Radar Waveform Design based on Optimization Theory (Radar, Sonar and Navigation) 1785619438, 9781785619434

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Radar Waveform Design Based on Optimization Theory

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Radar Waveform Design Based on Optimization Theory Edited by Guolong Cui, Antonio De Maio, Alfonso Farina and Jian Li

The Institution of Engineering and Technology

Published by SciTech Publishing, an imprint of The Institution of Engineering and Technology, London, United Kingdom The Institution of Engineering and Technology is registered as a Charity in England & Wales (no. 211014) and Scotland (no. SC038698). © The Institution of Engineering and Technology 2020 First published 2020 This publication is copyright under the Berne Convention and the Universal Copyright Convention. All rights reserved. Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may be reproduced, stored or transmitted, in any form or by any means, only with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licences issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publisher at the undermentioned address: The Institution of Engineering and Technology Michael Faraday House Six Hills Way, Stevenage Herts, SG1 2AY, United Kingdom www.theiet.org While the authors and publisher believe that the information and guidance given in this work are correct, all parties must rely upon their own skill and judgement when making use of them. Neither the authors nor publisher assumes any liability to anyone for any loss or damage caused by any error or omission in the work, whether such an error or omission is the result of negligence or any other cause. Any and all such liability is disclaimed. The moral rights of the authors to be identified as authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988.

British Library Cataloguing in Publication Data A catalogue record for this product is available from the British Library

ISBN 978-1-78561-943-4 (hardback) ISBN 978-1-78561-944-1 (PDF)

Typeset in India by MPS Limited Printed in the UK by CPI Group (UK) Ltd, Croydon

To the love of my life Luisa—Antonio To my beloved Franca, always—Alfonso To my loving wife Huifang—Guolong

Contents

About the editors Foreword Notation 1 On recent advances of binary sequence designs and their applications Ronghao Lin and Jian Li 1.1 Introduction 1.2 Algebraic methods 1.2.1 Barker sequences 1.2.2 Legendre sequences 1.2.3 m-Sequences 1.2.4 Gold sequences 1.2.5 Almost perfect autocorrelation sequences 1.2.6 Summary 1.3 Computation algorithms 1.3.1 Iterative twisted approximation 1.3.2 CD algorithm 1.3.3 CAN(PeCAN) family of algorithms 1.3.4 Summary 1.4 Conclusions References 2 Quadratic optimization for unimodular sequence synthesis and applications Guolong Cui, Xianxiang Yu, Goffredo Foglia, Yongwei Huang, and Jian Li 2.1 Introduction 2.2 Problem formulation 2.3 Iterative algorithms for both the continuous and discrete phase cases 2.3.1 Iterative algorithm for continuous phase case 2.3.2 Iterative algorithm for discrete phase case 2.3.3 Power method-like approaches for both the continuous and discrete phase cases

xiii xv xxi

1 1 2 2 3 4 6 8 9 10 10 14 17 30 33 34

37 37 39 40 40 42 45

viii Radar waveform design based on optimization theory 2.4 Numerical examples 2.4.1 Code design to optimize radar detection performance 2.4.2 Spectrally compatible waveform design 2.5 Conclusions Acknowledgments References 3 A computational design of phase-only (possibly binary) sequences for radar systems Mohammad Alaee-Kerahroodi, Augusto Aubry, Mohammad Mahdi Naghsh, Antonio De Maio, and Mahmoud Modarres-Hashemi 3.1 Introduction 3.1.1 Background and previous works 3.1.2 Contribution and organization 3.2 Problem formulation 3.3 CD code optimization 3.3.1 Continuous phase code design 3.3.2 Discrete phase code design 3.4 Numerical examples 3.4.1 Sequence design with good PSL 3.4.2 Sequence design with good ISL 3.4.3 Pareto-optimized solution and designing binary sequences 3.5 Conclusions AppendixA Proof of Lemma 3.1 Appendix B Derivation of the feasibility set Appendix C Proof of Lemma 3.2 References 4 Constrained radar code design for spectrally congested environments via quadratic optimization Marco Piezzo, Yongwei Huang, Augusto Aubry, and Antonio De Maio 4.1 Introduction 4.2 System model 4.3 Figures of merit and constraints 4.3.1 Detection probability 4.3.2 Energy and similarity constraints 4.3.3 Spectral compatibility constraint 4.3.4 Bandwidth priority constraint 4.4 QCQP’s solution methods via rank-one matrix decomposition 4.5 Radar waveform design in a spectrally crowded environment under similarity and spectral coexistence constraints 4.5.1 Code design optimization problem 4.5.2 Performance analysis 4.6 Radar waveform design in a spectrally crowded environment under similarity, energy modulation, and spectral coexistence constraints

46 46 51 58 58 58 63

63 64 65 66 67 69 72 73 76 79 82 85 86 88 89 89 93

93 95 96 97 97 98 99 100 103 103 106

109

Contents 4.6.1 Code design optimization problem 4.6.2 Performance analysis 4.7 Radar waveform design under similarity, bandwidth priority, and spectral coexistence constraints 4.7.1 Code design optimization problem 4.7.2 Performance analysis 4.8 Conclusions A.1 Proof of Theorem 4.1 A.2 Proof of Theorem 4.2 A.3 Proof of Theorem 4.3 A.4 Proof of Proposition 4.1: SDP relaxation tightness for (4.36) References 5 Robust transmit code and receive filter design for extended targets detection in clutter Seyyed Mohammad Karbasi, Augusto Aubry, Antonio De Maio, Mohammad Hassan Bastani, and Alfonso Farina 5.1 Introduction 5.2 Target and signal model 5.2.1 Target model 5.2.2 Signal model 5.3 Problem formulation 5.3.1 Filter matrix optimization 5.3.2 Code matrix optimization 5.4 Filter and code synthesis 5.4.1 Filter synthesis 5.4.2 Code synthesis 5.5 Special case of practical importance: spherical uncertainty set 5.6 Numerical results 5.6.1 TAA uncertainty set size analysis 5.6.2 TAA uncertainty set for different target types 5.6.3 Spherical uncertainty set 5.7 Conclusions AppendixA Proof of Lemma 5.1 Appendix B Proof of Proposition 5.1 References 6 Optimizing radar transceiver for Doppler processing via non-convex programming Augusto Aubry, Mohammad Mahdi Naghsh, Ehsan Raei, Mohammad Alaee-Kerahroodi, and Bhavani Shankar Mysore 6.1 Introduction 6.2 Radar system operation 6.2.1 Transmit waveform 6.2.2 Receiver processing and signal model

ix 109 112 115 115 116 118 119 120 121 122 123

127

127 129 129 130 132 135 136 137 137 137 138 141 143 145 145 148 149 150 151

155

156 157 157 157

x

Radar waveform design based on optimization theory 6.2.3 Clutter and signal independent disturbance characterization 6.2.4 Performance metric for Doppler processing 6.3 Problem formulation and design issues 6.3.1 Constraints and optimization problem 6.3.2 Filter bank optimization: solution to problem Pw (n) 6.3.3 Radar code optimization: solution to problem Ps (n) 6.3.4 Transmit–receive system design: optimization procedure 6.4 Performance analysis 6.4.1 Monotonicity of the proposed method and the impact of similarity constraint 6.4.2 Impact of colored interference 6.4.3 Effect of target Doppler shift interval 6.4.4 Impact of receive filter bank size 6.4.5 Impact of sequence length on performance 6.4.6 Performance comparison 6.5 Conclusions AppendixA Proof of Proposition 6.1 Appendix B Proof of Proposition 6.2 Appendix C Proof of Lemma 6.1 References

7 Radar waveform design via the majorization–minimization framework Linlong Wu and Daniel P. Palomar 7.1 Introduction 7.2 Preliminaries: the MM method 7.2.1 The vanilla MM method 7.2.2 Convergence analysis 7.2.3 Acceleration schemes 7.2.4 Extension to the maximin case 7.3 Joint design of transmit waveform and receive filter 7.3.1 System model and problem formulation 7.3.2 MM-based method for joint design with multiple constraints 7.3.3 Numerical experiments 7.4 Robust joint design for the worst-case SINR maximization 7.4.1 Problem formulation 7.4.2 MM-based method for robust joint design 7.4.3 Numerical experiments 7.5 Conclusion AppendixA Proof of Lemma 7.1 Appendix B Proof of Lemma 7.4 Appendix C Proof of Lemma 7.5 Acknowledgment References

159 160 160 160 163 164 166 167 168 169 169 170 172 173 174 176 178 180 181

185 185 186 186 187 188 189 190 191 192 199 202 204 205 211 213 215 216 216 217 217

Contents 8 Lagrange programming neural network for radar waveform design Junli Liang, Yang Jing, Hing Cheung So, Chi Sing Leung, Jian Li, and Alfonso Farina 8.1 Introduction 8.2 Basics of LPNN 8.2.1 Problem statement 8.2.2 Lagrange programming neural network 8.3 LPNN for waveform design with spectral constraints 8.3.1 Problem statement 8.3.2 Algorithm development 8.3.3 LPNN stability analysis 8.4 LPNN for designing waveform with low PSL 8.4.1 Problem statement 8.4.2 Algorithm description 8.4.3 LPNN stability analysis 8.4.4 Summary of proposed algorithm 8.5 Numerical examples 8.5.1 Experiment 1: Flat spectrum waveform design 8.5.2 Experiment 2: Spectrally constrained waveform design for radar 8.5.3 Experiment 3: Region of interest around main lobe 8.5.4 Experiment 4: Region of interest on one side of main lobe 8.5.5 Experiment 5: Low-sidelobe autocorrelation level 8.6 Conclusions A.1 Positive definiteness of Hessian matrix of (8.48) A.2 Solution to (8.58) A.3 Adaptive selection scheme of C0 A.3.1 On positive definiteness of ∇θ2θ Lx¯ A.3.2 On positive definiteness of Z 0 A.3.3 On positive definiteness of Hessian matrix H  of (8.49) References

xi 221

221 223 223 224 228 228 229 231 233 233 234 237 239 240 241 245 249 250 252 253 253 254 255 255 256 257 257

9 Cognitive local ambiguity function shaping with spectral coexistence and experiments Guolong Cui, Jing Yang, Xiangxiang Yu, and Lingjiang Kong

263

9.1 Introduction 9.2 Problem formulation 9.2.1 Weighted integrated sidelobe level 9.2.2 Spectral coexistence 9.2.3 Optimization problem 9.3 Iterative sequential quadratic optimization algorithm 9.4 Numerical results 9.4.1 Simulation results 9.4.2 Experimental results

263 264 264 265 265 266 270 270 277

xii Radar waveform design based on optimization theory 9.5 Conclusions AppendixA Proof of Proposition 9.1 Appendix B Proof of (9.17) Appendix C Proof of Proposition 9.2 Acknowledgments References 10 Relative entropy-based waveform design for MIMO radar Bo Tang and Jun Tang 10.1 Introduction 10.2 Signal model and problem formulation 10.2.1 Signal model 10.2.2 Problem formulation 10.3 Two-stage algorithm design 10.3.1 Synthesis of energy-constrained waveforms 10.3.2 Convergence and computational complexity analysis 10.3.3 Extension to the synthesis of constant-modulus waveforms 10.3.4 Extension to the synthesis of similarity-constrained waveforms 10.4 One-stage algorithm design 10.4.1 Minorizing Part A 10.4.2 Minorizing Part B 10.4.3 Minorizing Part C 10.4.4 The minorized problem at the (k + 1)th iteration 10.4.5 Convergence and computational complexity analysis 10.4.6 Extension to include other constraints 10.4.7 Accelerated schemes for the one-stage methods 10.5 Numerical examples 10.6 Concluding remarks AppendixA Proof of (10.19) Appendix B Proof of Lemma 10.1 Appendix C An introduction to minorization–maximization Acknowledgment References Index

281 281 282 283 283 284 287 287 289 289 290 292 293 297 298 300 301 302 303 303 304 305 306 307 309 313 314 314 316 317 317 321

About the editors

Guolong Cui is a professor with the University of Electronic Science and Technology of China, Chengdu. His current research interests include waveform optimization theory and algorithms, adaptive signal processing, and statistical signal processing with emphasis on radars. Antonio De Maio is a professor with the University of Naples Federico II. His research interest is in the field of statistical signal processing, with emphasis on radar detection and optimization theory applied to radar signal processing. Dr. De Maio is a fellow member of IEEE and the recipient of the 2010 IEEE Fred Nathanson Memorial Award as the young (less than 40 years of age) AESS Radar Engineer 2010. Alfonso Farina (fellow of IEEE, FREng, FIET, fellow of EURASIP), Selex ES (retired) is a consultant in Roma, Italy. Dr. Farina is the author of more than 800 peerreviewed technical publications and of books and monographs (published worldwide). Awards he has received recently include the IET Achievement Medal for outstanding contributions to radar system design, signal, data and image processing, and data fusion (2014) and Christian Hülsmeyer Award from the German Institute of Navigation (DGON) (2019). Jian Li is a professor with the Department of Electrical and Computer Engineering, University of Florida, Gainesville, USA. In fall 2007, she was on sabbatical leave at MIT, Cambridge, Massachusetts. Her current research interests include spectral estimation, statistical and array signal processing, and their applications. Dr. Li is a fellow of IEEE and a fellow of IET.

Foreword

“Se comprendere é impossibile, conoscere é necessario.”∗ Primo Levi†

This book deals with innovative mathematical techniques, based on modern optimization theory, to design waveforms for radar and gather information from targets on the surveilled environment. One celebrated and most famous radar waveform is the Chirp, though the math techniques described in the book will, by necessity, go also beyond the chirp. Chirps seem ubiquitous in Nature (e.g., bird song, bat emitted signals, etc.), as well as in technology (commonly used in radar, sonar, laser, and spread-spectrum communications‡ ). In radar, its conception can be marked by the paper by J. H. Klauder, A. C. Pierce, S. Darlington, and W. J. Albersheim in 1960, “The theory and design of chirp radars” [1]. This new technique was called Chirp at Bell Telephone Laboratories. Today we know that chirp appears in its full glory in the fascinating and “gorgeous” astrophysics events related to the detection of gravitational waves, so relevant to broaden the knowledge about the Universe. One of the editors, Alfonso Farina, of this book was so lucky to attend to the fascinating banquet speech at the recent IEEE Radar Conference, the 24 April 2019, Boston (MA). The speech entitled “Gravitational waves—The New Cosmic Messenger,” was given by Dr. David H. Shoemaker [2], Senior Research Scientist, Leader, Advanced LIGO, MIT Kavli Institute (USA). Dr. Shoemaker [2] narrated the story and the gigantic scientific and technology efforts of so many devoted people and related source funds that enabled the first direct observation of gravitational waves made on 14 September 2015 and announced by the LIGO (Laser Interferometer Gravitational-Wave Observatory) and Virgo collaborations on 11 February 2016 [3]. Dr. Shoemaker [2] explained that binary stellar-mass black hole [4] systems merged at the boundary of the Universe and distorted the space-time fabric provoking gravitational waves, in accordance to A. Einstein’s theory of general relativity. The signal detected by the LIGO was a chirp! ∗

“If understanding is impossible, knowing is necessary.” https://en.m.wikipedia.org/wiki/Primo_Levi?wprov=sfla1. ‡ In Latin, spectrum means “image” or “apparition”. In the 17th century, the word “spectrum” was introduced into optics by Isaac Newton, referring to the range of colors observed when white light was dispersed through a prism. Newton I. “A letter of Mr. Isaac Newton … containing his new theory about light and colours …”. Philosophical Transactions of the Royal Society of London. 1671;6(80):3075–3087. doi: 10.1098/rstl.1671.0072. The word “spectrum” to describe a band of colors that has been produced, by refraction or diffraction, from a beam of light first appears on p. 3076. https://en.wikipedia.org/wiki/ Spectrum. †

xvi Radar waveform design based on optimization theory “Un cinguettio” we would say in Italian. A dramatic event at the boundary of the Universe, and a chirp perceived on the Earth. Gathering chirps from various sources and various “messenger media,” appear interesting in their somewhat noticeable diversity and, nonetheless, “all sharing the common capability to carry information.” Following this lead, the book will move on to introduce novel mathematical techniques to design radar waveforms which aim to improve the capability to extract information from the environment surrounding the radar. The editors would like to express their sincere gratitude to Dr. David H. Shoemaker [2] for his kind gesture to help in shaping the above introducing paragraphs of the book Foreword. As a matter of fact, radar waveform design is a hot topic within scientific and industrial communities due to the advent of new computing architectures, high-speed and off-the-shelf processors, arbitrary digital waveform generators, modern phasedarray with multiple transmit and receive channels, solid-state transmitters, etc. Indeed, suitable radar waveform synthesis and diversification can provide unique capabilities to enhance radar performance in terms of detection, classification, identification, and localization above all in severe interference scenarios characterized by the presence of clutter, jammer, and coexisting radiators. Transmit adaptivity is also at the base of a cognitive radar paradigm [5–7] to implement the perception-action cycle as aimed at playing a key role in the stimulation of the environment. In many situations the radar waveform synthesis is obtained as the solution to a constrained optimization problem where the constraints are dictated by both endogenous and exogenous information possibly gathered as the result of the perception cycle or stored in the transceiver as a priori memory. Radar waveform design algorithms are also fundamental to build waveform libraries from which the most appropriate ones can be selected on the “fly.” Here again, optimization theory turns out useful to address the specific performance metric optimization (objective function) as for instance the Integrated Sidelobe Level (ISL) or Peak to Sidelobe Level (PSL). As a matter of fact, there has been in the last 15 years a proliferation of innovative techniques for radar signal design heavily exploiting mathematical results developed in the context of optimization theory and operations research. This book attempts to provide an overview of some radar signals obtained as the result of an optimization process. Moreover, it tries to cover some of the most challenging application fields including binary waveform design, adaptation in the presence of signal-dependent interference, waveform operation in contested and congested environments, radar ambiguity function design, information-theoretic signal synthesis for Multiple-Input Multiple-Output (MIMO) radar. Innovative and sophisticated instruments from optimization theory and operations research such as the Coordinate Descent (CD) technique, semidefinite relaxation, rank-one matrix decomposition, Lagrange duality theory, fractional programming, Dinkelbach’s procedure, Pareto optimization, and Majorization–Minimization (MM)§ will be framed in the context of radar waveform design. §

Notice that MM is used to denote both: Majorization-Minimization and Minorization-Maximization. The actual meaning depends on the specific optimization context.

