us to convert CS radar detection to a problem that is similar to classical radar detection. This key step then enabled us to design detection architectures based on compressive measurements and combine them with conventional CFAR processing techniques for CS radar detection. We presented extensive simulation and experimental studies to show the efficacy of our theoretical results in practice.

References [1] L. Anitori, “Compressive sensing and fast simulations: Applications to radar detection,” PhD dissertation, Technical University of Delft, 2013. [2] L. Anitori, A. Maleki, M. Otten, R. Baraniuk, and P. Hoogeboom, “Design and analysis of compressive sensing radar detectors,” IEEE Trans. Signal Process., vol. 61, no. 4, pp. 813– 827, Feb. 2013. [3] M. A. Herman and T. Strohmer, “High-resolution radar via compressed sensing,” IEEE Trans. Signal Process., vol. 57, no. 6, pp. 2275–2284, Jun. 2009. [4] L. Anitori, M. Otten, and P. Hoogeboom, “Compressive sensing for high resolution radar imaging,” in Proc. IEEE Asia-Pacific Microwave Conf. (APMC), 2010. [5] S. M. Song, W. M. Kim, D. Park, and Y. Kim, “Estimation theoretic approach for radar pulse compression processing and its optimal codes,” Electronic Letters, vol. 36, no. 3, pp. 250– 253, Feb. 2000. [6] S. D. Blunt and K. Gerlach, “Adaptive pulse compression via MMSE estimation,” IEEE Trans. Aerosp. Electron. Syst., vol. 42, no. 2, pp. 572–583, Apr. 2006. [7] L. C. Potter, E. Ertin, J. T. Parker, and M. Cetin, “Sparsity and compressed sensing in radar imaging,” Proc. IEEE, vol. 98, no. 6, pp. 1006–1020, Jun. 2010. [8] M. I. Skolnik, Radar Handbook. McGraw-Hill, 1970. [9] M. A. Richards, Fundamentals of Radar Signal Processing. McGraw-Hill, 2005. [10] L. R. Varshney and D. Thomas, “Sidelobe reduction for matched filter range processing,” in Proc. IEEE Radar Conf., 2003. [11] J. V. DiFranco and W. L. Rubin, Radar Detection. Artech House, 1980. [12] S. M. Kay, Fundamentals of Statistical Singnal Processing: Detection Theory. PrenticeHall, 1998. [13] H. L. V. Trees, Detection, Estimation and Modulation Theory: Part III. John Wiley & Sons, 2001. [14] P. Stoica, J. Li, and M. Xue, “Transmit codes and receive filters for radar,” IEEE Signal Process. Mag., vol. 25, no. 6, pp. 94–109, Nov. 2008. [15] F. F. J. Kretschmer and K. Gerlach, “Low sidelobe radar waveforms derived from orthogonal matrices,” IEEE Trans. Aerosp. Electron. Syst., vol. 27, no. 1, pp. 92–102, Jan. 1991. [16] R. L. Frank, “Polyphase codes with good nonperiodic correlation properties,” IEEE Trans. Inf. Theory, vol. 9, no. 1, pp. 43–45, 1963. [17] A. Divito, A. Farina, G. Fedele, G. Galati, and F. Studer, “Synthesis and evaluation of phase codes for pulse compression radar,” Rivista Tecnica Selenia, vol. 9, no. 2, pp. 12–24, 1985. [18] Y. I. Abramovich and M. B. Sverdlik, “Synthesis of a filter which maximizes the signalto-noise radio under additional quadratic constraints,” Radio Eng. Electron. Phys., vol. 15, pp. 1977–1984, Nov. 1970.

Compressed CFAR Techniques

133

[19] M. H. Ackroyd and F. Ghani, “Optimum mismatched filters for sidelobe suppression,” IEEE Trans. Aerosp. Electron. Syst., vol. 9, no. 2, pp. 214–218, Mar. 1973. [20] S. Zoraster, “Minimum peak range sidelobe filters for binary phase-coded waveforms,” IEEE Trans. Aerosp. Electron. Syst., vol. 16, no. 1, pp. 112–115, Jan. 1980. [21] A. Zejak, E. Zentner, and P. Rapajic, “Doppler optimised mismatched filters,” IET Electronics Letters, vol. 27, no. 7, pp. 558–560, Mar. 1991. [22] C. Candan, “On the design of mismatched filters with an adjustable matched filtering loss,” in Proc. IEEE Radar Conf., 2010. [23] B. Steenson, “Detection performance of a mean-level threshold,” IEEE Trans. Aerosp. Electron. Syst., vol. 4, no. 4, pp. 529–534, Jul. 1968. [24] H. M. Finn and R. S. Johnson, “Adaptive detection mode with threshold control as a function of spatially sampled clutter-level estimates,” RCA Review, vol. 29, pp. 414–464, Sept. 1968. [25] G. M. Dillard and C. E. Antoniak, “A practical distribution-free detection procedure for multiple-range-bin radar,” IEEE Trans. Aerosp. Electron. Syst., vol. 6, no. 5, pp. 629–635, Sept. 1970. [26] P. P. Gandhi and S. Kassam, “Analysis of CFAR processors in homogeneous background,” IEEE Trans. Aerosp. Electron. Syst., vol. 24, no. 4, pp. 427–445, Jul. 1988. [27] A. D. Vito and G. Moretti, “Probability of false alarm in CA-CFAR device downstream from linear-law detector,” IET Electronics Letters, vol. 25, no. 25, pp. 1692–1693, Dec. 1989. [28] R. S. Raghavan, “Analysis of CA-CFAR processors for linear-law detection,” IEEE Trans. Aerosp. Electron. Syst., vol. 28, no. 3, pp. 661–665, Jul. 1992. [29] G. B. Goldstein, “False alarm regulation in log-normal and Weibull clutter,” IEEE Trans. Aerosp. Electron. Syst., vol. 9, no. 1, pp. 84–92, Jan. 1973. [30] E. Conte, M. Lops, and A. M. Tulino, “Hybrid procedure for CFAR in non-Gaussian clutter,” IEEE Proc. Radar, Sonar, and Navig., vol. 144, no. 6, pp. 361–369, Dec. 1997. [31] R. Ravid and N. Levanon, “Maximum-likelihood for Weibull background,” IEEE Proc. Part F (London), vol. 139, no. 3, pp. 256–264, Jun. 1992. [32] H. Rohling, “Radar CFAR thresholding in clutter and multiple target situations,” IEEE Trans. Aerosp. Electron. Syst., vol. 19, no. 4, pp. 608–621, Jul. 1983. [33] M. A. Khalighi and M. H. Bastani, “Adaptive CFAR processor for nonhomogeneous environments,” IEEE Trans. Aerosp. Electron. Syst., vol. 36, no. 3, pp. 889–897, Jul. 2000. [34] M. Sekine, T. Musha, Y. Tomita, and T. Irabu, “Suppression of Weibull-distributed clutters using a cell-averaging LOG/CFAR receiver,” IEEE Trans. Aerosp. Electron. Syst., vol. 14, no. 5, pp. 823–826, Sept. 1978. [35] R. Nitzberg, “Constant-false-alarm-rate signal processors for several types of interference,” IEEE Trans. Aerosp. Electron. Syst., vol. 8, no. 1, pp. 27–34, Jan. 1972. [36] S. R. Babu and R. Srinivasan, “Analysis of envelope detected mean level CFAR processors using importance sampling,” in Proc. IEEE Radar Conf., 2000. [37] E. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory, vol. 52, no. 2, pp. 489–509, Feb. 2006. [38] E. Candès, J. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Comm. Pure Appl. Math., vol. 59, no. 8, pp. 1207–1223, Aug. 2006. [39] D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory, vol. 52, no. 4, pp. 1289– 1306, Apr. 2006. [40] R. Tibshirani, “Regression shrinkage and selection via the LASSO,” J. Roy. Stat. Soc., Series B, vol. 58, no. 1, pp. 267–288, 1996.

134

Anitori and Maleki

[41] S. Chen, D. Donoho, and M. Saunders, “Atomic decomposition by basis pursuit,” SIAM J. on Sci. Computing, vol. 20, no. 1, pp. 33–61, 1998. [42] E. Candès and T. Tao, “Near optimal signal recovery from random projections: Universal encoding strategies?” IEEE Trans. Inf. Theory, vol. 52, no. 12, pp. 5406–5425, Dec. 2006. [43] A. Maleki and D. L. Donoho, “Optimally tuned iterative thresholding algorithm for compressed sensing,” IEEE J. Sel. Topics Sig. Proc., vol. 4, no. 2, pp. 330–341, Apr. 2010. [44] A. Maleki, L. Anitori, Y. Zai, and R. Baraniuk, “Asymptotic analysis of complex LASSO via complex approximate message passing (CAMP),” IEEE Trans. Inf. Theory, vol. 59, no. 7, pp. 4290–4308, July 2013. [45] D. L. Donoho, A. Maleki, and A. Montanari, “Message passing algorithms for compressed sensing,” Proc. Natl. Acad. Sci., vol. 106, no. 45, pp. 18914–18919, 2009. [46] D. L. Donoho, A. Maleki, and A. Montanari, “Message passing algorithms for compressed sensing: I. Motivation and construction,” in IEEE Proc. Inform. Theory Work. (ITW), 2010. [47] A. Maleki and A. Montanari, “Analysis of approximate message passing algorithm,” in Proc. IEEE Conf. Inform. Science and Systems (CISS), 2010. [48] D. L. Donoho, A. Maleki, and A. Montanari, “The noise-sensitivity phase transition in compressed sensing,” IEEE Trans. Inf. Theory, vol. 57, no. 10, pp. 6920–6941, Oct. 2011. [49] A. Maleki, “Approximate message passing algorithm for compressed sensing,” PhD dissertation, Stanford University, 2010. [50] M. Bayati and A. Montanari, “The dynamics of message passing on dense graphs, with applications to compressed sensing,” IEEE Trans. Inf. Theory, vol. 57, no. 2, pp. 764–785, Feb. 2011. [51] C. A. Metzler, A. Maleki, and R. G. Baraniuk, “From denoising to compressed sensing,” IEEE Transactions on Information Theory, vol. 62, no. 9, pp. 5117–5144, 2016. [52] A. Mousavi, A. Maleki, R. G. Baraniuk et al., “Consistent parameter estimation for lasso and approximate message passing,” Ann. Statist., vol. 45, no. 6, pp. 2427–2454, 2017. [53] M. Bayati and A. Montanari, “The LASSO risk for Gaussian matrices,” IEEE Trans. Inf. Theory, vol. 58, no. 4, pp. 1997–2017, Apr. 2012. [54] S. Wang, H. Weng, and A. Maleki, “Which bridge estimator is optimal for variable selection?” arXiv preprint arXiv:1705.08617, 2017. [55] J. H. G. Ender, “On compressive sensing applied to radar,” J. Signal Process., vol. 90, no. 5, pp. 1402–1414, May 2010. [56] L. Anitori, A. Maleki, W. van Rossum, R. Baraniuk, and M. Otten, “Compressive CFAR radar detection,” in Proc. IEEE Radar Conf., 2012. [57] S. Shah, Y. Yu, and A. Petropulu, “Step-frequency radar with compressive sampling SFR-CS,” in Proc. IEEE Int. Conf. Acoust., Speech, and Signal Process. (ICASSP), 2010. [58] F. J. Massey, “The Kolmogorov–Smirnov test for goodness of fit,” J. Am Statist. Assoc., vol. 46, no. 253, pp. 68–78, 1951. [59] M. A. Davenport, M. B. Wakin, and R. Baraniuk, “The compressive matched filter,” Rice Univ., ECE Dept., Tech. Rep. TREE-0610, Nov. 2006. [60] D. P. Meyer and H. A. Mayer, Radar Target Detection: Handbook of Theory and Practice. Academic Press Inc., 1973. [61] L. Anitori, M. Otten, and P. Hoogeboom, “Detection performance of compressive sensing applied to radar,” in Proc. IEEE Radar Conf., 2011. [62] D. Beker, W. van Rossum, S. Jacobs et al., “On pre-whitening and accuracy in doa estimation by sparse signal processing on beamformed data,” in Proc. Int. Work. on Compressed Sensing on Radar, Sonar, and Remote Sensing (CoSeRa), 2016.

5

Sparsity-Based Methods for CFAR Target Detection in STAP Random Arrays Haley H. Kim and Alexander M. Haimovich

5.1

Introduction Ground moving target indicator (GMTI) radar is an airborne radar mounted on an aircraft that detects the presence of targets on the ground. One of the main challenges faced by GMTI radars is the detection of slow-moving targets in the presence of ground clutter interference. Space–time adaptive processing (STAP) implementation with antenna arrays has been a classical approach to clutter cancellation in airborne radar [1]. One of the challenges with STAP is that the minimum detectable velocity (MDV) of targets is a function of the baseline of the antenna array: the larger the baseline (i.e., the narrower the beam), the lower the MDV. Unfortunately, increasing the baseline of a uniform linear array (ULA) entails a commensurate increase in the number of elements. Instead of using a large ULA and localizing targets by beamforming [2], one may consider a smaller ULA, but use more sophisticated localization algorithms, such as Capon’s method [3], MUSIC [4], or ESPRIT [5]. All three methods are capable of resolving targets within the Rayleigh resolution limit, whereas conventional beamformering cannot. MUSIC and ESPRIT, however, require knowledge of the number of targets. This information is rarely known to a radar and must be obtained by other means, such as using the Akaike information criteria (AIC) or the minimum description length (MDL) [6,7]. Unfortunately, methods such as the AIC or the MDL do not allow one to control the false alarm rate, a basic requirement in radar. In addition, all three methods require a large number of snapshots, which usually are not available in STAP applications. An alternative approach to increasing the resolution of a radar, but without using a large number of sensors is to use a large, but sparsely populated array. In a sparse array, the sensors are placed across a large array with interelement spacing greater than half a wavelength in a nonuniform manner to avoid grating lobes. There are typically two methods used to determine the sensor positions of a sparse array. The first method involves solving an optimization problem to determine the positions of the sensors such that the resulting beampattern meets some specifications [8,9]. Another method is to simply decide the sensor positions randomly. In random arrays [10,11], sensors are randomly placed across a large array aperture. Since the resolution of the radar depends mostly on the size of the aperture [10], a radar utilizing a sparse array may achieve a high angular resolution with significantly fewer sensors than a ULA. Unfortunately, sparse arrays do not come without drawbacks. Due to the spatial under-sampling, the array 135

136

Kim and Haimovich

beampattern suffers from high sidelobes. During the beamforming stages of STAP, these high sidelobes may cause a significant increase in false alarms [12]. It was shown in [13] that sparse arrays for which sensor locations are determined either by optimization or randomly yield similar peak sidelobe levels. In this chapter, we focus our attention only to random arrays. Moreover, one may envision applications in which the array elements cannot be controlled, for example, an array constituted of unmanned aerial vehicles (UAV) [14]. In [15], Carin demonstrates that measurements from random arrays are consistent to projection measurements that can be utilized by compressive sensing (CS) [16]. This suggests that the user may reap the full benefits of a large random array without worrying that the high sidelobes unnecessarily increase the false alarm rate. The goal of CS is to recover the signal of interest x, given the received data vector y and a linear model y = Ax + e, where A is a measurement matrix and e is an interference vector. If the signal x is known to be sparse (i.e., contains K nonzero elements where K is much smaller than the number of entries in x), the K sparse solution (a solution with at most K nonzero entries) may be found solving the nonconvex optimization problem miny − Ax22 x

subject to x0 ≤ K,

(5.1)

where x0 counts the numbers of nonzero elements in x. The optimization problem in (5.1) is nonconvex, and therefore only approximate solutions can be obtained. One approach to obtaining an approximate solution is to first convert the optimization problem (5.1) into the following optimization problem min y − Ax22 + λx0 . x

(5.2)

To transform (5.2) into a convex optimization problem, the nonconvex term x0 is replaced by the convex term x1 . Here, λ is a regularization parameter that controls the sparsity of the solution in x. For a specific choice of λ, numerous algorithms in literature, such as [16–21], are capable of solving (5.2) in polynomial time. These algorithms are often referred to as basis pursuit (BP). In [22,23] the authors use this approach to solve the sparse localization problem. Nevertheless, without the ability to control false alarms, this approach does not lend itself to radar applications. The authors in [24] argue that a constant false alarm rate (CFAR) radar may be obtained by properly designing the regularization parameter λ. However, in [25], the authors point out that the output noise distribution is unknown and unpredictable, which makes BP unsuitable for designing CFAR radars. Another approach to solving (5.1) is to use matching pursuit (MP) algorithms [26–30]. MP belong to the class of greedy algorithms, which search iteratively one by one for components of the unknown vector x. Components of x detected by MP iterations are removed from subsequent iterations to reduce interference to components of x yet to be detected. In this sense, MP implements a form of successive intereference cancellation (SIC). Although the MP approach generally has weaker guarantees than BP, it has been shown empirically that it often performs similarly, and in some applications

Sparsity-Based Methods for CFAR Target Detection in STAP Random Arrays

137

it outperforms BP [31]. The most substantial advantage of MP algorithms over BP algorithms is their lower computational complexity [32]. In fact, when applied to the radar problem, MP algorithms have a computational complexity comparable to that of a beamformer [33]. A large body of literature exists on compressive sensing applications to radar, but the literature on applying MP to CFAR radar is scarce, with some exceptions, e.g., [29,34]. In particular, [34] does not account for colored Gaussian noise and unknown interference covariance matrix. In this chapter, we extend the work in [34] and propose new detection algorithms for airborne radar, which combine the strengths of random arrays with the ability of sparsity based algorithms to handle under-sampling effects. We propose two sparsitybased CFAR detection algorithms, referred to as MP-CFAR and multibranch MP-CFAR (MBMP-CFAR), respectively. MP-CFAR consists of a target localization stage followed by a target detection stage. MBMP-CFAR generalizes MP-CFAR by maintaining multiple sets of candidate targets. In addition, we present an analysis of the performance of the new sparsity-based radar. In the analysis, the covariance matrix of the noise is not assumed to be known. The main results can be summarized as in the following: 1.

2.

3.

4.

5.2

Show that the number of element of a random array required to maintain a certain level of peak sidelobes scales with the logarithm of the array aperture, in contrast with a ULA, where the number of elements scales linearly with the array aperture. Formulate the problem of sparse target detection given space–time observations from random arrays. The observations are obtained in the presence of Gaussian colored noise of unknown covariance matrix, but for which secondary data is available for its estimation. Develop a CFAR detector for detecting targets by random arrays in unknown colored noise. The detector cancels previously detected targets from the observations to reduce interference between targets. The detector explicitly accounts for the number of spatial resolution cells, the number of array elements, and the number of training samples. Develop the performance analysis for the new sparsity-based radar detector, including expressions for the probability of false alarm and the probability of detection.

STAP Radar Concepts In this section, we introduce the STAP radar signal model and discuss properties of random arrays in STAP radar. In particular, we discuss the average sidelobe and, more importantly, the average peak sidelobe levels exhibited by a random array. We also discuss the clutter rank of the random array in STAP.

5.2.1

Signal Model Consider a radar system mounted on an aircraft, in which Na elements collect returns of a transmitted signal consisting of an Np -pulse coherent waveform with

138

Kim and Haimovich

q

up Np pulses

Aperture Z

Figure 5.1 STAP random array radar system model.

pulse-repetition-interval Tr . The radar operating carrier wavelength is λ, and the airborne platform velocity is vp , where the velocity vector is assumed aligned with the array axis. The Na receive sensor locations z1,z2,. . .,zNa are assumed to be chosen randomly within an aperture of length Z, where the sensor locations and the aperture length are expressed in units of the wavelength λ. For concreteness, it is assumed that the positions of receive elements are drawn from a uniform distribution. An example of an array is shown in Figure 5.1. Let u = sin θ denote the spatial frequency associated with the azimuth angle measured with respect to the normal to the array. The Na × 1 array response vector c(u), is defined T 1 j 2πz1 u j 2πz2 u e ,e ,. . .,ej 2πzNa u . (5.3) c(u) = √ Na Since by applying the vector c∗ (u) the array is steered to spatial frequency u, c(u) is also known as a steering vector. The pattern of a random array β (ω) is a stochastic process defined as the response of an array steered to spatial frequency (u − ω) to a target at spatial frequency u. The array pattern is given by β (ω) = |cH (u − ω)c(u)|2 .

(5.4)

The mainbeam is defined as the array patten in the region |ω| ≤ 1/Z, while the sidelobe region is |ω| > 1/Z. The peak sidelobe is defined μ = max|ω|>1/Z β (ω). The Doppler shift caused by a target moving at velocity vt relative to the normal to the array is fd = 2vt /λ. The normalized Doppler frequency v is the Doppler shift fd normalized to the sampling frequency 1/Tr , where Tr is the pulse repetition interval,

Sparsity-Based Methods for CFAR Target Detection in STAP Random Arrays

139

v = fd Tr . The Np × 1 temporal steering vector g(v) of a target with normalized Doppler frequency v is given by T 1 j 2πv 1,e ,. . .,ej 2π(Np −1)v . g(v) = Np

(5.5)

For notational convenience, let N = Na Np , then the N × 1 space–time steering vector of a target with spatial frequency u and Doppler v is given by a(u,v) = g(v) ⊗ c(u),

(5.6)

where ⊗ represents the Kroneckor product. The N × 1 baseband y signal received at the array from a target with steering vector a and complex amplitude x is given by y = ax + e,

(5.7)

where e = ec + ew is the interference vector consisting of the ground clutter contributions ec and complex-valued white Gaussian noise ew . We treat ground clutter and thermal noise as uncorrelated processes, and therefore the N × N interference and noise covariance matrix is given by R = E (ec + ew )(ec + ew )H = Rc + Rw . (5.8) Here Rw is the covariance matrix of the thermal noise given by Rw = σ2 I, where σ2 is the power of thermal noise. A typical model for the clutter covariance matrix Rc [35] is 1 Rc = s(u)a (u,ξu) aH (u,ξu) du, (5.9) −1

where ξ = 4vp Tr /λ and s(u) is the power of a clutter patch at spatial frequency u and normalized Doppler frequency ξu. The signal model for K targets is given by y = Ax + e.

(5.10)

Here, A is the N ×G measurement matrix whose columns are steering vectors associated with a grid of possible target locations on the angle-Doppler map, and x is a G × 1 vector of complex target amplitudes. The vector x contains only K G nonzeros. In later sections, we apply optimization algorithms that operate on a grid. To this end, we 2 discretize the angle-Doppler map into G = G grid points, where G is the number of grid points in each of the two domains. The G grid points serve as resolution cells. Typically the dimensionality of the signal space N is much smaller than the number of resolution cells, N G. The G × 1 vector of target gains x is assumed to be sparse, in the sense that it has K G nonzero entries. In STAP, the covariance matrix R is typically unknown, but can be estimated from secondary data. The secondary data is assumed to consist of independent identically distributed vectors with a covariance matrix common with the cell under test. Let L be the number of secondary data vectors and q(l) a secondary data vector, the maximum likelihood estimate (MLE) of the covariance matrix is the sample covariance matrix

140

Kim and Haimovich

1 / R= q(l)q(l)H . L L

(5.11)

l=1

In subsequent sections of this chapter, we will make use of the inverse of the sample covariance matrix. In order to ensure that / R−1 exists, we make the assumption that L > N.

5.2.2

Properties of Random Arrays In random arrays, antenna elements are placed at random between the end points of an array. Since the goal is to obtain a thinned array, the average spacing between antenna elements is larger than a half-wavelength. Thus, the term random arrays refers to arrays that are thinned relative to a filled ULA.

Average sidelobe levels of a random array Note that the beam pattern of a filled ULA with aperture Z and uniform illumination is given by [36] (sin (πZω) . βU LA (ω) = [Z sin (πω)])2 From this expression, it is seen that the main beam is the region |ω| ≤ 1/Z, while the sidelobes are |ω| > 1/Z. The number of sidelobes in the visible region |ω| < 1 is 2 (Z − 2) . Of interest are relatively large arrays, in which case the number of sidelobes may be approximated by 2Z. Given an array of Na elements placed at random over an aperture Z, it has been shown that the shape of the mainbeam β (ω), |ω| ≤ 1/Z, follows that of a filled ULA with little variation between instantiations of array elements. In Figure 5.2, we show an example that demonstrate the width of the mainbeam of a random array compared to the mainbeam of ULAs. In the figure, it is seen that the small ULA with 20 elements (which corresponds to an array aperture of 10λ) has a wider mainbeam compared to the random array and the large ULA both with aperture sizes of 15λ. The figure also shows that the random array and the large ULA have the same mainbeam width. Thus with significantly fewer elements, a random array provides the advantage of a narrow and stable mainbeam of a filled array. While there is no impact on the mainbeam, random arrays have higher sidelobes than filled arrays. By the Central Limit theorem, for a sufficiently large number of elements Na and a fixed value ω, b (ω) = cH (u − ω)c(u) = (1/Na )

Na

ej 2πzn ω,

n=1

is a complex-valued Gaussian random variable with mean φ (ω) = (1/Na )

Na E ej 2πzn ω = E ej 2πzω , n=1

Sparsity-Based Methods for CFAR Target Detection in STAP Random Arrays

141

0

Beampattern (dB)

ULA – 10 ULA – 15 RA – 15

–5

–10

–15 –0.25

–0.2

–0.15 –0.1 –0.05

0

0.05

0.1

0.15

0.2

0.25

Spatial frequency

Figure 5.2 Beampattern of a small ULA with 20 elements (10λ array), a random array with an array aperture of Z = 15λ with N = 20 elements, and a large ULA with 30 elements (15λ array).

and variance

2 var |b (ω)|2 = E |b (ω)|2 − φ (ω) .

It is shown in [10] that in the sidelobe region E Re (b (ω))2 ≈ E Im (b (ω))2 ≈ 1/2Na . Therefore the mean level of the beam pattern sidelobes is E |b (ω)|2 = E β (ω) ≈ 1/Na . Thus, the sidelobes of a random array are dominated by the term 1/Na rather than the sidelobes of the associated filled array.

Peak sidelobe level of a random array Next, we are interested in the statistics of the peak sidelobe μ = max β (ω) . |ω|>1/Z

Viewed as a function of ω, the array pattern β (ω) is a stochastic process. In the sidelobe region, the stochastic process is approximately ergodic, meaning that statistical averages may be gleaned from averages across the spatial frequency variable ω [10]. Furthermore, values of the stochastic process β (ω) can be approximated as independent when the values of the spatial frequency ω are separated by a sidelobe or more [10]. As previously discussed, the number of sidelobes is approximately 2Z, where Z is the aperture size of the random array. To find the CDF of the peak sidelobe, let

142

Kim and Haimovich

μ 2Na μ. Since b (ω) ∼ CN (0,1/Na ), it β (ω) 2Na β (ω) = 2Na |b (ω)|2 and follows that β (ω) is a chi-square random variable with 2 degrees of freedom. Recall that the sidelobes are approximated to be independent from each other [10]. It is easy to verify that the cumulative distribution function (CDF) of β (ω) is given by −t/2 . β (t) = 1 − e

(5.12)

It follows that the CDF of the peak sidelobe variable μ is $ %2Z . μ (t) = Pr{β (ω1 ) ≤ t,. . .,β (ω2Z ) ≤ t} = β (t)

(5.13)

Using a known relation for nonnegative random variables, ∞ $ % E μ = 1 − μ (t) dt.

(5.14)

Substituting (5.12) and (5.13) in (5.14), ∞ '2Z & E μ = 1 − 1 − e−t/2 dt.

(5.15)

0

0

The integration in (5.15) is solved in [37], where it is shown ∞

0

2Z '2Z & 1 −t/2 . 1− 1−e dt = 2 k

(5.16)

k=1

1 For large Z, the sum 2Z k=1 k asymptotically approaches ln 2Z + γE , where γE = 0.577 1 is Euler’s constant [37]. Since, ln 2Z " γE , we approximate 2Z k=1 k ≈ ln 2Z. Substituting this back into (5.15), E[ μ] = 2 ln 2Z. Finally, recalling that the peak sidelobe μ is related to the random variable μ as μ = μ/2Na , the mean peak sidelobe is given by ln (2Z) . E μ = Na

(5.17)

It is observed that the mean peak sidelobe is larger than the mean sidelobe by the factor ln (2Z) . To illustrate (5.17), we plot in Figure 5.3, the beampattern of a random array with Na = 15 elements filling an array of size 20λ, we also show the computed average sidelobe level and the average peak sidelobe level. Also, to maintain a fixed mean peak sidelobe level, the number of elements of the random array has to scale with the logarithm of the aperture length. This is contrast with a filled ULA in which the number of elements scales linearly with the aperture length Z. Another point of view that demonstrates that the number of necessary elements in a random array scales with ln Z rather than Z, is to compute the number of elements for which the peak sidelobe μ is lower than a level η, with probability α,α = Pr μ ≤ η = μ (η). The CDF of the peak sidelobe μ (t) can be computed from (5.13) and (5.12). Recalling the relation μ = μ/2Na , we have % $ −Na η 2Z . α = μ (2Na η) = 1 − e

(5.18)

Sparsity-Based Methods for CFAR Target Detection in STAP Random Arrays

143

0

Beampattern (dB)

Random array – 15 Average sidelobe level Average peak sidelobe level

–5

–10

–15 –1

–0.8

–0.6

–0.4

–0.2

0

0.2

0.4

0.6

0.

Spatial frequency

Figure 5.3 Beampattern of a random array using Na = 15 elements filling a large aperture of Z = 20λ. The computed average sidelobe level and average peak sidelobe level is also shown.

Taking ln of both sides, and noting that the expected result is such that Na η " 1, we approximate the ln function with the first term in its Taylor expansion $ % ln 1 − e−Na η ≈ −e−Na η . (5.19) Note that this approximation holds true because η is reasonably higher than the mean value. Using (5.19) and after a little algebra, obtain Na =

% 1$ ln 2Z − ln ln α −1 , η

which links between the number of elements and confidence level that the sidelobes do not exceed a set value. A similar result without proof has been presented in [38].

5.2.3

Clutter Response of Random Arrays STAP relies on the fact that the rank of the clutter covariance matrix Rc (often referred to as clutter rank) is much lower than the dimensionality of the signal space. As a result, whitening of the clutter interference does not result in significant loss of target signalto-noise ratio (SNR). In a filled ULA, the clutter map (defined as aH (u,v)Rc a(u,v), with u and v sweeping through their domains |u| < 1, |v| < 1), forms a diagonal ridge above the uv plane. The width of the ridge along the spatial frequency u axis equals the beamwidth of the array. Thus the clutter ridge of a random array is expected to be narrower than the clutter ridge of a filled ULA with the same number of elements. This is illustrated in Figure 5.4. The panel on the left of Figure 5.4 shows the clutter map of a ULA with N = 10 elements, while the panel on the right shows the clutter map of a random array of the

Kim and Haimovich

–0.5

–0.5

–0.4

–0.4

–0.3

–0.3

–0.2

–0.2

Normalized Doppler

Normalized Doppler

144

–0.1 0 0.1 0.2

–0.1 0 0.1 0.2

0.3

0.3

0.4

0.4

0.5 –0.5 –0.4 –0.3 –0.2 –0.1 0 0.1 0.2 Spatial frequency

0.3

0.4

0.5

0.5 –0.5 –0.4 –0.3 –0.2 –0.1 0 0.1 0.2 Spatial frequency

0.3

0.4

0.5

Figure 5.4 (Left figure): Clutter map using a ULA with N = 10 elements, P = 20 pulses, and

β = 1. (Right figure): Clutter map using a random array with N = 10 elements, P = 20 pulses, and β = 1. The elements of the sparse random array are spread across an array of size 15λ.

same number of elements (10) spread over 15λ (rather than the 5λ ULA aperture), both the clutter maps are generated with β = 1. It is noticed that the clutter ridge of the random array is narrower, which leads to a lower MDV. Note that the clutter map of the sparse array also exhibits multiple, spurious clutter ridges due to higher sidelobes of the beampattern. The clutter rank of a filled ULA can be computed from Brennan’s rule rc = rank(Rc ) = Na + (Np − 1)ξ,

(5.20)

where recall that ξ = 4vp Tr /λ. Now, given a random array with aperture size Z, let f ull represent the number of sensors in a filled ULA configuration. From [39], the Na clutter rank of a random array is f ull

r c ≈ Na

+ (Np − 1)ξ.

(5.21)

It is noticed that the clutter rank of a random array depends on the aperture size Z, f ull since Na = 2Z. This means the random array will require the same number of degrees of freedom to suppress the clutter as a large ULA. However, the number of degrees of freedom available to the random array is less than that of the large ULA. Therefore, fewer degrees of freedom are left to supply gain for the target than for the filled ULA. The MDV depends on the width of the clutter ridge and therefore, the MDV depends on the aperture size of the array [40]. To illustrate this, the signal-to-interference-plusnoise ratio (SINR) vs. normalized Doppler for three array configurations: a small ULA with 10 elements (Z = 5λ), a large ULA with 30 elements (Z = 15λ), and a random array with 10 elements randomly positioned across an array of size Z = 15λ at u = 0 is shown in Figure 5.5. The SNR is defined as |x|2 /σ2 and the clutter-to-noise ratio (CNR) is defined as s(u)/σ2 , where s(u) is the power of the clutter at spatial frequency u and Doppler ξu. For simplicity, it is assumed that the clutter power is constant for all clutter patches and the power is given by σc2 . Then the CNR is also given by σc2 /σ2 . The SINR is defined as

Sparsity-Based Methods for CFAR Target Detection in STAP Random Arrays

145

20 18 16

SINR (dB)

14 12 10 8

ULA – 5 ULA – 15 RA – 15

6 4 2 0 –0.1 –0.08

–0.06 –0.04 –0.02

0

0.02

0.04

0.06

0.08

0.1

Normalized Doppler

Figure 5.5 SINR vs. normalized Doppler for three array configurations: a small ULA with 10

elements (Z = 5λ), a large ULA with 30 elements (Z = 15λ), and a random array with 10 elements randomly positioned across an array of size Z = 15λ. The SNR is 10 dB and the CNR is 30 dB and the spatial frequency u is set to 0. −1 SINRi = aH i R ai . From Figure 5.5, we see that both the large ULA and the random array achieve a higher SINR at lower Doppler than the small ULA. For example, at normalized Doppler = 0.02, the small ULA is about 4 dB SINR, but the random array is about 9 dB. This disparity points to a lower MDV.

5.3

STAP Detection Problem In this section, we introduce the detection problem and summarize a popular detection algorithm, the adaptive beamformer (also known as the adaptive matched filter in [41]). We also analyze the performance of the adaptive beamformer applied to a random array to motivate the need for an alternative algorithm.

5.3.1

Review of Adaptive Beamformer The goal of GMTI radar is to determine the number of targets present and their locations in the angle-Doppler domain. One common approach to this problem is to divide the angle-Doppler map into G resolution cells and perform G detection tests, one for each of the G grid points. The number of targets is determined by counting the number of cells that pass the detection test, and the locations of the targets are determined by the cells that pass the test. The binary hypothesis test for any of the resolution cells based on the model (5.7) is given by H0 : x = 0 H1 : x != 0.

146

Kim and Haimovich

To recap, we are posing the problem of testing a STAP resolution cell for the presence of a target of unknown amplitude observed in the presence of Gaussian colored noise with unknown covariance matrix, when secondary data is available for estimating the covariance matrix, this problem has been solved by Kelly in [42]. Kelly was able to show that the probability of false alarm for the GLRT detector does not depend on interference covariance matrix [42]. In fact it was shown that the probability of false alarm depends on the dimensionality of the signal N , the number of training samples used to estimate the interference covariance matrix R, and the threshold parameter γ. Since the probability of false alarm does not depend on any unknown parameters (γ is a design parameter chosen to achieve a desired false alarm probability), the detector is described as a CFAR detector. A simpler approach is suggested in [41], where the likelihood of the observation is maximized only over the unknown amplitude (separately for each hypothesis). In this approach, the covariance matrix is assumed known through the derivation of the test statistic. It is later substituted with the sample covariance matrix of the secondary data in the final expression of the test statistic since the covariance matrix is unknown. While this procedure is ad hoc, it is argued in [41] that the resulting test statistic differs from GLRT statistic in [42] only by a term that vanishes when the set of secondary data is large. The test for deciding H1 for a resolution cell defined by the steering vector a is given by [41] R−1 y|2 |aH / T = H −1 ≥ γ. (5.22) R a a / It is noted that the test statistic is essentially a beamfomer aH applied to whitened R−1 a, hence we refer to this observations / R−1 y and normalized by the product aH / approach as adaptive beamforming (ABF). An alternative form of the test statistic that is used for performance evaluation is found by observing that according to Robey et al.[41], the test (5.22) for deciding H1 may be expressed as a ratio of two independent random variables T =

|ζ|2 ≥ γ. hψ

(5.23)

It $ is shown % in [41] that ζ is distributed CN (0,1) when a target is not present and CN hρ,1 when a target with steering vector at is present. The effect of estimating ˆ are not the same) is captured in the loss the covariance matrix (the fact that R and R factor h discussed in (5.25). The target’s SNR is given by H −1 2 xa R at . (5.24) ρ= aH R−1 a The test statistic (5.23) tests for the presence of a target by applying a steering vector a, but the actual steering vector of the target is at . The denominator of (5.23), ψ, is a chi-squared random variable with 2(L + 1 − N ) degrees of freedom. Since the factor (L + 1 − N ) appears in several expressions in the sequel, for notational brevity, let

Sparsity-Based Methods for CFAR Target Detection in STAP Random Arrays

147

M (L + 1 − N ). The factor h, first proposed in [43], is a loss factor 0 ≤ h ≤ 1 that captures the effect of estimating the covariance matrix from the secondary data. It is shown in [44] that in the absence of a target, the probability density function (PDF) of the loss factor is the beta PDF (N + N − 1)! M (5.25) h (1 − h)N−2 . p(h) = pβ (h;M + 1,N − 1) = M! (N − 2)! When a target is present, it was shown in [45] the PDF of the loss factor is given by p(h) = e

−C sin2 (θ)

M+1 m=0

M +1 (N + M − 1)! (N + M − 1 + m)! m

(5.26)

× C m pβ (h;M + 1,N − 1 + m). Where the term C is defined as

−1 C = aH R a t 1− t

|aH R−1 at |2 . −1 (aH R−1 a)(aH t R at )

(5.27)

We are now interested in computing the probability of false alarm. The probability of false alarm is given by Pr{T ≥ γ} under H0 . It is more convenient however to compute

# the probability Pr{T # ≥ hγ}, where T # = |ζ| ψ . The ratio T is a ratio of two independent chi-square random variables. Normalizing the numerator and the denominator by the respective degrees of freedom and adjusting the threshold accordingly yields 2

T# =

|ζ|2 /2 ≥ hM γ. ψ/2M

(5.28)

To proceed, we temporarily fix the loss factor h and find the probability of false alarm conditioned on h. In other words, we are interested in obtaining an expression for Pr{T # ≥ hM γ|h}. When no target is present, T # follows the F distribution with parameters 2 and 2M, denoted F (2,2M) . The probability of false alarm conditioned on h is simply given by Pr{T ≥ hγ|h} = 1 − F (2,2M) (hM γ|h),

(5.29)

where F (·,·) denotes the CDF of the F distribution F (·,·). To obtain the probability of false alarm we integrate (5.29) over the random variable h 1 F (2,2M) (hM γ|h)p(h)dh, (5.30) PF A = 1 − 0

where p(h) is given by (5.25). Note that the probability of false alarm depends only on the dimensionality of the signal N and the term M, which depends on the number of ˆ secondary samples used to obtain the estimate of the covariance matrix R. We now turn our attention to the probability of detection again. For convenience we seek to compute the probability Pr{T # ≥ hγ} instead of Pr{T ≥ γ}. Similar to the probability of false alarm, we begin by fixing the random variable h and find the probability of detection conditioned on h. When a target is present, the test statistic follows the noncentral F distribution with parameters 2 and 2M, and noncentrality

148

Kim and Haimovich

$ % parameter hρ, denoted F 2,2M,hρ . The conditional probability of detection of a target with SNR ρ (5.24) is given by Pr{T ≥ hγ|h} = 1 − F (2,2M,hρ) (hM γ|h).

(5.31)

To obtain the probability of detection we integrate (5.32) over the random variable h 1 (5.32) F (2,2M,hρ) (hM γ|h)p(h)dh, PD = 1 − 0

where p(h) is given by (5.26).

5.4

Compressive Sensing CFAR Detection Detection by ABF is agnostic to the possible presence of multiple targets, which increases the number of false alarms seen by the radar [13]. In contrast, the model (5.10) accounts for multiple targets. As explained previously, the number of rows of A, N , is much smaller than the number of columns G. The problem of recovering x given y and A is then underdetermined, and hence does not have a unique solution. Instead, inspired by compressive sensing techniques, we solve the following optimization problem min y − Ax22 subject to x0 ≤ K, x

(5.33)

where x0 denotes the number of nonzero elements of x. As discussed in Section 5.1, problems involving the zero norm are generally nonconvex, and their solution, implemented by an exhaustive search among all combinations of non-zero indices of x, requires exponential complexity [31]. Matching pursuit (MP) is a practical complexity algorithm whose solution approximates the solution to (5.33). However, MP is not directly applicable to the radar problem for two reasons: (1) it does not take into account the presence of clutter, and (2) in radar, the number of targets K is not known a priori. Proposed solutions to address these problems as well as various enhancements are the presented in this section. In radar, clutter contributions are typically much stronger than the unknown targets and, if not suppressed, may severely interfere with target detection. A whitening operation is applied to the observed data and to the measurement matrix A. Specifically, let R−1/2 A, then optimization (5.33) becomes z=/ R−1/2 y and B = / min z − Bx22 subject to x0 ≤ K. x

(5.34)

Unfortunately, to solve (5.34) one requires the knowledge of the number of targets K, which of course is unknown a priori. To implement a CFAR radar that exploits target sparsity, we propose a two-stage MP-CFAR detection algorithm. Candidate targets are localized in the first phase; in the second phase, they are tested for detection. A detected target is then canceled from the data. The cancellation of detected targets from the data is intended to remove mutual interference between targets and thus address one of the flaws of detection by ABF. A block diagram of the MP-CFAR algorithm is shown in Figure 5.6.

Sparsity-Based Methods for CFAR Target Detection in STAP Random Arrays

149

If test passes: Set: k = k + 1

Initialize: S0 = 0, k = 1

If test fails: Output: S0 = 0, k = 0

Sk

Figure 5.6 Block diagram of the MP-CFAR algorithm.

5.4.1

MP-CFAR Stage 1: MP Localization

The first pass of the MP localization algorithm uses whitened data z = / R−1/2 y and R−1/2 aj , j = 1,. . .,G. The first candidate target is whitened steering vectors bj = / localized by the index m1 of the vector bj that has the largest data projection, m1 = arg max j

2 |bH j z|

(5.35)

bH j bj

for j = 1,. . .,G. The index m1 localizes the target in the angle-Doppler domains. This information is subsequently used by the detection stage, as described in relation with Stage 2 later in this section. Next, we describe the localization of the k-th candidate target, given that k − 1 targets have already been localized and passed the detection test. The observed and whitened data z is processed to cancel the contribution of targets detected previously. Let a matrix B be formed with the columns bj . Let Sk−1 be the set of indices of columns of B associated with detected targets, and let BSk−1 be the matrix formed by the columns projection% matrix orthogonal to the detected targets is given by indexed by Sk−1 . The $ H −1 B = I − B BSk−1 . Similarly, steering vectors orthogonal to the P⊥ S k−1 Sk−1 BSk−1 BS k−1

detected targets are formed as follows: wj = P⊥ / Sk−1 . The k-th target BSk−1 bj , for all j ∈ is localized according to mk = arg max j

2 |wH j z|

wH j wj

.

(5.36)

This process continues until a candidate target fails the detection test.

Stage 2: Detection We now derive a CFAR detector that is applied to candidate targets localized in Stage 1. The first candidate target is detected according to (5.22), rewritten here for convenience: T =

/−1 2 |aH m1 R y| ≥ γ, /−1 aH m R am1 1

(5.37)

150

Kim and Haimovich

where m1 is the index found in Stage 1. Note that the test (5.37) may also be expressed R−1/2 am1 , in terms of the whitened steering vectors bm1 = / T =

2 |bH m1 z|

bH m1 bm1

≥ γ.

(5.38)

Next we describe the detection of candidate target k, given that k − 1 targets have already been localized and passed the detection test. The signal model is given by the expression z = bmk xmk + BSk−1 xSk−1 + n = BSk xSk + n,

(5.39)

where mk is the index of the resolution cell of the k-th candidate target found in Stage 1 (5.36), Sk is formed by adding mk to the set Sk−1 , Sk = Sk−1 mk , the matrix BSk = [bmk ,BSk−1 ] is the matrix formed by columns with indices in Sk , T xSk = xmk ,xTSk−1 , and n = / R−1/2 e. This signal model leads to the following detection test: H0 : xmk = 0 H1 : xmk != 0. Here, the following problem is posed: detect a target located at a specified whitened steering vector bmk and having unknown amplitude, observed in the presence of interference and noise. The interference is of unknown gain xSk−1 , but belonging to a known subspace BSk−1 . The noise is Gaussian colored noise for which the covariance matrix is unknown, but secondary data is available for its estimation. To develop the test statistic for the detection problem, we start by expressing the likelihoods of the observations under the two hypotheses. As in the discussion leading to (5.22), the detector is a generalized likelihood ratio detector only in the sense that the liklihood under H1 is maximized over the unknown target amplitude. To simplify the detector, as in [41], it is assumed that the PDF’s of the test statistic under each hypothesis are based on the true covariance matrix. It is noted that the subsequent analysis relies on the properties of the estimated covariance matrix. Thus, z = / R−1/2 y is modeled as having a covariance matrix equal to the identity matrix. It follows that under H0 , the likelihood is 'H & ' & % $ 1 − z−BSk−1 xSk−1 z−BSk−1 xSk−1 , p z|H0 = N e π while under H1 the likelihood is p (z|H1 ) =

1 −$z−BS xS %H $z−BS xS % k k k k . e πN

The GLRT for deciding H1 is given by maxxSk p(z|xSk ) ≥ γ. T = ln maxxSk−1 p(z|xSk−1 )

(5.40)

Sparsity-Based Methods for CFAR Target Detection in STAP Random Arrays

151

To obtain a more convenient form of the test, we note that under hypothesis H0 , the MLE of the gain vector xSk−1 is found from / xSk−1 = min z − BSk−1 xSk−1 22 . xSk−1

(5.41)

Minimizing (5.41) with respect to the vector of complex gains xSk yields −1 H / xSk−1 = (BH Sk−1 BSk−1 ) BSk−1 z.

(5.42)

−1 H / xSk = (BH Sk BSk ) BSk z.

(5.43)

Similarly,

Inserting (5.42) and (5.43) into (5.40), xSk−1 22 − z − BSk/ xSk 22 T = z − BSk−1/ & ' = zH PBSk − PBSk−1 z,

(5.44)

$ %−1 B is a projection matrix that projects onto the subspace where PB = B BH B spanned by B. Note that the decision statistic is a difference between two quadratic forms, where the quadratic form zH PBSk−1 z is an interference term that is canceled. The test statistic (5.44) may be further simplified, which will be useful to obtain expressions for performance evaluation of the MP-CFAR detector. We make use of the following result from [46,47]. Let D and E be two subspaces, and let PD and P[D,E] be projection matrices that project onto the subspaces spanned by the matrices D and [D,E], respectively. Let F = P⊥ D E, then the difference between projection matrices P[D,E] − PD is given by [47] P[D,E] − PD = PF .

(5.45)

Now, identify D = BSk−1 and E = bmk (whitened steering vector). Then F = P⊥ DE ⊥ b is a vector, and let f P b . Note that f = b . Since by design, = P⊥ m k m 1 m k k 1 BSk−1 BSk−1 fk−1 is already orthogonal to all previous vectors f1,. . .,fk−2, we have the following recurrent relations fk = P⊥ fk−1 bmk .

(5.46)

From this expression, fk is the projection of the whitened steering vector bmk orthogonal to the previous k − 1 targets. We use this vector to remove the potential interference the other k − 1 targets provides through the sidelobes. Applying (5.45), we obtain P[BSk−1 ,bmk ] − PBSk−1 = Pfk ,

(5.47)

H where Pfk = fk fH k /fk fk . Noting that BSk = [BSk−1 ,bmk ] and substituting (5.47) into (5.44), the test for deciding H1 on the detection of the k-th target can be expressed as

T =

2 |fH k z|

fH k fk

≥ γ.

For k = 1, f1 = bm1 , and (5.48) reverts to (5.38), as it should.

(5.48)

152

Kim and Haimovich

Algorithm 1 CFAR-MP ˆ γ. 1: Input: y,A, R, 2: 3: 4: 5:

6:

ˆ −1/2 y, B = R ˆ −1/2 A, W = B, k = 1. Initialize: S0 = ∅, r = R H 2 Find: Search for the index l that maximizes the metric maxj |wH j r| /wj wj . Update set of targets: Sk = Sk−1 l. Check: If Tsi ≥ γ (test statistic to decide if xsi is nonzero) for all si ∈ Sk continue. Otherwise output Sk−1 as solution and terminate. & '−1 HB Generate: P⊥ = I − B BH B S S k k Sk Sk . BS k

Remove found targets: W = P⊥ BSk B. 8: Renormalize: If wi 2 = 0, set wi = 0. 9: Return to step 3. 7:

The test statistic (5.48) is applied to every candidate target included in the set Sk . If any of the k tests fails to exceed the threshold γ, the algorithm terminates and outputs the set Sk−1 , the set of k − 1 target locations. Otherwise, MP-CFAR increments the number of targets k by one and reruns MP with the new value of k. The psuedocode for the MP-CFAR algorithm is listed in Algorithm 1.

5.4.2

Performance of the MP-CFAR Detector In this section, we develop analytical expressions for the probability of false alarm and probability of detection of the MP-CFAR detector for some simple cases. We will consider the case when no target is present and when a single target is present in the field of view. To obtain an expression for the probability of false alarm when no target is present in the field of view, we manipulate the test statistic (5.48) to express it in the form (5.28). By assumption, no target has been detected yet, hence the test is for target index k = 1. For k = 1, and based on notation developed previously, the vector R−1/2 am1 and z = / R−1/2 y. Now recall that m1 is the index obtained fs1 = bm1 = / from (5.36). It follows that (5.48) may be written T = max j

/−1 2 |aH j R y| . R−1 aj aH /

(5.49)

j

Other than the max operator, the test statistic in (5.49) is of the form (5.22), hence it can be reduced to the form (5.28), T = max j , j

(5.50)

where j =

|ζ j |2 /2 . hψj /2M

The probability of false alarm is given by 5 6 PF A = 1 − Pr max j ≤ hM γ . j

(5.51)

Sparsity-Based Methods for CFAR Target Detection in STAP Random Arrays

153

Using the assumption that the random variables j (5.51) are independent and identically distributed, $ %G . PF A = 1 − Pr j ≤ hM γ As discussed in relation with (5.29), j follows an F distribution with CDF F (2,2M) , from which the expression for the probability false alarm is approximately G 1

PF A = 1 −

F (2,2M) (hM γ|h)p(h)dh

.

(5.52)

0

The probability of detection of the first target is given by the same expression as for the ABF (5.32). Note that it is assumed that the target is assumed to be recovered in the first pass of the MP-CFAR detector. If this is the case, the probability of detection follows that of (5.32). Otherwise, the probability of detection will decrease due to the orthogonal projection. When a target is present, the probability of false alarm includes the event that a target is detected at the incorrect resolution cell. That would increase the PF A in (5.52). An algorithm that mitigates this type of false alarm is presented next.

5.4.3

MBMP-CFAR The MP-CFAR algorithm localizes the first target according to (5.35), namely, it finds the column of the whitened measurement matrix with the largest projection on the whitened data z. A false alarm (localizing the target in the wrong resolution cell) increases the chance of further false alarms downstream, since according to (5.36), localizing subsequent candidate targets depends on the location of the first target (5.35). A more robust approach is to hedge bets by finding multiple candidates for the location of the first target. Each such candidate target serves as seed to the localization and detection of subsequent targets. When the process is completed, a metric is used to select the set of targets that provides the best fit to the data. This algorithm, which generalizes MP-CFAR, is referred to as MBMP-CFAR. We introduce some notation that facilitates the presentation of MBMP-CFAR. A localization solution is referred to as a branch. The set D = {d1,d2,. . .,dk } contains the number of branches per target. A path is a sequence of branches specified by their index numbers. For example, the path (i1,i2,. . .,ik ), 1 ≤ i1 ≤ d1,. . .,1 ≤ (i ,i ,...,i ) ik ≤ dk . A localization solution 1 ≤ mk 1 2 k ≤ G, where G is the number of resolution cells (see [5.10]), consists of a path and the index number of the resolution (i ) (i i ) (i ...,i ) (i ,...,i ) cell. The set Sk 1 k = m1 1 ,m2 1, 2 ,ldots,mk 1, k contains the localization solution associated with path (ı1,i2,. . .,ik ) . For k candidate targets, MBMP maintains d1 × d2 × · · · × dk such sets. The matrix BSk was defined to consist of the whitened steering vectors bj indexed by Sk . Similarly, we define the matrix B (i1,...,ik ) to consist of whitened steering vectors indexed by the set analogous to (5.46) (i ,...,ik )

fk 1

= P⊥ B

(i ,...,i ) Sk 1 k .

Sk (i1 ,...,ik ) \mk

The vector

bk .

Sk (i ,...,i ) fk 1 k

is defined (5.53)

154

Kim and Haimovich

The inputs to MBMP-CFAR are the whitened measurement vector z = / R−1/2 y, −1/2 / aj , j = 1,. . .,G, and a set of positive integers whitened steering vectors bj = R D = {d1,d2,. . .,dG }. Similar to MP-CFAR, the MBMP-CFAR algorithm proceeds in two stages. MBMP generates a tree in which each branch is associated with a candidate target. At a given level in the tree, the first branch represents the strongest target (according to [5.54]), the second branch represents the second-strongest target, and so on. The number of branches at a given level is user-selected. The children of a given branch are generated by projecting the received data away from the ancestor candidate target(s) (according to [5.57]), and again ranking a user selected number of candidate targets according to their strength. All children from a given target have been orthogonally projected to remove the effect of the ancestor.

Stage 1: MBMP Localization To localize the candidates for the first target, the algorithm finds the d1 indices (1) (2) (d ) m1 ,m1 ,. . .,m1 1 that produce the d1 largest projections of steering vectors bj on the data z. Specifically, the resolution cell index that localizes the first branch of the first target is found from (1) m1

= arg max j

2 |bH j z|

bH j bj

.

(5.54)

The ith branch of the first target, 1 ≤ i ≤ d1 , is found from (i) m1

= arg

2 |bH j z|

max

bH j bj

j∈ / {m1,...,mi−1 }

.

(5.55)

To generate the d1 d2 branches associated with the second target, define the modified (i) steering vectors wj = P⊥ b (i) bj , for 1 ≤ i ≤ d1 . The orthogonal projection prevents m1

(i)

interference from a target at m1 . The resolution cell index associated with the first branch of the second target is given by (1)H

(1,1) m2

= arg max j

|wj

(1)H

wj

z|2 (1)

(5.56)

,

wj

whereas the index of branch i2 , 1 ≤ i2 ≤ d2 of the second target, given the path (i1,i2 ), (i )H

(i ,i ) m2 1 2

= arg

max j∈ / m1,...,mi2 −1

|wj 1

(i )H

wj 1

z|2 (i )

wj 1

.

(5.57) (i ,...,ik )

Generalizing to k targets and the path (i1,i2,. . .,ik ), define the vector wj 1 = P⊥ bj . The index associated with the k-th target is given by B (i ,...,ik ) Sk 1

Sparsity-Based Methods for CFAR Target Detection in STAP Random Arrays

155

S0

S1(1)

S2(1,1)

{ m1(1) }

S1(2)

{m(2) 1 }

{m1(1),m2(1,1) }

S2(2,2)

S2(1,2)

{m1(1) ,m2(1,2) }

S2(2,1)

(2)

(2,1)

{ m1 ,m 2

{m1(2) ,m(2,2) } 2

}

Figure 5.7 Graph of MBMP algorithm for a branch vector d = [2, 2]T .

(i ,...,ik )

mk 1

= arg

6 5 max j∈ / m1,...,mik−1

(i1,...,ik−1 )H 2 z wj (i ,...,i )H (i ,...,i ) wj 1 k−1 wj 1 k−1

.

(5.58)

An example of MBMP localization with D = {2,2,1,. . .} is illustrated in Figure 5.7. From the figure it is seen that MBMP localization starts with the empty set corresponding to no targets detected. The algorithm then searches for d1 = 2 steering vectors that generates the largest projection on the data. The algorithm then performs a detection test on the d1 resolution cells (the detection test is detailed next in Stage 2: Detection). If the detection test passes, MBMP-CFAR searches for d2 resolution cells using (5.57) (1) (1) with wj = P⊥ b (1) bj . The d2 resolution cells form two paths that stem from m1 (see m1

(2)

Figure 5.7). This is repeated using m1 to create d1 d2 paths. The process continues until the detection test fails, as explained in relation with Stage 2.

Stage 2: Detection The MBMP localization processing yielded d1 candidate locations for the first target. 2 H The largest score relative to the objective function |bH j z| /bj bj is obtained by the (1)

steering vector index m1 , because 2 H max |bH j z| /bj bj ≥ j

max

j∈ / {m1,...,mi−1 }

2 H |bH j z| /bj bj ,

(1)

(see [5.54] and [5.55]). Note that the choice of m1 also minimizes the residual of 2 the objective function z − Bx22 (see [5.34]), m1(1) = arg minj P⊥ bj z2 . The test to (1)

determine whether a target is present in the resolution cell m1 is given by (5.48)

156

Kim and Haimovich

T =

|bH(1) z|2 m1

bH(1) bm(1) m1

≥ γ.

(5.59)

1

(i )

If the test (5.59) is met, d1 target sets are updated as follows S1 1 (i ) = m1 1 , 1 ≤ i1 ≤ d1 . If the test fails, then MBMP-CFAR declares that no targets exist, and the algorithm terminates. To test for the detection of the k-th target, assume that k−1 targets have been detected. The residual along the path (i1,i2,. . .,ik ) is computed from 4 42 4 ⊥ 4 4 z (5.60) R (i1,...,ik ) = 4 P 4 B i ,...,i 4 . ( Sk 1

k)

2

The path that yields the lowest residual is given by (i1,i2,. . .,ik ) = arg min R (j1,...,jk ) . (j1,...,jk )

(5.61)

(i ...,i ) The test to determine whether a target is present in the resolution cell mk 1, k is given by (see definition of vectors f in [5.53]). (i ,...,ik )

T =

|fk 1

z|2

(i ,...,ik )H (i1,...,ik ) fk

fk 1

≥ γ.

(5.62)

(i i ) (i ...,i ) (i ) All k resolution cells m1 1 ,m2 1, 2 ,. . .,mk 1, k are tested. In other words, the (i ) (i ,...,i ) test (5.62) is performed for the vectors f1 1 ,. . .,fk 1 k . (i ,...,i ) If all k tests (5.62) are met, d1 × . . . × dk target sets are updated as follows Sk 1 k (i1 ) (i1, i2 ) (i ...,i ) = m1 ,m2 ,. . .,mk 1, k , 1 ≤ i1 ≤ d1,. . .,1 ≤ ik ≤ dk . The MBMP-CFAR algorithm proceeds to the localization and detection of the (k + 1) target. If the detection test fails, MBMP-CFAR outputs as solution the path (i1,i2,. . .,ik−1 ) = arg

min R (j1,...,jk−1 ) . (j1,...,jk−1 )

To illustrate the MBMP-CFAR, we return to the example in the previous subsection. MBMP-CFAR first searches for d1 = 2 steering vectors as described in the previous subsection. It then performs the detection test (5.59). If the detection test fails, the algorithm declares no target exists and terminates. If the detection test passes, the algorithm forms two paths that stem from empty set (see Figure 5.7). MBMP-CFAR then searches for d1 d2 paths as described in the previous subsection and then searches for the path that minimizes the residual of the objective function using (5.61). Note that the path that minimizes the residual of the objective function using two targets does not necessarily stem from the branch that started with the largest statistic using a single target and allows the algorithm to move away from that branch. It then tests the resolution cells obtained from (5.61) using (5.62). The algorithm terminates if either of the two resolution cells fail the detection test, otherwise the process continues until a detection test fails. Intuitively, the MBMP-CFAR algorithm generalizes the MP-CFAR by allowing the consideration of resolution cells that do not maximize the metric in (5.36). Note that

Sparsity-Based Methods for CFAR Target Detection in STAP Random Arrays

157

the first iteration of MBMP-CFAR produces the same localization solution as the MPCFAR and also performs the same detection test. Hence when no targets exist, the performance of both MBMP-CFAR and MP-CFAR is the same. Also note the path that (1) (1,...,1) ) is the MP-CFAR solution. corresponds to the set of resolution cells (m1 ,. . . mk (1) (1,...,1) However, MBMP-CFAR does not always test the set (m1 ,. . . mk ), because it may not minimize the residual (5.60). For example, using Figure 5.7, in the first iteration (1) MBMP-CFAR will always test m1 , however in the next iteration MBMP-CFAR will (1,1) (2,1) or the set of resolution cells S2 . Note test either the set of resolution cells S2 (1,2) (2,2) or S2 because they present higher that the algorithm will never test the paths S2 (1,1) (2,1) and S2 , respectively. objective functions than S2

5.5

Numerical Results In this section, we present numerical results on the MP-CFAR and MBMP-CFAR algorithms and compare them with ABF. Unless stated otherwise, in figures presented in this section, the aperture of the random arrays is 12λ (Z = 12, where Z is expressed in units of wavelength). The number of elements in the random array is Na = 16, thus the mean spacing between elements of the random array is 12λ/16. The number of coherent pulses used by all arrays is Np = 25. The SNR, defined as |x|2 /σ2 , is set to SNR = 15.5 dB unless stated otherwise. The CNR is set to 30 dB. It was seen −1 that the SINR of the random array, defined as SINRi = aH i R ai , is roughly 15 dB with these parameters. The number of training samples used to estimate the covariance matrix for the random array is L = 2N . Reduced-rank methods may be applied to reduce the size of the training set [48], but this is not the emphasis of this work. The number of resolution cells on the angle-Doppler map is given by G = (2Z + 1)2 = 625. A random realization of a random array is generated and remains fixed throughout the Monte Carlo simulations for all figures unless otherwise stated. Let St be the true set of resolution cells that contain targets, and let Sˆ be the set of resolution cells found by a ˆ t != ∅, and a detection event detector to have targets. A false alarm event occurs if S\S 7 occurs if Sˆ St != ∅. The threshold γ was computed by selecting a desired false alarm probability, the equations in [49] were then used to find the appropriate value of γ. The probabilities of false alarm of the MP-CFAR and ABF detectors are studied in Figure 5.8, which plots the empirical probability of false alarm against the SINR of a target present with the angle-Doppler pair (5/Z,0). The detection threshold for the ABF detector is set using (5.30), such that PF A = 10−3 . Applying (5.52), the detection threshold for the MP-CFAR detector is also set to PF A = 10−3 . In this figure, the arrays compared are a 12λ ULA and an 12λ random array. The random array has Na = 16 sensors; the resolution cells for this experiment was spaced apart by 1/12λ. For each curve (excluding the line PF A = 10−3 ), the results of 104 Monte-Carlo experiments were averaged to obtain the curves, and the ABF tested every resolution cell on the angle-Doppler map. The probability of false alarm of a true CFAR detector should not change as a function of SNR of a target present somewhere in the search area. It is

Kim and Haimovich

10–1

PF

158

ABF MP-CFAR ABF (large ULA)

10–2

10–3 10

11

12

13

14

15

16

17

18

19

20

SINR

Figure 5.8 Probability of false alarm vs. SINR of a target for the ABF with a random array,

MP-CFAR with a random array, and the ABF with a large ULA.

observed from the figure that the 12λ ULA ABF and the random array MP-CFAR detectors have probabilities of false alarm that are little changed as a function of the SNR of a target. More specifically, at low SNR the MP-CFAR experiences a probability of false alarm of about 2 × 10−3 instead of PF A = 10−3 . This slight increase in the probability of false alarm occurs because at low SNR, the probability of correct recovery (the probability that MP-CFAR recovers the correct resolution cell to test) is less than one. As the SNR of the target increases, the probability of correct recovery increases, and the false alarm probability of MP-CFAR decreases to PF = 10−3 as intended. It is noticed that the ABF using a 12λ ULA experiences a slight increase in the probability of false alarm as the SNR of the interfering target increases. This is because although the peak sidelobe of a ULA is relatively small (roughly −13 dB), it is not zero and therefore will ultimately affect the probability of false alarm. In contrast, a random array using ABF cannot cope with energy leaked by high sidelobes, and as the strength of the target increases, the probability of false alarm increases. In Figure 5.9, shown are the receiver operating characteristic (ROC) curves of the ABF using a 12λ ULA, the ABF using a random array, and MP-CFAR using a random array, and a single target in the field of view. The target again has the angle-Doppler pair (5/Z,0). From the figure, the large ULA using ABF performs well as expected. Since the ULA array does not exhibit large sidelobes, the target does not significantly increase the probability of false alarm. In contrast, it is seen that the ABF with the random array performs considerably worse. The random array has large sidelobes, and since the ABF does not account for the large sidelobes, the radar experiences a high false alarm rate. The MP-CFAR with a random array on the other hand performs similarly to the ABF with a large ULA. The MP-CFAR unlike the ABF, accounts for detected targets and removes the targets before detecting more targets. Note that the MP-CFAR performs

Sparsity-Based Methods for CFAR Target Detection in STAP Random Arrays

0.95 0.9

159

ABF CFAR-MP ABF (Large ULA)

PD

0.85 0.8 0.75 0.7 0.65 0.6 10–4

10–3

10–2

10–1

PF

Figure 5.9 ROC curve for a single target for the ABF with a random array, MP-CFAR with a

random array, and the ABF with a large ULA. Parameters SNR = 15.5 dB and CNR = 30 dB.

ABF MP-CFAR MBMP-CFAR – [3 1 1 ..]

10–1

–3

PF

P F = 10

10–2

10–3 0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

Sparsity Ratio

Figure 5.10 Probability of false alarm vs. the sparsity ratio of an array for the ABF with a random array, the MP-CFAR with a random array, and the MBMP-CFAR with a random array with branch vectors D = [3,1,1,. . .]. Parameters: SNR = 15.5 dB and CNR = 30 dB.

similarly to the ABF with the large ULA using about 3/4s of the number of elements compared to the large ULA. This demonstrates the savings without loss of performance that are gained by random arrays and the proposed MP-CFAR detector. In Figure 5.10, shown is the probability of false alarm vs. the sparsity ratio of a f ull f ull random array. The sparsity ratio is defined as sparsity ratio = Na /Na , where Na is the number of elements required to fill a ULA for an array of the same size. There are

Kim and Haimovich

10–1

ABF MP-CFAR MBMP-CFAR – [6 1 1 ..]

PF

160

10–2

10–3

Number of Targets

Figure 5.11 Probability of false alarm vs. the number of targets in the field of view for the ABF with a random array, the MP-CFAR with a random array, and the MBMP-CFAR with a random array with branch vectors D = [3,1,1,. . .]. Parameters: SNR = 15.5 dB and CNR = 30 dB.

two targets in the field of view, with the angle-Doppler pairs (5/Z,0) and (−5/Z,0). The threshold parameter γ is set so that the desired probability of false alarm is PF = 10−3 for all methods. As expected, the false alarm rate decreases as the sparsity ratio increases for all three curves, since using more elements in a random array decreases the average sidelobe and average peak sidelobe levels. Although the false alarm rate for the ABF decreases as the sparsity ratio increases, the false alarm rate is still much larger than 10−3 as intended. In contrast, MP-CFAR experiences significantly less false alarms at all sparsity ratios, in addition, at sparsity ratio 0.9, the MP-CFAR detector experiences a false alarm rate of about 10−3 , as intended. MBMP-CFAR further decreases the false alarms at low sparsity ratios, and as expected also experiences a false alarm rate of about 10−3 as intended. MBMP-CFAR experiences lower false alarm rates at lower sparsity ratios, which can be attributed to the additional resolution cells that the algorithm tests. In Figure 5.11, shown is the probability of false alarm vs. the number of targets for the ABF, MP-CFAR, and the MBMP-CFAR with branch vector D = [6,1,1,. . .] using a random array. The targets have the following angle-Doppler pairs s1 = (5/Z,0), s2 = (5/Z,1/Z), s3 = (−4/Z,2/Z), s4 = (−3/Z,2/Z), s5 = (−6/Z,−1/Z), s6 = (6/Z,−1/Z). For K = 1, the target with angle-Doppler pair s1 is placed on the map; for K = 2, the targets s1,s2 are on the map, etc. Again, the threshold parameter γ is set so that the desired probability of false alarm is PF = 10−3 for all methods. From the figure it is seen that the ABF performs significantly worse than the MP-CFAR and the MBMP-CFAR as the number of targets increases. The false alarm rate for the MP-CFAR on the other hand increases slightly from PF = 10−3 for K = 0 to about PF = 1.4 × 10−3 for K = 3. For K = 4, it is noticed that targets s3 and s4 are placed in adjacent resolution cells in angle. This causes significant interference between the two

Sparsity-Based Methods for CFAR Target Detection in STAP Random Arrays

161

1 0.95 0.9

ABF MP-CFAR MBMP-CFAR [3 1 1 ..]

0.85

PD

0.8 0.75 0.7 0.65 0.6 0.55 0.5 10–3

10–2

10–1

100

PF

Figure 5.12 ROC curve for two closely spaced targets for the ABF with a random array, MP-CFAR with a random array, and the MBMP-CFAR with a random array. Parameters SNR = 15.5 dB and CNR = 30 dB.

targets and the false alarm rate increases to about PF = 7 × 10−3 . The false alarm rate for MP-CFAR does not change significantly for K ≥ 4. The remaining targets s5 and s6 are far apart from each other and do not significantly interfere with each other or the other 4 targets and hence does not drastically impact the false alarm rate. MBMP-CFAR performs identically to the MP-CFAR for K ≤ 3, it also experience a increase in the false alarm rate for K = 4 from PF = 1.4 × 10−3 to PF = 4 × 10−3. Note that the false alarm rate for the MBMP-CFAR at K ≥ 4 is about 4 × 10−3 instead of 7 × 10−3 . This decrease in the false alarm rate stems from the increase in the probability of correct recovery in the MBMP-CFAR algorithm. In Figure 5.12, the ROC curves for the ABF, MP-CFAR, and the MBMP-CFAR are shown with branch vector D = [3,1,1,. . .] using a random array for two targets spaced closely together. The two targets are placed in adjacent resolution cells, and the angle-Doppler pairs of the targets are (−5/Z,0) and (−4/Z,0). From the figure, the ABF performs poorly because it cannot cope with the high sidelobes of a random array and both sparsity-based radars significantly outperform the ABF. Comparing the two sparsity-based CFAR algorithms, the MBMP-CFAR algorithm sees a slight performance gain compared to the MP-CFAR. This increase in performance is due to the increase in the probability of correct recovery that the MBMP-CFAR algorithm enjoys by testing more resolution cells.

5.6

Summary In this chapter we propose using a random array with the MP-CFAR and MBMP-CFAR algorithms to solve the target detection problem in a STAP setting. The random array

162

Kim and Haimovich

is a large undersampled array that achieves high resolution due to the large aperture at the cost of high sidelobes. Although conventional beamforming cannot cope with the high sidelobes introduced by the random array, the proposed sparsity-based algorithms can cope with the high sidelobes, allowing one to enjoy the high resolution of the random array without the consequences of the high sidelobes. This was achieved using the proposed algorithms by iteratively detecting targets one by one and removing their contributions from the data. Numerical simulations show that the proposed algorithms outperform beamforming methods when a random array is employed. We show in simulations that both MP-CFAR and MBMP-CFAR outperform the popular beamformer when a random array is employed. In particular, we show that the beamformer experiences significantly higher false alarms compared to the proposed methods, and is not compatible with a random array. In contrast, the MP-CFAR and MBMP-CFAR algorithms are shown to be able to cope with high sidelobes and are compatible with a random array.