Foreword xvii It is worth pointing out that there are nowadays in the open literature many high quality books dealing with the hot topic of radar waveform design (see for instance [8–12]). Nevertheless, to the best of our knowledge, this is the first book focused on the specific topic of radar waveform design exploiting tools from modern optimization theory. The cross-fertilization between radar waveforms and optimization theory is indeed the common thread (leitmotiv) which makes the proposed book almost unique. The book balances a practical point of view with a rigorous mathematical approach corroborated with a wealth of numerical study cases and some real experiments. Additionally, the book has a cross-disciplinary approach as it leverages the cross-fertilization by the recent research and discoveries in optimization theory. The material is organized in ten chapters, each one comprising a very comprehensive list of references. Specifically, Chapter 1 addresses binary sequence design which is of great interest for many application areas including active sensing, communications, and medical imaging. In recent years, computational methods have been developed to synthesize arbitrary waveforms under various practical constraints, including the constant modulus and the spectral containment restrictions. In fact, as the hardware cost of arbitrary waveform generators decreases, arbitrary waveforms are becoming widely popular in diverse advanced systems especially those based on cognitive active sensing. However, for massive commercial markets, the active sensing systems, such as those pertaining to the automotive radar, must be rather inexpensive. Therefore, highly diverse binary sequences with arbitrary lengths are primary candidates for practical implementations. In Chapter 1, the authors review important issues concerning the state of the art in binary sequence design as well as the properties of some relevant family of codes. Besides, they also discuss recent advances in binary sequence synthesis and their applications. Chapter 2 considers unimodular sequence synthesis under a similarity constraint for both the continuous and the discrete phase code alphabet. A computationally efficient iterative algorithm for the continuous phase case is proposed to optimize sequentially the quadratic objective function. The resulting quadratic problem is handled via multiple one-dimensional optimization instances with closed-form solutions. For the discrete phase case, an iterative block optimization algorithm is presented. More specifically, the design variables are partitioned into K blocks, and then, each block is sequentially optimized via exhaustive search while fixing the remaining K − 1 blocks (alternate optimization). Finally, some radar applications are provided to examine the capabilities of the discussed algorithms. Results have shown that the methods exhibit low computational costs and interesting performance gains in comparison with power method-like and semi-definite relaxation-related techniques. The goal of Chapter 3 is to provide a comprehensive study on the design of unimodular sequences possessing good aperiodic correlation properties for radar systems. First, the state of the art about this long-standing topic is provided reviewing various computationally efficient techniques. Then, a recent and technically sound framework aimed at designing sets of continuous/discrete phase sequences with good aperiodic autocorrelation features (in terms of PSL and ISL) is proposed. Specifically, resorting to the Pareto optimization approach, the weighted sum of the ISL- and PSLbased metrics is considered as the objective function to optimize under the phase-only

xviii Radar waveform design based on optimization theory constraints on the probing waveforms. Hence, an iterative procedure based on the CD (Coordinate Descent) method is introduced to deal with the resulting Non Polynomial (NP)-hard optimization problems. Each iteration of the method requires the solution of a non-convex min-max problem. It is handled either through a bisection or a Fast Fourier Transform (FFT)-based method respectively for the continuous and the discrete phase constraint. Several numerical examples have been provided to illustrate the performance of the method in comparison with the state of the art. Chapter 4 addresses radar waveform design in a spectrally crowded environment via constrained quadratic programming techniques with the goal to provide a unifying and systematic overview of some recent results available in the open literature. A taxonomy of the most relevant performance measures to optimize and/or control the waveform design process is provided. Several radar waveforms design problems are also presented aimed at maximizing a specific criterion (for instance the Signal to Interference plus Noise Ratio (SINR)), while accounting for different combinations of the performance measures in terms of constraints. Regardless of the chosen objective function and constraints, waveform design is formulated as a non-convex Quadratically Constrained Quadratic Program (QCQP), in general quite difficult to solve. Hence, a specific methodology, leveraging on rank-one matrix decomposition tools, is exploited to prove the hidden convexity of some instances of the waveform optimization or to build bespoke solution procedures ensuring good quality radar waveforms. In Chapter 5, a joint design of the radar transmit code and the receive filter for an extended target scenario is addressed. As a worst case, over the Target Impulse Response (TIR), SINR is used as the figure of merit for waveform optimization under a Peak-to-Average power Ratio (PAR) constraint. In order to handle the resulting NPhard problem, first of all an equivalent reformulation is given in terms of transmit code and receive filter matrix variables subject to rank-one constraints. Then, an iterative procedure is devised to obtain high-quality solutions to a relaxed version of the original problem. Hence, a randomization step is applied to form an optimized radar waveform and receive filter feasible for the original waveform design problem. Several numerical examples are finally presented to highlight the effectiveness of the method. Intertwined optimization of transmitter and receiver radar architectures for Doppler processing applications is addressed in Chapter 6 assuming the presence of signal-dependent interference. With the purpose of dealing with unknown target Doppler shift, a bank of filters is used at the receiver side. Besides, the worst case SINR at the output of the filter bank is chosen as the figure of merit to be optimized under both a similarity and an energy constraint on the transmit signal. Based on a suitable reformulation of the original non-convex max-min problem, an optimization procedure which monotonically improves the worst-case SINR and converges to a stationary point is developed. Each iteration of the algorithm involves both a convex and a generalized fractional programming problem which can be globally solved via the generalized Dinkelbach’s procedure with a polynomial computational complexity. Several numerical examples and case studies are then reported. Chapter 7 discusses the general MM (Majorization–Minimization) algorithmic framework, which turns out very useful and flexible for nonconvex problems. Actually,

Foreword xix many existing algorithms can be interpreted from MM’s perspective which has been successfully exploited in many applications. This chapter presents, in a self-contained manner, the MM framework and its use for radar waveform design. Two radar signal problems are presented and solved based on this perspective. Through these design problems, several key issues of applying MM and some organic combinations of MM with other methods are illustrated as well. Simulation results show that the MM-based algorithms are time-efficient and meet the typical real-time requirements for radar waveform design. In Chapter 8, a constrained optimization solution technique known as the Lagrange Programming Neural Network (LPNN) is exploited to design radar probing waveforms under specific constraints related to spectral requirements, unit-modulus, and/or ambiguity function. In the LPNN, there are two types of neurons involved in the computation of the optimal solution, namely, variable and Lagrangian neurons, which are responsible for finding a minimum point of the cost function as well as the solution at an equilibrium point, and for leading the dynamic trajectory into the feasible region. Local stability conditions of the dynamic neuron model are also available and discussed whereas simulation results show that the LPNN-based approach is a competitive alternative for waveform design. Chapter 9 deals with the problem of the Local Ambiguity Function (LAF) shaping to refrain the sidelobes of a strong return from masking weak targets and to ensure spectral coexistence in the presence of narrowband spectral interferences (e.g., the communication signal and jamming). An Iterative Sequential Quartic Optimization (ISQO) algorithm sharing polynomial-time complexity is proposed to minimize the objective function, accounting for the Weighted Integrated Sidelobe Level (WISL) and the spectral stop-band energy, under an energy constraint and a PAR constraint on the probing waveform. In each iteration, the procedure decomposes the original nonconvex quartic problem into a tractable quadratic subproblem which can be further converted to a linear optimization problem with a closed-form solution. The numerical simulations are provided to verify the effectiveness of ISQO and the results show that it outperforms some available counterparts in the aspect of the convergence speed and the detectability of multiple high-speed targets. Finally, the real data, collected by the experiment system based on an arbitrary waveform generator, are provided and discussed. The last chapter of the book (Chapter 10) deals with an information-theoretic waveform design method for MIMO radar systems. The goal is to enhance the detection performance of the MIMO radar class in the presence of some disturbances (including clutter and colored noise). To this end, the relative entropy between the probability density functions (pdfs) of the observations under the two hypotheses (target presence and absence) is chosen as the design metric. Several algorithms are discussed to tackle the encountered optimization problems. Besides, some of the developed algorithms are extended via the addition of some practical constraints including the constant-modulus and similarity ones. Finally, some numerical examples are given to demonstrate the effectiveness of the waveform design procedures. Each chapter is self-contained and is written by renowned researchers in radar waveform design/optimization theory. The emphasis of the book is on both theoretical results and practical applications and will clearly show the potential benefits in radar

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Radar waveform design based on optimization theory

signal processing waveform design achievable using modern optimization theory. A common list of symbols (reported at the end of this foreword) and extensive cross referencing has been realized so that the related material can be easily found and connected, thus significantly enhancing the book’s value as a reference. This book is addressed to systems engineers and their managers in the aerospace and defense sector, technical staff in procurement agencies and their technical advisors, students at MSc and PhD levels in signal processing, electrical engineering, optimization theory, and people interested in applications of optimization theory to radar engineering. The Editors wish to warmly thank the professionality and friendship of the other authors who have collaborated in writing some of the chapters in this book. Their cooperation has been precious to shape and complete the book. Dear Readers your criticisms and comments are most welcome. The Editors hope that the “technical trip” from the chirp waveform and beyond will be enjoyable and useful for your work. Much remains to be done/discovered, because—as it is well known—R&D is an endless travel sometimes weary and hopefully often a fun. Guolong Cui, Antonio De Maio, Alfonso Farina, and Jian Li

References [1] [2] [3] [4]

[5] [6] [7] [8] [9] [10] [11] [12]

Klauder JR, Price AC, Darlington S, Albersheim WJ. The theory and design of chirp radars. The Bell System Technical Journal. 1960;39(4):745–808. Shoemaker DH. Gravitational waves—The new cosmic Messenger. Available from: https://dcc.ligo.org/LIGO-G1802274/public. Abbott BP, Abbott R, Abbott TD, et al. Observation of gravitational waves from a binary black hole merger. Physical Review Letters. 2016;116(6):061102. Abbott BP, Abbott R, Abbott TD, et al. A three-detector observation of gravitational waves from a binary black hole coalescence. Physical Review Letters. 2017;119(14):141101. Haykin S. Cognitive Dynamic Systems: Perception-action Cycle, Radar, and Radio. New York: Cambridge University Press; 2012. Guerci JR. Cognitive Radar: The Knowledge-aided Fully Adaptive Approach. Norwood, MA: Artech House; 2010. Farina A, De Maio A, Haykin S, editors. The Impact of Cognition on Radar Technology. London: The IET; 2017. Levanon N, Mozeson E. Radar signals. John Wiley & Sons; 2004. Gini F, De Maio A, Patton L, editors. Waveform Design and Diversity for Advanced Radar Systems. London: The IET; 2012. Wicks MC, Mokole EL, Blunt SD, Schneible RS, Amuso VJ. Principles of Waveform Diversity and Design. Raleigh, NC: Scitech Publishing; 2011. He H, Li J, Stoica P. Waveform Design for Active Sensing Systems: A Computational Approach. New York: Cambridge University Press; 2012. Pillai SU, Li KY, Selesnick I, Himed B. Waveform Diversity: Theory & Applications. New York: McGraw-Hill; 2011.

Notation

a A a(n) or an A(m, n) ( · )T ( · )∗ ( · )† tr ( · ) det ( · ) Rank ( · ) λmax ( · ) λmin ( · ) Range(X ) I 0 RN CN CN ×M HN Z N R≥0 R N, i=1 and the entries of G −1 (X ) are given by ⎛ ⎞ 2N −1   −1   −1    1 G (X ) k,l = G (Y ) k,l + ⎝N δk−1 − G −1 (Y M ) ⎠ , k i=1

  for all (k, l) such that G −1 (M ) k,l = 1 and zeros otherwise.

(1.27)

k,i

Next, the orthogonal projection of an element of AP on AP is found [12]. Theorem 1.3 ([12]). For any X ∈ AP , let Y be the optimal projection of X in AP . Then N Y = · vv† , (1.28) v 2 T  where v = v0 , 01×(N −1) (2N −1)×1 ∈ C(2N −1)×1 , v0 is the eigenvector corresponding to the largest eigenvalue of X 0 , X 0 is the n × n upper left submatrix of X . Since X is a real-valued symmetric matrix, its eigenvectors are real valued. To transform the real-valued sequences to binary ones, the following converging function is considered in [12] h(t, s) = sgn(t) · |t|e where sgn(t) =



−μs

1, t ≥ 0, −1, t < 0.

,

(1.29)

(1.30)

where h(t, s) maps a generic t to its closest binary value as s → +∞, with μ being a positive real-valued parameter. Note that the size of μ can affect the speed of

On recent advances

13

Algorithm 1: ITROX for aperiodic sequence design [12] Input: Consider an initial point X 0 = G(X0 ) ∈ AP . Let s = 0. Step 1: Compute the eigenvalue decomposition of X s and find Y s ∈ AP according to Theorem 1.2. s = h(Y , s). Step 2: Compute Y Step 3: Compute X s+1 according to Theorem 1.3. s s−1 (i, j) < ξ . Otherwise, update Iteration: Repeat Steps 1–3 until max  Y (i, j) − Y i,j

r(k)

s = s + 1 and continue the iterations. Output: Set the optimized sequence as [sgn(v(1)) , . . . , sgn(v(N ))]T , where v is the s . eigenvector corresponding to the largest eigenvalue of Y

100 90 80 70 60 50 40 30 20 10 0 –10 –20

ITROX

–50

0 Shift (k)

50

Figure 1.10 The aperiodic autocorrelation function of a length-100 sequence obtained by ITROX

convergence—the larger the value of μ, the faster the convergence of f (t, s). For any N × N matrix X , let h(X , s) be [h(X , s)]k,l = h(X (k, l) , s) .

(1.31)

Then the ITROX algorithm for aperiodic sequence design can be summarized in Algorithm 1 [12]. It is worthwhile to note that adding a converging function does not affect the general convergence guarantee of ITROX [12]. Taking N = 100 as an example to analyze the optimized sequence obtained by ITROX, Figure 1.10 shows the aperiodic autocorrelation function obtained by ITROX, and the PSL is 10.

14

Radar waveform design based on optimization theory

1.3.1.2 Periodic case ITROX can also be used for periodic binary sequence designs [12]. Consider the set P . The sequences corresponding to the set have the ideal zero periodic sidelobes. That is, if X0 = xx † , and X0 ∈ P , then x can satisfy the total energy constraint and the sidelobe values are all zero. The constraint for P is G −1 (X0 ) 1 = N e1 .

(1.32)

The definition of P is similar to that of AP , and this set guarantees that the matrix can be decomposed, i.e., if X ∈ P , then X is a rank-one matrix. For the periodic case, the ideal sequence should satisfy c(0) = N and c(i) = 0, i = 1, . . . , N − 1, i.e., xx † ∈ P ∩ P . However, such an ideal sequence hardly exists. Therefore, ITROX uses alternating iterations again [12], starting from X 0 ∈ P , to find the closest element to X 0 (based on the · F norm) in P denoted by Y 0 , which is called the optimal projection of X 0 on P . Next, the optimal projection of Y 0 on P is found and is denoted by X 1 . Repeating these projections, the distance between Y s in P and X s+1 in P monotonically decreases, and hence the convergence is guaranteed. The following two theorems demonstrate how to find the orthogonal projections on P and P [12]. Theorem 1.4 ([12]). For any Hermitian matrix Y , let X be the optimal projection of Y in P . Then the entries of G −1 (X ) are given by ⎞ ⎛ N   −1  −1  −1    1⎝ G (X ) k,l = G (Y ) k,l + (1.33) G (Y ) ⎠ . N δk−1 − N i=1 k,i

Theorem 1.5 ([12]). For any X ∈ P , let Y be the optimal projection of X in P . Then N Y = · vv† , (1.34) v 2 where v is the eigenvector corresponding to the largest eigenvalue of X . The converging functions (1.29) and (1.31) are used similarly in [12] to transform the complex-valued sequences to binary sequences. Then the ITROX algorithm for periodic sequence design can be summarized in Algorithm 2 [12]. Taking N = 100 as an example to analyze the optimized sequence obtained by ITROX, Figure 1.11 shows the periodic autocorrelation function obtained by ITROX, and the PSL is 12.

1.3.2 CD algorithm The coordinate-descent (CD) algorithms are iterative methods. Each iteration is performed by fixing most components of the variable vector at their most recent values from the previous iteration and optimizes the objective function with respect to the

On recent advances

15

Algorithm 2: ITROX for periodic sequence design [12] Input: Consider an initial point X 0 = G(X0 ) ∈ P . Let s = 0. Step 1: Compute the eigenvalue decomposition of X s and find Y s ∈ P according to Theorem 1.4. s = h(Y , s). Step 2: Compute Y Step 3: Compute X s+1 according to Theorem 1.5. s s−1 (i, j) < ξ . Otherwise, update Iteration: Repeat Steps 1–3 until max  Y (i, j) − Y i,j

c(k)

s = s + 1 and continue the iterations. Output: Set the optimized sequence as [sgn(v(1)) , . . . , sgn(v(N ))]T , where v is the s . eigenvector corresponding to the largest eigenvalue of Y

100 90 80 70 60 50 40 30 20 10 0 –10 –20

ITROX

–50

0 Shift (k)

50

Figure 1.11 The periodic autocorrelation function of a length-100 sequence obtained by ITROX

remaining components in the current iteration [13]. A CD-based method proposed in [14] can also be used to design binary sequences with good aperiodic autocorrelation properties, while paving the way for low periodic autocorrelation sequence design by modifying its descent coefficients through the iterations [14].

1.3.2.1 Aperiodic case As described in Chapter 3, binary sequences with good aperiodic autocorrelation properties are generated by choosing m = 2. Taking N = 100 as an example to analyze the optimized sequence obtained by the CD algorithm, Figure 1.12 shows the aperiodic autocorrelation function obtained by lp -norm + CD algorithm, and the PSL is 8.

Radar waveform design based on optimization theory

r(k)

16

100 90 80 70 60 50 40 30 20 10

lp-Norm+CD algorithm Starting sequence

0 –10 –20 –30 –50

0

50

Shift (k)

Figure 1.12 The aperiodic autocorrelation function of a length-100 sequence obtained by the proposed CD algorithm

1.3.2.2 Periodic case Define the objective function similarly:  fθP (x)

=

max

k=1,...,N −1

θ |c(k)| + (1 − θ ) 2

N −1 

 |c(l)|

2

,

(1.35)

l=1

 −1 |c(k)|2 when θ = 0, and where θ ∈ [0, 1] is the Pareto weight. fθ (x) = ISL = Nk=1 k=N −1 fθ (x) is equivalent to PSL = max {|c(k)|}k=1 when θ = 1. Therefore, constructing a sequence with good periodic autocorrelation properties can be formulated as the following optimization problem:  min fθP (x) , θ x (1.36) Q s.t. x ∈ , where

 = x = [x (1) , . . . , x (N )]T | x (n) = 1 or − 1,

n = 1, . . . , N .

The CD algorithm solves the following problem in each iteration: 

s−1 , min fθP xd ; x−d θ x d Qd, xs−1 s.t. xd ∈ {1, −1} .

(1.37)

(1.38)

By modifying the CD coefficients, namely, adk = xs−1 ([(d + k − 1) mod N ] + 1) + xs−1 ([(d − k − 1) mod N ] + 1) ,

(1.39)

c(k)

On recent advances 100 90 80 70 60 50 40 30 20 10 0 –10 –20 –30

17

CD algorithm Starting sequence

–50

0 Shift (k)

50

Figure 1.13 The periodic autocorrelation function of a length-100 sequence obtained by the proposed CD algorithm

cdk =

N 



xs−1 (i) xs−1 (i + k) ,

(1.40)

i=1, i =d, (d−k−1)modN +1

where the CD algorithm can be used to generate binary sequences with good periodic autocorrelation properties. Similarly, lp -norm minimization is performed to optimize PSL. Taking N = 100 as an example to analyze the optimized sequence obtained by the CD algorithm, Figure 1.13 shows the periodic autocorrelation function obtained by lp -norm + CD algorithm, and the PSL is 8.

1.3.3 CAN(PeCAN) family of algorithms The recently proposed methods of the cyclic algorithm-new (CAN) and periodic CAN (PeCAN) are known to be computationally efficient, particularly because they employ fast Fourier transform (FFT) operations to design unimodular sequences with good correlation properties [6,15–20]. However, these cyclic algorithms cannot be used directly to design binary sequences with good autocorrelation properties due to the extremely multimodal nature of the associated optimization objectives. We present two sets of algorithms for designing binary sequences based on CAN (PeCAN), which are implemented by relaxing the unimodularity constraint and incorporating convergence functions. The resulting algorithms are referred to as CANARY (PeCANARY) and 1bCAN (1bPeCAN). Below, we first introduce CAN [15] and PeCAN [18] briefly, which were originally proposed to design constant-modulus sequences with arbitrary phases.

18

Radar waveform design based on optimization theory

1.3.3.1 Cyclic algorithm-new Define the spectral function: N 2 

(ω) = x(n) e−jωn = n=1

N −1 

r(k)e−jωk ,

ω ∈ [0, 2π ) .

(1.41)

k=−(N −1)

Then the ISL metric can be written as ISL =

2N 2 1 

ωp − N , 4N p=1

ωp =

π p, N

p = 1, . . . , 2N .

(1.42)

Consequently, CAN attempts to solve the following optimization problem: ⎡ ⎤2 2 2N N   ⎣ min x(n) e−jωp n − N ⎦ , s.t. |x(n)| = 1, n = 1, . . . , N . {x(n)}N n=1 p=1

n=1

(1.43) Notice that the criterion in (1.43) is a quartic function of {x(n)}. To circumvent the difficulties arising from optimizing a quartic objective, CAN takes an approximate approach and optimizes a related almost equivalent quadratic optimization problem: N 2 2N   √ min x(n) e−jωp n − N e jψp , 2N N {x(n)}n=1 ;{ψp }p=1 p=1 n=1 (1.44) s.t. ψp ∈ [0, 2π ] , p = 1, . . . , 2N , |x(n)| = 1,

n = 1, . . . , N .

Briefly speaking, if the criterion in (1.44) takes on a small value then so does (1.43). Let   α †p = e−jωp , . . . , e−j2N ωp , (1.45) and

† ⎤ α1 ⎢ .. ⎥ ⎣ . ⎦.

⎡ 1 A† = √ 2N

(1.46)

† α 2N

Then A† is a unitary 2N × 2N discrete Fourier transform (DFT) matrix. Let z = [x(1) , . . . , x(N ) , 0, . . . , 0]T2N ×1 ,

(1.47)

T 1  v = √ e jψ1 , . . . , e jψ2N . 2

(1.48)

and

On recent advances

19

Algorithm 3: CAN  T Input: Let x 0 = x0 (1) , . . . , x0 (N ) be a vector with random initial phases; e.g., x0 (n) can be set to e j2π θ (n) , where θ(n) are independent random variables uniformly distributed in [0, 1). Let s = 1. ! !2  2N Step 1: Fix x s−1 and compute ψp p=1 by minimizing !A† z s−1 − vs ! . More precisely, set ψp = arg fp , p = 1, . . . , 2N , by letting f = A† z s−1 . ! !2  2N Step 2: Fix ψp p=1 and compute x s by minimizing !A† z s − vs ! = z s − Avs 2 : Set xs (n) = e j arg(gn ) , n = 1, . . . , 2N , by letting g = Avs . Let x s = [xs (1) , . . . , xs (N )]T . Iteration: Repeat Steps 1 and 2 until |Ds − Ds−1 | < ξ . Otherwise, update s = s + 1 and continue the iterations. Output: The optimized sequence x s .

Then CAN solves the following optimization problem iteratively, as presented in a matrix form: ! † ! !A z− v! min 2N {x(n)}N n=1 ;{ψp }p=1 (1.49) s.t. ψp ∈ [0, 2π ] , p = 1, . . . , 2N , |x(n)| = 1,

n = 1, . . . , N .

CAN minimizes (1.49) by alternating the optimization between x and v. Let z s = [xs (1) , . . . , xs (N ) 0, . . . , 0]T2N ×1 , and let Ds represent the value of ! † s ! !A z − vs !2 at iteration s. Then we have ! !2 ! !2 ! !2 Ds−1 = !A† z s−1 − vs−1 ! ≥ !A† z s−1 − vs ! ≥ !A† z s − vs ! = Ds .

(1.50)

In other words, the value of D!s decreases !2 monotonically with each iteration. Hence, the CAN algorithm can reduce !A† z− v! iteratively and converges to a locally optimal solution. The CAN procedure is summarized in Algorithm 3.

The CAN procedure results in a unique navigation through the waveform search space when used for aperiodic binary sequence design. Particularly, we can prove the following result [21]. Theorem 1.6. If x s ∈ (defined in (1.37)) and x s+1 is the sequence obtained after one more iteration of the CAN algorithm, then x s+1 ∈ . Interestingly, if the initial sequence is drawn from , then the iterations of CAN are exclusively performed on , and thus resulting in a sequence in at convergence.

20

Radar waveform design based on optimization theory

Note that while Theorem 1.6 shows that CAN is able to produce binary outcomes in its quest for sequences with low aperiodic autocorrelation sidelobes, it does not work effectively, as shown later (in Figures 1.18 and 1.19).

1.3.3.2 Periodic cyclic algorithm-new In the periodic case, we have c(k) = r(k) + r(N − k)∗ = r(k) + r(k − N ),

k = 1, . . . , N − 1,

(1.51)

and c(0) = r(0). Therefore, let σp = (2π/N )p. Then N −1 



σp =

r(k)e

−jσp k

k=−(N −1)

=

N −1 

=

N −1 

(r(k) + r(k − N )) e−jσp k + r(0)

k=1

(1.52)

c(k)e−jσp k .

k=0

It is easy to prove that ISL =

N 2 1 

σp − N , N p=1

σp =

2π p, N

p = 1, . . . , N .

(1.53)

Similarly, PeCAN attempts to solve the following optimization problem: ⎡ ⎤2 2 N N   ⎣ x(n) e−jσp n − N ⎦ , s.t. |x(n)| = 1, n = 1, . . . , N . min {x(n)}N n=1 p=1

n=1

(1.54) Notice that the criterion in (1.54) is a quartic function of {x(n)}. PeCAN also takes an approximate approach and optimizes a related almost equivalent quadratic optimization problem: N 2 N   √ min x(n) e−jσp n − N e jψp , N N {x(n)}n=1 ;{ψp }p=1 p=1 n=1 (1.55) s.t. ψp ∈ [0, 2π ] , p = 1, . . . , N |x(n)| = 1,

n = 1, . . . , N .

Briefly speaking, if the criterion in (1.55) takes on a small value then so does (1.54). Let   β †p = e−jσp , . . . , e−jN σp , (1.56) and

† ⎤ β1 1 ⎢ ⎥ F † = √ ⎣ ... ⎦ . N † βN

⎡

(1.57)

On recent advances

21

Algorithm 4: PeCAN  T Input: Let x 0 = x0 (1) , . . . , x0 (N ) be a vector with random initial phases; e.g., x0 (n) can be set to e j2π θ (n) , where θ(n) are independent random variables uniformly distributed in [0, 1). Let s = 1. ! !2  N Step 1: Fix x s−1 and compute ψp p=1 by minimizing !F † x s−1 − vs ! . More precisely, set ψp = arg fp , p = 1, . . . , N , by letting f = F † x s−1 . ! !2  N Step 2: Fix ψp p=1 and compute x s by minimizing !F † x s − vs ! = x s − Fvs 2 : Set xs (n) = e j arg(gn ) , n = 1, . . . , N , by letting g = Fvs . Let x s = [xs (1) , . . . , xs (N )]T . Iteration: Repeat Steps 1 and 2 until |Es − Es−1 | < ξ . Otherwise, update s = s + 1 and continue the iterations. Output: The optimized sequence x s .

Then F † is a unitary N × N DFT matrix. Let T  v = e jψ1 , . . . , e jψN .

(1.58)

Then the PeCAN optimization problem is equivalent to ! † ! !F x − v!2 min N {x(n)}N n=1 ;{ψp }p=1 s.t.

ψp ∈ [0, 2π ] , |x(n)| = 1,

p = 1, . . . , N ,

(1.59)

n = 1, . . . , N .

PeCAN minimizes (1.59) by alternating!the optimization between x and v. Simi!2 lar to CAN, the PeCAN criterion Es = !F † x s − vs ! is monotonically decreasing through the iterations, leading to its convergence. The PeCAN steps are summarized in Algorithm 4.

We also have the following theorem regarding the PeCAN algorithm when used for binary periodic sequence design: Theorem 1.7. If x 0 ∈ (defined in (1.37)) and x 1 is the waveform obtained after one more iteration of the PeCAN algorithm, then x 1 ∈ . Interestingly, if the initial waveform is drawn from , then the iterations of PeCAN are exclusively performed on , and thus resulting in a sequence in at convergence. Note that while Theorem 1.7 shows that PeCAN is able to produce binary outcomes in its quest for sequences with low periodic autocorrelation sidelobes, it does not work effectively, as shown later (in Figures 1.21 and 1.22).

22

Radar waveform design based on optimization theory

1.3.3.3 CANARY The CANARY algorithm was originally introduced by Soltanalian et al. in [22] to design complementary sets of sequences. It can also be used to design a binary sequence with good aperiodic autocorrelation properties by setting the number of sequences in the set to 1 [22]. Consider the optimization problem [22]: min 2N {x(n)}N n=1 ;{ψp }p=1 s.t.

N  ! † ! !A z − v!2 +λ |x(i)|2 − 1 i=0

ψp ∈ [0, 2π ] ,

p = 1, . . . , 2N ,

arg(x(n)) ∈ {0, π} ,

(1.60)

n = 1, . . . , N ,

where A, z, and v are defined in (1.46), (1.47), and (1.48), respectively. At the sth iteration, for given x s−1 (equivalent to fixing z s−1 ), CANARY first solves the following optimization problem [22]: N  ! † s−1 ! s !2 ! |x(i)|2 − 1 min A z − v +λ {ψp }2N i=0 p=1

s.t. ψp ∈ [0, 2π ] ,

(1.61)

p = 1, . . . , 2N ,

! !2 which is equivalent to minimizing !A† z s−1 − vs ! . By defining f = A† z s−1 , the solution to (1.61) takes the form: ψp = arg fp , p = 1, . . . , 2N . (1.62)  2N Next, for given ψp p=1 , which is equivalent to fixing vs , CANARY solves the following optimization problem [22]: min z s − Avs 2 +λ

{x(n)}N n=1

N  s 2 |x (i)| − 1 , i=0

s.t. arg(x(n)) ∈ {0, π } ,

(1.63)

n = 1, . . . , N .

!2 ! This is a useful objective for sequence optimization as !A† z s − vs ! = z s − Avs 2 . Note that solving (1.63) to obtain the solution x can be dealt with in an elementwise manner. Let g = Avs . Then (1.63) is equivalent to the following single variable optimization problem: |xs (i) − g(i)|2 +λ |xs (i)|2 − 1 , s.t. arg(xs (i)) ∈ {0, π } . (1.64) min s x (i)

Now let xs (i) = |x| e jϕx , g(i) = |g| e jϕg , and Q(ϕ) denote the closest element in the set of quantized levels in {0, π } for a given ϕ. It is easy to verify that the minimizer ϕx of (1.64) is given by ϕx = Q ϕg .

On recent advances

23

Given ϕx , we can rewrite the criterion in (1.64) as |xs (i) − g(i)|2 +λ |x(i)|2 − 1 2 = |x| − |g| e j(ϕg −ϕx ) + λ |x|2 − 1

2 = Const1 + |x| − |g| cos ϕg − ϕx + λ |x|2 − 1 . " #$ %

(1.65)

f(|x|)

A minimizer |x| of f (|x|) satisfying |x| > 1 should minimize

f (|x|) = (1 + λ) |x|2 − 2|x| |g| cos ϕg − ϕx + Const2 ,

(1.66)



which implies that |x| = (|g| cos ϕg − ϕx /(1 + λ)). Otherwise, a minimizer |x| of f (|x|) satisfying |x| < 1 should minimize

f (|x|) = (1 − λ) |x|2 − 2 |x| |g| cos ϕg − ϕx + Const3 ,

(1.67)



which implies that |x| = (|g| cos ϕg − ϕx /(1 − λ)). In summary, the minimization of (1.64) with respect to |x| satisfies results in the following type of soft-thresholding solution:

⎧ |g| cos ϕg − ϕx ⎪ ⎪ ⎪ , ⎪ ⎪ 1−λ ⎪ ⎪ ⎪ ⎪ ⎨ |x| = 1, ⎪ ⎪ ⎪ ⎪

⎪ ⎪ |g| cos ϕg − ϕx ⎪ ⎪ ⎪ , ⎩ 1−λ

|g|
0}: T

[S1] [S2]

[S3]

[S4]

[S5]

†††

Let A k = ∅; Find the real roots of p¯ (x): – If L = 0, then if the constant term e ≤ 0 exit. Conversely, update A k = R and exit; – If L ≥ 1, sort the real roots, set i = 1, and perform the remaining steps;  If p¯ (xi ) > 0, then p¯ (x) > 0 on the interval (xi , xi+1 ): update A k = A k ∪ (xi , xi+1 ), if i < L set i = i + 1 and repeat step [S3], otherwise exit. Else, go ahead;  If p¯ (xi ) < 0, then p¯ (x) > 0 on the interval (xi−1 , xi ): update A k = A k ∪ (xi−1 , xi ), if i < L set i = i + 1 and repeat step [S3], otherwise exit. Else, go ahead;  If p¯ (xi ) = 0, then xi is a stationary point:  – If p¯ (xi ) > 0, then xi is a local minimum: update A k = A k ∪ (xi−1 , xi ) ∪ (xi , xi+1 ), if i < L set i = i + 1 and repeat step [S3], otherwise exit. Else, go ahead;  – If p¯ (xi ) < 0, then xi is a local maximum: if i < L set i = i + 1 and repeat step [S3], otherwise exit. Else, go ahead;  – If p¯ (xi ) = 0:   ∗ If p¯ (xi )  = 0, then xi is an inflection point. If p¯ (xi ) > 0: update A k = A k ∪ (xi , xi+1 ), if i < L set i = i + 1 and repeat step [S3], oth erwise exit. Conversely if p¯ (xi ) < 0: update A k = A k ∪ (xi−1 , xi ),

Notice that, x0 = −∞ and xL+1 = +∞.

A computational design of phase-only sequences for radar systems

∗

89

if i < L set i = i + 1 and repeat step [S3], otherwise exit. Else, go ahead;  If p¯ (xi ) = 0: · If p¯ (4) (xi ) > 0, then xi is a local minimum: update A k = A k ∪ (xi−1 , xi ) ∪ (xi , xi+1 ), if i < L set i = i + 1 and repeat step [S3], otherwise exit. Conversely, if p¯ (4) (xi ) < 0 then xi is a local maximum: if i < L set i = i + 1 and repeat step [S3], otherwise exit.

In order to calculate the union of the different sets, the fast and simple “union-find” algorithm [41] is employed. Precisely, let (l1 , u2 ), (l2 , u2 ), . . . , (lM , uM ) be M different intervals where li is the lower extreme of each set and ui is the upper extreme. Let t ∈ R2M be the vector containing the sorted li and ui , i = 1, . . . , M in increasing order (if li = uk then uk is located first than li ). Now, define a counter Count initialized as Count = 1; then check if the second entry of t is a left extreme or a right extreme of one among M intervals. If it is a left extreme, Count = Count + 1 otherwise Count = Count − 1. Now, if Count = 0 an interval disjoint from the remaining part of the set is obtained and the process continues for the construction of the remaining part of the union set starting from the successive entry of t.

Appendix C Proof of Lemma 3.2 The M -point DFT of z dk is, ⎡

⎤

⎢ ⎢ DFT(z dk ) = ⎢ ⎣

⎥ ⎥ ⎥ ⎦

adk + cdk + bdk 2π 4π adk + cdk e−j M + bdk e−j M .. .

adk + cdk e−j

2π(M −1) M

+ bdk e−j

4π(M −1) M

Next, observe that ¯ ¯ ¯ r¯k (φ¯ m )e−jφm = adk + cdk e−jφm + bdk e−2jφm , m = 1, . . . , M

Since |¯rk (φ¯ m )e−jφ¯m | = |¯rk (φ¯ m )|, m = 1, . . . , M , * +T |DFT(z dk )| = |¯rk (φ¯ 1 )|, |¯rk (φ¯ 2 )|, . . . , |¯rk (φ¯ M )| .

(C.1)

(C.2)

References [1] Alaee-Kerahroodi M, Aubry A, De Maio A, et al. A Coordinate-Descent Framework to Design Low PSL/ISL Sequences. IEEE Transactions on Signal Processing. 2017;65(22):5942–5956. [2] Alaee-Kerahroodi M, Modarres-Hashemi M, Naghsh MM. Designing Sets of Binary Sequences for MIMO Radar Systems. IEEE Transactions on Signal Processing. 2019;67(13):3347–3360.