References [1] J. Ward, “Space-time adaptive processing for airborne radar,” DTIC document, Tech. Rep., 1994. [2] B. D. Van Veen and K. M. Buckley, “Beamforming: A versatile approach to spatial filtering,” IEEE ASSP Magazine, vol. 5, no. 2, pp. 4–24, 1988. [3] J. Capon, “High-resolution frequency-wavenumber spectrum analysis,” Proceedings of the IEEE, vol. 57, no. 8, pp. 1408–1418, 1969. [4] R. Schmidt, “Multiple emitter location and signal parameter estimation,” IEEE Transactions on Antennas and Propagation, vol. 34, no. 3, pp. 276–280, 1986. [5] R. Roy and T. Kailath, “Esprit-estimation of signal parameters via rotational invariance techniques,” IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. 37, no. 7, pp. 984–995, 1989. [6] J. Rissanen, “A universal prior for integers and estimation by minimum description length,” The Annals of Statistics, pp. 416–431, 1983. [7] M. Wax and T. Kailath, “Detection of signals by information theoretic criteria,” IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. 33, no. 2, pp. 387–392, 1985. [8] 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 Transactions on Antennas and Propagation, vol. 50, no. 12, pp. 1732– 1742, 2002. [9] L. Cen, Z. L. Yu, W. Ser, and W. Cen, “Linear aperiodic array synthesis using an improved genetic algorithm,” IEEE Transactions on Antennas and Propagation, vol. 60, no. 2, pp. 895–902, 2012. [10] Y. Lo, “A mathematical theory of antenna arrays with randomly spaced elements,” IEEE Transactions on Antennas and Propagation, vol. 12, no. 3, pp. 257–268, 1964. [11] B. D. Steinberg, Principles of Aperture and Array System Design: Including Random and Adaptive Arrays. Wiley Interscience, vol. 1, 1976, p. 374.

Sparsity-Based Methods for CFAR Target Detection in STAP Random Arrays

163

[12] F. Athley, C. Engdahl, and P. Sunnergren, “On radar detection and direction finding using sparse arrays,” IEEE Transactions on Aerospace and Electronic Systems, vol. 43, no. 4, pp. 1319–1333, 2007. [13] B. D. Steinberg, “Comparison between the peak sidelobe of the random array and algorithmically designed aperiodic arrays,” IEEE Transactions on Antennas and Propagation, vol. 21, no. 3, pp. 366–370, 1973. [14] S. Tonetti, M. Hehn, S. Lupashin, and R. D’Andrea, “Distributed control of antenna array with formation of UAVs,” IFAC Proceedings Volumes, vol. 44, no. 1, pp. 7848–7853, 2011. [15] L. Carin, “On the relationship between compressive sensing and random sensor arrays,” IEEE Antennas and Propagation Magazine, vol. 51, no. 5, pp. 72–81, 2009. [16] D. L. Donoho, “Compressed sensing,” IEEE Transactions on Information Theory, vol. 52, no. 4, pp. 1289–1306, 2006. [17] R. Tibshirani, “Regression shrinkage and selection via the lasso,” Journal of the Royal Statistical Society, Series B (Methodological), pp. 267–288, 1996. [18] S. S. Chen, D. L. Donoho, and M. A. Saunders, “Atomic decomposition by basis pursuit,” SIAM Review, vol. 43, no. 1, pp. 129–159, 2001. [19] J. M. Bioucas-Dias and M. A. Figueiredo, “A new twist: Two-step iterative shrinkage/thresholding algorithms for image restoration,” IEEE Transactions on Image Processing, vol. 16, no. 12, pp. 2992–3004, 2007. [20] D. L. Donoho, A. Maleki, and A. Montanari, “Message-passing algorithms for compressed sensing,” Proceedings of the National Academy of Sciences, vol. 106, no. 45, pp. 18914– 18919, 2009. [21] A. Beck and M. Teboulle, “A fast iterative shrinkage-thresholding algorithm for linear inverse problems,” SIAM Journal on Imaging Sciences, vol. 2, no. 1, pp. 183–202, 2009. [22] D. Malioutov, M. Çetin, and A. S. Willsky, “A sparse signal reconstruction perspective for source localization with sensor arrays,” IEEE Transactions on Signal Processing, vol. 53, no. 8, pp. 3010–3022, 2005. [23] I. W. Selesnick, S. U. Pillai, K. Y. Li, and B. Himed, “Angle-doppler processing using sparse regularization,” in 2010 IEEE International Conference on Acoustics, Speech and Signal Processing 2010, pp. 2750–2753. [24] L. Anitori, A. Maleki, M. Otten, R. G. Baraniuk, and P. Hoogeboom, “Design and analysis of compressed sensing radar detectors,” IEEE Transactions on Signal Processing, vol. 61, no. 4, pp. 813–827, 2013. [25] N. A. Goodman and L. C. Potter, “Pitfalls and possibilities of radar compressive sensing,” Applied Optics, vol. 54, no. 8, pp. C1–C13, 2015. [26] J. A. Tropp and A. C. Gilbert, “Signal recovery from random measurements via orthogonal matching pursuit,” IEEE Transactions on Information Theory, vol. 53, no. 12, pp. 4655– 4666, 2007. [27] W. Dai and O. Milenkovic, “Subspace pursuit for compressive sensing: Closing the gap between performance and complexity,” DTIC document, Tech. Rep., 2008. [28] D. Needell and J. A. Tropp, “Cosamp: Iterative signal recovery from incomplete and inaccurate samples,” Applied and Computational Harmonic Analysis, vol. 26, no. 3, pp. 301–321, 2009. [29] M. Rossi, A. M. Haimovich, and Y. C. Eldar, “Compressive sensing with unknown parameters,” in 2012 Conference Record of the Forty Sixth Asilomar Conference on Signals, Systems and Computers (ASILOMAR), 2012, pp. 436–440.

164

Kim and Haimovich

[30] M. E. Davies and Y. C. Eldar, “Rank awareness in joint sparse recovery,” IEEE Transactions on Information Theory, vol. 58, no. 2, pp. 1135–1146, 2012. [31] M. Elad, Sparse and Redundant Representations: From Theory to Applications in Signal and Image Processing. Springer, 2010. [32] J. A. Tropp and S. J. Wright, “Computational methods for sparse solution of linear inverse problems,” Proceedings of the IEEE, vol. 98, no. 6, pp. 948–958, 2010. [33] H. H. Kim, M. A. Govoni, and A. M. Haimovich, “Cost analysis of compressive sensing for mimo stap random arrays,” in 2015 IEEE Radar Conference (RadarCon), 2015, pp. 0980– 0985. [34] M. Rossi, A. M. Haimovich, and Y. C. Eldar, “Spatial compressive sensing in MIMO radar with random arrays,” in 46th Annual Conference on Information Sciences and Systems (CISS), 2012, pp. 1–6. [35] J. Ward, “Space-time adaptive processing for airborne radar,” in IEE Colloquium on SpaceTime Adaptive Processing (Ref. No. 1998/241), 1998, pp. 2/1–2/6. [36] H. L. Van Trees, Detection, Estimation, and Modulation Theory: Optimum Array processing. John Wiley & Sons, 2004. [37] B. Eisenberg, “On the expectation of the maximum of IID geometric random variables,” Statistics & Probability Letters, vol. 78, no. 2, pp. 135–143, 2008. [38] B. D. Steinberg, “The peak sidelobe of the phased array having randomly located elements,” IEEE Transactions on Antennas and Propagation, vol. 20, no. 2, pp. 129–136, 1972. [39] J. Ward, “Space-time adaptive processing with sparse antenna arrays,” in Conference Record of the Thirty-Second Asilomar Conference on Signals, Systems and Computers 1998, pp. 1537–1541. [40] R. Klemm, Principles of Space-Time Adaptive Processing. IET, 2002, no. 159. [41] F. C. Robey, D. R. Fuhrmann, E. J. Kelly, and R. Nitzberg, “A CFAR adaptive matched filter detector,” IEEE Transactions on Aerospace and Electronic Systems, vol. 28, no. 1, pp. 208– 216, 1992. [42] E. J. Kelly, “An adaptive detection algorithm,” IEEE Transactions on Aerospace and Electronic Systems, no. 2, pp. 115–127, 1986. [43] I. S. Reed, J. D. Mallett, and L. E. Brennan, “Rapid convergence rate in adaptive arrays,” IEEE Transactions on Aerospace and Electronic Systems, no. 6, pp. 853–863, 1974. [44] E. Kelly, “Performance of an adaptive detection algorithm; rejection of unwanted signals,” IEEE Transactions on Aerospace and Electronic Systems, vol. 25, no. 2, pp. 122–133, 1989. [45] D. M. Boroson, “Sample size considerations for adaptive arrays,” IEEE Transactions on Aerospace and Electronic Systems, no. 4, pp. 446–451, 1980. [46] R. T. Behrens and L. L. Scharf, “Signal processing applications of oblique projection operators,” IEEE Transactions on Signal Processing, vol. 42, no. 6, pp. 1413–1424, 1994. [47] L. L. Scharf and B. Friedlander, “Matched subspace detectors,” IEEE Transactions on Signal Processing, vol. 42, no. 8, pp. 2146–2157, 1994. [48] J. R. Guerci, J. S. Goldstein, and I. S. Reed, “Optimal and adaptive reduced-rank STAP,” IEEE Transactions on Aerospace and Electronic Systems, vol. 36, no. 2, pp. 647–663, 2000. [49] J. Liu, H. Li, and B. Himed, “Threshold setting for adaptive matched filter and adaptive coherence estimator,” IEEE Signal Processing Letters, vol. 22, no. 1, pp. 11–15, 2015.

6

Fast and Robust Sparsity-Based STAP Methods for Nonhomogeneous Clutter Xiaopeng Yang, Yuze Sun, Xuchen Wu, Teng Long, and Tanpan K. Sarkar

6.1

Introduction Space–time adaptive processing (STAP) can effectively suppress clutter and achieve better moving target detection, as compared with traditional nonadaptive methods [1]. However, in a practical nonhomogeneous environment, the clutter characteristics would change fast and dynamically, making it difficult to collect a sufficient number of independent and identically distributed (i.i.d.) training samples for effective estimation of the clutter covariance matrix. Meanwhile, the computation of high-dimensional covariance matrix inversion requires a prohibitive computational complexity [2–4]. Therefore it is important to develop STAP methods for robust clutter suppression and fast computation capabilities. In the past three decades, many suboptimal STAP algorithms have been developed [4–8]. The reduced-dimension STAP algorithms, such as the pulse repetition interval (PRI)-staggered algorithm [4] the extended factored algorithm (EFA) [5,6], and the joint-domain localized (JDL) algorithm [7] can reduce the requirement of training samples. However, the covariance matrix inversion in computing the adaptive weight vector is still a time-consuming task for clutter suppression. On the other hand, the reduced-rank STAP algorithms [9–13], such as the principle components (PCs) [9–11], conjugate gradient (CG) [12], and the projection approximation subspace tracking (PAST) [13,14], can also reduce the requirement of training samples. However, because the performance of reduced-rank STAP algorithms is sensitive to the variation of clutter environment, an inappropriate rank selection may lead to a severe clutter suppression performance loss. In the past decade, some sparsity-based STAP methods have been investigated that significantly reduce the requirement of training samples [15–25]. It has been explored that the intrinsic sparsity of the clutter spatial-temporal power spectrum and the space–time adaptive weight vectors can be well utilized for clutter suppression [20,23]. However, the conventional sparse recovery methods cannot obtain desirable sparse recovery accuracy and convergence speed in a practical environment. Therefore, it is necessary to study on the fast and robust sparsity-based STAP methods for the practical complex clutter environment. In this chapter, some sparsity-based STAP methods are developed by exploiting the intrinsic sparsity of the clutter spatial-temporal power spectrum and the space–time adaptive weight vectors. Firstly, the signal model of airborne phased array radar is introduced, and then the intrinsic sparsity of STAP is analyzed according to the clutter 165

166

Yang, Sun, Wu, Long, and Sarkar

spatial-temporal power spectrum and the space–time adaptive weight vectors. Secondly, based on the sparsity of the clutter spatial-temporal power spectrum, a robust and fast iterative sparse recovery method for STAP is proposed, which can not only alleviate the effect of noise and dictionary mismatch but also reduce the computational complexity by recursive inverse matrix calculation. Afterwards, based on the sparsity of space–time adaptive weight vectors, a fast STAP method based on PAST with sparse constraint is proposed, which can provide a more robust and stable estimation of clutter subspace for a small set of training samples. Based on the simulated and the actual airborne phased array radar data, it is verified that the proposed methods can provide better performance with small training sample support in a practical complex nonhomogeneous environment.

6.2

Signal Models The side-looking antenna array configuration of an airborne phased array radar is considered and the corresponding geometry is shown in Figure 6.1. The N-element uniform linear antenna array aligns with the velocity direction of the platform, and the interelement spacing is dA . During each coherent processing interval (CPI), M identical pulses are transmitted with a pulse repetition frequency (PRF) of fr . The operation wavelength is λ, the height of platform is h with a velocity denoted by va , and L fast time samples are collected to cover the detection region in each pulse repetition interval (PRI). Each CPI data of the received signal is stored as an N × M × L data-cube, as shown in

Figure 6.1 Geometry of a side-looking antenna array configuration.

Fast and Robust Sparsity-Based STAP Methods for Nonhomogeneous Clutter

167

Figure 6.2 One CPI data-cube of airborne phased array radar.

Figure 6.2. Each slice corresponding to a fast-time sample of the data cube is an N × M matrix, which is stacked as an N M × 1 vector according to the channel order. It is known that radar target detection is a binary hypothesis problem, where hypothesis H1 corresponds to target presence and hypothesis H0 corresponds to target absence H0 : x = xc + xn,H1 : x = xc + xt + xn,

(6.1)

when x is composed of clutter xc , the received target echo xt , and the zero-mean complex Gaussian, spatially and temporally white noise xn . According to the geometry shown in Figure 6.1, the target can be model as a strong scatter point, whose azimuth angle is θt and the elevation angle is ϕt , the relative velocity is vt . Therefore the target echo xt ∈ CN M×1 can be given as xt = ξ˜t (v(ωt ,ϑt )) = ξ˜t (b(ωt ) ⊗ a(ϑt )) ,

(6.2)

˜ where $ ϑ%t = dA cos (θt ) $ ξ% t denotes the target complex amplitude, the spatial frequency is sin ϕt /λ, the Doppler frequency is ωt = 2vt cos (θt ) sin ϕt /λfr , b(ωt ) = [1, exp(j 2πωt ),. . ., exp(j (M − 1)2πωt )]T ∈ CM×1 is the target temporal steering vector, and a(ϑt ) = [1, exp(j 2πϑt ),. . ., exp(j (N − 1)2πϑ t )]T ∈ CN×1 is the target spatial steering vector. Meanwhile, the clutter at each range cell is the superposition of Nc independent clutter patches, which are distributed in azimuth with angle interval ϕ = 2π/Nc . Each clutter patch can be denoted by the azimuth angle θ and elevation angle ϕ according to each range cell. The spatial frequency ϑc,i and the normalized Doppler frequency ωc,i of the ith clutter patch of CUT are respectively expressed as ϑc,i =

$ % $ % $ % $ % dA 2va cos θl,i sin ϕl . cos θl,i sin ϕl ,ωc,i = λ λfr

(6.3)

Therefore, the space–time steering vector of the ith clutter patch is expressed as v(ωc,i ,ϑc,i ) = b(ωc,i ) ⊗ a(ϑc,i ),

(6.4)

168

Yang, Sun, Wu, Long, and Sarkar

where b(ωc,i ) = [1, exp(j 2πωc,i ),. . ., exp(j (M − 1)2πωc,i )]T ∈ CM×1 is the temporal steering vectors and a(ϑc,i ) = [1, exp(j 2πϑc,i ),. . ., exp(j (N − 1)2πϑc,i )]T ∈ CN×1 is the spatial steering vectors, respectively. Thus, based on Melvin’s model in [4], the space–time clutter snapshot xc ∈ CN M×1 can be expressed as xc =

Nc

ξ˜i v(ωc,i ,ϑc,i ),

(6.5)

i=1

where ξ˜i denotes the random complex amplitude corresponding to the ith clutter patch. The conventional STAP weight vector w ∈ CN M×1 is derived with the minimum noise variance (MNV) principle [1], which is shown as the following constrained power minimization problem

4 42 4 4 (6.6) min J (w) = E 4wH x4 s.t. wH v(ωt ,ϑt ) = 1, w

2

where v (ωt ,ϑt ) is the steer vector of the target. Then by using the method of Lagrange multipliers, the optimal adaptive weighting vector is obtained as w=

R−1 v(ωt ,ϑt ) . −1 t ,ϑ t )R v(ωt ,ϑ t )

vH (ω

(6.7)

It is well known and analyzed in some reference that the space–time covariance matrix R ∈ CN M×N M is usually estimated from K i.i.d. training samples around the lth range cell under test (CUT) [1,2,4], i.e., R˜ = E xx H ≈

1 K −1

K−1

xk xkH = R˜ c + R˜ n .

(6.8)

k=1,k!=l

˜ c denotes the clutter covariance matrix, the noise covariance matrix is In (6.8), R 2 ˜ Rn = σ I, the noise power is σ2 and I is the N M × N M identity matrix.

6.3

Sparsity Principle Analysis of STAP

6.3.1

Sparsity of the Clutter Spatial-Temporal Spectrum It is well known that the eigenvalue decomposition (EVD) of the space–time covariance matrix R is obtained as [26] R = Uc c Uc + σ2 Un UH n,

(6.9)

where Uc denotes the clutter subspace spanned by the principal eigenvectors, c = diag (ς1,. . .,ς P ) consists of the P principal eigenvalues of R, and Un denotes the noise subspace. The clutter covariance matrix can be denoted by only P principal eigenvalues instead of Nc clutter space–time steering vector. The clutter is sparse with respect to

Fast and Robust Sparsity-Based STAP Methods for Nonhomogeneous Clutter

169

0.012 pi/2 pi/3 pi/6

Degree of correlation

0.01

0.008

0.006

0.004

0.002

0 0

0.1

0.2

0.3 0.4 Spatial angle/rad

0.5

0.6

Figure 6.3 Correlations between space–time steering vectors with spatial angle π/2, π/3, and π/6.

the system degrees of freedom (DoFs). In order to further demonstrate the sparsity of clutter, the space–time correlation coefficient is given as 4 4 4v(ωp,ϑp )H v(ωq ,ϑq )4 4 4 4, cor v(ωp,ϑp ),v(ωq ,ϑq ) = 4 4v(ωp,ϑp )4 4v(ωq ,ϑq )4 $

%

(6.10)

which describes the degree of correlation between different space–time steering vectors, and the correlation results corresponding to three different spatial angles are shown in Figure 6.3. It is found that the space–time steering vector is highly correlated to the vectors that are spatially adjacent. It means that the clutter has a high correlation with the spatial angle, and the space–time steering vector near the clutter ridge can replace all the vectors in the spatial-temporal plane to approximate the clutter [27]. Therefore the received space–time data shows high sparsity in the angle-Doppler domain [19,20]. As shown in Figure 6.4, the major components of clutter spectrum are distributed near the ridge. Thus, compared with the whole spatial-temporal plane, the complex amplitude of received spectrum in most area is rather small, so the received space–time data is sparse with respect to the whole angle-Doppler domain. Based on this property, the homogeneous clutter can be well estimated by the sparse recovery approaches. The spatial-temporal plane is discretized into a grid with Ns spatial bins and N % bins, where each grid point is associated with a space–time steering $ d Doppler vector v fd,i ,fs,j . Therefore the space–time snapshot of the lth range cell can be given as xl = xc + xn = ϒ + xn,

(6.11)

Yang, Sun, Wu, Long, and Sarkar

0.5

0

0.4

–5

0.3 Normalized Doppler

170

–10

0.2 –15

0.1

–20

0 –0.1

–25

–0.2 –30

–0.3

–35

–0.4 –0.5 –0.5

–0.25

0 0.25 Normalized angle

0.5

–40

Figure 6.4 Sparse distribution of spatial-temporal spectrum in angle-Doppler domain.

Figure 6.5 The procedure of clutter suppression based on sparsity of spatial-temporal spectrum.

where ϒ = γ˜1,1, γ˜1,2,. . ., γ˜Ns ,Nd ∈ CNs Nd ×1 is the complex amplitude of the spectral distribution. The space–time overcomplete dictionary matrix ∈ CN M×Ns Nd is given as the collection of all space–time steering vectors, i.e., $ % $ % $ % = v fd,1,fs,1 ,. . .,v fd,i ,fs,j ,. . .,v fd,Nd ,fs,Ns . (6.12) Therefore, when the complex amplitude of the spectral distribution and the corresponding space–time steering vectors are estimated effectively based on the property of sparse distribution, the space–time covariance matrix R can be well reconstructed for clutter suppression. As Ns Nd is much bigger than the system DoFs N × M, the space– time dictionary is overcomplete and highly correlated, so that (6.11) is underdetermined. However, based on the theory of sparse recovery [28,29], the ill-posed equation

Fast and Robust Sparsity-Based STAP Methods for Nonhomogeneous Clutter

171

can be solved effectively with limited number of training sample. The sparse recovery of clutter spectrum can be solved by the following L1 -norm optimization ˆ = arg minϒ1 s.t. xl − ϒ2 ≤ ε, ϒ

(6.13)

where L1 -norm guarantees the sparsity of complex amplitude ϒ, and the L2 -norm restrains the estimation error within ε. Equation (6.13) can also be given as ˆ = arg min xl − ϒ2 + λ γ ϒ1, ϒ ϒ

(6.14)

where λ γ is the regularization parameter. In order to obtain a better spectrum distribution of homogeneous clutter, the adjacent training data can be utilized in the same processing. Afterwards, the covariance matrix R can be reconstructed as R=

Ns P Nd $ % $ % 1 γ˜p,i,j 2 vp fd,i ,fs,j vH fd,i ,fs,j + σ2 I, P

(6.15)

p=1 i=1 j =1

$ % where P is the number of training data, vp fd,i ,fs,j is the space–time steering vector in the overcomplete dictionary corresponding to the recovery result of each training data.

6.3.2

Sparsity of Space–Time Adaptive Weight Vectors As mentioned in the previous section, the clutter subspace is spanned by the eigenvectors of the corresponding P largest eigenvalues, thus clutter suppression involves a rankdeficient problem. According to the Brennan’s rules [3], the rank of the clutter subspace is much smaller than the DoFs. In other words, the available length of the adaptive weight vector determined by the number of antenna channels and the slow-time pulses is much larger than the required length for clutter suppression, implying the sparsity of the adaptive weight vector that can be exploited. Sparse least mean square (LMS)-type algorithms and

Figure 6.6 The relationship between the full space–time dimension and clutter subspace.

172

Yang, Sun, Wu, Long, and Sarkar

recursive least square (RLS)-type algorithms applied to system identification are studied in [30,31], which results in a performance improvement for sparse systems, meanwhile, several novel STAP methods with sparse constraint have recently been proposed in [23,24]. Among these methods, the concept of the L1 -norm sample matrix inversion (SMI) method is reviewed in this section. By employing the sparse constraint to the MNV cost function, the problem is described as the following constrained optimization problem

4 4 4 H 42 (6.16) min J1 (w) = E 4w xl 4 + 2κ (w) s.t. wH vt (ωt ,ϑt ) = 1, w

2

where (w) is a term to characterize the sparsity of the weight vector, and κ is a positive scalar that provides a trade-off between the sparsity and the output power. A larger value of κ implies that more components will be shrunk to zero. Because of the convexity of the L1 -norm constraint, it is common practice to approximate the sparse constraint as (w) = w1 [24]. Therefore the adaptive weight vector w deduced by Yang et al. [23] is ˆ + κ −1 vt (ωt ,ϑt ) R , (6.17) w= −1 ˆ vH vt (ωt ,ϑt ) t (ωt ,ϑ t ) R + κ where = diag |w11|+ε , |w21|+ε ,. . ., |wNM1 |+ε and ε is a small positive constant and wi , i = 1,2,. . .,NM are the entries of the adaptive weighting vector w. Compared with the adaptive weight vector obtained in (6.7), it is obvious that the sparse constraint yields ˆ an additional term κ in the inversion of the estimated clutter covariance matrix R. However, the adaptive weight vector in (6.17) is not a closed-form solution, since is a function of w. Some iterative methods, such as L1 -norm RLS and L1 -norm recursive SMI methods, can be used to compute the adaptive weight vector [23,24].

6.4

Fast and Robust Sparsity-Based STAP Methods

6.4.1

Robust and Fast Iterative Sparse Recovery Methods for STAP In the past decade, by exploiting the intrinsic sparsity of the clutter in the angle-Doppler domain, some sparse-based STAP methods have been proposed to achieve the sparse recovery of the clutter spatial-temporal spectrum with limited training data [17–22]. A global matched filter (GMF) [17] is firstly applied to the STAP, which demonstrates that both target and clutter can be identified based on a single snapshot without prior estimation of the clutter covariance matrix. Then, by assuming the knowledge of the clutter ridge in the spatial-temporal plane, targets are estimated as the sparse solution in the angle-Doppler domain outside this clutter ridge [18]. In these methods, the sparse recovery of the clutter spatial-temporal spectrum is usually formulated as a regularized optimization problem, which can be solved by convex optimization [19]. The computational complexity of such an operation will increase beyond our capacity when the

Fast and Robust Sparsity-Based STAP Methods for Nonhomogeneous Clutter

173

dimension of convex optimization becomes large, thereby making STAP implementation very difficult. Then a series of fast approximation algorithms have also been proposed [19–22]. The FOcal Underdetermined System Solver (FOCUSS) is one of the typical fast approximation algorithms, which uses the weighted L2 -norm minimization to recursively achieve the approximate estimation of the extinct clutter profiles [19]. However, because the performance is heavily affected by the mismatch of the overcomplete dictionary, and the regularization parameter cannot be adjusted adaptively according to the environment, the performance of FOCUSS would degrade in practical application. In this section, by exploiting the intrinsic sparsity of the clutter in the angle-Doppler domain, a robust and fast iterative sparse recovery method for STAP is given [21]. In the proposed method, the sparse recovery of the clutter spatial-temporal spectrum and the calibration of the space–time overcomplete dictionary are executed iteratively. The robust solution of sparse recovery is derived by regularized processing and calculated recursively based on the block Hermitian matrix property; afterwards the mismatch of the space–time overcomplete dictionary is calibrated by minimizing the cost function. The proposed method is verified based on simulated and actual airborne phased array radar data. In an actual clutter environment, the clutter component would possibly be located between two grids rather than the exact gird point of the dictionary, so the mismatch between the space–time overcomplete and actual clutter distribution cannot be avoided. When the mismatch is considered, the space–time snapshot of lth range cell can be changed into

xl = xc + xn =

Nd Ns

% $ γ˜i,j v fd,i ,fs,j + xn = ϒ l + xn,

(6.18)

i=1 j =1

where = + denotes the actual overcomplete dictionary and is the mismatch matrix. Therefore, a robust and fast iterative sparse recovery method for STAP in practical environments is proposed, and the main procedures of the proposed method are mainly demonstrated in the following explorations.

A. Method Formulation a. Sparse Recovery Processing The effect of additive noise is not considered in the basic sparse recovery method [19]. However, the additive noise is inevitable in practical environments, which will increase the recovery error. Therefore, the regularized processing is employed to reduce the effect of additive noise in the sparse recovery. In this method, the fast approximation FOCUSS method is utilized for clutter sparse recovery. The sparse recovery problem in (6.14) can be converted as

min J (ϒ l ) = ϒl

N s Nd i=1

4 4 4 γl,i 4 s.t. xl − ϒ l 2 ≤ ε, p

(6.19)

174

Yang, Sun, Wu, Long, and Sarkar

then the cost function is given by the Lagrange multiplier method L (ϒ l ) = J (ϒ l ) + αxl − ϒ l 2,

(6.20)

where α denotes the Lagrange multipliers that match the noise level. By solving the gradient of ϒ l , we can get 8 % $ (6.21) ∇L (ϒ l ) = |p| (ϒ l ) ϒ l + α H ϒ l − H xl , p−2 p−2 % $ 9 , then the appropriate Hessian matrix where (ϒ l ) = diag γl,1 ,. . ., γl,Ns Nd can be obtained by 8 (6.22) ∇ 2 L (ϒ l ) = |p| (ϒ l ) + αH . Afterwards, by applying the quasi-Newton method, we can get (k+1)

γl

(k)

= γl

0 $ (k) %1−1 $ (k) % − ∇ 2 L γl · ∇L γl .

(6.23)

Then, by substituting (6.21) and (6.22) into (6.23), we can get (k+1)

ϒl

0 8$ 1−1 (k) % = δ ϒ l + H H xl ,

(6.24)

%−2 $ 9 $ (k) % and (k) = W (k) , where = ϒl where δ=1/α. By defining W (k) & ' p p 1− /2 (k) 1− /2 (k) W (k) = diag γl,1 ,. . ., γl,Ns Nd is the diagonal weighting matrix at the kth iteration, the solution at kth iterative can be derived as (k+1)

ϒl

$ %H 0 $ %H 1−1 = W (k) (k) xl . δI + (k) (k)

(6.25)

It is easily found from (6.25) that the matrix inversion is still needed to calculate complex amplitude, which will significantly influence the convergence of iteration. Although the adaptive subspace selection [19] can be applied to reduce the dimension of complex amplitude in the iterations, the direct inverse calculation still cannot be avoided. However, based on the mathematical analysis, it can be proved that the $ %H matrix T=δI+(k) (k) is a Hermitian matrix, therefore the matrix inversion can be calculated recursively based on the block Hermitian matrix property [33,34]. The low-resolution estimation based on the Fourier spectrum is employed as the ini(0) tial value of ϒ l , i.e., ϒ l = H xl , and then the calculation can be executed iteratively as (6.25). During the iterations, the prominent components in are gradually reinforced, while the remaining small components are suppressed until they become close to zero. (k) Finally, when the absolute difference of ϒ l is smaller than the convergence threshold, the sparse recovery result is obtained. From (6.25), it can be found that when the noise level is reduced to 0, i.e., δ → 0, the proposed method will degenerate to the FOCUSS method. Because the regularized processing is applied in the iteration, the proposed method can effectively improve the recovery performance under noise.

Fast and Robust Sparsity-Based STAP Methods for Nonhomogeneous Clutter

175

b. Mismatch Calibration Processing The mismatch between the space–time overcomplete dictionary and actual clutter distribution is ignored in conventional sparsity-based STAP methods, which would decrease the performance of clutter suppression. Therefore, the mismatch calibration processing (k) is investigated in this section. After ϒ l is obtained at the kth iteration, the estimation of can be obatined by 4 4 % 4 $ (k) 4 (6.26) (k) = arg min J (k) = 4(k) 42 + 4xl − (k) ϒ l 42 . (k)

(k)

(k)

By defining e(k) = xl − ϒ l , and y(k) = ϒ l , the cost function of (6.26) can be given as 4 4 4 % 4 $ (6.27) J (k) = 4(k) 42 + 4e(k) − (k) y(k) 42 . Then the mismatch can be calculated by solving gradient equation % $ & $ % ' $ %H ∂J (k) H (k) (k) (k) H y = + y − e(k) y(k) = 0. (k) ∂ Therefore the estimation of can be obtained '−1 $ %H & (k) $ (k) %H (k) = e(k) y(k) + H . y y

(6.28)

(6.29)

Afterwards, the space–time overcomplete dictionary is calibrated at the kth iteration by (k) = +(k) . It can be found that the mismatch of the space–time overcomplete dictionary can be calibrated gradually by minimizing the corresponding cost function, so that the mismatch between the dictionary and actual clutter distribution has been reduced effectively. When the following convergence condition is satisfied as ϒ (k+1) − ϒ (k) l l (6.30) ≤ ξ, (k+1) ϒl the iteration will stop, and then the sparse recovery is achieved. Then the reconstruction of space–time covariance matrix R and adaptive weighting vector can be calculated correspondingly.

B. Method Verification In this section, simulated data and two actual measured airborne phased array radar data sets (MCARM [34] and one other data) are used to verify the clutter suppression performance of the proposed methods, and compared with SMI, EFA [3], STAP using CVX [19], and FOCUSS [19] methods.

a. Simulated Data The simulation parameters are listed in Table 6.1, where ρs and ρ d are set to be 4 in the simulations, and all the results are averaged over 500 Monte Carlo runs. The output SINRs versus the number of snapshots based on SMI, EFA, STAP using CVX, FOCUSS, and the proposed method are investigated and the results are shown in Figure 6.7. It is found that the sparsity-based STAP methods can obtain desirable

Yang, Sun, Wu, Long, and Sarkar

Table 6.1 Simulation parameters. Parameter

Value

Parameter

Value

Number of spatial elements

8

8

Radar frequency Platform velocity Main beam look direction Target normalized Doppler frequency

450 MHz 200 m/s Side-looking 0.15

Number of temporal pulses in a CPI Pulse repetition frequency Height of platform Clutter-to-noise ratio (CNR) SNR

1200 Hz 12 km 40 dB 5 dB

24

22

20 Output Power [dB]

176

18

16

14 OPT SMI EFA FOCUSS STAP using CVX Proposed Method

12

10 124 8

16

32

48

64 72 Range Cells

96

128

Figure 6.7 Output SINRs versus number of snapshots.

SINR performance with small training sample support, which exhibits much faster convergence than conventional STAP methods. Meanwhile, the proposed method can provide better performance than other sparsity-based methods with the same number of training samples, because the calibration of space–time overcomplete dictionary and the regularization in the sparse recovery processing are applied in the proposed method. The output SINRs with four training samples are investigated correspondingly and the results are shown in Figure 6.8. It is found that the SMI and EFA methods cannot obtain desirable SINR with minimal training sample support, owing to the insufficient estimation of the clutter covariance matrix, so that these two methods could not provide desirable target detection performance in a practical clutter environments. However, the sparsity-based STAP methods could provide much better performance in the instance of good estimation of clutter distribution, so that the clutter covariance matrix can be well reconstructed. Moreover, the proposed method can obtain better sparse recovery performance than conventional sparsity-based STAP methods, because of the calibration

Fast and Robust Sparsity-Based STAP Methods for Nonhomogeneous Clutter

177

25

15

SINR [dB]

5

−5

−15

OPT SMI EFA FOCUSS STAP using CVX Proposed Method

−25

−35

−45 −0.5

−0.4

−0.3

−0.2

−0.1 0 0.1 0.2 Normalized Doppler

0.3

0.4

0.5

Figure 6.8 Output SINRs of SMI, EFA, STAP using CVX, FOCUSS, and our proposed method.

of the space–time overcomplete dictionary and regularization processing. The proposed method can effectively improve the SINR performance, especially in low Doppler frequency regions where the target located. In the following, the range detections with four training samples are also investigated correspondingly, and the results are shown in Figure 6.9. It accords with the results in Figure 6.8, in that the conventional SMI and EFA methods cannot suppress the clutter effectively with small training sample support, and this leads to undesirable target detection performance in practical clutter environments. However, the sparsitybased STAP methods can obtain desirable range detection performance, and moreover the proposed method can produce the larger difference of output power between the tested range cells and adjacent range cells than FOCUSS and CVX methods, which will be very useful for target detection in practical complex nonhomogeneous environments.

b. MCARM Data The MCARM data [34] is used to verify the STAP methods in this section. The array was an L-band phased array antenna using 22 elements arranged as a 2 × 11 configuration. The PRF of the radar is 1,984 Hz, 128 pulses are contained in one CPI, the platform velocity is 100 m/s, and the height of the platform is 3,078 m. We note that ρ s and ρ d are both set to 6 while 12 pulses and 8 elements data of MCARM are used, the target is located at the 299th range cell with −0.15 Doppler frequency, and 4 range cell data around the 299th range cell are selected as the training samples. The output SINRs are investigated and the results are shown in Figure 6.10. It is similar with the previous simulated results that SMI and EFA methods cannot obtain desirable SINR with minimal training sample support. Moreover, the proposed method

Yang, Sun, Wu, Long, and Sarkar

40 30

Output Power [dB]

20 10 0 −10 −20 −30

SMI EFA FOCUSS STAP using CVX Proposed Method

−40 −50 1

50

100

150

Range Cells

Figure 6.9 Range detections of SMI, EFA, STAP using CVX, FOCUSS, and our proposed

method. 25

10

SINR [dB]

178

0

−10 SMI EFA FOCUSS STAP using CVX Proposed Method

−20

−0.5

−0.4

−0.3

−0.2

−0.1 0 0.1 0.2 Normalized Doppler

0.3

0.4

0.5

Figure 6.10 Output SINRs of SMI, STAP using CVX, FOCUSS, and our proposed method based on MCARM data.

can obtain better SINR performance in both main-lobe and side-lobe regions than in conventional sparsity-based STAP methods. The range detections are also investigated and the results are shown in Figure 6.11. It is similar to the previous simulated results, in that the SMI and EFA methods cannot

Fast and Robust Sparsity-Based STAP Methods for Nonhomogeneous Clutter

179

0

Output Power [dB]

−10

−20

−30

SMI EFA FOCUSS STAP using CVX Proposed Method

−40

250

270

290 310 Range Cells

330

350

Figure 6.11 Range detections of SMI, EFA, STAP using CVX, FOCUSS, and our proposed method based on MCARM data.

detect the target effectively, while the sparsity-based STAP methods can provide desirable detection. The proposed method can also obtain a larger difference of the output power between the tested range cell and adjacent range cells than the FOCUSS method and STAP using CVX, so that the target can be detected correctly.

c. Actual Measured Airborne Radar Data The actual measured airborne radar data is also applied to verify the proposed method comparing with SMI, EFA, CVX, and FOCUSS methods in the section. The actual measured airborne radar data consists of 16 spatial channels and 128 temporal pulses in a CPI. As before, ρs and ρd are both set to 6, and 100 range snapshots of the first 8 channels and the first 12 pulses are used, a strong target is located in the 231th range cell with the normalized Doppler of about 0.07, and 4 range cell data around the 231th range cell are selected as the training samples. The output SINRs are also investigated and the results are shown in Figure 6.12. It is similar to the previous results that were based on simulated data and MCARM data, in that the SMI and EFA methods cannot obtain desirable SINR, while the sparsitybased STAP methods provide better performance with limited training samples, and the proposed method can obtain better performance, especially in the main-lobe region. The range detection results are shown in Figure 6.13. It is also similar to the previous results that were based on simulated data and MCARM data, in that SMI and EFA methods cannot detect the target effectively, while the sparsity-based STAP methods provide desirable detection, and the proposed method can obtain a greater disparity in output power between the tested range cell and adjacent range cells than FOCUSS and CVX methods.

Yang, Sun, Wu, Long, and Sarkar

25

Output SINR [dB]

10

0

−10 SMI EFA FOCUSS STAP using CVX Proposed Method

−20

−0.5

−0.4

−0.3

−0.2

−0.1 0 0.1 Normalized Doppler

0.2

0.3

0.4

0.5

Figure 6.12 Output SINRs of SMI, STAP using CVX, FOCUSS, and our proposed method based on actual measured airborne radar data.

85

75

Output Power [dB]

180

65

55

45

SMI EFA FOCUSS STAP using CVX Proposed Method

35 200

220

240 260 Range Cells

280

Figure 6.13 Range detections of SMI, EFA, STAP using CVX, FOCUSS, and our proposed method based on actual measured airborne radar data.

Fast and Robust Sparsity-Based STAP Methods for Nonhomogeneous Clutter

6.4.2

181

Fast PAST Methods with Sparse Constraint for STAP A. Method Formulation It is well known that the clutter covariance matrix has a low rank property, which means that fewer DoFs are required than those offered by the system, thus a high degree of sparsity can be exploited in the adaptive weight vector. Based on this property, several sparse constrained STAP methods have been proposed [23,24]. In these approaches, by imposing the sparse constraint to the conventional STAP cost function, the adaptive weight vector is altered to yield a performance improvement compared with the STAP methods that do not apply sparse constraint. However, L1 -norm RLS and L1 -norm SMI methods do not fully utilize the low-rank property, which means that extra training samples are needed to perform the STAP [23]. On other hand, the L1 -norm conventional conjugate gradient (L1 -norm CCG) method requires multiple iterations for each input data and thus leads to increased computational complexity [24]. In order to improve the clutter suppression capability with minimal training sample support, by further exploiting the low-rank property of the clutter covariance matrix, a fast STAP method–based PAST with the sparse constraint is given [25]. In the proposed method, the sparse constraint is imposed into the cost function of PAST, and the adaptive weight vector is then derived iteratively by using the RLS method and matrix inversion lemma to update the autocorrelation matrix and cross correlation terms. Because of the sparse constraint in PAST, the proposed method provides more robust estimation of the clutter subspace. Therefore the clutter suppression performance can be significantly improved effectively when a small training sample is used. Through simulated results and two sets of actual airborne phased array radar data, it is verified that the proposed method achieves nearly the same performance at lower computational complexity compared with existing sparsity-constrained STAP methods, and provides performance improvement compared with conventional STAP methods without sparse constraint. According to the eigencanceller [9,13], the corresponding weight vector can be obtained by wEV D =

$ % ˆ cU ˆH I−U c vt (ωt ,ϑ t ) . % $ ˆ ˆH vH t (ωt ,ϑ t ) I − Uc Uc vt (ωt ,ϑ t )

(6.31)

By reducing the required number of sample support from 2N K to 2P , the EVDbased method can improve the convergence of clutter suppression. However, because of the direct EVD processing, the computational complexity of the EVD-based method remains O((N M)3 ), which is impractical for real-time processing. To remedy this problem, the PAST technique is employed to reduce the computational complexity for clutter subspace acquisition [35,36]. Owing to the approximation in the iteration, the PAST method may suffer performance loss compared with the EVD method, especially when the environment is nonhomogeneous. In order to improve the clutter suppression performance when only a small training sample support is available, we propose the fast STAP method based on PAST with sparse constraint in this section.

182

Yang, Sun, Wu, Long, and Sarkar

The PAST method [35] is derived by minimizing the mean squares error between the space–time received data and its projection on the clutter subspace, which is denoted by 54 42 6 4 ˆH ˆ c ) = E 4xl − W ˆ cW J2 (W c xl 2 (6.32) % $ H % $ % $ H ˆW ˆ c + tr W ˆW ˆ c·W ˆ ˆc R ˆH ˆ − 2tr W ˆc R = tr R c Wc , ˆ denotes the ˆ c spans the clutter subspace as Uc , and tr(R) where the N M × P matrix W ˆ ˆ trace of the covariance matrix R. The cost function J2 (Wc ) obtains its global minimum ˆ Otherwise, all the stationary only when Us contains the principal eigenvectors of R. ˆ c ) are saddle points. Therefore, similar to the L1 -norm SMI method, by points of J2 (W ˆ c ), the cost function of the proposed method is imposing the sparse constraint to J2 (W described as 54 42 6 $ % 4 ˆH ˆ c ) = E 4xl − W ˆ cW ˆ J3 (W c xl 2 + 2κ Wc % $ % $ H (6.33) ˆW ˆc ˆ − 2tr W ˆc R = tr R % $ % $ H ˆW ˆ c·W ˆ ˆ ˆH ˆc R + tr W c Wc + 2κ Wc . The cost function in (6.33) can be represented in the following exponentially weighted form J3 (Wˆ c (k)) =

k

4 42 $ % 4 4 ρ K−i 4xi − Wˆ c (k)Wˆ cH (k)xi 4 + 2κ Wˆ c (k) 2

i=1

$ % $ % ˆ ˆ Wˆ c (k) = tr R(k) − 2tr Wˆ cH (k)R(k) % $ % $ ˆ Wˆ c (k) · Wˆ cH (k)Wˆ c (k) + 2κ Wˆ c (k) , + tr Wˆ cH (k)R(t)

(6.34)

where ρ is a forgetting factor to ensure that data in the distant past are downweighted, which can provides a trade-off between the tracking capability and evaluated error ˆ when the system operates in a nonstationary environment. In addition [11,37], R(k) k H H k−i ˆ − 1) + xk x denotes the exponentially weighted covariance = i=1 ρ xi xi = ρ R(k k ˆ ˆ c (k), iterative matrix. As J3 (Wc (k)) is a fourth-order function of the elements of W ˆ processing is thus necessary to minimize J3 (Wc (k)). The core idea of the PAST method ˆH ˆH is to employ W c (i − 1)xi to approximate the projection Wc (k)xi for 1 ≤ i ≤ k. Therefore, the cost function in (6.34) can be rewritten as J4 (Wˆ c (k)) =

k

4 42 & ' 4 4 ρ k−i 4xi − Wˆ c (k)Wˆ cH (k)xi 4 + 2κ Wˆ c (k) 2

i=1

=

k i=1

4 42 & ' 4 4 ρ k−i 4xi − Wˆ c (k)yi 4 + 2κ Wˆ c (k) ,

(6.35)

2

ˆH where yi = W c (i − 1)xi . The projection approximation changes the error performance ˆ c ). For stationary or slowly time-varying signals, the difference between surface of J3 (W ˆH ˆH W c (i − 1)xi and Wc (k)xi is small, and as the number of iterations increases, the ˆ c (k)) effect of the past input data gradually becomes insignificant. As a result, J4 (W

Fast and Robust Sparsity-Based STAP Methods for Nonhomogeneous Clutter

183

ˆ c ) and a desirable estimation of the clutter subspace can effectively approximate J3 (W ˆ c (k)). can be obtained by minimizing J4 (W ˆ c (k)) is a second-order function of the estimated clutter subspace Note that J4 (W ˆ c (k). Therefore, based on the reference in [23], the sparse constraint is approximation W % $ % $ ˆ c (k) is given as Wˆ c (k) = sign Wˆ c (k) , the sub-gradient of (6.35) with respect to W given as $ % ˆ ˆ ∇JW (6.36) ˆ c (k) = −rxy + Wc (k)Ryy + κsign Wc (k) , where

sign (·) is a component-wise sign function, which is defined as sign (x) x/|x| for x != 0 = [22,23], and rxy (k) and Ryy (k) are given as 0 for x = 0 rxy (k) =

k

H ρ k−i xi yH i = ρrxy (k − 1) + xk yk

i=1

Ryy (k) =

k

(6.37) H ρ k−i yi yH i = ρRyy (k − 1) + yk yk .

i=1

By equating the above gradient terms to zero, the clutter subspace can be estimated as $ $ %% ˆ c (k) = rxy (k) − κsign W ˆ c (k) R−1 W (6.38) yy (k). It is obvious that using (6.38) directly is computationally consuming, but the computational complexity can be effectively reduced by employing the RLS method. Let $ % ˆ c (k) . T (k) = rxy (k) − κsign W (6.39) Then, substituting (6.37) into the above equation, T (k) can be described by the following recursive equation 0 $ % $ %1 ˆ ˆ − κsign W (k) − ρκsign W (k − 1) . (6.40) T (k) = ρT (k − 1) + xk yH c c k Because the instantaneous error of the weight vector changes slowly in each time step, the sign of the weights does not change rapidly. As such, T (k) can be approximated by $ % $ % ˆ T (k) ≈ ρT (k − 1) + xk yH (6.41) k + κ ρ − 1 sign Wc (k − 1) . Therefore, the clutter subspace can be recursively estimated as ˆ c (k) = T (k) R−1 W yy (k).

(6.42)

Then by defining P(k) = R−1 yy (k) and using the matrix inversion lemma, we can get 2 3 P(k − 1)yk yH 1 k P(k − 1) P(k) = P(k − 1) − , (6.43) ρ ρ + yH k P(k − 1)yk which can be rewritten as P(k) = ρ−1 P(k − 1) − g(k)hH (k) ,

(6.44)

184

Yang, Sun, Wu, Long, and Sarkar

where h(k) = P (k − 1) yk , g(k) =

h(k) . ρ+yH k h(k)

By substituting (6.41) and (6.44) into

ˆ c (k − 1) can be obtained as ˆ c (k) and W (6.42), the iterative relationship between W H ˆ c (k) = W ˆ c (k − 1) + xk − W ˆ c (k − 1)yk g (k) W (6.45) % $ 1 ˆ c (k − 1) P (k − 1) − g (k) hH (k) . +κ 1− sign W ρ ˆ c (k) from the proposed method, the space– After estimating the clutter subspace W time adaptive weight vector can be obtained as $ % ˆ cW ˆH I−W c vt (ωt ,ϑ t ) . (6.46) wSC−P AST = % $ ˆ ˆH vH t (ωt ,ϑ t ) I − Wc Wc vt (ωt ,ϑ t ) Compared with the solution of the conventional PAST method [35], because of the sparse constraint, there is an additional term in (6.38). It has been proved in [38] that the mean squared error (MSE) of L1 -norm constraint RLS method is lower than that of RLS, which means that the L1 -norm constraint RLS method represents a significant performance improvement over the conventional RLS method. As in the proposed method, the RLS method is employed for estimating the clutter subspace, thus a more robust solution of the estimated clutter subspace can be obtained because of the sparse constraint. The following three important observations are in order. First, the initial value of P(k) ˆ c (k) should must be a Hermitian and positive-definite matrix, and the initial value of W be composed of P orthogonal vectors. Therefore, without loss of generality, P(0) is set ˆ c (0) are set to the P leading unit to the P × P identity matrix and the columns of W vectors of the N M × N M identity matrix. Second, in order to maintain the Hermitian symmetry, an operation Tri {·} denoted by the operator is employed in the processing, which indicates that only the upper (or lower) triangular part of P(k) is reserved, and its Hermitian transposed version is copied to the opposite triangular part. Third, as PAST may not converge to an orthonormal basis [35], an operation denoted by the operator Orth {·} is employed in each iteration to guarantee the orthogonality between the vectors ˆ c (k). in W

B. Method Verification In this section, simulated data and two actual measured airborne phased array radar data sets (MCARM [34] and one other data) are used to verify the clutter suppression performance of the proposed L1 -norm PAST method. The performance is compared with those obtained from the SMI, EVD, PAST, L1 -norm CCG [23,24], L1 -norm modified conjugate gradient (L1 -norm MCG) [23,24], and shrinkage operator STAP [11] methods.

a. Simulated Data The parameters of the simulated data are listed in Table 6.2. In the simulations, the SMI method is implemented with a loading factor of 10 dB above the noise power, the forgetting factor ρ is set to 0.97, and zero vectors are used to initialize the adaptive

Fast and Robust Sparsity-Based STAP Methods for Nonhomogeneous Clutter

185

Table 6.2 Simulation parameters. Value

Parameter

Value

Number of spatial elements Number of temporal pulses in a CPI Radar frequency Platform velocity Channel spacing

8 8

Main beam look direction Target normalized Doppler frequency Pulse repetition frequency Height of platform Clutter-to-noise Ratio (CNR)

Side-looking 0.2

450 MHz 200 m/s λ/2

70

70

60

60

Improvement Factor[dB]

Improvement Factor[dB]

Parameter

50

40

30 OPT SMI EVD PAST L1−norm PAST

20

10

20

40

60

80

100

120

Number of Samples

140

160

180

200

1200 Hz 12 km 40 dB

50

40

30 OPT L1−norm CCG L1−norm MCG Shrinkage Operator STAP L1−norm PAST

20

10

20

40

60

80

100

120

140

160

180

200

Number of Samples

Figure 6.14 IF performance of EVD, PAST, L1-norm PAST, L1-norm CCG, L1-norm MCG, and shrinkage operator STAP methods versus the number of training samples.

weight vector in the L1 -norm CCG and L1 -norm MCG methods. A scaled N M × N M identity matrix εI is used as the initialization matrix for the PAST, L1 -norm PAST, L1 -norm CCG, and L1 -norm MCG methods, respectively, where ε = 1 × 10−3 is used. All results are averaged over 500 Monte Carlo runs. The improvement factor (IF) performance of the proposed method with respect to the number of training samples is given and compared with the EVD, PAST, L1 -norm PAST, L1 -norm CCG, L1 -norm MCG, and shrinkage operator STAP methods. It is assumed that the selected rank is 15, and the value of sparse constraint parameter is set to κ = 1000. The results are shown in Figure 6.14. It is clear that the SMI method obtains the worst convergence performance, whereas the proposed L1 -norm PAST converges with the lowest number of training samples and outperforms the conventional PAST method. On the other hand, the EVD, L1 -norm CCG and shrinkage operator STAP methods achieve comparable performance. In addition, compared with the L1 -norm PAST, the L1 -norm CCG methods and the shrinkage operator STAP method, the L1 norm MCG method suffers from significant performance loss when the training sample support is small. The IF performance of different methods with respect to the target Doppler frequency using a small training samples support is shown in Figure 6.15. It is assumed that the selected rank is 15, the number of training samples is 40, and the sparse constraint

70

70

60

60

50

50

Improvement Factor [dB]

Improvement Factor [dB]

Yang, Sun, Wu, Long, and Sarkar

40

30

20

0 −0.5

40

30

20 OPT SMI EVD PAST L1−norm PAST

10

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

OPT L1−norm CCG L1−norm MCG Shrinkage Operator STAP L1−norm PAST

10

0 −0.5

0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Normalized Doppler

Normalized Doppler

Figure 6.15 IF performance of SMI, EVD, PAST, L1-norm PAST, L1-norm CCG, L1-norm MCG, and shrinkage operator STAP methods versus the target Doppler frequency. 50

50

40

40

Improvement Factor [dB]

Improvement Factor [dB]

186

30

30

SMI EVD

20

20

PAST L1−norm PAST

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

Normalized Doppler

0.2

0.3

0.4

−0.5

L1−norm CCG L1−norm MCG Shrinkage Operator STAP L1−norm PAST

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

Normalized Doppler

Figure 6.16 IF performance of SMI, EVD, PAST, L1-norm PAST, L1-norm CCG, L1-norm MCG, and shrinkage operator STAP methods based on 200 training samples of MCARM data.

parameter is κ = 1000. Similar to Figure 6.14, the SMI method yields the worst IF performance, while the EVD, the L1 -norm CCG, and shrinkage operator STAP methods perform the best. The proposed L1 -norm PAST obtains better IF performance than the conventional PAST method. On the other hand, the L1 -norm MCG method suffers from some performance loss for the underlying small training sample support case when compared with the L1 -norm PAST and L1 -norm CCG methods.

b. MCARM Data In the verifications, 12 pulses and 8 elements data of the MCARM are applied, and the target is located at the 299th range cell with a normalized Doppler frequency of −0.15. The rank is selected as 19, the SMI method is implemented with a loading factor of 10 dB above the noise power, the value of the sparse constraint parameter is chosen to be κ = 1000, and the initialization matrix for the PAST, L1 -norm PAST, L1 -norm CCG, and L1 -norm MCG methods are chosen to be 10−3 I. First, 201 training samples from range cells number 200 to number 400 are used, and the IF results and range detection are shown in Figures 6.16 and 6.17. It is

Fast and Robust Sparsity-Based STAP Methods for Nonhomogeneous Clutter

5

5 L1−norm CCG

L1−norm CCG

L1−norm MCG

0

L1−norm PAST

Shrinkage Operator STAP L1−norm PAST

−5

−10

−10 Output Power [dB]

Output Power [dB]

L1−norm MCG

0

Shrinkage Operator STAP

−5

−15 −20

−15 −20

−25

−25

−30

−30

−35

−35

−40 280

187

290

300

310

320

330

340

350

−40 280

360

290

300

310

Range Cell

320

330

340

350

360

Range Cell

Figure 6.17 Range detection of SMI, EVD, PAST, L1-norm PAST, L1-norm CCG, L1-norm MCG, and shrinkage operator STAP methods based on 200 training samples of MCARM data.

40

Improvement Factor [dB]

Improvement Factor [dB]

40

30

20

10

30

20

10

SMI

L1−norm CCG

EVD

L1−norm MCG

PAST

Shrinkage Operator STAP L1−norm PAST

L1−norm PAST

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Normalized Doppler

−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Normalized Doppler

Figure 6.18 IF performances of SMI, EVD, PAST, L1-norm PAST, L1-norm CCG, L1-norm MCG, and shrinkage operator STAP methods based on 20 training samples of MCARM data.

seen that, when a sufficient number of training samples are available, all the STAP methods achieve acceptable IF performance and the target can be detected correctly. Next, we only use 21 training samples from range cells number 260 to 280, and the yielding range detection and IF results are depicted in Figures 6.18 and 6.19. It is evident that, as the number of training samples becomes smaller, the SMI and L1 norm MCG methods no longer perform properly, whereas the EVD, PAST, L1 -norm PAST, L1 -norm CCG, and shrinkage operator STAP methods still provide satisfactory performance.

c. Actual Measured Airborne Radar Data The actual measured airborne radar data are collected using 16 spatial channels and consist of a CPI with 128 temporal pulses. We use the first 8 channels and the first 16 pulses for processing. A strong target is located in the 231th range cell with a normalized Doppler frequency of about 0.07, and the clutter rank is selected as 18. In addition, the

5

5

0

0

−5

−5

−10

−10

Output Power [dB]

Output Power [dB]

Yang, Sun, Wu, Long, and Sarkar

−15 −20 −25

−15 −20 −25

−30

−30

SMI

L1−norm CCG L1−norm MCG Shrinkage Operator STAP L1−norm PAST

EVD

−35

−35

PAST L1−norm PAST

−40 280

290

300

310

320

330

340

350

−40 280

360

290

300

310

Range Cell

320

330

340

350

360

Range Cel l

Figure 6.19 Range detection of SMI, EVD, PAST, L1-norm PAST, L1-norm CCG, L1-norm MCG, and shrinkage operator STAP methods based on 20 training samples of MCARM data.

60

60

50

50 Improvement Factor [dB]

Improvement Factor [dB]

188

40

30

30

20

20 SMI EVD PAST L1−norm PAST

10 −0.5

40

−0.4

−0.3

−0.2

−0.1

0

0.1

Normalized Doppler

0.2

0.3

0.4

0.5

L1−norm CCG L1−norm MCG Shrinkage Operator STAP L1−norm PAST

10 −0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Normalized Doppler

Figure 6.20 IF performances of SMI, EVD, PAST, L1-norm PAST, L1-norm CCG, L1-norm MCG, and shrinkage operator STAP methods based on 300 training samples of actual measured airborne radar data.

SMI method is implemented with a loading factor of 10 dB above the noise power, the value of the sparse constraint parameter is selected as κ = 1000, and 10−3 I is used as the initialization matrix for the PAST, L1 -norm PAST, L1 -norm CCG, and L1 -norm MCG methods. We first use 301 training samples from range cells number 120 to 400, and the yielding range detection and IF results are shown in Figures 6.20 and 6.21. Similar to the results based on the MCARM data, as the number of training samples is sufficient in this case, all the STAP methods achieve acceptable IF performance and the target can be detected correctly. We then reduce the number of training samples to 21 collected between range cells number 120 and 140, and the corresponding range detection and IF results are shown in Figures 6.22 and 6.23. Again, similar to the MCARM data case, as the number of the training samples is insufficient in this case, the SMI and the L1 norm MCG methods do not offer satisfactory performance, whereas the EVD, PAST, L1 -norm PAST, L1 -norm CCG, and shrinkage operator STAP methods still function well.

Fast and Robust Sparsity-Based STAP Methods for Nonhomogeneous Clutter

0

−10

−20

−30

−20

−30

−40

−40

−50

−50

150

180

210

240

270

L1−norm CCG L1−norm MCG Shrinkage Operator STAP L1−norm PAST

−10

Output Power [dB]

Output Power [dB]

0

SMI EVD PAST L1−norm PAST

300

189

150

180

210

Range Cells

240

270

300

Range Cells

60

60

50

50 Improvement Factor [dB]

Improvement Factor [dB]

Figure 6.21 Range detection of SMI, EVD, PAST, L1-norm PAST, L1-norm CCG, L1-norm MCG, and shrinkage operator STAP methods based on 300 training samples of actual measured airborne radar data.

40

30

40

30

20

20

L1−norm CCG

SMI EVD PAST L1−norm PAST

10 −0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

L1−norm MCG Shrinkage Operator STAP

10

L1−norm PAST

−0.5

0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Normalized Doppler

Normalized Doppler

Figure 6.22 IF performances of SMI, EVD, PAST, L1-norm PAST, L1-norm CCG, L1-norm MCG, and shrinkage operator STAP methods based on 20 training samples of actual measured airborne radar data. 0

0

−10

−10

Output Power [dB]

Output Power [dB]

−20

−30

−40

−20

−30

−40 −50

SMI EVD PAST L1−norm PAST

−60 150

180

210

240

Range Cells

270

300

L1−norm CCG L1−norm MCG Shrinkage Operator STAP L1−norm PAST

−50 150

180

210

240

270

Range Cells

Figure 6.23 Range detection of SMI, EVD, PAST, L1-norm PAST, L1-norm CCG, L1-norm MCG, and shrinkage operator STAP methods based on 20 training samples of actual measured airborne radar data.

300

190

Yang, Sun, Wu, Long, and Sarkar

6.5

Conclusions In this chapter, fast and robust sparsity-based STAP methods for practical environments have been developed by exploiting the intrinsic sparsity ofthe clutter spatial-temporal power spectrum and the space–time adaptive weight vectors. Firstly, the signal model of received space–time data for airborne phased array radar is introduced, and the intrinsic sparsity of STAP is analyzed according to the clutter spatial-temporal power spectrum and the space–time adaptive weight vectors. Secondly, based on the sparsity of clutter spatial-temporal power spectrum, a robust and fast iterative sparse recovery method for STAP is introduced, which can not only alleviate the effect of noise and dictionary mismatch, but also reduce computational complexity by the recursive inverse matrix calculation. Afterwards, based on the sparsity of space–time adaptive weight vectors, a fast STAP method based on PAST with sparse constraint is introduced, which provides a more robust and stable estimation of clutter subspace when there is only a small set of training samples is available. Based on the simulated and the actual airborne phased array radar data, it is verified that the proposed methods can provide better performance with minimal training sample support in practical complex nonhomogeneous environments.

References [1] J. R. Guerci, Space-Time Adaptive Processing for Radar. Artech House, 2003. [2] J. Ward, “Space-time adaptive processing for airborne radar,” Technical Report 1015, MIT Lincoln Laboratory, Dec. 1994. [3] W. Zhang, Z. He, and J. Li, “A method for finding best channels in beam-space post-Doppler reduced-dimension STAP,” IEEE Trans. Aerosp. Electron. Syst., vol. 50, pp. 254–264, 2013. [4] W. L. Melvin, “Space-time adaptive radar performance in heterogeneous clutter,” IEEE Trans. on Aerospace and Electronic Systems, vol. 36, pp. 621–633, 2000. [5] Y. L. Wang, Y. N. Peng, and Z. Bao, “Space-time adaptive processing for airborne radar with various array orientation,” IET Radar, Sonar Navigation, vol. 144, pp. 330–340, 1997. [6] W. Zhang, Z. He, and J. Li, “A method for finding best channels in beam-space post-Doppler reduced-dimension STAP,” IEEE Trans. on Aerospace and Electronic Systems, vol. 50, pp. 254–264, 2013. [7] H. Wang and L. Cai, “On adaptive spatial-temporal processing for airborne surveillance radar systems,” IEEE Trans. on Aerospace and Electronic Systems, vol. 30, pp. 660–670, 1994. [8] A. K. Shackelford, K. Gerlach, B. D. Blunt, “Partially adaptive STAP using the FRACTA algorithm,” IEEE Trans. on Aerospace and Electronic Systems, vol. 45, pp. 58–69, 2009. [9] T. Long, Y. Liu, X. Yang, and Y. Sun, “Improved eigenanalysis canceler based on dataindependent clutter subspace estimation for space-time adaptive processing,” Science China: Information Sciences, vol. 56, no. 10, pp. 1–10, 2013. [10] R. Fa and R. C. de Lamare, “Reduced-rank STAP algorithms using joint iterative optimization of filters,” IEEE Trans. on Aerospace and Electronic Systems, vol. 47, pp. 1668–1684, 2011.

Fast and Robust Sparsity-Based STAP Methods for Nonhomogeneous Clutter

191

[11] S. Sen, “Low-rank matrix decomposition and spatio-temporal sparse recovery for STAP radar,” IEEE Journal of Selected Topics in Signal Processing, vol. 9, no. 8, pp. 1510–1523, 2015. [12] Y. L. Wang, W. J. Liu, and W. C. Xie, “Reduced-rank space-time adaptive detector for airborne radar,” Science China: Information Sciences, vol. 57, no. 8, pp. 1–11, 2014. [13] W. J. Liu, W. C. Xie, R. F. Li, Z. T. Wang, and Y. L. Wang, “Adaptive detectors in the Krylov subspace,” Science China: Information Sciences, vol. 57, no. 10, pp. 1–11, 2014. [14] P. Parker, A. L. Swindlehurst, “Space-time autoregressive filtering for matched subspace STAP,” IEEE Trans. on Aerospace and Electronic Systems, vol. 39, no. 2, pp. 510–520, 2003. [15] M. A. Herman and T. Strohmer, “High-resolution radar via compressed sensing,” IEEE Trans. on Signal Processing, vol. 57, no. 6, pp. 2275–2284, 2009. [16] K. R. Varshney, M. Cetin, J. W. Fisher, and A. S. Willsky, “Sparse representation in structured dictionaries with application to synthetic apertureradar,” IEEE Trans. on Signal Processing, vol. 56, no. 8, pp. 3548–3561, 2008. [17] S. Maria and J. J. Fuchs, “Application of the global matched filter toSTAP data an efficient algorithmic approach,” in Proc. IEEE Int. Conf. Acoust. Speech Signal Process. Toulouse, France, May 2006, pp. 14–19. [18] I. W. Selesnick, S. U. Pillai, K. Y. Li, and B. Himed, “Angle-Doppler processing using sparse regularization,” in Proc. IEEE ICASSP, Dallas, TX, Mar. 2010, pp. 2750–2753. [19] K. Sun, H. Zhang, G. Li, H. Meng, and X. Wang, “A novel STAP algorithm using sparse recovery technique,” in Proc. IGARSS, Cape Town, South Africa, July 2009, pp. 336–339. [20] Z. Yang, X. Li, H. Wang, and L. Nie, “Sparsity-based space-time adaptive processing using complex-valued homotopy technique for airborne radar,” IET Signal Processing, vol. 8, no. 5, pp. 552–564, 2014. [21] X. Yang, Y. Sun, T. Zeng, and T. Long, “Robust and fast iterative sparse recovery method for space-time adaptive processing,” Science China: Information Sciences, vol. 59, no. 6, pp. 1–13, 2016. [22] Q. Wu, Y. D. Zhang, M. G. Amin, and B. Himed, “Space-time adaptive processing and motion parameter estimation in multi-static passive radar exploiting Bayesian compressive sensing,” IEEE Transactions on Geoscience and Remote Sensing, vol. 54, no. 2, pp. 944– 957, 2016. [23] Z. Yang, R. C. de Lamare, and X. Li, “L1-regularized STAP algorithms with a generalized sidelobe canceller architecture for airborne radar,” IEEE Trans. on Signal Processing, vol. 60, no. 2, pp. 674–686, 2012. [24] Z. Yang, R. C. de Lamare, and X. Li, “Sparsity-aware space-time adaptive processing algorithms with L1-norm regularisation for airborne radar,” IET Signal Processing, vol. 6, no. 5, pp. 413–423, 2012. [25] X. Yang, Y. Sun, T. Zeng, and T. Long, “Fast STAP method based on PAST with sparse constraint for airborne phased array radar,” IEEE Trans. On Signal Processing, vol. 64, no. 17, pp. 4550–4561, 2016. [26] A. Haimovich, “The eigencanceller: Adaptive radar by eigenanalysis methods,” IEEE Trans. on Aerospace and Electronic Systems, vol. 32, no. 2, pp. 532–542, 1996. [27] M. E. Tipping, “Sparse Bayesian shrinkage and selection learning andthe relevance vector machine,” Journal of Machine Learning Research, vol. 1, no. 9, pp. 211–244, 2001.

192

Yang, Sun, Wu, Long, and Sarkar

[28] D. L. Donoho, M. Elad, V. N. Temlyakov, “Stable recovery of sparse overcomplete representations in the presence of noise,” IEEE Trans. on Information Theory, vol. 5, no. 1, pp. 6–18, 2006. [29] Q. Q. Jia, R. B. Wu, “Space time adaptive parameter estimation of moving target based on compressed sensing,” Journal of Electronics and Information Technology, vol. 35, no. 11, pp. 2714–2720, 2013. [30] J. Jin, Y. Gu, and S. Mei, “A stochastic gradient approach on compressive sensing signal reconstruction based on adaptive filtering framework,” IEEE Journal of Selected Topics in Signal Processing, vol. 4, no. 2, pp. 409–420, 2010. [31] E. M. Eksioglu, “RLS adaptive filtering with sparsity regularization,” in Proc. 10th Int. Conf. Inf. Sci., Signal Process. Appl., 2010, pp. 550–553. [32] X. Yang, Y. Liu, T. Long, “Pulse-order recursive method for inverse covariance matrix computation applied to space-time adaptive processing,” Science China: Information Sciences, vol. 56, no. 4, pp. 1–12, 2013. [33] X. Yang, Y. Sun, Y. Liu, T. Zeng, T. Long, “Fast inverse covariance matrix computation based on element-order recursive method for space-time adaptive processing,” Science China: Information Sciences, vol. 58, no. 2, pp. 1–14, 2015. [34] B. N. S. Babu, J. A. Torres, and W. L. Melvin, “Processing and evaluation of multichannel airborne radar measurements (MCARM) measured data,” in Proc. IEEE Int. Symp. Phased Array Systems and Tech., Boston, MA, Oct. 1996, pp. 395–399. [35] R. Badeau, B. David, and G. Richard, “Fast approximated power iteration subspace tracking,” IEEE Trans. on Signal Processing., vol. 53, no. 8, pp. 2931–2941, 2008. [36] X. Yang, Y. Liu, Y. Sun, and T. Long, “Improved PRI-staggered space-time adaptive processing algorithm based on projection approximation subspace tracking subspace technique,” IET Radar Sonar Navigation, vol. 8, no. 5, pp. 449–456, 2014. [37] B. Yang, “Projection approximation subspace tracking,” IEEE Trans. on Signal Processing, vol. 43, no. 1, pp. 95–107, 1995. [38] B. Babadi, N. Kalouptsidis, V. Tarokh, “SPARLS: The sparse RLS algorithm,” IEEE Trans.on Signal Processing, vol. 58, no. 8, pp. 4013–4025, 2010.

7

Super-Resolution Radar Imaging via Convex Optimization Reinhard Heckel

A radar system emits probing signals and records the reflections. Estimating the relative angles, delays, and Doppler shifts from the received signals allows to determine the locations and velocities of objects. However, due to practical constraints, the probing signals have finite bandwidth B, the received signals are observed over a finite time interval of length T only, and a radar typically has only one or a few transmit and receive antennas. These constraints fundamentally limit the resolution up to which objects can be distinguished. Specifically, a radar cannot distinguish objects with delay and Doppler shifts much closer than 1/B and 1/T , respectively, and a radar system with NT transmit and NR receive antennas cannot distinguish objects with angels closer than 1/(NT NR ). As a consequence, the delay, Doppler, and angular resolution of standard radars is proportional to 1/B and 1/T , and 1/(NT NR ). In this chapter, we show that the continuous angle-delay-Doppler triplets and the corresponding attenuation factors can be resolved at much finer resolution, using ideas from compressive sensing. Specifically, provided the angle-delay-Doppler triplets are separated either by factors proportional to 1/(NT NR −1) in angle, 1/B in delay, or 1/T in Doppler direction, they can be recovered at a significantly smaller scale or higher resolution.