90 [3] [4]

[5] [6] [7] [8]

[9]

[10] [11] [12]

[13]

[14]

[15]

[16]

[17]

[18] [19]

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Baden JM, O’Donnell B, Schmieder L. Multiobjective Sequence Design via Gradient Descent Methods. IEEE Transactions on Aerospace and Electronic Systems. 2018;54(3):1237–1252. Li Y, Vorobyov SA. Fast Algorithms for Designing Unimodular Waveform(s) With Good Correlation Properties. IEEE Transactions on Signal Processing. 2018;66(5):1197–1212. Rowe W, Stoica P, Li J. Spectrally Constrained Waveform Design [sp Tips Tricks]. IEEE Signal Processing Magazine. 2014;31(3):157–162. Melvin W, Scheer J. Principles of Modern Radar: Advanced Techniques. London: The Institution of Engineering and Technology; 2012. Cohen MN, Baden JM, Cohen PE. Biphase Codes With Minimum Peak Sidelobes. In: IEEE National Radar Conference; 1989. p. 62–66. Coxson G, Russo J. Efficient Exhaustive Search for Optimal-Peak-Sidelobe Binary Codes. In: Proceedings of the IEEE Radar Conference; 2004. p. 438–443. Nunn CJ, Coxson GE. Best-Known Autocorrelation Peak Sidelobe Levels for Binary Codes of Length 71 to 105. IEEE Transactions on Aerospace and Electronic Systems. 2008;44(1):392–395. Nasrabadi MA, Bastani MH. A Survey on the Design of Binary Pulse Compression Codes with LowAutocorrelation. London: INTECH Open; 2010. Soltanalian M, Stoica P. Computational Design of Sequences With Good Correlation Properties. IEEE Transactions on Signal Processing. 2012;60(5): 2180–2193. Alaee-Kerahroodi M, Aubry A, De Maio A, et al. Design of binary sequences with low PSL/ISL. In: 2017 25th European Signal Processing Conference (EUSIPCO); 2017. p. 2211–2215. Stoica P, He H, Li J. On Designing Sequences With Impulse-Like Periodic Correlation. IEEE Signal Processing Letters. 2009;16(8):703–706. Aubry A, De Maio A, Jiang B, et al. Ambiguity Function Shaping for Cognitive Radar Via Complex Quartic Optimization. IEEE Transactions on Signal Processing. 2013;61(22):5603–5619. De Maio A, Huang Y, Piezzo M, et al. Design of Radar Receive Filters Optimized According to Lp -Norm Based Criteria. IEEE Transactions on Signal Processing. 2011;59(8):4023–4029. Chen B, He S, Li Z, et al. Maximum Block Improvement and Polynomial Optimization. SIAM Journal on Optimization. 2012;22(1):87–107. Deb K. Multi-Objective Optimization Using Evolutionary Algorithms. vol. 16. Chichester: John Wiley & Sons; 2001. Boyd S, Vandenberghe L. Convex Optimization. Cambridge: Cambridge University Press; 2004. De Maio A, Piezzo M, Farina A, et al. Pareto-Optimal Radar Waveform Design. Radar, Sonar & Navigation. 2011;5(4):473–482. De Maio A, Piezzo M, Iommelli S, et al. Design of Pareto-Optimal Radar Receive Filters. International Journal of Electronics and Telecommunications. 2011;57(4):477–481.

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Chapter 4

Constrained radar code design for spectrally congested environments via quadratic optimization Marco Piezzo1 , Yongwei Huang2 , Augusto Aubry3 , and Antonio De Maio3

4.1 Introduction The electromagnetic spectrum is becoming more and more crowded mainly due to accelerating demands of bandwidth driven by the mobile communications industry as well as the need for coexistence in the vicinity of spectrally overlaid telecommunication systems. It is thus mandatory for the development of advanced radar signals ensuring compatibility with the surrounding electromagnetic radiators, namely, keeping acceptable the mutual interference induced on frequency overlaid systems. Many papers in the open literature have dealt with the problem of devising radar signals with a suitable frequency allocation [1] so as to induce acceptable interference levels on frequency-overlaid systems, while optimizing radar performance in terms of range-Doppler resolution, low range/Doppler sidelobes, detection, and tracking capabilities. In [2], a waveform design technique is introduced to confer some desired spectral nulls to the radar signal. The idea is to perturb a stepped frequencymodulated pulse forcing an additional fast time polyphase code. The approach is extended in [3] to the case of continuous phase waveforms with nulls at specific frequencies. The effectiveness of both the mentioned strategies is assessed in [4], via an experimental analysis. An alternating projection algorithm for the construction of chirp-like constant-modulus signals with a single spectral null is proposed in [5], whereas in [6], its extension, addressing the synthesis of multiple notches, is established. Some iterative algorithms are introduced in [7] for the joint design of the transmit signal and the receive filter achieving frequency stop-band suppression and range sidelobes minimization. In [8,9], sparse frequency constant modulus radar signals with a low integrated sidelobe level (ISL) are built optimizing a suitable combination of the ISL metric and a penalty function accounting for the waveform 1

Elettronica S.p.A., Rome, Italy School of Information Engineering, Guangdong University of Technology, Guangzhou, China 3 Dipartimento di Ingegneria Elettrica e delle Tecnologie dell’Informazione, Universitá degli studi di Napoli “Federico II,” Naples, Italy 2

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frequency allocation. In [10], the authors develop an iterative procedure aimed at designing unimodular waveforms maximizing the ratio of the maximal stopband level to the minimal passband level. In [11], two algorithms are introduced to synthesize waveforms with desired spectral behavior according to a weighted least-squares fitting approach. Hopped-frequency waveforms with low range sidelobes under given spectral constraints are provided in [12]. Some experimental demonstrations on the effectiveness of the transmit notching method developed in [13] to reduce mutual interference with other spectrum users in the same band are carried out in [14]. Furthermore, assuming that both the radar and the communications systems utilize multicarrier waveforms, radar waveform optimization is performed in [15] considering the Cramer–Rao lower bound for target’s time delay estimation as a figure of merit and appropriately constraining the per-subcarrier power allocation. Other interesting and technically sound approaches are proposed in [16–30]. In the present chapter, we deal with the radar waveform design in spectrally crowded scenarios via constrained quadratic programming techniques, aimed at providing a unifying and systematic overview of some recent results available in the open literature [31–33]. To this end, we introduce a taxonomy of the most relevant performance measures to optimize and/or control at the waveform design process: ●

● ●

●

●

The signal-to-interference-plus-noise-ratio (SINR) which is tantamount to accounting for the target detection probability. The signal energy which is related to the maximum radar range. The similarity between the sought code and a reference one that exhibits some valuable features, in terms of amplitude variations, peak sidelobe level (PSL) and ISL, just to list a few. The interference level injected on the shared bandwidth aimed at assuring spectral compatibility with the licensed RF systems. The transmitted energy over the free spectrum portion promoting high-quality frequency usage in (but not limited to) the probing process.

The above figures of merit provide the key ingredients for the radar waveform optimization process. Indeed, we present several radar waveforms design problems aimed at maximizing a specific criterion (i.e., the SINR), while accounting for different combinations of the performance measures in terms of constraints. Regardless of the chosen objective function and constraints, the resulting waveform design is formulated as a nonconvex quadratically constrained quadratic program (QCQP) problem, in general quite difficult to solve. Hence, we exploit a specific methodology leveraging rank-one matrix decomposition tools to prove the hidden convexity of some instances of the waveform optimization or to build solution procedures ensuring good-quality radar waveforms. At the analysis stage, we assess the performance of the synthesized waveforms studying the trade-off among the achievable SINR, spectral shape, and autocorrelation. The results show that the proposed algorithms are capable of ensuring both spectral coexistence with the overlaid RF emitters and a satisfactory detection performance at the price of other desirable radar signal features, such as sidelobe levels and/or range resolution.

Constrained radar code design for spectrally congested environments

95

The remainder of the chapter is organized as follows. In Section 4.2, the model for the radar transmitted signal and the description of coexisting wireless systems are reported. In Section 4.3, the figures of merit and constraints to be involved in the optimal radar selection problem are presented. In Section 4.4, the solution procedure exploiting the hidden convexity of the optimization problems is presented. In Sections 4.5–4.7, specific waveform optimization instances are dealt with. In Section 4.8, the conclusions are drawn out, and proposals for possible future research tracks are discussed. Finally, in Appendix A, the proofs of the propositions and theorems within the chapter are reported.

4.2 System model Let c(t) be the baseband equivalent of the transmitted radar pulse, assumed to be composed of N linearly modulated sub-pulses; the code element for the ith sub-pulse is denoted by c(i) and the vector c = [c(1), . . . , c(N )]T ∈ CN represents the fasttime radar code. The waveform at the receiver end is down-converted to baseband, undergoes a sub-pulse matched filtering operation, and then is sampled. The N -dimensional column vector v = [v(1), . . . , v(N )]T ∈ CN of the fast time observations,∗ associated with the range-azimuth cell under test, can be expressed as v = αT c + n ,

(4.1)

with αT a complex parameter accounting for channel propagation and backscattering effects from the target within the range-azimuth bin of interest, and n the N -dimensional column vector containing the filtered disturbance echo samples. Specifically, the vector n accounts for both white internal thermal noise and interfering signals due to unknown and unlicensed, possibly hostile jammers as well as licensed overlayed telecommunication networks sharing the same frequencies as the radar of interest.∗∗ Moreover, n is modeled as a complex, zero-mean, circularly symmetric Gaussian random vector with covariance matrix E[nn† ] = M . As to the overlayed electromagnetic emitters, it is necessary to distinguish between two possible sets of (normalized) frequency bands. With reference to the licensed radiators coexisting with the radar of it is supposed that each of  interest,  them is working over a frequency band k = f1k , f2k , k = 1, . . . , K, where f1k and f2k denote the lower and upper normalized frequencies for the kth system, respectively. Additionally, the remaining K˜ frequency bands, free from cooperative transmissions, ˜ ˜ 0 = [0, f11 ] if ˜ k˜ = [f2k˜ , f1k+1 ], where k˜ = 1, . . . , K˜ − 1 plus  are defined as the sets  ˜ ˜ 1 K K ˜ K+1 = [f2 , 1] if f2  = 1. f1  = 0 and  From the radar perspective, the coexisting wireless networks exhibit a different importance, based for instance on their distance from the radar and their tactical

∗

For simplicity, in the following, we refer to ideal sinc-shaped sub-pulses. As a consequence, the code elements coincide with the samples (at Nyquist rate) of the continuous waveform. ∗∗ Notice that the vector n may also account for the clutter contributions.

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relevance (i.e., navigation systems, defense communications, and public services). Similarly, among the free frequency bands, some are preferable due to their quality. Evidently, for a smart and efficient spectrum allocation, it is of paramount importance for the sensing radar to have an accurate, reliable, and comprehensive radio environment awareness, in order to glean knowledge about the number K of coexisting licensed radiators and consequently the number K˜ of interference-free channels, emitters’ bandwidths (referred to as stop-band in the following and defined by the ˜ ˜ ˜ k˜ ), lower and upper normalized frequencies f1k and f2k for k , and f2k and f1k+1 for  and the level of purity of the usable bandwidths. Radio environment map (REM) [34] represents the key to gain the aforementioned spectrum awareness which is at the base of an intelligent and agile spectrum management. It can be defined as an integrated database, digitizing and indexing the available electromagnetic information. Figure 4.1 illustrates some REM multi-domain knowledge sources that are represented by geographical features, available services and spectral regulations, locations and activities of telecommunication networks, and previous radio experiences and measurements [35–37].

4.3 Figures of merit and constraints In this section, we focus on the key performance measures to optimize and control during the design of the radar code.

GIS

REM SERVICE & REGULATIONS Sensing module Sensing module

TELCOM. Net. ACTIVITY PROFILE

Sensing module

EXPERIENCE

Sensing fusion center

Waveform adaptation

Figure 4.1 A pictorial representation of the REM and its usage in radar

Constrained radar code design for spectrally congested environments

97

4.3.1 Detection probability According to the signal model in (4.1), the problem of establishing the presence or absence of a target in the cell under test can be formulated as the following binary hypotheses test:  H0 : v = n . (4.2) H1 : v = αT c + n Under the assumption of Gaussian interference with known positive-definite covariance matrix M , the generalized likelihood ratio test detector over αT for model (4.2), which coincides with the optimum test (according to the Neyman–Pearson criterion) if the phase of αT is uniformly distributed in [0, 2π) [38–40], is given by H1 |v† M −1 c|2 > < G, H0

(4.3)

where G is the detection threshold set according to a desired value of the false alarm probability (Pfa ). In the case of nonfluctuating target, the detection probability (Pd ), for a given value of Pfa , can be analytically expressed as    Pd = Q 2|αT |2 c† M −1 c, −2 ln Pfa , (4.4) where Q(·, ·) denotes the Marcum Q function of order 1. Equation (4.4) shows that, given Pfa , Pd depends on the radar code and the disturbance covariance matrix only through the SINR, defined as SINR = |αT |2 c† Rc ,

(4.5)

where R = M −1 . Notice that Pd is an increasing function of SINR; as a consequence, the maximization of Pd can be obtained optimizing the SINR over the radar code.

4.3.2 Energy and similarity constraints Designing a code which optimizes the detection performance does not provide any kind of control on the shape of the resulting coded waveform. Otherwise stated, an unconstrained optimization can lead to signals with significant modulus variations, poor range resolution, high PSLs, and more in general an undesired ambiguity function behavior. These drawbacks can be partially circumvented by constraining the sought radar code [41] to exhibit some similarity with a known unit energy reference code c0 ‡ : c − c0 2 ≤ ε ,

(4.6)

where the parameter ε ≥ 0 rules the size of the similarity region. Forcing (4.6) allows to indirectly control desirable autocorrelation features of the radar waveform; the smaller ε, the higher the degree of similarity between the designed radar code and c0 . ‡

c0 shares constant modulus, reasonable range resolution and ISL/PSL.

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An additional aspect to account for in the radar waveform design concerns the signal power, which is strictly related to the power available for transmission. In order to exploit as much as possible the transmitter potentialities, the energy of the devised waveform can be set to the maximum level by imposing the energy constraint: c2 = 1 .

(4.7)

On the other hand, in order to provide more degrees of freedom to the waveform optimization process, it is also possible to account for an energy modulation at the transmitter; this is tantamount to forcing the constraint: 1 − η ≤ c2  ≤ 1 ,

(4.8)

where η (0 ≤ η ≤ 1) is a design parameter which rules the maximum allowable decrease of the radar transmit power (hence, it can be set based on radar range equation argumentations or radar maximum operation range).

4.3.3 Spectral compatibility constraint To guarantee spectral compatibility with the overlayed telecommunication services, the radar has to control the amount of overall interfering energy produced on the shared frequency bands. From an analytical point of view, the amount of energy transmitted on the kth band can be computed as  fk 2 Sc (f ) df = c† RIk c, (4.9) f1k



2 −1 where Sc (f ) = Nn=0 c(n)e−j2π fn is the energy spectral density (ESD) of the fasttime code c and ⎧ k k m=l ⎪ ⎨ f2 − f1 k k , (m, l) ∈ {1, . . . , N }2 . (4.10) RI (m, l) = ej2πf2 (m−l) − ej2πf1k (m−l) ⎪ m  = l ⎩ j2π (m − l) For K radiators, the total interfering energy transmitted on the coexisting systems is given by IE =

K 

c† RIk c .

(4.11)

k=1

Hence, denoting by EI the amount of allowed interference, to overlay the radar with coexisting telecommunication networks, the transmitted waveform has to comply with the constraint c† RI c ≤ EI ,

(4.12)

where RI =

K  k=1

wk RIk .

(4.13)

Constrained radar code design for spectrally congested environments

99

In (4.13), the relative impact of the different emitters is quantified via the association to each of them of a suitable weight wk ≥ 0, k = 1, . . . , K, selected for instance based on the distance from the radar and the tactical importance (i.e., navigation systems, defense communications, and public services). Imposing the constraint (4.12) allows to quantify and control the amount of interference induced on coexisting telecommunication networks.

4.3.4 Bandwidth priority constraint It is possible to account for the quality of the bandwidths available for transmission via a specific constraint measuring their purity. This allows to prioritize the available frequency channels (free of cooperative transmissions), in the sense that the spectral energy injected can be tuned accordingly. ˜ k˜ , the By applying the same model as in (4.9) and (4.10) to the channels  amount of spectral energy injected by the radar in the non-congested bandwidths can be computed as I¯E = c†

K˜ 

˜

RIk,2 c ,

(4.14)

˜ k=1

where

˜ RIk,2 (m, l)

=

⎧ ˜ k+1 k˜ ⎪ ⎪ ⎨ f1 − f2 e ⎪ ⎪ ⎩

˜ j2πf1k+1 (m−l)

m=l ˜ j2π f2k (m−l)

−e j2π (m − l)

m = l

, (m, l) ∈ {1, . . . , N }2 . (4.15)

Thus, denoting by E¯ I the minimum energy level transmitted by the radars on these channels, the transmitted waveform has to comply with the constraint c† RI ,2 c ≥ E¯ I ,

(4.16)

with RI ,2 =

K˜ 

˜

w˜ k˜ RIk,2 ,

(4.17)

˜ k=1

where the coefficients w˜ k˜ ≥ 0, k˜ = 1, . . . , K˜ depend on the channel quality. For instance, bandwidths with low frequencies could be preferred for signal attenuation arguments; bandwidths which are close to an interferer characterized by a limited spectral purity could be undesired; finally, bandwidths belonging to emitters within a short distance from the radar could be penalized.

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Radar waveform design based on optimization theory

4.4 QCQP’s solution methods via rank-one matrix decomposition The above constraints and figures of merit represent the key ingredients for the radar waveform optimization process. Indeed, the synthesis of radar waveforms can be aimed at maximizing a specific criterion (i.e., the SINR), while accounting for different combinations of the remaining constraints. Regardless of the chosen objective function and constraints, the resulting waveform design is formulated as a nonconvex QCQP problem. Nevertheless, under suitable conditions, it is often possible to prove the hidden convexity, so that an optimal radar code can be found in polynomial time. This possibility is explored in this chapter via a solution procedure based on Semidefinite Programming (SDP) problem relaxation and some particular rank-one matrix decomposition techniques [42, Theorem 2.3], which are cited as Lemmas 4.1 and 4.2. Lemma 4.1. Let X be a nonzero N × N (N ≥ 3) complex Hermitian positive semidefinite matrix and {A1 , A2 , A3 , A4 } be Hermitian matrices, and suppose that (tr(YA1 ), tr(YA2 ), tr(YA3 ), tr(YA4 ))  = (0,0,0,0) for any nonzero complex Hermitian positive semidefinite matrix Y of size N × N . Then, ●

●

if Rank (X ) ≥ 3, one can find, in polynomial time, a rank-one matrix xx † such that x (synthetically denoted as x = D1 (X , A1 , A2 , A3 , A4 )) is in Range (X ) and x † Ai x = tr (XAi ), where i = 1, 2, 3, 4; if Rank (X ) = 2, for any z not in the range space of X , one can find a rank-one matrix xx † such that x (synthetically denoted as x = D2 (X , A1 , A2 , A3 , A4 )) is in the linear subspace spanned by {z} ∪ Range (X ) and x † Ai x = tr (XAi ), where i = 1, 2, 3, 4.

Lemma 4.2. Suppose that X is a N × N complex Hermitian positive semidefinite matrix of rank R, and A, B are two N × N given Hermitian matrices. Then, there is a rank-one decomposition of X , synthetically denoted as D3 (X , A1 , A2 ), such that

X = Rr=1 xr xr† , xr† Axr = tr (AX )/R, and xr† Bxr = tr (BX )/R, r = 1, . . . , R. In what follows, we study the solution methods for a QCQP. Consider the following general form homogeneous QCQP: min

x † Ax

s.t

bl ≤ x † Bl x ≤ al , x † Cm x = cm ,

x

x † Dn x ≤ dn ,

l = 1, . . . , L m = 1, . . . , M n = 1, . . . , N¯

,

(4.18)

where x ∈ CN , A, {Bl }, {Cm }, and {Dn } are Hermitian matrices. There are double-sided constraints, one-sided inequality constraint, and equality constraints in the QCQP. If

Constrained radar code design for spectrally congested environments

101

inhomogeneous quadratic functions are involved in the QCQP, namely, the following inhomogeneous QCQP is considered: min

x † Ax + 2Re(¯a† x)

s.t

0 ≤ x † Bl x + 2Re(b†l x) + bl ≤ al , x † Cm x + 2Re(cm† x) + cm = 0 ,

x

x † Dn x + 2Re(dn† x) + dn ≤ 0 ,

l = 1, . . . , L m = 1, . . . , M n = 1, . . . , N¯

(4.19)

,

then it can be converted equivalently into the homogenous QCQP:  †    x A a¯ x min † x, t t a¯ 0 t  †    Bl bl x x s.t. 0 ≤ ≤ al , l = 1, . . . , L † t t bl bl    †  Cm cm x x = 0, m = 1, . . . , M . t cm† cm t  †    x Dn d n x ≤ 0, n = 1, . . . , N¯ t t dn† dn  †    x 00 x =1 t 01 t

(4.20)

In  other words, problems (4.20) and (4.19) share the same optimal value, and if x is optimal for (4.20), then x  /t  is optimal for (4.19). Therefore, we need only t to focus on homogeneous QCQP problem (4.18), since any inhomogenous QCQP can be recast into a homogeneous QCQP (at a cost of one more variable and one more constraint introduced). The conventional SDP relaxation problem for (4.18) is 

min

tr (AX )

s.t.

bl ≤ tr (Bl X ) ≤ al ,

l = 1, . . . , L

tr (Cm X ) = cm ,

m = 1, . . . , M n = 1, . . . , N¯

X

tr (Dn X ) ≤ dn , X 0

(4.21)

,

and its dual problem is max

wl , w¯ l , ym , zn

s.t.

L



N¯ bl wl + Ll=1 al w¯ l + M m=1 cm ym + n=1 dn zn



N¯ A − Ll=1 (wl + w¯ l )Bl − M m=1 ym Cm − n=1 zn Dn 0 wl ≥ 0, w¯ l ≤ 0 , l = 1, . . . , L l=1

ym ∈ R , zn ≤ 0 ,

m = 1, . . . , M n = 1, . . . , N¯

(4.22) .

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Radar waveform design based on optimization theory

The optimality conditions (the complementary conditions) are wl (tr (Bl X ) − bl ) = 0, l = 1, . . . , L w¯ l (tr (Bl X ) − al ) = 0, l = 1, . . . , L , zn (tr (Dn X ) − dn ) = 0, n = 1, . . . , N¯ and

⎛⎛ tr ⎝⎝A −

L 

(wl + w¯ l )Bl −

l=1

M  m=1

ym C m −

(4.23) (4.24) (4.25)

N¯ 

⎞ ⎞ zn Dn ⎠ X ⎠ = 0.

(4.26)

n=1

We assume that both the SDP and its dual are solvable and (X  ; {wl  }, {w¯ l }, {ym }, is an optimal primal-dual pair, which satisfies the optimality conditions (4.23)– (4.26). If X  = u u† is of rank one, then u is an optimal solution for the QCQP problem (4.18). Now, we assume that the rank of X  is greater than one. Thus, we look for a sufficient condition such that SDP problem (4.21) has a rank-one optimal solution, and under that condition, the rank-one solution can be constructed via a certain rank-one matrix decomposition. {zn })

Theorem 4.1. Suppose that integers L, M , and N¯ are nonnegative, belonging to the set {0, 1, 2, 3}. Suppose that 1 ≤ L + M + N¯ ≤ 3. Then, there is a rank-one solution for SDP problem (4.21). Proof. See Appendix A.1. Now, we focus on the cases of (4.18) (or (4.21)) with four constraints. Theorem 4.2. Suppose that integers L, M , and N¯ are nonnegative, belonging to the set {0, 1, 2, 3, 4}. Suppose that L + M + N¯ = 4. Then, 1. 2.

there is a rank-one solution for SDP problem (4.21), if the rank of X  is greater than or equal to three; a suboptimal rank-one solution for SDP problem (4.21), if the rank of X  is two and the dimension of x in (4.18) is more than two.

Proof. See Appendix A.2. Next, we wish to present another solvable subclass of QCQPs with four constraints. Suppose that M ≥ 1 and N¯ ≥ 1; in other words, there are at least one equality constraint and one inequality constraint. Suppose further that C1  0 and c1 > 0. In this case, we show that the corresponding QCQPs are solvable. Theorem 4.3. Suppose that integers L, M , and N¯ are nonnegative, belonging to the set {0, 1, 2, 3, 4}. Suppose that L + M + N¯ = 4 with M ≥ 1 and N¯ ≥ 1. Assume that C1  0 and c1 > 0. Then, SDP problem (4.21) admits a rank-one optimal solution if tr (D1 X  ) < d1 . Proof. See Appendix A.3.

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We remark that if tr (C1 X ) = tr (X ) = 1 is assumed, then the double-sided constraint b1 ≤ tr (B1 X ) ≤ a1 reduces to one-sided constraint b1 ≤ tr (B1 X ) for a1 ≥ λmax (B1 ) or tr (B1 X ) ≤ a1 for b1 ≤ λmin (B1 ).

4.5 Radar waveform design in a spectrally crowded environment under similarity and spectral coexistence constraints 4.5.1 Code design optimization problem In this subsection, we focus on a waveform design technique which attempts to maximize the detection probability and at the same time provides a control both on the interfering energy produced in the stop-bands and on desirable features of the transmitted waveform. Precisely, we consider as figure of merit the SINR defined in (4.5) and impose an energy constraint (4.7) together with a similarity constraint (4.6). Finally, to ensure coexistence among the radar and wireless telecommunication infrastructures sharing the same spectrum, we require the transmitted waveform to comply with (4.12). The waveform design problem can be formulated as the nonconvex QCQP optimization problem ⎧ ⎧ max c† Rc max c† Rc ⎪ ⎪ ⎪ ⎪ c c ⎪ ⎪ ⎨ ⎨ s.t. c† c = 1 s.t. c† c = 1 e , (4.27) P1 = P2 † ⎪ ⎪ c† RI c ≤ EI c RI c ≤ EI ⎪ ⎪ ⎪ ⎪   ⎩ ⎩ c − c0 2 ≤ ε Re c† c0 ≥ 1 − ε/2 e

where the symbol = states that the two problems are equivalent.§ Notice that design procedures like those in [2], [3], [6], [43], [18] can be applied for the construction of a high-quality reference code, namely, a code exhibiting desired spectral properties as well as radar features. Hence, moving in a suitable Euclidean ball centered around the aforementioned code, we can improve the desired radar performance still keeping the spectral coexistence. The feasibility of problem P1 , which not only depends on the parameters EI and ε but also on the pre-fixed code c0 , is discussed in [31], where the concept of Interference/Similarity (I/S) achievable region associated with the radar code c0 is presented. For the sake of clarity, we show a notional example of the I/S achievable region, for E¯ I = 0.15 and ε¯ = 0.2, in Figure 4.2 with gray-stripes, for the chirp signal 1 2 c0 (n) = √ ej2π Ks (n/fs ) , N

n = 0, . . . , 99,

(4.28)

with N = 100, Ks = (350 × 103 )/(250 × 10−6 ) Hz/s, fs = 400 kHz. The interfering environment is characterized by the presence of K = 2 stop-bands, with w1 = w2 = 1, § Two problems Pi and Pj are said equivalent if, starting from an optimal solution xi of the former, it is possible to construct an optimal solution xj of the latter and vice versa, in polynomial time.

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0.2

e

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0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18

Ef

Figure 4.2 I/S achievable region, for the chirp similarity waveform (4.28) with K = 2, [ f11 , f21 ] = [0, 0.15], and [ f12 , f22 ] = [0.98, 0.99] located at the normalized frequencies [f11 , f21 ] = [0, 0.15] and [f12 , f22 ] = [0.98, 0.99]. Analytic details can be found in [31]. We now show how to compute, in polynomial time, an optimal solution to the nonconvex optimization problem P2 . First of all, let us observe that an optimal solution to P2 can be obtained from an optimal solution to the following enlarged quadratic problem (EQP): ⎧ max c† Rc ⎪ ⎪ ⎪ ⎨ c s.t. c† c = 1 , (4.29) P3 ⎪ c† RI c ≤ EI ⎪ ⎪     ⎩ Re2 c† c0 + Im2 c† c0 = c† c0 c0 † c ≥ δε where δε = (1 − ε/2)2 . In fact, since the feasible region of P3 is larger than that of P2 , every optimal solution to P3 , which is feasible for P2 , is also an optimal solution to P2 [44]. Thus, assume that c¯ is an optimal solution to P3 and let φ = arg (¯c† c0 ). It is easily seen that c¯ ejφ is still an optimal solution to P3 . Now, observing that † (¯cejφ )† c0 = |¯c† c0 |, c¯ ejφ is a feasible solution to P2 . In other words, c¯ ej arg (¯c c0 ) is optimal for both P2 and P3 . Now, we have to find an optimal solution to P3 and, to this end, we exploit the equivalent matrix formulation ⎧ max tr (CR) ⎪ ⎪ C ⎪ ⎪ ⎪ tr (C) = 1 ⎨ s.t. P3 (4.30) tr (CRI ) ≤ EI , ⎪ ⎪ ⎪ tr (CC0 ) ≥ δε ⎪ ⎪ ⎩ C = cc†

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Algorithm 1: Algorithm to solve the radar waveform design problem P1 Require: c0 , R, RI , EI , δε . Ensure: An optimal solution c to P1 . ¯ and the optimal value v(P4 ); 1: solve SDP P4 finding an optimal solution C ¯ 2: if Rank (C) = 1 then 3: set c¯ = λmax (¯c)v, with v being an eigenvector associated to the maximum eigenvalue of c¯ ; ¯ = 2 then 4: else if Rank (C)   ¯ R, I , RI , C0 ; 5: find c¯ = D2 C, 6: else   ¯ R, I , RI , C0 ; 7: find c¯ = D1 C, 8: end if † 9: output c = c¯ ej arg (¯c c0 ) .

where C0 = c0 c0 † . All the nonconvexity of problem P3 is now confined in the rankone constraint C = cc† . Problem (4.30) can be relaxed into a convex SDP optimization problem, neglecting the rank-one constraint [45]. By doing so, we obtain an EQP relaxed ⎧ max tr (CR) ⎪ ⎪ C ⎪ ⎪ ⎪ tr (C) = 1 ⎪ ⎨ s.t. P4 (4.31) tr (CRI ) ≤ EI . ⎪ ⎪ ⎪ tr (CC ) ≥ δ ⎪ 0 ε ⎪ ⎪ ⎩ C 0 Let us now observe that problem P4 is solvable; in fact, the feasible set ⎧ tr (C) = 1 ⎪ ⎪ ⎪ ⎨ tr (CRI ) ≤ EI tr (CC0 ) ≥ δε ⎪ ⎪ ⎪ ⎩ C 0

(4.32)

is a compact set (closed and bounded), and the objective function of P4 is continuous; furthermore, as shown in [31, Appendix A], assuming that problem P1 is strictly feasible, problem P4 is strictly feasible. This property can be of paramount importance from a numerical point of view, since it guarantees that at any optimal point the complementary conditions are satisfied [46] and interior point methods [44] can be used. In order to prove the hidden convexity of problem P3 , it is sufficient to apply Theorem 4.1. In fact, set (L, M , N¯ ) = (0, 1, 2), and it follows that SDP problem P4 has a rank-one solution. Similar to the steps in the proof of Theorem 4.1, we can summarize in Algorithm 1 the procedure leading to an optimal solution of P1 .

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The computational complexity connected with the implementation of the algorithm is polynomial as both the SDP problem and the decomposition of Lemma 4.1 can be performed in polynomial time. In fact, the amount of operations, involved in solving the SDP problem, is O(N 3.5 log (1/ζ )) [46, p. 250] and the rank-one decomposition requires O(N 3 ) operations.

4.5.2 Performance analysis In this subsection, we assess the performance of the proposed waveform design technique in terms of achievable SINR value, spectral shape, and autocorrelation features. Specifically, we study the I/S achievable region of problem P1 for a fixed interfering scenario and analyze the behavior of the waveforms devised according to Algorithm 1. We consider a case-study radar whose baseband equivalent transmitted signal has a two-sided bandwidth of 810 kHz, and it is sampled at fs = 810 kHz. As to the interference, we suppose that it is composed of unlicensed narrowband continuous jammers, white interference, and licensed coexisting telecommunication networks spectrally overlayed to the radar of interest. Specifically, we model the disturbance covariance matrix as KJ K  σI ,k k  RI + σJ ,k RJ ,k , (4.33) M = σ0 I +

fk k=1 k=1 where ● ● ●

●

● ●

●

σ0 = 0 dB is the thermal noise level; K = 7 is the number of licensed radiators; σI ,k accounts for the energy of the kth coexisting telecommunication network operating on the normalized frequency band [f2k , f1k ] (σI ,k = 10 dB, k = 1, . . . , K);

fk = f2k − f1k is the bandwidth associated with the kth licensed radiator, for k = 1, . . . , K; KJ = 2 is the number of active and unlicensed narrowband jammers; σJ ,k , k = 1, . . . , KJ , accounts for the energy of the kth active jammer (σJ ,1 dB = 50 dB, σJ ,2 dB = 40 dB); RJ ,k , k = 1, . . . , KJ , is the normalized disturbance covariance matrix of the kth † active unlicensed jammer, defined as RJ ,k = rJ ,k rJ ,k , with rJ ,k (n) = ej2π fJ ,k n/fs , where fJ ,k denotes the Doppler shift of the kth jammer (fJ ,1 /fs = 0.7, and fJ ,2 /fs = 0.75).

As to the overlayed and foreseen telecommunication systems, which spectrally coexist with the radar of interest, we consider the following baseband equivalent radar stop-bands [47]: 1 = [0.0000, 0.0617] , 2 = [0.0988, 0.2469] , 4 = [0.3086, 0.3827] , 5 = [0.4074, 0.4938] ,

3 = [0.2593, 0.2840] , (4.34)

6 = [0.5185, 0.5556] , 7 = [0.9383, 1.0000] . 

An SDP problem can be efficiently solved in polynomial time through interior point methods [44], namely, iterative algorithms which terminate once a prespecified accuracy ζ is reached. The number of iterations necessary to achieve convergence usually ranges between 10 and 100 [44].

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Furthermore, we suppose licensed/foreseen overlayed systems with the same relevance, namely, wk = 1 for k = 1, . . . , 7. Based on the assumed stop-bands (4.34) and weights wk s, we can compute the matrix RI , defined in (4.13), and enforce the interference energy constraint on the transmitted radar waveform. Notice that, the matrix RI does not depend on the frequencies of the unlicensed jammers and is the only function of the spectral bands (4.34) and weights associated to the licensed radiators. As to the reference waveform c0 , we model it as a unitary norm linear frequency-modulated pulse (see (4.28)) with a duration of 200 μs and a chirp rate Ks = (750 × 103 )/(200 × 10−6 ) Hz/s; it results in N = 162 samples due to the considered sampling frequency. Finally, for comparison purpose, with reference to the example shown later in Figure 4.4, we consider the transmit sequence c˜0 and the receive filter g˜0 devised through the procedure in [7]. Specifically, with reference to the soft-power constraint transmit waveform design technique [7]¶ , we assume, as starting sequence, the code c0 , and fix δ = 0.9, λT = 10−5 , and R(2) = RI , whereas, with reference to the receive filter design technique, we fix β = 0.05, λT = 10−5 , n = 10 and assume R(2) = M . (More details about the Lindenfeld’s algorithms, the aforementioned parameters, and their setting, can be found in [7].) The I/S achievable region for the considered scenario is represented in Figure 4.3. Notice that the radar designer can choose the pair (EI , ε), referred to in the sequel as operative point, to suitably trade-off spectral coexistence, desirable radar waveform characteristics, and achievable SINR of the system. For instance, with reference to Figure 4.3, considering (EI , ε) equal to the point A, the frequency coexistence of the radar with the overlayed telecommunication networks is emphasized, with respect to choosing (EI , ε) equal to the point B. In the latter case, other radar features, such as high-range Doppler resolution and/or low sidelobe levels, are privileged. 0.7 (0.066,0.264) (0.066,0.444) (0.066,0.654) (0.235,0.066) (0.216,0.264) (0.426,0.264)

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EI

Figure 4.3 I/S achievable region, for chirp similarity code with an interference environment accounting for K = 7 equivalent stop-bands ¶

As for the design parameters of the soft-power constraint, we adopt the same nomenclature as in [7].

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In Figure 4.4, we provide the ESD, the squared modulus of the autocorrelation function (ACF), and the normalized SINR of the waveforms devised for three operative points (upper triangle, star, and hexagram in Figure 4.3). Specifically, we consider the points (Eˆ I , εn ), n = 1, 2, 3, with Eˆ I = 0.066, ε1 = 0.264, ε2 = 0.444, ε3 = 0.654 (notice that (Eˆ I , ε1 ) corresponds to the point A in Figure 4.3), and compare the results with those achievable by the couple signal/receiver (c˜0 , g˜0 ). In Figure 4.4(a), we plot the ESD of the synthesized signals versus the normalized frequency, together with that of the reference code. The stop-bands in which the 20 15 10 0 ESD (dB)

–10 –20 –30

Alg.[7] Alg.1, (EI,e3) = (0.066,0.654)

–40

c0

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Alg.1, (EI,e1) = (0.066,0.264) Alg.1, (EI,e2) = (0.066,0.444)

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Figure 4.4 (a) ESD; (b) squared modulus of the ACF. Shaded light-gray rectangle: stop-bands; gray curve: reference code c0 ; black curve: Algorithm 1, EI = 0.066, ε1 = 0.264; black dashed curve: Algorithm 1, EI = 0.066, ε2 = 0.444; black dotted curve: Algorithm 1, EI = 0.066, ε3 = 0.654; black dash-dotted curve: Algorithm [7]; (c) normalized SINR versus ε; black curve: Algorithm 1, EI = 0.066; black dashed curve: Algorithm [7], c˜0 .