7.1

Introduction A traditional single-input single-output (SISO) pulse-Doppler radar system transmits a probing signal and receives the reflections from objects with a single antenna. By estimating the induced delays and Doppler shifts, the radar system can determine the relative distances and velocities of the objects. However – as with any imaging system – physics imposes a limit on how well objects can be resolved. The resolution of a radar system is determined by the bandwidth B of the probing signals and the time interval T over which the responses are observed. Specifically, the delay and Doppler resolution is proportional to 1/B and 1/T , meaning that objects closer than that are essentially impossible to distinguish under noise. Since both B and T cannot be made arbitrarily large due to physical limitations, those two constraints fundamentally limit the resolution of a SISO radar system. In contrast to SISO radar systems, mulitple-input, multipleoutput (MIMO) radar systems [1,2] use multiple antennas to transmit probing signals simultaneously and record the reflections from the objects with multiple receive antennas. A MIMO radar can thereby, in principle, resolve the relative angles in addition to 193

194

Heckel

the relative distances and velocities of objects with a single measurement. However, the angular resolution of a MIMO radar is 1/(NT NR ), where NT and NR are the number of transmit and receive antennas, and is thus again limited by a physical constraint, namely the number of antennas (see Section 7.6 for a detailed argument on the resolution). Even though objects that are simultaneously much closer than 1/(NT NR ) in angle, 1/B in delay, and 1/T in Doppler direction, are impossible to distinguish for real-world radar systems in general, it is still possible to determine the locations of the objects with a much higher degree of accuracy than the resolution limit of (1/(NT NR ),1/B,1/T ). In this chapter, we discuss signal recovery techniques for building super-resolution radar systems that can achieve localization accuracy below the resolution limit. In more detail, we study the problem of recovering the continuous delays and Doppler shifts in a SISO radar system, and the problem of recovering the angles, delays, and Doppler shifts in a MIMO radar system, in both cases from the responses to known and suitably selected probing signals. As we see later, those problems – termed the super-resolution radar and super-resolution MIMO radar problems – amount to recovering a signal that is sparse in a continuous dictionary from linear measurements, and can thus be viewed as a generalization of the traditional compressive sensing problem. If the objects may be assumed to lie on a sufficiently coarse grid, compressed sensing-based [3] approaches provably recover the delay-Doppler pairs for SISO radar system [4–6], and the angle-delay-Doppler triplets for MIMO radar systems [7,8]. However, to establish those results, the aforementioned papers assume that angles, delays, and Doppler shifts lie on a sufficiently coarse grid, specifically a grid with spacing 1/(NT NR ),1/B, and 1/T , in angle, delay, and Doppler direction, respectively (see Section 7.2.3). Since NT ,NR ,B, and T are physical problem parameters, they can in general not be made (arbitrarily) large in order to make the grid finer. In fact, the coarseness of the grid is required for the measurement matrix to be incoherent, therefore the aforementioned results cannot straightforwardly be extended to a grid with significantly finer spacing. In some special cases, however, off-the-grid recovery is possible with standard spectral estimation techniques. For example, for a single input antenna and either known and constant delays (see Section 7.2.4), or known and constant Doppler shifts, the super-resolution radar problem reduces to a standard 2D line spectral estimation problem [8, sec. 5]. For these special cases, the object locations can be recovered – off the grid – with standard spectral estimation techniques such as Prony’s method, MUSIC, and ESPRIT [9]. In general, however, the super-resolution radar problems cannot be reduced to the classical line spectral estimation problem. Therefore, traditional spectral estimation techniques are not directly applicable. Recently, an alternative, convex optimization based approach to solve the classical line spectral estimation has been proposed that is much more generally applicable than traditional line spectral estimation techniques. Specifically, the paper [10] shows that the frequency parameters, which are the unknowns in the line spectral estimation problem, can be perfectly recovered by solving a convex total-variation norm minimization program, provided they are sufficiently separated. Related convex programs have been studied for compressive sensing off the grid [11], denoising [12], signal recovery from short-time Fourier measurements [13], and the SISO and MIMO super-resolution radar

Super-Resolution Radar Imaging via Convex Optimization

195

problems [14,15], and the generalized line-spectral estimation problem [16]. The focus of this chapter is on explaining how this convex optimization based approach enables high resolution in radar. In particular we discuss the results in [14,15], showing that a convex program recovers the continuous angles, delays, and Doppler shifts perfectly, provided that they are sufficiently separated. Furthermore, we show that a simple convex 1 -minimization program recovers the angles and delay-Doppler shifts on an arbitrarily fine grid, again provided they are sufficiently separated. Finally, we provide numerical results demonstrating robustness to noise. The remainder of this chapter is organized as follows. In the first part we consider the SISO radar model. In more detail, Section 7.2 contains the radar model and formal problem statement, in Sections 7.3 and 7.3.2 we present the convex optimization based recovery approach and corresponding performance guarantees, and in Section 7.4 we show that 1 -minimization recovers the locations on an arbitrarily fine grid. In Section 7.4.2, we provide numerical results demonstrating that the approach is robust to noise and in Section 7.5.2 we outline the proof of the main technical statements. In the second part, Section 7.6, we explain how the results for SISO radar can be extended to the MIMO case. We conclude in Section 7.7 with a discussion on challenges in applying those ideas in practice, and current and future research directions.

7.2

Signal Model and Problem Statement A radar system with a single transmit and single receive antenna is typically modeled as a linear system. The response y recorded at the receive antenna, to a probing signal, x, emitted at the transmit antenna, is a weighted superposition of delayed and Dopplershifted versions of the probing signal x: (7.1) y(t) = s(τ,ν)x(t − τ)ei2πν t dνd τ. Here, s denotes the spreading function, which describes the scene being sensed, and τ and ν are the delays and Doppler shifts. Often, the moving objects are modeled by point scatterers. Mathematically, this means that the spreading function specializes to s(τ,ν) =

S

bj δ(τ − τ¯ j )δ(ν − ν¯j ).

j =1

Here, bj is the (complex-valued) attenuation factor associated with the delay-Doppler pair ( τ¯ j , ν¯j ). With the spreading function above, the input-output relation (7.1) reduces to y(t) =

S

bj x(t − τ¯ j )ei2πν¯ j t .

(7.2)

j =1

Thus, the received signal is a superposition of the reflections of the probing signal by the point scatterers. The relative distances and velocities of the S-many objects can be trivially obtained from the delay-Doppler pairs ( τ¯ j , ν¯j ). In order to locate the objects,

196

Heckel

we therefore need to estimate the delay-Doppler pairs and the corresponding attenuation factors bj from a single input–output measurement, i.e., from the response y to a known and suitably selected probing signal x. As we will see later, the particular choice of the probing signal is crucial for good localization performance.

7.2.1

Band- and Time-Limitation and Resolution The probing signal x can be controlled by the system engineer and is known. However, due to practical and technological constraints, it must be band-limited and approximately time-limited. Also, again due to practical constraints, we can only observe the response y over a finite time interval. For concreteness, we assume that i. ii.

we observe the response y over an interval of length T and that x has bandwidth B and is approximately supported on a time interval of length proportional to T .

The time- and band-limitation determines the “natural” resolution of the system, which is the accuracy up to which the delay-Doppler pairs can be identified. A standard pulseDoppler radar that samples the received signal at its Nyqist rate, and performs digital matched filtering, estimates the parameters up to accuracy 1/B and 1/T in delay (τ) and Doppler (ν) directions, respectively, and therefore only identifies the delay-Doppler pairs up to the natural solution. From the input–output relation (7.2), it is evident that band and approximate time limitation of the input signal x implies that the response y is band- and approximately time-limited as well – provided that the delay-Doppler pairs are compactly supported. In radar, due to path loss and finite velocity of the objects in the scene this is indeed the case [17]. Throughout, we will therefore assume that the delay-Doppler pairs (τ¯ j , ν¯j ) lie in the region : ; : ; −T T −B B , × , . 2 2 2 2 This is not a restrictive assumption as this region can have area BT " 1, which is typically very large. In fact, for many applications, it is reasonable to assume that the delay-Doppler pairs lie in a region of area significantly smaller than one [18–20], an assumption often referred to as the linear system being “underspread”. We do not make or require this assumption here. By the 2W T -Theorem [21], band and approximate time limitation of the response y implies that y is essentially characterized by on the order of BT coefficients. We therefore sample y in the interval [−T /2,T /2] at rate 1/B, so as to collect L := BT samples, denoted by yp := y(p/B) (for simplicity we assume in the following that L = BT is an odd integer). As detailed in [14, sec. 5], those samples are given by yp =

S j =1

bj [Fνj Tτj x]p,

p = −N,. . .,N,

N :=

L−1 , 2

(7.3)

Super-Resolution Radar Imaging via Convex Optimization

where N 1 [Tτ x]p := L

2

k=−N

N

x e

−i2π k L

197

3 pk

e−i2πkτ ei2π L

(7.4)

=−N

and [Fν x]p := xp ei2πpν . Here, we defined the time-shifts τj := τ¯ j /T and frequency shifts νj := ν¯j /B. To avoid ambiguity, from here onwards we refer to (τ¯ j , ν¯j ) as a delay-Doppler pair and to (τj ,νj ) as a time–frequency shift. From (τ¯ j , ν¯j ) ∈ [−T /2,T /2] × [−B/2,B/2] we have (τj ,νj ) ∈ [−1/2,1/2]2 . Since Tτ x and Fν x are 1-periodic in τ and ν, we assume in the remainder of the chapter without loss of generality that (τj ,νj ) ∈ [0,1]2 . The operators Tτ and Fν have an interesting interpretation as fractional time and frequency shift operators in CL . In fact, if the parameters τ and ν lie on a (1/L,1/L) grid, the operators Fν and Tτ reduce to the “natural” time frequency shift operators in CL , i.e., [Tτ x]p = xp−τL and [Fν x]p = xp ei2πpν . The definition of a time shift in (7.4) as taking the Fourier transform, modulating the frequency, and taking the inverse Fourier transform is a very natural definition of a continuous time-shift τj ∈ [0,1] of a discrete vector x = [x0,. . .,xL−1 ]T . Finally, note that to obtain the input–output relation (7.3) (see [14, sec. 5]) from (7.2), a periodic sinc function is approximated with a finite sum of sinc functions (this is where partial periodization of x becomes relevant). Thus, if we take the probing signal to be essentially time-limited, then equality in (7.3) does not hold exactly. However, in [14, sec. 5] it is shown that for a random probing signal, as considered in this chapter, the √ incurred relative 2 -error decays as 1/ L and is therefore negligible for large L. It is confirmed numerically in the same paper that the approximation error made in this process is negligible. Moreover, if we took x to be T -periodic, then the input–output relation (7.3) becomes exact, but at the cost of the probing signal x not being timelimited.

7.2.2

Formal Problem Statement From the discussion in the previous section we conclude that identification of the objects under the constraints that the probing signal x is band-limited and the response y to the probing signal is observed over a finite time interval, reduces to the estimation of the triplets {(bj ,τj ,νj )}Sj=1 from the samples {yp }N p=−N . Thus, in this chapter, we consider the problem of recovering those triples from the samples {yp }N p=−N in (7.3). We call this the super-resolution radar problem, as recovering the exact time–frequency shifts {(τj ,νj )}Sj=1 “breaks” the natural resolution limit of (1/B,1/T ) achieved by a standard pulse-Doppler radar. Alternatively, one can view the super-resolution radar problem as that of recovering a signal that is S-sparse in the continuous dictionary of time–frequency shifts of an

198

Heckel

L-periodic sequence x . In order to see this, and to better understand the super-resolution radar problem, it is instructive to consider two special cases.

7.2.3

Time–Frequency Shifts on a Grid Suppose the delay-Doppler pairs ( τ¯ j , ν¯j ) lie on a ( B1 , T1 )-grid. As a consequence the time–frequency shifts (τj ,νj ) lie on a ( L1 , L1 )-grid, which in turn implies that τj L and νj L are integers in {0,. . .,L − 1}. Thus, the super-resolution radar problem reduces to a sparse signal recovery problem with a Gabor measurement matrix. To see this, note that under the aforementioned assumption, the input–output relation (7.3) reduces to yp =

S

bj xp−τj L ei2π

(νj L)p L

,

p = −N,. . .,N .

j =1

Writing this equation in vector-matrix form gives y = Gx b. 2

Here, the vector y contains as entries the samples yp , Gx ∈ CL×L is the Gabor matrix with window x, defined by kp

[Gx ]p,(k,) := xp− ei2π L ,

k,,p = −N,. . .,N,

(7.5)

2

and b ∈ CL is a sparse vector with the j -th nonzero entry given by bj and indexed by (τj L,νj L). Thus, recovery of the triplets {(bj ,τj ,νj )}Sj=1 amounts to recovering the S-sparse vector b from the measurement vector y. A – by now standard – recovery approach is to solve a convex 1 -norm-minimization program. From [22, thm. 5.1] we know that, provided the x are i.i.d. sub-Gaussian random variables, and provided that S ≤ cL/(log L)4 for a sufficiently small numerical constant c, with high probability, all S-sparse vectors b can be recovered from y via 1 -minimization. Note that the result [22, thm. 5.1] only applies to the Gabor matrix Gx and therefore does not apply to the super-resolution problem where the “columns” Fν Tτ x are highly correlated. In fact, the two problems are conceptually very different: [22, thm. 5.1] shows that the columns of the Gabor matrix Gx are nearly orthogonal, while the “columns” Fν Tτ x are extremely correlated for two pairs of time–frequency shifts that are close.

7.2.4

Only Time or Only Frequency Shifts Next, suppose we only have time- or only frequency shifts. In both cases, recovery of the unknowns {(bj ,τj )} and {(bj ,νj )}, respectively, is equivalent to the recovery of a weighted superposition of spikes from low-frequency samples. Specifically, if we only have frequency shifts, and therefore τj = 0 for all j , the input–output relation (7.3) reduces to

Super-Resolution Radar Imaging via Convex Optimization

yp = xp

S

bj ei2πpνj ,

p = −N,. . .,N .

199

(7.6)

j =1

Note that the samples {yp } above are samples of a mixture of S complex sinusoids, and estimation of the coefficients {(bj ,νj )} corresponds to determining the magnitudes and the frequency components of these sinusoids. Estimating the coefficients {(bj ,νj )} from the samples {yp } is known as a line spectral estimation problem and can be solved using classical approaches, such as Prony’s method [23, ch. 2], as well as convex programming based approaches [10]. An analogous situation arises when there are only time shifts (νj = 0 for all j ) as taking the discrete Fourier transform of yp yields a relation exactly equal to (7.6).

7.3

Atomic Norm Minimization and Associated Performance Guarantees We next present a convex optimization based recovery algorithm. Even though the corresponding convex program can be solved in polynomial time, standard solvers are currently computationally very expensive, limiting the practical applicability. However, in Section 7.4 we will discuss a very closely related convex program that has a significantly better computational efficiency at the cost of making a small griding error that is due to a discretization step. Since the results and intuition for both approaches are nearly the same, we start by discussing the continuous case here.

7.3.1

Atomic Norm Minimization We first define for convenience the vector rj := [τj ,νj ], and write the input–output relation (7.3) in matrix-vector form: y = Gx FH z,

z=

S

bj f(rj ).

(7.7)

j =1 L2 ×L2

Here, FH ∈ C is the (inverse) 2D discrete Fourier transform matrix with the entry qk+r in the (k,)-th row and (r,q)-th column given by [FH ](k,),(r,q) := L12 ei2π L , and the entries of the vector f are given by [f(r)](r,q) := e−i2π(r τ+q ν ) , k,,q,r = −N,. . .,N (here, and in the following we use for convenience a two- or three-dimensional index 2 to refer to entries of vectors and matrices). Moreover, Gx ∈ CL×L is the Gabor matrix defined in (7.5). The significance of the representation in (7.7) is that recovery of the unknowns {(bj ,rj )} from z is a 2D line spectral estimation problem that can be solved with standard spectral estimation techniques such as Prony’s method [9]. Therefore, we only 2 need to recover z ∈ CL from y ∈ CL . To do so, we use that z is a sparse linear combination of atoms in the set A := {f(r),r ∈ [0,1]3 }. A regularizer that promotes such a sparse linear combination is the atomic norm induced by these signals [24], defined as

200

Heckel

zA :=

inf

bk ∈C,rk ∈[0,1]2

|bk | : z =

k

bk f(rk ) .

k

We estimate z by solving the basis pursuit type atomic norm minimization problem problem 4 4 (7.8) AN(y) : minimize 4z˜ 4A subject to y = A˜z. z˜

To summarize, we estimate the attenuation factors bk and time–frequency shifts rk from y by i. ii. iii.

solving AN(y) in order to obtain z, estimating the rk from z by solving the corresponding 2D-line spectral estimation problem, and solving the linear system of equations y = S−1 k=0 bk Af(rk ) for the bk .

We remark that the rk may be obtained more directly from a solution to the dual of (7.8) [14, sec. 6]; see also [12, sec. 3.1], [10, sec. 4], [11, sec. 2.2] for details on this approach as it is applied to related problems. Since computation of the atomic norm involves taking the infimum over infinitely many parameters, finding a solution to AN(y) may appear to be daunting. For the one-dimensional case (i.e., only time or frequency shifts), the atomic norm can be characterized in terms of linear matrix inequalities [11, prop. 2.1]. This characterization is based on the Vandermonde decomposition lemma for Toeplitz matrices, and allows us to formulate the atomic norm minimization program as a semidefinite program that can be solved in polynomial time. While this lemma generalizes to higher dimensions [25, thm. 1], it fundamentally comes with a rank constraint that appears to prohibit an straightforward characterization of the atomic norm in terms of linear matrix inequalities. Nevertheless, based on [25, thm. 1], one can obtain a semidefinite programming relaxation of AN(y), which can be solved in polynomial time. Similarly, a solution of the dual of AN(y) can be found with a semidefinite programming relaxation. Since the computational complexity of the corresponding semidefinite programs is quite large, we will not dive into the details of those semidefinite programming relaxations. As mentioned before, instead, we show in Section 7.4 that the parameters {rj } can be recovered on an arbitrarily fine grid via 1 -minimization. While this leads to a gridding error, the grid may be chosen sufficiently fine for the gridding error to be negligible compared to the error induced by additive noise, and in practice, there is always some additive noise.

7.3.2

Recovery Guarantees for Atomic Norm Minimization We consider a random probing signal by taking the samples of the probing signal x in (7.3) to be i.i.d. Gaussian random variables. More generally, the result presented in Theorem 7.6 later in the chapter continues to hold if we choose the samples as subGaussian random variables, for example as random signs. Note that the probing signal can be chosen by the radar engineer, therefore, choosing the coefficients at random is not

Super-Resolution Radar Imaging via Convex Optimization

201

problematic and can be done in practice. Theorem 7.6 shows that, with high probability, the triplets {(bj ,τj ,νj )} can be recovered perfectly from the samples by solving a convex program, provided that the number of time–frequency shifts is sufficiently smaller than the number of measurement, and provided that the following minimum separation condition holds: definition 7.1 (Minimum separation condition) We say the time–frequency shifts (τj ,νj ) ∈ [0,1]2,j = 1,. . .,S satisfy the minimum separation condition if 2.38 (7.9) , for all j != j #, N where |τj − τj # | is the wraparound distance on the unit circle. For example, |3/4 − 1/2| = 1/4 but |5/6 − 1/6| = 1/3 != 2/3. max(|τj − τj # |,|νj − νj # |) ≥

Note that the time–frequency shifts must not be separated in both time and frequency, for example the minimum separation condition can hold even when τj = τj # for some j != j # . The main result on recovery via atomic norm minimization from the paper [14] is stated next. theorem 7.2 Assume that the samples of the probing signal x ∈ CL are i.i.d. N (0,1/L) random variables. Consider a signal where the sign of the attenuation factors {bj }Sj=1 is i.i.d. uniform on {−1,1} or the complex unit disc, and suppose that the time–frequency shifts {(τj ,νj )}Sj=1 obey the minimum separation condition. Furthermore, choose δ > 0 and assume that the number of nonzero attenuation factors, S, and the number of measurements, L, obey S≤c

L 3

log (L/δ)

,

where c is a numerical constant. Then, with probability at least 1− δ, z = is the unique minimizer of AN(y), y = Gx FH z.

S

j =1 bj f(rj )

This result is essentially optimal in terms of the allowed sparsity level, as the number S of unknowns can be linear – up to a logarithmic-factor – in the number of observations L. Even when we are given the time–frequency shifts (τj ,νj ), we can only hope to recover the corresponding attenuation factors bj by solving the linear system of equations in (7.3), provided that S ≤ L. Since the complex-valued coefficients bj in the radar model describe the attenuation factors, it is natural to assume that the phases of different bj are independent from each other and are uniformly distributed on the unit circle of the complex plane. Indeed, standard models for wireless communication channels and radar [26], assume the coefficients {bj } to be complex Gaussian distributed. Nevertheless, we believe that the random sign assumption is not necessary for Theorem 7.2 to hold. In Section 7.7, we discuss a closely related problem, which does not require the random sign assumption, and thus provides a basis for the claim of the random sign assumption not being necessary. Finally, we would like to point out that Theorem 7.2 continues to hold for sub-Gaussian sequences x , for example random signs.

202

Heckel

7.3.3

Necessity of Minimum Separation Theorem 7.2 imposes a minimum-separation condition, and indeed some form of separation between the time–frequency shifts is necessary for stable recovery. To be specific, we consider the simpler problem of line spectral estimation (see Section 7.2.4) that is obtained from our setup by setting τj = 0 for all j . Clearly, any condition necessary for the line spectral estimation problem is also necessary for the super-resolution radar problem. # # # If there are S # frequencies {νk }Sk=1 in an interval of length smaller than 2S L , and S is large, then in the presence of even a tiny amount of noise, stable recovery of the attenuation factors and time–frequency shifts even from z=

S

bj f(rj ),

j =1

where we set, with a slight abuse of notation, [f(νj )] = ei2πj νj , is not possible (see [27, thm. 1.1] and [10, sec. 1.7]). Condition (7.9) allows us to have 0.4 S # time–frequency # shifts in an interval of length 2S L , which is optimal up to the constant 0.4. This argument illustrates that for stable recovery, it is relevant whether a number of frequencies, say S many, cluster together in a small interval of size smaller than S/L. To illustrate this point further, consider again the simpler problem of line S spectral estimation, i.e., recovery or the frequencies from z = j =1 bj f(νj ), with [f(νj )] = ei2πj νj . Consider the following (Vandermonde) matrix parameterized by S and : V = [f(0),f((1 − )/L),. . .,fL ((2S − 1)(1 − )/L)]. We next state a theorem, which provides a lower bound on the condition number of V. The lower bound implies that there are signals with S many frequencies in an interval smaller than S/L, that are indistinguishable even under a tiny amount of additive noise. 1 theorem 7.3 [28, thm. 1.3] Fix some ∈ (0,1), let K = 1− L, and let O(S) . S = O(log(L/(1 − ))). Then the matrix V has condition number at least e

The theorem implies that there exists a vector b with unit norm that obeys Vb2 ≤ e−O(S) . As a consequence, ⎛ ⎞2 N ⎝ bj ei2πνj + bj ei2πνj ⎠ = Vb22 ≤ e−O(S), =−N

j odd

j even

which means that there are two sets of S many point sources each, with separation 2(1−) L , but telling them apart requires an exponentially small additive error. To obtain intuition on the constants involved in the statement, we plot the condition number of V for different values of S and in Figure 7.1. While those two arguments show that some form of separation between the time– frequency shifts is necessary, the exact form of separation required in (7.9) may not be necessary for stable recovery and less restrictive conditions may suffice. Indeed, in

Super-Resolution Radar Imaging via Convex Optimization

S=2 S=4 S=8 S = 16 S = 32

1 1/κ

203

0.5

0 0.6

0.8

1

1− Figure 7.1 Inverse of the condition number κ of the matrix V with entries [V]pq = e−i2πpq(1−)/L , L = 200, and q = 1,. . .,S, for different values of the number of sources S as a function of the separation between frequencies of (1 − )/L.

the simpler problem of line spectral estimation, Donoho [27] showed that stable superresolution is possible via an exhaustive search algorithm, even when condition (7.9) is # violated locally, as long as every interval of the ν-axis of length 2S L contains less than S # /2 frequencies and S # is small (in practice, think of S # 10).

7.3.4

Implications for the Detection Accuracy of Radar Systems Translated to the continuous setup, Theorem 7.2 implies that with high probability, the triplets (bj , τ¯ j , ν¯j ) can be identified perfectly provided that | τ¯ j − τ¯ j # | ≥

4.77 4.77 or |ν¯j − ν¯j # | ≥ , B T

for all j != j #,

(7.10)

and provided that S ≤ cBT /log3 (BT ). Since we can exactly identify the delay-Doppler pairs (τ¯ j , ν¯j ), as opposed to only localizing them on a grid, this result offers a significant improvement in resolution over conventional radar techniques. Specifically, with a standard pulse-Doppler radar, which samples the received signal and performs digital matched filtering in order to detect the objects, the delay-Dopper shifts ( τ¯ j , ν¯j ) can only be estimated up to an uncertainty of about (1/T ,1/B). We hasten to add that in the radar literature, the term super-resolution is often used for the ability to resolve objects that are very close – even closer than the Rayleigh resolution limit [29] that is proportional to 1/B and 1/T for delay and Doppler resolution, respectively. The norm minimization approach discussed here permits identification of each object with a precision that is much higher than 1/B and 1/T as long as the other objects are not too close. Specifically, other objects should be separated by a constant multiple of the Rayleigh resolution limit as formalized by the minimum separation condition (7.10). Recall that, however, any method that attempts to recover objects closer than the resolution limit can only succeed if there are very few objects below that limit, since resolving many objects that are all below the resolution limit is in general impossible, as discussed previously in Section 7.3.3. Finally, recall that the approach discussed here allows the delay-Doppler pairs (τ¯ j , ν¯j ) to lie in [−T /2,T /2] × [−B/2,B/2] so the delay-Doppler pairs can lie in a rectangle

204

Heckel

of area L = BT " 1. The ability to handle a potentially large region in which delayDoppler pairs can lie in is important in radar applications, since we might need to resolve objects with large relative distances and relative velocities.

7.3.5

Can Standard Nonparametric Methods Yield Similar Performance? We finally note that standard nonparametric estimation methods such as the MUSIC algorithm can in general not be applied directly to the super-resolution radar problem. The reason is that MUSIC relies on multiple measurements (often referred to as snapshots) [9, sec. 6.3], whereas we assume only a single measurement {yp }N p=−N to be available. In our context, multiple measurements would amount to carrying out multiple, independent input–output measurements. However, by choosing the probing signal x in (7.3) to be periodic, a single measurement can be transformed into multiple measurements, and for that case, such algorithms as, for example, the MUSIC algorithm may be applied. However, this approach, discussed in detail in [14, appendix H], the time-shifts {τj } to lie in a significantly requires the frequencies {νj } to be distinct, √ smaller range than [0,1], and S < L, as opposed to the much milder condition S < L/ log3 (L/δ) required by the convex program. In addition, applying MUSIC in that way is (significantly) more sensitive to noise than the convex programmingbased approach discussed in this chapter. If multiple measurements are available, for example by observing distinct paths of a signal by an array of antennas, the situation might be different. For that case, subspace methods have been studied for delay-Doppler estimation in [30].

7.4

Super-Resolution Radar on a Fine Grid A practical approach to estimating the triplets {(bj ,τj ,νj )} from the received signal y in the input–output relation (7.3) is to suppose that the time–frequency shifts lie on a fine grid, and solve the problem on that grid. In general this leads to a gridding error, which, however, is minimized as the grid grows finer [31]. We next discuss the corresponding (discrete) sparse signal recovery problem. Suppose the time–frequency shifts lie on a fine grid with spacing (1/K,1/K), where 2 K is an integer obeying K ≥ L. Let b ∈ CK be the signal with each entry bm,n corresponding to one of the grid points, with nonzeros equal to the attenuation factors bj for the time–frequency shifts (τj ,νj ),j = 1,. . .,S. See Figure 7.2 for an illustration. With this assumption, the input–output relation (7.3) becomes: yp =

K−1

bm,n [Fm/K Tn/K x]p,

m,n=0

Writing this relation in matrix-vector form yields y = Rb,

p = −N,. . .,N .

Super-Resolution Radar Imaging via Convex Optimization

205

1 K

ν

1 L

1 L

τ Figure 7.2 Time frequency shifts that lie on a grid: (1/L,1/L) is the “natural” grid, and

(1/K,1/K) is the fine grid. Each dot corresponds to a potential nonzero, and the larger dots correspond to the actual nonzeros {bj }. 2

where, as before, y is the vector containing as entries the values yp , and R ∈ CL×K , is the matrix with (m,n)-th column given by Fm/K Tn/K x. The matrix R contains as columns “fractional” time–frequency shifts of the sequence x . If K = L, R contains as columns only “whole” time–frequency shifts of x and R is equal to the Gabor matrix Gx , defined by (7.5). In this sense, K = L is the natural grid (see Section 7.2.3) and the ratio SRF := K/L can be interpreted as a super-resolution factor. The super-resolution factor determines by how much the (1/K,1/K) grid is finer than the original (1/L,1/L) grid. A standard approach to the recovery of the sparse signal b from the underdetermined linear system of equations y = Rb is to solve the following convex program: 4 4 4 4 ˜ (7.11) L1(y) : minimize 4b˜ 4 subject to y = Rb. b˜

1

The following theorem is the main result from [14] for recovery on the fine grid. theorem 7.4 Assume that the samples of the probing signal x are i.i.d. N (0,1/L) random variables, L = 2N + 1, and that L = 2N + 1 ≥ 1024. Consider a signal b supported on S ⊆ {0,. . .,K − 1}2 , and suppose that it satisfies the minimum separation condition min

(m,n),(m#,n# )∈S : (m,n)!=(m#,n# )

1 2.38 max(|m − m# |,|n − n# |) ≥ . K N

Moreover, suppose that the nonzeros of b are i.i.d. uniform on {−1,1} or the complex unit disk. Choose δ > 0, let y = Rb be the measurement corresponding to b, and suppose that the number of nonzeros of b is sufficiently smaller than the number of samples L S≤c

L 3

log (L/δ)

,

where c is a numerical constant. Then, with probability at least 1 − δ, b is the unique minimizer of L1(y), y = Rb.

206

Heckel

Note that Theorem 7.4 does not impose any restriction on K, in particular it can be arbitrarily large. The proof of Theorem 7.4, discussed in Section 7.5.2, is closely linked to that of Theorem 7.2.

7.4.1

Implementation Details The matrix R has dimension L × K 2 , thus as the grid becomes finer, i.e., K becomes larger, the complexity of solving L1(y) increases. The complexity of solving L1(y) can be managed as follows. First, the complexity of first-order convex optimization algorithms (such as TFOCS [32]) for solving L1(y) is dominated by multiplications with the matrices R and RH . Due to the structure of R, those multiplications can be done very efficiently by utilizing the fast Fourier transform. Second, in practice we have ( τ¯ j , ν¯j ) ∈ [0,τmax ] × [0,νmax ], which means that 0 τ 1 0 ν 1 max max (τj ,νj ) ∈ 0, × 0, . (7.12) T B νmax K It is therefore sufficient to consider the restriction of R to the τmaxBT = τmax νmax L · 2 SRF many columns corresponding to the time–frequency shifts (τj ,νj ) satisfying (7.12). Since typically τmax νmax BT = L, this results in a significant reduction of the problem size. 2

7.4.2

Numerical Results and Robustness We next discuss numerical results that show that the convex optimization-based superresolution approach is robust to noise. Consider the following modification of 1 -norm minimization, which accounts for noise and the gridding error: 4 4 4 42 4 4 4 4 L1-ERR : minimize 4b˜ 4 subject to 4y − Rb˜ 4 ≤ δ. b˜

1

2

The parameter δ is chosen on the order of the noise variance or the expected gridding error. The paper [14] considered a synthetic problem with L = 201, where each problem instance is generated by drawing√S = 10 time–frequency shifts (τj ,νj ) uniformly at random from the interval [0,2/ 201]2 . This amounts to drawing the corresponding delay-Doppler pairs ( τ¯ j , ν¯j ) from the interval [0,2] × [0,2]. The attenuation factors bj corresponding to the time–frequency shifts were drawn uniformly at random from the complex unit disc, independently across j . Measurements were then obtained according to the input–output relation (7.3). Figure 7.6 depicts the average resolution error versus the super-resolution factor SRF . = K/L. The resolution error is defined as the average over j = 1,. . .,S of L ( τˆ j − τj )2 + (νˆj − νj )2 , where the ( τˆ j , νˆj ) are the estimates of the time–frequency shifts extracted from a solution of L1-ERR, obtained with the SPGL1 solver [33]. Note that the resolution attained at SRF = 1 corresponds to the resolution attained by matched filtering and by the compressive sensing radar architecture [4] that was discussed in Section 7.2.3.

Super-Resolution Radar Imaging via Convex Optimization

207

2 -resolution error

0.4 0.3

Standard/CS Radar

0.2 10 dB SNR

0.1

30 dB SNR

Noiseless 0

1

5

10

15

Super-resolution factor 1 T 1 B

Figure 7.3 Super-resolution radar uniformly provides better resolution error than standard

or CS radar. The plot show the resolution error for the recovery of S = 10 time–frequency shifts from the observation y with and without additive Gaussian noise n of a certain signal-to-noise ratio . SNR := y2 /n2 . The resolution error is defined as the average over

L ( τˆ j − τj )2 + (νˆj − νj )2 , where (τj ,νj ) are the original time–frequency shifts, and the ( τˆ j , νˆj ) are the time–frequency shifts on the grid, obtained by solving L1-ERR, for different super-resolution factors. The different super-resolution factors are illustrated below the plot.

As mentioned before, there are two error sources that were incurred by this approach. The first is the gridding error obtained by assuming that the points lie on a fine grid with grid constant (1/K,1/K), which decays in K. The second is the additive noise error, which is constant. The figure shows that for SRF larger than 1, the resolution is significantly improved using the super-resolution radar approach. Moreover we see that for small super-resolution factors SRF, the gridding error dominates, while for large values of SRF, the additive noise error dominates. In this experiment, the gridding error approximately decays as 1/SRF. The experiment demonstrates that in practice solving the super-resolution radar problem on a fine grid is essentially as good as solving it on the continuum – provided the super-resolution factor is chosen sufficiently large, so that the gridding error becomes negligible relative to the error due to additive noise.

7.5

Proof Outline In this section, we discuss the proofs of theorems 7.2 and 7.4 from [14], which are closely linked. Specifically, both theorems 7.2 and 7.4 follow from the existence of certain dual certificates, and the certificate for proving theorem 7.4 is obtained directly from the certificate constructed to prove theorem 7.2.

208

Heckel

7.5.1

Proof of Theorem 7.2 Theorem 7.2 is proven by constructing an appropriate dual certificate; the existence of this certificate guarantees that the solution to AN(y) is z, as formalized by Proposition 1, as we will see shortly. Proposition 1 is a consequence of strong duality, and is wellknown for the discrete setting from the compressed sensing literature [3]. The proof is standard, see for example [11, proof of prop. 2.4]. For convenience, in this section, we set A = Gx FH , so that y = Az. proposition 1 Let y = Az with z = Sj=1 bj f(rj ). Suppose there exists a function, called dual certificate, of the form Q(r) = q,Af(r), parameterized by the complex coefficients q ∈ CL , such that Q(rj ) = sign(bj ), for all j , and |Q(r)| < 1 for all r ∈ [0,1]2 \ {r1,. . .,rS }.

(7.13)

Moreover, suppose that the vectors {Af(rj )}Sj=1 are linearly independent. Then z is the unique minimizer of AN(y). A condensed argument showing that Proposition 1 is true follows. Suppose4for4 con # # 4 #4 tradiction that there exists another optimal solution z# = j bj f(rj ) with z A = # # # j |bj | and {rj } != {rj }. First, suppose that {rj } ⊆ {rj }. Then, linear independence of # the vectors {Af(rj )} contradicts that of z != z. Next, suppose that {r#j } !⊆ {rj }. We then have that 4 #4 i 4z 4 − zA = |bj# | − |bj | > Re Q∗ (r#j )bj# − Re Q∗ (rj )bj A j

j

j

ii

= Re

j

j

qH Af(r#j )bj# − Re

qH Af(rj )bj

j

iii

= 0.

Here, inequality (i) follows from the dual polynomial interpolating the sign pattern, and from |Q(r#j )| < 1 for at least one r#j , which in turn follows from {r#j } !⊆ {rj } and |Q(r)| < 1 for all r ∈ / {rj }, by assumption. Inequality (ii) follows from the definition of the dual polynomial, and inequality (iii) follows from Az# = Az, by assumption. This contradicts that z# is an optimal solution. We now turn to the construction of a dual certificate obeying the conditions of Proposition 1, which concludes the proof of Theorem 7.2. The construction of the dual certificate Q is inspired by the construction of related certificates in [10,11]. First, recall that the entries of f(r),r = [τ,ν], are given by [f(r)](k,p) = ei2π(kτ+pν ) . From = < Q(r) = q,Af(r) = AH q,f(r) , it is seen that Q is a two-dimensional trigonometric polynomial in the variables τ and ν with coefficient vector AH q. To build the certificate, we therefore need to construct a two-dimensional trigonometric polynomial that satisfies the interpolation and boundedness condition (7.13), and whose coefficients are constraint to be of the form AH q. Since

Super-Resolution Radar Imaging via Convex Optimization

209

the matrix A is a function of the random probing signal x, Q is a random trigonometric polynomial. We construct Q explicitly. It is instructive to first consider the construction of a deterministic two-dimensional trigonometric polynomial > ? ¯ ¯ Q(r) = q,f(r) , with unconstrained, deterministic coefficients q¯ ∈ CL , that satisfies the interpolation and boundedness conditions (7.13), but whose coefficient vector q¯ is not constraint to be of the form AH q. Such a construction has been established – provided that the parameters {rk } obey the minimum separation condition in Definition 7.9 – by Candès and Fernandez-Granda [10, prop. 2.1, prop. C.1] for the one- and two-dimensional case. ¯ [10] interpolate the signs {sign(bj )} with a fast-decaying To construct the polynomial Q, kernel ¯ := F (τ)F (ν), G(r) and slightly adopt this interpolation near the parameters {rj } with the partial derivatives ¯ (n1,n2 ) (r) := ∂ n1 /∂ τ n1 ∂ n2 /∂ν n2 G(r) ¯ G to ensure that local maxima are achieved at the rj : ¯ Q(r) =

S

¯ − rj ) + α¯ 1j G ¯ (1,0) (r − rj ) + α¯ 2j G ¯ (0,1) (r − rj ). α¯ j G(r

(7.14)

j =1

Here, F is the squared Fejér kernel, which is a particular trigonometric polynomial with coefficients gj , i.e., F (t) =

N

gj ei2πtj .

j =−N

Shifted versions of the polynomial F (i.e., F (t −t0 ) for some t0 ∈ R) and the derivatives of the polynomial F are also one-dimensional trigonometric polynomials of degree N . ¯ its partial derivatives, and shifted versions thereof are two-dimensional Therefore G, ¯ is concluded by ¯ trigonometric polynomials of the form q,f(r). The construction of Q showing that the coefficients α¯ j , α¯ 1j , α¯ 2j , α¯ 3j , can be chosen such that Q¯ reaches global maxima at the parameters {rj }. The construction of Q in the paper [14] for proving Theorems 1 follows a similar program. Specifically, the polynomial Q is constructed such that > ? it interpolates the signs sign(bj ) at rj with the functions Gn (r,rj ) = Agn (rj ),Af(r) . Here, gn (r),n = (n1,n2 ) is the vector with (v,k)-th coefficient given by gk gp (i2πk)n1 (i2πp)n2 e−i2π(τk+ν p), where the {gj } are the coefficients of the squared Fejér kernel F , just defined. With this definition, we have ¯ n (r − rj ). E Gn (r,rj ) = G This follows from

1 0 1 0 E AH A = FH E GH x Gx F = I,

210

Heckel

where we used E GH x Gx = I, shown at the end of this section. Moreover, Gn (r,rk ) ¯ n (r − rk ). concentrates around G Now, Q is constructed by interpolating the signs sign(bj ) at rj with G(0,0) (r,rj ), j = 1,. . .,S, and slightly adopting this interpolation near the points {rj } with linear combinations of the functions G(1,0) (r,rj ) and G(0,1) (r,rj ), in order to ensure that local maxima of Q are achieved exactly at the rj . Specifically, we set Q(r) =

S

α j G(0,0) (r,rj ) + α1j G(1,0) (r,rj ) + α2j G(0,1) (r,rj ).

(7.15)

j =1

Note that Q(r) is a linear of the functions Gm (r,rj ), and by definition > H combination ? of Gm (r,rj ) it obeys A q,f(r) , for some q, as desired. The proof is concluded by showing that, with high probability, there exists a choice of coefficients αj ,α1j , and α 2j such that Q reaches global maxima at the rj and Q(rj ) = uj , for all j . For this argument to work, the particular choice of Gm (r,rj ) is crucial; the main ingredients ¯ − rj ), and certain for the argument to work are that Gm (r,rj ) concentrates around G(r ¯ ¯ properties of the deterministic functions G and Q.

Proof of E Gx H Gx = I:

By definition of the Gabor matrix in (7.5), the entry in the (k,)-th row and (k #,# )-th column of GH x Gx is given by N

[GH x Gx ](k,),(k #,# ) =

kp

∗ xp− xp−# e−i2π L ei2π

k# p L

.

p=−N

# Noting that E [x ] = 0, we conclude that E [GH x Gx ](k,),(k #,# ) = 0 for != . For ∗ # = , using the fact that E[xp− xp− ] = 1/L, we arrive at E[[GH x Gx ](k,),(k #,# ) ] =

N 1 i2π (k# −k)p L e . L p=−N

The is equal to 1 for k = latter H E Gx Gx = I.

7.5.2

k#

and 0 otherwise. This concludes the proof of

Proof of Theorem 7.4 The following proposition, which is standard in the compressed sensing literature (see e.g., [3]) states that the existence of a dual polynomial guarantees that L1(y) recovers b from the measurement y = Gb. The proposition is the discrete analogue of Proposition 1 from earlier in the chapter. proposition 2 Let y = Rb and let S denote the support (i.e., the set of nonzero elements) of b. Assume that the columms of R corresponding to S are linearly independent. If there exists a vector v in the row space of R with vS = sign(bS )

and

vS c ∞ < 1,

(7.16)

Super-Resolution Radar Imaging via Convex Optimization

211

then b is the unique minimizer of L1(y). Here, vS its the vector consisting of the entries of v indexed by S, and likewise vS c consists of the entries not indexed by S, i.e., the entries indexed by the complement of S. Theorem 7.4 now follows directly from the existence of the polynomial Q that was constructed in the previous section. To see this, define v as [v](m,n) = Q([m/K,n/K]) and note that v satisfies (7.16), since Q([m/K,n/K]) = sign(b(m,n) ) for (m,n) ∈ S and |Q([m/K,n/K])| < 1 for (m,n) ∈ / S.

7.6

MIMO Radar In this section, we discuss super-resolution imaging in the context of MIMO radar. A MIMO radar uses multiple transmit antennas to send – typically orthogonal – probing signals simultaneously, and records the reflections from the objects with multiple receive antennas. As shown in this section, a MIMO radar can thereby, in principle, resolve the relative angles in addition to the relative distances and velocities of objects with a single measurement. To illustrate the principle of a MIMO radar, first consider a radar system with a single transmit and multiple receive antennas, and consider a object that lies in the far field of the radar, so that the reflections of the objects that arrive at the receiver are essentially parallel, as illustrated in Figure 7.4. The reflection from the object must travel an additional distance of dR sin(θ) between the signals received at two adjacent receive antennas. Thus, from an estimate of the angle of arrival we can determine the relative position of a object. The angular resolution that can be achieved as well as the number of objects that can be distinguished increases linearly in the number of receive antennas. As a consequence, for doubling the angular resolution or doubling the number of objects to be distinguishable, a MISO radar needs to double its number of transmit antennas. As we discuss next, using multiple transmit antennas in addition to multiple

× θ object dR

Figure 7.4 Principle of MISO radar: × and

correspond to transmit and receive antennas. The transmit antenna sends a probing signal, and the reflections of the object are received by three receive antennas. Estimating the relative delays of the probing signal allows to estimate the angle of the object relative to the antenna array, in addition to distance and velocity.

212

Heckel

Reflection from object 1 dR

r = 0,j = 0

× ×

dT

×

θ Object 1 d

Figure 7.5 Principle of MIMO radar: × and

correspond to transmit and receive antennas. Throughout, we assume the spacing of the NT transmit and NR receive antennas to be dT = 2fc

T and dR = cN 2fc , where fc is the carrier frequency.

c

receive antennas can give a much larger angular resolution from far fewer antennas. Specifically, by arranging NT transmit and NR receive antennas in a particular way (see Figure 7.5), a MIMO radar can obtain the same resolution obtained by a MISO (or SIMO) radar with NT NR uniformly spaced receive (or transmit) antennas. This is often called a MIMO virtual array. In this section, we discuss a MIMO radar model, and show that the fundamental limit for resolving the angle-delay-Doppler triplets is (1/(NT NR ),1/B,1/T ). We furthermore show that this limit can be overcome in the sense that triplets can be resolved on a much finer grid, provided they are sufficiently separated.

7.6.1

MIMO Signal Model and Problem Statement We consider a MIMO radar with NT transmit and NR receive antennas that are colocated and lie in a plane along with S objects, see Figure 7.5 for an illustration. The technical results presented in this section generalize to the more general setup where objects lie in three-dimensional space and the transmit and receive antennas lie in a two-dimensional plane. We consider the simpler two-dimensional setup since the generalization to three dimensions are straightforward. As in the previous section, we assume that the objects are located in the far field of the array. As a consequence, propagating waves appear planar and the angles between the object and each antenna are (approximately) the same. We let the transmit and receive antennas be uniformly spaced with spacings NT , respectively, where fc is the carrier frequency of the probing dT = 2f1 c and dR = 2f c signals. This spacing yields a uniformly spaced virtual array with NT NR antennas, and thus maximizes the number of virtual antennas achievable with NT transmit and NR

Super-Resolution Radar Imaging via Convex Optimization

213

receive antennas [34,35]. As explained in Section 7.6.2, the (baseband) signal yr (t) at continuous time t received by antenna r = 0,. . .,NR − 1, consists of the superposition of the reflections from the objects of the transmitted probing signals xj (t),j = 0,. . ., NT − 1, and is given by yr (t) =

S

bk ei2πrNT βk

N T −1

ei2πj βk xj (t − τ¯ k )ei2πν¯ k t .

(7.17)

j =0

k=1

Here, bk ∈ C is the attenuation factor, βk ∈ [0,1] the angle or azimuth parameter, and τ¯ k and ν¯k are the delay and Doppler shift, all associated with the k-th object. The parameters βk , τ¯ k , ν¯k determine the angle (β = − sin(θ)/2 see Figure 7.5), distance, and velocity of the k-th object relative to the radar. Locating the object therefore amounts to estimating the continuous parameters bk ,βk , τ¯ k , ν¯k from the responses yr ,r = 0,. . .,NR − 1, to known and suitably selected probing signals xj . As discussed in Section 7.2, due to practical constraints, the probing signals must be band-limited and approximately time-limited, and the responses to the probing signals can only be observed over a finite time interval. We make the same assumption on the probing signals as well as on the received signals as we made in Section 7.2; in particular, we assume that the received signals yr are observed over an interval of length T and that the probing signals xj have bandwidth B and are approximately supported on a time interval proportional to T . As explained in Section 7.2, it follows from the band limitation of the probing signals xt and the limited observation time of the received signals yr that the received signals are characterized by the samples [yr ]p =

S k=1

bk e

i2πrNT β k

N T −1

ei2πj βk [Fνk Tτk xj ]p .

(7.18)

j =0

Here, yr contains the samples of the received signal yr taken at rate 1/B (i.e., [yr ]p = yr (p/B) in the interval p/B ∈ [−T /2,T /2]), and xj is the vector containing the samples of the probing signal ([xj ]p := xj (p/B)). We have reduced the problem of identifying the locations of the objects under the constraints that the probing signals xj are band-limited and the responses yr are observed over a finite time interval only, to the estimation of the parameters bk ∈ C, (βk ,τk ,νk ) ∈ [0,1]3,k = 1,. . .,S, from the samples [yr ]p,r = 0,. . .,NR − 1,p = −N,. . .,N , in the input–output relation (7.18). We call this the super-resolution MIMO radar problem.

7.6.2

Derivation of the MIMO Input–Output Relation In this section, we derive the MIMO input–output relation (7.17). Towards this goal, we first consider a single object. The j -th antenna transmits the signal xj (t)ei2πfc t , where fc is the carrier frequency. This signal propagates to the object, which we assume to be a point scatterer, gets reflected, and propagates back to the r-th receiver. From Figure 7.5, we see that the corresponding delay is, as a function of the angle between antennas and the object, θ, distance to the object, d, and speed of light, c, given by

214

Heckel

τ˜ :=

sin(θ)(dT j + dR r) 2(dT j + dR r) 2d + = τ¯ − β . c c c

For the second equality, we defined the angle parameter β := − sin(θ)/2 and the delay τ¯ := 2d c . Taking the Doppler shift into account, the reflection of the j -th probing signal received by the r-th receive antenna is given by ˜ j (t − τ)e ˜ i2π(fc +ν¯ )(t−τ˜ ) . bx

(7.19)

Here, b˜ ∈ C is the attenuation factor associated with the object, and ν¯ := 2v c fc is the Doppler shift, which is a function of the relative velocity, v, of the object. By choosing T the antenna spacing as dT = 2fc c and dR = cN 2fc , the reflection of the j -th probing signal received by the r-th receive antenna in (7.19) becomes ¯ i2π(fc +ν¯ )β ˜ j (t − τ)e ˜ i2π(fc +ν¯ )(t−τ) e bx

j +rNT fc

¯ i2πβ(j +rNT ) ˜ j (t − τ)e ¯ i2π(fc +ν¯ )(t−τ) ≈ bx e .

Here, the approximation follows by the Doppler shift ν¯ being much smaller than the ¯ If follows that the reflection of the carrier frequency fc , therefore fcf+c ν¯ ≈ 1, and τ˜ ≈ τ. j -th probing signal received by the r-th receive antenna, after demodulation, is ¯ i2πν¯ t ei2πβ(j +rNT ), bxj (t − τ)e ˜ −i2πν¯ τ¯ . Next, consider S objects with parameters (bk ,βk , where we defined b = be τ¯ k , ν¯k ). Since, for S objects, the (demodulated) signal yr received by antenna r consists of the superposition of the reflections of the probing signals xj ,j = 0,. . .,NT − 1, transmitted by the transmit antennas, we obtain the input-output relation in (7.17) simply by summing over the reflections given by bk xj (t − τ¯ k )ei2πν¯ k t ei2πβk (j +rNT ) .

7.6.3

MIMO Atomic Norm Minimization Recall that our goal is to recover the unknown parameters (bk ,βk ,τk ,νk ) from the measurements {yr }. Toward this goal, we proceed analogously as in Section 7.2.1. We start by defining for convenience the vector r := [β,τ,ν], and write the input–output relation (7.18) in matrix-vector form: y = Az,

z=

S

bk f(rk ).

(7.20)

k=1 2N

Here y is obtained by stacking the vectors yT0 ,. . .,yTNR −1 , and the vector f(r) ∈ CL has entries [f(r)](v,k,p) = ei2π(vβ+kτ+pν ),v = 0,. . .,NT NR − 1,

T NR

k,p = −N,. . .,N .

Similarly as before, we use for convenience a three dimensional index to refer to entries 2 of a vector. Finally, the matrix A ∈ CNR L×NR NT L is defined as follows. The expression wr,p := e

i2πrNT β

N T −1 j =0

ei2πj β [Fν Tτ xj ]p,

Super-Resolution Radar Imaging via Convex Optimization

215

in (7.3) can be written as wr,p =

N T −1

N

ap,k,j ei2π(kτ+pν +(j +NT r)β),

j =0 k=−N

with ap,k,j =

N k 1 [xj ] ei2π(−p) L . L =−N

Let fp,j ∈ CL be the vector with kth entry [fp,j ]k = ap,k,j , k = −N,. . .,N , 2 and let Aj ∈ CL×L be the block-diagonal matrix with fTp,j on its pth diagonal, p = −N,. . .,N . With this notation, A is defined as the block-diagonal matrix with the 2 matrix [A0,. . .,ANT −1 ] ∈ CL×NT L on its diagonal, for all NR blocks on the diagonal. With this notation, (7.3) becomes (7.20). Similarly as for the SISO radar problem, recovery of the unknowns bk ,rk = [βk ,τk ,νk ] from the measurement z = Sk=1 bk f(rk ) is a 3D line spectral estimation problem that can be solved with standard spectral estimation techniques. In order to recover the vector z from the measurement y, we use that z is a sparse linear combination of atoms in the set A := {f(r),r ∈ [0,1]3 }, and estimate z by solving the basis pursuit type atomic norm minimization problem 4 4 AN(y) : minimize 4z˜ 4A subject to y = A˜z, z˜

where

zA :=

inf

bk ∈C,rk ∈[0,1]3

k

|bk | : z =

bk f(rk ) .

k

To summarize, as for the SISO radar problem, we estimate the parameters bk ,rk from y by: i. ii. iii.

7.6.4

solving AN(y) in order to obtain z, estimating the rk from z by solving the corresponding 3D-line spectral estimation problem, and solving the linear system of equations y = S−1 k=0 bk Af(rk ) for the bk .

Recovery Guarantees for MIMO Atomic Norm Minimization As before, we take the probing signals to be random by choosing its samples, i.e., the entries of the xj as i.i.d. Gaussian (or sub-Gaussian) zero-mean random variables with variance 1/(NT L). Moreover, we again require a minimum separation condition to be satisfied. definition 7.5 (MIMO minimum separation condition) We say the triplets (βj ,τj ,νj ) ∈ [0,1]2,j = 1,. . .,S satisfy the minimum separation condition if for all j,j # : j != j # ,

216

Heckel

|β j − βj # | ≥

10 NT NR − 1

or

|τj − τj # | ≥

5 N

or

|νj − νj # | ≥

5 . N

(7.21)

As before, |τj − τj # | is the wraparound distance on the unit circle. Note that the triplets must not be separated in angle, time, and frequency simultaneously; for the MIMO minimum separation condition to be satisfied, it is sufficient if they are separated in at least one of those dimensions. theorem 7.6 Assume that the samples of the probing signals xj ,j = 0,. . .,NT − 1, are i.i.d. zero-mean Gaussian random variables with variance 1/(NT L), and let L = 2N + 1 ≥ 1024 and NT NR ≥ 1024. Consider a signal where the signs of the attenuation factors {bj }Sj=1 are i.i.d. uniform on {−1,1} or the complex unit disc, and suppose that the triplets {(βj ,τj ,νj )}Sj=1 obey the MIMO minimum separation condition. Furthermore, choose δ > 0 and assume that the number of nonzero attenuation factors, S, obeys S≤c

min(L,NT NR ) log3 (L/δ)

(7.22)

,

where c is a numerical constant. Then, with probability at least 1 − δ, z = is the unique minimizer of AN(y), y = Az.

S

k=1 bk f(rk )

Theorem 7.2 guarantees that, with high probability, the attenuation factors and location parameters can be recovered perfectly from the observation y by solving a convex program (recall that the parameters bk ,rk can be obtained from z), provided that the locations rk = [β k ,τk ,νk ] are sufficiently separated in either angle, time, or frequency, and provided that the total number of objects satisfies condition (7.22). Note that, translated to the physical parameters τ¯ k , ν¯k , the MIMO minimum separation condition becomes: For all k,k # : k != k # , |βk − βk # | ≥

10 NT NR − 1

or

| τ¯ k − τ¯ k # | ≥

10.01 B

or

|ν¯k − ν¯k # | ≥

10.01 . T

Theorem 7.2 is essentially optimal in the number of objects that can be located, since S can be linear – up to a log-factor – in min(L,NT NR ), and S ≤ min(L,NT NR ) is a necessary condition to uniquely recover the attenuation factors bk even if the locations rk are known. To see this, note that for the linear system of equations (7.20) to have a unique solution, the vectors Af(rk ) must be linearly independent. If βk = 0, for all k, or if τk = 0 and νk = 0, for all k, the vectors Af(rk ),rk = (βk ,τk ,νk ),k = 0,. . .,S − 1 can only be linearly independent provided that S ≤ L and S ≤ NT NR , respectively. This is seen from ⎡ ⎤ T −1 i2πj β e F T x ei2π0β N ν τ j j =0 ⎢ ⎥ .. ⎥. Af(r) = ⎢ . ⎣ ⎦ T −1 i2πj β ei2πNT (NR −1)β N e F T x ν τ j j =0 We finally note that Theorem 7.2 is proven by constructing a dual certificate in a similar manner to our certificate construction for the SISO in Section 7.7.

Super-Resolution Radar Imaging via Convex Optimization

7.6.5

217

MIMO Super-Resolution Radar on a Fine Grid As discussed before for the SISO radar setup, a practical approach to estimate the parameters rk from the received signals, is to suppose the angle-time-frequency triplets lie on a fine grid, and solve the recovery problem on that grid. In general this leads to a gridding error, that, however, decreases as the grid becomes finer. We next discuss the corresponding (discrete) sparse signal recovery problem. Suppose the parameters (βk ,τk ,νk ) lie on a grid with spacing (1/K1,1/K2,1/K3 ), where K1,K2,K3 are integers obeying K1 ≥ NT NR , K2,K3 ≥ L = 2N + 1. With this assumption, the super-resolution MIMO radar problem reduces to the recovery of the sparse (discrete) signal b ∈ CK1 K2 K3 from the measurement y = Rb, where R ∈ CNR L×K1 K2 K3 is the matrix with (n1,n2,n3 )-th column given by Af(rn ),

rn = (n1 /K1,n2 /K2,n3 /K3 ).

Note that the nonzeros of b and its indices correspond to the attenuation factors bk and the locations rk on the grid. A standard approach to the recovery of the sparse signal b from the underdetermined linear system of equations y = Rb is to solve the following convex program: 4 4 4 4 ˜ (7.23) L1(y) : minimize 4b˜ 4 subject to y = Rb. b˜

1

What follows is the main result for recovery on the fine grid. theorem 7.7 Assume L = 2N + 1 ≥ 1024, NT NR ≥ 1024, and suppose we observe y = Rb, where b is a sparse vector with nonzeros indexed by the support set S ⊆ [K1 ] × [K2 ] × [K3 ], [K] := {0,. . .,K − 1}. Suppose that those indices satisfy the following minimum separation condition: for all triplets (n1,n2,n3 ),(n#1,n#2,n#3 ) ∈ S, |n1 − n#1 | 10 ≥ K1 NT NR − 1

or

|n2 − n#2 | 5 ≥ K2 N

or

|n3 − n#3 | 5 ≥ . K3 N

Moreover, we assume that the signs of the nonzeros of b are chosen independently from symmetric distributions on the complex unit circle. Choose δ > 0 and assume S≤c

min(L,NT NR ) log3 (L/δ)

,

where c is a numerical constant. Then, with probability at least 1 − δ, s is the unique minimizer of L1(y) in (7.23). Note that Theorem 7.7 does not impose any restriction on K1,K2,K3 , in particular they can be arbitrarily large. The proof of Theorem 7.7 is closely linked to that of Theorem 7.6; specifically, similarly to the SISO case, the existence of a certain dual certificate guarantees that b is the unique minimizer of L1(y). The dual certificate is obtained directly from the dual certificate for the continuous case, which, as mentioned

218

Heckel

before, is constructed for the MIMO case in a similar way as the certificate for the SISO case has been constructed in Section 7.7.

Numerical Results and Robustness Paralleling the discussion in Section 7.4.2 for SISO radar, we next briefly numerically evaluate the resolution obtained by the norm minimization approach to enable superresolution in a MIMO radar, and demonstrate robustness to noise. We discuss a synthetic experiment from [15]. In that experiment, a problem instance was generated by setting NT = 3,NR = 3, L = 41, and S = 5.√Object locations (βk ,τk ,νk ) were drawn uniformly at random from [0,1] × [0,2/ L]2 . Moreover, we choose K1 = SRFNT NR ,K2 = SRFL, and K3 = SRFL, where SRF ≥ 1 can be interpreted as a super-resolution factor as it determines by how much the (1/K1,1/K2,1/K3 ) grid is finer than the original, coarse grid (1/(NT NR ),1/L,1/L). To account for additive noise, as before, we solve the following modification of L1(y) in (7.23) 4 4 4 42 4 4 4 4 L1-ERR : minimize 4b˜ 4 subject to 4y − Rb˜ 4 ≤ δ, b˜

2

1

with δ chosen on the order of the noise variance. There are two error sources incurred by this approach: the gridding error obtained by assuming the points lie on a grid with spacing (1/K1,1/K2,1/K3 ), which decreases in SRF and becomes negligible, and the additive noise error, which is constant. The results of the simulations, depicted in Figure 7.6, show that the object resolution of the super-resolution approach is significantly better than that of the compressed sensing-based approach [7,8] corresponding to recovery on the coarse grid, i.e., SRF = 1. Moreover, the results show that the approach

Resolution error

7.6.6

SNR = 5dB SNR = 10dB SNR = 20dB Noiseless

0.6 0.4 0.2 1

2

3

4

5

6

SRF Figure 7.6 Resolution error for the recovery of S = 5 objects from the samples y with and without additive Gaussian noise n of a certain signal-to-noise ratio SNR = y22 /n22 , for

varying super-resolution factors (SRFs). The resolution error is defined as the average over (NT2 NR2 (βˆ k − βk )2 + L2 ( τˆ k − τk )2 + L2 (νˆk − νk )2 )1/2 , k = 1,. . .,S, where (βˆ k , τˆ k , νˆk ) are the locations obtained by solving L1-ERR.

Super-Resolution Radar Imaging via Convex Optimization

Resolution error

0.15

219

IAA L1-ERR

0.1 0.05 0 30

20

10

SNR in dB Figure 7.7 Resolution error (smaller is better) of L1-ERR and IAA applied to y + n, where n ∈ CNR L is additive Gaussian noise, such that the signal-to-noise ratio is SNR := y22 /n22 . As before, the resolution error is defined as (NT2 NR2 (βˆ k − βk )2 + L2 ( τˆ k − τk )2 + L2 (νˆk − νk )2 )1/2 , where (βˆ k , τˆ k , νˆk ) are the

locations obtained by solving L1-ERR.

is robust to noise and that even under noise, the localization accuracy is significantly improved over a standard approach to radar. We next compare our approach to the iterative adaptive approach (IAA) [36], proposed for MIMO radar in [37]. IAA is based on weighted least squares and has been proposed in the array processing literature. IAA can work well even with only one snapshot only and can therefore be directly applied to the MIMO super-resolution problem. However, to the best of our knowledge, no analytical performance guarantees are available in the literature that attest IAA similar performance than the 1 -minimizationbased approach. We compare the IAA algorithm [36, table II, entitled “The IAA-APES Algorithm”] to L1-ERR, for a problem with parameters NT = 3,NR = 3, and L = 41, as before, but with SRF = 3 and (βk ,τk ,νk ) = (k/(NR Nt ),k/L,k/L),k = 1,. . .,S, so that the location parameters lie on the fine grid and are separated. As before, we draw the corresponding attenuation factors bk i.i.d. uniformly at random from the complex unit disc. Our results, depicted in Figure 7.7, show that L1-ERR performs better in this experiment than IAA, in particular for small signal-to-noise ratios.

7.7

Discussion and Current and Future Research Directions In this section we discuss a class of signal recovery problems that are closely related to the SISO and MIMO radar problem, corresponding results, and open theoretical research problems, and comment on computational challenges in applying the methods discussed here in practical radar systems. The SISO and MIMO radar problems discussed in this chapter are versions of a more general problem, namely that of recovering a signal that is sparse in a continuously indexed dictionary, with the index corresponding to the locations and velocities of objects. In contrast, traditional compressive sensing research has focused on the recovery of signals that are sparse in discretely indexed dictionaries via convex programs [3]

220

Heckel

amongst other methods. As discussed in this chapter in the context of radar, signals that are sparse in continuously indexed dictionaries can be recovered via a convex program either by solving an atomic norm minimization problem, or by discretizing the continuous parameter space. However, the discretization step induces a gridding error. While in practice – provided the grid is chosen sufficiently fine – the gridding error is negligible, fine discretization leads to dictionaries with extremely correlated, i.e., coherent, columns, and the theory of compressive sensing, and many practical algorithms, rely on the dictionary to be incoherent and therefore does not apply to fine grids. The primary difficulty with recovering signals in such dictionaries is that the elements of the dictionaries are very close to each other – which is the case both for continuously indexed dictionaries as well as for finely discretized signals. Stable recovery of signals that are sparse in such dictionaries requires excluding signals that are supported on elements of the dictionary that are very close to each other. Such signals are excluded here by imposing the minimum separation condition. More specifically, the SISO and MIMO radar problems belong to a class of signal recovery problems where the goal is to recover unknown coefficients {bj } and location parameters {rj } from the measurement y=

S

bj Af(rj ).

j =1

Here, f(r) is a vector containing complex exponentials, and r is a d-dimensional location parameter. For example, if r is a one-dimensional location parameter then [f(r)]r = e−i2πr τ , and if r is two-dimensional location vector, as in the SISO radar problem, then [f(r)](r,q) = e−i2π(r τ+q ν ) . The matrix A is a problem-dependent matrix that parameterizes the dictionary; in the SISO case it is equal to A = Gx FH ; see (7.7). In other words, radar signals are sparse in a continuously dictionary that is parameterized by a matrix A. We hasten to add that there are a number of interesting signal recovery problems in continuously indexed dictionaries that are not of this particular form: the deconvolution problem in the paper [38] is such an example, and the computational imaging problem in [39] is another.

7.7.1

Stability to Noise In this section, we discuss analytical results on the stability to noise of the atomic norm minimization framework. While currently there are no formal results for the SISO and MIMO radar problem, we discuss statements pertaining to two closely related problems. The first is the classical line spectral estimation problem where A is the identity matrix, and the second is a generalized line spectral estimation problem, where A is a Gaussian random matrix. As mentioned before, for A = I, the sparse recovery problem reduces to the classical line spectral estimation problem studied for the noisy and noiseless case in [10, 12,40,41]. This problem is well understood, and atomic norm minimization succeeds under very general conditions. Specifically, the paper [10] shows that a convex program

Super-Resolution Radar Imaging via Convex Optimization

221

provably recovers the coefficients {bj } and location parameters {rj } perfectly, provided that the minimum separation condition holds (see Section 7.3.2). While standard spectral estimation techniques such as Prony’s method, MUSIC, and ESPRIT [9] also provably succeed for the noiseless case, even without requiring a separation condition, an advantage of the convex program is that it does not require knowledge of S, and perhaps more importantly is provably robust [12,40,42]. Specifically, Tang et al. [42] show that the atomic norm regularized least squares estimate enables near-optimal denoising of z from a noisy measurement y = z + e, S where z is a signal of the form z = j =1 f(νj ), and e is zero-mean Gaussian noise 2 with variance σ I. Specifically, provided the minimum separation condition holds, the atomic norm regularized least squares estimate zˆ obeys 4 4 4zˆ − z42 ≤ cσ2 S log(L), 2 with high probability. This result is essentially optimal; to see this, note that even if we knew the location parameters {νj } exactly, the best bound we could achieve would be σ2 S [43], only by a logarithmic factor short of the result. In addition, the paper [42] shows that the corresponding estimator localizes the frequencies up to a certain (small) error, provided that the number of samples is sufficiently large. Next, suppose that A is a M × L Gaussian random matrix with i.i.d. N (0,1/M) + iN (0,1/M) entries, with M typically much smaller than L. Assume we are given a noisy measurement y = Az + e, with z a signal of the form z = Sj=1 f(νj ) and e is noise. To estimate the signal z from such a measurement one can use an atomic norm optimization problem of the form 1 zˆ := arg min y − A¯z22 2 z¯

subject to ¯zA ≤ τ,

(7.24)

with τ a tuning parameter. Theorem 4 in the paper [16] shows that, as long as M ≥ cS log(L), with c a fixed numerical constant, the minimizer of (7.24) with τ = zA obeys 4 4 4zˆ − z42 ≤ cS log(L)σ2, 2 with high probability. Again, this results is essentially optimal both with respect to the number of measurements M required relative to the number of unknowns S, as well as with respect to the best bound we can achieve for estimation under noise. Moreover, there results do not make any assumptions on the coefficients bj . Unfortunately, the corresponding proof strategy does not carry over to the case where A is a structured random matrix, as in the SISO and MIMO radar problems considered here. However, numerical simulations suggest that for a number of structured random matrices, including the matrices parametrizing the SISO and MIMO radar problems, the performance of the nuclear norm minimization program, as well as 1 -norm regularized least squares, is similar.

222

Heckel

7.7.2

Computational Challenges A challenge in applying the convex optimization-based super-resolution methods in the context of radar is their computational complexity. Specifically, in the context of SISO radar, if we estimate the time–frequency shifts based on solving (the dual of) the atomic norm minimization problem, then the optimization variable of the corresponding convex program has dimensions L2 × L2 . Thus, an algorithm that solves or approximates the atomic norm minimization problem has computational complexity at least L4 , which is infeasible for real-world problems. As discussed in Section 7.4, what comes to our rescue is that in practice, we can solve the super-resolution radar problem on a fine grid, and recover the signal by solving a 1 -minimization problem. The complexity of numerically solving the corresponding program with a standard iterative algorithm such as FISTA [44] depends on the dimension of the matrix (determined by the problem size [BT and number of antennas]), as well as the corresponding super-resolution factor (see Section 7.4.1) and the conditioning of the matrices involved. Increasing the superresolution factor leads to both a larger problem size (the number of columns in the SISO radar problem increases quadratically in the super-resolution factor), which results in a larger iteration complexity, as well as in general to a slower convergence of the iterative algorithm, since the conditioning of the matrices involved become worse. Thus, an interesting research direction is to develop computationally efficient algorithms for the recovery of signals in continuously indexed dictionaries in general, and for the SISO and MIMO radar problem in particular. See the papers [45,46] for some recent work in this direction.

References [1] D. W. Bliss and K. W. Forsythe, “Multiple-input multiple-output (MIMO) radar and imaging,” in Asilomar Conf. on Signals, Syst. and Comput., 2003, pp. 54–59. [2] J. Li and P. Stoica, “MIMO radar with colocated antennas,” IEEE Signal Process. Mag., vol. 24, no. 5, pp. 106–114, 2007. [3] E. J. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory, vol. 52, no. 2, pp. 489–509, 2006. [4] M. Herman and T. Strohmer, “High-resolution radar via compressed sensing,” IEEE Trans. Signal Process., vol. 57, no. 6, pp. 2275–2284, 2009. [5] R. Baraniuk and P. Steeghs, “Compressive radar imaging,” in IEEE Radar Conf., 2007, pp. 128–133. [6] R. Heckel and H. Bölcskei, “Identification of sparse linear operators,” IEEE Trans. Inf. Theory, vol. 59, no. 12, pp. 7985–8000, 2013. [7] D. Dorsch and H. Rauhut, “Refined analysis of sparse MIMO radar,” J. Fourier Anal. Appl., vol. 23, 2017. [8] T. Strohmer and H. Wang, “Adventures in compressive sensing based MIMO radar,” in Excursions in Harm. Anal., ser. Appl. Num. Harm. Anal., 2015, pp. 285–326. [9] P. Stoica and R. L. Moses, Spectral Analysis of Signals. Pearson Prentice Hall, 2005.

Super-Resolution Radar Imaging via Convex Optimization

223

[10] E. J. Candès and C. Fernandez-Granda, “Towards a mathematical theory of superresolution,” Comm. Pure Appl. Math., vol. 67, no. 6, pp. 906–956, 2014. [11] G. Tang, B. N. Bhaskar, P. Shah, and B. Recht, “Compressed sensing off the grid,” IEEE Trans. Inform. Theory, vol. 59, no. 11, pp. 7465–7490, 2013. [12] B. N. Bhaskar, G. Tang, and B. Recht, “Atomic norm denoising with applications to line spectral estimation,” IEEE Trans. Signal Process., vol. 61, no. 23, pp. 5987–5999, 2013. [13] C. Aubel, D. Stotz, and H. Bölcskei, “A theory of super-resolution from short-time Fourier transform measurements,” J. Fourier Anal. Appl., vol. 24, 2018. [14] R. Heckel, V. I. Morgenshtern, and M. Soltanolkotabi, “Super-resolution radar,” Inf. Inference, vol. 5, no. 1, pp. 22–75, 2016. [15] R. Heckel, “Super-resolution mimo radar,” in IEEE International Symposium on Information Theory, 2016, pp. 1416–1420. [16] R. Heckel and M. Soltanolkotabi, “Generalized line spectral estimation via convex optimization,” IEEE Trans. Inf. Theory, vol. 64, pp. 4001–4023, 2017. [17] T. Strohmer, “Pseudodifferential operators and Banach algebras in mobile communications,” Appl. Comput. Harmon. Anal., vol. 20, no. 2, pp. 237–249, 2006. [18] G. Tauböck, F. Hlawatsch, D. Eiwen, and H. Rauhut, “Compressive estimation of doubly selective channels in multicarrier systems,” IEEE J. Sel. Topics Signal Process., vol. 4, no. 2, pp. 255–271, 2010. [19] W. Bajwa, A. Sayeed, and R. Nowak, “Learning sparse doubly-selective channels,” in Proc. of 46th Allerton Conf. on Commun., Control, and Comput., Monticello, IL, 2008, pp. 575–582. [20] W. U. Bajwa, K. Gedalyahu, and Y. C. Eldar, “Identification of parametric underspread linear systems and super-resolution radar,” IEEE Trans. Signal Process., vol. 59, no. 6, pp. 2548–2561, 2011. [21] D. Slepian, “On bandwidth,” Proc. IEEE, vol. 64, no. 3, pp. 292–300, 1976. [22] F. Krahmer, S. Mendelson, and H. Rauhut, “Suprema of chaos processes and the restricted isometry property,” Commun. Pur. Appl. Math., vol. 67, no. 11, pp. 1877–1904, 2014. [23] A. Gershman and N. Sidiropoulos, Eds., Space-Time Processing for MIMO Communications. John Wiley & Sons, 2005. [24] V. Chandrasekaran, B. Recht, P. A. Parrilo, and A. S. Willsky, “The convex geometry of linear inverse problems,” Found. Comput. Math., vol. 12, no. 6, pp. 805–849, 2012. [25] Z. Yang, L. Xie, and P. Stoica, “Vandermonde decomposition of multilevel Toeplitz matrices with application to multidimensional super-resolution,” IEEE Trans. Inform. Theory, vol. 62, 2015. [26] P. A. Bello, “Characterization of randomly time-variant linear channels,” IEEE Trans. Commun. Syst., vol. 11, no. 4, pp. 360–393, 1963. [27] D. L. Donoho, “Superresolution via sparsity constraints,” SIAM J. on Math. Anal., vol. 23, no. 5, pp. 1309–1331, 1992. [28] A. Moitra, “Super-resolution, extremal functions and the condition number of vandermonde matrices,” in Proc. of the Forty-seventh Annual ACM Symposium on Theory of Computing, 2015, pp. 821–830. [29] A. Quinquis, E. Radoi, and F. C. Totir, “Some radar imagery results using superresolution techniques,” IEEE Trans. Antennas Propag., vol. 52, no. 5, pp. 1230–1244, 2004. [30] A. Jakobsson, A. L. Swindlehurst, and P. Stoica, “Subspace-based estimation of time delays and Doppler shifts,” IEEE Trans. Signal Process., vol. 46, no. 9, pp. 2472–2483, 1998.