Constrained radar code design for spectrally congested environments

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foreseen systems are transmitting are shaded in light gray. The curves highlight the capability of the proposed technique to suitably control the amount of energy produced over the shared frequency bands. In fact, for each ε, the energy transmitted in the stop-bands is lower than the allowed level, thus ensuring the coexistence with other transmitting systems. However, we observe that, given the considered setting for the design parameters, the transmit signal c˜0 is capable of ensuring an even greater suppression of the interference at the stop-bands than the devised codes c. Nevertheless, this behavior is quite expected as the signal design technique of [7] only focuses on the coexistence problem. In other words, the figure of merit is represented by the interference reduction, and no additional constraints are forced neither on the shape of the sought waveform (whose auto-ambiguity properties are unpredictable) nor on the SINR at the receiver side. On the contrary, the aim of the proposed algorithm is to maximize the attainable SINR, offering at the same time a control over the total amount of interference produced at certain frequencies, as well as on the resulting signal shape. Additionally, we notice that increasing the similarity parameter ε, namely, increasing the available degrees of freedom, smarter and smarter distributions of the useful energy are achieved. Otherwise stated, Algorithm 1 allows the designer to exactly choose the amount of energy suppression to force over the stop-bands, which is instead not possible for the procedure of [7]. We also observe a progressive reduction of the radar emission in correspondence of the shared frequencies, as well as an enhancement of the unlicensed jammer rejection capabilities. As a result, higher and higher SINR values can be achieved. In Figure 4.4(b), we provide a performance analysis in terms of autocorrelation properties of the designed waveforms. Evidently, better SINR values, spectral compatibility, and interference rejection are swapped for worse and worse range resolutions and/or ISLs/PSLs. It can be also observed that, as the auto-ambiguity properties of the transmit waveform of [7] are not directly under control, higher ISLs/PSLs values and, more in general, worse range sidelobe profiles than those of the proposed approach can be experienced. Nevertheless, the smoother behavior of our signals agrees with our design criterion, since the optimization problem itself involves a compromise between the desire of lowering the transmitted energy in the stop-bands as well as in correspondence of the jammer central frequencies and the need of keeping under control the ambiguity features of the sought signals.

4.6 Radar waveform design in a spectrally crowded environment under similarity, energy modulation, and spectral coexistence constraints 4.6.1 Code design optimization problem In this subsection, we consider again the SINR (4.5) as a figure of merit. Also, we enforce the spectral compatibility constraint (4.12), so as to account for the overlayed cooperative radiators, and the similarity constraint (4.6) to control features of the south waveform. Finally, we impose the relaxed energy constraint (4.8); this gives

110

Radar waveform design based on optimization theory

more degrees of freedom to the waveform optimization problem and permits to deal with scenarios where the maximum energy constraint is incompatible with at least one between the similarity and the spectral compatibility requirements. Based on the aforementioned guidelines, the waveform design problem can be formulated as the nonconvex QCQP problem: ⎧ max ⎪ ⎪ ⎪ c ⎨ s.t. P5 ⎪ ⎪ ⎪ ⎩

c† Rc 1 − η ≤ c† c ≤ 1 c† RI c ≤ EI c − c0 2 ≤ ε

.

(4.35)

More details concerning the I/S achievable region of problem P5 can be found in [32]. The technique adopted to solve problem P5 is now presented. First of all, notice that P5 has the following equivalent homogeneous QCQP reformulation with four constraints:

P6

⎧ max ⎪ ⎪ x ⎪ ⎪ ⎪ s.t. ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

tr (Q0 X ) 1 − η ≤ tr (Q1 X ) ≤ 1 tr (Q2 X ) ≤ EI tr (Q3 X ) ≤ 0

.

(4.36)

tr (Q4 X ) = 1 X = xx † , x = [cT , t]T

The matrices Qi are defined as follows:  Q0 =  Q3 =

R 0

     0 I 0 RI 0 , Q1 = , , Q2 = 0 0 0 00

   I −c0 00 , Q = . 4 † 01 −c0 1 − ε

(4.37)

(4.38)

Precisely, the optimal values of P5 and P6 are equal, i.e., v(P5 ) = v(P6 ).

(4.39)

In fact, assume that c solves P5 . It follows that [c , 1]T is feasible to P6 , and thus v(P5 ) = c † Rc ≤ v(P6 ). On the other hand, suppose that x  = [c , t  ] is optimal to P6 . Then, it is seen that c /t  is feasible to P5 , and thus, v(P6 ) = x  † Q0 x  = (c /t  )† R(c /t  ) ≤ v(P5 ). According to (4.39), we can solve

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P6 , in order to get an optimal solution to P5 . To this end, let us introduce the SDP relaxation of P6 ,

P7

⎧ max ⎪ ⎪ X ⎪ ⎪ ⎪ s.t. ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

tr (Q0 X ) 1 − η ≤ tr (Q1 X ) ≤ 1 tr (Q2 X ) ≤ EI

,

tr (Q3 X ) ≤ 0 tr (Q4 X ) = 1

(4.40)

X 0

which is also viewed as the SDP relaxation of P5 , considering (4.39). Now, we can show that the SDP relaxation P7 is tight (i.e., v(P7 ) = v(P6 )), exploiting the observation that the first constraint function and the third constraint function in (4.35) share the same Hessian matrix I . In other words, the QCQP P5 is hidden convex and can be solved polynomially. To proceed with the proof, denoted by  X =

C  x x † 1

 (4.41)

an optimal solution to problem P7 , and let the optimal energy δ1 = tr (Q1 X  )( = tr (C  )). Let us formulate the new waveform design problem: ⎧ max ⎪ ⎪ c ⎪ ⎨ s.t. P8 ⎪ ⎪ ⎪ ⎩

c† Rc c† c = δ1 c† RI c ≤ EI

,

(4.42)

c − c0 2 ≤ ε

which can be transformed equivalently into the homogeneous QCQP:

P9

⎧ max ⎪ ⎪ c ⎪ ⎪ ⎪ ⎪ ⎨ s.t. ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

tr (RC) tr (C) = δ1 tr (RI C) ≤ EI ,  δ1 +1−ε 2 tr (C0 C) ≥ 2 C = cc†

(4.43)

via certain phase rotation (cf. [48, page 5621]), i.e., v(P8 ) = v(P9 ).

(4.44)

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Radar waveform design based on optimization theory

Algorithm 2: Algorithm to solve the radar waveform design problem P5 Require: R, RI , c0 , η, EI , ε; Ensure: An optimal solution c to problem P5 ; 1: solve problem P7 , getting a solution X  (as in (4.41)); 2: if Rank (C  ) = 1 then 3: perform C  = c c† , getting c ; 4: else if Rank (C  ) = 2 then 5: find c = D2 (C  , R, I , RI , C0 ); 6: else 7: find c = D1 (C  , R, I , RI , C0 ); 8: end if † 9: return c ej arg (c c0 ) . †

Further, if c is optimal to P9 , then c ej arg (c c0 ) is optimal to P8 . Evidently, the SDP relaxation of P9 is ⎧ max tr (RC) ⎪ ⎪ C ⎪ ⎪ ⎪ ⎪ s.t. tr (C) = δ1 ⎨ tr (RI C) ≤ EI P10 . (4.45) ⎪   ⎪ 2 δ +1−ε ⎪ ⎪ tr (C0 C) ≥ 1 2 ⎪ ⎪ ⎩ C 0 To proceed with the solution technique, the following proposition is necessary. Proposition 4.1. Suppose that X  is an optimal solution (as in (4.41)) to problem P7 , and let δ1 = tr (C  ). Then, 1. 2. 3.

the optimal values of problems P8 , P9 , and P10 are equal, namely, v(P7 ) = v(P6 ) = v(P5 ) = v(P8 ) = v(P9 ) = v(P10 ); the solution C  (as in (4.41)) is optimal to problem P10 ; † problem P10 has always a rank-one optimal solution c c† , and c ej arg (c c0 ) is optimal to problem P8 and problem P5 .

Proof. See Appendix A.4. The procedure leading to an optimal solution to problem P5 is summarized in Algorithm 2, whose computational complexity is the same as Algorithm 1.

4.6.2 Performance analysis In this subsection, we analyze the behavior of the waveforms devised according to Algorithm 2; in doing so, we deal also with cases in which the technique shown in Section 4.5 cannot be applied (i.e., operative points not feasible to Algorithm 1). We consider the same simulation setup as in Section 4.5.2. Furthermore, we assume η = 0.5 as energy modulation parameter.

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The I/S achievable region for the analyzed scenario is reported in Figure 4.5, together with that of Algorithm 1 (which corresponds to η = 0). As expected, the energy modulation at the transmitter allows to enlarge the I/S achievable region, providing more and more radar operative points (EI , ε). Nevertheless, the higher the η, the lower the radar maximum operation range; otherwise stated, more demanding similarity and spectral constraints than those of Algorithm 1 can be enforced at the price of a reduction in the search volume. In Figure 4.6, we provide the ESD, the squared modulus of the ACF, and the normalized SINR for the waveforms devised through Algorithm 2 in correspondence of the operative points (EI ,n , εˆ ), n = 1, 2, 3, 4, with EI ,1 = 0.037, EI ,2 = 0.066, EI ,3 = 0.216, EI ,4 = 0.426, and εˆ = 0.264 (left-triangle, dot, diamond, and square of Figure 4.5, respectively), and compare the results with those obtained using the technique in Section 4.5 (i.e., Algorithm 1, for which only the points (EI ,2 , εˆ ), (EI ,3 , εˆ ), and (EI ,4 , εˆ ) are feasible). Specifically, in Figure 4.6(a), we plot the ESD of the synthesized transmit sequences versus the normalized frequency, together with that of the reference code c0 (the stop-bands are shaded in light gray). For the operative points shared with the I/S achievable region of Algorithm 1, the waveforms devised through both Algorithms 1 and 2 exhibit similar spectral features. Nevertheless, the proposed approach allows to satisfy more demanding spectral requirements with a fixed similarity level: in fact, the pair (EI ,1 , εˆ ) = (0.037, 0.264) cannot be applied with Algorithm 1. This effect is shown in Figure 4.6(c), where the normalized SINR of the devised waveform is plotted versus EI , assuming εˆ = 0.264. As expected, Algorithm 2 outperforms Algorithm 1; besides, Algorithm 2 provides satisfactory SINR levels, also for challenging spectral constraints that are infeasible to Algorithm 1. 0.7

h = 0.5 h=0 (0.037,0.264) (0.216,0.264) (0.426,0.264) (0.066,0.195) (0.066,0.264) (0.066,0.444) (0.066,0.654)

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Figure 4.5 I/S achievable region, for chirp similarity code with an interference environment accounting for K = 7 equivalent stop-bands; black line: Algorithm 2; black-dashed line: Algorithm 1

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Radar waveform design based on optimization theory 10 5 0

ESD (dB)

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Alg.2, (EI,1,e) = (0.037,0.264)

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Alg.1, (EI,4,e) = (0.426,0.264) Alg.1, (EI,3,e) = (0.216,0.264)

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–2

0.07

0.08

0.09

Alg.2, (h, e) = (0.5, 0.264)

–2.5

Alg.1, e = 0.264 0.05

(c)

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

EI

Figure 4.6 (a) ESD; (b) squared modulus of the ACF. Shaded light-gray rectangles: stop-bands; gray curve: reference code c0 ; gray dashed curve: Algorithm 2, EI ,1 = 0.037, εˆ = 0.264; black dashed curve: Algorithm 2, EI ,3 = 0.216, εˆ = 0.264; black o-marked dash-dotted curve: Algorithm 2, EI ,2 = 0.066, εˆ = 0.264; gray dot-marked dotted curve: Algorithm 1, EI ,2 = 0.066, εˆ = 0.264; black curve: Algorithm 2, EI ,4 = 0.426, εˆ = 0.264; black x-marked dashed curve: Algorithm 1, EI ,4 = 0.426, εˆ = 0.264; black dash-dotted curve: Algorithm 1, EI ,3 = 0.216, εˆ = 0.264. (c) Normalized SINR versus ε; black dashed curve: Algorithm 2, η = 0.5, εˆ = 0.264; black curve: Algorithm 1, εˆ = 0.264. Finally, in Figure 4.6(b), we provide a performance analysis in terms of autocorrelation properties of the devised waveforms. The results highlight the ability of the enforced similarity constraint to control some relevant features of the radar code, such as range resolution and PSL. Besides, higher SINR values are swapped for worse ISL values.

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4.7 Radar waveform design under similarity, bandwidth priority, and spectral coexistence constraints 4.7.1 Code design optimization problem In this subsection, we consider the synthesis of radar waveforms maximizing the achievable detection probability while ensuring spectral compatibility with the overlayed electromagnetic radiators. However, unlike the previous design approaches, we now account for the quality of the bandwidths available for transmission via the specific constraint (4.16). By doing so, we can prioritize the available frequency channels (free of cooperative transmissions), in the sense that the spectral energy injected in each channel can be tuned accordingly. We assume, as a figure of merit, the SINR defined in (4.5); furthermore, we impose the similarity constraint (4.6) and the energy constraint (4.7). Besides, we impose the spectral compatibility constraint (4.12). Finally, we force the bandwidth priority constraint (4.16). The resulting waveform design problem can be formulated as the nonconvex QCQP problem: ⎧ ⎧ max c† Rc max c† Rc ⎪ ⎪ ⎪ ⎪ c c ⎪ ⎪ ⎪ ⎪ † ⎪ ⎪ s.t. c† c = 1 s.t. c c = 1 ⎪ ⎪ ⎨ ⎨ e c† RI c ≤ EI . (4.46) P11 = P12 c† RI c ≤ EI ⎪ ⎪ ⎪ ⎪ † † ⎪ ⎪ ¯ ¯ c R c ≥ E c R c ≥ E ⎪ ⎪ I ,2 I I ,2 I ⎪ ⎪ ⎪ ⎪ ⎩ ⎩ † c − c0 2 ≤ ε Re(c0 c) ≥ 1 − ε/2 Observe that, in order to get an optimal solution to problem P12 , it is sufficient to solve the following QCQP problem P13 : ⎧ max c† Rc ⎪ ⎪ c ⎪ ⎪ ⎪ c† c = 1 ⎪ ⎨ s.t. c† RI c ≤ EI . (4.47) P13 ⎪ ⎪ † ⎪ ¯ c RI ,2 c ≥ EI ⎪ ⎪ ⎪ ⎩ † |c0 c| ≥ 1 − ε/2 Precisely, as for problem P2 and P3 , given an optimal solution c to P13 , we can † construct a solution c¯  = exp ( − j arg (c0 c ))c optimal to P12 . To proceed further, we consider the convex SDP relaxation of P13 : ⎧ max tr (RC) ⎪ ⎪ C ⎪ ⎪ ⎪ s.t. tr (C) = 1 ⎪ ⎪ ⎪ ⎨ tr (RI C) ≤ EI P14 . (4.48) ⎪ tr (RI ,2 C) ≥ E¯ I ⎪ ⎪ ⎪ ⎪ ⎪ tr (C0 C) ≥ δε ⎪ ⎪ ⎩ C 0

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Let C  be an optimal solution to problem P4 . If it is of rank-one, then the code c such that C  = c c† is also optimal to problem P3 . Otherwise, given an arbitrary rank solution, we can find a rank-one solution to problem P3 by suitably applying the rank-one decomposition procedures described in [42,49] (see Lemmas 4.1 and 4.2). Indeed, the following propositions hold true: Proposition 4.2. Suppose that C  is optimal to problem P14 with rank three or more. Then, a rank-one optimal solution to P14 can be always constructed. Proof. To show it, it suffices to employ Theorem 4.2 with (L, M , N¯ ) = (0, 1, 3). Proposition 4.3. Suppose that C  is optimal to problem P14 and such that one of the three inequality constraints at C  is strict. Then, there is a rank-one optimal solution to problem P14 . Proof. To prove it, it suffices to apply Theorem 4.3 with (L, M , N¯ ) = (0, 1, 3). As to the case where the rank of an optimal solution for problem P14 is two and all the constraints become equal at the solution, we try to find a suboptimal solution to problem P14 , as indicated by the second statement ofTheorem 4.2. The procedure to produce an optimal or suboptimal solution to problem P11 is briefly summarized into Algorithm 3. The total computational cost of the algorithm is dominated by solving the SDP relaxation problem, whose worst-case complexity is O(N 3.5 log (1/ζ )).

4.7.2 Performance analysis In this subsection, we assess the performance of Algorithm 3. As to the parameters setting for the radar transmit signal, the reference code c0 , and the interference, we assume the same simulation setup as in Section 4.5.2 with the exception: ● ●

●

●

ε ∈ [0.85, 2] (this range of values ensures the feasibility of the problem). K = 4, σJ ,1 dB = 40 dB, and σJ ,2 dB = 50 dB (according to model (4.33) for the disturbance covariance matrix). 1 = [0.14, 0.26], 2 = [0.39, 0.46], 3 = [0.55, 0.65], and 4 = [0.82, 0.87], with wk1 = 1 for k = 1, . . . , K. ˜ 1 = [0, 0.14] , ˜ 2 = [0.26, 0.39] ,  ˜ 3 = [0.46, 0.55] ,   (4.49) ˜ 4 = [0.65, 0.82] ,  ˜ 5 = [0.87, 1] . 

Besides, we suppose (based on some a-priori information) that, due to their characteristics, the first two channels are more appropriate for radar transmissions than the remaining three, which exhibit a poor spectral quality. Hence, we set the weights ˜ w˜ 1 = w˜ 2 = 1, while w˜ k˜ = 0 for k˜ = 3, . . . , K. First, we provide the analysis in terms of ESD of the waveforms devised assuming EI = 20, E¯ I = 0.99, and ε ∈ {0.85, 1, 1.5, 2}. Notice that, when ε = 2, no similarity

Constrained radar code design for spectrally congested environments

117

Algorithm 3: An optimal or suboptimal solution procedure for the radar waveform design problem P11 Require: ε, R, RI , RI ,2 , EI , E¯ I , c0 ; Ensure: An optimal or suboptimal solution c¯  to problem P11 ; 1: solve the SDP problem P14 , finding C  ; let δ1 = tr (RI C  ), δ2 = tr (RI ,2 C  ), and † δ3 = c0 C  c0 ; 2: if C  is of rank one, then get c from C  = c c† and go to step (8); 3: if tr (RI C  ) < EI , then perform the rank-one decomposition D3 (C  , RI ,2 − † δ2 I , C0 − δ3 I ) getting {c1 , . . . , cR }, among them choose c such that c cR† cI c < EI , output c = c/c; ¯ I , then perform the rank-one decomposition D3 (C  , RI − 4: if tr (RI ,2 C  ) > E c† R c δ1 I , C0 − δ3 I ) getting {c1 , . . . , cR }, among them choose c such that c†Ic,2 > E¯ I , output c = c/c; 5: if tr (C0 C  ) > δε , then perform the rank-one decomposition D3 (C  , RI − †

c† c c c

δ1 I , RI ,2 − δ2 I ) getting {c1 , . . . , cR }, among them choose c such that c0† c0 > δε , output c = c/c; 6: if C  is of rank three or more, then perform the rank-one decomposition D1 (C  , I , RI , RI ,2 , C0 ) getting c = c, and output it; 7: if C  is of rank two, then randomly pick up a vector z ∈ / Range (C  ), and find c in  the linear subspace spanned by z and Range (C ), such that c† c = 1, c† RIi c = δi , † i = 1, 2, and c† c0 c0 c = δ3 , and output c = c. †  8: set c¯ = exp ( − j arg (c0 c ))c .

is accounted at the code design stage. In Figure 4.7(a), we plot the ESD of the synthesized signals versus the normalized frequency, together with that of the reference code. The stop-bands correspond to the regions in shaded gray, while the remaining portions of the spectrum (white regions) denote the frequencies where no cooperative system is transmitting. The devised transmit waveforms exhibit, in correspondence of the cooperative stop-bands, a limited energy in comparison with that of c0 . This is more and more evident as the similarity constraint becomes less and less tighten. We remark that the energy allocation in the remaining channels is fully compatible with the assigned quality level; indeed, the last three available free bands are hindered with respect to the first two, where much energy is evidently injected. We also highlight the presence of two spectral nulls in correspondence of the hostile emitters’ central frequencies, as per design. The depth of these nulls increases with ε; in fact, the consequent availability of additional DOF leads to an improved energy allocation for the sought radar signals. Nevertheless, as shown in the curves of Figure 4.7(b), better energy allocation, spectral compatibility, and interference rejection are traded-off with a worse range resolution and/or sidelobe levels. Specifically, the devised waveforms get farther and farther from the reference signal as the parameter ε increases. This leads to radar codes with less attractive auto-ambiguity features.

118

Radar waveform design based on optimization theory 20 0

ESD (dB)

–20 –40 –60 c0

–80

Alg.3, e = 0.85 Alg.3, e = 1 Alg.3, e = 1.5 Alg.3, e = 2

–100 –120 0

0.1

0.2

0.3

0.4

(a)

0.5

0.6

0.7

0.8

0.9

1

Frequency 0

2

|ACF| (dB)

–10 –20 –30 –40 –50

є

–60 –150

–100

–50

(b)

0

50

100

150

Delay bin 0.5 0 –3

SINR (dB)

–0.5

5

× 10

0

–1

–5

–1.5

–10

c0

–2 –15

0.85

0.9

0.95

1

Alg.3, (EI,EI) = (20, 0.99)

1.05

–2.5 –3 0.8

1

(c)

1.2

1.4

1.6

1.8

2

e

Figure 4.7 (a) ESD; (b) squared modulus of the ACF. Shaded light gray regions: stop-bands; white regions: free bands; EI = 20, E¯ I = 0.99; gray curve: reference code c0 ; black o-marked dash-dotted curve: optimal code c¯  for ε = 0.85; black dotted curve: optimal code c¯  for ε = 1; black dashed curve: optimal code c¯  for ε = 1.5; black dot-marked curve: optimal code c¯  for ε = 2. (c) normalized SINR versus ε; black curve: Algorithm 3 for EI = 20, E¯ I = 0.99; gray curve: reference code c0 .

4.8 Conclusions In the present chapter, we have addressed radar waveform optimization in spectrally crowded scenarios via constrained quadratic programming techniques. First of all, we have defined a taxonomy of performance measures which are to be optimized and/or controlled in the waveform synthesis: SINR, signal energy, similarity constraint,

Constrained radar code design for spectrally congested environments

119

interference constraint for spectral compatibility, and quality constraint for bandwidth selection. Then, we have formulated the radar waveform design as a nonconvex QCQP problem and have exploited a specific methodology leveraging rank-one matrix decomposition tools to prove the hidden convexity of some instances of the waveform optimization or to build solution procedures ensuring good quality radar waveforms. At the analysis stage, we have assessed the performance of the devised waveforms studying the trade-off among the achievable SINR, spectral shape, and autocorrelation. The results have highlighted that the proposed algorithms are able to realize a good compromise between the spectral coexistence and the achievable SINR at the price of other desirable characteristics of the radar signal, for instance, maximum range, sidelobe levels, and/or range/Doppler resolution. Before concluding, we want to remark the methodological relevance of the chapter from the optimization theory point of view in the sense that the framework presented in this chapter is general and can be applied to a multitude of radar problems involving quadratic objective functions and a limited number of quadratic constraints.

Appendix A A.1 Proof of Theorem 4.1 Proof. Suppose that L + M + N¯ = 3. For example, we consider (L, M , N¯ ) = (1, 1, 1) (the discussion for the other cases, say (L, M , N¯ ) = (3, 0, 0), is similar). Suppose that (tr (B1 X  ), tr (C1 X  ), tr (D1 X  )) is nonzero. Otherwise, the optimal value is either zero or negative infinity, which is a trivial case. Suppose without loss of generality that tr (B1 X  )  = 0. Let β1 = tr (B1 X  ) and δ1 = tr (D1 X  ). Therefore, we have    c1 tr C1 − B1 X  = 0, (A.1) β1    δ1 tr D1 − B1 X  = 0. (A.2) β1 It follows from the specific

rank-one matrix decomposition Lemma 4.2 that there is a decomposition X  = Rr=1 ur u†r (where R is the rank of X  ) such that    c1 † tr C1 − B1 ur ur = 0, r = 1, . . . , R, (A.3) β1    δ1 tr D1 − B1 ur u†r = 0, r = 1, . . . , R. (A.4) β1

Since β1 = tr (B1 X  ) = tr (B1 Rr=1 ur u†r ), hence there is r0 ∈ {1, . . . , R}, say r0 = 1, such that tr (B1 u1 u†1 ) and β1 have the same sign. In other words, †

tr (B1 u1 u1 ) > 0. β1

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Radar waveform design based on optimization theory

Let λ=

β1 †

tr (B1 u1 u1 )

> 0.

From (A.3) and (A.4), it follows that †

†

†

tr (B1 (λu1 u1 )) = β1 , tr (C1 (λu1 u1 )) = c1 , tr (D1 (λu1 u1 )) = δ1 . †

†

In other words, λu1 u1 is feasible for (4.21). This also implies that λu1 u1 , together with {w1  , w¯ 1 , z1 }, fulfills (4.23)–(4.25). Observe that u1 ∈ Range (X  ), which means that † † we conclude that λu1 u1 λu1 u1 , together with {w1  , w¯ 1 , y1 , z1 }, satisfies (4.26). Hence, √ is optimal to (4.21) with (L, M , N¯ ) = (1, 1, 1), namely, λu1 is optimal to (4.18) with (L, M , N¯ ) = (1, 1, 1). Suppose that L + M + N¯ = 2. For example, we consider (L, M , N¯ ) = (1, 1, 0) (the discussion for the other cases, say (L, M , N¯ ) = (2, 0, 0), is similar). Then, we have tr (B1 X  ) = β1 , tr (C1 X  ) = c1 . Applying Lemma 4.2 gives rise to a matrix decomposition X  =

R r=1

ur u†r such that

tr (B1 (Rur u†r )) = β1 , tr (C1 (Rur u†r )) = c1 , r = 1, . . . , R. This means that any Rur u†r is not only feasible for (4.21) with (L, M , N¯ ) = (1, 1, 0), but also optimal for it (due to Rur u†r together with (w1  , w¯ 1 , y1 ), satisfying the complementary conditions (4.23)–(4.26)). Suppose that L + M + N¯ = 1. For example, we consider (L, M , N¯ ) = (1, 0, 0) (the discussion for the other cases, say (L, M , N¯ ) = (0, 1, 0), is similar). In other words, SDP problem (4.21) has only one constraint. Suppose that v is the optimal value i.e., v = tr (AX  ). Again, apply Lemma 4.2, and we obtain

R for (4.21),  † X = r=1 ur ur such that tr (A(Rur u†r )) = v , tr (B1 (Rur u†r )) = β1 , r = 1, . . . , R. This means that any Rur u†r is not only feasible but also optimal to (4.21) with (L, M , N¯ ) = (1, 0, 0), since the objective function value at Rur u†r is equal to the optimal value.

A.2 Proof of Theorem 4.2 Proof. Suppose that (L, M , N¯ ) = (1, 1, 2) (the discussion for the other cases, say (L, M , N¯ ) = (4, 0, 0), is similar). Assume that the rank of X  is greater than or equal to three. It follows from the first statement of Lemma 4.1 that a point x ∈ Range (X  ) can be constructed in polynomial time complexity such that (x † B1 x, x † C1 x, x † D1 x, x † D2 x) = = (tr (B1 X  ), tr (C1 X  ), tr (D1 X  ), tr (D2 X  )).

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Then, it is not hard to verify that x, together with (w1  , w¯ 1 , y1 , z1 , z2 ), satisfies the complementary conditions (4.23)–(4.26). Let x  = x. Thus, x  x † is optimal to (4.21) with (L, M , N¯ ) = (1, 1, 2). Assume that the rank of X  is equal to two (namely, the dimension of Range (X  ) is two). Again, it follows from the second statement of Lemma 4.1 that there is x ∈ Span (Range (X  ) ∪ {v}), where v ∈ / Range (X  ) (such v does exist due to the  dimension of X is more than two), such that (x † B1 x, x † C1 x, x † D1 x, x † D2 x) = = (tr (B1 X  ), tr (C1 X  ), tr (D1 X  ), tr (D2 X  )) holds. Let x  = x. It is easily seen that x  x † is feasible but not optimal to (4.21) due to the fact that the complementary condition (4.26) cannot be satisfied. Therefore, x  x † is treated as a suboptimal rank-one solution for (4.21).

A.3 Proof of Theorem 4.3 Proof. We assume that (L, M , N¯ ) = (1, 1, 2) (the discussion for the other cases is similar, say (L, M , N¯ ) = (2, 1, 1) or (1, 2, 1)), and that tr (D1 X  ) < d1 . Since C1  0 and c1 > 0, therefore the constraint tr (C1 X ) = c1 can be specified to tr (X ) = 1. Let β1 = tr (B1 X  ), δi = tr (Di X  ), i = 1, 2. Thus, we have tr ((B1 − β1 I )X  ) = 0, tr ((D2 − δ2 I )X  ) = 0. It follows from Lemma 4.2 that there is a decomposition X  =   ur u†r = β1 , r = 1, . . . , R, tr B1 ur 2   ur u†r = δ2 , r = 1, . . . , R. tr D2 ur 2

R r=1

ur u†r such that (A.5) (A.6)

Clearly,  tr

ur u†r ur 2

 = 1, r = 1, . . . , R.

We claim that there an r0 ∈ {1, 2, . . . , R}, say r0 = 1, such that   † u1 u1 tr D1 ≤ δ1 < d1 . u1 2 Otherwise,   ur u†r tr D1 > δ1 , r = 1, . . . , R, ur 2

(A.7)

(A.8)

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Radar waveform design based on optimization theory

and this implies that tr (D1 X  ) > δ1 , which is a contradiction. Based on (A.5)– † (A.8), we conclude that (u1 u1 /u1 2 ) is feasible for (4.21) with (L, M , N¯ ) = (1, 1, 2). Furthermore, it can be shown easily that (u1 u†1 /u1 2 ) fulfills the complementary conditions (4.23)–(4.26), with (w1  , w¯ 1 , y1 , z1 , z2 ). Therefore, (u1 u†1 /u1 2 ) is optimal to (4.21).

A.4 Proof of Proposition 4.1: SDP relaxation tightness for (4.36) Proof. (i) We show the inequality chain: v(P7 ) ≥ v(P6 ) = v(P5 ) ≥ v(P8 ) = v(P9 ) = v(P10 ) ≥ v(P7 ), considering v(P6 ) = v(P5 ) (as in (4.39)) and v(P8 ) = v(P9 ) (as in (4.44)). It is evident that v(P7 ) ≥ v(P6 ),

(A.9)

since problem P7 is a relaxation for problem P6 . Since δ1 ∈ [1 − η, 1], we have v(P5 ) ≥ v(P8 ).

(A.10)

Similar to [48, Sec. IV-A], we can claim that the SDP relaxation problem P10 for problem P9 is tight, namely, v(P9 ) = v(P10 ).

(A.11)

As a consequence, from (4.39), (4.44), (A.9)–(A.11), we have v(P7 ) ≥ v(P6 ) = v(P5 ) ≥ v(P8 ) = v(P9 ) = v(P10 ). Thus, in order to show that all the optimal values are equal, it is sufficient to prove that v(P10 ) ≥ v(P7 ).

(A.12)

Since any optimal solution X  of problem P7 (as in (4.41)) is feasible for the problem itself, the block C  (of X  ) complies with the first constraint function and the second constraint function of (4.43). From the fact that X  satisfies the third constraint function of (4.36), it follows that Re(x † c0 ) ≥ δ1 + 1 − ε/2. Moreover, C  x  x † (due † to X  0), and tr (C  c0 c0 ) ≥ |x † c0 |2 ≥ (δ1 + 1 − ε/2)2 , which implies that C  also fulfills the third constraint function of (4.43). Therefore, C  is feasible to problem P10 . This means that any optimal solution X  to problem P7 gives a feasible solution C  to problem P10 , and hence (A.12) holds true. (ii) Suppose that X  is an optimal solution to problem P7 (as in (4.41)). We have just showed that C  is feasible to problem P10 , with tr (RC  ) = v(P7 ) = v(P10 ). Thus C  is optimal to problem P10 . (iii) Apply Theorem 4.1 with (L, M , N¯ ) = (0, 1, 2), and we obtain an optimal solu† tion c c† for P10 . It is not hard to show that c ej arg (c c0 ) is optimal to problem P8 .

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[46]

[47]

[48]

[49]

Chapter 5

Robust transmit code and receive filter design for extended targets detection in clutter Seyyed Mohammad Karbasi1 , Augusto Aubry2 , Antonio De Maio2 , Mohammad Hasan Bastani1 , and Alfonso Farina3

5.1 Introduction Modern advances delivered to the radar technology, such as solid-state transmitters [1, Chapters 6, 11, and 25], novel computing architectures, real-time and off-the-shelf processors [2, Chapter 10], as well as digital arbitrary signal generators [3,4], have boosted up the surveillance systems with the capability of selecting the transmit waveforms and the receive filters adaptively and dynamically, in response to the actual operative scenario. As a matter of fact, this flexible adjustment of the probing signal and receive processing architecture, as required by the next cognitive radar generation, is envisioned as the key enabler for radar performance enhancement, in terms of target detection, identification, and tracking [5–9]. Joint radar waveform-filter design has attracted a lot of attention during the last decades in radar community [5–26]. Many papers have addressed the joint synthesis considering a point-like target model, where the target falls in a specific range cell [6, 7,9,19]. However, employing high-range resolution (HRR) radars, the range cell size may become far smaller than the target physical extent, and it is no longer appropriate to model the target as a point-like scatterer [27,28]. In such a context, the scattering behavior of the target is characterized by the target impulse response (TIR) [11], which is a function of the target physical structure, the target aspect angle (TAA) with respect to the radar line-of-sight (LOS), and the radar-operating frequency [29]. Thus, in the context of waveform design, different TIR models have been adopted in the open literature assuming a variety of optimization criteria [30,31]. Some papers have considered a deterministic TIR, which is obtained through previous measurements, available track files, and external databases [10–18,20–24]. An alternative approach is to employ a partially known TIR, or to characterize it as a random process [18,23,25].

1

Department of Electrical Engineering, Sharif University of Technology, Tehran, Iran Dipartimento di Ingegneria Elettrica e Tecnologie dell’Informazione, Università degli Studi di Napoli “Federico II,” Napoli, Italy 3 Selex ES (retired), Roma, Italy 2

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The problem of transmit waveform design for extended targets has been of great attention in the open literature during the last decades [10–18,20–23]. In [11], two waveform design approaches have been introduced for noise-only environment. In particular, the authors have addressed the maximization of a signal-to-noise ratio (SNR) assuming a deterministic TIR. Moreover, modeling the TIR as a random Gaussian process, the mutual information (MI) between the received signal and a Gaussian ensemble of the TIR has been considered as the objective function for waveform optimization. In [15], design algorithms, based on either the SNR or MI criteria, have been used to improve the performance of a closed-loop radar system for target recognition purposes. Therein, the targets have been modeled with known TIR and additive white Gaussian noise has been assumed for the interference. More realistic and challenging models have been studied, considering reverberating environments, i.e., clutter. In [24], illumination signals matched to both deterministic and stochastic extended targets have been designed assuming the presence of signal-dependent interference. Several scenarios, including SNR-based waveform design for stochastic targets and SNR/MI-based design for a known target in signal-dependent interference, have been addressed. Furthermore, a relation between MI and SNR has been investigated in the context of waveform design for stochastic targets. Additionally, in [13], following the signal-to-interference-plus-noise ratio (SINR)-based approach, an iterative procedure has been addressed to optimize the detection probability of a known TIR target embedded in clutter environment. This procedure has been applied in [32,33], for target identification. Moreover, in [17], the framework of [13] has been exploited to improve multiple-input–multiple-output radars performance. Unfortunately, the iterative techniques in [13,17] do not guarantee the SINR improvement in each iteration step and hence cannot guarantee the convergence [18]. In [18], a joint transmit waveform and receive filter technique has been proposed, which sequentially improves the achieved SINR for a known TIR in signal-dependent interference, under an energy constraint. Specifically, in each iteration, the algorithm finds a local maximum, which is also a global maximum along the dimension of the transmit waveform and the dimension of the receive filter separately. Finally, in [25,34–38] other interesting and technically sound waveform design strategies for extended radar target can be found. Remarkably, most of the aforementioned works just force an energy constraint on the sought probing waveform and assume simplified TIR models so as to ensure mathematical tractability to the resulting design problem. In this chapter, assuming signal-dependent interference, a novel design technique is described aimed at jointly optimizing the transmit code and the receive filter in terms of the SINR perceived from an extended target [23]. The focus of the chapter is on an already-tracked target whose TIR is assumed to be partially known (namely, it belongs to a preassigned uncertainty set) based on the available tracking information. In fact, due to the high sensitivity of the TIR to the target parameters such as its TAA, it is unrealistic to suppose an exact TIR knowledge. Hence, to make our design strategy robust with respect to TIR uncertainty, the worst case optimization paradigm is considered. In other words, the worst case SINR is optimized, at the receive filter output, with respect to all the possible TIRs belonging to the given uncertainty set. To this end, the set of likely TAAs for the extended target of interest can be obtained

Robust transmit code and receive filter design 129 leveraging previous measurements and/or tracking files so as to define the TIR uncertainty set in terms of their corresponding TIRs. Moreover, as an alternative approach, the uncertainty set can be defined as a spherical set centered at a prescribed TIR. Remarkably, the proposed approach is useful to continue tracking or to classify an already-tracked target, adapting suitably the probing signal based on previous target observations. In addition to an energy constraint on the transmit waveform, an upper bound on the signal peak-to-average-power ratio (PAR) is also considered, which makes the devised signal practically implementable when the transmitter operates close to saturation. The resulting problem belongs to the class of non-convex max– min optimization problems, which is hard to solve. Hence, the relaxation techniques are used to tackle the resulting relaxed optimization problem resorting to the alternating optimization paradigm. Each iteration involves the solution of two semidefinite programming (SDP) problems, and the overall procedure monotonically improves the relaxed SINR objective, with proved convergence. Besides, the randomization approaches in [39] and [40] are utilized, to synthesize the transmit code and the receive filter sharing a robust response. Finally, at the analysis stage the performance of the proposed algorithm is assessed, providing several case studies. The results reveal that the new algorithm can address the joint design of transmit waveform and receive filter, enforcing PAR constraint and ensuring robustness against prior knowledge inaccuracies about the TIR. The rest of the chapter is organized as follows. In Section 5.2, the target and signal models as well as the statistical characteristics of the interference are introduced and described. In Section 5.3, the design problem is formulated first and the iterative algorithm for the relaxed problem is introduced. Section 5.4 is devoted to the receive filter and transmit code synthesis based on the solution to the relaxed problem. Moreover, the problem with spherical uncertainty set is discussed in Section 5.5. Numerical results are given in Section 5.6. Finally, concluding remarks are provided in Section 5.7 together with some possible future research tracks.