224

Heckel

[31] G. Tang, B. N. Bhaskar, and B. Recht, “Sparse recovery over continuous dictionaries-just discretize,” in Asilomar Conf. on Signals, Syst. and Comput., Pacific Grove, CA, 2013, pp. 1043–1047. [32] S. R. Becker, E. J. Candès, and M. C. Grant, “Templates for convex cone problems with applications to sparse signal recovery,” Math. Prog. Comp., vol. 3, no. 3, pp. 165–218, 2011. [33] E. van den Berg and M. P. Friedlander, “Probing the pareto frontier for basis pursuit solutions,” SIAM J. Sci. Comput., vol. 31, no. 2, pp. 890–912, 2008. [34] B. Friedlander, “On the relationship between MIMO and SIMO radars,” IEEE Trans. Signal Process., vol. 57, no. 1, pp. 394–398, Jan. 2009. [35] T. Strohmer and B. Friedlander, “Analysis of sparse MIMO radar,” Appl. Comput. Harm. Anal., vol. 37, no. 3, pp. 361–388, 2014. [36] T. Yardibi, J. Li, P. Stoica, M. Xue, and A. B. Baggeroer, “Source localization and sensing: A nonparametric iterative adaptive approach based on weighted least squares,” IEEE Trans. Aerosp. Electron. Syst., vol. 46, no. 1, pp. 425–443, 2010. [37] W. Roberts, P. Stoica, J. Li, T. Yardibi, and F. Sadjadi, “Iterative adaptive approaches to MIMO radar imaging,” IEEE J. Sel. Topics Signal Process, vol. 4, no. 1, pp. 5–20, 2010. [38] B. Bernstein and C. Fernandez-Granda, “Deconvolution of point sources: A sampling theorem and robustness guarantees,” Commun. Pur. Appl. Math., to appear, 2017. [39] N. Antipa, G. Kuo, R. Heckel et al., “Diffusercam: Lensless single-exposure 3D imaging,” Optica, vol. 5, no. 1, p. 19, 2018. [40] E. J. Candès and C. Fernandez-Granda, “Super-resolution from noisy data,” J. Fourier Anal. Appl., vol. 19, no. 6, pp. 1229–1254, 2014. [41] C. Fernandez-Granda, “Super-resolution of point sources via convex programming,” Information and Inference, vol. 5, pp. 251–303, 2016. [42] G. Tang, B. N. Bhaskar, and B. Recht, “Near minimax line spectral estimation,” IEEE Trans. Inf. Theory, vol. 61, no. 1, pp. 499–512, 2015. [43] F. Bunea, A. Tsybakov, and M. Wegkamp, “Sparsity oracle inequalities for the Lasso,” Electron. J. Stat., vol. 1, pp. 169–194, 2007. [44] A. Beck and M. Teboulle, “A fast iterative shrinkage-thresholding algorithm for linear inverse problems,” SIAM Journal on Imaging Sciences, vol. 2, no. 1, p. 183202, 2009. [45] N. Boyd, G. Schiebinger, and B. Recht, “The alternating descent conditional gradient method for sparse inverse problems,” SIAM J. Optimiz., vol. 27, no. 2, pp. 616–639, 2017. [46] N. Rao, P. Shah, and S. Wright, “Forward-backward greedy algorithms for atomic norm regularization,” IEEE Trans. Signal Process., vol. 63, no. 21, pp. 5798–5811, 2015.

8

Adaptive Beamforming via Sparsity-Based Reconstruction of Covariance Matrix Yujie Gu, Nathan A. Goodman, and Yimin D. Zhang

Traditional adaptive beamformers are very sensitive to model mismatch, especially when the training samples for adaptive beamformer design are contaminated by the desired signal. In this chapter, we reconstruct a signal-free interference-plus-noise covariance matrix for adaptive beamformer design. Exploiting the sparsity of sources, the interference covariance matrix can be reconstructed as a weighted sum of the outer products of the interference steering vectors, and the corresponding parameters can be estimated from a sparsity-constrained covariance matrix fitting problem. In contrast to classical compressive sensing and sparse reconstruction techniques, the sparsityconstrained covariance matrix fitting problem can be effectively solved as a modified least-squares solution by using the a priori information on the array structure. Extensive simulation results demonstrate that the proposed adaptive beamformer almost always provides near-optimal output performance, regardless of the input signal power.

8.1

Introduction Adaptive beamforming is an effective spatial filtering technique that adjusts the beamforming weight vector to increase the strength of the signal of interest while suppressing interference and noise. As a ubiquitous task in array signal processing, adaptive beamforming has been widely used in radar, sonar, wireless communications, radio astronomy, seismology, speech processing, medical imaging, and many other areas (see, for example, [1–5] and the references therein). Unlike conventional data-independent beamformers (e.g., fixed or switched beamformers), adaptive beamformers depend on the array received data and hence are expected to provide better capabilities for interference suppression and signal enhancement. Nevertheless, it is also well known that adaptive beamformers are extremely sensitive to model mismatch, especially when the training samples used for the calculation of the beamforming weight are contaminated by the desired signal. In practice, such model mismatch commonly occurs. For example, the data covariance matrix cannot be accurately estimated due to the limited number of training samples, and the steering vector of the desired signal may also be imprecise or even unknown due to look direction error, imperfect calibration, and other effects. Whenever model mismatches exist, classical adaptive beamformers (e.g., Capon beamformer [6]) will suffer severe performance degradation. To this end, adaptive beamformer design with robustness against model mismatch has been an intensive research 225

226

Gu, Goodman, and Zhang

topic in the past decades, and various robust adaptive beamforming techniques have been proposed (see, for example, [4,7,8] and the references therein). Based on the principle of adaptive beamforming, these robust adaptive beamformers can be classified into two major categories. In the first category, robust adaptive beamforming techniques process the sample covariance matrix, because the exact interference-plus-noise covariance matrix is usually unavailable in practical applications. The sample covariance matrix is a maximum likelihood estimate of the data covariance matrix, and thus leads to the optimal output of the resulting adaptive beamformer when the sample size tends to infinity. Unfortunately, the sample size is often limited in practice, thus resulting in significant performance degradation, especially when the desired signal is present in the training samples [9,10]. The most popular robust beamforming technique in this category is the diagonal loading technique [9,11–13], which adds a scaled identity matrix to the sample covariance matrix to reduce the conditional number. A major problem with diagonal loading is that there is no clear rule to choose the optimal diagonal loading factor in different scenarios. In order to choose the diagonal loading factor adaptively, rather than in an ad hoc way, several user parameter–free adaptive beamforming algorithms were proposed (see, for example, [14] and the references therein). The shrinkage estimation approach [15] in the sense of minimizing mean squared error (MSE) can automatically compute the diagonal loading levels without the need to specify any user parameters. However, this approach leads to an estimate of the statistical covariance matrix of the array received data rather than the required interference-plus-noise covariance matrix. In such a case, the performance degradation becomes severe with the increase of the desired signal power, even when the desired signal steering vector is exactly known. The eigenspace decomposition technique [16,17] is another popular approach for robust adaptive beamforming that is applicable to an arbitrary steering vector mismatch case. The key idea of this technique is to use the projection of the presumed steering vector onto the sample signal-plusinterference subspace. This approach requires the knowledge of the dimension of the signal-plus-interference subspace. It is known that this approach suffers severe performance degradation from the subspace swap1 when the signal-to-noise ratio (SNR) is low [18,19]. It also suffers from the signal self-nulling problem, especially at high SNR levels. A sparsity-based iterative adaptive approach (IAA) [20] can iteratively update the spatial power estimates in the whole observation field and subsequently update the covariance matrix used for adaptive beamformer design. Although it does improve the power estimate, the IAA beamforming algorithm is not robust against direction-ofarrival (DOA) mismatch because its weight is simply that of the scanning grid point corresponding to the assumed DOA of the desired signal. In the second category, robust adaptive beamforming techniques process the presumed desired signal steering vector because the exact knowledge of the steering vector is not easy to obtain in practice. In practical situations, steering vector mismatch can 1 A subspace swap occurs when the measured data is better approximated by some components of the noise

subspace than by some components of the signal subspace, i.e., there is a switch of vectors between the estimated signal and noise subspaces.

Adaptive Beamforming via Sparsity-Based Reconstruction of Covariance Matrix

227

easily occur due to look-direction errors [21,22] or imperfect array calibration and distorted antenna shape [23]. Besides these, other common causes leading to steering vector mismatches include array manifold mismodeling because of source wavefront distortions that result from environmental inhomogeneities [24,25], near-far problem [26], source spreading and local scattering [27–30], as well as other effects [10]. In this category, the linear constrained minimum variance (LCMV) beamformer [31] is most commonly used. It provides robustness against uncertainty in the signal look direction by broadening the main lobe of the beampattern. However, the additional imposed constraints reduce the degrees of freedom (DOFs) of the resulting adaptive beamformer. More importantly, the LCMV beamformer becomes less robust when any other types of steering vector mismatch beyond the look-direction errors become dominant. To improve the robustness of adaptive beamformers against arbitrary unknown steering vector mismatches, the worst-case performance optimization-based technique [32–34] makes explicit use of an uncertainty set of the signal steering vector. This method requires that the upper bound of the norm of the mismatch vector is a priori unknown. Moreover, the worst operating conditions may not always occur. Hence, this adaptive beamforming technique is also an ad hoc approach and will suffer from performance degradation whenever the upper bound of the norm of the mismatch vector is either overestimated or underestimated. Another representative technique in this category is to estimate the desired signal steering vector by maximizing the beamformer output, under the constraint that the convergence of the steering vector estimate to any interference steering vector or their combinations is prohibited [35,36]. However, the imposed norm constraint on the steering vector is too strict to be satisfied, particularly when there exist local scattering encountered in, e.g., mobile communications and indoor speech signal processing. In such cases, gain perturbations in different sensors cannot be ignored, and then the norm constraint no longer holds. As mentioned previously, these two categories of adaptive beamforming techniques were developed almost independently in the past four decades. Obviously, these adaptive beamforming techniques are not optimal, because they respectively assume that either the desired signal steering vector or the interference-plus-noise covariance matrix is exactly known. Since the pioneering work of Vorobyov et al. [37], adaptive beamforming has been required to be jointly robust against covariance matrix uncertainty and steering vector mismatch [38–42]. It is worth noting that, in [40], the interference-plusnoise covariance matrix is reconstructed by integrating the outer products of interference steering vectors weighted by the Capon spatial spectrum over a region separated from the desired signal direction, thus removing the desired signal component from the covariance matrix used for adaptive beamformer design. The reconstructed interferenceplus-noise covariance matrix is then used to correct the presumed signal steering vector in order to maximize the beamformer output power under the only constraint that the corrected steering vector does not converge to any interference steering vector or their combinations. Based on the reconstructed interference-plus-noise covariance matrix and the estimated desired signal steering vector, the resulting adaptive beamformer provides a near-optimal output performance with a fast convergence rate. However, the computational complexity of covariance matrix reconstruction is high due to the

228

Gu, Goodman, and Zhang

integral operation. In addition, there is a certain level of performance loss when the number of training samples is small, because the source power obtained from the Capon spatial spectrum is underestimated and, as a result, the estimated interference-plus-noise covariance matrix is inaccurate. In this chapter, we will elaborate adaptive beamforming via sparsity-based reconstruction of the interference-plus-noise covariance matrix [43]. By exploiting the sparsity of sources in the observed spatial domain, the interference covariance matrix is reconstructed as a linear combination of the outer products of the interference steering vectors weighted by their individual power, which can be estimated from a sparsityconstrained covariance matrix fitting problem. As such, the proposed technique provides a signal-free interference-plus-noise covariance matrix to enable robust adaptive beamformer design that avoids the signal self-nulling problem. It requires low computational complexity, as there is no matrix inversion or eigen-decomposition involved in the sparsity-constrained covariance matrix fitting problem. Hence, the proposed adaptive beamforming technique is suitable for an arbitrary number of training samples [44]. When the number of training samples is larger than the number of array sensors, the formulated sparsity-constrained covariance matrix fitting problem can be effectively solved by using the known array structure, i.e., estimate the directions of sources and their power in turn. The proposed adaptive beamformer is compared to existing stateof-the-art adaptive beamformers in terms of computational complexity, output signalto-interference-plus-noise ratio (SINR) performance, and convergence rate. Numerical simulations clearly demonstrate the near-optimal output performance and faster convergence rate of the proposed adaptive beamforming algorithm exploiting the sparsity of sources in the spatial domain.

8.2

Adaptive Beamforming Criterion In this section, we first build the narrowband array signal model, then briefly review adaptive beamforming criteria and classical adaptive beamformers.

8.2.1

Array Signal Model Consider a narrowband array consisting of M omnidirectional sensors, depicted in Figure 8.1. The baseband received signal of the array at the time instant k, x(k) = [x1 (k),. . .,xM (k)]T ∈ CM , can be represented as x(k) = x s (k) + x i (k) + n(k),

(8.1)

where x s (k), x i (k), and n(k) are statistically independent components of the desired signal, interference, and noise, respectively. Here, ( · )T denotes the transpose operator. Among them, the desired signal vector x s (k) is expressed as x s (k) = a s s(k),

(8.2)

where s(k) is the desired signal waveform, and a s ∈ CM is the corresponding signal steering vector. Ideally, the steering vector is a function depending on the array geometry

Adaptive Beamforming via Sparsity-Based Reconstruction of Covariance Matrix

229

Figure 8.1 System block diagram of the adaptive beamformer.

as well as source direction, e.g., a s a(θs ), where θs is the direction of the desired signal impinging on the array. For example, the ideal steering vector of a uniform linear array (ULA) has the form of 1T 0 2π 2π (8.3) a(θ) = 1,e−j λ d sin θ,. . .,e−j λ (M−1)d sin θ , √ where θ is the DOA of the source, j = −1 is the imaginary unit, λ is the wavelength of the narrowband signal, and d = λ/2 is the inter-element spacing of the array. Similarly, the steering vector of the interference has the similar form with a different source direction. In contrast, there is no such form for the additive noise because noise does not have a fixed direction.

8.2.2

Adaptive Beamforming Criteria The objective of adaptive beamforming is to design a data-dependent beamforming weight vector w = [w1,. . .,wM ]T ∈ CM , such that the beamformer output y(k) = wH x(k)

(8.4)

is the best estimate of the desired signal waveform s(k), where ( · )H denotes the Hermitian transpose. To this end, a number of adaptive beamforming criteria have been developed in the past decades. Among them, maximum SINR [6] is the most popular one. Other feasible adaptive beamforming criteria include minimum MSE [3], minimum least-squares error [45], and minimum mutual information [46]. The interested readers are referred to references [3,4,47] for the detailed performance tradeoffs among different adaptive beamforming criteria. In this chapter, the maximum SINR criterion will be mainly considered for adaptive beamformer design.

230

Gu, Goodman, and Zhang

The beamformer output for the SINR maximization problem, defined as 2 E wH x s (k) max SINR 2 w E wH (x i (k) + n(k)) 2 σs2 wH a s , = H w R i+n w

(8.5)

is mathematically equivalent to the minimum variance distortionless response (MVDR) problem [6] as min wH R i+n w w

s.t. wH a s = 1,

where σs2 E |s(k)|2 is the desired signal power, and R i+n E (x i (k) + n(k)) (x i (k) + n(k))H ∈ HM ,

(8.6)

(8.7)

is the interference-plus-noise covariance matrix. Here, E [ · ] denotes the statistical expectation, and HM denotes the M × M Hermitian matrix. Using the Lagrange multiplier method, the solution of the MVDR problem wMVDR =

R −1 i+n a s −1 aH s R i+n a s

,

(8.8)

is easily obtained. The MVDR beamformer is sometimes referred to as the Capon beamformer, which maximizes the output SINR. Substituting the data covariance matrix R = E x(k)x H (k) = σs2 a s a H s + R i+n,

(8.9)

into (8.6) in lieu of the generally unavailable interference-plus-noise covariance matrix R i+n , the corresponding solution, wMPDR =

R −1 a s −1 aH s R as

,

(8.10)

is referred to as the minimum power distortionless response (MPDR) beamformer. Using the matrix inversion lemma, the MPDR beamformer is proven to be equivalent to the MVDR beamformer as $ %−1 1 wMPDR = R i+n + σs2 a s a H as s −1 H as R as H −1 R −1 1 i+n a s a s R i+n −1 = − R as i+n −1 −1 aH σs−2 + a H s R as s R i+n a s = αwMVDR, (8.11) & ' −1 H R −1 a 1−a H R −1 a /σ −2 +a H R −1 a where the scalar coefficient α = a H R a /a s s s s s s s s s i+n i+n i+n does not affect the adaptive beamformer performance in terms of the output SINR. Hence, the MPDR beamformer is also referred to as an MVDR beamformer in the majority of the early literature.

Adaptive Beamforming via Sparsity-Based Reconstruction of Covariance Matrix

231

In practical array applications including radar, however, the data covariance matrix cannot be accurately estimated due to the limited training samples, and the signal steering vector may not be precisely known because of the imperfect knowledge of the source location, propagation environment and/or array calibration. In such cases, the MPDR beamformer suffers severe performance degradation, which becomes obvious with the increase of input signal power. Hence, the MPDR beamformer underperforms the MVDR beamformer in practical applications.

8.2.3

Adaptive Beamformer Design Limited by the size of training samples, the exact data covariance matrix R is not easy available in practical applications, not to mention the signal-free interference-plus-noise covariance matrix R i+n . It is usually replaced by the sample covariance matrix K ˆ = 1 x(k)x H (k), R K

(8.12)

k=1

where K is the number of snapshots (i.e., training samples). The resulting adaptive beamformer, wSMI =

ˆ −1 a¯ s R ˆ a¯ H s R

−1

a¯ s

,

(8.13)

is called the sample matrix inversion (SMI) beamformer [48], where a¯ s = a(θs ) is the presumed signal steering vector. Whenever there exists a desired signal in the array received signal x(k), the SMI beamformer is in essence an MPDR beamformer (8.10) ˆ will converge to R, and rather than an MVDR beamformer (8.8). As K → ∞, R the corresponding output SINR will approach the optimal value under stationary and ˆ and R is ergodic assumptions. However, when K is small, the large gap between R known to dramatically affect the output performance of the SMI beamformer, especially when there is a desired signal in the training samples [9,10]. In order to reduce the sensitivity of the SMI beamformer to model mismatches, many different beamforming algorithms have been developed in the past decades and successfully applied in a wide range of areas (see, for example, [3,4,7,14] and the references therein). In the following, several classical adaptive beamforming algorithms are briefly reviewed.

Diagonal Loading Beamforming Diagonal loading is the most popular adaptive beamforming approach that is robust to ˆ in the SMI beamthe data uncertainty [9,13]. Replacing the sample covariance matrix R ˆ former (8.13) by a diagonally loaded sample covariance matrix R + ξI , the resulting beamformer, $ % ˆ + ξI −1 a¯ s R (8.14) wDL−SMI = $ %−1 , ˆ a¯ H a¯ s s R + ξI is referred to as the diagonal loading SMI (DL-SMI) beamformer, where ξ is a diagonal loading factor, and I is an identity matrix.

232

Gu, Goodman, and Zhang

The performance of the DL-SMI beamformer depends on the diagonal loading factor ξ. It is usually chosen in an ad hoc way, typically about ten times the noise power, i.e., ξ = 10σn2 , where the noise power σn2 is assumed to be known [9]. Obviously it is not optimal, nor is the instantaneous noise power easy to know. In order to adaptively choose the loading factor, several user parameter–free approaches have been proposed for adaptive beamforming [14]. However, this method leads to an estimate of the data covariance matrix rather than that of the interference-plus-noise covariance matrix. Regardless of the value of the chosen diagonal loading factor, the performance loss of the DL-SMI beamformer is inevitable, and this degradation becomes more severe with the increase of the desired signal power [49]. The main reason is that the desired signal component is always active in any kind of diagonal loading beamformer and its effect becomes more pronounced with the increase of input SNR [40].

Eigenspace Decomposition Beamforming Motivated by the success of DOA estimation [50], the idea of eigen-decomposition has also been introduced for adaptive beamformer design [16,17]. Replacing the presumed steering vector a¯ s in (8.13) by the projection of a¯ s onto the sample signal-plusinterference subspace, the resulting eigenspace beamformer is given by ˆ −1 P E a¯ s = E−1 E H a¯ s , wEIG = R

(8.15)

where P E = EE H is the orthogonal projection matrix onto the signal-plus-interference subspace. Here, the matrix E contains the signal-plus-interference subspace eigenvecˆ and the diagonal matrix contains the corresponding eigenvalues. tors of R, The eigenspace beamformer is robust to arbitrary steering vector mismatch. However, this approach does not work well at low SNR as well as at high signal-to-interference ratio (SIR) cases. In the former case, the estimation of the projection matrix onto the signal-plus-interference subspace breaks down because of the high probability of subspace swaps. In the latter case, the desired signal component denominates the sample covariance matrix, thus degrading the performance of the adaptive beamformer. Furthermore, the eigenspace beamformer also does not work well when the dimension of the signal-plus-interference subspace is high and/or difficult to determine.

Worst-Case Beamforming The worst-case performance optimization-based adaptive beamforming [32] guarantees a distortionless response for all possible steering vectors in a predetermined set. The worst-case adaptive beamforming problem can be formulated as ˆ min wH Rw w

s.t.

max |wH (a¯ s + es )| ≥ 1,

es 2 ≤ε

(8.16)

where es = a s − a¯ s denotes the mismatch vector between the actual signal steering vector a s and the presumed signal steering vector a¯ s , and ε is the upper bound of the norm of the mismatch vector es . Here, · 2 denotes the 2 -norm, also called the Euclidean norm. Because the constraint condition is nonlinear and nonconvex, the worst-case adaptive beamforming problem (8.16) is a semi-infinite nonconvex quadratic

Adaptive Beamforming via Sparsity-Based Reconstruction of Covariance Matrix

233

program, and is NP-hard2 . By using the special structure of the objective function and the constraints, the nonconvex optimization problem can be reformulated as a secondorder cone programming (SOCP) problem ˆ min wH Rw w

s.t.

wH a¯ s ≥ ε w2 + 1, % $ Im wH a¯ s = 0,

(8.17)

which is convex and can be efficiently solved in polynomial time using the wellestablished interior point methods. Here, Im ( · ) denotes the imaginary part of a complex number. The worst-case beamformer is robust to arbitrary unknown signal steering vector mismatch with an upper-bounded norm. However, in practical applications, neither the mismatch vector nor its upper bound is a priori known. Either overestimation or underestimation of the upper bound of the norm of the steering vector mismatch will degrade the performance of the worst-case beamformer. In addition, the worst-case beamformer also suffers the signal self-nulling problem because it uses the sample covariance matrix ˆ rather than the interference-plus-noise covariance matrix R i+n . R

Iterative Adaptive Beamforming The IAA algorithm [20] is a kind of sparse approach to beamforming by iteratively updating the spatial spectrum estimation and beamforming weighting vectors based on a weighted least-squares approach. Considering that the IAA depends on the unknown spatial spectrum distribution, it must be implemented in an iterative way. The initialization is done by a delay-and-sum (DAS) beamformer, which is a spatial matched filter with a data-independent weight vector wDAS = a(θ) M , as sˆl (k) = a H (θl )x(k)/M, l = 1,. . .,L, k = 1,. . .,K, from which the power estimates sl (k)|2,l = 1,. . .,L. Here, L is the number of potential are given by pˆ l = K1 K k=1 |ˆ source locations in the observed field (or the number of scanning points), which is usually much larger than the true number of sources. Then, the IAA algorithm repeats the following iterative process ¯ = A(θ)diag(p)A ˆ H (θ) R for l = 1,. . .,L ¯ −1 a(θl ) R wl = ¯ −1 a(θl ) a H (θl )R ¯ pˆ l = wH l Rwl end for

(8.18)

to converge, where A(θ) = [a(θ1 ),a(θ2 ),. . .,a(θL )] ∈ CM×L is the array steering L matrix, pˆ = [pˆ 1, pˆ 2,. . ., pˆ L ]T ∈ RL + is the estimated spatial spectrum. Here, R+ denotes the set of L-dimensional vectors of nonnegative real numbers. 2 In optimization theory, NP-hard problems represent a class of extremely difficult problems that cannot be

solved in polynomial time.

234

Gu, Goodman, and Zhang

As such, the IAA algorithm can achieve the signal waveform (and hence signal power) estimation by way of sparse signal representation. It performs well when there is no model mismatch. However, when there is a slight model mismatch on the signal steering vector, e.g., signal-look direction mismatch, performance degradation would occur, and the degradation becomes severe with the increase of the input SNR.

8.3

Covariance Matrix Reconstruction-Based Adaptive Beamforming In order to avoid, or at least mitigate, the signal self-nulling phenomenon prevalent in adaptive beamformers, in this section, we will elaborate a covariance matrix sparse reconstruction method to provide an estimate of the signal-free interference-plus-noise covariance matrix for adaptive beamformer design. In such a case, the performance of the resulting adaptive beamformer will always approach the optimal value in terms of the output SINR. Moreover, the proposed adaptive beamformer has a faster convergence rate than classical adaptive beamformers.

8.3.1

Interference-Plus-Noise Covariance Matrix Reconstruction Similar to the signal covariance matrix in (8.9), i.e., R s = σs2 a(θs )a H (θs ), the interference-plus-noise covariance matrix has the form of R i+n =

Q

σi2q a(θiq )a H (θiq ) + σn2 I,

(8.19)

q=1

where Q is the number of interferers, a(θiq ) is the steering vector of the q-th interference impinging from the DOA θiq , and σi2q is the corresponding interference power. Hence, in order to have an accurate estimate of the interference-plus-noise covariance matrix R i+n , we need to know the steering vectors of all interferers via DOAs and their individual power, together with the noise power. When these pieces of information are unavailable, the interference-plus-noise covariance matrix can be reconstructed as [40] ˆ i+n = pCapon (θ)a(θ)a H (θ)d θ, (8.20) R ¯

where a(θ) is the steering vector associated with a hypothetical direction θ, pCapon (θ) =

1 ˆ −1 a(θ) a H (θ)R

(8.21)

¯ is the complement sector of . Here, is the Capon spatial spectrum estimator, and is a known or estimated angular sector in which the desired signal is located. Hence, the ˆ i+n collects all interference and noise in the out-of-sector covariance matrix estimator R ¯ , which effectively excludes the desired signal component.

Adaptive Beamforming via Sparsity-Based Reconstruction of Covariance Matrix

235

Correspondingly, the adaptive beamformer based on interference-plus-noise covariance matrix reconstruction wRecon =

ˆ −1 ¯s R i+n a

(8.22)

−1

ˆ ¯s a¯ H s R i+n a

can dramatically improve the performance regardless of the desired signal power (see [40] and accompanying simulations). Nevertheless, the estimation accuracy of ˆ i+n in (8.20) is poor because the Capon estimator (8.21) underestimates the true R power, especially when the number of snapshots is limited. On the other hand, the computational complexity is high because the covariance matrix reconstruction process introduces the unnecessary integral operation, where the number of interferers is actually countable.

8.3.2

Sparsity-Based Interference-Plus-Noise Covariance Matrix Reconstruction Because of the DOF requirement, the number of array sensors is typically larger than the true number of sources. Hence, besides the low-rank characteristic of the array covariance matrix, the target sources in the observed field have the sparse nature. In such a case, this sparsity can be leveraged to reconstruct the interference-plus-noise covariance matrix R i+n , which will provide better estimation accuracy and simplify the ¯ integral operation of (8.20) over the entire complement sector . According to (8.19), the interference-plus-noise covariance matrix is a function of the directions and power of interferers, as well as the noise power. The estimation accuracy of these parameters will affect the performance of the adaptive beamformer via the reconstructed interference-plus-noise covariance matrix. To estimate the parameters of both the desired signal and interferers, we formulate a sparsity-constrained covariance matrix fitting problem according to (8.9) as 4 4 4ˆ 4 p0 = Q + 1, min 4R s.t. − AP AH − σn2 I 4 p,σn2

F

p ≥ 0, σn2 > 0,

(8.23)

where p ∈ RL + is the spatial spectrum distribution on the sample grids of the observed ¯ } is the correspatial domain (e.g., {θ1,θ2,. . .,θ ∈ RL×L + L ∈ ∪ ), P = diag(p) M×L is the array manifold sponding diagonal matrix, A = a(θ1 ),a(θ2 ),. . .,a(θL ) ∈ C matrix, and · F and · 0 , respectively, denote the Frobenius norm of a matrix and the 0 “norm” of a vector. Note that, although it does not satisfy the positive homogeneity, the 0 “norm”, which counts the number of nonzero elements in a vector, is an ideal measure of sparsity. According to the sparse observation, the number of potential sources is much larger than the true number of sources, i.e., L " Q + 1. The idea behind (8.23) is intuitive in the sense that it tries to find the sparsest spatial spectrum distribution p and the noise power σn2 such that the difference between the resulting ˆ is minimized. covariance matrix AP AH + σn2 I and the sample covariance matrix R

236

Gu, Goodman, and Zhang

However, the true number of sources is a priori unknown. Even if known, it is understood that (8.23) is a difficult combinatorial optimization problem due to the nonconvex 0 “norm” constraint, and is intractable even for moderately sized problems. In the past decades, many approximation methods have been proposed to solve this nonconvex optimization problem, such as greedy approximations [51,52] and lp (p ≤ 1) convex relaxations [53,54]. When the solution p is sufficiently sparse, the 0 “norm” can be approximately replaced by the 1 -norm. By introducing the 1 -norm convex relaxation, the nonconvex optimization problem (8.23) can be formulated as a convex one: 4 4 4ˆ 4 min 4R − AP AH − σn2 I 4

F

p,σn2

s.t.

p1 ≤ σs2 +

K

σi2k + σn2 + δ,

k=1

p ≥ 0, σn2 > 0,

(8.24) K

where the 1 -norm of p equals the power sum of all sources (i.e., σs2 + k=1 σi2k + σn2 ), and a small number δ > 0 is added to the power constraint in order to allow a space for the optimization algorithm to search for p. However, in practical applications, the true number of sources is not easy to know, not to mention their power. Alternatively, the convex optimization problem in (8.24) can be reformulated as a basis pursuit denoising (BPDN) problem [55] as 4 4 4ˆ 4 p ≥ 0, min 4R − AP AH − σn2 I 4 + γp1 s.t. p,σn2

F

σn2 > 0,

(8.25)

where γ is a regularization parameter controlling the trade-off between the sparsity of the spatial spectrum and the residual norm of covariance matrix fitting. The optimization problem is convex and can be solved using standard and highly efficient interior point methods. Besides the BPDN, the least absolute shrinkage and selection operator (LASSO) [56] is another popular formulation based on the 1 -norm relaxation. Note that, there is no matrix inversion or eigen-decomposition required in the proposed sparsity-constrained covariance matrix fitting problem. Hence, it is suitable for arbitrary number of snapshots from one to infinity. However, the obtained solution is not absolutely sparse because of the 1 -norm relaxation. In addition, the regularization parameter γ is difficult to determine in different scenarios. Either overestimation or underestimation will sacrifice the balance between data fidelity and sparsity, which subsequently leads to performance degradation of the resulting adaptive beamformer. When the number of snapshots is larger than the number of array sensors, we can decompose the sparsity-constrained covariance matrix fitting problem into two associated subproblems: (1) a source localization problem to find the DOA support of sources; and (2) a power estimation problem operating on the DOAs that were estimated in the first subproblem. The combination of these two subproblems represents an approximation to the solution of the sparsity-constrained covariance matrix fitting problem. Compared to adaptive beamforming, DOA estimation is a more mature array processing technique, and there are many sophisticated methods available (see, for example,

Adaptive Beamforming via Sparsity-Based Reconstruction of Covariance Matrix

237

[3,53] and the references therein). In general, the DOAs are estimated either from a spectral search algorithm (see, for example, [50,57]) or from a search-free polynomial rooting algorithm (e.g., [58] and the references therein). For convenience, here we simply use the classical Capon spatial spectrum p Capon (θ) in (8.21) to estimate the DOAs of sources. The estimated DOAs provide the support of the sparse vector defined in the proposed sparsity-constrained covariance matrix fitting problem. Let p denote the set of directions corresponding to the peaks of p Capon on the entire ¯ for which the cardinality is usually greater than observed spatial domain (i.e., ∪ ), the true number of sources because of the spurious peaks (i.e., | p | = p0 > Q + 1). Here, | · | denotes the cardinality of a set. In order to minimize the 0 “norm” p0 to find the sparsest solution of (8.23), a common method is to remove the spurious peaks by setting a threshold, such as the noise power, which can be approximately$ estimated as the % ˆ ) [59], where minimum eigenvalue of the sample covariance matrix (i.e., σˆ n2 = λmin R λ min ( · ) denotes the minimum eigenvalue of a matrix. In theory, there are M − Q − 1 eigenvalues that equal the actual noise power σn2 . However, in practical applications with a limited number of snapshots, the minimum eigenvalue of the sample covariance matrix is always smaller than the noise power. Hence, if the value of a peak in the Capon spatial spectrum pCapon is lower than the threshold, it will be regarded as a spurious peak and its corresponding direction will be removed from the set p . After removing all the ˜ p = { θ˜ p,1,. . ., θ˜ ˜ } with cardinality spurious peaks, the residual set is denoted as p, Q ˜ ≤ | p |. In such a case, p0 = Q ˜ ≥ Q + 1. ˜ p| = Q | After finding the DOA support θ˜ p = [ θ˜ p,1,. . ., θ˜ ˜ ]T , the sparsity-constrained p, Q

covariance matrix fitting problem (8.23) degenerates into an inequality-constrained least-squares problem: 4 4 ˆ − A( θ˜ p )P ( θ˜ p )AH ( θ˜ p ) − σn2 I 4 min 4R F

p(θ˜ p ),σn2

s.t.

p( θ˜ p ) > 0, σn2 > 0,

(8.26)

where P ( θ˜ p ) = diag(p( θ˜ p )) is a diagonal matrix with the power distribution p( θ˜ p ) ∈ ˜ Q˜ R on the DOA support θ˜ p , and A( θ˜ p ) = a(θ˜ p,1 ),. . .,a( θ˜ ˜ ) ∈ CM×Q is the cor++

Q˜

p, Q

˜ responding array manifold matrix. Here, R++ denotes the set of Q-dimensional vectors of positive real numbers. The strict inequality constraint enforced here indicates that the signal power on the found DOA support θ˜ p are always positive. The optimization problem (8.26) is convex and can be solved using highly efficient interior point methods. It is noted that covariance matrix reconstruction-based adaptive beamformers are not very sensitive to the estimation error in the noise power. Therefore, for the sake of simplicity, the optimization variable of noise power σn2 is taken to be the miniˆ which leads to a simplified inequality-constrained least-squares mum eigenvalue of R, problem: 4 4 ˆ − A( θ˜ p )P ( θ˜ p )AH ( θ˜ p )4 ˆ − λmin (R)I min 4R F

p(θ˜ p )

s.t. p( θ˜ p ) > 0.

(8.27)

Gu, Goodman, and Zhang

Using the vectorization property, this can be further simplified as 4 4 ˆ − λmin (R)I ˆ ) − (A( θ˜ p ) A(θ˜ p ))p( θ˜ p )4 s.t. p( θ˜ p ) > 0, (8.28) min 4vec(R 2 p(θ˜ p )

where vec ( · ) denotes the vectorization operator, and denotes the Khatri–Rao product. Without the inequality constraint, the closed-form solution to (8.27) is given by −1 p( θ˜ p ) = GH G GH r,

(8.29)

where G = A( θ˜ p ) A( θ˜ p ) = a(θ˜ p,1 ) ⊗ a( θ˜ p,1 ),. . .,a( θ˜ p, Q˜ ) ⊗ a( θ˜ p, Q˜ ) 2 ˜ = vec(a( θ˜ p,1 )a H ( θ˜ p,1 )),. . .,vec(a( θ˜ p, Q˜ )a H ( θ˜ p, Q˜ )) ∈ CM ×Q is obtained by stackˆ ) ∈ CM 2 ˆ ing the outer products of the sources steering vectors, and r = vec(R−λ min (R)I is vectorized from the sample covariance matrix subtracted by an estimated noise covariance matrix. Here, ⊗ denotes the Kronecker product. Then, the estimated spatial ˜ spectrum of (8.23) is Q-sparse and is expressed as

p(θ) =

p( θ˜ p ) 0

˜ p, θ∈ ˜ p. θ∈ /

(8.30)

˜ entries of the estimated spatial spectrum p(θ) are nonzero and all Namely, only Q ˜ other L − Q entries are zero. An example of spatial spectrum comparison between the proposed sparse spectrum (8.30) and the Capon spectrum (8.21) is illustrated in Figure 8.2, where three sources impinge from DOAs of −50◦ , −20◦ , and 5◦ with the SNR of 30 dB, 30 dB, and 20 dB, respectively. It is clear that the proposed method achieves a more accurate estimate of the signal power. 30

Capon spectrum Sparse spectrum

25 20 15 Power (dB)

238

10 5 0 –5

–10 –15 –90

–60

–30

0 θ( )

Figure 8.2 Spatial spectrum comparison.

30

60

90

Adaptive Beamforming via Sparsity-Based Reconstruction of Covariance Matrix

239

However, when some sources are very weak, there may be negative entries in p(θ˜ p ) (8.29), which is obtained by discarding the inequality constraint in (8.27). Without loss of generality, assume that the q-th ˜ entry of p( θ˜ p ) is negative, i.e., p( θ˜ p, q˜ ) < 0. In such a case, the inequality constraint in (8.26) will not be satisfied, and the closedform solution in (8.29) should be modified. A simple method is to force p( θ˜ p, q˜ ) to be a small positive value δ > 0 (for example, δ = 10−5 is used in our simulations), and the ˜ ˜ ˜ ,. . ., θ˜ ˜ ]T ∈ RQ−1 , power estimation of other sources, θ¯ p = [θ˜ p,1,. . ., θ˜ p, q−1 ˜ , θp, q+1 p, Q will be modified as ¯ H G] ¯ −1 G ¯ H r¯ , ¯ θ¯ p ) = [G p(

(8.31)

H ˜ ˜ ˜ )a H ¯ = vec(a(θ˜ p,1 )a H ( θ˜ p,1 )),. . .,vec(a( θ˜ p, q−1 where G ˜ )a ( θp, q−1 ˜ )),vec(a( θp, q+1 2 ˜ H ˜ M ×(Q−1) , and r¯ = vec(R ˜ ˆ − λmin (R)I ˆ ( θ˜ p, q+1 ˜ )),. . .,vec(a( θ ˜ )a ( θ ˜ )) ∈ C p, Q

p, Q

2 −δa( θ˜ p, q˜ )a H ( θ˜ p, q˜ )) ∈ CM . In other words, we recalculate the source powers after fixing the power of weak sources, thus resulting in a modified spatial spectrum as ⎧ ⎪ ¯ θ¯ p ) θ ∈ θ¯ p, ⎨ p( p(θ) = (8.32) δ θ = θ˜ p, q˜ , ⎪ ⎩ 0 ˜ θ∈ / θp .

˜ Using the Q-sparse spatial spectrum p(θ), the interference-plus-noise covariance matrix can be sparsely reconstructed as ˆ i+n = p(θiq )a(θiq )a H (θiq ) + σˆ n2 I, (8.33) R ¯ ˜p θiq ∈ ∩

where a(θiq )a H (θiq ) is the outer product of the q-th interference steering vector a(θiq ). ˜ elements in the set ¯ ∩ ˜ p , the integral operation in (8.20) Because there are at most Q is effectively simplified to be a summation operation (8.33) by using the sparse characteristics of sources in the observed spatial domain. Note that there is no desired signal component in the reconstructed interference-plus-noise covariance matrix. Considering the possible look direction mismatch, the DOA of the desired signal can be located by searching for the peak of pCapon in , i.e., θ˜ s = arg maxθ∈ pCapon (θ), ˜p and the corresponding steering steering vector is denoted as a˜ s = a( θ˜ s ). When ∩ is empty, which is common at low SNRs, we simply use the presumed signal steering vector for adaptive beamformer design, i.e., a˜ s = a¯ s , even if there is a look direction mismatch. ˆ i+n and Substituting the reconstructed interference-plus-noise covariance matrix R the estimated signal steering vector a˜ s into the MVDR beamformer (8.8) together, we can propose the adaptive beamformer as w=

ˆ −1 ˜s R i+n a ˆ −1 ˜ s a˜ H s R i+n a

.

(8.34)

The proposed adaptive beamforming algorithm based on sparse reconstruction of the interference-plus-noise covariance matrix is summarized in Table 8.1.

240

Gu, Goodman, and Zhang

Table 8.1 Adaptive beamforming algorithm based on sparse reconstruction of interference-plus-noise covariance matrix. Step 1: Step 2: Step 3: Step 4:

Estimate the DOAs of the sources by, e.g., searching for the peaks of the Capon spatial spectrum p Capon (θ) (8.21); ˜ Solve the least-squares problem (8.27) to obtain the Q-sparse spatial spectrum p(θ) (8.30) or (8.32); ˆ i+n (8.33) and Reconstruct the interference-plus-noise covariance matrix R estimate the signal steering vector a˜ s ; Compute the proposed adaptive beamformer w (8.34).

The computational complexity of the proposed adaptive beamforming algorithm is O(LM 2 ) with L " M, which is mainly dominated by the spectral search. If a searchfree DOA estimation technique [58] is adopted, the computational complexity can be ˜ 2 M 2 )), where O(M 3 ) is the complexity of DOA estifurther decreased to O(max(M 3, Q 2 2 ˜ M ) is the complexity of power estimation. Therefore, the proposed mation and O(Q adaptive beamforming algorithm has complexity slightly higher than the DOA estimation algorithm. Meanwhile, the computational complexity of the & covariance 'matrix ¯ | | 2 . Note reconstruction-based adaptive beamforming algorithm [40] is O | ∪ | LM ¯ however that, if the spatial estimate of the sources in the entire region is desired, the SMI beamformer has the complexity of O(LM 2 ) as well.

8.4

Simulation Results In our simulations, a ULA with M = 10 omnidirectional sensors spaced half wavelength apart is considered. It is assumed that there is one desired signal from the presumed direction θ¯ s = 5◦ and two uncorrelated interferers from −50◦ and −20◦ , respectively. The interference-to-noise ratio (INR) at each sensor is equal to 30 dB. The additive noise is modeled as a complex circularly symmetric Gaussian zero-mean spatially and temporally white process. When comparing the performance of the adaptive beamforming algorithms with respect to the input SNR, the number of snapshots is fixed to be K = 30. In the performance comparison of mean output SINR versus the number of snapshots, the SNR in each sensor is set to be fixed at 20 dB. For each data point (SNR or number of snapshots), 500 Monte Carlo trials are performed. The proposed interference-plus-noise covariance matrix sparse reconstructionbased beamformer (8.34) is compared to the SMI beamformer [48], the DL-SMI beamformer [9], the eigenspace decomposition-based beamformer [16], the worst-case performance optimization-based beamformer [32], the IAA beamformer [20], and the interference-plus-noise covariance matrix reconstruction-based beamformer [40]. All the tested beamformers are adaptive beamformers, i.e., their weight vectors depend on the received array data. The diagonal loading factor ξ in the DL-SMI beamformer (8.14) is assumed to be ten times the noise power, where the noise power is regarded as a priori known. The eigenspace-based beamformer (8.15) is assumed to know the exact number

Adaptive Beamforming via Sparsity-Based Reconstruction of Covariance Matrix

241

of interference sources. In the worst-case beamformer (8.16), the upper bound of the mismatched vector is ad hoc chosen to be ε = 0.3M, as suggested in [32]. Without loss of generality, the angular sector covering the direction of the desired signal in the reconstruction-based beamformers is set to be = [θ¯ s − 5◦, θ¯ s + 5◦ ] (namely, ¯ = [−90◦, θ¯ s − 5◦ ) ∪ ( θ¯ s + 5◦,90◦ ] [0◦,10◦ ]), and the corresponding out-of-sector is ◦ ◦ ◦ ◦ ¯ with 0.1◦ (namely, [−90 ,0 ) ∪ (10 ,90 ]). The sampling grid is uniform in ∪ increment between adjacent grid points. As a benchmark, the optimal SINR (8.5) is also shown in all figures, which is calculated from the exact interference-plusnoise covariance matrix and the actual desired signal steering vector. Considering that the output performance of the interference-plus-noise covariance matrix (sparse) reconstruction-based beamformers is very close to the optimal SINR regardless of the input signal power [40,43], we also compare the output performance in terms of deviation from the optimal SINR. For fair comparison, the actual steering vector of the desired signal is normalized so that a22 = M(= 10) [32,33]. The CVX software [60] is used to solve the related convex optimization problems.

8.4.1

Example 1: Exactly Known Signal Steering Vector In our first example, we consider an ideal scenario where the steering vectors of both the desired signal and the interferers are exactly known. Namely, there is no steering vector mismatch. Note that, even in this ideal case, the presence of the desired signal in the training samples may still substantially degrade the output performance of adaptive beamformers as compared with the signal-free training case [3,32,40]. However, it can be seen from Figure 8.3(a) that the output performance of adaptive beamformers based on interference-plus-noise covariance matrix reconstruction is almost always equal to the optimal SINR for all SNR values between −30 dB and 50 dB (i.e., SIR ranges from −60 dB to 20 dB), which illustrates the high dynamic range. In particularly, the output SINR of the proposed interference-plus-noise covariance matrix sparse reconstructionbased adaptive beamformer is approximated as SINR ≈ a22 SNR = M × SNR,

(8.35)

which achieves the design goal of the adaptive beamformer and outperforms the other tested beamformers. From Figure 8.3(b), the average SINR performance loss of the proposed adaptive beamformer is about 0.002 dB. In contrast, there is an average performance loss of 0.158 dB for the interference-plus-noise covariance matrix reconstruction-based adaptive beamformer [40], which is because the Capon spatial spectrum estimator underestimates the interferences power and reduces the estimation accuracy of the interference-plus-noise covariance matrix. It should be noted that the signal power is 100 times higher than the interference power in the case of SNR = 50 dB, which can be used to illustrate the situation when the SIR approximately approaches to infinity. Figure 8.3(c) shows the convergence rates of the tested adaptive beamformers versus the number of snapshots K. It is clear that the adaptive beamformer based on interference-plus-noise covariance matrix (sparse) reconstruction converges much faster than the other tested adaptive beamformers.

Gu, Goodman, and Zhang

60 50

Output SINR (dB)

40 30 20

Optimal SINR SMI DLSMI Eigenspace Worst-Case IAA Reconstruction Sparsity

10 0

–10 –20 –30 –30

–20

–10

0 10 20 Input SNR (dB)

30

40

50

(a) 2 SMI DLSMI Eigenspace Worst-Case IAA Reconstruction Sparsity

Deviations from optimal SINR (dB)

1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 –30

–20

–10

0 10 20 Input SNR (dB)

30

40

50

(b) 30 25 20

Output SINR (dB)

242

15 10

Optimal SINR SMI DLSMI Eigenspace Worst-Case IAA Reconstruction Sparsity

5 0 –5

–10 10

20

30

40

50 60 Snapshots

70

80

90

100

(c) Figure 8.3 First example: exactly known steering vectors. (a) output SINR versus input SNR; (b) deviations from optimal SINR versus input SNR; (c) output SINR versus number of snapshots.

Adaptive Beamforming via Sparsity-Based Reconstruction of Covariance Matrix

243

20 10

Beampattern (dB)

0 –10 –20 –30 Optimal SMI DL-SMI EIG WorstCase

–40 –50 –60 –90

–60

–30

0 θ( )

30

60

90

20 10

Beampattern (dB)

0 –10 –20 –30 –40

Optimal IAA Reconstruction Sparsity

–50 –60 –90

–60

–30

0 θ( )

30

60

90

Figure 8.4 First example: beampattern comparison.

In Figure 8.4, we compare the beampattern of the proposed beamformer with those of the other tested beamformers for K = 30 and SNR = 20 dB, where the vertical solid line denotes the direction of the desired signal and the vertical dashed lines denote the directions of interference. It is evident that the beampattern of the proposed adaptive beamformer almost exactly matches that of the optimal one.

8.4.2

Example 2: Fixed Signal DOA Mismatch In the second example, a scenario with fixed signal DOA mismatch is considered. We assume that the actual DOA of the desired signal is 8◦ , while the assumed one

244

Gu, Goodman, and Zhang

is 5◦ . Correspondingly, there is a fixed DOA mismatch of 3◦ for the desired signal. By comparing Figure 8.5(a) to Figure 8.3(a), we can see that, when the input SNR is 20 dB, there is about 18 dB of performance loss for both the SMI beamformer and the DL-SMI beamformer. There is no obvious performance change for the worst-case beamformer, while the eigenspace-based beamformer suffers clear performance loss when the signal power is higher than the interference power. Compared with the performance loss (about 2 dB) of the interference-plus-noise covariance matrix reconstructionbased beamformer, there is almost no performance loss for the proposed adaptive beamformer, based on covariance matrix sparse reconstruction when the input SNR is above 0 dB. When the SNR is lower than 0 dB, the performance loss is mainly because of the DOA estimation accuracy of the Capon spatial spectrum. The output performance of the proposed beamformer can be further improved by introducing more sophisticated DOA estimation methods, especially for low SNR cases. Similarly, as shown in Figure 8.5(c), the output performance of the proposed adaptive beamformer is close to the optimal SINR when the number of snapshots is larger than the number of array sensors.

8.4.3

Example 3: Random Sources of DOA Mismatch In the third example, a more practical scenario with random DOA mismatches is considered. More specifically, random DOA mismatches of both the desired signal and the interferers are assumed to be uniformly distributed in [−4◦,4◦ ]. That is to say, the actual DOA of the desired signal is uniformly distributed as U [ θ¯ s − 4◦, θ¯ s + 4◦ ] (i.e., U [1◦,9◦ ]), and the DOAs of the interferers are uniformly distributed as U [−54◦, − 46◦ ] and U [−24◦, − 16◦ ], respectively. Note that, random DOAs of the signal and interferers change from trial to trial but remain fixed from snapshot to snapshot. It can be seen from Figure 8.6(a) that the output performance of the proposed beamformer is much closer to the optimal SINR than other tested beamformers. When the input SNR is less than −10 dB, there is an approximately 0.6 dB performance loss because there may be no peak in the angular sector for the Capon spectrum or the peak’s value is less than the threshold; therefore, there presents a random DOA mismatch for the desired signal by using the presumed DOA θ¯ s , the center of the desired signal sector . In addition, due to the limited sampling grid, the performance of the proposed beamformer does not exactly converge to the optimal one when the input SNR is higher than 0 dB. In detail, the maximum estimation error of source DOAs is 0.05◦ , which is half of the grid increment of 0.1◦ . Such DOA estimation error will degrade the output performance of the proposed beamformer because both the reconstructed ˆ i+n and the modified signal steering vecinterference-plus-noise covariance matrix R tor a˜ s depend on the DOA estimation. The possible solutions to mitigate the effect of grid limitation include the grid refinement method [53] and the off-grid direction estimation method [61–64]. In addition to achieving a faster convergence rate than the interference-plus-noise covariance matrix reconstruction-based beamformer [40], the proposed interference-plus-noise covariance matrix sparse reconstruction-based beamformer offers a stable output performance with the increase of SNR while others do not, as shown in Figure 8.6(c).

Adaptive Beamforming via Sparsity-Based Reconstruction of Covariance Matrix

60 50

Output SINR (dB)

40 30 20

Optimal SINR SMI DLSMI Eigenspace Worst-Case IAA Reconstruction Sparsity

10 0

–10 –20 –30 –30

–20

–10

0 10 20 Input SNR (dB)

30

40

50

(a) 2 DLSMI Eigenspace Worst-Case IAA Reconstruction Sparsity

Deviations from optimal SINR (dB)

1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 –30

–20

–10

0 10 20 Input SNR (dB)

30

40

50

(b) 30 25

Output SINR (dB)

20 15 10 5

Optimal SINR SMI DLSMI Eigenspace Worst-Case IAA Reconstruction Sparsity

0 –5

–10 –15 –20 10

20

30

40 50 60 70 Number of snapshots

80

90

100

(c) Figure 8.5 Second example: fixed signal DOA mismatch. (a) output SINR versus input SNR;

(b) deviations from optimal SINR versus input SNR; (c) output SINR versus number of snapshots.

245

Gu, Goodman, and Zhang

60 50

Output SINR (dB)

40 30 20

Optimal SINR SMI DLSMI Eigenspace Worst-Case IAA Reconstruction Sparsity

10 0

–10 –20 –30 –30

–20

–10

0 10 20 Input SNR (dB)

30

40

50

(a) 2 DLSMI Eigenspace Worst-Case IAA Reconstruction Sparsity

Deviations from optimal SINR (dB)

1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 –30

–20

–10

0 10 20 Input SNR (dB)

30

40

50

(b) 30 25 20

Output SINR (dB)

246

15 10

Optimal SINR SMI DLSMI Eigenspace Worst-Case IAA Reconstruction Sparsity

5 0 –5

–10 10

20

30

40 50 60 70 Number of snapshots

80

90

100

(c) Figure 8.6 Third example: random sources look direction mismatch. (a) output SINR versus input SNR; (b) deviations from optimal SINR versus input SNR; (c) output SINR versus number of snapshots.

Adaptive Beamforming via Sparsity-Based Reconstruction of Covariance Matrix

8.4.4

247

Example 4: Coherent Local Scattering In the fourth example, we consider a scenario where the spatial signature of the desired signal is distorted by local scattering effects. Specifically, the desired signal is assumed to be a plane wave with the presumed steering vector a¯ s , whereas the actual steering vector a s is formed as the superposition of five signal paths, including four coherent scattered paths, as a s = a¯ s +

4

ej ψt a(θt ),

(8.36)

t=1

where a(θt ),t = 1,2,3,4, correspond to coherently scattered paths. The steering vector of the t-th path, a(θt ), can be modeled as a plane wave from the direction of θt . The DOAs of scattered paths follow independent normal distribution θt ∼ N ( θ¯ s ,4◦ ), t = 1,2,3,4, and the phases of scattered paths follow independent uniform distribution ψt ∼ U [0,2π) ,t = 1,2,3,4. Note that the tested adaptive beamformers are implemented in a block adaptive manner, which means that both θt and ψt ,t = 1,2,3,4 change from run to run but do not change from snapshot to snapshot. From Figure 8.7, the output performance loss of the proposed beamformer is less than 0.7 dB, which is much smaller than what is suffered by the other tested beamformers.

8.4.5

Example 5: Wavefront Distortion In the fifth example, we consider the situation where the desired signal spatial signature is distorted by wave propagation effects in an inhomogeneous medium. We assume independent-increment phase distortions of the desired signal wavefront [25,65]. In each Monte Carlo run, each of these phase distortions is independently drawn from a Gaussian random generator N (0,0.04), which remains fixed from snapshot to snapshot. From Figure 8.8, it is clear that the proposed beamformer provides more stable and nearoptimal output performance than the other tested beamformers regardless of the input signal power or the number of snapshots.

8.4.6

Example 6: Incoherent Local Scattering In the sixth example, we assume incoherent local scattering of the desired signal, which is common in array applications due to the multipath scattering effects caused by the presence of local scatters. In such a case, the desired signal is assumed to have a timevarying spatial signature as [32,40] a s (k) = s0 (k)a¯ s +

4

st (k)a(θt ),

(8.37)

t=1

where st (k) ∼ N (0,1),t = 0,1,2,3,4, are independent and identically distributed (i.i.d.) zero-mean complex Gaussian random variables that change from snapshot to snapshot, θt ∼ N ( θ¯ s ,4◦ ),t = 1,2,3,4, are the random DOAs changing from run to run while remaining fixed from snapshot to snapshot. This corresponds to the case of incoherent

Gu, Goodman, and Zhang

60 50

Output SINR (dB)

40 30 20

Optimal SINR SMI DLSMI Eigenspace Worst-Case IAA Reconstruction Sparsity

10 0 –10 –20 –30 –30

–20

–10

0 10 20 Input SNR (dB)

30

40

50

(a) 2 DLSMI Eigenspace Worst-Case IAA Reconstruction Sparsity

Deviations from optimal SINR (dB)

1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 –30

–20

–10

0 10 20 Input SNR (dB)

30

40

50

(b) 30 25 20

Output SINR (dB)

248

15 10 Optimal SINR SMI DLSMI Eigenspace Worst-Case IAA Reconstruction Sparsity

5 0 –5

–10 –15 10

20

30

40 50 60 70 Number of snapshots

80

90

100

(c) Figure 8.7 Fourth example: coherent local scattering. (a) output SINR versus input SNR; (b) deviations from optimal SINR versus input SNR; (c) output SINR versus number of snapshots.

Adaptive Beamforming via Sparsity-Based Reconstruction of Covariance Matrix

249

60 50

Output SINR (dB)

40 30 20

Optimal SINR SMI DLSMI Eigenspace Worst-Case IAA Reconstruction Sparsity

10 0

–10 –20 –30 –30

–20

–10

0 10 20 Input SNR (dB)

30

40

50

(a) 2 SMI DLSMI Eigenspace Worst-Case IAA Reconstruction Sparsity

Deviations from optimal SINR (dB)

1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 –30

–20

–10

0 10 20 Input SNR (dB)

30

40

50

(b) 30 25

Output SINR (dB)

20 15 10 5

Optimal SINR SMI DLSMI Eigenspace Worst-Case IAA Reconstruction Sparsity

0 –5

–10 –15 –20 10

20

30

40 50 60 70 Number of snapshots

80

90

100

(c) Figure 8.8 Fifth example: wavefront distortion. (a) output SINR versus input SNR; (b) deviations

from optimal SINR versus input SNR; (c) output SINR versus number of snapshots.

250

Gu, Goodman, and Zhang

local scattering [30], where the rank of the signal covariance matrix R s is higher than one. In the general-rank case, the output SINR should be rewritten as [10] SINR =

wH R s w , wH R i+n w

(8.38)

which is maximized by [10] w = P{R −1 i+n R s },

(8.39)

where P { · } stands for the principal eigenvector of a matrix. It can be seen from Figure 8.9(a) that the proposed beamformer outperforms all other tested beamformers especially at high SNR. The performance loss of the proposed beamformer is less than 0.1 dB. In contrast, there is about 7.5 dB performance loss for the interference-plus-noise covariance matrix reconstruction-based beamformer. The ¯ due to the incomain reason is that the signal of interest leaks into the out-of-sector herent local scattering, and then the reconstructed interference-plus-noise covariance ˆ i+n (8.20) is contaminated by the leaked desired signal component. matrix R

8.4.7

Discussion From the extensive simulation results illustrated for different scenarios in this chapter, it is clear that the proposed adaptive beamformer, based on interference-plus-noise covariance matrix sparse reconstruction, consistently enjoys the best performance as compared to other tested beamformers. More specifically, having benefitted from the interference covariance matrix reconstruction, which excludes the desired signal component, the output performance of the proposed beamformer is always close to or equal to the optimal SINR regardless of the input SNR. In contrast, there is a slight output performance degradation for the interference-plus-noise covariance matrix reconstructionbased beamformer because the Capon spatial spectrum estimator underestimates the power of interferers and thus decreases the estimation accuracy of the interference-plusnoise covariance matrix. On the other hand, other tested beamformers use the signalcontaminated covariance matrix, and thus degrade the output performance particularly when the input SNR is high. Another significant advantage of the proposed adaptive beamformer is that it achieves faster convergence, and only requires the number of snapshots to be slightly higher than the number of array sensors in order to approach the optimal SINR. In contrast, if an average performance loss of less than 3 dB is required for the SMI beamformer, the number of signal-free snapshots must be at least twice the number of array sensors [48]. However, in the underlying problem where the snapshots are contaminated by the desired signal, the convergence becomes much slower, i.e., a much higher number of snapshots are required [17]. Note that, although the ULA is adopted in our simulations, the idea of covariance matrix sparse reconstruction proposed in this chapter for the adaptive beamformer design can be generalized to arbitrary arrays, e.g., the coprime array [66,67]. Having benefitted from the larger array aperture due to the sparse deployment, the coprime array adaptive beamforming algorithm proposed in [67] achieves high robustness

Adaptive Beamforming via Sparsity-Based Reconstruction of Covariance Matrix

60 50 40 Output SINR (dB)

30 20

Optimal SINR SMI DLSMI Eigenspace Worst-Case IAA Reconstruction Sparsity

10 0 –10 –20 –30 –30

–20

–10

0 10 20 Input SNR (dB)

30

40

50

(a) 2 SMI DLSMI Eigenspace Worst-Case IAA Reconstruction Sparsity

Deviations from optimal SINR (dB)

1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 –30

–20

–10

0 10 20 Input SNR (dB)

30

40

50

(b) 30 25

Output SINR (dB)

20 15 10

Optimal SINR SMI DLSMI Eigenspace Worst-Case IAA Reconstruction Sparsity

5 0 –5 –10 10

20

30

40 50 60 70 Number of snapshots

80

90

100

(c) Figure 8.9 Sixth example: incoherent local scattering. (a) output SINR versus input SNR;

(b) deviations from optimal SINR versus input SNR; (c) output SINR versus number of snapshots.

251

252

Gu, Goodman, and Zhang

against model mismatches with a significant reduction in the number of antennas and the associated radio frequency chains. It is also worth noting that the extension of the proposed narrowband adaptive beamforming technique to the broadband adaptive beamforming is straightforward [3]. For example, applying fast Fourier transform (FFT) to the broadband signal yields narrowband components, which can then be independently processed using the proposed (narrowband) covariance matrix sparse reconstruction-based adaptive beamforming technique. Subsequently, the time-domain broadband beamformer output is obtained by applying an inverse FFT to the output of the individual narrowband beamformers.

8.5

Conclusion In this chapter, we proposed a simple, effective robust adaptive beamforming algorithm based on interference-plus-noise covariance matrix sparse reconstruction. Specifically, by exploiting the sparsity of sources distributed in the observed spatial domain, accurate interference-plus-noise covariance matrix reconstruction can be achieved by estimating the sparse spatial spectrum distribution from a sparsity-constrained covariance matrix fitting problem, which provides a signal-free interference-plus-noise covariance matrix for the beamformer design. The formulated sparsity-constrained covariance matrix fitting problem can be effectively solved with a priori information of the estimated source DOAs rather than 1 -norm relaxation-type approximations. Simulation results evidently demonstrate the effectiveness of the proposed algorithm. Compared to the existing techniques, the performance of the proposed method is nearly optimal over a wide range of input SNR and various error conditions. In addition, the proposed technique also has low computational complexity.

References [1] B. D. Van Veen and K. M. Buckley, “Beamforming: A versatile approach to spatial filtering,” IEEE ASSP Mag., vol. 5, no. 2, pp. 4–24, Apr. 1988. [2] H. Krim and M. Viberg, “Two decades of array signal processing research: The parametric approach,” IEEE Signal Process. Mag., vol. 13, no. 4, pp. 67–94, July 1996. [3] H. L. Van Trees, Optimum Array Processing: Part IV of Detection, Estimation, and Modulation Theory. John Wiley & Sons, 2002. [4] J. Li and P. Stoica, Eds., Robust Adaptive Beamforming. John Wiley & Sons, 2005. [5] W. Liu and S. Weiss, Wideband Beamforming: Concepts and Techniques. Wiley, 2010. [6] J. Capon, “High-resolution frequency-wavenumber spectrum analysis,” Proc. IEEE, vol. 57, no. 8, pp. 1408–1418, Aug. 1969. [7] S. A. Vorobyov, “Principles of minimum variance robust adaptive beamforming design,” Signal Process., vol. 93, no. 12, pp. 3264–3277, Dec. 2013. [8] K. Yang, T. Ohira, Y. Zhang, and C.-Y. Chi, “Super-exponential blind adaptive beamforming,” IEEE Trans. Signal Process., vol. 52, no. 6, pp. 1549–1563, June 2004.

Adaptive Beamforming via Sparsity-Based Reconstruction of Covariance Matrix

253

[9] H. Cox, R. M. Zeskind, and M. H. Owen, “Robust adaptive beamforming,” IEEE Trans. Acoust. Speech Signal Process., vol. 35, no. 10, pp. 1365–1376, Oct. 1987. [10] A. B. Gershman, “Robust adaptive beamforming in sensor arrays,” Int. J. Electron. Commun., vol. 53, no. 6, pp. 305–314, Dec. 1999. [11] W. F. Gabriel, “Spectral analysis and adaptive array superresolution techniques,” Proc. IEEE, vol. 68, no. 6, pp. 654–666, June 1980. [12] Y. I. Abramovich, “Controlled method for adaptive optimization of filters using the criterion of maximum SNR,” Radio Eng. Electron. Phys., vol. 26, no. 3, pp. 87–95, Mar. 1981. [13] B. D. Carlson, “Covariance matrix estimation errors and diagonal loading in adaptive arrays,” IEEE Trans. Aerosp. Electron. Syst., vol. 24, no. 4, pp. 397–401, July 1988. [14] L. Du, T. Yardibi, J. Li, and P. Stoica, “Review of user parameter-free robust adaptive beamforming algorithms,” Digital Signal Process., vol. 19, no. 4, pp. 567–582, July 2009. [15] P. Stoica, J. Li, X. Zhu, and J. R. Guerci, “On using a priori knowledge in space-time adaptive processing,” IEEE Trans. Signal Process., vol. 56, no. 6, pp. 2598–2602, June 2008. [16] L. Chang and C.-C. Yeh, “Performance of DMI and eigenspace-based beamformers,” IEEE Trans. Antennas Propagat., vol. 40, no. 11, pp. 1336–1347, Nov. 1992. [17] D. D. Feldman and L. J. Griffiths, “A projection approach for robust adaptive beamforming,” IEEE Trans. Signal Process., vol. 42, no. 4, pp. 867–876, Apr. 1994. [18] J. K. Thomas, L. L. Scharf, and D. W. Tufts, “The probability of a subspace swap in the SVD,” IEEE Trans. Signal Process., vol. 43, no. 3, pp. 730–736, Mar. 1995. [19] M. Hawkes, A. Nehorai, and P. Stoica, “Performance breakdown of subspace-based methods: Prediction and cure,” in Proc. IEEE Int. Conf. Acoust., Speech, Signal Process., Salt Lake City, UT, May 2001, pp. 4005–4008. [20] T. Yardibi, J. Li, P. Stoica, M. Xue, and A. B. Baggeroer, “Source localization and sensing: A nonparametric iterative adaptive approach based on weighted least squares,” IEEE Trans. Aerosp. Electron. Syst., vol. 46, no. 1, pp. 425–443, Jan. 2010. [21] L. C. Godara, “The effect of phase-shift errors on the performance of an antenna-array beamformer,” IEEE J. Ocean. Eng., vol. 10, no. 3, pp. 278–284, July 1985. [22] J. W. Kim and C. K. Un, “An adaptive array robust to beam pointing error,” IEEE Trans. Signal Process., vol. 40, no. 6, pp. 1582–1584, June 1992. [23] N. K. Jablon, “Adaptive beamforming with the generalized sidelobe canceller in the presence of array imperfections,” IEEE Trans. Antennas Propagat., vol. 34, no. 8, pp. 996–1012, Aug. 1986. [24] A. B. Gershman, V. I. Turchin, and V. A. Zverev, “Experimental results of localization of moving underwater signal by adaptive beamforming,” IEEE Trans. Signal Process., vol. 43, no. 10, pp. 2249–2257, Oct. 1995. [25] J. Ringelstein, A. B. Gershman, and J. F. Böhme, “Direction finding in random inhomogeneous media in the presence of multiplicative noise,” IEEE Signal Process. Lett., vol. 7, no. 10, pp. 269–272, Oct. 2000. [26] Y. J. Hong, C.-C. Yeh, and D. R. Ucci, “The effect of a finite-distance signal source on a far-field steering applebaum array – two dimensional array case,” IEEE Trans. Antennas Propagat., vol. 36, no. 4, pp. 468–475, Apr. 1988. [27] K. I. Pedersen, P. E. Mogensen, and B. H. Fleury, “A stochastic model of the temporal and azimuthal dispersion seen at the base station in outdoor propagation environments,” IEEE Trans. Veh. Technol., vol. 49, no. 2, pp. 437–447, Mar. 2000. [28] J. Goldberg and H. Messer, “Inherent limitations in the localization of a coherently scattered source,” IEEE Trans. Signal Process., vol. 46, no. 12, pp. 3441–3444, Dec. 1998.

254

Gu, Goodman, and Zhang

[29] D. Astely and B. Ottersten, “The effects of local scattering on direction of arrival estimation with MUSIC,” IEEE Trans. Signal Process., vol. 47, no. 12, pp. 3220–3234, Dec. 1999. [30] O. Besson and P. Stoica, “Decoupled estimation of DOA and angular spread for a spatially distributed source,” IEEE Trans. Signal Process., vol. 48, no. 7, pp. 1872–1882, July 2000. [31] O. L. Frost, “An algorithm for linearly constrained adaptive array processing,” Proc. IEEE, vol. 60, no. 8, pp. 926–935, Aug. 1972. [32] 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. Signal Process., vol. 51, no. 2, pp. 313–324, Feb. 2003. [33] J. Li, P. Stoica, and Z. Wang, “On robust Capon beamforming and diagonal loading,” IEEE Trans. Signal Process., vol. 51, no. 7, pp. 1702–1715, July 2003. [34] R. G. Lorenz and S. P. Boyd, “Robust minimum variance beamforming,” IEEE Trans. Signal Process., vol. 53, no. 5, pp. 1684–1696, May 2005. [35] A. Hassanien, S. A. Vorobyov, and K. M. Wong, “Robust adaptive beamforming using sequential quadratic programming: An iterative solution to the mismatch problem,” IEEE Signal Process. Lett., vol. 15, pp. 733–736, Nov. 2008. [36] A. Khabbazibasmenj, S. A. Vorobyov, and A. Hassanien, “Robust adaptive beamforming based on steering vector estimation with as little as possible prior information,” IEEE Trans. Signal Process., vol. 60, no. 6, pp. 2974–2987, June 2012. [37] S. A. Vorobyov, A. B. Gershman, Z.-Q. Luo, and N. Ma, “Adaptive beamforming with joint robustness against mismatched signal steering vector and interference nonstationarity,” IEEE Signal Process. Lett., vol. 11, no. 2, pp. 108–111, Feb. 2004. [38] Y. J. Gu, W.-P. Zhu, and M. N. S. Swamy, “Adaptive beamforming with joint robustness against covariance matrix uncertainty and signal steering vector mismatch,” Electronics Lett., vol. 46, no. 1, pp. 86–88, Jan. 2010. [39] Y. Gu and A. Leshem, “Robust adaptive beamforming based on jointly estimating covariance matrix and steering vector,” in Proc. IEEE Int. Conf. Acoust., Speech, Signal Process., Prague, Czech Republic, May 2011, pp. 2640–2643. [40] Y. Gu and A. Leshem, “Robust adaptive beamforming based on interference covariance matrix reconstruction and steering vector estimation,” IEEE Trans. Signal Process., vol. 60, no. 7, pp. 3881–3885, July 2012. [41] L. Huang, J. Zhang, X. Xu, and Z. Ye, “Robust adaptive beamforming with a novel interference-plus-noise covariance matrix reconstruction method,” IEEE Trans. Signal Process., vol. 63, no. 7, pp. 1643–1650, Apr. 2015. [42] J. Yang, G. Liao, J. Li, Y. Lei, and X. Wang, “Robust beamforming with imprecise array geometry using steering vector estimation and interference covariance matrix reconstruction,” Multidim. Syst. Signal Process., vol. 28, no. 2, pp 451–469, Apr. 2017. [43] Y. Gu, N. A. Goodman, S. Hong, and Y. Li, “Robust adaptive beamforming based on interference covariance matrix sparse reconstruction,” Signal Process., vol. 96, Part B, pp. 375–381, Mar. 2014. [44] Y. Gu and Y. D. Zhang, “Single-snapshot adaptive beamforming,” in Proc. IEEE Sensor Array Multichannel Signal Process. Workshop, Sheffield, UK, July 2018. [45] Y. C. Eldar, A. Nehorai, and P. S. La Rosa, “An expected least-squares beamforming approach to signal estimation with steering vector uncertainties,” IEEE Signal Process. Lett., vol. 13, no. 5, pp. 288–291, May 2006.

Adaptive Beamforming via Sparsity-Based Reconstruction of Covariance Matrix

255

[46] K. Kumatani, T. Gehrig, U. Mayer et al., “Adaptive beamforming with a minimum mutual information criterion,” IEEE Trans. Audio Speech Lang. Process., vol. 15, no. 8, pp. 2527– 2541, Nov. 2007. [47] Y. Rong, Y. C. Eldar, and A. B. Gershman, “Performance tradeoffs among adaptive beamforming criteria,” IEEE J. Sel. Top. Signal Process., vol. 1, no. 4, pp. 651–659, Dec. 2007. [48] I. S. Reed, J. D. Mallett, and L. E. Brennan, “Rapid convergence rate in adaptive arrays,” IEEE Trans. Aerosp. Electron. Syst., vol. 10, no. 6, pp. 853–863, Nov. 1974. [49] L. Du, J. Li, and P. Stoica, “Fully automatic computation of diagonal loading levels for robust adaptive beamforming,” IEEE Trans. Aerosp. Electron. Syst., vol. 46, no. 1, pp. 449– 458, Jan. 2010. [50] R. O. Schmidt, “Multiple emitter location and signal parameter estimation,” IEEE Trans. Antennas Propag., vol. 34, no. 3, pp. 276–280, Mar. 1986. [51] J. A. Tropp and A. C. Gilbert, “Signal recovery from random measurements via orthogonal matching pursuit,” IEEE Trans. Inf. Theory, vol. 53, no. 12, pp. 4655–4666, Dec. 2007. [52] A. J. Miller, Subset Selection in Regression. Chapman and Hall, 2002. [53] D. Malioutov, M. Çetin, and A. S. Willsky, “A sparse signal reconstruction perspective for source localization with sensor arrays,” IEEE Trans. Signal Process., vol. 53, no. 8, pp. 3010–3022, Aug. 2005. [54] Y. C. Eldar and G. Kutyniok, Eds., Compressed Sensing: Theory and Applications. Cambridge University Press, 2012. [55] S. Chen, D. Donoho, and M. Saunders, “Atomic decomposition by basis pursuit,” SIAM J. Sci. Comput., vol. 20, no. 1, pp. 33–61, 1998. [56] R. Tibshirani. “Regression shrinkage and selection via the lasso,” J. Roy. Stat. Soc. B, vol. 58, no. 1, pp. 267–288, 1996. [57] R. Roy and T. Kailath, “ESPRIT-estimation of signal parameters via rotational invariance techniques,” IEEE Trans. Acoust. Speech Signal Process., vol. 37, no. 7, pp. 984–995, July 1989. [58] A. B. Gershman, M. Rubsamen, and M. Pesavento, “One- and two-dimensional directionof-arrival estimation: An overview of search-free techniques,” Signal Process., vol. 90, no. 5, pp. 1338–1349, May 2010. [59] K. Harmanci, J. Tabrikian, and J. L. Krolik, “Relationships between adaptive minimum variance beamforming and optimal source localization,” IEEE Trans. Signal Process., vol. 48, no. 1, pp. 1–12, Jan. 2000. [60] M. Grant, S. Boyd, and Y. Y. Ye, “CVX: MATLAB software for disciplined convex programming,” Dec. 2017. Available: http://cvxr.com/cvx/ [61] G. Tang, B. N. Bhaskar, P. Shah, and B. Recht, “Compressed sensing off the grid,” IEEE Trans. Inf. Theory, vol. 59, no. 11, pp. 7465–7490, Nov. 2013. [62] Y. Li and Y. Chi, “Off-the-grid line spectrum denoising and estimation with multiple measurement vectors,” IEEE Trans. Signal Process., vol. 64, no. 5, pp. 1257–1269, Mar. 2016. [63] C. Zhou, Y. Gu, X. Fan et al., “Direction-of-arrival estimation for coprime array via virtual array interpolation,” IEEE Trans. Signal Process., vol. 66, no. 22, pp. 5956–5971, Nov. 2018. [64] C. Zhou, Y. Gu, Z. Shi, and Y. D. Zhang, “Off-grid direction-of-arrival estimation using coprime array interpolation,” IEEE Signal Process. Lett., vol. 25, no. 11, pp. 1710–1714, Nov. 2018.

256

Gu, Goodman, and Zhang

[65] O. Besson, F. Vincent, P. Stoica, and A. B. Gershman, “Approximate maximum likelihood estimators for array processing in multiplicative noise environments,” IEEE Trans. Signal Process., vol. 48, no. 9, pp. 2506–2518, Sept. 2000. [66] C. Zhou, Z. Shi, and Y. Gu, “Coprime array adaptive beamforming with enhanced degreesof-freedom capability,” in Proc. IEEE Radar Conf., Seattle, WA, May 2017, pp. 1357–1361. [67] C. Zhou, Y. Gu, S. He, and Z. Shi, “A robust and efficient algorithm for coprime array adaptive beamforming,” IEEE Trans. Veh. Tech., vol. 67, no. 2, pp. 1099–1112, Feb. 2018.

9

Spectrum Sensing for Cognitive Radar via Model Sparsity Exploitation Augusto Aubry, Vincenzo Carotenuto, Antonio De Maio, and Mark A. Govoni

9.1

Introduction Radio frequency (RF) electromagnetic spectrum is a limited natural resource necessary for an ever-growing number of services and systems. Indeed, both high-quality/highrate wireless services as well as accurate and reliable remote-sensing capabilities call for increased amounts of bandwidth [1], posing a challenge to the overall usability of noncooperative systems. Not surprisingly, the RF spectrum congestion problem has attracted the interest of many scientists and engineers in the last few years and it is currently becoming one amongst the hot topics in both regulation and research field [2]. In the RF spectrum congestion context, spectral cognizance is envisioned as a key enabler to the next-generation cognitive radars, which take advantage of the perceptionaction cycle [3]. Based on the characteristics of frequency overlaid emitters and relying on the waveform diversity paradigm, a primary objective of cognitive radar is to utilize probing waveforms that optimize radar performance while guaranteeing spectral coexistence [4–9]. This requires electromagnetic awareness of the operative scenario, gathered via the dynamic estimation of the spectrum occupancy, which is mandatory for a flexible and efficient utilization/management of the frequency resources [2]. In this context, a multitude of spectrum-sensing algorithms has been developed mainly in the field of communication networks to counter spectrum scarcity in some frequency bands and to increase the degree of utilization of certain spectrum portions whose occupancy varies sharply from time to time and location to location [10,11]. The available techniques can be clustered in two main classes [12]. On one side, there are algorithms known as supervised procedures, which exploit intrinsic and specific emitter characteristics by correlating actual measurements with known signal patterns [11]. On the other side, there are unsupervised methods that rely on appropriate statistics of the collected measurements, such as cyclostationary parameters [13] or data sample covariance matrix attributes [14] (for example the condition number, the largest eigenvalue, and the trace). Other approaches, which take advantage of some degrees of freedom available at the receiver, have been also proposed to accomplish multi-domain spectral sensing. For instance, in [15,16] two-dimensional (2-D) sensing strategies relying on the cooperation between multiple single channel sensors are proposed to improve detection performance. In [17–19], array signal-processing techniques are developed to detect a possible primary user in a given bandwidth. An adaptive cyclostationary 257

258

Aubry, Carotenuto, De Maio, and Govoni

beamforming-based spectrum sensing method for a multiple-antenna sensor is designed in [20]. Moreover, in [21], multidimensional spectrum data are described via a spectrum tensor model and an algorithm is developed to construct spectrum maps jointly exploiting tensor completion and prediction scheme. In this chapter, focusing on the modern cognitive radar context, we deal with 2-D spectrum sensing, namely, the space–frequency characterization of the radio environment surrounding the radar of interest, without any statistical assumptions on the sources’ signals (see the discussion in Section 9.3 about the interest toward emitters’ angular information). To this end, we suppose a sensor, which, unlike usual 5G sensing devices, is equipped with multiple receive antennas (many modern radars show more than 500 receive channels, often more than 1,000) as well as high-speed processors and develop a formal discrete-time sensing signal model. Hence, we describe two different signal-processing strategies to get accurate space–frequency awareness via block sparsity exploitation at the recovery stage. •

•

The first technique [22] leans on the iterative adaptive algorithm (IAA) [23–25], which is a sequential procedure aimed at enhancing the spectrum estimate, suitably reducing the leakage effect suffered by the conventional filter bank, and includes a Bayesian information criterion (BIC)-based stage [26] to promote block-sparsity in the recovery process; The second approach [27] retrieves the space–frequency profile as solution to a regularized maximum likelihood (RML) estimation problem, where a term promoting the block-sparsity of the 2-D profile is directly included in the objective. This penalty function is intertwined with the “lq -norm” (0 < q ≤ 1) of the vector and contains the space–frequency source energies pushing for a sparse 2-D profile estimate. Since this procedure extends the sparse learning via iterative minimization (SLIM) algorithm that was developed in [28] to the block-sparsity scenario, in the following, it is referred to as block SLIM (BSLIM).

At the analysis stage, some case studies are illustrated to assess the effectiveness of the proposed 2-D spectrum sensing strategies. Specifically, the capabilities of the different signal processing techniques to recover the actual space–frequency occupancy maps are evaluated for both simulated and measured data (acquired via the software-defined radio (SDR) device “RTL-SDR R820T2 RTL2832U 1PPM TCXO”). The results highlight that both BSLIM and IAA may provide a reliable space–frequency electromagnetic cognizance with significant performance improvements as compared with the classic filter bank at the price of increased computational complexity. To summarize, the main contributions of this chapter are: •

The introduction of a 2-D sensing signal model exploiting the data collected at the end of each radar pulse repetition interval (PRI) to address the spectrum-sensing task for a modern (potentially cognitive) radar equipped with a phased-array antenna and several TX-RX modules. These data are already used in modern radars to establish (with heuristic tools) the “less disturbed frequency” and accomplish frequency diverse transmissions.

Spectrum Sensing for Cognitive Radar via Model Sparsity Exploitation

•

•

•

259

A formal connection between the 2-D spectrum-sensing task and undetermined linear models with a sparse vector of unknowns. This paves the way for the exploitation of a plethora of technically sound algorithms that are able to recover the unknown parameters of interest, i.e., angular location and spectral support of the emitters. The application of theoretically grounded criteria, borrowed from statistical signal processing (i.e., matched filters, IAA, IAA plus BIC, BSLIM), to perform sensing task in the context of a cognitive radar. The performance assessment (in terms of 2-D spectrum recovery capabilities) of the aforementioned strategies both on simulated and measured data highlighting the practical effectiveness of BSLIM and IAA BIC-based approach to provide valuable and reliable information.

The chapter exploits the results in the papers [22,27], and it is organized as follows. Section 9.2 introduces the system model for the considered spectrum-sensing scenario. In Section 9.3, some signal-processing algorithms to perform the recovery of the 2-D radio environmental map are described. Section 9.4 is devoted to the performance analysis of these procedures for both simulated and real data. Finally, Section 9.5 concludes the chapter and highlights some possible future research.

9.2

System Model and Problem Formulation Let us consider a sensor that is equipped with M antennas1 that collects the signals transmitted by K sources located in specific, but unknown, angular directions θ¯ 1,. . ., θ¯ K , with given, but unknown, spectral extensions, i.e., bandwidths.2 Figure 9.1 provides a pictorial representation of the considered sensing scenario, where the goal is to characterize the space–frequency features of the radio environment surrounding the radar of interest. The baseband discrete-time signal at the output of the receiving array (obtained by sampling the continuous-time signals according to the Nyquist rate for the sensing bandwidth B) can be written as y(n) =

K

s( θ¯ h )xh (n) + w(n), n = 1,. . .N,

(9.1)

h=1

where N is the number of available snapshots in the observation time or data-window, y(n) ∈ CM , n = 1,. . .N, is the n-th observed snapshot, s(θ) ∈ CM is the spatial steering vector associated with the angular location θ (depending on the array configuration), xh (n) ∈ C, n = 1,. . .N, refers to the signal emitted by the source located at angle θ¯ h , h = 1,. . .,K, and w(n) ∈ CM , n = 1,. . .N, are independent and 1 A large number of channels/antennas is generally assumed to be the sensing task performed by a modern

(possibly cognitive) radar. 2 Symbols presenting the overbar usually refer to the true parameters.

260

Aubry, Carotenuto, De Maio, and Govoni

Figure 9.1 Sensing scenario: a sensor equipped with multiple receive antennas is used to collect

signals emitted by multiple transmit sources. ©[2018] IEEE. Reprinted, with permission from [“Two-dimensional spectrum sensing for cognitive radar,” 2018 IEEE Radar Conference (RadarConf18), Oklahoma City, OK, USA, 2018.].

identically distributed (i.i.d.) zero-mean circularly symmetric white Gaussian vectors, i.e., E[w(n)w(m)H ] = σ2 I if n = m, or otherwise E[w(n)w(m)H ] = 0. In order to highlight the spectral characteristics of the different sources, let us resort to the frequency representation of each signal xh (n), h = 1,. . .,K, i.e., x h = [xh (1),xh (2),. . .,xh (N )]T =

NF

s F (ωk )a¯ h,k ,

(9.2)

k=1

where NF ≥ N is the number of frequency bins, 1 s F (ωk ) = √ [1, exp(j 2πωk ),. . ., exp(j 2π(N − 1)ωk )]T ∈ CN , N √ k−1 with ωk = NF , and a¯ h,k is proportional, via N /NF , to the Fourier transform of the signal samples emitted by the h-th source (during the observation time) evaluated at the frequency ωk . Hence, (9.1) can be written as y = [y(1)H ,y(2)H ,. . .,y(N )H ]H =

NF K

s¯ ( θ¯ h,ωk )a¯ h,k + w,

(9.3)

h=1 k=1

where • •

w = [w(1)H ,w(2)H ,. . .,w(N)H ]H ∈ CN M is the overall noise vector; s¯ (θ,ωk ) = s F (ωk ) ⊗ s(θ), is the steering vector associated with the angle θ and k-th frequency bin.

Spectrum Sensing for Cognitive Radar via Model Sparsity Exploitation

261

Based on (9.3), in the presence of N1 data-windows each composed of N snapshots, the baseband discrete-time signal at the output of the receiving array for the h-th space–time observation, h = 1,. . .,N1 , can be expressed as yh =

NF K

s¯ ( θ¯ m,ωk )a¯ m,k,h + wh,

(9.4)

m=1 k=1

where •

•

w1,. . .,wN1 ∈ CN M are the interference vectors affecting the different datawindows, modeled as i.i.d. zero-mean circularly symmetric white Gaussian random vectors, with covariance matrix σ2 I ; a¯ m,k,h is proportional to the Fourier transform of the signal emitted by the m-th source (during the h-th data-window), evaluated at the frequency ωk .

1 In order to proceed, let {θi }K i=1 be a grid of angular locations that are assumed fine enough such that the true positions of the existing sources lie on the grid, i.e., 1 3 { θ¯ 1,. . ., θ¯ K } ⊆ {θi }K i=1 . Under this mild assumption , the signal model (9.4) can be recast as K1 NF (9.5) s¯ (θi ,ωk )ai,k,h + wh, yh =

i=1 k=1

where s¯ (θi ,ωk ), i = 1,. . .,K1 , k = 1,. . .,NF , define the overall dictionary and ai,k,h , i = 1,. . .,K1 , k = 1,. . .,NF , represent the overall space–frequency profile at the hth snapshot. Notice that the space–frequency profile in general depends on h, since the signal transmitted by each emitter changes with the data-window, due to both data modulation and channel condition state. However, if the space–frequency pair (i ,k ) does not belong to the space–frequency support of the emitters at the h -th snapshot (in the following referred to as an inactive pair at the h -th snapshot), then it is assumed that ai ,k ,h = 0 for all the snapshots. This is tantamount to requiring that both the angular location as well as the spectral support of the emitters are stationary along the overall data acquisition time. We write this formally as (i ,k ) is inactive at h ⇒ ai ,k ,h = 0, ∀h = 1,. . .,N1 .

(9.6)

Based on (9.4) and (9.5), 2-D spectrum sensing is tantamount to recovering the overall space–frequency profile. Precisely speaking, the space–frequency occupancy map can be obtained from the estimated profile starting from the set of angle-frequency bins whose energy is different from zero, i.e., N1

|ai ,k ,h |2 > 0.

h=1

3 In the presence of mismatches, some refinements can be accomplished, for instance resorting to the

RELAX algorithm [25,29,30].

262

Aubry, Carotenuto, De Maio, and Govoni

To shed light on the intrinsic structure of the considered signal model (9.5), let us introduce the following compact vectorial representation y h = H x h + wh, h = 1,. . .,N1,

(9.7)

where • •

x h = [a1,1,h,. . .,a1,NF ,h,a2,1,h,. . .,aK1,NF ,h ]T ∈ CK1,NF , is the vector containing the space–frequency profile for the h-th snapshot; H ∈ CN M,K1 NF is the model matrix defined as H = [¯s (θ1,ω1 ),. . ., s¯ (θ2,ω1 ),. . ., s¯ (θK1 ,ωNF )] = [h1,. . .,hK1 NF ].

(9.8)

Hence, in the presence of multiple data-windows, the overall signal snapshots can be cast as follows Y = H X + W,

(9.9)

Y = [y 1,. . .,y N1 ] ∈ CMN,N1 ,

(9.10)

X = [x 1,. . .,x N1 ] ∈ CK1 NF ,N1 ;

(9.11)

W = [w1,. . .,wN1 ] ∈ CMN,N1 .

(9.12)

where

According to (9.9), the received signal for each data-window is the superposition of the weighted angular-frequency components associated with the different sources. Such configuration, usually referred to as the subspace signal model [31,32], is ever-present in signal processing applications including direction of arrival evaluation and spectral analysis where the functional form of the steering vectors is application-dependent. Regardless of the actual steering vector structure, several algorithms have been proposed in the open literature to estimate the unknown model parameters [24,33–35]. In the following section, some advanced recovery strategy exploiting the block-sparsity of the unknown matrix X induced by condition (9.6) are presented. Before concluding this section, some interesting remarks about the importance of 2-D spectrum awareness for cognitive radars are now provided. Precisely: •

Cognitive radars equipped with multiple-input multiple-output (MIMO) capability embrace the possibility of realizing adaptive beamforming at both the transmission and the receiver end [36]. This is a relevant feature as in the numerator of the radar equation there are both the transmit and receive gains. Hence, overthrowing a sidelobe direction both at the transmitter and the receiver could be much more effective for interference cancellation purposes than doing this only at the receiver side. Besides, transmit beampattern shaping can be very helpful to mitigate the interference induced by the radar on spectrally overlaid communication systems. Now, in order to shape the beampattern on transmit, exogenous information on angular locations of jammers and/or coexisting

Spectrum Sensing for Cognitive Radar via Model Sparsity Exploitation

•

•

263

networks is necessary to force the necessary angular constraints [37–41]. This can be achieved via 2-D spectrum sensing, whose output can be used to shape the transmit waveform in the frequency domain and to overthrow desired sidelobe directions. The neat result is an improvement in the coexistence between radar and overlaid communication systems. Furthermore, a space–time waveform design accounting for specific space–frequency constraints can be also conceived [42]. In this case, leveraging the information provided by the 2-D spectrum sensing the space–time transmit waveform can be synthesized so as to exhibit an appropriate frequency behavior in specific angular directions. Another important motivation of the 2-D spectrum sensing stems from the possibility of exploiting the frequency and angle of a specific emitter to recover with a suitable beamformer and frequency filter the signal of interest for additional analysis, i.e., classification purposes, namely, continuous wave (CW) vs. pulsed, modulation characteristics, cooperative vs. noncooperative emitters, and so on. Last but not least, the proposed 2-D approach paves the way to more advanced forms of environmental awareness. Precisely, multiple sensors (with the related 2-D recovered maps) can be employed to acquire the knowledge of the spatial coordinates and hence the power of emitters via triangulation on each active bandwidth. By doing so, the coverage area of any emitter can be predicted and highly appropriate space–frequency constraints can be forced so as to meet compatibility requirements.

Finally, it is worth observing that, in the presence of multipath, virtual sources will appear at the receiver side, i.e., K is larger than the actual number of emitters. In such a case, under the no-restrictive assumption4 that K < M, the 2-D spectrum sensing shares the potentiality of recovering not only the line of sight (LOS) components but also the virtual sources. Once recovered, it is also conceivable that specific postprocessing aimed at detecting the paths associated with the same sources via simple cross-correlation approaches or via advanced clustering after a specific feature extraction process.

9.3

2-D Radio Environmental Map Recovery Strategies In this section, two adaptive signal processing techniques with the ability to recover the space–frequency occupancy map via block-sparsity exploitation are described: BIC-based IAA processing and BSLIM .

9.3.1

BIC-based IAA Processing A sequential procedure that jointly exploits the profile estimate provided by the IAA strategy [24,25] and the BIC framework [26] is adopted at the recovery stage to reliably evaluate the space–frequency occupancy map [22]. The main idea is to select the mini4 As already mentioned, modern phased array radars exploit several receive channels (often more than 500).

264

Aubry, Carotenuto, De Maio, and Govoni

mum number of space–frequency sources able to appropriately fit the available data by means of a smart successive interference cancellation approach. In order to proceed, let us introduce the IAA technique. It is a sequential iterative algorithm that tries to evaluate the unknown 2-D profile iteratively refining the estimates provided by the conventional filter bank approach usually suffering the so-called leakage effect [25]. Focusing on the ith steering vector, the idea is to consider all the other contributions as interference. In this respect, assuming that the phases of xi,h , i ∈ {1,. . .,i − 1,i + 1,. . .,K1 NF }, h = 1,. . .,N1 , are i.i.d. random variables uniformly distributed over [0,2π]5 , the average (over the N1 data-windows) covariance matrix of the interference experienced by the source corresponding to the ith steering vector is Qi =

K 1 NF

2 Pl hl hH l + σ I,

i = 1,. . .,K1 NF ,

(9.13)

l=1 l!=i

with Pl =

1 2 N1 X l 2 ,

l = 1,. . .K1 NF , and T X k = Xk,1,Xk,2,. . .,Xk,N1 ∈ CN1 ,

is the k-th row of the matrix X. In other words, Pl is the average power (over the different data-windows) of the l-th source. As shown in [24], the best linear unbiased estimator of Xi (notice that the different data-windows are statistically uncorrelated) is given by −1 hH i Qi Y

ˆi = X

−1 hH i Qi hi

,

i = 1,. . .,K1 NF

(9.14)

Now, letting R=

K 1 NF

2 Pl hl hH l + σ I,

(9.15)

l=1

it follows that Qi = R − Pi hi hH i .

(9.16)

Therefore, applying the matrix inversion lemma [24,34] to (9.16) −1 hH i Qi =

−1 hH i R

−1 1 − Pi hH i R hi

,

i = 1,. . .,K1 NF ,

(9.17)

consequently, the estimator of X i can be written as ˆi = X

−1 hH i R Y

−1 hH i R hi

,

i = 1,. . .,K1 NF

(9.18)

5 This is in accordance with the key references [24,25,43], which lay the theoretical background of the

IAA approach. Nevertheless, at the analysis stage arbitrary signals, i.e., not necessarily characterized by frequency bin weights with i.i.d. phases, are considered.

Spectrum Sensing for Cognitive Radar via Model Sparsity Exploitation

265

Algorithm 1 IAA for 2-D Spectrum Sensing 1: 2: 3: 4: 5: 6:

(0)

ˆi = Initialization. Set p = 0 and X repeat p = p + 1. (p) (p−1) 2 Pˆi = N11 Xi 2 . K 1 NF (p) 2 ˆ = Pˆi hi hH R i + σ I. i=1 ˆ −1 Y hH R = Hi −1 , ˆ hi hi R K N 1 F (p)

ˆ (p) X i

hH i hi 2 Y ,

i = 1,. . .,K1 NF .

i = 1,. . .,K1 NF .

ˆ ˆ (p−1) ¯ and p ≤ p¯ 2 > . |X i 2 − X i

7:

until

8:

ˆi =X ˆ , i = 1,. . .,K1 NF . Output. Estimated 2-D profile X

i=1

p

Now, since R depends on the unknowns X i 22 , i = 1,. . .,K1 NF , the IAA iteratively ˆ i 2 in (9.15) and initializes the process with replaces Pi with the estimates Pˆi = N11 X 2 the filter bank outputs [22]. Otherwise stated, at each step the IAA predicts the matrix ˆ say, using in (9.15) the source contributions X ˆ that were estimated at the previous R, R step, according to (9.18). This procedure is iterated until a convergence condition is reached, e.g., a maximum number of iterations p¯ is performed, or K 1 NF

(p) ˆ ˆ (p−1) ¯ X i 2 − X ≤ , 2 i

i=1

where p > 0 is the iteration step and ¯ is a maximum distance among two successive profile estimates. The IAA process is summarized in Algorithm 1. Finally, the per-iteration computational complexity of the IAA [24,25] is O(NF K1 N 2 M 2 ). Some computationally efficient implementations can be also conceived see [43–46]. Let us now promote block sparsity in the recovery process jointly using the profile estimate obtained by Algorithm 1 and the BIC framework. To this end, let us denote by •

• •

K¯ an upper bound to the actual number of space–frequency sources (resulting from some upper bounds on the number of sources K and their frequency support); I(k) = {h˜ 1,. . ., h˜ k−1 } the set of the k − 1 space–frequency sources chosen up to the iteration k − 1; 4 ⎛4 4 4 4 4 ⎜4 4 ⎜ 4 ˜ ˆ hi X i 4 BICk (h) = N MN1 log ⎝4Y − 4 5 6 4 4 ˜ 4 4 i∈ I (k)∪{h}

2

+ 4k log (2N MN1 ) ,

2⎞

⎟ ⎟ ⎠

(9.19)

266

Aubry, Carotenuto, De Maio, and Govoni

where N MN1 is the size of the observation data, i.e., the product between the number of available snapshots, antennas, and data-windows, and the factor 4 in the second term of (9.19) accounts for the number of unknowns for each source, i.e., its complex valued amplitude, angle, and frequency. ¯ the space–frequency source with index Then, at step k, 1 ≤ k ≤ K, ˜ h˜ k = arg min BICk (h) h!˜ ∈I (k)

is selected as new profile entry, namely h˜ k is included in the index set defining the updated space–frequency profile (I(k + 1) = I(k) ∪ {h˜ k }). As to the procedure initialization, I(1) = ∅ , i.e., at step k = 1 the sources indices define the empty set. In a nutshell, at the k-th iteration of the algorithm a new source is selected such that the updated profile minimizes (9.19). This procedure is repeated until ¯ Hence, denoting by k ≤ K. $ % k = arg min BICk h˜ k , ¯ k∈{1,..., K}

the profile recovery is obtained from I(k ). In particular,

ˆ i if i ∈ I(k ) C X ˆ BI . = X i 0 otherwise

(9.20)

¯ It is worth pointing out that the procedure requires the storage of the two K-dimensional real-valued vectors (ordered sets) [h˜ 1, h˜ 2,. . ., h˜ K¯ ],

BIC1 (h˜ 1 ),BIC2 (h˜ 2 ),. . .,BICK¯ (h˜ K¯ ) .

9.3.2

BSLIM Approach In this subsection, a space–frequency profile recovery based on the RML estimation paradigm is presented [27]. Precisely, the following regularized minimization problem for block-sparse signal reconstruction is considered min N MN1 log(σ2 ) + σ12 H X − Y 22 + f1 (X) X,σ 2 P , (9.21) 2 s.t. σL2 ≤ σ2 ≤ σU where f1 (X) =

K 1 NF k=1

1 %q/2 2 0$ −1 Xk 22 + q

is the block-sparsity promoting penalty term, with T X k = Xk,1,Xk,2,. . .,Xk,N1 ∈ CN1

(9.22)

Spectrum Sensing for Cognitive Radar via Model Sparsity Exploitation

267

the k-th row of the matrix X and > 0 a smoothing factor making (9.22) differentiable. 2 are respectively a lower bound and an upper bound for the white In (9.21), σL2 and σU 2 interference. σL can be evaluated characterizing the power level associated with the 2 can be obtained via measureisolated operation of the receiver components, whereas σU ments in stressing conditions (for instance, in terms of device operating temperatures) and accounting for a conservative confidence level on the estimate. Remarkably, when q = 1 and = 0, (9.22) boils down to

2

K N 1 F

Xk 2 −

k=1

2 N1 NF , q

which is equivalent to the objective function of the mixed l2 / l1 -optimization program (L-OPT) that was proposed in [47–49] to develop a block-sparsity recovery algorithm for noiseless measurements. Furthermore, if the noise power level is fixed, (9.21) becomes the group version of the basis pursuit denoising algorithm presented in [50] to perform the recovery of block-sparse signals in the presence of noisy data. It is also worth pointing out that the regularized minimization problem P extends the recovery approach developed in [28] to a block-sparsity situation. Additionally, unlike [50,51], (9.21) accounts for an unknown noise power level at the recovery stage, providing a more general framework to the regularized version of the unconstrained smoothed l2 / lp minimization approach proposed in [50,51]. Problem P is a nonconvex optimization problem (the objective is a nonconvex function) and the framework proposed in [52,53] is exploited to systematically solve it and obtain high quality solutions to the formulated block-sparse recovery problem. Specifically, two independent variable blocks are considered: the former is the noise variance σ 2 while the latter is the space–frequency profile X. Thus, denoting by g(X,σ2 ) = NMN1 log(σ 2 ) + σ12 H X − Y 22 + f1 (X), the procedure developed in [27] and summarized in Algorithm 2 can be used to solve P.

Algorithm 2 BSLIM for 2-D Spectrum Sensing 1: 2: 3: 4: 5: 6: 7:

2 , > 0, δ > 0, and q ∈ ]0,1]. Input. σL2 , σU (0)

Initialization. Set n = 0, σ2 = σL2 , and X(0) = diag(h1 2,. . .,hK1 NF 2 )−1 HHY repeat n = n + 1.$ $ (n−1) 2 %1− q % 2 , i = 1,. . .,K N . D 1 = diag d¯1,. . ., d¯K1 NF , with d¯i = Xi 2 + 1 F $ % −1 (n) H H (n−1) X = D1H H D1H σ I Y, % 2% $ $ (n) 1 , with σˆ 2 = N MN σ2 = min max σL2 , σˆ 2 ,σU H X(n) − Y 22 1 (n−1)

(n)

until g(X (n−1),σ2 ) − g(X(n),σ2 ) > δ. ˆ = X(n) . 9: Output. Estimated 2-D profile X 8:

268

Aubry, Carotenuto, De Maio, and Govoni

Remarkably, as shown in [27], & ' (n) the sequence of points X (n),σ2 generated by Algorithm 2 decreases the • objective function in P; ' & (n),σ 2 (n) is a Karush–Kuhn– any cluster point of the produced sequence X • Tucker (KKT) point to P. To gain more insights on the effectiveness of the proposed recovery approach, let us observe that when q → 0, (9.22) converges to K 1 NF

$

% X k 2 + .

(9.23)

k=1

As a consequence, the penalty term (9.23) promotes block-sparsity in the profile recovery, since small values of Xk 2 lead to very low values of the objective to minimize. Otherwise stated, (9.21) pushes for multiple null rows in the recovered matrix X. Before proceeding, it is worthwhile to noting that the BSLIM approach could have also been framed in the context of Bayesian estimation. Specifically, the solution to (9.21) defines the maximum a posteriori (MAP) estimate of X,σ2 for the following Bayesian model $ % y i |X,σ2 ∼ N H x i ,σ2 I ,i = 1,. . .,N1 y i |X,σ2 , i = 1,. . .,N1,are statistically independent random vectors, % $ 2 σ ∼ U σL2 ,σU

: K8 1 NF 'q/2 ; 4 2 &4 4Xk 42 + fX (X) ∝ exp − , q 2

(9.24)

k=1

where σ2 and X are statistically independent quantities, whereas f (X) is a blocksparsity promoting prior for X. Indeed, the MAP estimate of X and σ 2 is given by the optimal solution to the following optimization problem max X,σ2

fY |X,σ2 (Y |X,σ 2 )fX (X)fσ2 (σ2 ) ,

(9.25)

which is equivalent P (it is enough to consider the negative logarithm of the objective 2 → ∞, a so called improper prior is assumed for σ 2 at the in (9.25)). Notice that, as σU recovery stage [54,55], tantamount to assuming that σ2 has equal probability over the range [σL2 ,∞].

Adaptive Selection of the Parameter q To make the developed BSLIM algorithm user parameter–free, an adaptive computation of q is reported (the approach can be easily extended to account also for the smoothing factor > 0). Precisely, inspired by [28], a procedure that jointly exploits the profile estimates provided by the BSLIM strategy for different values of q and the BIC framework [24,26] is described.

Spectrum Sensing for Cognitive Radar via Model Sparsity Exploitation

269

Let BSLIM be run with q = q¯ and let Rq¯ be the set of selected active row indices (see equation (9.27) and the related description for more details on its evaluation); h(q) ¯ = |Rq¯ |, namely the number of selected active rows; q¯ ¯ X be the least-squares estimate of X associated with the selected active rows Rq¯ .

• • •

Hence, denoting by Iq ⊆ ]0,1] the discrete set of the considered q¯ values, the overall

ˆ BSLI M = X ¯ q , where q = arg min BIC(q), space–frequency profile is recovered as X q∈Iq

with the BIC-based objective function BIC(q), q ∈ Iq , defined as $ % ¯ q − Y 22 + (2N1 + 2)h(q) log (2N MN1 ) . BIC(q) = 2N MN1 log H X

(9.26)

In equation (9.26), N MN1 is the size of the observation data, i.e., the product between the number of available snapshots, antennas, and data-windows, while the factor (2N1 + 2) in the second term of (9.26) represents the number of unknowns for each source, i.e., the N1 complex valued amplitudes, the angle, and the frequency. ˆ q¯ be the As to the evaluation of Rq¯ , again a BIC-based strategy is employed. Let X ˆ qo¯ be the matrix obtained from X ˆ q¯ sorting profile recovered by Algorithm 2 as q = q, ¯ X ¯ ¯ ¯ ˆ qo,1 ˆ qo,2 ˆ qo,N its rows so that X 22 ≥ X 22 ≥ . . ., ≥ X 2, namely, the per-row energy of 1 2 q¯ ˆ o is arranged in decreasing order, and r qo¯ = [roq¯ (1),roq¯ (2),. . .,roq¯ (K1 NF )]T ∈ NK1 NF X

¯ ˆ qo,i ˆ q¯q¯ , be the vector containing the corresponding ordered row indices, i.e., X = X r (i) i = 1,. . .,K1 NF . Then q¯ q¯ (9.27) Rq¯ = ro (1),. . .,ro (kq ¯ ) ,

where kq ¯ = arg min BICq¯ (k), ¯ k∈{1,... K}

with

& ' k q¯ k ˆ q¯ 2 BICq¯ (k) = 2N MN1 log H o X o − Y 2 + (2N1 + 2)k log (2N MN1 ) , ¯ k = 1,. . ., K.

(9.28)

In (9.28) q¯ k

q¯

q¯

• •

H o ∈ CMN,k is the matrix containing the first k columns of H o , with H o the matrix obtained from H sorting its columns according to the permutation induced q¯ by the vector r o ; q¯ k ˆ qo¯ ; ˆ o is the matrix containing the first k rows of X X K¯ is an upper bound to the actual number of space–frequency sources (result-

•

ing from some upper bounds on the number of sources K and their frequency support); the second term of (9.28) represents the BIC-based penalty.

•

270

Aubry, Carotenuto, De Maio, and Govoni

9.4

Performance Analyses In this section, the capability to perform 2-D spectrum sensing of both BSLIM and BIC-based IAA processing algorithms is assessed on both simulated and measured data scenarios.

Simulated Data A spectrum-sensing bandwidth of B = 500 MHz around the carrier frequency f0 = 2.4 GHz is considered. Moreover, the sensor is equipped with a uniform linear array of M = 10 antennas where the spacing between the antennas is d = λ 0 /2 (λ0 the operating wavelength) and σ2 = 1. In the analyzed scenario K = 4 emitters with θ¯ 1 = π/(10 K1), θ¯ 2 = π/40, θ¯ 3 = π/5, and θ¯ 4 = −π/10 are present. The first three emitters are communication sources operating on [−0.1B − 2T1 s (1 + β), − 0.1B + 2T1 s (1 + β)], [0.2B − 2T1 s (1 + β),0.2B + 1 1 1 2Ts (1 + β)], and [− 2Ts (1 + β), 2Ts (1 + β)], respectively, with β = 0.5 and Ts = 10/B. The fourth emitter is a jammer radiating a zero-mean circularly symmetric Gaussian signal with a flat spectrum over [−0.1B,0.1B]. As to the communication sources, they transmit data via a quadrature phase shift keying (QPSK) modulation employing a rootraised-cosine filter with roll-off parameter β and symbol rate Ts as reference pulse. Finally, the signal-to-noise ratio (SNR) of the different emitters is 10 dB. In the following, K1 = 40 uniformly spaced discrete angles over the interval [−π/2,π/2] are considered. N1 = 10 independent data-windows are processed each with N = 100 snapshots and NF = 2N , i.e., twice the frequency resolution induced by the number of available samples. Figure 9.2 displays the nominal space–frequency profile of the analyzed scenario, obtained evaluating for each angle bin the average, over the N1 data-windows, energy spectral density (ESD) of the signal received on 90 80 70 60 50 40 30 20 10 0 –10 –20 –30 –40 –50 –60 –70 –80 –90 –2.5–2.3–2.1–1.9–1.7–1.5–1.3–1.1–0.9–0.7–0.5–0.3–0.1 0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9 2.1 2.3 2.5

q (deg)

9.4.1

f(Hz)

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

×108

Figure 9.2 Space–frequency profile of the analyzed scenario. The gray scale is proportional to the

energy in each space–frequency bin.

Spectrum Sensing for Cognitive Radar via Model Sparsity Exploitation

271

q (deg)

(a) 90 80 70 60 50 40 30 20 10 0 –10 –20 –30 –40 –50 –60 –70 –80 –90 –2.5–2.3–2.1–1.9–1.7–1.5–1.3–1.1–0.9–0.7–0.5–0.3–0.1 0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9 2.1 2.3 2.5

f(Hz)

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

8

×10

(b) Figure 9.3 Space–frequency occupancy map, first trial: (a) BSLIM; (b) BIC-based IAA processing. The detected bins correspond to the hotter pixels.

the considered angle direction, normalized to the maximum value. The plot clearly highlights the space–frequency portions occupied by the K emitters. Figures 9.3 and 9.4 show the space–frequency occupancy maps recovered via the BLSIM technique with adaptive selection of q and the BIC-based IAA processing for two independent trials. In Algorithm 1, the exit condition based on the maximum number of iterations p¯ = 4 is considered and the BIC-based IAA processing assumes 2 = 10. Also, the K¯ = 180. In Algorithm 2 δ = 10−1 , = 10−6 , σL2 = 1, and σU ¯ adaptive selection of q supposes K = 180 and Iq = {0.01,0.12,0.23,0.34,0.45,0.56, 0.67,0.78,0.89,1}. The obtained maps clearly show that both BLSIM and BIC-based IAA are able to provide almost exact recovery localizing accurately the sources in both angle and frequency domains. However, BSLIM exhibits fewer false alarms than the counterpart (there is no

Aubry, Carotenuto, De Maio, and Govoni

(a) 90 80 70 60 50 40 30 20 10 0 –10 –20 –30 –40 –50 –60 –70 –80 –90 –2.5–2.3–2.1–1.9–1.7–1.5–1.3–1.1–0.9–0.7–0.5–0.3–0.1 0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9 2.1 2.3 2.5

q (deg)

272

f(Hz)

×10

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

8

(b) Figure 9.4 Space–frequency occupancy map, second trial : (a) BSLIM; (b) BIC-based IAA processing. The detected bins correspond to the hotter pixels.

angular dithering). More important, BSLIM presents a computational complexity lower than the BIC-based IAA processing. Finally, due to the dictionary redundancy, both approaches detect just a subset of frequency bins for each emitter. Indeed, the considered temporal directions are linearly dependent vectors (due to the frequency oversampling) and both BLSIM and BIC-based IAA processing are devised to automatically pick up an essential subset of temporal signatures to reconstruct the signal. Otherwise stated, the frequency representation of the signals is not unique and the results highlight the ability of the considered approaches to retrieve a sparse signal description. To further grasp insights on the performance of the considered recovery strategies, in Table 9.1 the empirical false alarm rate, PF A , and the empirical detection probability, PD , are reported assuming the same simulation setup as in Figure 9.2. Specifically,

Spectrum Sensing for Cognitive Radar via Model Sparsity Exploitation

273

Table 9.1 Empirical false alarm rate, PFA , and empirical detection probability, PD , for BSLIM and BIC-based IAA processing for the sensing scenario of Figure 9.2. Recovery Algorithm

PF A

PD

BSLIM

5.0852 × 10−5

0.9812

BIC-based IAA processing

0.0027

0.9197

Figure 9.5 Empirical ROC for the filter bank approach (solid curve), IAA algorithm (dashed

curve), BSLIM q = 1 (dashed-dotted curve), and BSLIM q = 0.45 (dotted curve).

for both BSLIM and BIC-based IAA techniques, a moving average filtering (with 2 equal weights) along the frequency dimension is performed over the recovered profile (containing in each space–frequency bin the estimated average energy) so as to handle the intrinsic on-off behavior of the map. Hence, PF A is evaluated counting the detections (over 20 independent trials, each composed of N1 = 10 independent data-windows6 ) in the space–frequency bins where no emitters are present. Analogously, PD is obtained counting the number of detections, over the same 20 independent trials, where space– frequency sources are located. Inspection of the table confirms previous considerations and highlight that BSLIM can provide, for the analyzed situations, better performance than BIC-based IAA. Finally, in Figure 9.5 the empirical receiver operating characteristic (ROC) curves of the filter bank approach, IAA processing, and BLSIM (with q equal to either 0.45 or 1) are reported, assuming the same simulation setup as in Figure 9.2, but with SNR = 5dB. Specifically, for each approach, the threshold is set so as to ensure the desired empirical 6 This setup corresponds to 157,320 space–frequency bins, which are emitters free and 2,080 bins occupied

by RF sources, where some guard cells for the communication transmitters have been included (the 3-dB bandwidth measure is considered for each emitter).

274

Aubry, Carotenuto, De Maio, and Govoni

PF A over 20 independent trials. Hence, PD is evaluated comparing the actual threshold with the final estimated map. The latter is obtained performing a noncoherent energy integration of the N1 estimated space–frequency profiles, i.e., for any space–frequency bin the mean of the square modulus for the available N1 estimates is evaluated regardless of the approach. In addition, BSLIM employs a moving average filtering (with 2 equal weights) along the frequency dimension for both q’s. The results clearly corroborate the superiority of BSLIM and IAA over the filter bank approach, especially for low values of the false alarm rate. This result reflects the leakage effect suffered by the filter bank. Finally, BSLIM uniformly outperforms IAA when q = 1.

9.4.2

Measured Dataset In this subsection, the performance of the developed algorithms on measured data is assessed via the SDR device “RTL-SDR R820T2 RTL2832U 1PPM TCXO.” It works over the frequency range 24-1766 MHz and allows to sense an arbitrary bandwidth belonging to the mentioned interval. The discrete-time signal is obtained first downconverting the received continuous-time signal at the intermediate frequency (IF) of 3.57 MHz and using a sampling frequency of 28 MHz. Then the baseband discrete-time signal at the Nyquist rate for the sensing bandwidth of interest is collected via digital processing. Additional information about this device are available at [56]. The experimental setup herein considered is illustrated in Figure 9.6 where a logperiodic antenna is connected to the SDR device to sense the environment. In this context, no angle discrimination can be performed among the RF sources since just one spatial channel is available at the receiver. Otherwise stated, the 2-D spectrum sensing reduces to the conventional 1-D problem where the goal is to establish which frequency bins are occupied by RF emitters. Hence, in the following, the radio environment state characterization is described via the frequency occupancy map associated with a reference angle, i.e., θ = 0 without loss of generality. The first conducted experiment refers to a sensing bandwidth lying in the range of frequencies occupied by terrestrial trunked radio (TETRA) communications [57]. Precisely, a center frequency f0 = 393 MHz, a one-sided bandwidth of B = 2 MHz, and a sampling frequency of 2 complex Mega-samples per second are considered. As to the parameters used by the filter bank approach, the IAA-based algorithms, and the BSLIM techniques, it is assumed: • • • •

M = 1 receive antenna; N = 100 samples per snapshot; N1 = 10 snapshots; NF = 2N frequency bins.

As in Subsection 9.4.1, Algorithm 1 involves a maximum number of iterations given by p¯ = 4 and the BIC-based IAA processing uses K¯ = 180. Additionally, in Algorithm 2, 2 = 10. The lower bound to the noise δ = 10−1 , = 10−6 , σL2 = 0.004, and σU power level is estimated via a measured data spectrum where no RF sources are present,

Spectrum Sensing for Cognitive Radar via Model Sparsity Exploitation

275

Figure 9.6 Experimental setup for measured data acquisition via SDR device.

and this value is also used to evaluate the interference covariance matrix in the IAAbased procedures. The upper bound is higher than 10 times the power of the available data. Moreover, the adaptive selection of q assumes K¯ = 180 and Iq = {0.01,0.12, 0.23,0.34,0.45,0.56,0.67,0.78,0.89,1}. For comparison purposes also the behavior of the filter bank technique, Algorithm 1 without BIC-based stage, and Algorithm 2 for a fixed q is analyzed. To this end, it is assumed that they perform a noncoherent energy integration of the N1 = 10 estimated profiles, and the resulting value in each bin is compared with a threshold, i.e., the noise power level times a multiplicative factor (empirically set). In Figure 9.7, the results corresponding to the 1,000 collected samples are displayed. In particular, Figure 9.7.a illustrates the square modulus of the acquired data Fourier transform versus frequency, i.e., the data spectrum, while Figures 9.7(b), 9.7(c), 9.7(d), 9.7(e), and 9.7(f) illustrate the frequency occupancy maps obtained via the filter bank approach, the IAA algorithm, the BIC-based IAA processing, the BSLIM algorithm with adaptive selection of q, and Algorithm 2 with q = 1, respectively. Specifically, the vertical yellow lines in these maps identify the frequency bins where RF sources are detected. Inspection of the maps highlights the superiority of the IAA-based strategies and BSLIM techniques over the conventional filter bank approach. Indeed, the latter pro-

276

Aubry, Carotenuto, De Maio, and Govoni

(a)

(b)

(c)

(d)

(e)

(f)

Figure 9.7 Results associated with the first set of 1,000 samples collected in the TETRA

bandwidth. (a) data spectrum. Frequency occupancy map: (b) filter bank approach; (c) IAA algorithm; (d) BIC-based IAA processing; (e) BSLIM (q = 1); (e) BSLIM (adaptive q selection). The detected frequency bins correspond to the hotter pixels.

vides an unreliable recovery of the frequency occupancy profile mainly due to the energy spillover effect induced by the multiple spectral contributions. On the other hand, both IAA-based algorithms and BSLIM strategies are able to accurately retrieve the actual frequency bins occupied by RF sources. However, BLIM shares a lower computational complexity than the counterpart. Notice that a possible missed detected emitter could be present around the frequency −0.9 MHz. However, it can not be a priori claimed if it is an emitter or a noise spike. Remarkably, BSLIM algorithm with adaptive selection of q and the BIC-based IAA processing are totally adaptive and do not require any ad hoc threshold selection, which is a valuable feature from a practical point of view.

Spectrum Sensing for Cognitive Radar via Model Sparsity Exploitation

(a)

(b)

(c)

(d)

(e)

(f)

277

Figure 9.8 Results associated with the second set of 1,000 samples collected in the TETRA

bandwidth. (a) data spectrum. Frequency occupancy map: (b) filter bank approach; (c) IAA algorithm; (d) BIC-based IAA processing; (e) BSLIM (q = 1); (e) BSLIM (adaptive q selection). The detected frequency bins correspond to the hotter pixels.

Figure 9.8 reports the results of the described spectrum-sensing algorithms for another dataset still composed of 1,000 samples, assuming the same setup as in Figure 9.7. The preceding remarks hold true also in this case, namely the IAA-based strategies and BSLIM techniques prove effective to recover the frequency occupancy maps. In the second experiment the focus is on a sensing bandwidth belonging to the spectrum occupied by global system for mobile (GSM) communications [58]. Specifically, a center frequency f0 = 943 MHz, a one-sided bandwidth of B = 2.8 MHz, and a sampling frequency of 2.8 complex Mega-samples per second are considered.

278

Aubry, Carotenuto, De Maio, and Govoni

(a)

(b)

(c)

(d)

(e)

(f)

Figure 9.9 Results associated with the first set of 1,000 samples collected in the GSM bandwidth.

(a) data spectrum. Frequency occupancy map: (b) filter bank approach; (c) IAA algorithm; (d) BIC-based IAA processing; (e) BSLIM (q = 1); (e) BSLIM (adaptive q selection). The detected frequency bins correspond to the hotter pixels.

In this analysis, the parameters used by all the procedures are the same as in the first experiment7 but for σL2 = 0.01 and the threshold used after noncoherent energy integration in filter bank technique, Algorithm 1 without BIC-based stage, and Algorithm 2 with q = 1. Indeed, a different multiplicative factor to the noise power level is considered. Figures 9.9 and 9.10 show the results associated with two different experiments, each corresponding to a dataset of 1,000 samples. Like the previous analysis, both the

7 Specifically, the adopted parameters are: p¯ = 4, K ¯ = 180,Iq = {0.01,0.12,0.23,0.34,0.45,0.56,0.67, 2 = 10. 0.78,0.89,1},δ = 10−1, = 10−6 , and σU

Spectrum Sensing for Cognitive Radar via Model Sparsity Exploitation

(a)

(b)

(c)

(d)

(e)

(f)

279

Figure 9.10 Results associated with the second set of 1,000 samples collected in the GSM bandwidth. (a) data spectrum. Frequency occupancy map: (b) filter bank approach; (c) IAA algorithm; (d) BIC-based IAA processing; (e) BSLIM (q = 1); (e) BSLIM (adaptive q selection). The detected frequency bins correspond to the hotter pixels.

acquired data spectrum and the obtained frequency occupancy maps are provided for each dataset. The obtained maps confirm the ability of the IAA-based and BSLIM strategies to correctly detect and identify frequency bins occupied by RF sources. Finally, the conventional filter bank approach exhibits performance close to the other algorithms for the datasets in the GSM bandwidth, unlike the other analyses. As final and concluding remark, the reported analysis has highlighted the effectiveness of the introduced approaches to endow spectrum awareness. Moreover, accounting for both computational complexity and recovery ability, BSLIM appears to be the best choice.

280

Aubry, Carotenuto, De Maio, and Govoni

9.5

Conclusions 2-D spectrum sensing has been considered to gather real-time space–frequency scenario awareness, assuming a sensor equipped with multiple receive channels. A formal discrete-time sensing signal model has been developed and two adaptive signal processing algorithms have been introduced for recovering the space–frequency occupancy map via block-sparsity exploitation. The former employs the IAA technique and incorporates a BIC-based stage to promote block-sparsity in the recovery process. The latter applies the RML estimation paradigm to automatically push for block-sparsity in the 2-D profile evaluation. At the analysis stage both simulated and measured data scenarios are considered to evaluate the capability of the proposed procedures to retrieve the actual space–frequency occupancy maps. The reported results clearly show the effectiveness of the developed tools and their superiority over the conventional filter bank approach. Precisely, both BIC-based IAA and BSLIM with adaptive q selection are able to grant an accurate and reliable space–frequency occupancy map recovery without requiring an ad hoc threshold selection. Furthermore, better performance than the classic filter bank is obtained at the price of a higher computational complexity. Finally, considering both computational complexity and recovery ability, it is reasonable to use BSLIM for 2-D spectrum sensing. Future research might concern the use of constant false alarm rate (CFAR) strategies to pick up the most relevant space–frequency contributions within the estimated profile, as well as the analysis of different model order selection rules both in the IAA context and for the adaptive selection of the parameter q.

Acknowledgment The work of Drs. Augusto Aubry and Antonio De Maio was sponsored by the US Army RDECOM International Technology Center – Atlantic in partnership with the US Army Research Laboratory under USAITC-A Seedling Project W911NF-17-2-0134.

References [1] M. Wicks, “Spectrum crowding and cognitive radar,” in 2010 2nd International Workshop on Cognitive Information Processing, Italy, June 2010, pp. 452–457. [2] H. Griffiths, L. Cohen, S. Watts et al., “Radar spectrum engineering and management: Technical and regulatory issues,” Proceedings of the IEEE, vol. 103, no. 1, pp. 85–102, 2015. [3] A. Farina, A. De Maio, and S. Haykin, Eds., The Impact of Cognition on Radar Technology. Schitech Publishing, 2017. [4] H. He, P. Stoica, and J. Li, “Waveform design with stopband and correlation constraints for cognitive radar,” in 2010 2nd International Workshop on Cognitive Information Processing, Elba, Italy, June 2010.

Spectrum Sensing for Cognitive Radar via Model Sparsity Exploitation

281

[5] M. A. Govoni and R. A. Elwell, “Qualitative analysis of interference on receiver performance using advanced pulse compression noise (APCN),” in SPIE Defense, Security, and Sensing Conference, Baltimore, MD, May 2015. [6] M. A. Govoni, “Enhancing spectrum coexistence using radar waveform diversity,” in IEEE Radar Conference, Philadelphia, PA, May 2016. [7] A. Aubry, A. De Maio, M. Piezzo et al., “Cognitive radar waveform design for spectral coexistence in signal-dependent interference,” in IEEE Radar Conference, Cincinnati, OH, May 2014. [8] A. Aubry, V. Carotenuto, A. De Maio, A. Farina, and L. Pallotta, “Optimization theory-based radar waveform design for spectrally dense environments,” IEEE Aerospace and Electronic Systems Magazine, vol. 31, no. 12, pp. 14–25, 2016. [9] A. Aubry, V. Carotenuto, and A. De Maio, “Forcing multiple spectral compatibility constraints in radar waveforms,” IEEE Signal Processing Letters, vol. 23, no. 4, pp. 483– 487, 2016. [10] Y. Zhao, J. Gaeddert, K. Bae, and J. Reed, “Radio environment map-enabled situation-aware cognitive radio learning algorithms,” in Proceedings of Software Defined Radio Technical Conference, Orlando, FL, November 2006. [11] H. Tang, “Some physical layer issues of wide-band cognitive radio systems,” in IEEE International Symposium on New Frontiers in Dynamic Spectrum Access Networks, Baltimore, MD, November 2005. [12] T. Yucek and H. Arslan, “A survey of spectrum sensing algorithms for cognitive radio applications,” IEEE Communications Surveys & Tutorials, vol. 11, no. 1, pp. 116–130, 2009. [13] A. Fehske, J. Gaeddert, and J. Reed, “A new approach to signal classification using spectral correlation and neural networks,” in IEEE International Symposium on New Frontiers in Dynamic Spectrum Access Networks, Baltimore, MD, November 2005. [14] D. Guimaraes, R. A. de Souza, and A. Barreto, “Performance of cooperative eigenvalue spectrum sensing with a realistic receiver model under impulsive noise,” Journal of Sensor and Actuator Networks, vol. 2, pp. 46–69, 2013. [15] Q. Wu, G. Ding, J. Wang, and Y. D. Yao, “Spatial-temporal opportunity detection for spectrum-heterogeneous cognitive radio networks: Two-dimensional sensing,” IEEE Transactions on Wireless Communications, vol. 12, no. 2, pp. 516–526, 2013. [16] G. Ding, J. Wang, Q. Wu, F. Song, and Y. Chen, “Spectrum sensing in opportunityheterogeneous cognitive sensor networks: How to cooperate?” IEEE Sensors Journal, vol. 13, no. 11, pp. 4247–4255, 2013. [17] P. Wang, J. Fang, N. Han, and H. Li, “Multiantenna-assisted spectrum sensing for cognitive radio,” IEEE Transactions on Vehicular Technology, vol. 59, no. 4, pp. 1791–1800, 2010. [18] R. Zhang, T. J. Lim, Y. C. Liang, and Y. Zeng, “Multi-antenna based spectrum sensing for cognitive radios: A GLRT approach,” IEEE Transactions on Communications, vol. 58, no. 1, pp. 84–88, 2010. [19] A. Taherpour, M. Nasiri-Kenari, and S. Gazor, “Multiple antenna spectrum sensing in cognitive radios,” IEEE Transactions on Wireless Communications, vol. 9, no. 2, pp. 814– 823, 2010. [20] K. L. Du and W. H. Mow, “Affordable cyclostationarity-based spectrum sensing for cognitive radio with smart antennas,” IEEE Transactions on Vehicular Technology, vol. 59, no. 4, pp. 1877–1886, 2010.

282

Aubry, Carotenuto, De Maio, and Govoni

[21] M. Tang, G. Ding, Q. Wu, Z. Xue, and T. A. Tsiftsis, “A joint tensor completion and prediction scheme for multi-dimensional spectrum map construction,” IEEE Access, vol. 59, no. 4, pp. 8044–8052, 2016. [22] A. Aubry, A. De Maio, and M. Govoni, “Two-dimensional spectrum sensing for cognitive radar,” in IEEE Radar Conference, Oklahoma City, OK, April 2018. [23] S. D. Blunt and K. Gerlach, “A novel pulse compression scheme based on minimum mean-square error reiteration,” in Proceedings of the International Conference on Radar, Adelaide, Australia, September 2003. [24] W. Roberts, P. Stoica, J. Li, T. Yardibi, and F. A. Sadjadi, “Iterative adaptive approaches to mimo radar imaging,” IEEE Journal of Selected Topics in Signal Processing, vol. 4, no. 1, pp. 5–20, 2010. [25] T. Yardibi, J. Li, P. Stoica, M. Xue, and A. B. Baggeroer, “Source localization and sensing: A nonparametric iterative adaptive approach based on weighted least squares,” IEEE Transactions on Aerospace and Electronic Systems, vol. 46, no. 1, pp. 425–443, 2010. [26] P. Stoica and Y. Selen, “Model-order selection: A review of information criterion rules,” IEEE Signal Processing Magazine, vol. 21, no. 4, pp. 36–47, 2004. [27] A. Aubry, V. Carotenuto, A. De Maio, and M. Govoni, “Multi-snapshot spectrum sensing for cognitive radar via block-sparsity exploitation,” IEEE Transactions on Signal Processing, vol. 67, no. 6, pp. 1396–1406, 2019. [28] X. Tan, W. Roberts, J. Li, and P. Stoica, “Sparse learning via iterative minimization with application to mimo radar imaging,” IEEE Transactions on Signal Processing, vol. 59, no. 3, pp. 1088–1101, 2011. [29] J. Li and P. Stoica, “Efficient mixed-spectrum estimation with applications to target feature extraction,” IEEE Transactions on Signal Processing, vol. 44, no. 2, pp. 281–295, 1996. [30] J. Li, P. Stoica, and D. Zheng, “Angle and waveform estimation via relax,” IEEE Transactions on Aerospace and Electronic Systems, vol. 33, no. 3, pp. 1077–1087, 1997. [31] S. M. Kay, Ed., Fundamentals of Statistical Signal Processing, Volume II: Detection Theory. Prentice Hall, 1998. [32] L. L. Scharf, Ed., Statistical Signal Processing. Addison-Wesley Reading, 1991. [33] M. A. Richards, J. A. Scheer, and W. A. Holm, Eds., Principles of Modern Radar: Basic Principles. SciTech Publishing, 2010. [34] P. Stoica and R. L. Moses, Eds., Spectral Analysis of Signals. Pearson Prentice Hall, 2006. [35] S. D. Blunt, K. Gerlach, and T. Higgins, “Aspects of radar range super-resolution,” in IEEE Radar Conference, Boston, MA, April 2007. [36] J. R. Guerci, Ed., Cognitive Radar: The Knowledge-Aided Fully Adaptive Approach. Artech House Inc., 2010. [37] D. R. Fuhrmann and G. San Antonio, “Transmit beamforming for mimo radar systems using signal cross-correlation,” Transactions Aerospace Electronic System, vol. 44, no. 1, pp. 171– 186, 2008. [38] N. Shariati, D. Zachariah, and M. Bengtsson, “Minimum sidelobe beampattern design for mimo radar systems: A robust approach,” in IEEE International Conference on Acoustics, Speech, and Signal Processing, Florence, Italy, May 2014. [39] A. Aubry, A. De Maio, and Y. Huang, “MIMO radar beampattern design via PSL/ISL optimization,” IEEE Transactions on Signal Processing, vol. 64, no. 15, pp. 3955–3976, 2016. [40] A. Konar and N. D. Sidiropoulos, “Hidden convexity in qcqp with toeplitz-hermitian quadratics,” IEEE Signal Processing Letters, vol. 10, no. 22, pp. 1623–1627, 2015.

Spectrum Sensing for Cognitive Radar via Model Sparsity Exploitation

283

[41] A. Aubry, V. Carotenuto, and A. De Maio, “New results on generalized fractional programming problems with Toeplitz quadratics,” IEEE Signal Processing Letters, vol. 10, no. 22, pp. 1623–1627, 2015. [42] O. Aldayel, V. Monga, and M. Rangaswamy, “Successive qcqp refinement for mimo radar waveform design under practical constraints,” IEEE Transactions on Signal Processing, vol. 14, no. 64, pp. 1623–1627, 2015. [43] M. Xue, L. Xu, and J. Li, “IAA spectral estimation: Fast implementation using the Gohberg– Semencul factorization,” IEEE Transactions on Signal Processing, vol. 59, no. 7, pp. 3251– 3261, 2011. [44] G. O. Glentis and A. Jakobsson, “Efficient implementation of iterative adaptive approach spectral estimation techniques,” IEEE Transactions on Signal Processing, vol. 59, no. 9, pp. 4154–4167, 2011. [45] G. O. Glentis and A. Jakobsson, “Superfast approximative implementation of the iaa spectral estimate,” IEEE Transactions on Signal Processing, vol. 60, no. 1, pp. 472–478, 2012. [46] W. Sun, H. C. So, Y. Chen, L. T. Huang, and L. Huang, “Approximate subspace-based iterative adaptive approach for fast two-dimensional spectral estimation,” IEEE Transactions on Signal Processing, vol. 62, no. 12, pp. 3220–3231, 2014. [47] Y. C. Eldar and M. Mishali, “Robust recovery of signals from a structured union of subspaces,” IEEE Transactions on Information Theory, vol. 55, no. 11, pp. 5302–5316, 2009. [48] Y. C. Eldar, P. Kuppinger, and H. Bolcskei, “Block-sparse signals: Uncertainty relations and efficient recovery,” IEEE Transactions on Signal Processing, vol. 58, no. 6, pp. 3042–3052, 2010. [49] Y. C. Eldar and G. Kutyniok, Eds., Compressed Sensing: Theory and Applications. Cambridge University Press, 2012. [50] Y. Wang, J. Wang, and Z. Xu, “On recovery of block-sparse signals via mixed l2 / lq (0 < q ≤ 1) norm minimization,” EURASIP Journal on Advances in Signal Processing, vol. 76, pp. 1–17, 2013. [51] M. Lai, Y. Xu, and W. Yin, “Improved iteratively reweighted least squares for unconstrained smoothed lq minimization,” SIAM Journal on Numerical Analysis, vol. 51, pp. 927–957, 2013. [52] M. Razaviyayn, M. Hong, and Z.-Q. Luo, “A unified convergence analysis of block successive minimization methods for nonsmooth optimization,” SIAM Journal on Optimization, vol. 23, no. 2, pp. 1126–1153, 2013. [53] A. Aubry, A. D. Maio, A. Zappone, M. Razaviyayn, and Z. Luo, “A new sequential optimization procedure and its applications to resource allocation for wireless systems,” IEEE Transactions on Signal Processing, vol. 66, no. 24, pp. 6518–6533, 2018. [54] A. Gelman, J. B. Carlin, H. S. Stern, and D. B. Rubin, Eds., Bayesian Data Analysis. Boca Raton, FL: Chapman & Hall/CRC, 2004. [55] C. P. Robert and G. Casella, Eds., Monte Carlo Statistical Methods.Springer Science + Business Media, 2004. [56] R. W. Stewart, K. W. Barlee, D. S. Atkinson, and L. H. Crockett, Software Defined Radio Using MATLAB & Simulink and the RTL-SDR. Strathclyde Academic Media, 2015. [57] J. Dunlop, D. Girma, and J. Irvine, Digital Mobile Communications and the TETRA System. John Wiley & Sons, 2013. [58] M. Rahnema, “Overview of the GSM system and protocol architecture,” IEEE Communications Magazine, vol. 31, no. 4, pp. 92–100, 1993.

10

Cooperative Spectrum Sharing between Sparse Sensing-Based Radar and Communication Systems Bo Li

10.1

∗

and Athina P. Petropulu

∗∗

Introduction Radio spectrum has been essential to a wide array of technologies, including communications, position and navigation systems, and radars. Until recently, radar and communication systems jointly consumed most of the spectrum below 6 GHz, with commercial and noncommercial (i.e., military radar) uses assigned on distinct bands. However, recent studies have shown that large chunks of spectrum designated for radar applications are underutilized [1], while there is spectrum congestion in bands devoted to commercial wireless communications. Further, as the number of connected devices grows, the spectrum congestion problem will worsen. The nonuniform spectrum utilization is clearly illustrated in the data collected in downtown Berkeley, California, as shown in Figure 10.1 [2]. To address the need for more efficient use of spectrum, US government agencies have been examining the possibility of allowing wireless broadband systems to operate in the 3500–3650 MHz band, previously used exclusively by high-powered shipborne, airborne, and ground-based radar systems operated by the Department of Defense [3,4]. On examining the viability of coexistence of radar and communication systems in that band, the National Telecommunications and Information Administration (NTIA) proposed the use of exclusion zones [5], which would protect base stations from radar interference. However, those zones cover large US metropolitan areas, and such an approach would not solve the problem of improving spectral efficiency. The main problem with different systems using the same spectrum is the interference that one system exerts to the other. According to a 2006 NTIA report [6], levels of interference to noise ratio between −9 dB and −2 dB, which are below the thermal noise floor level of the radar receiver, can reduce the probability of target detection. Also, the interference generated by the radar reduces the throughput of a communication system operating nearby. Spectrum-sharing methods are focused on enabling radar and communication systems to share the spectrum efficiently by minimizing interference effects. The

∗ Bo Li is now with Aurora Innovation, Inc. This work was supported by NSF under Grant ECCS-1408437. ∗∗ This work was supported by NSF under Grant ECCS-1408437.

284

Cooperative Spectrum Sharing

285

Figure 10.1 Spectrum utilization measurement in 0–6 GHz band [2].

spectrum-sharing literature for controlling interference includes works that explore large physical separation [7–9], or dynamic spectrum access methods [10–14] using orthogonal frequency division multiplexing (OFDM) signals and optimally allocating subcarriers between the two systems [15–17]. Methods that use specially designed radar waveforms have also been considered [18–21]. There are also works that explore the spatial degrees of freedom enabled by the use of multiple antennas at both systems [22–28]. Multiple-input multiple-output (MIMO) radars offer spatial degrees of freedom and have been used for spectrum sharing with communication systems. In earlier MIMO radar works, the interference mitigation was considered either for the communication system [22–25], or for the radar [28], but not for both. For example, in the well studied null space projection based scheme of [22–27], the radar eliminates its interference to the communication system by projecting its waveforms onto the null space of the interference channel. However, in those works, the interference generated by the communication system to the radar was not addressed. By applying spatial multiplexing to each system in isolation, one may miss out on the potential performance improvement stemming from a coordinated operation of the two systems. Indeed, recent advances in cognitive radio and cloud technology, provide a framework that can be leveraged for a coordinated operation. A related emerging technology is the dual-function radarcommunication system, which achieves the objectives of two systems by using the same transmitter or receiver resources. Readers can refer to [29] for an overview of dual-function radar-communication systems. In this chapter, we discuss some approaches that rely on spatial multiplexing and sparse sensing, and exploit a cooperative framework for the use of spectrum [30–36]. In particular, we study cooperative spectrum sharing between MIMO radars that employ sparse sensing and matrix completion (MIMO-MC), and MIMO communication systems. As will be discussed in this chapter, MIMO-MC radars are particularly well suited for spectrum sharing [30,31]. This is because each radar receive antenna uses a time-varying sparse sampling scheme, which effectively modulates the communicationradar interference channel and increases its null space. This gives the opportunity to the communication system to transmit along that null space and thus avoid interfering with the radar. Also, a cooperative design has the potential to improve spectrum utilization due to the increased number of design degrees of freedom. Co-design requires access to physical-layer information on both systems. For example, both systems would have to share physical-layer information with a node designated as the control center, which would optimally design the signaling schemes of each system. Obviously, this requires

286

Li and Petropulu

the systems’ willingness to cooperate, with payback reduced interference for the radar and higher throughput for the communication system. Of course, the amount of information that can be shared and the privacy issues involved would have to be evaluated in each case. Examples of radar systems that could be amenable to such cooperation include radar for autonomous vehicles, weather monitoring, etc. The remainder of this chapter is organized as follows. Section 10.2 provides some background on MIMO-MC radars. Section 10.3 presents the coexistence model between a MIMO-MC radar and a MIMIO communication system, and outlines all assumptions made. Section 10.4 formulates the problem of spectrum sharing and provides efficient algorithms for solving it. Finally, Section 10.5 presents simulation results demonstrating the performance gains of cooperation, and Section 10.6 offers some concluding remarks.

10.2

MIMO Radars Using Sparse Sensing MIMO radars transmit different waveforms from their transmit (TX) antennas. Their receive (RX) antennas forward their measurements to a fusion center, where a data matrix is populated with the information received by each RX antenna. This matrix is then used in array processing methods to estimate target information. For a relatively small number of targets as compared to the number of TX and RX antennas, the data matrix is low rank [37], and thus can be reconstructed with provable accuracy (under certain conditions) based on a small set of its entries via matrix completion (MC) [38,39]. This observation is the basis of MIMO-MC radars. In MIMO-MC radars, each RX antenna forwards to the fusion center a small number of pseudo-randomly subNyquist sampled measurements, along with their sampling times, and partially fills a row of the data matrix. Under certain conditions [39], the full data matrix, corresponding to Nyquist sampled data, can be stably recovered via MC techniques, with reconstruction error proportional to the noise level. The recovered matrix can be subsequently used for target detection via standard array processing methods [37]. The sub-sampling at the antennas avoids the need for high-rate analog-to-digital converters, and the reduced number of samples translates into power and bandwidth savings over the antenna-fusion center link. Because MIMO-MC radars recover all missing entries of the data matrix, they do not suffer from signal-to-noise ratio (SNR) loss due to data sub-sampling. MIMO-MC radars can achieve the high resolution of traditional MIMO radars with significantly fewer samples and reduced hardware complexity. We should note that compressive sensing (CS)–based MIMO radars [40–42] are another sample-reduction approach in the literature. Compared to MIMO-CS radars, MIMO-MC radars avoid basis mismatch problems. Let us consider a collocated MIMO radar system using uniform linear arrays (ULA) for both transmission and reception. Let Mt,R and Mr,R be the number of TX and RX antennas, respectively, and dt and dr the inter-element spacing at the transmit and receive arrays, respectively. The radar transmits L pulses, with pulse repetition interval (PRI) TPRI and carrier wavelength λc . The radar operates in two phases; in the first phase the TX antennas transmit waveforms and the RX antennas receive target returns, while

Cooperative Spectrum Sharing

287

in the second phase, the RX antennas forward their measurements to a fusion center. The K far-field targets are with distinct angles {θk }, target reflection coefficients {βk }, and Doppler shifts {νk }, and are assumed to fall in the same range bin. In the absence of clutter, the noisy data matrix at the fusion center can be formulated as [37,39,43] Y R = V r V Tt P S + W R ,

(10.1)

where the m-th row of Y R ∈ CMr,R ×L contains the L fast-time raw samples forwarded by the m-th antenna [44]; S = [s(1),. . .,s(L)], is the waveform matrix, with s(l) = [s1 (l),. . .,sMt,R (l)]T being the l-th snapshot across the transmit antennas; the transmit waveforms are assumed to be orthogonal, i.e., it holds that SS H = I Mt,R [37]; W R denotes additive noise; and P ∈ CMt,R ×Mt,R denotes the transmit precoding matrix. V t [vt (θ1 ),. . .,vt (θK )] and V r [vr (θ1 ),. . .,vr (θK )] respectively denote the transmit and receive steering matrix and vr (θ) ∈ CMr,R is the receive steering vector defined as r r T (10.2) vr (θ) e−j 2π0ϑ ,. . .,e−j 2π(Mr,R −1)ϑ , where ϑr = dr sin(θ)/λc denotes the spatial frequency with respect to the receive array. vt (θ) ∈ CMt,R is the transmit steering vector and is respectively defined. Matrix is defined as diag([β1 ej 2πν1 ,. . .,β K ej 2πνK ]). D V r V Tt is the target response matrix. At the fusion center, Y R passes through matched filters, after which target estimation is performed via standard array processing methods [45]. When K is smaller than Mr,R and L, the noise-free data matrix, M DP S, is low rank and under certain conditions can be provably recovered based on a subset of its entries. This observation gave rise to MIMO-MC radars [37,39,43], where each RX antenna sub-samples the target returns and forwards the samples to the fusion center. The partially filled data matrix at the fusion center can be mathematically expressed as follows (see [37, scheme I]): Y R = (M + W R ),

(10.3)

where denotes the Hadamard product; is the sub-sampling matrix containing 0’s and 1’s. The sub-sampling rate p equals 0 /(LMr,R ), where 0 denotes the number of 1’s in . When p equals 1, the matrix is filled with 1’s, and the MIMO-MC radar is identical to the traditional MIMO radar. At the fusion center, the completion of M can be achieved by the following nuclear norm minimization problem [38] min M∗ M

s.t. M − Y R F ≤ δ,

(10.4)

where ·∗ denotes the matrix nuclear norm; δ > 0 is a parameter determined by the sampled entries of the noise matrix, i.e., W R . It is shown in [38] that the recovery of M is stable against noise. The matrix recovery error is proportional to the noise level δ, while the recovery conditions are given in terms of the coherence of M. [38]

288

Li and Petropulu

The coherence parameters of M, (μ0,μ1 ), are defined as

μ0 ≥ max(μ(U ),μ(V )) K

K

μ1

Mr,R L

≥

U ·k V H ·k ∞,

(10.5) (10.6)

k=1

where U ∈ CMr,R ×K and V ∈ CL×K contain the left and right singular vectors of M; U ·k denotes the k-th column vector of U ; · ∞ denotes the maximum entry of the matrix. ; and μ(V ) is the coherence of subspace spanned by basis matrix V , defined as : ; L L 2 max V l· ∈ 1, , (10.7) μ(V ) K 1≤l≤L K where V l· denotes the l-th row vector of V . The coherence of subspace U is similarly defined. According to [46], assuming that matrix M is sampled uniformly at random at m points, there exist constants C and c, such that if 5 6 1/2 m ≥ C max μ21,μ0 μ1,μ0 n1/4 nK β log n (10.8) for some β > 2, the minimizer of the nuclear norm problem is unique and equal to M with probability at least 1 − cn−β . The condition in (10.8) implies that the smaller the coherence parameters, the fewer samples are needed for recovering the full matrix M. Upper bounds of the coherence parameters can give us an idea of how small those parameters can be. The coherence parameters of M for the case P = I Mt,R are given in the following theorem [43]. theorem 10.1 (Coherence of M when P = I Mt,R ) Let the minimum spatial frequency separation of the K targets be ξt and ξr with respect to the transmit and receive 2 arrays. On denoting the Fejér kernel by Fn (x) = n1 sin 2(πnx) , and for dt = dr = λc /2 sin (πx) and Mr,R Mt,R K ≤ min , , (10.9) FMr,R (ξr ) FMt,R (ξt ) it holds that μ(U ) ≤

Mr,R

Mr,R μr0 .

− (K − 1) FMr,R (ξr )

(10.10)

Further, if every snapshot of the waveforms S ·l ≡ s(l) satisfies the following equation: 0 π π1 Mt,R ,θ ∈ − , , ∀l ∈ N+ , (10.11) |S T·l vt (θ)|2 = L L 2 2 then μ(V ) is upper bounded by μ(V ) ≤

Mt,R

Mt,R μt0 .

− (K − 1) FMt,R (ξt )

(10.12)

Cooperative Spectrum Sharing

289

Consequently, the matrix M has coherence parameters μ0 max{μr0,μt0 } and μ1 √ K μ0 . The bounds in Theorem 10.1, along with the orthogonality property of the radar waveforms in (10.11), were used to design waveforms with good incoherence properties. The work of [43] involves numerical optimization on the complex Stiefel manifold, which has high computational complexity. However, radar waveforms need to be updated frequently as security against adversaries, which makes the issue of computational cost more severe. It was also observed in [43] via simulations that using a random unitary matrix [47] as the waveform matrix results in performance very close to that of the optimum waveform, indicating that a random unitary matrix might be a good approximation of the optimal waveform. The matrix completion performance degrades when the signal-tointerference-plus-noise ratio (SINR) drops below 10 dB [37], which suggests that along with “good” radar waveforms, a precoder, designed to mitigate interference would be very important. In the following subsection, we consider a MIMO-MC radar that uses a random unitary matrix as the waveform matrix S, and a nontrivial precoder maxtrix P [35,36].

10.2.1

MIMO-MC Radar Using Random Unitary Matrix A random unitary matrix [47] can be obtained through Gram–Schmidt orthogonalization of a random matrix with entries distributed as independent and identically distributed (i.i.d.) Gaussian. This means that such waveforms can be generated easily. The following theorem provides an upper bound of the coherence μ(V ) of M when a random unitary waveform is used and a nontrivial radar precoder P is employed. theorem 10.2 Consider the MIMO-MC radar presented in Section 10.2, using a random unitary waveform matrix S, and with M as defined in (10.3). For any transmit precoder P such that for K0 = Rank(M) it holds that K0 ≤ K, and for an arbitrary transmit array geometry and target angles, the coherence of the right singular vector subspace of M is bounded as √ K0 + 2 3K0 ln L + 6 ln L μ(V ) ≤ μ˜ t0, K0 with probability 1 − L−2 , and the coherence of subspace U obeys μ(U ) ≤ μr0 is defined in Theorems 10.1. Proof

(10.13) K r K0 μ0 ,

where

The following lemma is used in the proof [48].

lemma 10.3 each t > 0

Let SN be a χ 2 random variable with N degrees of freedom. Then for √ 2 P SN − N ≥ t 2N + t 2 ≤ e−t /2 .

(10.14)

290

Li and Petropulu

It is clear that K0 is not larger than K. Recall that M has a compact singular-value decomposition (SVD) given as M = U V H ,

(10.15)

where U ∈ CMr,R ×K0 and V ∈ CL×K0 contain the left and right singular vectors of M; ∈ RK0 ×K0 is diagonal containing the singular values. Consider the QR decomposition of V r and S T P T V t : V r = Qr R r ,

(10.16)

S P V t = Qt R t , T

T

where Qr ∈ CMr,R ×K and Qt ∈ CL×K0 are with orthonormal columns, R r ∈ CK×K is upper triangular, and R t ∈ CK0 ×K has an upper staircase form. The matrix R r R Tt ∈ CK×K0 is full column rank with a compact SVD given by U 1 1 V H 1 , where U 1 ∈ H H K×K K ×K 0 0 0 , V1 ∈ C , U 1 U 1 = V 1 V 1 = I K0 , and 1 is diagonal, containing the C singular values of R r R Tt . Therefore, we have T ∗ H M = Qr U 1 1 V H 1 Qt = Qr U 1 1 (Qt V 1 ) ,

(10.17)

which is a valid SVD of M. The uniqueness of the singular values of a matrix indicates that ≡ 1 . Therefore, we can choose U = Qr U 1 and V = Q∗t V 1 . We have μ(U ) =

Mr,R K0

sup (Qr )m· U 1 22

m∈N+ M

r,R

Mr,R ≤ K0

sup

m∈N+ M

(Qr )m· 22

K r = μ , K0 0

(10.18)

r,R

where μr0 is the upper bound on μ(U ) defined in Theorem 10.1. We also have μ(V ) =

L L sup (Q∗t )l· V 1 22 = sup (Qt )l· 22 . K0 l∈N+ K0 l∈N+ L

(10.19)

L

If K0 is strictly smaller than K, we cannot represent Qt in terms of S T P T V t and R t because of the singularity of R t . To mitigate this issue, we apply column permutations F on R t to bring forward the first nonzero elements in each row R t F = (R 1 R 2 ) such that R 1 ∈ CK0 ×K0 is square, upper triangular, and invertible. The QR decomposition S T P T V t can be rewritten as S T P T V t F = Qt (R 1 R 2 ).

(10.20)

Qt = S T P T V t F K0 R −1 1 ,

(10.21)

We can represent Qt as

Cooperative Spectrum Sharing

291

where F K0 denotes the first K0 columns of F . Substituting Qt into μ(V ), we obtain μ(V ) =

4$ % 42 L 4 4 sup 4 S T l· P T V t F K0 R −1 1 42 K0 l∈N+ L

$ % $ −1 %H H H ∗ ∗ L R1 = sup S T l· P T V t F K0 R −1 F K0 V t P (S )· 1 K0 l∈N+

(10.22)

L

It holds that

$ −1 %H $ H %−1 = R1 R1 R −1 1 R1 %−1 $ H = RH 1 Qt Qt R 1 %−1 $ H ∗ ∗ T T = FH K0 V t P S S P V t F K0 %−1 $ H ∗ T = FH , K0 V t P P V t F K0

(10.23)

where the last equality holds because SS H = I Mt,R . Consider the QR decomposition of P T V t F K0 given by P T V t F K0 = Qa R a ,

(10.24)

where Qa ∈ CMt,R ×K0 contains orthonormal columns, and R a ∈ CK0 ×K0 is upper triangular and full rank. Substituting (10.23) and (10.24) into (10.22), we have μ(V ) =

$ %−1 H ∗ L sup s T R a R H Ra s l a Ra K0 l∈N+ l L

L L = sup s T s ∗ = sup s l 22, K0 l∈N+ l l K0 l∈N+ L

(10.25)

L

where s l QTa S ·l , and the second equality holds because R a is invertible. Based on [49, theorem 3], if Mt,R = O(L/ ln L), the entries of S can be approximated by i.i.d. Gaussian random variables with distribution CN (0,1/L). Since Qa has orthonormal columns, s l ∈ CK0 ,∀l ∈ N+ L also contains i.i.d. Gaussian random variable with distri2 . Based on Lemma 10.3 bution CN (0,1/L), and Ls l 22 is distributed according to χK 0 √ and setting t = 6 ln L, it holds that

P Ls l 22 ≥ K0 + 2 3K0 ln L + 6 ln L ≤ L−3 . (10.26) Applying the union bound, we have that ⎡ ⎤ √ + 2 3K ln L + 6 ln L K 0 0 ⎦ ≤ L−2 . P ⎣ sup s l 22 ≥ L + l∈N

(10.27)

L

Combining (10.25) and (10.27) gives √ : ; K0 + 2 3K0 ln L + 6 ln L P μ(V ) ≥ ≤ L−2 . K0

(10.28)

292

Li and Petropulu

From the derivation, the bound on μ(V ) holds for any target angles, array geometry, and precoding matrix P as long as P T V t F K0 has full column rank K0 . This completes the proof of Theorem 10.2. Based on Theorem 10.2, we have the following theorem for the coherence parameters of M. theorem 10.4 (Coherence of M with random unitary waveform matrix) Consider the MIMO-MC radar presented in Section 10.2 with S being random unitary. For dr = λc /2, arbitrary transmit array geometry, and for K≤

Mr,R , FMr,R (ξr )

(10.29)

the matrix M has coherence parameters

K r t μ , μ˜ μ0 max K0 0 0

μ 1 K0 μ 0

(10.30) (10.31)

with probability 1 − L−2 , where μ˜ t0 is defined in Theorem 10.2. μr0 is the upper bound on μ(U ) given in Theorem 10.1. The above result holds for any precoding matrix P such that the rank of M is K0 . Proof The theorem can be proved by substituting the bounds on μ(U ) and μ(V ) in Theorem 10.1 with the bounds derived in Theorem 10.2. Some comments are in order. First, if K0 is O(ln L), the upper bound μ˜ t0 > 1 is a small constant O(1); this can be seen from the definition of μ˜ t0 in (10.13) by choosing K0 > ln L. Therefore, based on (10.13), the coherence parameter of M is close to 1, which means that M has good coherence property. A similar bound was provided on the coherence of the subspaces spanned by a random orthogonal basis in [46]. Second, unlike the results in [43, theorem 2], the probabilistic bound on μ(V ) is independent of the target angles and array geometry. Third, the results in Theorem 10.4 hold for any random unitary matrix S. The radar waveform can be changed periodically, which would be good for security reason, without affecting the matrix completion performance. Fourth, the probabilistic bound on μ(V ) in Theorem 10.2 is independent of P . This means that we can design P , for example for the purpose of transmit beamforming and interference suppression, without affecting the incoherence property of M. This key observation validates the feasibility of precoding for spectrum sharing between MIMOMC radar and communication systems, a topic that will be discussed next. Note that this chapter focuses on the design of radar precoder in the spatial domain, not the waveform. The radar precoder will not affect the waveform ambiguity properties in the time and Doppler domains.

Cooperative Spectrum Sharing

10.3

293

Coexistence System Model We consider a coexistence scenario as shown in Figure 10.2, where a MIMO-MC radar system and a MIMO communication system operate using the same carrier frequency. Note that the coexistence model also applies to MIMO radar, because when full sampling is adopted the MIMO-MC radar becomes equivalent to a traditional MIMO radar. In the coexistence system, H ∈ CMr,C ×Mt,C denotes the communication channel, where Mr,C and Mt,C denote respectively the number of RX and TX antennas of the communication system. G1 ∈ CMr,C ×Mt,R and G2 ∈ CMr,R ×Mt,C denote the interference channels between the communication and radar systems. The radar operates in pulsed mode; in each pulse, it first transmits a short pulse waveform, and then listens for target echoes for a much longer period. The duration of these two phases comprises the PRI. Figure 10.3 shows the radar-communication coexistence signal model during two periods of one radar PRI. At the communication receiver, radar interference is present only during the radar transmit period. On the other hand, the communication interference at the radar receiver is present during the entire radar PRI. In this chapter, we focus on joint radar and communication waveform design during Period 1. Readers can refer to [33] for communication waveform design schemes that can reduce the interference to the radar receiver during the entire radar PRI.

… … Collocated MIMO radar

… Communication TX

… Communication RX

Figure 10.2 A MIMO communication system sharing spectrum with a collocated MIMO radar system.

Figure 10.3 Radar-communication coexistence signal model during one radar PRI.

294

Li and Petropulu

Figure 10.4 The spectrum-sharing architecture. The cooperation is coordinated by the control center, a node with high computing power that also serves as the radar fusion center. The control center collects information from radar and communication systems, computes jointly optimal signaling schemes for both systems and sends each scheme back to the corresponding system.

Cooperative spectrum sharing can be implemented via the system architecture of Figure 10.4. The coordination of the cooperation is conducted by a control center, which collects information from the two systems, formulates and solves an optimization problem, and passes to each systems its optimal parameters. The control center can be thought of as an enhanced spectrum access system (SAS) used in the FCC release [50], and is connected to the radar/communication system via either a wireless link, or a backhaul channel. The control center can also integrate the functionality of the radar fusion center, i.e., target detection, estimation and tracking, and specifically for the MIMO-MC radar, also matrix completion. There are several advantages in having a control center that encompasses the radar fusion center. First, a powerful all-in-one center greatly simplifies the complexity of the overall network. Second, radar operators, especially in military applications, are not willing to share information directly with civilian cellular systems out of security concerns. In such cases the control center can be operated by the radar, and enable cooperation while maintaining the isolation of the radar and communication systems. Third, the radar and communication systems only need communication interfaces with the control system. The coexistence model considered here relies on the following assumptions.

Transmitted Signals It is assumed that the two systems transmit narrowband waveforms with the same symbol period. To evaluate the feasibility of radar and communication systems having the same symbol period, let us consider an S-band search and acquisition radar with range resolution equal to 300 m (a typical range resolution is between 100 m and 600 m [51,52]). The corresponding radar sub-pulse duration is 2 μs. Communication symbol

Cooperative Spectrum Sharing

295

duration of 2 μs is quite typical in model cellular systems [53]. The transmitted signal is narrowband if the channel coherence bandwidth is larger than the signal bandwidth [53– 55]. In a macro-cell, typical values for the channel coherence bandwidth are of the order of 1 MHz [56,57], which is larger than the signal bandwidth of 0.5 MHz (or symbol interval 2 μs). Thus, the narrowband assumption is typically valid. If higher signal bandwidth is needed, OFDM signaling can be used for both radar [15,17] and the communication system [56,57]. Our coexistence model can still be valid on each OFDM carrier, over which the signal can be considered as narrowband.

Fading We assume that H , G1 , and G2 are flat fading, which is valid when the transmitted signals are narrowband. The flat-fading assumption is common practice in the radarcommunication system coexistence literature [22–26]. In addition, all channels are assumed to be block fading over the radar PRI. For a radar with medium pulse repetition frequency, the PRI is usually between 30 μs and 0.3 ms. The typical channel coherence time for 2.5 GHz and 5.8 GHz carrier frequency ranges from 2 ms up to 200 ms [58]. The channel coherence time is much larger than the radar PRI. As for the moving targets, the resulting Doppler shifts are usually assumed to be constant during one PRI [44,59]. Therefore, channel block fading is a reasonable assumption.

Channel State Information (CSI) The channel H is assumed to be perfectly known at the communication transmitter. The channels G1 and G2 are also assumed to be perfectly known at the radar. CSI estimation can be achieved using pilot channels [22,60] scheduled by the control center in timedivision multiplexing (TDM) fashion. As a simple example, based on Figure 10.4, the communication transmitter, i.e., the base station (BS), transmits a reference signal in pilot burst A, and this is used by the radar to estimate G2 . The communication receiver, i.e., a user entity, transmits a reference signal in pilot bursts B, and this is used by the BS and the radar to estimate H and G1 , respectively, based on channel reciprocity [61]. All estimated CSI is sent to the control center by the radar and the BS, where it is used to jointly optimize the spatial multiplexing. Note that CSI estimation and feedback can be scheduled based on the channel coherence time, which is much larger than the radar PRI. Figure 10.5 shows a simplified schematic diagram for CSI estimation/feedback and receiving design results from the control center based on TDM. Existing techniques in cognitive radios and multiuser MIMO (MU-MIMO) [62–68] can also be applied to reduce the overhead for CSI feedback.

The Radar Mode of Operation We consider the target tracking scenario, in which the radar searches for targets with unknown radar cross section (RCS) variances, in particular directions of interest, given by set {θk }, and at a particular range bin of interest [69,70]. In such scenarios, the target parameters have typically been obtained from previous tracking cycles, and are used to optimize the transmission for better SINR performance [44].

296

Li and Petropulu

Figure 10.5 TDM-based CSI estimation and feedback and reception of design results from the control center.

Under the aforementioned assumptions on transmitted signals, fading, CSI, and the radar mode of operation, let us consider a target scene at a particular range bin as in Section 10.2 but with clutter. The baseband signal received by the radar receivers during L symbol durations in one radar PRI can be expressed as % $ Y R = DP S + G2 X2 + W R . F GH SI + CP F GH I FGHI signal

interference

(10.32)

noise

The signal received by the communication receivers can be expressed as H X + G1 P S1 + W C . Y C = FGHI F GH I FGHI signal

interference

(10.33)

noise

Y R , D, P , S, W R , and appearing in the above equations are as defined in Section 10.2. Note that the delay in the radar signal model is assumed a to be known and thus appropriately compensated for. The waveform-dependent interference term CP S contains interferences from point scatterers (clutter or interfering objects). If there are Kc point scatterers at angles {θkc }, and reflection coefficients {β ck } within the same range c c c T c bin as the targets, then C K k=1 β k vr (θk )vt (θk ) denotes the clutter response matrix. Y C and W C denote the received signal and additive noise at the communication RX antennas, respectively. X [x(1),. . .,x(L)] is a matrix whose columns, x(i)’s are codewords from the codebook of the communication system. W R/C contains i.i.d. random 2 ). i ,i ∈ {1,2} are diagonal matrices containing the entries distributed as CN (0,σR/C j α random phase offset e il between the MIMO-MC radar and the communication system at the l-th symbol. These phase offsets are time-varying and arise due to the random phase jitters of the oscillators between the radar transmitter and the communication receiver [31,71]. Note that the Doppler shift will not be an issue in the design considered in this chapter. This is because the radar signal model in (10.32) is for the fast-time samples received in one radar pulse, and during a pulse, the Doppler shift is usually assumed to be constant [41,44,59,72]. In that case, the Doppler shift can be absorbed into the target RCS, and does not affect our design. The control center aims to protect the radar system and maximize the spectrum efficiency. In the following, we present a joint design of the communication and radar transmissions, which will be implemented at the control center, so that the interference at the radar RX antennas is minimized, thus allowing for successful matrix completion, while certain communication system requirements are met [36].

Cooperative Spectrum Sharing

10.4

297

Cooperative Spectrum Sharing In this section, we formulate the MIMO-MC radar and MIMO communication spectrum sharing problem and present an algorithm to solve it [36]. For the communication system, the covariance of interference plus noise is given by 2 R Cin = G1 GH 1 + σC I

(10.34)

where P P H /L is positive semidefinite. For l ∈ N+ L , the instantaneous information rate is unknown because the interference plus noise is not necessarily Gaussian due to the random phase offset ej αil . However, a lower bound of the rate equals [73] H . (10.35) C(R xl ,) log2 I + R −1 Cin H R xl H The above bound is achieved when the codeword x(l), l ∈ N+ L is distributed as CN (0,R xl ). The average communication rate over L symbols is as follows 1 C(R xl ,), L L

Cavg ({R xl },)

(10.36)

l=1

where {R xl } denotes the set of all R xl ’s. The MIMO-MC radar only partially samples Y R . Therefore, only the sampled target signal and sampled interference determine the matrix completion performance. Therefore, it would make sense to define the effective signal power (ESP) and effective interference power (EIP) at the radar RX node, referring to the sampled entries only, i.e., ''6 5 & & ESP E tr (DP S) (DP S))H 1 0 166 5 50 βk (D k P S)) βk (D k P S)H = E tr k k 0 166 5 5 βk βj (D k P S) (D j P S)H = E tr k j 0 16 5 H H E{βk βj }

l D k P E{s l s H }P D

= tr l l j k j l 5 0 16 (a) (10.37) = tr σ2

l D k D H k l k βk l & ' & ' (b) = tr σβ2k D k D H σβ2k D H

D k = tr k k k k ' & 2 ∗ H σβk vt (θk )vr (θk ) vr (θk )vTt (θk ) = tr k & ' (c) = pLMr,R tr σβ2k v∗t (θk )vTt (θk ) k

= pLMr,R tr (D t ) '6 5 & EIP E tr (CP S) ( (CP S))H '6 5 & + E tr (G2 X2 ) ( (G2 X2 ))H = pLMr,R tr (C t ) +

L l=1

&

'

tr G2l R xl GH 2l ,

(10.38)

298

Li and Petropulu

Figure 10.6 The sub-sampling at the radar receiver modulates the interference channel from the communication transmitter to the radar receiver G2 . As shown in the left figure, the null space of G2 is typically empty; thus the communication system transmission would always introduce interference to the radar. Due to the random sub-sampling, the null space of the modulated interference channel G2l shown in the right figure becomes nonempty, and thus, it is possible for the communication system to introduce zero EIP to the radar receiver if it transmits in the null space of G2l .

where D k vr (θk )vTt (θk ) for k ∈ N + K , s l s(l), and l = diag(·l ). From (10.37), (a) follows from the fact that E{βk βj } = δj k σβ2k , where δj k denotes the Kronecker delta; (b) follows from the fact that l = l l and = L l=1 l ; (c) follows from (θ ) v (θ ) = = pLM . Additionally, we have the following the fact that vH k r k 1 r,R r Kc 2 ∗ c T c 2 v∗ (θ )vT (θ ), C = c σ σ v (θ definitions: D t = K c k t k t k=1 β k t k )vt (θk ), σβ k and σ β k k=1 β t k

denote the standard deviation of β k and βck , respectively; G2l l G2 . The derivation for EIP is similar to that for ESP and is omitted for brevity. The derivation in (10.37) and (10.38) assumes that the target and clutter reflection coefficients are independent complex Gaussian with zero mean, which are typical assumptions in the literature [70,74,75]. The sub-sampling at the radar receiver modulates the interference channel G2 (see Figure 10.6). At sampling instance l, only the interference at radar RX antennas corresponding to 1’s in ·l is sampled. Thus, the effective interference channel during the l-th symbol duration is G2l . To match the interference channel variation, the communication system should use adaptive transmission with symbol-dependent covariance matrix R xl [31]. This would be the optimal approach, however, it would involve high computational cost. A suboptimal alternative would be constant rate communication transmission, i.e., R xl ≡ R x ,∀l ∈ N+ L , outlined in Section 10.4.3. Incorporating the expressions for effective target signal, interference and additive noise, the effective radar SINR becomes ESINR =

tr (C t ) +

tr (D t ) . % $ H 2 l=1 tr G2l R xl G2l /(pLMr,R ) + σR

L

One can see that the joint design of the communication TX covariance matrices {R xl }, the radar precoder P (embedded in ), and the radar sub-sampling scheme is necessary to maximize the ESINR. In Theorem 10.4, we prove that the radar precoder P can be designed without affecting the incoherence property of M.

Cooperative Spectrum Sharing

299

At the control center, the spectrum sharing problem can be formulated as follows: (P1 )

max

{R xl }0,0,

ESINR ({R xl },,) ,

s.t. Cavg ({R xl },) ≥ C, L

tr (R xl ) ≤ PC ,Ltr () ≤ PR ,

(10.39a) (10.39b)

l=1

tr (V k ) ≥ ξtr (),∀k ∈ N+ K,

(10.39c)

is proper,

(10.39d)

where V k v∗t (θk )vTt (θk ). The constraint of (10.39a) restricts the communication rate to be at least C, in order to support reliable communication and avoid service outage. The constraints of (10.39b) restrict the total communication and radar transmit power to be no larger than PC and PR , respectively. The constraints of (10.39c) restrict the power of the radar probing signal in directions of interest to be no smaller than I , i.e., vTt (θk )v∗t (θk ) ≥ the power achieved by the uniform precoding matrix trM() t,R

ξvTt (θk ) trM() I v∗t (θk ) = ξtr (). It holds that ξ ≥ 1, which is used to control the t,R beampattern at the target angles of interest. The purpose of this constraint is to ensure fairness across the multiple targets. The constraint in (10.39d) imposes restrictions on the radar sub-sampling matrix such that it corresponds to a fixed sub-sampling rate p and has large spectral gap. In the matrix completion literature, is either a uniformly random sub-sampling matrix [38], or the adjacency matrix of a regular bipartite graph with large spectral gap [76]. The spectral gap of a matrix is defined as the difference between the largest and the second largest singular values [76]. In order for the control center to formulate and solve the problem of (10.39) it needs: (i) (ii)

the communication and radar system CSI; estimation and feedback of CSI are discussed in Section 10.3. target angles, and clutter parameters {σβ2c } and {θkc }. Since the control center k integrates the radar fusion center functionality, the target angles obtained from the previous tracking cycle will be available. In practice, the clutter parameters could be estimated when the targets are absent [74]. If the parameters {σβ2c } are k

(iii)

not known, we can use a single value, σ02 , for all the targets. With such choice, the objective treats all target directions equally. One possible choice for σ02 is the smallest target RCS variance that could be detected by the radar. Note that the solution of (P1 ) is independent on the specific value of σ02 . all parameters in the constraints. Parameters like power budget and required communication rate could be provided by the radar and communication systems.

Problem (P1 ) is nonconvex with respect to optimization variable triplet ({R xl },,). In Subsection 10.4.1 we present an algorithm to find a local solution via alternating optimization, while in Subsection 10.4.2, we discuss the feasibility and solution properties of (P1 ) [36].

300

Li and Petropulu

10.4.1

Solution to the Spectrum Sharing Problem Using Alternating Optimization In this section we discuss the alternating iterations with respect to {R xl }, , and .

The Alternating Iteration with Respect to {R xl }

We first solve for {R xl } while setting and to be equal to the solution from the previous iteration, i.e., we formulate the following problem: (PR )

min

{R xl }0

L % $ tr G2l R xl GH 2l l=1

s.t. Cavg ({R xl },) ≥ C,

L

(10.40) tr (R xl ) ≤ PC .

l=1

Problem (PR ) is convex and involves multiple matrix variables, the joint optimization with respect to which requires high computational complexity. The semidefinite matrix 2 real scalar variables, which would result in a complexity variables {R xl } have LMt,C 2 3.5 of O((LMt,C ) ) if an interior-point method [77] was used. An efficient algorithm for solving Problem (PR ) can be implemented based on the Lagrangian dual decomposition [77]. The Lagrangian of (PR ) can be written as L({R xl },λ1,λ2 ) =

L % $ tr G2l R xl GH 2l l=1

+ λ1

L

tr (R xl ) − PC

$ % + λ2 C − Cavg ({R xl }) ,

l=1

where λ1 ≥ 0 and λ2 ≥ 0 are the dual variables associated with the transmit power and the communication rate constraints, respectively. The dual problem of (PR ) is (PR -D)

max g(λ1,λ2 ),

λ 1,λ 2 ≥0

where g(λ 1,λ2 ) is the dual function defined as g(λ 1,λ2 ) =

inf L({R xl },λ1,λ2 ).

{R xl }0

The domain of the dual function, i.e., dom g, is λ1,λ2 ≥ 0 such that g(λ1,λ2 ) > −∞. The problem is called dual feasible if (λ1,λ2 ) ∈ dom g. The dual function g(λ 1,λ2 ) can be obtained by solving L independent subproblems, each of which can be written as follows && ' ' (PR -sub) min tr GH 2 l G2 + λ 1 I R xl R xl 0 (10.41) H . H R H − λ2 log2 I + R −1 xl wl Before giving the solution of (PR -sub), let us first state some observations. Observation 1) The average capacity constraint should be active at the optimal point. This means that the achieved capacity is always C and λ2 > 0. To show this, let us

Cooperative Spectrum Sharing

301

assume that the optimal point {R ∗xl } achieves Cavg ({R ∗xl }) > C. Then we can always shrink {R ∗xl } until the average capacity reduces to C, which would also reduce the objective. Thus, we end up with a contradiction. By complementary slackness, the corresponding dual variable is positive, i.e., λ2 > 0. $ % + Observation 2) GH 2 l G2 + λ 1 I is positive definite for all l ∈ NL . This can be shown H by contradiction. Suppose that there exists l such that G2 l G2 + λ1 I is singular. Then it must hold that GH 2 l G2 is singular and λ 1 = 0. Therefore, we can always find a nonzero vector v lying in the null space of GH 2 l G2 . At the same time, it holds that −1/2 R wl H v != 0 with very high probability, because H is a realization of the random channel. If we choose R xl = αvvH and α → ∞, the Lagrangian L({R xl },0,λ2 ) will be unbounded from below, which indicates that λ1 = 0 is not dual feasible. This means that λ1 is strictly larger than 0 if GH 2 l G2 is singular for any l. Thus, the claim is proven. Based on these observations, we have the following lemma. lemma 10.5 [63,64] For given feasible dual variables λ1,λ2 ≥ 0, the optimal solution of (PR -sub) is given by −1/2

R ∗xl (λ1,λ2 ) = l

−1/2

U l l U H l l

(10.42)

,

−1/2 −1/2 ˜ ; where l GH 2 l G2 + λ 1 I ; U l is the right singular matrix of H l R wl H l 2 + l = diag(βl1,. . .,β lr ) with βli = (λ2 − 1/σli ) , r and σli ,i = 1,. . .,r, respectively, ˜ l . It also holds that being the rank and the positive singular vales of H

r ∗ H H R H log2 I + R −1 = xl wl

i=1

&

'+

log(λ2 σli2 )

.

(10.43)

Based on Lemma 10.5, the solution of (PR ) can be obtained by finding the optimal dual variables λ∗1,λ∗2 . The cooperative spectrum sharing problem (PR ) can be solved via the procedure outlined in Algorithm 1. The convergence of Algorithm 1 is guaranteed by the convergence of the ellipsoid method [78]. The complexity of the dual decomposition based algorithm is only linearly dependent on L.

Algorithm 1 Algorithm for the alternating iteration (PR ) 1: 2: 3: 4: 5: 6: 7: 8:

Input: H,G1,G2,,Pt ,C,σC2 Initialization: λ1 ≥ 0,λ2 ≥ 0 repeat Calculate R ∗xl (λ1,λ2 ) according to (10.42) with the λ1 and $ given % λ2 ; L ∗ Compute the subgradient of g(λ1,λ2 ) as l=1 tr R xl (λ1,λ2 ) − Pt and C − Cavg ({R ∗xl (λ1,λ2 )}) respectively for λ1 and λ2 ; Update λ1 and λ2 accordingly based on the ellipsoid method [78]; until λ1 and λ2 converge to a prescribed accuracy. Output: R ∗xl = R ∗xl (λ1,λ2 )

302

Li and Petropulu

The Alternating Iteration with Respect to Via simple algebraic manipulations, the EIP from the communication transmission can be reformulated as L

& ' T tr G2l R xl GH 2l ≡ tr ( Q),

l=1

where the l-th column of Q contains the diagonal entries of G2 R xl GH 2 . With fixed {R xl } and , we can solve via min tr (T Q) s.t. is proper.

(10.44)

Recall that the sampling matrix is required to have large spectral gap. However, it is difficult to incorporate such conditions in the optimization problem (10.44). Based on the fact that row and column permutation of the sampling matrix would not affect its singular values and thus the spectral gap, a suboptimal approach is to search the best sampling scheme by permuting rows and columns of an initial sampling matrix 0 , i.e., min tr (T Q)

s.t. ∈ ℘(0 ),

(10.45)

where ℘(0 ) denotes the set of matrices obtained by arbitrary row and/or column permutations. 0 is generated with binary entries and pLMr,R ones, where x denotes the largest integer smaller or equal to x. Therefore, the constraint on the number of 1’s in can also be satisfied. One good candidate for 0 would be a uniformly random sampling matrix, as such matrix exhibits large spectral gap with high probability [76]. Multiple trials with different 0 ’s can be used to further improve the choice of . However, the search space is very large since the cardinality of ℘(0 ) is Mr,R ! L!, where operator ! denotes the factorial. One can reduce the search space as follows [36]: min tr (T Q) ≡ tr (QT ) s.t. ∈ ℘r (0 ),

(10.46)

where ℘r (0 ) denotes the set of matrices obtained by arbitrary row permutations. The search space in (10.46) equals Mr,R !, i.e., the cardinality of ℘r (0 ), which is greatly reduced compared to that in (10.45). Furthermore, the following proposition shows that such reduction of search space comes without any performance loss. proposition 10.6 For any 0 , searching for an in ℘r (0 ) can achieve the same EIP as searching in ℘(0 ). Proof We can prove the proposition by showing that the EIP achieved by any 1 ∈ ℘(0 ) can also be achieved by a certain 2 ∈ ℘r (0 ). For the pair (1,{R xl }), the ˜ xl }), where same EIP can be achieved by the pair (2,{R • •

2 is constructed by performing on 0 the row permutations performed from 0 to 1 , and ˜ xl } is a permutation of {R xl } according to the column permutations performed {R from 0 to 1 .

Cooperative Spectrum Sharing

303

In other words, the column permutations on is unnecessary because {R xl } will be automatically optimized to match the column pattern of . The claim is proven. The problem in (10.46) aims to find the best one-to-one match between the rows of 0 and the rows of Q. Let us construct a cost matrix C r ∈ RMr,R ×Mr,R with [C r ]ml 0m· (Ql· )T . The problem turns out to be a linear assignment problem with 3 ) using the cost matrix C c , which can be solved efficiently in polynomial time O(Mr,R Hungarian algorithm [79].

The Alternating Iteration with Respect to For the optimization of with fixed {R xl } and , the constraint in (10.39a) is noncon¯ is given as vex with respect to . The first order Taylor expansion of C(R xl ,) at $ % ¯ − tr Al ( − ) ¯ , C(R xl ,) ≈ C(R xl , )

(10.47)

where Al is defined as Al − =

∂C(R xl ,) ∂Re()

H GH 1 [(G1 G1

T

+

¯ = 2 −1 σC I )

2 H −1 ¯. − (G1 GH 1 + σC I + H R xl H ) ]G1 =

(10.48) The sequential convex programming technique is applied to solve by repeatedly solving the following approximate optimization problem (P ) max 0

tr (D t ) , tr (C t ) + ρ

˜ s.t. tr () ≤ PR /L,tr (A) ≤ C,

(10.49)

tr (V k ) ≥ ξtr () ,∀k ∈ N+ K, where C˜ =

L $

% ¯ + tr (A ¯ l) − C , C(R xl , )

l=1

A=

L

Al ,

(10.50)

l=1

ρ=

L % $ 1 2 tr R xl GH 2 l G2 + σR . pLMr,R l=1

¯ is updated as the solution C˜ and ρ are real positive constants with respect to , and of the previous repeated problem. Problem (10.49) could be equivalently formulated as a semidefinite programming problem (SDP) via the Charnes–Cooper transformation [74,80].

304

Li and Petropulu

Algorithm 2 The overall spectrum sharing algorithm. 1: 2: 3: 4: 5: 6: 7: 8:

2 ,δ Input: D t ,C t ,H,G1,G2,PC/R ,C,σC/R 1 PR Initialization: = LM I , = 0 ; t,R repeat Update {R xl } by solving (PR ) using Algorithm 1 with fixed and ; Update by solving (10.46) with fixed {R xl } and ; Update by solving a sequence of approximated SDP problem (10.49) with fixed {R xl } and ; until ESINR increases by √ amount smaller than δ1 Output: {R xl },,P = L1/2

max

˜ 0,φ>0

˜ t ), tr (D

˜ t ) = 1 − φρ s.t. tr (C ˜ ˜ ˜ ≤ φPR /L,tr (A) ≤ φ C, tr () % $ ˜ k − ξI ) ≥ 0,∀k ∈ N+ . tr (V

(10.51)

K

∗

˜ ,φ∗ ), can be obtained by using any The optimal solution of (10.51), denoted by ( 2 )3.5 ). standard interior point method–based SDP solver with a complexity of O((Mt,R ∗ ˜ /φ∗ . In each alternating iteration with respect The solution of (10.49) is given by to , it is required to solve several iterations of SDP due to the sequential convex programming. It is easy to show that the objective function, i.e., ESINR, is nondecreasing during the alternating iterations of {R xl }, and , and is upper bounded. According to the monotone convergence theorem [81], the alternating optimization is guaranteed to converge. The cooperative spectrum sharing algorithm in the presence of clutter maximizing the effective radar SINR is summarized in Algorithm 2.

10.4.2

Insight on the Feasibility and Solutions of the Spectrum-Sharing Problem In this subsection, we provide some key insights on the feasibility of (P1 ) and the rank of the solutions obtained by Algorithm 2.

Feasibility A necessary condition on C for the feasibility of (P1 ) with respect to {R xl } is C ≤ Cmax (PC ) where L 1 log2 I + σC−2 H R xl H H , {R xl }0 L

Cmax (PC ) max

l=1

s.t.

L l=1

tr (R xl ) ≤ PC .

(10.52)

Cooperative Spectrum Sharing

305

The above optimization problem is convex and has a closed-form solution based on water-filling [54]. It can be shown that Cmax (PC ) is essentially the largest achievable communication rate when there is no interference from radar transmitters to the communication receivers. Note that C = Cmax (PC ) will generate a nonempty feasible set for {R xl } only if G1 GH 1 = 0 (omitting the trivial case = 0), i.e., the radar transmits in the null space of the interference channel G1 to the communication receivers. A necessary condition on ξ for the feasibility of (P1 ) with respect to is ξ ≤ ξmax , where ξmax max

0,ξ≥0

ξ,

s.t. tr (V k ) ≥ ξtr (),∀k ∈ N+ K.

(10.53)

Note that the above optimization problem is independent of tr (). Without loss of generality, we assume that tr () = 1, based on which we have the following equivalent SDP formulation ξmax

max

0,ξ≥0

ξ, s.t. tr () = 1, tr (V k ) ≥ ξ,∀k ∈ N+ K.

(10.54)

It is easy to check that ξmax ≥ 1, which can be achieved by set (,ξ) to be (I /Mt,R ,1). The following proposition provides a sufficient condition for the feasibility of (P1 ). proposition 10.7 If C,ξ,PC > 0,PR > 0 are chosen such that C < Cmax (PC ) and ξ ≤ ξmax , then (P1 ) is feasible. Proof If C < Cmax (PC ), the feasible set for {R xl } determined by constraints in (10.39a) and (10.39b) F{Rxl } is nonempty as long as tr () is sufficiently small. If ξ ≤ ξmax , the feasible set for determined by constraints in (10.39c) F1 is nonempty and has no restriction on tr (). If ∈ F1 , then α ∈ F1,∀α > 0. The overall feasible set for , F , is the intersection of feasible sets determined by (10.39a), (10.39b), and (10.39c). F is nonempty as long as F1 and F{Rxl } are nonempty because we can choose any ∈ F1 and scale it down to make (P1 ) feasible. The claim is proven.

The Rank of Solutions We are also particularly interested in the rank of , obtained using Algorithm 2. Since the sequential convex programming technique is used for finding , it suffices to focus on the rank of the solution of (P ). To achieve this goal, we first introduce the following SDP problem ˜ min tr () s.t. tr (A) ≤ C, 0

tr (D t ) ≥ γ, tr (C t ) + ρ

tr (V k ) ≥ 0,∀k ∈

(10.55)

N+ K,

where γ is a real positive constant. The following proposition relates the optimal solutions of problems (10.49) and (10.55).

306

Li and Petropulu

proposition 10.8 If γ in (10.55) is chosen to be the maximum achievable SINR of (10.49), denoted as SINRmax , the optimal of (10.55) is also optimal for (10.49). Proof Denote ∗1 and ∗2 the optimal solutions of (10.49) and (10.55), respectively. It is clear that ∗1 is a feasible point of (10.55). This means that tr (∗2 ) ≤ tr (∗1 ) ≤ PR . Therefore, ∗2 is a feasible point of (10.49). It holds that SINRmax ≡

tr (∗1 D t ) tr (∗2 D t ) ≥ ≥ SINRmax . tr (∗1 C t ) + ρ tr (∗2 C t ) + ρ

(10.56)

It is only possible when all the equalities hold. In other words, ∗2 is optimal for (10.49). This completes the proof. In order to characterize the optimal solution of (10.55), we need the following key lemma: lemma 10.9 Matrix Al defined in (10.48) is positive semidefinite. In addition, A= L l=1 Al is also positive semidefinite. 2 Proof For simplicity of notation, we denote that X G1 GH 1 + σC I 0 and Y H H −1 −1 H R xl H 0. Let us rewrite Al as Al = G1 [X − (X + Y ) ]G1 . It is clear to see that Al is Hermitian because both X −1 and (X + Y )−1 are Hermitian. It is sufficient to show that Z X−1 − (X + Y )−1 is positive semidefinite. We have that

X−1 − (X + Y )−1 = X−1 Y (X + Y )−1,

(10.57)

which could be shown by right multiplying (X + Y ) on both sides of the equality. Since X, Y , and Z are Hermitian, we have Z = X −1 Y (X + Y )−1 = (X + Y )−1 Y X −1 .

(10.58)

Since (X + Y )−1 is invertible, there exists a unique positive definite matrix V , such that (X + Y )−1 = V 2 . Simple algebra manipulation shows that V −1 ZV −1 = (V −1 X−1 V −1 )(V Y V ) = (V Y V )(V −1 X−1 V −1 ),

(10.59)

i.e., V −1 ZV −1 is a product of two commutable positive semidefinite matrices V −1 X−1 V −1 and V Y V . Therefore, V −1 ZV −1 and thus Z is positive semidefinite. We prove that Al is semidefinite. Further, A is also semidefinte because it is the sum of L semidefinite matrices. Based on Lemma 10.9, we prove the following result by following the approach in [80]: proposition 10.10 Suppose that (10.55) is feasible when γ is set to SINRmax . Then, the following claims hold: 1) Any optimal solution of (10.55) has rank at most K. 2) All rank-K solutions ∗K of (10.55) have the same range space.

Cooperative Spectrum Sharing

307

3) Any solution ∗K − with rank less than K has range space such that Range (∗K − ) ⊂ Range (∗K ). 4) (10.49) and (10.51) always have solutions with rank at most K and with the same range space properties as that for (10.55). Proof Problem (10.55) is an SDP, whose Karush–Kuhn–Tucker (KKT) conditions [77] are given as + λ2 Dt +

K

νk V k = I + λ1 A + λ2 γC t +

k=1

K

νk ξI

(10.60a)

k=1

= 0

(10.60b)

0, 0,λ1 ≥ 0,λ2 ≥ 0,{νk } ≥ 0

(10.60c)

tr (D t ) ≥ γtr (C t ) + γρ tr (V k ) ≥ 0,∀k ∈

(10.60d)

N+ K,

(10.60e)

where 0,λ1 ≥ 0,λ2 ≥ 0, and {νk } ≥ 0 are dual variables. We can rewrite (10.60a) as follows K Rank () + Rank λ2 D t + νk v∗t (θk )vTt (θk ) ≥ Rank

k=1

I + λ1 A + λ2 γC t +

K

(10.61)

νk ξI .

k=1

2 ∗ T ∗ T Recall that D t = k σβ k vt (θk )vt (θk ). It holds that λ 2 D t + k νk vt (θk )vt (θk ) has rank at most K. Since A and C t are positive semidefinite, the matrix on right hand side of (10.61) has full rank. Therefore, Rank () is not smaller than Mt,R − K. From (10.60b) and (10.60d) we conclude that any optimal solution must have rank at most K. The second claim asserts that if there are multiple solutions with rank K, they all have the same range space. This can be proved using contradiction. Suppose that ∗1 and ∗2 are rank-K solutions of (10.55) and Range (∗1 ) != Range (∗2 ). Based on convex theory, any convex combination of ∗1 and ∗2 , saying ∗3 α∗1 + (1 − α)∗2,∀α ∈ (0,1), is also a solution of (10.55). However, ∗3 is with rank at least K + 1, which contradicts the fact that any solution must have rank at most K. The third claim could also be proved using contradiction. Suppose that ∗1 and ∗2 are respectively rank-K solution and solution with rank smaller than K, and Range (∗2 ) \ Range (∗1 ) is nonempty. Then any convex combination of ∗1 and ∗2 , saying ∗3 α∗1 + (1 − α)∗2,∀α ∈ (0,1), is also a solution of (10.55). However, ∗3 is again with rank at least K + 1, which contradicts the fact that any solution must have rank at most K. The last claim on the solutions of (10.49) and (10.51) follows from Proposition 10.8. Proposition 10.10 indicates that the rank of the optimal precoding matrix will not be larger than the number of the targets.

308

Li and Petropulu

10.4.3

Constant-Rate Communication Transmission for Spectrum Sharing Adaptive communication transmission for spectrum sharing methods involves high complexity. A suboptimal transmission approach of constant rate, i.e., R xl ≡ R x , ∀l ∈ N+ L , has lower implementation complexity. In such case, the spectrum sharing problem can be reformulated as (P#1 )

max

R x 0,0

ESINR# (R x ,,),

(10.62)

s.t. C(R x ,) ≥ C, Ltr (R x ) ≤ PC ,Ltr () ≤ PR , tr (V k ) ≥ 0,∀k ∈ N+ K, where ESINR# =

tr (D t ) % $ 2 tr (C t ) + tr G2 R x GH 2 /(pLMr,R ) + σR

(10.63)

and = L l=1 l is diagonal and with each entry equal to the number of 1’s in the corresponding row of . Similar techniques in Algorithm 2 can be used to solve (P#1 ). We can see that (P#1 ) has much lower complexity because there is only one matrix variable for the communication transmission. However, the drawback of the constantrate communication is that R x cannot adapt to the variation of the effective interference channel G2l . On the other hand, the adaptive communication transmission considered in (P1 ) can fully exploit the channel diversity introduced by the radar sub-sampling procedure. Another consequence is that the ESINR# depends on only through . Since is searched among the row permutations of a uniformly random sampling matrix, the number of 1’s in each row of is close to pL, or equivalently, will be very close to the scaled identity matrix pLI . To further reduce the complexity, the optimization with respect to in (P#1 ) is omitted because all row permutations of will result in a very similar ESINR# . From a different perspective, if the radar sub-sampling matrix is not available for the radar and communication cooperation, we can safely replace

with pLI in the ESINR# . The discussion in this paragraph asserts that, for the case of constant-rate communication transmission almost no performance degradation occurs due to the absence of the knowledge of .

10.4.4

Traditional MIMO Radars for Spectrum Sharing The traditional MIMO radars without sub-sampling can be considered as a special case with p = 1, and thus there is no need for the matrix completion. In such case, the constant-rate communication transmission becomes the optimal scheme because the interference channel G2 stays as a constant for the period of L symbol time due to

Cooperative Spectrum Sharing

309

the block fading assumption. The spectrum sharing problem has the same form as (P#1 ) with the objective function being SINR =

tr (D t ) . % $ 2 tr (C t ) + tr G2 R x GH 2 /Mr,R + σR

(10.64)

Note that SINR ≈ ESINR# because ≈ pLI . Therefore, traditional MIMO radars can achieve approximately the same spectrum sharing performance as MIMO-MC radars when the communication system transmits at a constant rate. However, for MIMO-MC radars, the adaptive communication transmission and the radar subsampling matrix can be designed to achieve significant radar SINR increase over the traditional MIMO radars. This advantageous flexibility is introduced by the sparse sensing (i.e., sub-sampling) in MIMO-MC radars. Performance results comparing MIMO-MC radars with different p values against the traditional MIMO radars are provided in Section 10.5.4.

10.5

Numerical Results In this section, we provide simulation examples to quantify the performance of the jointly designed spectrum-sharing method described in this chapter for the coexistence of the MIMO-MC radars and communication systems. Unless otherwise stated, we use the following default values for the system parameters. The MIMO radar system consists of collocated Mt,R = 16 TX and Mr,R = 16 RX antennas, respectively forming transmit and receive half-wavelength uniform linear arrays. The radar waveforms are chosen from the rows of a random orthonormal matrix [30]. We set the length of the radar waveforms to L = 16. The wireless communication system consists of collocated Mt,C = 4 TX and Mr,C = 4 RX antennas, respectively forming transmit and receive half-wavelength uniform linear arrays. For the communication capacity and power constraints, we take C = 16 bits/symbol and PC = 6400 (the power is normalized by the additive noise power). The radar transmit power budget is PR = 1000 × PC , which is typical for radar systems; high power is needed to to combat path loss associated with far-field targets [44]. The additive white Gaussian 2 = 1. There are three stationary targets with RCS variance noise variances are σC2 = σR 2 = 0.5, located in the far field with path loss of 30 dB. Clutter is generated by four σβ0

point scatterers, all having the same RCS variance, σβ2 ; the variance is determined by the

2 . Based on these numbers, the possible range clutter-to-noise ratio (CNR) 10 log σβ2 /σR of SNR at the communication receiver is between 12 dB and 26 dB, which is supported by LTE systems [82,83]. The radar power budget corresponds to a per-receive-antenna SNR of about 23 dB when only additive noise is considered. For a typical radar system with a single antenna, operating with probability of detection of 0.9 and probability of false alarm of 10−6 , the required SNR is about 13.2 dB [44]. However, the actual SNR may be much smaller because spatial degrees of freedom are used to mitigate clutter and interference from the communication systems.

310

Li and Petropulu

The channel H is modeled as Rayleigh fading, i.e., it contains independent entries, distributed as CN (0,1). The interference channels G1 and G2 are modeled as Rician fading. The power in the direct path is 0.1, and the variance of Gaussian components contributed by the scattered paths is 10−3 . The performance metrics considered include the following: • • • •

The radar effective SINR, i.e., the objective of the spectrum sharing problem; / F /MF , The matrix completion relative recovery error, defined as M − M / where M is the completed data matrix at the radar fusion center; The radar transmit beampattern, i.e., the transmit power for different azimuth angles vTt (θ)P v∗t (θ); The MUSIC pseudo-spectrum and the relative target RCS estimation RMSE / obtained using the least-squares estimation on the completed data matrix M.

Monte Carlo simulations with 100 independent trials are carried out to get an average performance.

10.5.1

The Radar Transmit Beampattern and the MUSIC Spectrum In this subsection, we present an example demonstrating the advantages of the above described jointly

designed radar precoding scheme as compared to uniform precodLPR /Mt,R I , and null space projection (NSP) precoding, i.e., P = ing, i.e., P =

H LPR /Mt,R V V , where V contains the basis of the null space of G1 [25]. For the joint design–based scheme of (10.39), we choose ξ = ξmax . The target angles with respect to the array are respectively −10◦ , 15◦ , and 30◦ ; the four-point scatterers are , and 45◦ . The CNR is 30 dB. In this simulation, the direct at angles −45◦ , −30◦ , 10◦√ ◦ path in G1 is generated as 0.1vt (φ)vH t (φ), where φ = 15 , with vt (φ) being defined in (10.2). In other words, the communication receiver is taken at the same azimuth angle as the second target. Recall that the NSP technique projects the radar waveform onto the null space of the interference channel G2 in order to avoid creating interference to the communication receiver. Because the null space and row space of a matrix are orthogonal to each other, there will be no radar power radiated along the null space of G2 , thus, targets in those locations will be missed. The precoding approach presented here does not suffer from such problem, because the precoding is computed via a joint design method instead of projecting to the null space of G2 . The radar transmit beampattern and the spatial pseudo-spectrum obtained using the MUSIC algorithm are shown in Figure 10.7. The achieved ESINR, MC relative recovery error, and relative target RCS estimation RMSE are listed in Table 10.1. We observe that the jointly designed precoding scheme achieves significant improvement in ESINR, MC relative recovery error, and target RCS estimation accuracy. As expected, the uniform precoding scheme just spreads the transmit power uniformly in all directions. The NSP precoding scheme achieves a similar beampattern as the uniform precoding scheme, with the exception of the deep null that the NSP places in the direction of the communication receiver. The null means that the transmit power towards the second target is severely attenuated and thus the

Cooperative Spectrum Sharing

311

Table 10.1 The radar ESINR, MC relative recovery errors, and the relative target RCS estimation RMSE for MIMO-MC radar and communication spectrum sharing.

Precoding schemes

ESINR

MC Relative Recovery Errors

Relative RCS Est. RMSE

Joint-design precoding

31.3 dB

0.038

0.028

−44.3 dB

1.00

1.000

NSP based precoding

−46.3 dB

1.00

0.995

Radar TX Beampattern (dB)

Uniform precoding

Jointly Designed Precoding Scheme Uniform Precoding Scheme Null Space Projection Scheme

60

40

20

0

–20 –50

0

50

0

50

Azimuth Angle

Spatial Spectrum in dB

0 –5 –10 –15 –20 –25 –30 –50

Azimuth Angle

Figure 10.7 The radar transmit beampattern and the MUSIC spatial pseudo-spectrum for MIMO-MC radar and communication spectrum sharing. The true positions of the targets and clutters are labeled using solid and dashed vertical lines, respectively.

probability of missing the second target is increased. We should note that neither the uniform nor the NSP precoding schemes have any capability of clutter mitigation. From Figure 10.7, we observe that the jointly designed precoding scheme successfully focuses the transmit power toward the three targets and nullifies the power toward the point scatterers. The three targets can be accurately estimated from the pseudo-spectrum obtained by the joint design. Meanwhile, the communication system can still achieve

312

Li and Petropulu

the required rate by aligning its transmission along a channel subspace that does not interfere with the radar emissions. This significant advantage is enabled by the joint design of radar and communication transmissions.

Comparison of Different Levels of Cooperation In this subsection, we compare several algorithms with different levels of radar and communication cooperation. The compared algorithms include: Uniform radar precoding and selfish communication: the radar transmit anten nas use the trivial precoding, i.e., P = LPR /Mt,R I ; and the communication system minimizes the transmit power to achieve certain average capacity without any concern about the interference it exerts to the radar system. This algorithm involves no radar and communication cooperation. NSP based radar precoding and selfish communication: the radar transmit anten nas use the fixed precoding, i.e., P = LPR /Mt,R V V H , while the selfish communication scheme is the same with the previous case. Uniform radar precoding and joint design of R xl and to minimize the effective interference to the radar receiver. Design of P and selfish communication: only the radar precoding matrix P is designed to maximize the radar ESINR. Joint design of P , R xl , and in (10.39).

•

•

• • •

We use the same values for all parameters as in the previous simulation except that the radar transmit power budget PR changes from 51,200 to 2.56 × 106 . Figure 10.8 shows

20 10

Uniform Precoding + Selfish Comm. NSP Precoding + Selfish Comm. Uniform Precoding + Design R xl&W Design P + Selfish Comm. Jointly Designed Precoding

0 –10

Relative RCS Estimation RMSE

Rada r T X P ower Budget P R

MC Relative Recovery Error

1 30

ESINR in dB

10.5.2

0.8 Uniform Precoding + Selfish Comm. NSP Precoding + Selfish Comm. Uniform Precoding + Design R xl&W

0.6

Design P + Selfish Comm. Jointly Designed Precoding

0.4

0.2

× 10 5

Radar TX Power Budget P R

× 10 5

1 0.8 0.6 0.4

Uniform Precoding + Selfish Comm. NSP Precoding + Selfish Comm. Uniform Precoding + Design R xl&W Design P + Selfish Comm. Jointly Designed Precoding

0.2

Radar TX Power Budget P R

× 10 5

Figure 10.8 Comparison of spectrum sharing with different levels of cooperation between the MIMO-MC radar and the communication system under different PR .

Cooperative Spectrum Sharing

313

the achieved ESINR, the MC relative recovery error, and the relative target RCS estimation RMSE. The algorithms that use trivial uniform and NSP-based radar precoding perform poorly because the point scatterers are not properly mitigated. The scheme that designs P only could mitigate the scatterers but the interference from the communication transmission is not controlled. The joint design of P , R xl , and simultaneously addresses the clutter and the mutual interference between the radar and the communication systems, and thus achieves the best performance amongst all the algorithms. The performance gains come from high-level cooperation between the two systems.

10.5.3

Comparison between Adaptive and Constant-Rate Communication Transmissions In this subsection, we evaluate the performance of two communication transmission schemes, namely, adaptive transmission with different R xl ’s for all l ∈ N+ L , and constant-rate transmission with R x across all pulses. We use the following parameter setting: Mt,R = 16,Mr,R = Mt,C = 8,Mr,C = 2, C = 10 bits/symbol, PC = 64 2 and and PR = 1000 × PC . For the G1 and G2 , Rayleigh fading is used with fixed σG 1 2 varying σG2 . The results of ESINR, MC relative recovery error and the relative target 2 are shown in Figure 10.9. The value RCS estimation RMSE for different values of σG 2 2 of σG2 varies from 0.05 to 0.5, which effectively simulates different distances between the communication transmitter and the radar receiver. It is clear that the adaptive communication transmission outperforms the constant-rate counterpart under various values of interference channel strength. As discussed in Section 10.4.3, the adaptive

Figure 10.9 Comparison of spectrum sharing with adaptive and constant-rate communication transmissions under different levels of variance of the interference channel from the communication transmitter to the radar receiver.

314

Li and Petropulu

communication transmission can fully exploit the channel diversity of G2l introduced by the radar sub-sampling procedure. The price for the performance advantages is high complexity. The average running times on a laptop with Intel Core i5 dual-core 2.4 GHz CPU for the adaptive and constant-rate communication transmissions are respectively 15.6 and 4.8 seconds. The choice between these two transmission schemes can be made depending on the available computing resources.

10.5.4

Comparison between MIMO-MC Radars and Traditional MIMO Radars In this subsection, we present a simulation to show the advantages of MIMO-MC radars compared to the traditional full-sampled MIMO radars. The parameters are the same as 2 = 0.3 and σ 2 = 1, which those in the simulation in Section 10.5.3, but with fixed σG G2 1 indicates strong mutual interference, especially interference from the communication transmitter to the radar receiver. The radar transmit power budget PR is taken to be equal to 10 × PC . We consider two targets; one is randomly located and the other is taken to be 25◦ away. We also consider 4 randomly located point scatterers. Figure 10.10 shows the results under different MIMO-MC sub-sampling rates p. Note that full sampling is used for the traditional MIMO radar. The MC relative recover error for the traditional radar is actually the output distortion-to-signal ratio. A smaller distortionto-signal ratio corresponds to a larger output SNR. For ease of comparison, a black dashed line is used for the traditional MIMO radar. We observe that the MIMO-MC radar achieves better performance in ESINR than the traditional radar. This is due to the fact that the communication system can effectively prevent its transmission from

Figure 10.10 Comparison of spectrum sharing with traditional MIMO radars and MIMO-MC

radars with different sub-sampling rates p.

Cooperative Spectrum Sharing

315

interfering the radar system when the number of actively sampled radar RX antennas is small, i.e., sub-sampling is small. In addition, the larger ESINR of the MIMO-MC radar results in a larger output SINR than that of the traditional radar. Furthermore, the MIMO-MC radar achieves better target RCS estimation accuracy than the traditional radar if its sub-sampling rate is between 0.4 and 0.7. For p larger than 0.7, the target RCS estimation accuracy achieved by the MIMO-MC radar is worse than that achieved by the traditional radar because small ESINRs for p ≥ 0.7 introduce high distortion in the completed data matrix. The results in Figure 10.10 could be used to help the selection of radar sub-sampling rate p. For the best target RCS estimation accuracy, p = 0.6 is the best choice, while for the biggest savings in terms of samples and similar performance as traditional radars, p = 0.4 is the best choice. Since there is no closed-form solution for the joint design problem, it is difficult to provide a theoretical justification. Based on these results, we conclude that MIMO-MC radars can coexist with communication systems and achieve better target RCS estimation than traditional radars, while saving up to 60% of data samples. Such significant advantage is introduced by sparse sensing (i.e., sub-sampling) in MIMO-MC radars, as discussed in Section 10.4.4.

10.6

Conclusions In this chapter, we have considered the coexistence of a MIMO-MC radar and a wireless MIMO communication system by sharing a common carrier frequency. The radar transmits random unitary waveforms, and both radar and communication systems use precoders. The precoders and the radar sub-sampling scheme have been jointly designed by the control center to maximize the radar SINR while meeting certain rate and power constraints for the communication system. Random unitary waveforms can be easily generated and updated for waveform security. We should note that the presented joint design–based spectrum sharing method can also be applied to traditional MIMO radars, which is a special case of MIMO-MC radars with 100% sub-sampling rate, i.e., p = 1. The jointly designed spectrum sharing scheme has been evaluated via extensive simulations. Specifically, we have shown that cooperative design brings a significant performance advantage as compared to noncooperative design. The jointly designed spectrum sharing scheme successfully focuses the transmit power towards the targets and nullifies the power towards the clutter. It achieves significant improvement in ESINR, MC relative recovery error, and target RCS estimation accuracy. We have also compared the performance and complexity of the adaptive and the constant-rate communication transmission schemes for radar-communication spectrum sharing. Finally, we have provided simulation-based comparison of MIMO-MC radars and traditional MIMO radars coexisting with communication systems. We have observed that the MIMO-MC radar achieves better performance in terms of ESINR and output SNR. Our simulations suggest that MIMO-MC radars can coexist with communication systems and achieve better target RCS estimation than traditional radars, while saving up to 60% in data samples. Of course these advantages come at increased computations for MC.

316

Li and Petropulu

We should note that the constraint requiring that the number of targets is smaller than the number of radar antennas results in an inefficient usage of the MIMO radar degrees of freedom. However, the high resolution of traditional MIMO radar is retained by MIMO-MC radars with a great reduction of sample and hardware complexity. The considered signal model is for narrow-band waveforms. Broadband MIMO systems typically use OFDM waveforms [16]. In such case, the joint design still applies on individual component carriers. This would substantially expand the application scenarios of the results presented in this chapter.

References [1] “Realizing the full potential of government-held spectrum to spur economic growth,” The Presidents Council of Advisors on Science and Technology (PCAST), technical report, July 2012. [Online]. www.dtic.mil/dtic/tr/fulltext/u2/a565091.pdf. [2] D. Cabric, I. D. O’Donnell, M. S. Chen, and R. W. Brodersen, “Spectrum sharing radios,” IEEE Circuits and Systems Magazine, vol. 6, no. 2, pp. 30–45, 2006. [3] “FCC proposes innovative small cell use in 3.5 GHz band,” Federal Communications Commission (FCC), news release, December 2012. [Online]. apps.fcc.gov/edocs_public/ attachmatch/DOC-317911A1.pdf. [4] G. Locke and L. E. Strickling, “An assessment of the near-term viability of accommodating wireless broadband systems in the 1675–1710 MHz, 1755–1780 MHz, 3500–3650 MHz, and 4200–4220 MHz, 4380–4400 MHz bands,” US Dept. of Commerce, the National Telecommunications and Information Administration, technical report TR-13-490, 2012. [5] E. Drocella, J. Richards, R. Sole, F. Najmy, A. Lundy, and P. McKenna, “3.5 GHz exclusion zone analyses and methodology,” US Dept. of Commerce, the National Telecommunications and Information Administration, technical report TR-15-517, 2015. [6] F. H. Sanders, R. L. Sole, B. L. Bedford, D. Franc, and T. Pawlowitz, “Effects of RF interference on radar receivers,” US Dept. of Commerce, the National Telecommunications and Information Administration, technical report TR-06-444, 2006. [7] A. Lackpour, M. Luddy, and J. Winters, “Overview of interference mitigation techniques between WiMAX networks and ground based radar,” in 20th Annual Wireless and Optical Communications Conference, April 2011, pp. 1–5. [8] F. H. Sanders, R. L. Sole, J. E. Carroll, G. S. Secrest, and T. L. Allmon, “Analysis and resolution of RF interference to radars operating in the band 2700–2900 MHz from broadband communication transmitters,” US Dept. of Commerce, the National Telecommunications and Information Administration, technical report TR-13-490, 2012. [9] M. R. Bell, N. Devroye, D. Erricolo, T. Koduri, S. Rao, and D. Tuninetti, “Results on spectrum sharing between a radar and a communications system,” in 2014 International Conference on Electromagnetics in Advanced Applications (ICEAA), 2014, pp. 826–829. [10] Q. Zhao and B. M. Sadler, “A survey of dynamic spectrum access,” IEEE Signal Processing Magazine, vol. 24, no. 3, pp. 79–89, May 2007. [11] E. Hossain, D. Niyato, and Z. Han, Dynamic Spectrum Access and Management in Cognitive Radio Networks. Cambridge University Press, 2009. [12] L. S. Wang, J. P. McGeehan, C. Williams, and A. Doufexi, “Application of cooperative sensing in radar-communications coexistence,” IET Communications, vol. 2, no. 6, pp. 856– 868, July 2008.

Cooperative Spectrum Sharing

317

[13] S. S. Bhat, R. M. Narayanan, and M. Rangaswamy, “Bandwidth sharing and scheduling for multimodal radar with communications and tracking,” in IEEE Sensor Array and Multichannel Signal Processing Workshop, June 2012, pp. 233–236. [14] R. Saruthirathanaworakun, J. M. Peha, and L. M. Correia, “Opportunistic sharing between rotating radar and cellular,” IEEE Journal on Selected Areas in Communications, vol. 30, no. 10, pp. 1900–1910, 2012. [15] S. C. Surender, R. M. Narayanan, and C. R. Das, “Performance analysis of communications & radar coexistence in a covert UWB OSA system,” in IEEE Global Telecommunications Conference, 2010, pp. 1–5. [16] S. Gogineni, M. Rangaswamy, and A. Nehorai, “Multi-modal OFDM waveform design,” in IEEE Radar Conference, April 2013, pp. 1–5. [17] A. Turlapaty and Y. Jin, “A joint design of transmit waveforms for radar and communications systems in coexistence,” in IEEE Radar Conference, 2014, pp. 0315–0319. [18] A. Aubry, D. M. A., M. Piezzo, and A. Farina, “Radar waveform design in a spectrally crowded environment via nonconvex quadratic optimization,” IEEE Transactions on Aerospace and Electronic Systems, vol. 50, no. 2, pp. 1138–1152, 2014. [19] A. Aubry, A. De Maio, Y. Huang, M. Piezzo, and A. Farina, “A new radar waveform design algorithm with improved feasibility for spectral coexistence,” IEEE Transactions on Aerospace and Electronic Systems, vol. 51, no. 2, pp. 1029–1038, April 2015. [20] K. Huang, M. Bica, U. Mitra, and V. Koivunen, “Radar waveform design in spectrum sharing environment: Coexistence and cognition,” in IEEE Radar Conference, 2015, pp. 1698–1703. [21] M. Bica, K. W. Huang, V. Koivunen, and U. Mitra, “Mutual information based radar waveform design for joint radar and cellular communication systems,” in 2016 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), March 2016, pp. 3671–3675. [22] S. Sodagari, A. Khawar, T. C. Clancy, and R. McGwier, “A projection based approach for radar and telecommunication systems coexistence,” in IEEE Global Telecommunication Conference, December 2012, pp. 5010–5014. [23] A. Babaei, W. H. Tranter, and T. Bose, “A practical precoding approach for radar/communications spectrum sharing,” in 8th International Conference on Cognitive Radio Oriented Wireless Networks, July 2013, pp. 13–18. [24] S. Amuru, R. M. Buehrer, R. Tandon, and S. Sodagari, “MIMO radar waveform design to support spectrum sharing,” in IEEE Military Communication Conference, November 2013, pp. 1535–1540. [25] A. Khawar, A. Abdel-Hadi, and T. C. Clancy, “Spectrum sharing between S-band radar and LTE cellular system: A spatial approach,” in IEEE International Symposium on Dynamic Spectrum Access Networks, April 2014, pp. 7–14. [26] C. Shahriar, A. Abdelhadi, and T. C. Clancy, “Overlapped-MIMO radar waveform design for coexistence with communication systems,” in IEEE Wireless Communications and Networking Conference, 2015, pp. 223–228. [27] A. Khawar, A. Abdelhadi, and T. C. Clancy, MIMO Radar Waveform Design for Spectrum Sharing with Cellular Systems: A MATLAB Based Approach. Springer, 2016. [28] H. Deng and B. Himed, “Interference mitigation processing for spectrum-sharing between radar and wireless communications systems,” IEEE Transactions on Aerospace and Electronic Systems, vol. 49, no. 3, pp. 1911–1919, July 2013.

318

Li and Petropulu

[29] A. Hassanien, M. G. Amin, Y. D. Zhang, and F. Ahmad, “Signaling strategies for dualfunction radar-communications: An overview,” IEEE Aerospace and Electronic Systems Magazine, vol. 31, no. 10, pp. 36–45, October 2016. [30] B. Li and A. P. Petropulu, “Spectrum sharing between matrix completion based MIMO radars and a MIMO communication system,” in IEEE International Conference on Acoustics, Speech and Signal Processing, April 2015, pp. 2444–2448. [31] B. Li, A. P. Petropulu, and W. Trappe, “Optimum co-design for spectrum sharing between matrix completion based MIMO radars and a MIMO communication system,” IEEE Transactions on Signal Processing, vol. 64, no. 17, pp. 4562–4575, September 2016. [32] B. Li and A. P. Petropulu, “Radar precoding for spectrum sharing between matrix completion based MIMO radars and a MIMO communication system,” in IEEE Global Conference on Signal and Information Processing, December 2015, pp. 737–741. [33] B. Li, H. Kumar, and A. P. Petropulu, “A joint design approach for spectrum sharing between radar and communication systems,” in IEEE International Conference on Acoustics, Speech and Signal Processing, March 2016, pp. 3306–3310. [34] B. Li and A. P. Petropulu, “MIMO radar and communication spectrum sharing with clutter mitigation,” in IEEE Radar Conference, May 2016, pp. 1–6. [35] B. Li and A. P. Petropulu, “Matrix completion based MIMO radars with clutter and interference mitigation via transmit precoding,” in 2017 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), March 2017, pp. 3216–3220. [36] B. Li and A. P. Petropulu, “Joint transmit designs for coexistence of MIMO wireless communications and sparse sensing radars in clutter,” IEEE Transactions on Aerospace and Electronic Systems, vol. 53, no. 6, pp. 2846–2864, December 2017. [37] S. Sun, W. Bajwa, and A. P. Petropulu, “MIMO-MC radar: A MIMO radar approach based on matrix completion,” IEEE Transactions on Aerospace and Electronic Systems, vol. 51, no. 3, pp. 1839–1852, July 2015. [38] E. J. Candès and Y. Plan, “Matrix completion with noise,” Proceedings of the IEEE, vol. 98, no. 6, pp. 925–936, June 2010. [39] D. S. Kalogerias and A. P. Petropulu, “Matrix completion in colocated MIMO radar: Recoverability, bounds and theoretical guarantees,” IEEE Transactions on Signal Processing, vol. 62, no. 2, pp. 309–321, Jan 2014. [40] C. Chen and P. P. Vaidyanathan, “Compressed sensing in MIMO radar,” in Asilomar Conference on Signals, Systems and Computers, 2008, pp. 41–44. [41] Y. Yu, A. P. Petropulu, and H. V. Poor, “MIMO radar using compressive sampling,” IEEE Journal of Selected Topics in Signal Processing, vol. 4, no. 1, pp. 146–163, February 2010. [42] M. G. Amin, Compressive Sensing for Urban Radar. CRC Press, 2014. [43] S. Sun and A. P. Petropulu, “Waveform design for MIMO radars with matrix completion,” IEEE Journal of Selected Topics in Signal Processing, vol. 9, no. 8, pp. 1400–1414, December 2015. [44] M. A. Richards, Fundamentals of Radar Signal Processing. McGraw-Hill, 2005. [45] H. Krim and M. Viberg, “Two decades of array signal processing research: The parametric approach,” IEEE Signal Processing Magazine, vol. 13, no. 4, pp. 67–94, 1996. [46] E. J. Candès and B. Recht, “Exact matrix completion via convex optimization,” Foundations of Computational Mathematics, vol. 9, no. 6, pp. 717–772, 2009. [47] K. Zyczkowski and M. Kus, “Random unitary matrices,” Journal of Physics A: Mathematical and General, vol. 27, no. 12, p. 4235, 1994.

Cooperative Spectrum Sharing

319

[48] B. Laurent and P. Massart, “Adaptive estimation of a quadratic functional by model selection,” Annals of Statistics, vol. 28, no. 5, pp. 1302–1338, 2000. [49] T. Jiang, “How many entries of a typical orthogonal matrix can be approximated by independent normals?” The Annals of Probability, vol. 34, no. 4, pp. 1497–1529, 2006. ˘ Zs ´ rules with regard to commercial operations in the [50] “Amendment of the commissionâA 3550–3650 MHz band,” Federal Communications Commission (FCC), technical report, April 2015. [Online]. https://apps.fcc.gov/edocs_public/attachmatch/FCC-15-47A1.pdf. [51] C. Kopp, “Search and acquisition radars (S-band, X-band),” Air Power Australia, technical report APA-TR-2009-0101, 2009. [Online]. www.ausairpower.net/APAAcquisition-GCI.html. [52] “Radar performance,” Radtec Engineering Inc., technical report, 2015. [Online]. http:// radar-sales.com/PDFs/Performance_RDR%26TDR.pdf. [53] T. Rappaport, Wireless Communications Principles and Practice. Prentice Hall, 2001. [54] D. Tse and P. Viswanath, Fundamentals of Wireless Communication. Cambridge University Press, 2005. [55] A. Goldsmith, Wireless Communications. Cambridge University Press, 2005. [56] R. P. Jover, “LTE PHY fundamentals,” technical report, 2015. [Online]. www.slideshare.net/ PrashantSengar/lte-phy-fundamentals-50510450. [57] “LTE in a nutshell: The physical layer,” Telesystem Innovations Inc., white paper, 2010. [58] J. G. Andrews, A. Ghosh, and R. Muhamed, Fundamentals of WiMAX: Understanding Broadband Wireless Networking. Prentice Hall, 2007. [59] B. Li and A. P. Petropulu, “Distributed MIMO radar based on sparse sensing: Analysis and efficient implementation,” IEEE Transactions on Aerospace and Electronic Systems, vol. 51, no. 4, pp. 3055–3070, October 2015. [60] M. Filo, A. Hossain, A. R. Biswas, and R. Piesiewicz, “Cognitive pilot channel: Enabler for radio systems coexistence,” in 2nd International Workshop on Cognitive Radio and Advanced Spectrum Management, May 2009, pp. 17–23. [61] R. Rogalin, O. Y. Bursalioglu, and H. Papadopoulos, “Scalable synchronization and reciprocity calibration for distributed multiuser MIMO,” IEEE Transactions on Wireless Communications, vol. 13, no. 4, pp. 1815–1831, 2014. [62] R. Zhang and Y. Liang, “Exploiting multi-antennas for opportunistic spectrum sharing in cognitive radio networks,” IEEE Journal of Selected Topics in Signal Processing, vol. 2, no. 1, pp. 88–102, February 2008. [63] R. Zhang, Y. Liang, and S. Cui, “Dynamic resource allocation in cognitive radio networks,” IEEE Signal Processing Magazine, vol. 27, no. 3, pp. 102–114, May 2010. [64] S. J. Kim and G. B. Giannakis, “Optimal resource allocation for MIMO ad hoc cognitive radio networks,” IEEE Transactions on Information Theory, vol. 57, no. 5, pp. 3117–3131, May 2011. [65] L. Lu, X. Zhou, U. Onunkwo, and G. Y. Li, “Ten years of research in spectrum sensing and sharing in cognitive radio.” EURASIP J. Wireless Comm. and Networking, vol. 2012, p. 28, 2012. [66] K. T. Phan, S. A. Vorobyov, N. D. Sidiropoulos, and C. Tellambura, “Spectrum sharing in wireless networks via QoS-aware secondary multicast beamforming,” IEEE Transactions on signal processing, vol. 57, no. 6, pp. 2323–2335, 2009. [67] H. Du and T. Ratnarajah, “Robust utility maximization and admission control for a MIMO cognitive radio network,” IEEE Transactions on Vehicular Technology, vol. 62, no. 4, pp. 1707–1718, 2013.

320

Li and Petropulu

[68] X. Hou and C. Yang, “How much feedback overhead is required for base station cooperative transmission to outperform non-cooperative transmission?” in 2011 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2011, pp. 3416–3419. [69] P. Stoica, J. Li, and Y. Xie, “On probing signal design for MIMO radar,” IEEE Transactions on Signal Processing, vol. 55, no. 8, pp. 4151–4161, 2007. [70] G. Cui, H. Li, and M. Rangaswamy, “MIMO radar waveform design with constant modulus and similarity constraints,” IEEE Transactions on Signal Processing, vol. 62, no. 2, pp. 343– 353, 2014. [71] R. Mudumbai, G. Barriac, and U. Madhow, “On the feasibility of distributed beamforming in wireless networks,” IEEE Transactions on Wireless Communications, vol. 6, no. 5, pp. 1754–1763, 2007. [72] C. Chen and P. P. Vaidyanathan, “MIMO radar ambiguity properties and optimization using frequency-hopping waveforms,” IEEE Transactions on Signal Processing, vol. 56, no. 12, pp. 5926–5936, 2008. [73] S. N. Diggavi and T. M. Cover, “The worst additive noise under a covariance constraint,” IEEE Transactions on Information Theory, vol. 47, no. 7, pp. 3072–3081, November 2001. [74] Z. Chen, H. Li, G. Cui, and M. Rangaswamy, “Adaptive transmit and receive beamforming for interference mitigation,” IEEE Signal Processing Letters, vol. 21, no. 2, pp. 235–239, February 2014. [75] C. Chen and P. P. Vaidyanathan, MIMO Radar Spacetime Adaptive Processing and Signal Design. John Wiley & Sons, 2008, pp. 235–281. [76] S. Bhojanapalli and P. Jain, “Universal matrix completion,” in Proceedings of The 31st International Conference on Machine Learning, 2014, pp. 1881–1889. [77] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge University Press, 2004. [78] R. G. Bland, D. Goldfarb, and M. J. Todd, “The ellipsoid method: A survey,” Operations Research, vol. 29, no. 6, pp. 1039–1091, 1981. [79] H. W. Kuhn, “The Hungarian method for the assignment problem,” Naval Research Logistics Quarterly, vol. 2, no. 1-2, pp. 83–97, 1955. [80] Q. Li and W.-K. Ma, “Optimal and robust transmit designs for MISO channel secrecy by semidefinite programming,” IEEE Transactions on Signal Processing, vol. 59, no. 8, pp. 3799–3812, 2011. [81] J. Yeh, Real Analysis: Theory of Measure and Integration. World Scientific, 2006. [82] “3GPP TS LTE evolved universal terrestrial radio access (E-UTRA) physical layer procedures,” 3rd Generation Partnership Project (3GPP), Technical Specification TS 36.213 V8.0, 2009. [83] M. T. Kawser, B. Hamid, N. Hasan, M. S. Alam, and M. M. Rahman, “Downlink SNR to CQI mapping for different multiple antenna techniques in LTE,” International Journal of Information and Electronics Engineering, vol. 2, no. 5, p. 757, 2012.

11

Compressed Sensing Methods for Radar Imaging in the Presence of Phase Errors and Moving Objects Ahmed Shaharyar Khwaja,∗ Naime Ozben Onhon, and Mujdat Cetin

11.1

Introduction and Outline of the Chapter Compressed sensing (CS) is a useful tool for processing sparse signals, i.e., signals that have a few large values and many small or zero values. In [1], the authors state that if an object has a sparse representation in a basis, a lesser number of nonadaptive measurements contains enough information to reconstruct the object via basis pursuit. A dictionary for sparse representation can be implicitly defined through a transform or explicitly defined through a collection of signals that can represent an object or a signal in a compressed form, e.g., a piece-wise smooth object can be represented by only a few coefficients after taking a wavelet transform, or a narrow-band signal can be described in terms of a few entries of a dictionary consisting of exponential signals with varying frequencies. In the former case, CS reconstruction can be described as a dictionary-less approach, whereas in the latter case, it can be described as a dictionary-based approach. In [2], the authors explain that a sparse or piece-wise constant signal can be recovered from a random number of its partial Fourier coefficients. Candès and Tao show in [3] that a measured coded signal corrupted by errors can be recovered using linear programming, provided the signal is sparse and there are certain constraints on the coding matrix. References [4,5] provide a review of CS, as well as identify potential application areas for data compression, channel coding, inverse problems, data acquisition, single-pixel imaging, etc. The purpose of radar imaging is to extract an underlying signal present in the electromagnetic data received by an antenna. The underlying signal can be the image of an area or of an object being observed by the antenna. In many cases, these signals are sparse, thus making them suitable for applications of CS. References [6] introduces a new approach to radar imaging based on CS where a low-dimensional, nonadaptive, linear projection is used to acquire an efficient representation of a compressible signal directly, using just a few measurements. The use of this approach can result in the design of new radar systems, where a matched filter is not needed at the receiver and the analog-to-digital conversion bandwidth can be reduced. Reference [7] demonstrates the advantage of CS for radar imaging by achieving higher resolution and artifact-free

∗ Ahmed Shaharyar Khwaja is supported by The Scientific and Technological Research Council of Turkey

(TUBITAK) - 2236 Co-Funded Brain circulation grant.

321

322

Khwaja, Onhon, and Cetin

images from randomly undersampled data. A new radar imaging system based on CS is presented in [8]. In this chapter, we present CS and its applications in radar imaging, specifically synthetic aperture radar (SAR) and inverse synthetic aperture radar (ISAR) imaging, focusing on scenarios involving unknown motion. This chapter is organized as follows: • •

•

•

We first provide the relevant mathematical expressions for CS and SAR, and then formulate the problem of CS SAR imaging. Thereafter, we consider the case where there are unknown motion errors present during the SAR acquisition process. The compensation of these errors blindly is called autofocus. We formulate the problem, present a general autofocus-based CS solution, and then review existing literature on this topic. In the next section, we formulate the problem of SAR moving target imaging, discuss the types of existing CS-based solutions and present a survey of existing methods in the literature. In the next section, we present CS ISAR imaging, followed by a literature review of this topic.

We would like to mention that in this chapter, all equations that represent functions with arguments show matrices or vectors obtained using each combination of values of their arguments, e.g., S(k,y) = exp{−j 2ky} represents a function. The arguments of this function are vectors k and y having lengths equal to Nt and Ny , respectively. This function results in the creation of a two-dimensional (2D) matrix S whose number of rows and columns are equal to Ny and Nt , respectively. The qth row of S is obtained by using all entries in k and the qth entry of y, i.e., exp{−j 2kyq }. All other equations not representing functions with arguments, e.g., z = α are obtained by following the normal matrix multiplication rules.

11.2

Compressed Sensing and Radar Imaging Compressed sensing has been shown to be a useful tool in applications of radar imaging [9,10]. Reference [11] demonstrates the advantage of CS for achieving higher resolution compared to traditional processing techniques. In [12] and [13], the authors validate the applications of CS with real data by showing that even after many missing samples in the raw data, an image can still be reconstructed without any loss of resolution. Examples of traditional reconstruction with full and limited data, and CS-based reconstruction using limited data, are shown in Figure 11.1. These examples have been taken from [9]. It can be observed that CS-based reconstruction can generate a high resolution image even from limited data. The concept of CS states that a sparse unknown signal can be recovered from incomplete sets of linear measurements by a specifically designed nonlinear recovery algorithm, offering the possibility of signal compression, hardware simplification, as well as emphasizing certain features in an image [14]. The CS theory mainly involves formulating a system model first and, subsequently, the incorporation of the model into a

Compressed Sensing, Radar Imaging, Phase Errors, and Moving Objects

(a)

(b)

323

(c)

Figure 11.1 The reconstructions of the Slicy target from the MSTAR data set. (a) The reference

image reconstructed from high-bandwidth data. (b) The conventional image reconstructed from limited-bandwidth data. (c) Image reconstructed from limited-bandwidth data using compressed sensing. (Taken from [9] with permission.)

least squares–based solution regularized by l0 -norm minimization to encourage sparse solutions. Recent work on compressed sensing has shown that to make the solution scalable for large problems, the l0 norm can be replaced by an lp -norm, with 0 < p ≤ 1 [15,16].

11.2.1

Compressed Sensing We first provide the basic expressions involved in CS. Let z be a signal that is sampled and y be the measured samples of the signal. The size of z is N × 1 and the size of y is K × 1, where K N . Let A be a K × N measurement matrix used to acquire the measured samples. In many cases, the signal z is compressible or sparse in some dictionary of dimensions N × M,M ≥ N . The signal z can be written as z = α. The measured samples y can then be written in terms of A and as y = Aα +

(11.1)

y = α + ,

(11.2)

or

where denotes noise and = A. If the number of measurements is less than the number of unknown variables, the recovery of α from y is hampered by the fact that there can be many solutions to this under-determined problem. However, if satisfies a property known as the restricted isometric property (RIP), α can be recovered by solving the following problem: minz y − α22 + λα0 .

(11.3)

The RIP means that the columns of should not be very similar, as this would ensure a sufficient number of linearly-independent measurements. The above problem is a combinatorial optimization problem and is solved by either using a relaxed version

324

Khwaja, Onhon, and Cetin

of the problem that replaces the l0 -norm with an lp -norm, e.g., with p = 1, or using greedy algorithms. The first term in (11.3) represents the closeness of the solution to the observations, while the second term represents a priori sparse information. The term λ is used to balance these two terms and is known as a hyper-parameter. Its choice can influence the solution. Note that (11.3) can also be obtained by assuming a Laplacian prior for α and Gaussian distribution for the noise, and subsequently solving for α using maximum a posteriori estimation.

11.2.2

Synthetic Aperture Radar Imaging A flat scene, divided into nr × ny scatterers in the range (across-track) and azimuth (along-track) directions, respectively, is considered. A SAR antenna moves along a ¯ certain path in azimuth direction with velocity V . The antenna emits a chirp pulse p(t) from every azimuth position, also called aperture position. % $ t ¯ = rect (11.4) p(t) cos 2πfc t + πKt 2 . Tp This pulse has a length Tp , a chirp-rate K, and t is the across-track time sampled at with Nt being the total number of acrossa frequency ft , i.e., t = 0, f1t , f2t ,. . ., Nft −1 t track time samples. The received pulse is a delayed version of the transmitted pulse and this delay is given by the two-way distance between the antenna and the scatterer. The total received raw data are given by an integration of the received pulse over the whole scene. However, as a discretized scene is considered, the raw data are considered as a sum of signals received from each of the scatterers observed by the antenna at different positions in the azimuth direction, and can be described by the following equation after demodulation: N

σ(rn,yn )Pn (t,y),N = nr × ny ,

(11.5)

Pn (t,y) = p (t − 2dn (y)/c) ,n = {1,2,. . .,N }

(11.6)

S(t,y) =

n=1

where

and y is the along-track distance sampled at a frequency fτ /V , i.e., y = 0, fVτ , 2V fτ ,. . ., V (Ny −1) . fτ

The total number of along-track time samples is given by Ny and

& t − 2dn (y)/c exp − j 4πfc (dn (y)/c) Tp ' +j πK(t − 2dn (y)/c)2 .

p (t − 2dn (y)/c) = rect

(11.7)

The backscattering coefficient of the nth scatterer is represented by σ(rn,yn ). These backscattering coefficients constitute a reflectivity map σ(r,y), whose discretized version is defined as follows:

Compressed Sensing, Radar Imaging, Phase Errors, and Moving Objects

N

(r,y) =

σ(rn,yn )δ(r − rn,y − yn ),

325

(11.8)

n=1

where δ(.,.) represent a 2D Dirac pulse. This reflectivity map represents the intensity of scene. The sensor-target distance is given as dn (y) =

each scatterer in an observed 2 2 rn + (y − yn ) , where rn = xn2 + H 2 is the slant-range, xn is the ground-range, and H is the height of the SAR antenna above the flat scene. Different SAR processing algorithms, e.g., the wavefront reconstruction algorithm [17], the omega-k algorithm [18], range-Doppler algorithm [19], or the chirp scaling algorithm [20] can be used to generate images from the raw data. The expression for raw data in a one-dimensional (1D) wavenumber domain calculated using the principle of stationary phase (POSP) [19] is given as

. σ(rn,yn ) exp −j 2k rn2 + (y − yn )2 , (11.9) S(k,y) = p( k) n

where p( k) is the Fourier transform (FT) of p(t), k is the range wavenumber, i.e.,

k = {−Nt /2×2πft /(Nt c),(−Nt /2+1)×2πft /(Nt c),. . .,(Nt /2−1)×2πft /(Nt c)}, k = kc + k, and kc = 2πfc /c.

11.2.3

Compressed Sensing SAR Imaging As a first instance, we note that the received raw data can be considered as a weighed sum of pulses Pn (.), given by (11.5). We reshape these pulses in one column of size Nt Ny × 1 using a function ξ(.) that converts a matrix into a column, i.e., pn = ξ(Pn (t,y)),n = {1,2,. . .,N }

(11.10)

and subsequently arranges them as different columns of a matrix of size Nt Ny × N as follows: = [p1,p2,. . .,pN ].

(11.11)

Similarly, converting the reflectivity as a column of size N × 1 as follows: σ = [σ1,σ2,. . .,σN ]T ,

(11.12)

where σn = σ(rn,yn ),n = {1,2,. . .,N}, we can write the received data s as a column of size Nt Ny × 1 as follows: s = σ + ,

(11.13)

where a nonzero value of the reflectivity leads to the selection of raw data of a column from . This selected column corresponds to the position of the nonzero value of the reflectivity. In case of data loss or undersampling in the azimuth and/or range, we can consider different rows or columns to be missing in , thus we can rewrite (11.13) as s = σ + ,

(11.14)

326

Khwaja, Onhon, and Cetin

Figure 11.2 Equivalence between CS formulation and SAR imaging.

where = A can be considered as an undersampled basis, which is equivalent to the undersampling operator/measurement matrix A multiplied by the basis. The undersampling operator A can be considered as a downsampling operator in the across-track direction or along-track direction. It can either represent an analog-to-digital converter for across-track sampling or a sampling strategy in along-track direction such that the RIP is satisfied. The measurement matrix has a size K × Nt Ny , where K < Nt Ny . This model is shown in Figure 11.2. The figure shows pulses being emitted and received at different positions in the along-track direction. The positions are given by the small circles placed vertically and the observed scene is represented by a big circle. The pulses varying according to radar-target distance form the dictionary. The received data consist of these pulses weighed by the reflectivity of the observed scene. The raw data are collected from all the positions in the along-track direction and then stored after analogto-digital conversion. To recover the reflectivity from this under-determined problem, the following conditions should be met: •

•

•

The reflectivity vector σ is sparse or compressible, which is a valid assumption in CS SAR imaging when there are a few strong reflecting points in a scene. In [10], the authors show the compressibility of scenes observed by SAR, which underlines the suitability of CS for solving SAR imaging problems. An example from [10] can be seen in Figure 11.3, where standard reconstruction is compared to CS reconstruction using p = 1 and p = 0.8. For a sparsity level sl , the number of linearly independent measurements M should be M = O(sl log2 (Nt Ny )) [21]. Normally, CS has been successfully applied to SAR scenes using around 50% of the measurements, which underlines the utility of CS applied to SAR imaging. The matrix should observe the RIP. This property is satisfied by random Gaussian matrices, Fourier matrices, chirp function matrices [22], etc. The latter two matrices are directly involved in CS SAR imaging and justify the use of CS for SAR imaging. In [23], the authors present examples of Fourier domain sampling strategies that acquire lower number of samples compared to those required by

Compressed Sensing, Radar Imaging, Phase Errors, and Moving Objects

(a)

(b)

327

(c)

Figure 11.3 SAR images of a vehicle from 360◦ aperture. (a) Standard Fourier image. Compressed sensing-based solution using lp -norm instead of l0 -norm in (11.3), where (b) p = 1 and (c) p = 0.8. (Taken from [10] with permission.)

(a)

(b)

(c)

(d)

Figure 11.4 Original resolution: 0.3 m: (a) Conventional image. (b) Compressed sensing reconstruction. Original resolution: 0.6 m: (c) Conventional image. (d) Compressed sensing reconstruction. (Taken from [14] with permission)

standard Nyquist sampling. The authors further demonstrate that the data obtained by these patterns can be used for successful CS SAR reconstruction. The conditions for CS reconstruction can be satisfied for the SAR imaging problem when there are a few strong scatterers embedded in a weak background. In this case, the reflectivity reconstruction can be carried out by solving the following minimization problem: σˆ = argmins − σ22 + λσ0 . σ

(11.15)

This equation can be solved using different recovery methods: linear programming [2], orthogonal matching pursuit (OMP) [24], iterative shrinkage/thresholding [25], modified quasi-Newton method [14], etc. The reconstructed reflectivity vector σˆ is then reshaped into a 2D form as follows: ˆ ˆ (r,y) = ξ −1 ( σ),

(11.16)

which shows the estimated value of the reflectivity for different range and azimuth positions. This estimation does not suffer from artifacts or reduction in resolution, which may arise if the undersampled data are used for reconstruction using traditional SAR imaging techniques. Examples of CS reconstruction are shown in Figure 11.4, taken from [14].

328

Khwaja, Onhon, and Cetin

11.3

Synthetic Aperture Radar Autofocus and Compressed Sensing In SAR imaging, the platform is assumed to be following a straight trajectory. However, that may not be the case in airborne SAR systems due to human errors or atmospheric turbulence. Similarly, in the case of satellite SAR systems, the ionosphere can cause a change in the propagation path, which results in degradation in image quality [26]. Such errors, if unaccounted for during SAR imaging, will lead to appearance of image artifacts in the processed images. A measure of the trajectory deviations of the platform is available from the onboard GPS that can be used to compensate for these errors in the acquired raw data [27]. However, in many cases these measurements are only available to a certain accuracy level, and residual motion errors still remain, even after carrying out traditional motion compensation using these measurements. This type of error causes phase errors in the SAR data, and the compensation of these unknown phase errors, known as autofocus, cause defocusing in the reconstructed image. The traditional autofocus techniques perform post processing, i.e., they use conventionally reconstructed [28,29] defocused images for phase error estimation. The best known of these techniques is phase gradient autofocus (PGA) [30], which estimates phase errors from the defocused images by isolating many single defocused targets via center-shifting and windowing operations. The aim of the windowing operation is to preserve the information contained in the blur footprints of the center-shifted targets and at the same time to leave out all of the contributions from other surrounding targets with weak reflectivities [30].

11.3.1

Synthetic Aperture Radar Autofocus The system model for SAR imaging can be modified to take into account these unknown motion errors. Assuming residual motion errors ren (y) in range direction for the nth scatterer, and ye (y) in the azimuth direction, the radar-target distance can now be written as . (11.17) den (y) = (rn − ren (y))2 + (y − yn − ye (y))2 . The presence of superscript n in ren (y) indicates that these errors are dependent on the range position of a scatterer, whereas the absence of n in ye (y) means that these errors are only dependent on the aperture positions. Please note that this model does not make a narrow-beamwidth assumption, but considers the variation of motion errors within the aperture for each point that is being imaged. Now the expression of raw data in the 1D wavenumber domain can be written as . (11.18) Sn (k,y) = p( k)σ(rn,yn ) exp(−j 2k (rn2 + (y − yn )2 ) Sen (k,y), where

Sen (k,y)

. . n 2 2 2 2 = exp −j 2k (rn − re (y)) − (y − yn − ye (y)) + rn + (y − yn ) (11.19)

is the error term due to the residual motion errors.

Compressed Sensing, Radar Imaging, Phase Errors, and Moving Objects

329

Considering the case where the residual motion errors do not create any significant major shift in the range position, we can write the motion errors as . . n n 2 2 2 2

se (kc,y) = exp −j 2kc (rn − re (y)) + (y − yn − ye (y)) − rn + (y − yn ) (11.20) and the raw data can be written as

$ % Sen (t,y) = σ(rn,yn )p t − 2den (y)/c sen (kc,y),

(11.21)

which differs from the raw data in case of no motion errors by the term sen (kc,y).

11.3.2

Synthetic Aperture Radar Autofocus in a Compressed Sensing Framework In [31], the authors investigate the effects of the phase errors caused by radar platform motion, in the CS reconstruction. The experimental results show that for quadratic and sinusoidal phase errors, the smearing in the defocused image reconstructed by CS is not symmetrical as in the conventional defocused SAR image due to the nonlinear effects of CS reconstruction. Therefore, when the PGA is applied to refocus the CS image, although the responses of prominent points can be refocused, some background noise remains. To take into account unknown motion errors, we define motion errors for the nth target as a diagonal matrix of size Nt Ny × Nt Ny , given as

n = diag[ sen (kc,y), sen (kc,y),. . ., sen (kc,y)].

(11.22)

We first consider the case where the raw data for the nth point can be isolated from total raw data. This is possible after the normal SAR imaging process that assumes a straight trajectory, as the point targets in the SAR image will be partially focused. Hence, a small patch around the partially focused target can be used to extract the target response and raw data can be generated from it using inverse processing [32]. In this case, the raw data can be written as s˜n = A n σ + ,

(11.23)

where s˜n is the raw data for the nth target of size Nt Ny × 1. The solution to the above problem can be obtained as follows: ˆ n = argmin˜sn − A n σ22 + λσ0, σˆ n, σ, n

(11.24)

where now the solution consists of estimating the reflectivity and motion errors for the nth target. Equation (11.24) can be solved using the method called sparsity-driven autofocus (SDA) proposed in [33]. This method solves for joint SAR imaging and phase-error correction. Phase errors are incorporated in the problem as model errors, and phase error correction is performed during the image formation process. Example results from [33] are shown in Figure 11.6. The proposed method handles the problem as an optimization problem in which the cost function is composed of a data fidelity term that depends on the phase errors and a regularization term, which is the l1 -norm of the reflectivity.

330

Khwaja, Onhon, and Cetin

The given cost function is minimized jointly with respect to the reflectivity and the phase error using coordinate descent technique. The algorithm is an iterative two-step algorithm, which cycles through steps of image formation and phase error estimation and compensation. 1.

In the first step, the cost function is minimized with respect to the reflectivity as follows: σˆ n = argmin˜sn − A n σ22 + λσ1 . σ, n

2.

(11.25)

In the second step, the estimated reflectivity is used to solve for the unknown motion errors. ˆ n = argmin˜sn − A n σ22 .

(11.26)

n

The above two steps are repeated until there is no significant change in the values of the estimates. The solution from this method is dependant on an appropriate choice of the variable λ. As the method does not require creating a dictionary of phase errors, it can be seen as a dictionary-less approach. Consider another case where the motion errors are not range- and azimuth position– dependent, but only depend on the aperture position, i.e., the trajectory errors are given only by ye (y). Given these motion phase errors as φe (yq ) for the qth aperture position, the problem can be formulated as s = A σ + ,

(11.27)

= diag[ se (kc,y), se (kc,y),. . ., se (kc,y)]

(11.28)

where

is a diagonal matrix of size Nt Ny × Nt Ny and

se (kc,y) = [exp{−j kc φe (y1 ), exp{−j kc φe (y2 ),. . ., exp{−j kc φe (yNy )}]

(11.29)

is a vector of size 1 × Ny . Figure 11.5 shows the system model for CS SAR autofocus. The figure shows that in the presence of trajectory deviations, the received data have a different form compared to the case of no trajectory deviations. This difference comes from a “perturbing” term . This term is a result of radar-target distance that is different in the presence of trajectory deviations compared to the radar-target distance in the absence of trajectory deviations. In this case, the solution can be written as follows: ˆ = argmins − A σ22 + λσ1 . ˆ σ,

σ,

(11.30)

This solution does not require making any sub-patches and can be obtained using the SDA method. Besides the SDA method, other methods have also been proposed to solve the CS SAR autofocus problem. A brief overview of some of these references is given in the following paragraphs. In [34], the authors propose a sparsity-based SAR autofocus (SBA) method that corrects phase errors within the image reconstruction process like the SDA. Actually,

Azimuth

Compressed Sensing, Radar Imaging, Phase Errors, and Moving Objects

331

y

Ψ den (Y) : radar-target distance

A

S = AΔΨΨσ

A/D

t Range Figure 11.5 CS SAR imaging in the presence of trajectory deviations.

(a)

(b)

(c)

Figure 11.6 Images reconstructed by (a) conventional imaging method, (b) sparsity-driven imaging method, (c) sparsity driven autofocus method. The reconstructed image obtained using sparsity-driven autofocus method is focused. This can be seen by highly localized four points in the center of the figure. The images obtained using conventional imaging and sparsity-driven imaging methods are defocused, as seen by multiple artifacts apparent throughout the corresponding figures. (Taken from [33] with permission)

both methods aim to solve approximately the same problem within a block relaxationbased framework. Different from the SDA, the proposed method incorporates an additional surrogate parameter of the SAR reflectivity field into the optimization problem to make the algorithm stable and to guarantee the convergence to an accumulation point or a connected set of accumulation points. Experimentally, it is also shown that this formulation results in a faster convergence. Reference [35] proposes a sparsity-based SAR imaging algorithm, called perturbed autofocus SAR (PA-SAR), which jointly solves for autofocus and off-grid target errors, i.e., for scatterers that are not on the discrete grid. The proposed algorithm uses a variant of the OMP algorithm called a parameter perturbation-based orthogonal matching pursuit (PPOMP) algorithm for efficient estimation of off-grid scatterer locations. The location of a target, which is not exactly on the grid, is described using the closest grid node coordinates with an additive unknown perturbation. Since off-grid targets with strong reflectivities may adversely effect the reconstruction of neighboring targets with weaker reflectivities, the off-grid oriented structure of PA-SAR provides an advantage in terms of the resolvability of these targets.

332

Khwaja, Onhon, and Cetin

Reference [36] uses a different sparse reconstruction approach called the expectation maximization-based matching pursuit (EMMP) algorithm [37] for solving the SAR autofocus problem. The EMMP algorithm treats the compressive measurements as incomplete data and constructs through iterative expectation and maximization (EM) steps the complete data corresponding to a set of SAR data for each strong target. The EM iterations provide more accurate and efficient estimation of the individual target parameters as well as enable the estimation of unknown phases for each complete data component. In conclusion, the proposed EMMP-based SAR imaging algorithm is described as being more greedy, computationally less complex, and having lower reconstruction errors compared to l1 -norm minimization. In [38], the authors compare the results of SDA [33], SBA [34], [36] EMMP with a parameter-free variant of EMMP called the OMP-based autofocus algorithm (AOMP), in terms of mean-square error (MSE), entropy, target-to-background ratio (TBR), and signal-to-noise ratio (SNR). Furthermore, these sparsity-based techniques are compared to the well-known PGA [30]. Comparison results show that all of the sparsity-based techniques form focused and sparse images with some slight qualitative variations. The performance of these techniques depends highly on the selection of the hyperparameters. Moreover it is shown that, in terms of phase error estimation performance, SBA, SDA, and AOMP work better compared to the PGA and EMMP. As expected, considering the run time, the performance of the PGA is much faster than the sparsitybased techniques. Reference [39] presents a method called autofocusing iteratively re-weighted augmented Lagrangian method (AIRWALM), which is based on an iteratively re-weighted augmented Lagrangian method (IRWALM) [40] and a sparsity-driven autofocus (SDA) method [33]. The proposed method optimizes over the reflectivities and phase errors jointly to solve a constrained formulation of the sparsity driven autofocus problem with an lp -norm, p ≤ 1 cost function. Instead of solving the unconstrained problem in (11.30), the AIRWALM solves the following problem: argmin σ,

p

σp

s.t. s − A σ2 ≤ .

(11.31)

The motivation for this problem formulation is that it is easier to determine the error bound , rather than choosing a regularization parameter. Moreover, the use of p-norms further enhances the sparsity, which may result in a better phase error estimation. This formulation offers a reduced computation time compared to the SDA algorithm. In [41], the authors propose a novel signal processing algorithm for joint SAR image formation and autofocus in a synthesis dictionary-based sparse representation framework. The proposed algorithm can be applied broadly to scenes that exhibit sparsity with respect to any dictionary. This is done by extending the SBA imaging framework from [34] to joint SAR image formation and autofocus. Phase error vector is estimated using a MAP estimator and compensated through an iterative algorithm to produce focused images.

Compressed Sensing, Radar Imaging, Phase Errors, and Moving Objects

333

The work in [42] uses a parametric sparse representation to compensate for platform motion errors in SAR that improves the imaging quality compared to other autofocus methods. The imaging consists of estimating the reflectivities of the scatterers, followed by calculating azimuth velocity errors and range acceleration errors.

11.4

Synthetic Aperture Radar Moving Target Imaging and Compressed Sensing Compressed sensing was initially applied to scenes that were assumed to contain only static targets. However, an observed scene can have moving targets such as vehicles, boats, etc. For this case assuming the absence of moving targets can lead to the appearance of artifacts in the processing results, similar to the case when a scene was processed without taking into account residual motion errors. Therefore, CS SAR processing has to be modified to take into account moving targets. In the following sections, we describe the moving target imaging problem, give details about CS SAR processing for moving targets, and review existing literature on this topic.

11.4.1

Synthetic Aperture Radar Moving Target Imaging The radar-target distance for an nth moving target having a constant radial/range velocity vnr and constant azimuth velocity vny can be modified as n (y) dm

=

&

rn − vnr

y '2 & y '2 + y − yn − vny V V

(11.32)

that can be approximated as & n 1− v n (y) ≈ rn − r (y − yn ) + dm V

' vny 2 (y V

− yn )2

2rn

.

(11.33)

The resulting raw data can then be written as 2d n (y) n Sm (t,y) = σ(rn,yn )p t − m . c

(11.34)

This movement of the target leads to two effects after processing: • •

A shift in azimuth position due to the range velocity vnr creating an azimuth vn wavenumber shift of 2kc Vr . A blurring effect caused by the azimuth velocity vny .

These effects can be further seen in Fig. 11.7, taken from [43], where two moving vehicles are shown. One of the vehicles moving in range direction is shifted from its original position on the road, while the second vehicle moving in azimuth direction is blurred.

334

Khwaja, Onhon, and Cetin

Figure 11.7 Effects of target motion on processed image. Image courtesy Artemis, Inc.

For multiple moving targets, the expression for raw data is given as a weighted sum of data from individual targets as follows: Sm (t,y) =

Nm

σ(rn,yn )Pmn (t,y),Nm = nr × ny × nvr × nvy ,

(11.35)

n=1

where

2d n (y) Pmn (t,y) = p t − m ,n = {1,2,. . .,Nm } c

(11.36)

and nvr and nvy are the total number of range and azimuth velocities considered. After processing, each target will undergo a different amount of azimuth shift and defocusing due to different velocities. These velocities should be taken into account during processing to compensate for these effects. Therefore, the velocities should be estimated first so that the estimated values could be used for focusing of the images.

11.4.2

Synthetic Aperture Radar Moving Target Imaging in a Compressed Sensing Framework Compressed sensing can provide a convenient way of estimation of motion parameters as well as the reflectivities and original positions of moving targets. Considering constant velocities, the reflectivity can be considered as a four-dimensional matrix

Compressed Sensing, Radar Imaging, Phase Errors, and Moving Objects

335

as follows: (r,y,vr ,vy ) =

Nm

σ(rn,yn,vnr,vny )δ(r − rn,y − yn,vr − vnr,vy − vny ).

(11.37)

n=1

Note that if range and/or azimuth acceleration is considered, the reflectivity will have higher than four dimensions. This matrix shows the reflectivity of a target corresponding to its range and azimuth positions, and having certain range and azimuth velocities. The matrix is rearranged in a vector of size Nm × 1 as follows: 1 2 Nm T σm = [σm ,σm ,. . .σm ] ,

(11.38)

n = σ(r ,y ,vn,vn ),n = {1,2,. . .,N }. where σm n n r y m Traditionally a fixed dictionary–based approach was used for CS-based moving-target imaging, where the dictionary would consist of data for all possible range and azimuth velocities for all the points in a considered scene. Stojanovic and Karl [44] use a fixed dictionary–based approach for moving target imaging and showed that CS can be used to estimate velocities and positions of moving objects considering a single scatterer in each observation pixel for mono- and multistatic SAR configurations. The authors considered a high signal-to-clutter ratio (SCR), i.e., a scenario in which the moving targets’ amplitude are much higher than the stationary targets. In [45], a fixed dictionarybased CS processing method is used to focus targets moving in range direction only, considering a low SCR, where a clutter cancellation filter was used to increase the SCR. A fixed dictionary is given as 1 2 Nm ,pm ,. . .pm ], m = [pm

(11.39)

n is the response of the nth moving target given as a N N × 1 vector: where pm t y n (y) 4πfc dm 2d n (y) 2 n = exp −j pm + j πK t − m ,n = {1,2,. . .,Nm }. (11.40) c c

Using the undersampling operator A, the data corresponding to moving targets can be expressed as sm = Am σm + .

(11.41)

The dictionary becomes very large if a large number of velocity parameters are considered. The OMP algorithm is typically used as recovery algorithm for motion parameter estimation due to low computational complexity and ease of implementation for reconstruction of reflectivities. Moreover, the process of selecting basis vectors one by one can be useful in cases where correlation exists amongst the basis vectors. The algorithm solves the following problem: σˆ m = argminsm − Am σm 22 + λσm 0 . σm

(11.42)

The steps involved in the OMP algorithm consist of iteratively: 1) correlating the received data with each column of the sub-sampled dictionary, 2) finding the column

336

Khwaja, Onhon, and Cetin

with the maximum correlation, 3) estimating the reflectivity using the selected column via a least-squares solution, and 4) removing the contribution of the estimation from the received data. The solution shows the amplitude for each combination of motion parameter values. The reconstructed reflectivity vector σˆ can be rearranged as a four-dimensional matrix as follows: ˆ m (r,y,vr ,vy ) = f −1 ( σˆ m ). (11.43) The four-dimensional matrix can be further divided into 2D reflectivity maps showing estimated reflectivity for range and azimuth positions corresponding to different values of velocities. Example results from [45] are shown in Figure 11.8. In [46], the authors apply the CS SAR imaging approach to estimate the target reflectivities and motion parameters of targets having rotational and vibrational motions based on fixed dictionaries. In [47], a fixed dictionary–based approach is used for indication of human motion with through-the-wall imaging. Different approaches were also suggested in the literature to reduce the dictionary size and enable the applications of CS SAR moving target imaging to realistic scenes. In [48], a fixed dictionary–based CS imaging approach is used to focus data moving in azimuth direction, where the range velocity is estimated before this step using a Radon transform approach. Using such an approach can help in decreasing the dictionary size as the fixed dictionary would consist of only azimuth velocities. The fixed dictionary– based method is also applied to motion parameter estimation and focusing of the processed images in [49], where the relative localization of the defocused targets provides an opportunity to generate the locations in the dictionary only around a limited region of the scene. Results obtained using this method are shown in Figure 11.9. References [49] and [50] examine the CS imaging performance in the presence of dictionary mismatch, i.e., when the discrete parameters used to generate the dictionary elements are different from the parameters in the data. In this case, the data are assumed # to be generated from a dictionary m , and hence the reconstruction is carried out as follows: #

#

σˆ m = argmins − Am σm 22 + λσ0 . σ

(11.44)

The reconstructed reflectivity in this case is different from the actual reflectivity and this difference depends on the correlation between the actual and the assumed dictionaries. The reconstructed reflectivity will become #

#

σˆ m = m m σ.

(11.45)

These references also showed that to deal with this problem an oversampled dictionary can be created, and the oversampling requirement is more severe for range velocity, i.e., the dictionary has to be created such that the range velocity spacings are very small. This can be seen in Figure 11.10, where MSE versus dictionary mismatch in positions and velocities can be seen. It can be noted that a small mismatch causes a sudden increase of MSE to a maximum value. The reason is that a small mismatch in range velocity causes the position of the reconstructed reflectivity to be different from

Compressed Sensing, Radar Imaging, Phase Errors, and Moving Objects

Scene containing Multiple Moving Points in a Pixel

Reconstructed Scene (Velocity = 3 m/s)

4

2.5 2

Amplitude

3

Amplitude

337

2

1

1.5 1 0.5

0 40

0 40

30

15 20

30

15

10 10 0

10

5 0

Azimuth Bins

Range Bins

(a) Simulated scene

0

Range Bins

(b) Reconstructed scene

Reconstructed Scene (Velocity = 4 m/s)

Reconstructed Scene (Velocity = 5 m/s)

3

2.5

2.5

2

Amplitude

Amplitude

10

5 0

Azimuth Bins

20

2 1.5 1

1.5 1 0.5

0.5 0 40

0 40 30

30

15 20

Azimuth Bins

15 20

10 10

10 10

5 0

0

0

Azimuth Bins

Range Bins

(c) Reconstructed scene

5 0

Range Bins

(d) Reconstructed scene Reconstructed Scene (Velocity = −3 m/s)

2.5

Amplitude

2 1.5 1 0.5 0 40 30

15 20

10 10

Azimuth Bins

5 0

0

Range Bins

(e) Reconstructed scene Figure 11.8 Simulated and reconstructed scene. The reconstructed scenes show estimated reflectivities for different values of velocities. (Taken from [45] with permission)

the actual position, as dictionary elements with different combinations of range velocity and azimuth positions can be highly correlated. The technique in [51] aims to recover moving targets up to a membership in an equivalence class. These equivalence motion classes involve different combinations of starting positions and velocities that correspond to the same SAR-to-target range history.

Khwaja, Onhon, and Cetin

Image containing two moving targets 1

Focussed Image at 7.9 m/s 1

0.9

100 200

Road

50

0.8

100

0.7

150

0.9 0.8

300

0.7

Azimuth bins

Azimuth bins

200 0.6

400

0.5 500 0.4 600

0.6

250 300

0.5

Road

350

0.4

400

0.3

0.3

700

450 0.2

0.2 500

800 0.1

0.1

550

900 2

4

6

8

10

12

14

16

18

600

0

20

2

4

6

8

10

12

14

16

18

20

0

Range bins

Range bins

(a) Difference of cross-polarized channels.

(b) Focused result at velocity 7.9 m/s.

Focussed Image at 8.9 m/s 0.45 50 0.4

100

0.35

150

Azimuth bins

200

0.3

250 0.25 300

Road

350

0.2

400

0.15

450 0.1 500 0.05

550 600

2

4

6

8

10

12

14

16

18

20

0

Range bins

(c) Focused result at velocity 8.9 m/s. Figure 11.9 Comparison of input image and images with retrieved motion parameters. (Taken from [49] with permission) Effects of Basis Mismatch 0.01 Mismatch in range velocity Mismatch in range position Mismatch in azimuth position Mismatch in azimuth velocity

0.009 0.008 0.007 0.006

MSE

338

0.005 0.004 0.003 0.002 0.001 0

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Mismatch

Figure 11.10 Mean-square error versus dictionary mismatch. (Taken from [49] with permission)

Compressed Sensing, Radar Imaging, Phase Errors, and Moving Objects

339

The proposed technique is based on minimization of a cost function to estimate an image of the stationary background and the equivalence motion classes. The sparsity of moving targets is incorporated into the cost function with an l1 -norm regularization term reflecting the assumption that the number of moving targets in the scene is relatively small compared to the stationary points. In [52], an off-grid CS method is applied to the problem of indication of ground moving targets from SAR images. The proposed method uses a variant of CS [53], called “continuous basis pursuit.” With the use of “continuous basis pursuit,” its aim is to reconstruct the exact velocity of every target by using tools such as Taylor expansion. Another approach is to use a dictionary-less approach, as presented in [54]. Here, it is assumed that either the moving targets have azimuth motion only, or any range motion has been compensated. The approach handles phase errors resulting from target motion as errors on the observation model of a static scene. The proposed method is an extension of the SDA algorithm for the moving target imaging problem. It is based on minimization of a cost function, which involves regularization terms imposing sparsity on the reflectivity field to be imaged, as well as on the spatial structure of the motionrelated phase errors, reflecting the assumption that only a small percentage of the entire scene contains moving targets. An advantage of this approach is that it can handle nonconstant velocities. In this approach, first, the CS SAR moving target imaging problem is formulated as s = A( m )σm + ,

(11.46)

where m is a matrix of size Nt Ny × Nm containing the phase errors due to motion and represents element-by-element multiplication. The matrix is given as m

m = [ ψ1m, ψ 2m,. . ., ψN m ],

(11.47)

where the individual vectors making up the matrix are

ψ nm = [exp{−j kc φen (y1 )},. . ., exp{−j kc φen (y1 )}, exp{−j kc φen (y2 )},. . ., exp{−j kc φen (y2 )}. . . exp{−j kc φen (yNy )},. . ., exp{−j kc φen (yNy )}]T . (11.48) In this case, the solution can be written as follows: σˆ m = argmins − A( m )σm 22 + λ1 σm 1 + λ2 β − 11,s.t.|β(k)| = 1∀k, σm,β

(11.49) where β T = [β1 T ,β2 T ,. . .,βNy T ]

(11.50)

and βn = [exp(−j kc φe1 (yn ), exp(−j kc φe2 (yn ),. . ., exp(−j kc φeNm (yn ))]T .

(11.51)

The extra term is used for facilitating the solution. As the term β represents the phase errors due to the moving points, and there are only a few moving points in a scene,the

340

Khwaja, Onhon, and Cetin

50

50

100

100

150

150

200

200

250

250

300

300

350

350

400

400

450

450

500

500 50

100

150

200

250

300

350

400

450

500

50

100

150

200

(a)

250

300

350

400

450

500

300

350

400

450

500

(b)

50

50

100

100

150

150

200

200

250

250

300

300

350

350

400

400

450

450 500

500 50

100

150

200

250

(c)

300

350

400

450

500

50

100

150

200

250

(d)

Figure 11.11 (a) Original scene. (b) Defocused image obtained using traditional processing. (c)

Image reconstructed by sparsity-driven imaging assuming a stationary scene. (d) Image reconstructed by the sparsity-driven moving object imaging approach. (Taken from [59] with permission)

term β−1 becomes sparse, and is hence represented as β−11 . An example of imaging carried out using this approach is shown in Figure 11.11. In [55], the authors extend the applicability of sparsity-driven moving target– focusing methods to very low signal-to-clutter ratio environments. First, SAR raw data are divided into subapertures in the azimuth direction. Subsequently, low-rank and sparse decomposition is applied using the multiple subapertures data to accomplish the separation of moving targets from the stationary SAR background. In [56], the authors use a variant of fractional FT as a basis to estimate the Doppler rate from undersampled data, and hence the azimuth velocity. They use Radon transform to estimate range velocity. In [57], moving targets in a processed imaged are refocused by first extracting a ROI sub-image and then following a two-step iterative procedure.

Compressed Sensing, Radar Imaging, Phase Errors, and Moving Objects

341

A sparse image is first obtained using an initial phase compensation parameter. Then the phase compensation parameter is estimated. In [58], the smeared images of moving targets are removed from a processed image to obtain a clear SAR image from multi-channel SAR images. The presented approach uses a fixed-dictionary CS-based method to separate SAR data from moving targets and noise. The fixed dictionary is based on frequency shifts for different range velocities and angle between flight path and a moving object that has dependency on both range and azimuth velocity of the moving target.

11.5

Inverse Synthetic Aperture Radar Imaging and Compressed Sensing Inverse synthetic aperture radar imaging consists of a moving target observed by a static antenna. The target’s motion creates a synthetic aperture, which can be used to obtain a high resolution image. However, a target can have an unknown maneuvering motion, which is similar to the case of SAR imaging in the presence of residual motion errors. This unknown maneuvering motion needs to be estimated and compensated to obtain a high-resolution artifact-free image. The moving target in a static background can be considered as a sparse scene, therefore, CS can be applied for ISAR imaging. In the following, we formulate the problem of ISAR imaging and CS application for ISAR imaging, followed by a literature review of existing CS ISAR imaging approaches.

11.5.1

Inverse Synthetic Aperture Radar Imaging A moving object is considered to have a range velocity vnr and a rotational motion given by an angle θn (τ) as a function of along-track time τ. The azimuth time-varying radar-target distance dn (τ) for the n-th point moving with the range-velocity vnr and rotating with the angle θn (τ) is given as follows [60]: dn (τ) = rn + vnr τ + xn cos( θn (τ)) + yn sin( θn (τ)),

(11.52)

where τ is the along-track time sampled at a frequency fτ , i.e., τ = 0, f1τ , f2τ ,. . ., Nfτ −1 , τ with Nτ being the total number of along-track time samples. The slant-range distance to the rotation center of the observed moving point is represented by rn ; xn and yn denote the range and azimuth distance of the point from the rotation center, respectively. The rotational motion is responsible for creating a high-resolution in the along-track direction. Note that it is normally assumed that the whole object is rotating with the same rotation angle, i.e., θn (τ) = θ0 (τ)∀n. Generally, ISAR imaging problems consider range-compressed and range-aligned ISAR data, where any movement across range cells due to translational motion is assumed to be eliminated. Therefore, the data are aligned along the azimuth direction for each range bin. Such data, corresponding to a range bin and a point n, can be written as sn (τ) = σn pn (τ).

(11.53)

342

Khwaja, Onhon, and Cetin

The reflectivity of the point n is given by σn , and pn (τ) describes the range-aligned signal based on the rotational motion of the target as follows: % $ τ (11.54) pn (τ) = rect exp −j kc dan (τ) , Tτ & ' where rect Tττ is a limit on the pulse size according to pulse-width Tτ arising from the total target observation time in azimuth direction, and dan (τ) = xn cos( θn (τ)) + yn sin( θn (τ)) is the radar-target distance after range-alignment. The total rangecompressed and aligned raw data for a total number of Nm moving points in a single range bin can then be written as a sum of received data from each point as follows: s(τ) =

Nm

σn pn (τ).

(11.55)

n=1

The purpose of ISAR imaging is to obtain a focused image for each range bin from the raw data. For ISAR, if the rotational motion is not properly accounted for in the imaging process, it will result in blurring in the processed image in the along-track direction. Therefore, the rotational rate should be estimated and compensated to get a focused image.

11.5.2

Inverse Synthetic Aperture Radar Moving Target Imaging in a Compressed Sensing Framework Compressed sensing can be used to estimate motion parameters as well as the reflectivities and original positions of moving targets for ISAR, similar to SAR moving target imaging. Similar to traditional CS SAR moving target imaging, a fixed dictionary-based approach was used initially for CS ISAR imaging. To carry out ISAR imaging using a fixed dictionary–based approach, a model for the rotational motion has to be assumed. First, it is assumed that cos( θn (τ)) ≈ 1 and sin( θn (τ)) ≈ θn (τ). This is the smallangle approximation and can be valid for high-frequency radar imaging [60]. Then,

θn (τ) is further approximated. In order to take into account realistic highly maneuvering motion, θn (τ) can be approximated until a third-order term, as in [61]. Accordingly, the time-varying rotationangle is given as 1 1

θn (τ) ≈ yn ωn τ + yn ω˙ n τ 2 + yn ω¨ n τ 3, 2 6

(11.56)

where ωn , ω˙ n , and ω¨ n are the rotational rate, rotational acceleration, and the rate of rotational acceleration, respectively. Let α n = kc yn ωn be defined as the rotation rate phase term. It can be further written as αn = 2πdn f , where dn gives the azimuth frequency/Doppler pixel position of the scatterer corresponding to the phase term αn , and f = Nfττ is the frequency resolution. The goal of CS ISAR imaging is to estimate dn to form a processed image showing the position of the scatterer in the azimuth frequency/Doppler location. The rotational acceleration phase term is further defined

Compressed Sensing, Radar Imaging, Phase Errors, and Moving Objects

343

as βn = k2c yn ω˙ n , and γn = k6c yn ω¨ n is defined as the rotational acceleration rate phase term. Then, the pulse pn (τ) for a point n can be redefined as ' & τ (11.57) pn (τ,α n,βn,γn ) = rect exp −j αn τ − j βn τ 2 − j γn τ 3 . Tτ As in Section 11.4, a reflectivity vector σm of size Nm × 1 is defined such that its n = σ(α ,β ,γ ), n = {1,2,. . .,N }, where N = N × N × entries have the form σm n n n m m α β Nγ . The total number of considered points for rotational rate, rotational acceleration, and rotational acceleration rate are Nα,Nβ and Nγ , respectively. The dictionary is defined as Nm 1 2 3 m = [pm ,pm ,pm ,. . .,pm ], n is the response of the nth rotating target given as where pm ' & τ n pm exp −j αn τ − j βn τ 2 − j γn τ 3 ,n = {1,2,. . .,Nm }. = rect Tτ

(11.58)

(11.59)

The dictionary m has a size Nτ × Nm . Making use of the dictionary definition, the received raw data for one range bin can be written as s = Am σm + ε.

(11.60)

In Figure 11.12, the model based on (11.60) is illustrated. Equation 11.60 can be solved for each range bin using the fixed dictionary–based approach as follows: 2 σˆ m = argmins − Am σm 2 + λσm 0 . σm

(11.61)

The reconstructed reflectivity σˆ m for each range bin is then converted to a 1D form, which shows the reflectivity estimate versus the Doppler pixel position. In [61] and [62], it is further shown that to avoid the degradation of performance caused by dictionary mismatch, the fixed dictionary should be sufficiently upsampled in rotational acceleration and rotational acceleration rate. Figure 11.13 gives one such example, where the degradation of performance with respect to mismatch in rotational

Figure 11.12 Compressed sensing ISAR imaging.

Khwaja, Onhon, and Cetin

MSE vs. β and γ mismatch 1 0.9

MSE

344

1

0.8

0.8

0.7

0.6

0.6

0.4

0.5 0.4

0.2

0.3

0 5

0.2 0.95 0.7 0.45 0.2

0.2

γ mismatch

0.45

0.7

0.95

5 0.1

β mismatch

Figure 11.13 MSE versus β mismatch and γ mismatch. The circled region shows variation of MSE with fixed β mismatch and varying γ mismatch, and vice versa. (Taken from [61] with permission)

acceleration and rotational acceleration rate can be seen. The performance degradation is evaluated in terms of MSE between actual and reconstructed reflectivity in the presence of mismatch. Therefore, to avoid any performance degradation, the dictionary should be constructed with very fine spacing of rotational acceleration and the rotational acceleration rate. A parametric dictionary or a dictionary-less approach can also be used to avoid the high upsampling required for a fixed dictionary–based approach. In the parametric dictionary approach, the motion parameters of the moving targets are estimated as part of the solution directly. In this case, the reconstruction problem can be defined as 2 σˆ m = argmin s − Am (α,β,γ)σ2 + λσm 0, σm ,α,β,γ

(11.62)

where the term m (α,β,γ) now shows that dictionary elements are generated based on the motion model given in (11.56), and the parameters are found such that the resulting dictionary elements are well matched to the received raw data. One possible approach to solve this problem is as part of the OMP algorithm, where the dictionary elements are iteratively generated such that they are best matched to received data. This can be expressed as αˆ k , βˆ k , γˆk = argmax| < s,(Am (11.63) (α k ,β k ,γk )) > |, α k ,β k ,γk

where k is the iteration number and | < .,. > | represents correlation. An example of results obtained using a parametric dictionary-based approach is shown in Figure 11.14. This figure was taken from [61].

Compressed Sensing, Radar Imaging, Phase Errors, and Moving Objects

Comparison of reference image and reconstructed image

250

250

200

200

150

150

Azimuth

Azimuth

Reference image

100

100

50

50

10

20

30

40

50

Reference image Reconstructed image

60

10

20

Range

Comparison of reference image and reconstructed image

200

200

150

150

Azimuth

Azimuth

50

60

Comparison of reference image and reconstructed image 250

Reference image Reconstructed image

100

100

50

50

30

40

(b)

250

20

30

Range

(a)

10

345

40

50

60

Range

Reference image Reconstructed image

10

20

30

40

50

60

Range

(c)

(d)

Figure 11.14 (a) Reference image. Note that the reference and reconstructed images are

overlapped for comparing the reconstruction performance: a successful reconstruction will result in the overlap looking like the reference image, whereas an unsuccessful reconstruction will result in the overlap having points around the reference image. (b) Overlap of reference and reconstructed images using a parametric dictionary approach with 50% downsampled data. (c) Overlap of reference image and reconstructed image using a technique based on integrated high-order matched phase transform with 50% data; partial reconstruction can be seen as the technique was proposed for imaging with fully sampled data. (d) Overlap of reference image and reconstructed image using a dictionary without a rotational acceleration rate phase term with 50% data. The resulting reconstruction using this dictionary fails to generate a discernible image. (Taken from [61] with permission)

Other approaches in existing literature on CS ISAR imaging approximate θ(τ) until a first-order term, i.e.,

θ(τ) ≈ yn ωn τ,

(11.64)

where it is assumed that the target is moving with a uniform rotational velocity, or only such a small number of continuous samples in azimuth direction are acquired that the target can be considered to have uniform motion during the acquisition. Such an approximation can be seen as a small-angle approximation, or a uniform rotation

346

Khwaja, Onhon, and Cetin

approximation. The focusing can be carried out using a FT as part of a range-Doppler approach (RDA), which results in a low resolution image. Compressed sensing can be used to improve resolution in such a case. The pulse making up the dictionary is given as

n pm

τ = rect Tτ

exp (−j α n τ) ,n = {1,2,. . .,Nα }.

(11.65)

The work in [63] proposes a 2D CS-based reconstruction for a target containing translational motion and a small-angle rotation with limited motion, by decoupling translational and rotational motions and carrying out CS imaging in range and azimuth directions separately. The work in [64] shows that the CS formulation directly in 2D instead of the traditional 1D stacking approach can generate high-resolution images when data have a few missing samples, or is gapped with a large number of missing samples. The authors assume small rotation and a dictionary based on Fourier matrices. The work in [65] applies Bayesian CS and uses a parametric dictionary for CS ISAR imaging in the presence of uniform rotation. The work in [66] uses a 2D CS ISAR imaging that considered uniform rotational motion, and uses dictionaries based on nonuniform Fourier transform for both range and azimuth directions. The work in [67] proposes CS ISAR image representation based on a sparsity prior and nonlocal total variation that reconstructs strong scatterers in an observed object and maintains the overall shape of the object, respectively. The authors assume a uniform rotation rate and a Fourier basis matrix. The work in [68] proposes sparse imaging to compensate for range migration in the presence of uniform rotational motion, making use of Bayesian sparse representation. The proposed imaging is divided into two steps: coarse imaging carried out based on the minimum entropy criterion, and a residual phase error correction performed using a 2D Fourier dictionary. The work in [69] proposes an autofocus-based technique to compensate for phase errors arising due to imperfect translational motion compensation based on sparse Bayesian learning. The target is assumed to be rotating uniformly, and these phase errors due to imperfect translational motion compensation could affect the focusing in azimuth direction. The technique is based on a two-step process where the parameters of the scatterer coefficient are calculated in the first step, and noise parameter and phase error due to imperfect translational motion is estimated in the second step. In [70], the author makes use of a segment from available data in the frequency domain and generates a prediction of what frequency domain data for other segments would look like. By comparison of the phases of the measurements and the predictions it should be possible to derive information about the motion compensation errors. Data are processed over small segments, such that the phase error history is assumed to be linear. The small-angle approach is simple in its implementation, but can result in a loss of discernibility of the image. Therefore, other approaches based on more complicated motion models were proposed. One of these approaches assumes a uniform rotation motion, but considers cos( θn (τ)) ≈ 1−( θn (τ))2 /2. These techniques are described

Compressed Sensing, Radar Imaging, Phase Errors, and Moving Objects

347

in [71–73], where the object is undergoing uniform rotational motion, resulting in a second-order phase term that is directly dependent on the first-order phase term

θn (τ) ≈ yn ωn τ −

xn 2 2 ω τ . 2 n

The dictionary is composed of pulses of the following form: % $ τ n exp − j α n τ − j ζn τ 2 ,n = {1,2,. . .,Nα × Nζ }, pm = rect Tτ

(11.66)

(11.67)

where ζn = k2c xn ωn2,n = {1,2,. . .,Nα × Nζ }. The authors use a parametric dictionary for reconstruction. Such a model also allows determination of the actual azimuth position of the moving targets. In [74], a similar motion model is used, and the phase error after translational motion compensation is considered. A two-step imaging process is used, where reflectivity is estimated first using compressed sensing reconstruction and starting with a phase error estimate, then the phase errors are estimated using entropy minimization method. In [68], the authors deal with translational motion phase error compensation and a target undergoing motion according to the model given in (11.70). The proposed imaging approach involves a two-step process: 1) A coarse error compensation is achieved using the minimum entropy criterion; 2) a sparsity-driven optimization using a 2D Fourier-based dictionary is carried out, where the residual phase errors are treated as model error and removed to achieve a fine correction. In other approaches, the rotation motion is assumed to consist of nonuniform rotation up to a second-order term as follows:

θn (τ) ≈ yn ωn τ +

yn ω˙ n τ 2 . 2

The dictionary is composed of the following terms: % $ τ n = rect exp − j α n τ − j βn τ 2 ,n = {1,2,. . .,Nα × Nβ }. pm Tτ

(11.68)

(11.69)

In [62], a dictionary is presented that can deal with both first-order rotational velocity and second-order rotational acceleration phase terms. An analysis is also carried out to quantify the effects of spacing of different parameters in the dictionary on the imaging performance. In [75], the authors consider rotational velocity and uniform acceleration. The work in [76] consider uniform and nonuniform rotational motion in the presence of unavailable or corrupted data and demonstrate the application of quadratic time frequency–based representations for CS imaging when rotational acceleration is present in the data. In [77], the authors consider rotational velocity and uniform acceleration. Most of the autofocus approaches lose their efficacy in subaperture cases, because the vacant apertures break the FT relationship between the range-compressed data and the RD image. Eigenvector-based autofocus is an exception. Rotational acceleration is compensated by searching, then the residual motion errors are calculated by eigenvector-based autofocus.

348

Khwaja, Onhon, and Cetin

The authors in [78] consider uniformly accelerated rotation targets. The maneuvering signal model is formulated as chirp code and represented using a chirp-Fourier basis. Then sparse representation is applied to realize range-Doppler imaging from the sparse apertures, where the superposition of chirp parameters is acquired using the modified discrete chirp Fourier transform (MDCFT). After preprocessing involving sample selection, rotation center determination, and noise reduction, the chirp parameters are used to estimate the parameters of rotational motion using the weighted least square (WLS) method. Finally, a high-resolution scaled-ISAR image is achieved by rescaling the acquired RD image using the estimated rotational velocity. In [79], the authors propose a weighted eigenvector-based phase correction method to correct for unknown phase errors, followed by using a partial Fourier matrix to achieve CS ISAR imaging. In [80], it is assumed that the translational motion compensation has been successfully accomplished and only rotational motion is considered. Rotational acceleration is considered and a dictionary based on scaled nonuniform Fourier transform is used. The reflectivity is modeled as a complex Gaussian distribution. In [81], the authors further extend the technique in [68] to the maneuvering target model undergoing uniform rotational acceleration. In this case, the rotational motion is approximated as yn xn (11.70)

θn (τ) ≈ yn ωn τ + ω˙n τ 2 − ωn2 τ 2, 2 2 and the dictionary is composed of % $ τ n (11.71) pm exp − j α n τ − j ηn τ 2 ,n = {1,2,. . .,Nα × Nη }, = rect Tτ where ηn = k2c (yn ω˙n − xn ωn2 ),n = {1,2,. . .,Nα × Nη }. Some other techniques do not assume any particular form of motion errors, and can deal with random phase errors. In [82], the authors propose an iterative two-step method: in the first step, an estimate of the reflectivity is obtained using an expansioncompression variance component–based method. In the second step, the phase errors are estimated making use of maximum likelihood method, similar to the method proposed in [33] for SAR autofocus. The whole process is repeated until no significant improvement can be seen. The method can deal with any type of motion errors. In [83], the authors propose a novel sparse Bayesian ISAR imaging algorithm with a newly proposed logarithmic Laplacian prior, which is achieved by putting a logarithm on the exponent of the Laplacian prior. Compared to the Gaussian scale mixture and Laplacian priors, the proposed logarithmic Laplacian prior has a narrower main lobe and higher tail values, and performs better on sparseness representation. Then, the logarithmic Laplacian prior-based ISAR image is reconstructed by MAP estimation, and arbitrary phase errors are estimated based on the minimum entropy criterion during the iterative process of sparse signal recovery. In [84], the authors impose continuity in range by imposing a Bayesian model, and propose an autofocus algorithm to compensate for random phase errors. In [85], the authors assume a target only having a range velocity and micro-Doppler motion. They use a joint sparsity model to separate micro-Doppler from the main target

Compressed Sensing, Radar Imaging, Phase Errors, and Moving Objects

349

body’s signal in the translational motion compensated data. Micro-Doppler is caused by nonstationary parts of moving targets and is represented by a sinusoidal phase in the azimuth direction, whereas the main body’s signal can be represented as a linear phase in the azimuth direction. Due to the linear phase, the main targets’ signals can be considered to be jointly sparse in the azimuth direction. The authors further assume a Bernoulli–Gaussian distribution for the signal in each azimuth bin. In [86], the authors propose CS to compensate for rotating parts in an observed object, which cause a micro-Doppler effect. A CS-based short-time Fourier transform is used to identify and remove data due to rotating parts from the entire data. The resulting data that do not contain the micro-Doppler effect are focused using CS and a dictionary that assumes uniform motion.

11.6

Conclusions In this chapter, we presented CS-based synthetic aperture radar autofocus, synthetic aperture radar moving target imaging, and inverse synthetic aperture radar imaging. We presented the theory behind each approach, formulated the problem mathematically, discussed general state-of-the-art solutions, and gave a review of existing approaches presented in the literature to deal with these problems. While CS autofocus approaches are usually dictionary-less approaches, SAR moving target imaging and ISAR imaging can be divided into fixed dictionary–based and parametric dictionary–based or dictionary-less approaches. The former class of approaches was used in initial work for SAR moving target imaging and ISAR imaging. These dictionaries are based on a fixed motion model and each entry of the dictionary is generated for one combination of the motion model considered. The parametric dictionary approach was used in later work and has the advantage of not being limited to a fixed motion model, and does not require the creation of a huge dictionary.

References [1] D. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory, vol. 52, no. 4, pp. 1289–1306, 2006. [2] E. Candès and T. Tao, “Decoding by linear programming,” IEEE Trans. Inf. Theory, vol. 51, no. 12, pp. 4203–4215, 2005. [3] E. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory, vol. 52, no. 2, pp. 489–590, 2006. [4] E. Candès and M. Wakin, “An introduction to compressive sampling,” IEEE Signal Process. Mag., vol. 25, no. 2, pp. 21–30, 2008. [5] R. Baraniuk, “Compressed sensing,” IEEE Signal Process. Mag., vol. 24, no. 4, pp. 14–20, 2007. [6] R. Baraniuk and P. Steeghs, “Compressive radar imaging,” in Proc. IEEE Radar Conference, 2007, pp. 128–133.

350

Khwaja, Onhon, and Cetin

[7] V. Patel, G. Easley, D. Healy, and R. Chellappa, “Compressed synthetic aperture radar,” IEEE J. Sel. Topics Signal Process., vol. 4, no. 2, pp. 244–254, 2010. [8] K. Aberman and Y. C. Eldar, “Sub-nyquist SAR via fourier domain range-doppler processing,” IEEE Trans. Geosci. Remote Sens., vol. 55, no. 11, pp. 6228–6244, 2017. [9] M. Cetin, I. Stojanovic, N. Onhon et al. “Sparsity-driven synthetic aperture radar imaging: Reconstruction, autofocusing, moving targets, and compressed sensing,” IEEE Signal Process. Mag., vol. 31, no. 4, pp. 27–40, 2014. [10] L. Potter, E. Ertin, J. Parker, and M. Cetin, “Sparsity and compressed sensing in radar imaging,” Proc. IEEE, vol. 98, no. 6, pp. 1006–1020, 2010. [11] M. Herman and T. Strohmer, “High-resolution radar via compressed sensing,” IEEE Trans. Signal Process., vol. 57, no. 6, pp. 2275–2284, 2009. [12] M. T. Alonso, P. Lopez-Dekker, and J. J. Mallorqui, “A novel strategy for radar imaging based on compressed sensing,” IEEE Trans. Geosci. Remote Sensing, vol. 48, no. 12, pp. 4285–4295, 2010. [13] X. Dong and Y. Zhang, “A novel compressive sensing algorithm for SAR imaging,” IEEE J. Select. Topics Appl. Earth Observ. Remote Sensing, vol. 7, no. 2, pp. 708–729, 2013. [14] M. Cetin and W. C. Karl, “Feature-enhanced synthetic aperture radar image formation based on nonquadratic regularization,” IEEE Trans. Image Processing, vol. 10, no. 4, pp. 623–631, 2001. [15] D. L. Donoho and M. Elad, “Optimally sparse representation in general (nonorthogonal) dictionaries via l1 minimization,” Proceedings of the National Academy of Sciences, vol. 100, no. 5, pp. 2197–2202, 2003. [16] D. M. Malioutov, M. Cetin, and A. S. Willsky, “Optimal sparse representations in general overcomplete bases,” Proc. IEEE International Conference on Acoustics, Speech, and Signal Processing, pp. 793–796, 2004. [17] M. Soumekh, Synthetic Aperture Radar Signal Processing. Wiley, 1999. [18] C. Cafforio, C. Prati, and F. Rocca, “SAR data focusing using seismic migration techniques,” IEEE Trans. Aerosp. Electron. Syst., vol. 27, no. 2, pp. 194–207, 1999. [19] I. Cumming and F. Wong, Digital Processing of Synthetic Aperture Radar Data. Artech House, 2005. [20] R. K. Raney, H. Runge, R. Bamler, I. Cumming, and F. Wong, “Precision SAR processing using chirp scaling,” IEEE Trans. Geosci. Remote Sens., vol. 32, no. 4, pp. 786–799, 1994. [21] R. Baraniuk, V. Cevher, M. Duarte, and C. Hegde, “Model-based compressive sensing,” IEEE Trans. Inf. Theory, vol. 56, pp. 1982–2001, 2010. [22] L. Applebaum, S. Howard, S. Searle, and R. Calderbank, “Chirp sensing codes: Deterministic compressed sensing measurements for fast recovery,” Appl. Comput. Harmon. Anal., vol. 26, pp. 283–290, 2009. [23] I. Stojanovic, M. Cetin, and W. C. Karl, “Compressed sensing of monostatic and multistatic SAR,” IEEE Geosci. Remote Sens. Lett., vol. 10, no. 6, pp. 1444–1448, 2013. [24] J. Tropp and A. Gilbert, “Signal recovery from random measurements via orthogonal matching pursuit,” IEEE Trans. Inf. Theory, vol. 53, no. 12, pp. 4655–4666, 2007. [25] M. Elad and M. Zibulevsky, “Iterative shrinkage algorithms and their acceleration for l1l2 signal and image processing applications,” IEEE Signal Process. Mag., vol. 27, no. 3, pp. 78–88, 2010. [26] D. P. Belcher and N. C. Rogers, “Theory and simulation of ionospheric effects on synthetic aperture radar,” IET Radar, Sonar & Navigation, vol. 5, no. 5, pp. 541–551, 2009. [27] G. Franceschetti and R. Lanari, Synthetic Aperture Radar Processing. CRC Press, 1999.

Compressed Sensing, Radar Imaging, Phase Errors, and Moving Objects

351

[28] W. G. Carrara, R. M. Majewski, and R. S. Goodman, Spotlight Synthetic Aperture Radar: Signal Processing Algorithms. Artech House, 1995. [29] J. Walker, “Range-doppler imaging of rotating objects,” IEEE Trans. Aerosp. Electron. Syst., vol. AES-16, pp. 23–52, 1980. [30] D. Wahl, P. Eichel, D. Ghiglia, and C. Jakowatz, “Phase gradient autofocus: A robust tool for high resolution SAR phase correction,” IEEE Trans. Aerosp. Electron. Syst., vol. 30, no. 3, pp. 827–835, 1994. [31] T. Jihua, S. Jinping, H. Xiao, and Z. Bingchen, “Motion compensation for compressive sensing SAR imaging with autofocus,” in Proc. 2011 6th IEEE Int. Conf. Ind. Electron. Appl., 2011, pp. 1564–1567. [32] A. S. Khwaja, L. Ferro-Famil, and E. Pottier, “Efficient SAR raw data generation for anisotropic urban scenes based on inverse processing,” IEEE Geosci. Remote Sens. Lett., vol. 6, no. 4, pp. 757–761, 2009. [33] O. Onhon and M. Cetin, “A sparsity-driven approach for joint SAR imaging and phase error correction,” IEEE Trans. Image Process., vol. 21, no. 4, pp. 2075–2088, 2012. [34] S. Kelly, M. Yaghoobi, and M. Davies, “Sparsity-based autofocus for undersampled synthetic aperture radar,” IEEE Trans. Aerosp. Electron. Syst., vol. 50, no. 2, pp. 972–986, 2014. [35] S. Camlica, A. C. Gurbuz, and O. Arikan, “Autofocused spotlight SAR image reconstruction of off-grid sparse scenes,” IEEE Trans. Aerosp. Electron. Syst., vol. 53, no. 4, pp. 1880– 1892, 2017. [36] S. Ugur, O. Arikan, and A. Gurbuz, “SAR image reconstruction by expectation maximization based matching pursuit,” Digital Signal Processing, vol. 37, pp. 75–84, 2015. [37] A. C. Gurbuz, M. Pilanci, and O. Arikan, “Expectation maximization based matching pursuit,” in Proc. IEEE International Conference on Acoustics, Speech and Signal Processing, 2012. [38] S. Camlica, H. E. Guven, A. C. Gurbuz, and O. Arikan, “Analysis of sparsity based joint SAR image reconstruction and autofocus techniques,” in Proc. 3rd International Workshop on Compressed Sensing Theory and its Applications to Radar, Sonar and Remote Sensing (CoSeRa), 2015 pp. 99–103. [39] A. Gungor, M. Cetin, and E. Guven, “An augmented lagrangian method for autofocused compressed SAR imaging,” in Proc. 2015 3rd IEEE Int. Workshop Compressed Sens. Theory Appl. Radar Sonar Remote Sensing, 2015, pp. 1–6. [40] H. E. Guven and M. Cetin, “An augmented lagrangian method for sparse SAR imaging,” in Proc. 10th European Conference on Synthetic Aperture Radar, 2014. [41] M. J. Hasankhan, S. Samadi, and M. Cetin, “Sparse representation-based algorithm for joint SAR image formation and autofocus,” Signal, Image and Video Processing, vol. 11, no. 4, pp. 589–596, 2015. [42] Y. C. Chen, G. Li, Q. Zhang, Q. J. Zhang, and X. G. Xia, “Motion compensation for airborne SAR via parametric sparse representation,” IEEE Trans. Geosci. Remote Sens., vol. 55, no. 1, pp. 551–562, 2017. [43] M. T. Crockett, “Target motion estimation techniques for single-channel SAR,” master’s thesis, Brigham Young University, 2014. [44] I. Stojanovic and W. Karl, “Imaging of moving targets with multi-static SAR using an overcomplete dictionary,” IEEE J. Sel. Topics Signal Process., vol. 4, no. 1, pp. 164–176, 2010.

352

Khwaja, Onhon, and Cetin

[45] A. S. Khwaja and J. Ma, “Applications of compressed sensing for SAR moving target velocity estimation and image compression,” IEEE Trans. Instrum. Meas., vol. 60, no. 8, pp. 2848–2860, 2011. [46] S. Zhu, A. M. Djafari, H. Wang et al., “Parameter estimation for SAR micromotion target based on sparse signal representation,” EURASIP J. Adv. Sig. Proc., vol. 2012, 2012. [47] F. Ahmad and M. Amin, “Through-the-wall human motion indication using sparsity-driven change detection,” IEEE Trans. Geosci. Remote Sens., vol. 51, no. 2, pp. 881–890, 2013. [48] Q. Wu, M. Xing, C. Qiu, B. Liu, Z. Bao, and T.-S. Yeo, “Motion parameter estimation in the SAR system with low PRF sampling,” IEEE Geosci. Remote Sens. Lett., vol. 7, no. 3, pp. 450–454, 2010. [49] A. S. Khwaja and X. P. Zhang, “Motion parameter estimation and focusing from SAR images based on sparse reconstruction,” IEEE Geosci. Remote Sens. Lett., vol. 11, no. 8, pp. 1350–1354, 2011. [50] A. S. Khwaja, M. Naeem, and A. Anpalagan, “Analysis of moving object imaging from compressively sensed SAR data in the presence of dictionary mismatch,” Int. J. Antenn. Propag., vol. 2013, 2013. [51] J. Gunther, J. Hunsaker, H. Anderson, and T. Moon, “Sparse reconstruction of equivalence classes of moving targets using single-channel synthetic aperture radar,” in Proc. IEEE International Conference on Acoustic, Speech and Signal Processing, 2014. [52] L. Prunte, “Gmti from multichannel SAR images using compressed sensing under off-grid conditions,” in Proc. International Radar Symposium, 2013. [53] C. Ekanadham, D. Tranchina, and E. P. Simoncelli, “Recovery of sparse translation-invariant signals with continuous basis pursuit,” IEEE Trans. on Signal Processing, vol. 59, no. 10, pp. 4735–4744, 2011. [54] N. O. Onhon and M. Cetin, “SAR moving target imaging in a sparsity-driven framework,” in Proc. SPIE Optics+Photonics, Wavelets and Sparsity XIV, 2011, pp. 8138–8139. [55] M. Yasin, M. Cetin, and A. S. Khwaja, “SAR imaging of moving targets by subaperture based low-rank and sparse decomposition,” in Proc. IEEE Signal Processing and Communications Applications Conference, 2017. [56] X. Zhang, G. Liao, S. Zhu, D.Yang, and W. Du, “Efficient compressed sensing method for moving-target imaging by exploiting the geometry information of the defocused results,” IEEE Geosci. Remote Sens. Lett., vol. 12, no. 3, pp. 517–521, 2015. [57] Y. Chen, Q. Zhang, G. Li, and J. Sun, “Refocusing of moving targets in SAR images via parametric sparse representation,” Remote Sensing, vol. 9, no. 8, pp. 1–15, 2017. [58] L. Prunte, “Compressed sensing for removing moving target artifacts and reducing noise in SAR images,” in Proc. European Conference on Synthetic Aperture Radar, 2016. [59] N. O. Onhon and M. Cetin, “SAR moving object imaging using sparsity imposing priors,” EURASIP J. Adv. Sig. Proc., vol. 2017, 2017. [60] V. Chen and H. Ling, Time-Frequency Transforms for Radar Imaging and Signal Analysis. Artech House, 2002. [61] A. S. Khwaja and M. Cetin, “Compressed sensing ISAR reconstruction considering highly maneuvering motion,” Electronics, vol. 6, no. 1, 2017. [62] A. S. Khwaja and X. P. Zhang, “Compressed sensing ISAR reconstruction in the presence of rotational acceleration,” IEEE J. Sel. Top. Appl. Earth Observ., vol. 7, no. 7, pp. 2957–2970, 2014. [63] Z. Liu, X. Wei, and X. Li, “Decoupled ISAR imaging using rsfw based on twice compressed sensing,” IEEE Trans. Aerosp. Electron. Syst., vol. 50, no. 4, pp. 3195–3211, 2014.

Compressed Sensing, Radar Imaging, Phase Errors, and Moving Objects

353

[64] S. Tomei, A. Bacci, E. Giusti, M. Martorella, and F. Berizzi, “Compressive sensing-based inverse synthetic radar imaging imaging from incomplete data,” IET Radar Sonar Navig., vol. 10, no. 2, pp. 386–397, 2016. [65] B. Wang, S. Zhang, and W. Q. Wang, “Bayesian inverse synthetic aperture radar imaging by exploiting sparse probing frequencies,” IEEE Antennas Wirel. Propag. Lett., vol. 14, pp. 1698–1701, 2015. [66] S. Li, G. Zhao, W. Zhang, Q. Qiu, and H. Sun, “ISAR imaging by two-dimensional convex optimization-based compressive sensing,” IEEE Sens. J., vol. 16, no. 19, pp. 7088–7093, 2016. [67] X. Zhang, T. Bai, H. Meng, and J. Chen, “Compressive sensing based ISAR imaging via the combination of the sparsity and nonlocal total variation,” IEEE Geosci. Remote Sens. Lett., vol. 11, no. 5, pp. 990–994, 2014. [68] G. Xu, M. Xing, X. G. Xia et al., “High-resolution inverse synthetic aperture radar imaging and scaling with sparse aperture,” IEEE J. Sel. Top. Appl. Earth Observ., vol. 8, no. 8, pp. 4010–4027, 2015. [69] L. Zhao, L. Wang, G. Bi, and L. Yang, “An autofocus technique for high-resolution inverse synthetic aperture radar imagery,” IEEE Trans. Geosci. Remote Sensing, vol. 52, no. 10, pp. 6392–6403, 2014. [70] J. Ender, “Autofocusing ISAR images via sparse representation,” in Proc. European Conf. on Synthetic Aperture Radar, 2012. [71] W. Rao, G. Li, X. Wang, and X.-G. Xia, “Adaptive sparse recovery by parametric weighted l1 minimization for ISAR imaging of uniformly rotating targets,” IEEE J. Select. Topics Appl. Earth Observ. Remote Sensing, vol. 6, no. 2, pp. 942–952, 2013. [72] G. Li, H. Zhang, X. Wang, and X. G. Xia, “ISAR 2-d imaging of uniformly rotating targets via matching pursuit,” IEEE Trans. Aerosp. Electron. Syst., vol. 48, pp. 1838–1846, 2012. [73] B. Jiu, H. Liu, H. Liu et al., “Joint ISAR imaging and cross-range scaling method based on compressive sensing with adaptive dictionary,” IEEE Trans. Antennas Propag, vol. 63, no. 5, pp. 2112–2122, 2015. [74] H. R. Hashempour and M. A. Masnadi-Shirazi, “Inverse synthetic aperture radar phase adjustment and cross-range scaling based on sparsity,” Digit. Signal Proc., vol. 68, pp. 93– 101, 2017. [75] G. Xu, M. Xing, and Z. Bao, “High-resolution inverse synthetic aperture radar imaging of manoeuvring targets with sparse aperture,” Electron. Lett., vol. 51, no. 3, pp. 287–289, 2015. [76] L. Stankovic, “ISAR image analysis and recovery with unavailable or heavily corrupted data,” IEEE Trans. Aerosp. Electron. Syst., vol. 51, no. 3, pp. 2093–2106, 2015. [77] L. Zhang, J. Duan, Z. Qiao, M. Xing, and Z. Bao, “Phase adjustment and ISAR imaging of maneuvering targets with sparse apertures,” IEEE Trans. Aerosp. Electron. Syst., vol. 50, no. 3, pp. 1955–1973, 2014. [78] G. Xu, M. Xing, L. Zhang, J. Duan, Q. Chen, and Z. Bao, “Sparse apertures ISAR imaging and scaling for maneuvering targets,” IEEE J. Sel. Topics Appl. Earth Observ. Remote Sens., vol. 7, no. 7, pp. 2942–2956, 2014. [79] J. Duan, L. Zhang, and M. Xing, “A weighted eigenvector autofocus method for sparseaperture ISAR imaging,” EURASIP J. on Adv. Sig. Proc., vol. 92, 2013. [80] G. Xu, L. Yang, L. Zhao, and G. Bi, “ISAR maneuvering targets imaging and motion estimation from parametric sparse Bayesian learning,” in Proc. IEEE International Geoscience and Remote Sensing Symposium, 2016.

354

Khwaja, Onhon, and Cetin

[81] G. Xu, L. Yang, G. Bi, and M. Xing, “Maneuvering target imaging and scaling by using sparse inverse synthetic aperture,” Signal Process., vol. 137, pp. 149–159, 2017. [82] W. Su, Y. Qin, H. Wang, and Q. Yang, “Joint ISAR imaging and phase error correction based on sparse bayesian learning,” Int. J. Sig. Proc. Sys., vol. 4, no. 6, pp. 487–493, 2016. [83] S. Zhang, Y. Liu, X. Li, and G. Bi, “Logarithmic Laplacian prior based Bayesian inverse synthetic aperture radar imaging,” Sensors, vol. 16, no. 5, 2016. [84] L. Zhao, L. Wang, G. Bi et al. “Structured sparsity-driven autofocus algorithm for highresolution radar imagery,” Signal Process., vol. 125, pp. 376–388, 2016. [85] L. Sun, X. Lu, and W. Chen, “Joint sparsity-based ISAR imaging for micromotion targets,” IEEE Geosci. Remote Sens. Lett., vol. 13, no. 11, pp. 1734–1738, 2016. [86] Q. Hou, Y. Liu, and Z. Chen, “Reducing micro-Doppler effect in compressed sensing ISAR imaging for aircraft using limited pulses,” Electron. Lett., vol. 51, no. 12, pp. 937–939, 2015.

Index

abiotic processing, stages of, 467–468 abiotic reactions, 430, 457, 467 abiotic sulfurization, 483–484 abyssal peridotite, 455–456 accretionary cycle, 277 accumulation curve, 631 acetate, in ultramaﬁc systems, 495 acetogenesis, 483 active volcanoes emissions from, 194–197, 216 temporal variability of, 208–209 adiabatic mantle, 166–167 afﬁnity, 590 aldehyde disproportionation reactions, 433–434 aldol reactions, 434 Alfred P. Sloan Foundation, 1 aliphatic chains, 462 alkalinity, cycle of, biological evolution and, 296 alkanes, 424, 426–427 alkenes, 424 alloy–silicate melt partitioning, 29 coefﬁcients, 16–17 Dcalloy/silicate and, 25 hydrogen and, 25–26 of LEVEs, 20–21 LEVEs and, 25 American–Antarctic Ridge, 255–257 anabolic reactions, 607–608 anaerobic methane-oxidizing archaea (ANME), 530–531, 534 anhydrous MORBs, 135–136 animation, as substitution reaction, 430–433 ANME. See anaerobic methane-oxidizing archaea Anthropocene, 627 antigorite, 285–286 aqueous electrolytes, 368 in conﬁned liquids, 372 aquifers, 191–192 aragonite, 74, 137 archaea anaerobic methane-oxidizing, 530–531, 534 in subsurface biome, 533–534

Archean, 283 Ashadze, 494 asthenospheric mantle, 70–72, 78–80 atmosphere loss, MO and, 17–19 atmospheric recycling, of sulfur, 100–102 ATP, 588 Aulbach, S., 68 axial diffuse vents basalts and, 492 oceanic rocky subsurface and, 492 axial high temperature basalts, 488–492 oceanic rocky subsurface and, 488–492 Azores, 240–242 Bagana, 217 Baltic Sea, 527–528 basalts. See also mid-ocean ridge-derived basalts axial diffuse vents and, 492 axial high temperature, 488–492 carbon content of, 4–5 carbon dioxide and, 144 ocean islands and, 144 benzaldehyde, 433–434 benzene, 426 Berner’s model, 336 bicarbonate ions, 19 bioavailability, of OC, 505 bioﬁlm-based metabolisms, 504 biogeochemical cycling, 480 reaction rate controls, 505 biogeochemistry, of deep life, 561–562 biological evolution, 299–300 alkalinity, cycle of and, 296 dioxygen cycle and, 294–296 subduction and, 294 biomass, 587–588 deep biosphere, 588 energy limits and, 588 bioorthogonal non-canonical amino tagging (BONCAT), 563 biotic recycling, of sulfur, 100–102

653

Downloaded from https://www.cambridge.org/core. IP address: 5.188.219.59, on 06 Oct 2019 at 16:31:36, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108677950.021

654

Index

bipartite networks, 642–643 Birch–Murnaghan equation of state (BMEOS), 171 Birch’s law, 50 BMEOS. See Birch–Murnaghan equation of state Boltzmann constants, 395 BONCAT. See bioorthogonal non-canonical amino tagging Brazilian diamonds, 103 bridgmanite, 75–76, 90–91 Brillouin scattering, 77 Brønsted acid catalysis, 422 BSE. See Bulk Silicate Earth bulk rock investigations, 449–451 bulk silicate, 112 Bulk Silicate Earth (BSE), 322, 325 carbon in, 10–14, 25–26 C/H ratio of, 14, 19 chondrites, 6–7 C/N ratios of, 14, 19 C/S ratio of, 16–18 D/H ratio of, 7–9 equilibrium accretion and budget of, 19–21 hydrogen in, 10–12 LEVE budgets of, 14–19, 25–26 magma ocean differentiation and budget of, 19–21 nitrogen in, 10–12 S/N ratio of, 16–18 sulfur in, 10–12 volatile budget of, 19–21 Bureau, H., on diamond formation, 106–108 CaCO3, 56. See also carbonates deep carbon stored as, 74 in dolomite, 73–74 calcite, 74 calcium silicate perovskite, 112 Ti-poor, 112–113 Ti-rich, 112–113 calderas emissions and, 201–206, 209 temporal variability of, 209 unrest in, 203–204 Calvin–Benson–Bassham cycle, 564–565 CaMg(CO3), 73–74. See also carbonates Canary Islands, 143, 149–150, 240–242 Candidate Phyla Radiations, 556–565 Cannizzaro reactions, 433–434 Cape Vede, 149–150 carbide, 29, 461 atomic scale structure of, 47 crystalline, 47 in Fe(Ni) alloys, 72 graphite–Fe, 8 molten iron, 47 carbide inner core model, 41–42 carbon. See also deep carbon; organic carbon abundance, 11–14 abundance of, in mantle, 67–73

in basalts, 4–5 baseline, 347–348 in BSE, 10–14, 25–26 in chondritic building blocks, 12 across CMB, 55–56 in continental lithosphere, 70–72 of continental subsurface, 500–501 in convecting mantle, 70–72, 254–257 in core formation, 20 in core over time, 55–56 in core–mantle segregation, 24–25 deﬁning, 40 dissolved inorganic, 480–489 distribution of, 80–81, 276 on Earth, 4–5 in E-chondrites, 11–12 estimates of abundance of, 66–67 in exogenic systems, 347–348 extraction of, from mantle, 67–73 feedbacks, 299 in Fe(Ni) alloys, 72 forms of, 11–14, 66 fractionation of, 18–19 inheritance of, in mantle, 68 isotopic composition of, 8–9 as light element in core, 27–28, 40, 55 in mantle, 238 mantle melting and, 257–262 melting points and, 53–54 in meteorites, 11–14 mineral ecology, 630–633 movement of, 1 outgassed from volcanoes, 211–215 oxidation of, 418–419 oxidized form, 70–72 in partial melting, 258 perturbations in ﬂux of, 277–278 polymorphs, 73–74 ratios, 11–14 in redox reactions, 80–81 reduced form, 70–72 residence time of, 277–278 sedimentary, 133 solidus and, 264 solubility, 18, 20–21 sources of, 211–215 speciation of, from mantle, 67–73 stability of, 70–72 in subconduction zones, 133 temporal distribution of, 628–629 in ultramaﬁc systems, 494 in ureilites, 16 volcanic, 215 carbon budgets, 4–5 constraints on, 56–57 of core, 40–41 from core accretion, 66–67 from core–mantle differentiation, 66–67

Downloaded from https://www.cambridge.org/core. IP address: 5.188.219.59, on 06 Oct 2019 at 16:31:36, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108677950.021

Index ingassing, 66 of Moon, 8–11 outgassing, 66 volcanic carbon and, 216–217 carbon cycle, 57 cadence of, 278–279 carbon deposition centers and, 276–278 carbonate melts in, 129 components of, 277 continental, 499 contingency at, 299–300 deep water and, 105–106 diamonds and, 94–95 longevity of, 278–279, 299 long-term, 276–278 mantle transition zone and, 102–103 non-steady-state dynamic of, 299 organic chemistry of, 416–420, 438–439 pace of, 278–279 prediction of, 280–281 pulse of, 278–279, 292, 299 self-stabilizing feedbacks, 299 subduction, 276 subduction zones and, 416–417 supercontinent assembly and, 625–626 surface processes and, 279–283 tectonic, 279, 293–295 carbon deposition centers, carbon cycle and, 276–278 carbon dioxide, 455–456. See also emissions, carbon dioxide atmospheric plumes of, 188–189 basalts and, 144 bulk, 135–136 decadal averages of, 196–197 degassing, 238–264 diffuse emissions, 191–192, 201–206 direct measurement of, 191 dissolution, 21–22 eclogites and, 146–147 experimental containers and, 457–458 during explosive eruptions, 197–198 ﬂux, 195–196, 242–243, 250 global emission rates of, 193–194 groundwater and, 191–192 incipient melting and, 165–166, 170–171, 177 indirect measurement of, 190–191 in magmas, 238–264 as magmatic volatile, 188 in mantle, 67–68, 177 mantle plumes and, 251–254 melt density and, 170–171 methanation, 367–375 methane and, 95–96 in mid-ocean ridge system, 242–243 in MORBs, 243, 250–251 OIB, 252 partial melting and, 165–166 peridotite and, 146–147

655

primary magma, 253 saturation, 238–264 solubility of, 238 volcanic, 190 carbon distribution, in core formation, 22–25 carbon dynamics, at subduction/collision transition, 292–293 carbon ﬂux, 150–151, 339–341 carbonate precipitation and, 331–333 carbonate weathering and, 330–331 metamorphic inputs, 329–330 OC weathering and, 330–331 outputs, 331–334 silicate weathering and, 331–333 subduction zones and, 281 volcanic inputs, 328–329 Carbon in Earth, 1 carbon isotopes composition, 103 of diamonds, 103 in ﬂuid-buffered systems, 97 fractionation, 97 of PDAs, 109 redox-neutral formation and, 96–98 carbon neutrality, of subduction zones, 283–284 carbon phase diagram, 595 oxygen fugacity of, 76–77 carbon reservoirs, 323–324 deep, 74–75 sizes of, 41 carbon solubility, 18 in magmas, 67–68 subduction zone and, 284–285 carbon speciation in MO, 22 oxy-thermobarometry and, 76–77 carbon transformation pathways, in subduction zones, 277–278 carbon transport, 133 in cratonic lithospheric mantle, 142 under nanoconﬁnement, 363–364 in subduction zone, 289 carbonaceous chondrites, LEVEs in, 15–29 carbonaceous matter (CM) abiotic formation of, 466–468 accumulation of, 466 composition of, 466–467 experimental occurrences of, 461–465 formation of, 464 future research on, 469–470 in hydrothermal experiments, 462–463 in hydrothermally altered mantle-derived rocks, 449 limits to knowledge about, 469–470 oxygen fugacity and, 462–463 carbonate basalt, 113 carbonate ions, 19

Downloaded from https://www.cambridge.org/core. IP address: 5.188.219.59, on 06 Oct 2019 at 16:31:36, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108677950.021

656 carbonate melts in carbon cycle, 129 compositions, 137–138 in cratonic lithospheric mantle, 138–139, 142–143 from diapirs, 287–288 extraction of, 148 from hot slabs, 287–288 importance of, 150 incipient melting of, 166–168 in intraplate settings, 143 under mid-ocean ridges, 147–148 migration of, 129, 132 at ocean islands, 143 silicate melts and, 168–169 stability ﬁelds of, 147 structure of, 168–169 with subconduction zones, 132–134 in upper mantle, 129 in various geodynamic settings, 166–168 carbonate stability constraints on, 130–131 oxygen fugacity and, 130 carbonate weathering, carbon ﬂux and, 330–331 carbonated basalts bulk compositions, 137 melt stability of, 179 carbonated MORBs, 135–136 carbonated sediment melting of, 134–138 potassium in, 137–138 solidus of, 135–136 in transition zone, 134–138 in upper mantle, 134–138 carbonates assimilation, 70 breakdown of, 80–81 in cratonic lithospheric mantle, 140–142 experimental calibrations, 140–142 Fe-bearing, 75 formation of, 133 in mantle, 72–73, 143–145 mineral dissolution of, 133–134 Na-carbonate, 137 pelagic, 296–299 phase at solidus, 135–136 precipitation, 331–333 pump, 279–280 redox constraints on, 140–142 seismic detectability of, 77 silicate and, 142–143 solubility of, 285 stability ﬁelds of, 147 structure of, 282 thermoelastic properties of, 77–78 carbonate–silicate melts, formation of, 69 carbonatite melts, 173 mobility of, 168–178

Index carbonatites, 133–134 abundance of, 149 classiﬁcation of, 148 crustally emplaced, 147–150 deep, 144 emplacement of, 149 evolution of, 149 formation of, 144, 149 limits to knowledge and, 150–151 magmas, 130 magmatism of, 149 ocean islands and, 149 in subconduction zones, 134 carbon-bearing ﬂuids complexity in, 358, 360 ﬂuid–ﬂuid interactions, 366 guest molecules in, 366–372 nanoconﬁnement and, 363 carbon-bearing phases limits to knowledge about, 81–82 stable forms of, 66 carbon-bearing reactants, in experiments, 458–461 carbonite peridotite in mantle, 131–132 melting of, 131–132 CARD. See catalyzed reporter deposition Carnegie Institute of Science, 388–389 CaSiO3-perovskite, 112–113 retrogressed, 114–115 CaSiO3-walstromite, 91–92 catabolic reactions, 599–601 catalyzed reporter deposition (CARD), 562–563 cathodoluminescence, of Marange diamond, 95 CaTiO3-perovskite, 112–113 C-bearing phases, in E-chondrites, 11–12 cell counts, 586 cellular bioenergetics, 566–567 cementite. See Fe3C Cenozoic, 276, 293 Census of Deep Life, 558 C/H ratios of BSE, 14, 16–19 bulk weight, 13–14 subchondritic, 19 C–H species, stabilization of, 19 Le Chatelier’s Principle, 420 chlorite, 285–286 chondrites BSE, 6–7 CI, 6–7 E-chondrites, 6–7, 11–12, 15 EH, 15 EL, 15 ordinary, 15–16 chondritic building blocks, carbon in, 12 CI, chondrites, 6–7

Downloaded from https://www.cambridge.org/core. IP address: 5.188.219.59, on 06 Oct 2019 at 16:31:36, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108677950.021

Index Circulation Obviation Retroﬁt Kit, 528–530 Claisen–Schmidt condensation, 434 climate, 188, 215 stability of Earth, 4 climatic drivers in elemental cycling, 319–321 negative feedbacks and, 319–321 clinopyroxene, 136–137 CLIPPIR diamonds, 114 inclusions in, 114–115 silicates in, 114–115 closed-system volcanoes, 209 CM. See carbonaceous matter CMB. See core-mantle boundary C/N ratios of BSE, 14, 19 bulk weight, 13–14 superchondritic, 19 Coast Range Ophiolite Microbial Observatory (CROMO), 532 C–O–H ﬂuids, 134 cold oxic basement, subsurface biome of, 530 compositional expansion coefﬁcients, 41–50 compressional-wave velocity, 53 conductive geotherms, 132 conﬁned liquids aqueous electrolytes in, 372 reactivity and, 374–375 solubility and, 369–370 volatile gas solubility in, 370–372 continent. See supercontinent assembly continental crust, 326–327 continental lithosphere, carbon in, 70–72 continental lithospheric mantle, 326–327 continental rifts, emissions and, 201–206 continental subsurface, 497–498 biomes, 524–527 carbon content of, 500–501 carbon cycling, 499 deep bedrock in, 503–504 deep coal beds in, 503 environments, 498–501 hydrocarbon reservoirs in, 502–503 continental weathering, 310 convecting mantle, 237 carbon in, 70–72, 254–257 limits of knowledge about, 263–264 melting, 257–262 plumes, 251–254 sampling, 240–242 convective geotherms, 132 core, 4–5 carbon as light element in, 27–28, 40, 55 carbon budgets of, 40–41 carbon in, over time, 55–56 composition, 57 Fe7C3 at, 55

657

limits to knowledge about, 57–58 pressure of, 44 recovery, 237 core accretion carbon budget from, 66–67 multistage, 25–26 simulation of, 24–25 core formation carbon distribution in, 22–25 carbon in, 20 Dcalloy/silicate in, 22–25 disequilibrium, 26–27 LEVEs in, 20 multistage, 23–26, 28 proto-Earth, 26–27 single-stage, 28 sulﬁde segregation and, 16–18 core–mantle boundary (CMB) carbon across, 55–56 chemical equilibrium at, 55–56 pressure for, 54–55 temperature at, 53 core–mantle differentiation, carbon budget from, 66–67 core–mantle fractionation, 23–24 core–mantle segregation, 5, 22–23 carbon in, 24–25 cracking reactions, 496–497 cratonic lithospheric mantle carbon transport in, 142 carbonate melts in, 138–139, 142–143 carbonates in, 140–142 deep, 142–143 kimberlite in, 138–139 metasomatism of, 142–143 oxygen fugacity in, 141 reduction of, 141 Cretaceous Peninsular Ranges, 344–345 CROMO. See Coast Range Ophiolite Microbial Observatory crustally emplaced carbonatites, 147–150 cryptic methane cycle, 405–406 C/S ratios of BSE, 16–18 bulk weight, 13–14 of fumaroles, 193 temporal variability and, 210–211 C/X ratios, 5 cycloalkanes, 426 cyclohexane, 423–426 cyclohexanol, 426 Darcy’s law, 177 Dcalloy/silicate alloy–silicate melt partitioning and, 25 in core formation, 22–25 sulfur in, 25

Downloaded from https://www.cambridge.org/core. IP address: 5.188.219.59, on 06 Oct 2019 at 16:31:36, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108677950.021

658

Index

DCO. See Deep Carbon Observatory deamination rates, 431–432 as substitution reaction, 430–433 DECADE. See Deep Earth Carbon Degassing decompression melting, 257 deep bedrock, in continental subsurface, 503–504 deep biosphere adaptations for survival in, 539, 568–569 biomass, 588 energetics, 585 limits to knowledge about, 505–506 locations, 481 metabolism, 562–565 similarities across, 504–505 deep carbon as CaCO3, 74 organic chemistry of, 416–420, 438–439 reservoir, magnesite as, 74–75 science, emergence of, 1 subduction, 288–289 Deep Carbon Observatory (DCO), 1, 90, 388–389 Carbon Mineral Challenge, 632 data and, 620 DMGC and, 115–116 Integrated Field Site Initiatives, 641 on volcanism, 189–190 deep carbonatites, 144 deep coal beds, in continental subsurface, 503 Deep Earth Carbon Degassing (DECADE), 195, 206, 217–218 on volcanism, 189–190 deep life, 539–541 biogeochemistry of, 561–562 deep mantle oxy-thermobarometry of, 76–77 redox freezing in, 111–114 Deep Sea Drilling Program (DSDP), 250 deep water carbon cycle and, 105–106 diamonds and, 105 in ringwoodite, 106 degassing diffuse, 199–201, 204 MO, 5 passive, 197, 206–207 dehydration, 436 aqueous alcohol, 421–422 as elimination reaction, 420–423 dehydrogenation reactions, 423–427 depleted MORB mantle (DMM), 211–215 Desulfovibrio indonesiensis, 570–571 devolatization pattern, 285–286 D/H ratio, of BSE, 7–9 diagenesis, 430 DIAL. See differential absorption LIDAR diamantiferous peridotite, 70

diamonds Brazilian, 103 Bureau on, 106–108 carbon cycle and, 94–95 carbon isotope composition of, 103 carbonates in, 135 CLIPPIR, 114–115 crystallization from single carbon ﬂuid species, 97–98 deep water and, 105 defects in, 92 depth of formation, 91–92 diagnostic tools for, 107 experiments for studying, 106–108 Frost on, 106–108 FTIR maps and, 92–94 future research on, 115–116 geobarometry of, 91 HDF migration and, 99–100 history of, 93–94 inclusion entrapment, 106–108 isochemical precipitation, 97 Jagersfontein, 103 Kankan, 103 limits to knowledge about, 115–116 lithospheric, 89–90 mantle metasomatism and formation of, 99–100 from mantle transition zone, 103–104 Marange, 95 metasomatic ﬂuids and formation of, 98–100 Monastery, 103 monocrystalline growth of, 107 natural growth media, 107 Northwest Territories Canadian, 99–100 obtaining, 89–90 platelets in, 93–94 polycrystalline formation of, 108–109 precipitation of, and methane, 95–96 Proterozoic lherzolitic formation, 110–111 redox freezing and, 111–114 redox-neutral formation of, 96–98 scanning electron microscope images of, 107–108 sublithospheric, 89–90 super-deep, 105 synthesizing, 106–107 thermal modelling of, 92–94 trapping of inclusions in, 92 Diamonds and the Mantle Geodynamics of Carbon (DMGC) DCO and, 115–116 goals of, 115–116 research areas of, 90 on super-deep diamonds, 105 diapirs, carbonate melt from, 287–288 DIC. See dissolved inorganic carbon dielectric constants, in nanoconﬁnement, 372–374 differential absorption LIDAR (DIAL), 190–191 differential equations, ﬁrst order, 317

Downloaded from https://www.cambridge.org/core. IP address: 5.188.219.59, on 06 Oct 2019 at 16:31:36, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108677950.021

Index diffuse degassing, 199–201, 204 emissions from, 207–208 diffuse emissions, 191–192, 201–207 diffusion pore, 364–366 surface, 364–365 viscosity-diffusion, 171–172 diffusion-sink experiments, 176 dioxygen cycle, biological evolution and, 294–296 disequilibrium core formation, 26–27 disproportionation reactions, 433–434 aldehyde, 433–434 dissolution, of siderites, 462–463 dissolved inorganic carbon (DIC), 480–489 dissolved organic carbon (DOC), 480–489 solubilization of, 484 DMGC. See Diamonds and the Mantle Geodynamics of Carbon DMM. See depleted MORB mantle DOC. See dissolved organic carbon dolomite CaCO3 in, 73–74 crystal structure of, 73–74 high-pressure polymorphs and, 73–74 iron and, 73–74 MgCO3 in, 73–74 dolomitic carbonite, 132 Dorado Outcrop, 492–493 dormancy, 588–589 dormant volcanoes, emissions from, 198 down-going slab materials, 133 DSDP. See Deep Sea Drilling Program E. coli, 570–571 EAR. See East African Rift Earth. See also Bulk Silicate Earth carbon on, 4–5 climate stability of, 4, 313 life on, 4 mantle reservoir of, 4–5 organic chemistry and, 415–416 proto-Earth core formation, 26–27 structure of, 4–5 surface temperature of, 313 whole-Earth carbon cycle, 315–316, 338–341 Earth Microbiome Project, 641 EarthChem Library, 240–242, 623 EAS. See electrophilic aromatic substitution East African Rift (EAR), 149, 205–206, 217, 328–329 East Paciﬁc Rise, 179, 240–241 Ebelman reaction, 292 EC. See Eddy covariance E-chondrites carbon in, 11–12 C-bearing phases in, 11–12 LEVEs and, 15 model, 6–7

659

eclogite, 70 carbon dioxide and, 146–147 in mantle, 144 melting, 215 eclogite-derived melts, 146–147 eclogitic lithospheric diamonds, 90 Eddy covariance (EC), 191 Eger Rift, 206 EH chondrites, 15 elastic geobarometry, 91–92 electrical conductivity anomalous, 181 enhancement, 176–177 incipient melting and, 173–174, 179 melt mobility and, 179 in olivine matrix, 174 electrophilic aromatic substitution (EAS), 433–434 elemental cycling basic concepts of, 315 climatic drivers in, 319–321 negative feedback in, 319–321 residence time in, 315–319 steady state in, 315–319 elimination reactions dehydration as, 420–423 hydration as, 420–423 EM1 OIB, 102 EMFDD reaction, 131 emissions, carbon dioxide from active volcanoes, 194–197, 216 calderas and, 201–206, 209 constraints, 207 continental rifts and, 201–206 cumulative, 201–202 data distribution, 203 decadal averages of, 196–197 diffuse, 191–192, 201–207, 216 from diffuse degassing, 207–208 from dormant volcanoes, 198 estimation of, 197 during explosive eruptions, 197–198 fumaroles and, 198–201 global of carbon dioxide, 193–194 hydrothermal systems, 201–206 measurement of, 199–201 next iteration of, 206–208 over geologic time, 215 plume gas, 188, 201–203 quantifying, 215–217 synthesis of, 215–217 temporal variability of, 208–209 vent, 216 EMOD buffers, 96–97 endogenic systems, 314–315 energy limits, 585 anabolism and, 607–608 biomass and, 588

Downloaded from https://www.cambridge.org/core. IP address: 5.188.219.59, on 06 Oct 2019 at 16:31:36, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108677950.021

660

Index

energy limits (cont.) density and, 603–605 maintenance in, 586–587 microbial states and, 586–589 time and, 606–607 Enermark ﬁeld, 526–527 entropy, 590 changes in, 590 deﬁning, 590 enzyme evolution, 635–636 equilibrium accretion, BSE budget and, 19–21 eruption forecasting, temporal variability and, 209–211 eukaryotes, in subsurface biome, 535–536 eutectic composition, 41–42 of Fe–O binary system, 42–43 of Fe–S binary system, 42–43 of Fe–Si binary system, 42–43 exogenic reservoirs, 327–328 exogenic systems, 314–315 carbon ﬂux, 331 carbon in, 347–348 experimental containers, carbon dioxide and, 457–458 extreme cellular biophysics, 570–572 extreme molecular biophysics, in subsurface environment, 567–570 Fe3C density of, 47–48 inner core phase and, 50–52 natural form of, 44–48 near iron end member, 48 orthorhombic, 44–48 Fe7C3 constraints from, 52 at core, 55 electrical resistivity of, 55 sound velocities of, 52 Fe-bearing carbonates, melting of, 75 Fe–C alloy constraints from, 52 elasticity parameters for, 45 liquid, 49, 52 melting temperatures of, 53–55 near iron end member, 52 slab-derived, 56 sound velocities of, 50 Fe–C binary system, 41–42 densities of, 44, 48 FeCO3, 78–79 feedback loops, 317–318 Fe–H, sound velocities of, 53 Fe–light element alloys melting curve parameters, 52 sound velocities of, 52–53 Fe(Ni) alloys carbide in, 72 carbon in, 72 precipitation curve, 70–71

Fe–Ni–C alloys, solidus temperature ranges in, 72 Fe–O binary system characterizing, 44 eutectic composition, 42–43 melting temperatures, 55 sound velocities of, 53 ferropericlase, 76 Fe–S binary system characterizing, 42 eutectic composition, 42–43 eutectic point of, 55 melting temperatures, 55 sound velocities of, 53 Fe–Si binary system characterizing, 42–44 eutectic composition, 42–43 melting temperatures, 55 sound velocities of, 53 FISH. See ﬂuorescent in situ hybridization Fisher–Tropsch process, 460–461 ﬂank gas emission, 188 ﬂuid addition, 215 ﬂuid inclusions, in oceanic lithosphere, 456–464 ﬂuid–ﬂuid interactions, 366 ﬂuorescent in situ hybridization (FISH), 562–563 ﬂux melting, 144 formaldehyde, 459 formate, in ultramaﬁc systems, 495 founder effect, 540 Fourier-transform infrared spectroscopy (FTIR) maps, 190–191, 238, 451–452 diamonds and, 92–94 Friedel–Crafts reaction, 434 Frost, D. J., 69 on diamond formation, 106–108 FTIR. See Fourier transform infrared spectroscopy maps FTT reactions, 457–458 magnetite and, 464 fumaroles, 213–214 C/S ratios of, 193 emission rates and, 198–201 G protein-coupled receptors (GPCRs), 568 Gakkel Ridge, 240–241, 250 Galapagos Spreading Center, 248–249 Garrett melt inclusion, 246–247 gas giants, growth of, 10 generalized inverse Gauss–Poisson (GIGP), 631–632 genetic drift, 540 Genomic Standard Consortium, 641 geobarometry of diamonds, 91 elastic, 91–92 geo–bio interactions, 640–643 geochemical tracers, 68 geologic time emissions over, 215 volcanic carbon and, 215

Downloaded from https://www.cambridge.org/core. IP address: 5.188.219.59, on 06 Oct 2019 at 16:31:36, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108677950.021

Index geological cycle, 294 GeoMapApp, 241–242 geomimicry, 439 geotherms conducive, 132 convective, 132 Gibbs energy, 589–599 changes in, 591 composition and, 599–601 densities, 604–605 molal, 604 pressure and, 599–601 standard state, 592–595 surveying, 601–603 temperature and, 599–601 GIGP. See generalized inverse Gauss–Poisson global emission rates, of carbon dioxide, 193–194 Global Volcanism Program (GVP), 197 Volcanoes of the World, 194 GOSAT. See Greenhouse Gases Observing Satellite GPCRs. See G protein-coupled receptors grain boundaries, 361–362 Grand Tack scenario, 8–11 graphite, 29 exhausting, 259–260 formation, 465–466 in mantle, 259–260 thermodynamic predictions, 465 graphite–Fe–carbide, 8 graphitization, 282–283 green chemistry, 439 greenhouse conditions, 342 Greenhouse Gases Observing Satellite (GOSAT), 192–193 greenhouse intervals, 342–343 groundwater carbon dioxide and, 191–192 Vesuvio, 191–192 Guaymas Basin, 527–528 guest molecules, in carbon-bearing ﬂuids, 366–372 Gulf of Mexico, 527–528 Gutenberg discontinuity, 164–165 GVP. See Global Volcanism Program Halicephalobus mephisto, 535 harzburgite, 132 Hashin–Shtrikman upper-bound (HS+) model, 174–176 Hauri, E. H., 189–190, 248–249, 264, 323 Hawaii melt inclusions, 263 Hazen, R. M., 630–632 HDF microinclusions, in lithospheric diamonds, 99 HDF migration, diamonds and, 99–100 heat ﬂux, from hotspots, 252 Helgeson–Kirkham–Flowers (HKF) equations, 596 helium, 213–215, 244–245 hematite-magnetite, 457 hematite–magnetite–pyrite, 457

661

heteroatoms, 456 HFSE. See high-ﬁeld-strength element high-ﬁeld-strength element (HFSE), 98–99 highly siderophile element (HSE), 15–29 sulﬁde segregation and, 16–18 HIMU OIB, 102 histone-like nucleoid structuring proteins (HNS), 568 HKF equations. See Helgeson–Kirkham–Flowers equations Holocene, 194 hot slabs, carbonate melt from, 287–288 hothouses, 345–346 hot spots, 240–242, 257 heat ﬂux from, 252 HS+ model. See Hashin–Shtrikman upper-bound model HSE. See highly siderophile element hydration, as elimination reaction, 420–423 hydraulic fracturing, 526–527 hydrocarbon reservoirs, in continental subsurface, 502–503 hydrocarbons, short-chain, 495 hydrogen alloy–silicate melt partitioning and, 25–26 in BSE, 10–12 fractionation of, 18–19 isotopic composition of, 8–9 methane and, 459–460 hydrogenation reactions, 423–427 Hydrogenophaga, 531–532 hydrogenotrophic methanogenesis, 483 hydrolyzable amino acids, 495 hydrothermal carbon pump, 279–280, 283 circulation, 495–496 experiments, CM in, 462–463 petroleum, 484–497 reactions, 436–437 hydrothermal systems abundance of, 204 emissions and, 201–206 sedimented, 496–497 volcanism and, 204 hydrothermally altered mantle-derived rocks, CM in, 449 ICB. See inner core boundary ICDP. See International Continental Drilling Programs icehouse conditions, 342 icehouse drivers, 344–345 Iceland, 240–242 igneous aquifers, 499–502 IMLGS. See Index to Marine and Lacustrine Geological Samples incipient melting carbon dioxide and, 165–166, 170–171, 177 of carbonate melt, 166–168 composition, 167

Downloaded from https://www.cambridge.org/core. IP address: 5.188.219.59, on 06 Oct 2019 at 16:31:36, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108677950.021

662 incipient melting (cont.) deﬁning, 163–165 density, 170–171 electrical conductivity and, 173–174, 179 interconnectivity, 175–176 limits of knowledge about, 182 mantle convection and, 181–182 melt mobility of, 177–179 origins, 164 of peridotite, 179 proﬁles, 167–168 of silicate melt, 166–168 stability ﬁelds in, 165–166 transport properties, 171–172 types of, 177–178 viscosity-diffusion, 171–172 water and, 165–166, 170–171 Index to Marine and Lacustrine Geological Samples (IMLGS), 240–241 inner core Fe3C and, 50–52 late veneer, 8–11 phase, 50–52 sound velocities in, 50–51 inner core boundary (ICB), 41–42 insoluble organic molecules (IOMs), 11 Integrated Ocean Drilling Program (IODP), 250, 527 International Continental Drilling Programs (ICDP), 641 International Ocean Discovery Program (IODP), 641 interphase boundaries, 361–362 Interunion Commission on Biothermodynamics, 596–597 intraplate settings, carbonate melts in, 143 inverse Monte Carlo simulations, 26–27 IODP. See Integrated Ocean Drilling Program, International Ocean Discovery Program IOMs. See insoluble organic molecules iron carbon alloys, 40–41 dolomite and, 73–74 melting point, 53–54 redox capacity of, 107–108 spin state, 78–79 iron end member Fe3C near, 48 Fe–C alloy near, 52 iron–light element systems binary phase relations, 41–42 phase relations of, 41–44 isotope clumping, 388 kinetics, 393–399 isotopic reservoirs, 401–405 Jagersfontein diamonds, 103 Jagersfontein kimberlite, 76

Index Juan de Fuca Ridge, 240–241, 248–249, 493, 528–530 warm anoxic basement of, 528–530 Jupiter, 8–11 Kaapvaal cratons, 69, 101, 103 Kankan diamonds, 103 karpatite, 450–451 Kerguelen Islands, 143 kerogen, 282–283 Kidd Creek, 400–401 Kilauea, 253 kimberlite, 89–90, 106–107, 139–140 in cratonic lithospheric mantle, 138–139 eruption dates, 93, 111 genesis of, 140 group 1, 139 group 2, 139 Jagersfontein, 76 magmatism, 110–111 origins of, 139 oxygen fugacity and, 130–131 parental magma composition, 139–140 kinetic array, 399–401 kinetic inhibition, 419–420 kinetic minimum, 293 kinetic rate constants, 340 kinetics isotope clumping, 393–399 Michaelis–Menten, 393–399 Kokshetav, 292 LAB. See lithosphere–asthenosphere boundary labile amino acids, 497 large igneous provinces (LIPs), 254 large ion lithophile element (LILE), 98–99 large number of rare events (LNRE), 630–631 late accretion, 14–16 LEED. See low-energy electron diffraction LEVEs. See life-essential volatile elements Lewis acid catalysis, 422 lherzolite, 132 LIDAR. See Light Detection and Ranging life, records of, 294 life-essential volatile elements (LEVEs), 4, 28–29 alloy–silicate melt partitioning and, 25 alloy–silicate partitioning of, 20–21 budgets of BSE, 14–19, 25–26 in carbonaceous chondrites, 15–29 constraints from isotopes of, 7–8 in core formation, 20 delivery timing of, 17–18 distributions of, 5–6 E-chondrites and, 15 initial distributions of, 20 isotopic compositions of, 5 limits of knowledge, 29 origins of, 11, 19–20

Downloaded from https://www.cambridge.org/core. IP address: 5.188.219.59, on 06 Oct 2019 at 16:31:36, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108677950.021

Index solubility data for, 19 unknowns involving, 29 Light Detection and Ranging (LIDAR), 190–191 light elements, 49 carbon as, in core, 27–28, 40 lignin phenols, 481–483 Ligurian Tethyan ophiolites, 453–454 LILE. See large ion lithophile element LIPs. See large igneous provinces liquid Fe–C alloy, 49 constraints from, 52 elasticity parameters for, 46 sound velocities of, 52 liquid outer core, oxygen in, 44 lithophile elements, 6–7 lithosphere–asthenosphere boundary (LAB), 164 deﬁning, 181 geophysical discontinuities, 181 thermal, 167–168 lithospheric diamonds, 89–90 classiﬁcation of, 90 composition of, 90 eclogitic, 90 formation of, 90 HDF microinclusions in, 99 peridotitic, 90 reduced mantle volatiles in, 94–96 refertilization in, 110 lithospheric mantle, continental, 326–327 lithospheric reservoir, 348 LNRE. See large number of rare events Logatchev hydrothermal ﬁelds, 449–450, 494 Loihi, 253 longevity, of carbon cycle, 278–279 Lost City, 404–405 low energy states, 589 low-velocity zone (LVZ), 164–165, 181 limits of knowledge about, 182 low-energy electron diffraction (LEED), 452–453 Lucky Strike segment, 240–241 LVZ. See low-velocity zone macrofauna, 481 magma ocean (MO) atmosphere interactions, 17–19 BSE budget and, 19–21 carbon speciation in, 22 degassing, 5 magmas, carbon dioxide in, 238–264 magnesite, as deep carbon reservoir, 74–75 magnesium budgets, 492–493 magnetite, 459 FTT and, 464 MAGs. See metagenome-assembled genomes Maier–Kelley formulation, 596 Main Ethiopian Rift (MER), 205–206

663

maintenance in energy limits, 586–587 measurements of, 587 Manam, 217 mantle. See also convecting mantle; cratonic lithospheric mantle; deep mantle; upper mantle abundance of carbon in, 67–73 adiabatic, 166–167 asthenospheric, 70–72, 78–80 carbon dioxide in, 67–68, 177 carbon in, 238 carbonate in, 143–145 carbonate minerals in, 72–73 carbonite peridotite in, 131–132 convection, 181–182 deep, 76–77, 111–114 degassing, 339–342 eclogite in, 144 extraction of carbon from, 67–73 graphite in, 259–260 incipient melting and, 181–182 ingassing, 339–342 inheritance of carbon at, 68 oxidation of, 258 oxidized carbon in, 77–78 peridotite in, 113–114, 144 resistive lids, 164–165 slab-derived ﬂuids in, 134 speciation of carbon from, 67–73 sulfur in, 100–102 mantle geodynamics. See Diamonds and the Mantle Geodynamics of Carbon mantle melting regime, 164 carbon and, 257–262 mantle metasomatism, 100–101 characterizing, 163 deﬁning, 163–165 diamond formation and, 99–100 mantle plumes carbon dioxide and, 251–254 convecting, 251–254 mantle reservoirs of Earth, 4–5 modern, 322–326 primitive, 322–326 mantle transition zone carbon cycle and, 102–103 diamonds from, 103–104 hydration state of, 105 MAR. See Mid-Atlantic Ridge Marange diamonds, 95 cathodoluminescence of, 95 methane and, 95–96, 98 RIFMS for, 98 Mars, 26–27, 259–260, 321 Masaya, 209 mass-independent fractionation (MIF), 100–101

Downloaded from https://www.cambridge.org/core. IP address: 5.188.219.59, on 06 Oct 2019 at 16:31:36, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108677950.021

664 MED. See Mineral Evolution Database melt, incipient. See incipient melting melt composition, melt mobility and, 177–179 melt density calculation of, 170–171 carbon dioxide and, 170–171 curve, 170–171 water and, 170–171 melt inclusions data sets, 240–242 Garrett, 246–247 glassy, 253 Hawaii, 263 isotopic heterogeneity in, 246–248 MORB, 244–248 OIBs and, 252–253 Siqueiros, 246–247 volumes, 242 melt mobility electrical conductivity and, 179 of incipient melts, 177–179 melt composition and, 177–179 melt stability, of carbonated basalts, 179 melts. See speciﬁc types Menez Gwen, 494 MER. See Main Ethiopian Rift Mesozoic, 276 metagenome-assembled genomes (MAGs), 558–560 metamorphic inputs, carbon ﬂux, 329–330 metamorphism, deﬁning, 188 metasomatic ﬂuids, diamond-forming, 98–100 metasomatism. See also mantle metasomatism of cratonic lithospheric mantle, 142–143 overprints, 142 metatranscriptomics, 560 meteorites, carbon in, 11–14 methanation, carbon dioxide, 367–375 methane, 388–389, 447–448, 459, 489 biogenic, 503 carbon dioxide and, 95–96 cycling, 504 in diamond precipitation, 95–96 formation, 403, 465–466 hydrogen and, 459–460 limits to knowledge about, 409 in Marange diamonds, 95–96, 98 oxidation, 405–409 production of, 459–460 synthesis of, 95–96 thermodynamic equilibrium and, 388–389 in ultramaﬁc systems, 494–495 methanogenesis differential reversibility of, 406 reversibility of, 394 methanol, 459 formation of, 459 methylcyclohexanol, 435–436

Index MgCO3, 56. See also carbonates in dolomite, 73–74 Michaelis–Menten kinetics, 393–399 microbial array, 399–401 microbial ecosystems, 640–643 microbial metabolism, in subsurface environment, 562–565 microbial states, energy limits and, 586–589 micro-Raman spectroscopy, 91–92 microscale, in situ investigations at, 451–464 Mid-Atlantic Ridge (MAR), 240–241, 494 mid-ocean ridge system carbon dioxide in, 242–243 carbonate melts under, 147–148 mid-ocean ridge-derived basalts (MORBs), 112–113, 213, 237 anhydrous, 135–136 bulk compositions of, 135–136 carbon dioxide in, 243, 250–251 carbonated, 135–136 chemistry of, 135–136 compositions, 137, 248–251 eruption of, 243 melt inclusions, 244–248 oxidation of, 69 oxygen fugacity and, 69 samples, 243–244 solubility in, 243–244 vapor-undersaturated, 246 variations in, 248–251 MIF. See mass-independent fractionation Mineoka ophiolite complex, 455–456 Mineral Evolution Database (MED), 621 Miyakejima volcano, 195 MO. See magma ocean modern mantle reservoirs, 322–326 molecular lubrication, pore diffusion and, 365–366 Momotombo, 209 Monastery diamonds, 103 montmorillonites, 464–465 Moon carbon budgets of, 8–11 formation of, 11, 26–27 MORBs. See mid-ocean ridge-derived basalts Mount Etna, 208, 328–329 Multi-Gas measurements, 190–191 Murowa, 93 Na-carbonate, at solidus, 137 Nankai Trough, 527–528 nanoconﬁnement carbon transport under, 363–364 carbon-bearing ﬂuids and, 363 dielectric constants in, 372–374 nanoporosity, 359–360, 362–363 features of, 360–363 NanoSIMS, 562–563

Downloaded from https://www.cambridge.org/core. IP address: 5.188.219.59, on 06 Oct 2019 at 16:31:36, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108677950.021

Index National Centers for Environmental Information (NCEI), 240–241 National Oceanographic and Atmospheric Association, 240–241 NBO/T approach, 21–22 NCEI. See National Centers for Environmental Information negative feedback, 317–318, 338 climatic drivers and, 319–321 in elemental cycling, 319–321 Neoproterozoic, 346–347 network analysis, 640–643 Newer Volcanics of Victoria, 132 Nibelungen, 494 nitrogen aggregation, 93 in BSE, 10–12 depletion, 18–19 fractionation of, 18–19 isotopic composition of, 8–9 as siderophile elements, 25 nitrogen cycle, mantle transition zone and, 102–103 nominal oxidation state of carbon (NOSC), 587–588 non-ideal conditions, 598 Northwest Territories Canadian diamonds, 99–100 NOSC. See nominal oxidation state of carbon novel genes, 564–565 Nuna, 629 Nyiragongo volcano, 195, 201–206 OC. See organic carbon Ocean Drilling Program (ODP), 492–493 Hole 735B, 455 Leg 201, 527–528 ocean island basalt (OIB), 237 carbon dioxide in, 252 chemistry of, 135–136 melt inclusions and, 252–253 sulﬁdes from, 101–102 ocean islands basalts and, 144 carbonate melts beneath, 143 carbonatites and, 149 oceanic crust, 487–488 axial diffuse vents and, 492 axial high temperature and, 488–492 characteristics, 491 ﬂuid inclusions in, 456–464 recharge water and, 487–488 ridge ﬂanks and, 492–493 subsurface biome of, 528 ultramaﬁc systems and, 493–495 warm anoxic basement, subsurface biome of, 528–530 OCO-2. See Orbiting Carbon Observatory ODP. See Ocean Drilling Program OET. See oxygen exposure time OIB. See ocean island basalt

665

Oldoinyo Lengai, 198–199 oligomer dissociation, 569 olivine, 132 carbonation of, 462 olivine matrix, electrical conductivity in, 174 Olmani Cinder cone, 132 OMI. See Ozone Monitoring Instrument Opalinus Clay, 526–527 orangeites, 139 Orbiting Carbon Observatory (OCO-2), 192–193 ordinary chondrites, 15–16 organic carbon (OC), 282 anaerobic breakdown of, 483 bioavailability of, 505 burial rate, 333 carbon ﬂux and, 330–331 dissolved, 480–489 oxidation of, 481 particulate, 480–489 weathering, 330–331 organic chemistry bonds in, 415–416 of carbon cycle, 416–420, 438–439 of deep carbon, 416–420, 438–439 Earth and, 415–416 organic matter preservation, in sedimentary subsurface, 484–485 organic oxidations, 427–429 orthopyroxene, 132 oxidation aqueous, 428 of carbon, 418–419 methane, 405–409 organic, 427–429 of organic carbon, 481 oxidized carbon, in mantle, 77–78 oxygen, in liquid outer core, 44 oxygen exposure time (OET) models of, 486–487 sedimentary subsurface and, 486 oxygen fugacity, 17–18, 21–22, 150–151 of carbon phases, 76–77 carbonate stability and, 130 CM and, 462–463 in cratonic lithospheric mantle, 141 kimberlite and, 130–131 magnitude of, 131 MORBs and, 69 oxy-thermobarometry carbon speciation and, 76–77 of deep mantle, 76–77 Ozone Monitoring Instrument (OMI), 193, 206–207 data sets, 197 pace, of carbon cycle, 278–279 PAH. See polycyclic aromatic hydrocarbon Paleocene–Eocene thermal maximum (PETM), 319

Downloaded from https://www.cambridge.org/core. IP address: 5.188.219.59, on 06 Oct 2019 at 16:31:36, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108677950.021

666

Index

partial melting carbon dioxide and, 165–166 carbon in, 258 particulate organic carbon (POC), 480–489 microorganisms accessing, 484 PDAs. See polycrystalline aggregates Pearson correlation coefﬁcients, 246–247 pelagic carbonates, in subduction zone, 296–299 periclase, 113–114 peridotite, 130, 144–145 abyssal, 455–456 carbon dioxide and, 146–147 incipient melting of, 179 in mantle, 113–114, 144 solidus of, 258 peridotitic lithospheric diamonds, 90 permeability, 177, 203–204 perturbations, 277–278 Peru Margin, 527–528 petit spot volcanism, 179, 238 PETM. See Paleocene–Eocene thermal maximum petrogenic carbon, 481–483 Phanerozoic, 149, 281–282 phase relations, of iron–light element systems, 41–44 Photobacterium profundum, 570–571 piezolyte, 570 Pitcairn, 253 Planck constants, 395 planetary embryos, 28–29 sulfur in, 26 plume gas emissions, 188, 201–203 POC. See particulate organic carbon Poisson’s ratio, 77–78 polycrystalline aggregates (PDAs), 108–109 absolute ages of, 109 carbon isotope values of, 109 formation of, 109 polycrystalline diamond formation, 108–109 polycyclic aromatic hydrocarbon (PAH), 450–451, 497 pore diffusion molecular lubrication and, 365–366 steric effects and, 364–365 porosity. See nanoporosity potassium, in carbonated sediment, 137–138 predictive reaction-rate models, 432–433 PREM model, 50 pressure–temperature plot, silicate melts and, 144–145 primary magma carbon dioxide, 253 primitive mantle reservoirs, 322–326 process end members, 401–402 propanoic acid, 428 protein expression, 635–636 protein unfolding, 569 Proterozoic, 283 Proterozoic lherzolitic diamond formation, 110–111 through time, 110–111

protoplanetary bodies, 20 P–T trajectories, 285–287, 289 subduction zone, 289–290 pulse, of carbon cycle, 278–279 pumps carbonate, 279–280 hydrothermal carbon, 279–280 soft-tissue, 279–280 pyrite–pyrrhotite–magnetite, 457 QFM buffer, 258 quartz–fayalite–magnetite, 457 radiogenic isotopes, 237–238 rare biosphere, 525–526 rare earth elements (REE), 129 Rayleigh isotopic fractionation in multi-component systems (RIFMS), 97, 109 for Marange diamonds, 98 reactivity, conﬁned liquids and, 374–375 recharge water, oceanic rocky subsurface and, 487–488 recycling processes, 164 Redoubt Volcano, 204 redox capacity of iron, 107–108 of sulﬁdes, 107–108 redox constraints, on carbonates, 140–142 redox freezing in deep mantle, 111–114 deﬁning, 113–114 diamonds and, 111–114 redox processes, in subduction zone, 290–291 redox reactions, 75, 81 carbon in, 80–81 redox-neutral formation carbon isotope fractionation and, 96–98 of diamonds, 96–98 reduced mantle volatiles in lithospheric diamonds, 94–96 in sublithospheric diamonds, 94–96 REE. See rare earth elements refractory elements, constraints from isotopes of, 6–7 refractory garnet peridotites, 111 remineralization, 481 reservoirs carbon, 41, 323–324 deep carbon, 74–75 exogenic, 327–328 hydrocarbon, 502–503 isotopic, 401–405 lithospheric, 348 mantle, 4–5, 322–326 Solar, 8–9 residence time deﬁning, 318–319 in elemental cycling, 315–319 response time, deﬁning, 318–319

Downloaded from https://www.cambridge.org/core. IP address: 5.188.219.59, on 06 Oct 2019 at 16:31:36, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108677950.021

Index Rhine Graben, 206 ribosomal gene sequencing, 558 ridge ﬂanks advective ﬂow through, 492 oceanic rocky subsurface and, 492–493 RIFMS. See Rayleigh isotopic fractionation in multicomponent systems ringwoodite, deep water in, 106 Rio Grande Rift, 206 rocks. See speciﬁc types Rodinia, 620, 629 supercontinent assembly of, 623–625 Rotorua, 201–204 RRUFF Project, 633 S isotopic systematics, 100–101 in sulﬁde inclusions, 101 Sabatier reaction, 395–398 Saccharomyces cerevisiae, 570–571 SAGMEG. See South African Gold Mine Miscellaneous Euryarchaeal Group SAGs. See single-cell ampliﬁed genomes sapropels, 484 scanning electron microscope images, of diamonds, 107–108 scanning transmission X-ray microscope, 485 Schoell plot, 402 seaﬂoor dredging, 237–238 seaﬂoor weathering feedback, 338 secondary ion mass spectrometry (SIMS), 238 sedimentary aquifers, 499–502 sedimentary carbon, 133 subduction, 280–281 sedimentary subsurface, 481 chemical composition of, 481–484 organic matter preservation in, 484–485 oxygen exposure time and, 486 sorption in, 485–486 sedimented hydrothermal systems, 496–497 selective preservation, 483–484 serpentinized oceanic rocks, 451–452 Serpentinomonas, 531–532 shear-wave velocity, 53 Shimokita Peninsula, 527–528 siderites, 461 dissolution of, 462–463 siderophile elements, 6–7 nitrogen as, 25 silicate, 4, 310. See also Bulk Silicate Earth carbonate and, 142–143 silicate melt, 21–22 carbonate melts and, 168–169 extraction of, 148 formation of, 143–144 incipient melting of, 166–168 pressure–temperature plot and, 144–145 stability ﬁelds of, 147

667

structure of, 168–169 in upper mantle, 143–144 in various geodynamic settings, 166–168 viscosity-diffusion and, 172 silicate weathering carbon ﬂux and, 331–333 feedback, 334–338 global rates of, 336–337 silicates, in CLIPPIR diamonds, 114–115 SIMS. See secondary ion mass spectrometry single-carbon species, 459 single-cell ampliﬁed genomes (SAGs), 558–560 single-species ecosystems, 525–526 SiO2 bulk, 135–136 in subduction zone, 291 SIP. See stable isotope probing Siqueiros Fracture Zone, 245–246 Siqueiros melt inclusion, 246–247 Siqueiros Transform, 240–241 slab-derived ﬂuids, in mantle, 134 slave cratons, 101 SLiMEs. See subsurface lithoautotrophic microbial ecosystems small polar compounds, 496 small volcanic plumes, 198–201 smectite clays, 464–465 S/N ratio, of BSE, 16–18 snowballs, 346–347 soft-tissue pump, 279–280 Solar reservoir, 8–9 solidus carbon and, 264 carbonate phase at, 135–136 of carbonated sediment, 135–136 curves, 136 Na-carbonate at, 137 of peridotite, 258 solubility conﬁned liquids and, 369–370 of DOC, 484 sorption, in sedimentary subsurface, 485–486 sound velocities of Fe–C alloy, 50 in inner core, 50–51 South African Gold Mine Miscellaneous Euryarchaeal Group (SAGMEG), 534 Southwest Indian Ridge, 255–257 spin transition, 77–78 diagram, 78–79 spot measurements, 195–196 SRB. See sulfate-reducing bacteria stability ﬁelds, in incipient melting, 165–166 stable isotope probing (SIP), 562–563 steady state in elemental cycling, 315–319 transition to new, 321–322

Downloaded from https://www.cambridge.org/core. IP address: 5.188.219.59, on 06 Oct 2019 at 16:31:36, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108677950.021

668 steric effects pore diffusion and, 364–365 surface diffusion and, 364–365 Stromboli, 208 S-type asteroids, 7–9 subaerial volcanic budget, 206–207 sub-arc depths, 133–134 subconduction zone carbon in, 133 carbonate melts with, 132–134 carbonatites in, 134 cross-section of, 134 subduction, 215, 311 biological evolution and, 294 carbon cycling, 276 cycle, 277 deep carbon, 288–289 ﬂux, 334 sedimentary carbon, 280–281 shelf carbon, 276–278 subduction zones, 300 carbon cycle and, 416–417 carbon ﬂux and, 281 carbon neutrality of, 283–284 carbon solubility and, 284–285 carbon transformation pathways in, 277–278 carbon transport in, 289 dissolution in, 285–287 models of, 310–312 pelagic carbonates in, 296–299 P–T trajectories, 289–290 redox processes in, 290–291 SiO2 in, 291 sources and sinks, 279–280 tectonic building blocks at, 292–293 thermal anomalies in, 289 water in, 291–292 subduction/collision transition, carbon dynamics at, 292–293 sublithospheric diamonds, 89–90 formation of, 90–91 inclusions in, 96 reduced mantle volatiles in, 94–96 study of, 90–91 sub-seaﬂoor sediments, subsurface biomes, 527–528 substitution reaction animation as, 430–433 deamination as, 430–433 subsurface biome, 524–526, 572–573. See also continental subsurface; deep biosphere adaptations for survival in, 539, 568–569 archaea in, 533–534 of cold oxic basement, 530 continental, 524–527 deep life in, 539–541 deﬁning, 524–525 diffusivity in, 537

Index ecology in, 536 eukaryotes in, 535–536 evolution of, 536 extreme cellular biophysics in, 570–572 extreme molecular biophysics in, 567–570 genetic potential of, 558–561 global trends in study of, 533 habitable zones, 525–526 interactions in, 534–535 isolates, 534–535 microbial metabolism in, 562–565 of oceanic crust, 528 of other environments, 532–533 pH of, 537–538 pressure effects in, 567 salinity in, 538 sub-seaﬂoor sediments, 527–528 temperature of, 538–539 of ultra-basic sites, 530–532 viruses in, 536 of warm anoxic basement, 528–530 subsurface lithoautotrophic microbial ecosystems (SLiMEs), 499–502 sulfate-reducing bacteria (SRB), 526–527 sulﬁde segregation HSEs and, 16–18 post-core formation, 16–18 sulfur abundance of, 100–101 atmospheric recycling of, 100–102 biotic recycling of, 100–102 in BSE, 10–12 in Dcalloy/silicate, 25 fractionation of, 18–19 isotope composition, 8 isotope measurements, 101 as magmatic volatile, 188 in mantle, 100–102 in planetary embryos, 26 solar nebula condensation temperature, 14 sulfurization, abiotic, 483–484 sulﬁde inclusions, 100 S isotopic systematics in, 101 sulﬁdes from OIB, 101–102 redox capacity of, 107–108 supercontinent assembly, 621 carbon cycle and, 625–626 of Rodinia, 623–625 super-deep diamonds discovery of, 105 DMGC on, 105 surface diffusion, steric effects and, 364–365 surface processes, carbon cycle and, 279–283 Taupo Volcanic Zone (TVZ), 201–204, 217 Tavurvur, 217 TDLS. See tunable diode laser spectrometers

Downloaded from https://www.cambridge.org/core. IP address: 5.188.219.59, on 06 Oct 2019 at 16:31:36, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108677950.021

Index tectonic building blocks, 292 at subduction zone, 292–293 tectonic carbon cycle, 279, 293–295 temporal variability, 208–209 of active volcanoes, 208–209 of calderas, 209 C/S ratios and, 210–211 of emissions, 208–209 eruption forecasting and, 209–211 terrestrial building blocks, 6–7 tertiary alcohols, 436 tetracarbonates, 80–81 TGA. See thermogravimetric analyses theoretical modeling, constraints from, 6–7 thermal anomalies, in subduction zone, 289 thermochronometer, 92 thermodynamics equilibrium, 388–389 graphite, 465 methane and, 388–389 predictions, 457, 465–466 thermogravimetric analyses (TGA), 451–452 time, energy limits and, 606–607 Titan, 632 titanium, 457–458 transition zone, carbonated sediment in, 134–138 Tropospheric Ozone Monitoring Instrument (TROPOMI), 193 tunable diode laser spectrometers (TDLS), 192 tunneling, 399–400 Turrialba Volcano, 209 TVZ. See Taupo Volcanic Zone UAVs. See unmanned aerial vehicles ultra-basic sites, subsurface biome of, 530–532 ultramaﬁc systems acetate in, 495 carbon in, 494 formate in, 495 methane in, 494–495 oceanic rocky subsurface and, 493–495 United States Geological Survey (USGS), 623 unmanned aerial vehicles (UAVs), 192, 198 upper mantle carbonate melts in, 129 carbonated sediment in, 134–138 schematic representations of, 141 silicate melt in, 143–144 ureilites, carbon in, 16 Urey reaction, 284–285 USGS. See United States Geological Survey vapor bubble volumes, 242 vent emissions, 216 Venus, 321 Vesuvio groundwater, 191–192 Vinet equation of state, 171

669

viruses, in subsurface biome, 536 viscosity-diffusion changes in, 172 incipient melting, 171–172 silicate melt and, 172 volatile elements. See life-essential volatile elements volatile gas solubility, in conﬁned liquids, 370–372 volcanic arcs, 284 volcanic carbon carbon budget and, 216–217 ﬂux of, 215 geologic time and, 215 limits to knowledge about, 217–218 volcanic carbon dioxide, 190 advances in, 192–193 volcanic inputs, carbon ﬂux, 328–329 volcanoes and volcanism active, 194–197, 208–209, 216 carbon outgassed from, 211–215 closed-system, 209 DCO on, 189–190 DECADE on, 189–190 deﬁning, 188 dormant, 198 hydrothermal systems and, 204 petit spot, 179, 238 small volcanic plumes, 198–201 subaerial volcanic budget, 206–207 warm anoxic basement, subsurface biome of, 528–530 water. See also dehydration deep, 105–106 incipient melting and, 165–166, 170–171 as magmatic volatile, 188 melt density and, 170–171 recharge, 487–488 in subduction zone, 291–292 weathering carbonate, 330–331 continental, 310 organic carbon, 330–331 seaﬂoor weathering feedback, 338 silicate, 331–338 wehrlite, 132 whole-Earth carbon cycle box model, 315–316 modeling, 338–341 World Energy Council, 204 xenoliths, 66–67 X-ray diffraction, 91–92 X-ray emission spectroscopy, 77 X-ray microscope, scanning transmission, 485 Yellowstone, 217 Zimbabwe, 95

Downloaded from https://www.cambridge.org/core. IP address: 5.188.219.59, on 06 Oct 2019 at 16:31:36, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108677950.021

Downloaded from https://www.cambridge.org/core. IP address: 5.188.219.59, on 06 Oct 2019 at 16:31:36, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/9781108677950.021