5.2 Target and signal model 5.2.1 Target model In HRR radars, the target range extent is far larger than the radar range resolution, so that it is no longer suitable to utilize the point-like scatterer target model. Specifically, using the scattering center model, the large extent targets are described as a set of multiple dominant scattering centers [41,42] (when utilizing fine range resolutions via high bandwidth waveforms); i.e., portions of the target that produce a significant return are projected into a finite number of isolated range cells along the radar LOS direction [43]. In this context, the target is modeled as a linear system with a finite impulse response that can be parametrized by the target shape and its direction toward the radar LOS [44]. Figure 5.1 displays the construction of the TIR using the projection of the scattering centers on the radar LOS. Evidently, the TIR is highly sensitive with respect to small changes in the target orientation [1,45].

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t(n) LOS Range

Figure 5.1 TIR construction of a sample from extended target obtained via projection of the target scattering centers on the radar LOS

5.2.2 Signal model A monostatic radar system is considered that transmits the coded waveform s(t), given by s(t) =

N −1 

s(k)g(t − kT ),

k=0

where g(t) is the radar sub-pulse, T is the sub-pulse interval length, and s(k) is the kth element of the N -dimensional transmit code s = [s(0), . . . , s(N − 1)]T ∈ CN . The discrete-time baseband equivalent representation of the observations from the extended target of interest, which is embedded in a signal-dependent interfering environment (see Figure 5.2), can be expressed as r(n) = [tθ (n) + c(n)]  s(n) + v(n) =

N −1 

[tθ (n − k) + c(n − k)] s(k) + v(n),

(5.1)

k=0

where n is the discrete-time index, tθ (n) is the TIR (corresponding to the TAA θ), sharing a support interval of length Q∗ , which means, without loss of generality, that tθ (n) = 0 unless n ∈ {0, . . . , Q − 1}, c(n) denotes the clutter impulse response (CIR), and v(n) represents the receiver noise. In addition, the TIR vector is defined as t θ = [tθ (0), . . . , tθ (Q − 1)]T ∈ CQ . Hence, collecting all the samples in

∗ The parameter Q is determined according to the maximum target dimension of interest and the radar range resolution.

Robust transmit code and receive filter design 131 Extended target of interest θ tθ (n) Monostatic radar system

s(n) +

tθ (n) S(n) Transmit signal

C(n)

s(n)

C(n) v(n) Received signal from the target of interest

Additive noise

r(n)

Clutter

+

Figure 5.2 A pictorial representation of the received signal from an extended target embedded in clutter r(n) containing information from the extended target of interest within a single vector r = [r(0), . . . , r(M − 1)]T ∈ CM , where M = Q + N − 1 is the total number of received discrete-time samples containing target information, the received samples of interest in (5.1) can be expressed as r = T θ s + Cs + v, where v = [v(0), . . . , v(M − 1)]T is the additive noise contribution, Tθ =

Q−1 

tθ (n)J n ,

(5.2)

n=0

and C=

M −1 

c(n)J n ,

n=−N +1

are, respectively, the TIR and the CIR matrices, with J n being the M × N -dimensional shift matrix:  1, if 1 − 2 = n , (5.3) J n (1 , 2 ) = 0, if 1 − 2  = n 1 ∈ {1, . . . , M }, 2 ∈ {1, . . . , N }. As to the statistical characterization of the signal-independent interference, the noise vector v is assumed to be zero-mean, i.e., E[v] = 0, and circularly symmetric with

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covariance matrix  v = E[vv† ] = σv2 I , where σv2 is the variance of each filtered noise sample. With reference to the clutter contribution, instead, c(n) is supposed to be a zero-mean random process and is denoted by rc (n, n ) = E[c(n)c∗ (n )]

(5.4)

the correlation between the clutter scatterers at nth and n th range bins. Furthermore, the clutter and the noise are assumed to be uncorrelated random processes. As to the TIR, the assumption of its exact knowledge is far from reality due to high sensitivity with respect to TAA. A possibility to circumvent this drawback is to suppose that the actual TIR belongs to a finite uncertainty set, say I , obtained by sampling the true TIR over a sector of possible TAAs. Furthermore, another possibility to model the lack of a-priori knowledge relies on the use of a spherical uncertainty set, namely, to assume that the actual TIR lies in a norm ball centered around a nominal TIR (the size of the ball is ruled by its radius that represents a design degree of freedom). For both the situations, the aim is to optimize the transmit–receive pair in order to maximize the worst case SINR over the TIR’s uncertainty set. In Section 5.3, the mentioned worst case optimization problem over a finite uncertainty set for the TIR is formulated and a procedure to tackle its relaxed version is described. Section 5.4 introduces a procedure to synthesize an optimized waveform-filter pair, while in Section 5.5, the overall framework is specialized to the spherical uncertainty set.

5.3 Problem formulation In this section, it is formulated the waveform-filter pair design problem aimed at maximizing the worst case SINR while forcing some practical constraints on the shape of the code. Precisely, suppose that the received vector r is filtered through w = [w(0), . . . , w(M − 1)]T ∈ CM . Hence, the resulting SINR at the filter output† for a specific TAA θ is given by  †  w T θ s2  SINRθ (s, w) =  2  2  . E w† Cs + E w† v

(5.5)

The worst case SINR over all the possible TAAs of interest collected in the discrete set I is considered as figure of merit SINR(s, w) = min SINRθ (s, w), θ ∈I

in order to robustify system behavior with respect to inaccuracies on the target state information. As to the probing waveform requirements, in addition to a constraint

†

Obviously, it is assumed that w  = 0.

Robust transmit code and receive filter design 133 on the transmitted energy, i.e., s2 = 1, the upper bound γ  is forced on the PAR, namely, maxn=0,...,N −1 |s(n)|2 = max N |s(n)|2 ≤ γ  n=0,...,N −1 (1/N )s2

(5.6)

which is tantamount to considering |s(n)|2 ≤ γ =

γ N

n = 0, . . . , N − 1.

Summarizing, the robust transmit code s and the receive filter w joint design, accounting for the energy and PAR constraints, can be formulated as‡ ⎧ ⎪ min SINRθ (s, w) ⎨max s,w θ ∈I . P s.t. s2 = 1 ⎪ ⎩ 2 |s(n)| ≤ γ , n = 0, . . . , N − 1 Remark 5.1. Notice that setting γ = 1/N , the special case of constant modulus codes, i.e., |s(n)| = 1/N , for all n = 0, . . . , N − 1 is obtained. Conversely, setting γ = 1 is tantamount to an unconstrained situation. These two case studies are used as benchmark in the performance analysis in order to assess the effectiveness of the developed design framework. In order to proceed with the development of the optimization procedure adopted to tackle problem P, let us define the rank-one code matrix as S  ss† ∈ HN and the rank-one filter matrix as W  ww† ∈ HM . Additionally, two equivalent expressions for SINRθ in (5.5) are given in the following technical lemma. Lemma 5.1. The SINR (5.5) at the output of the receive filter can be expressed in two equivalent forms: SINRθ (s, w) =

¯ θ ,W)s w† (Tθ , S)w s† (T = , † † ¯ cv (W)s w  cv (S)w s

where †

(Tθ , S) = Tθ STθ ,  cv (S) = E[CSC† ] + σv2 I, ¯ θ ,W) = (T ¯ cv (W) = E[C†WC] + σv2 tr (W)I.  † TθWTθ ,

Proof. See Appendix A.

‡

A related problem has been considered in [19][26, Chapter 6].

(5.7) (5.8) (5.9) (5.10)

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¯ cv (W ) can be calculated utilizing the Observe that the functions  cv (S) and  clutter second-order statistics, defined in (5.4). According to Lemma 5.1, problem P can be equivalently expressed, as a function of the matrix pair (S,W ), as ⎧ ⎪ ⎪ max min ⎪ ⎪ S,W θ ∈I ⎪ ⎪ ⎨ s.t. ¯ P ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

tr ((T θ , S)W ) tr ( cv (S)W ) tr (S) = 1 tr (E n S) ≤ γ , rank(S) = 1, rank(W ) = 1,

n = 1, . . . , N S0 W 0

,

where E n denotes the N × N matrix with the value 1 at its (n, n)th entry and 0 at all ¯ neither the objective function nor the the other entries. Notice that, in problem P, ¯ obtained by removing the rank-one constraints are convex. The SDP relaxation of P, rank-one constraints on the matrices S and W , is then given by ⎧ ⎪ ⎪ ⎪max min ⎪ ⎨ S,W θ ∈I s.t. P1 ⎪ ⎪ ⎪ ⎪ ⎩

tr ((T θ , S)W ) tr ( cv (S)W ) tr (S) = 1 tr (E n S) ≤ γ , S  0,W  0

n = 1, . . . , N

.

 Hereafter, SINR(S,W ) denoted the worst case relaxed objective function in P1 , defined as  SINR(S,W ) = min θ ∈I

 ¯ θ ,W )S tr (T tr ((T θ , S)W )  . = min ¯ cv (W )S θ ∈I tr  tr ( cv (S)W )

 ) over the positive semidefinite Note that SINRθ (s, w) is the restriction of SINR(S,W rank-one matrices. Let us observe that P1 is still non-convex and hard to solve. In order to find an optimized solution for P1 , an alternating optimization procedure is introduced. Specifically, at the mth step of the algorithm, starting from a code matrix S (m−1) , the matrix W (m) , solving P1 , given the code matrix S = S (m−1) , is found. Once the filter matrix W (m) is obtained, a suitable code matrix S (m) that solves P1 is searched, given the filter matrixW = W (m) , and so on. To trigger the algorithm, a feasible code matrix should be assigned for S (0) . From an analytical point of view, the matrices W (m) and S (m) are, respectively, optimal solutions to problems Pw(m) and Ps(m) , defined as  minθ ∈I tr (T θ , S (m−1) )W  max Pw(m) , W tr  cv (S (m−1) )W ⎩ s.t. W  0 ⎧ ⎨

Robust transmit code and receive filter design 135 and

 ¯ θ ,W (m) )S minθ ∈I tr (T  ¯ cv (W (m) )S tr  . tr (S) = 1 tr (E n S) ≤ γ , n = 1, . . . , N S0

⎧ ⎪ ⎪ max ⎪ ⎪ ⎨ S (m) Ps s.t. ⎪ ⎪ ⎪ ⎪ ⎩

The proposed algorithm has some special properties outlined in the following proposition. Proposition 5.1. Assume that Pw(m) and Ps(m) are solvable.§ Let {(S(m) ,W(m) )} be a sequence of points obtained through the proposed sequential optimization procedure and is denoted by (m)

 SINR

 (m) ,W(m) ). = SINR(S

 (m)  The sequence SINR is monotonic increasing and converges to a finite value. Proof. See Appendix B. The next subsections are devoted to the study of the optimization problems Pw(m) and Ps(m) whose solutions are required to implement the proposed iterative optimization algorithm.

5.3.1 Filter matrix optimization For a fixed code matrix S, the following proposition paves the way for an efficient solution to Pw(m) . Proposition 5.2. ⎧ max ⎪ ⎪ ⎪ ⎨ W,α s.t. Pw ⎪ ⎪ ⎪ ⎩

Let us define the SDP problem α

 tr  cv (S(m−1) )W = 1 tr (Tθ , S(m−1) )W ≥ α, W0

∀θ ∈ I

.

Then, problems Pw(m) and Pw are equivalent. Additionally, any optimal solution to Pw is an optimal solution to Pw(m) . Proof. The proof can be obtained exploiting the scale invariance of the problem and introducing an auxiliary variable α (accounting for the worst case useful energy).

§

“Solvable,” means that the problem is feasible and bounded, and the optimal value is attained, see [46, p. 13].

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Radar waveform design based on optimization theory

Algorithm 1: Algorithm for the relaxed optimization problem Require: σv2 , rc (n, n ), I , γ , N , s0 , and ζ . Ensure: A solution (S  ,W  ) to P2 . † (0) 1: Set m = 0, S = s0 s0 ; 2: repeat 3: m := m + 1; 4: Construct the matrices (T θ , S (m−1) ) for all θ ∈ I , according to (5.7); 5: Construct the matrix  cv (S (m−1) ) according to (5.8); 6: Solve problem Pw finding an optimal solution W (m) ; ¯ θ ,W (m) ) for all θ ∈ I , according to (5.9); 7: Construct the matrices (T ¯ 8: Construct the matrix  cv (W (m) ) according to (5.10);  9: Solve problem Ps finding an optimal solution (S¯ , α¯  , ξ  ); hence, construct ¯  for the pair (S (m) ,W (m) ); S (m) = Sξ and calculate the value of the SINR (m)

 = SINR;  Let SINR (m) (m−1)  − SINR  11: until |SINR |≤ζ  (m) 12: Output S = S and W  = W (m) .

10:

Being Pw a solvable convex SDP problem, its optimal solution can be found in polynomial time [47, p. 168].

5.3.2 Code matrix optimization The following proposition lays the ground for an efficient solution to Ps(m) . Proposition 5.3. Let us define the SDP problem Ps , considering the fixed filter matrix W(m) , given by ⎧ α max ⎪ ⎪  ,α  ,ξ ⎪ S ⎪  ⎪ (m)  ⎪ ¯ )S = 1 ⎪ ⎨ s.t. tr  cv (W (m) ¯ θ ,W )S ≥ α  , ∀θ ∈ I tr (T Ps . ⎪ tr S = ξ ⎪ ⎪ ⎪ ⎪ tr En S ≤ γ ξ , n = 1, . . . , N ⎪ ⎪ ⎩ S  0, ξ ≥ 0. Problems Ps(m) and Ps are equivalent. Additionally, given an optimal solution   (S¯ , α¯  , ξ¯ ) to Ps , (S¯ /ξ¯ ) is an optimal solution to Ps(m) . Proof. The proof can be developed introducing an auxiliary variable α (accounting for the worst case useful energy) and using the Charnes–Cooper transformation [48]. Observe that problem Ps is a solvable convex SDP problem and thus its optimal solution can be obtained in polynomial time. Algorithm 1 summarizes the proposed

Robust transmit code and receive filter design 137 iterative optimization procedure for the relaxed transmit code and receive filter matrices design. From a practical point of view, the algorithm requires a condition so as to stop the iterations. For this purpose, a lower bound threshold is set, say ζ ,  on the improvement between two consecutive values of the partial optimized SINR (i.e., forcing an iteration gain constraint). The computational burden of Algorithm 1 is proportional to the number of iterations N¯ as well as the computational cost of each iteration. Finally, each iteration comprises the solution of two SDP problems Pw and Ps , which requires a complexity effort in the order of O(N 4.5 log (1/η)) and O(M 4.5 log (1/η)), respectively (see [39]), with η being the required solution precision.

5.4 Filter and code synthesis As discussed earlier, the proposed sequential optimization procedure in Algorithm 1 provides an optimized relaxed transmit code and receive filter pair (S  ,W  ) (see Proposition 5.1). The final step is to synthesize the transmit code s and the receive filter w , based on (S  ,W  ). In the special case where the obtained matrices are rank-one; i.e., S  = s s† or W  = w w† , the optimized solution to P is directly given by (s , w ). However, in general, rank-one approximation techniques based on randomization methods [39,40] can be used as discussed in the following subsections.

5.4.1 Filter synthesis Starting from the solution (S  ,W  ) and employing the method in [39], a rank-one approximate solution can be found via the randomized samples wk ∼ NC (0,W  ), for k = 1, . . . , K, where K is the number of generated random vectors. Specifically, let us first perform the minimization wk (T θ , S  )wk †

αk = min θ ∈I

wk  cv (S  )wk †

.

Then, the vector wk is picked out, which exhibits the largest value αk . The synthesis procedure for the filter w is summarized in Algorithm 2.

5.4.2 Code synthesis Exploiting the technique described in [40], first of all, L random feasible code samples s ,  = 1, . . . , L, are generated, which satisfy both the energy and PAR constraints. Then, the best sample, in terms of achieved worst case SINR, is picked up as the synthesized radar code. Precisely, let us define d as the vector whose ith entry is  (5.11) d(i) = S  (i, i) i = 1, . . . , N , and d − as the vector whose ith entry is  1 , if d(i) > 0 d − (i) = d(i) i = 1, . . . , N . 1, if d(i) = 0

(5.12)

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Radar waveform design based on optimization theory

Algorithm 2: Algorithm for filter synthesis Require: (S  ,W  ), K, and I . Ensure: A randomized approximate solution w for P.  1: Draw random vectors w k ∈ CM from the complex normal distribution NC (0,W ), k = 1, . . . , K; 2: Compute † wk (T θ , S  )wk , αk = min † θ ∈I wk  cv (S  )wk for all k = 1, . . . , K; Pick up maximum value over {α1 , . . . , αK }, say αi , and select wi as optimized solution; 4: Output w  = w i . 3:

Furthermore, define the matrices D = Diag (d),

D− = Diag (d − ).

(5.13)

Now, produce L different random vectors s¯  drawn from the complex Gaussian distribution, namely,    s¯  ∼ NC 0, D− S  + I − D− D D− ,  = 1, . . . , L. Moreover, for each vector s¯  , compute s , setting s (i) = d(i)ej arg (¯s (i)) , i = 1, . . . , N , and the objective function β = min θ ∈I

†¯ ˜ s (T θ , W )s , † ¯ ˜ s  cv (W )s



with W˜ = w w† , and select the code sample s that shares the maximum value β , say si , as the optimized solution to P. The procedure providing the optimized code s is summarized in Algorithm 3. Finally, Algorithm 4 defines the overall joint waveform-filter design procedure.

5.5 Special case of practical importance: spherical uncertainty set In many scenarios, an estimated TIR of the illuminated target of interest is available using side-information obtained through previous scans and employing cognitive methods [5,6]. As already mentioned, due to the high sensitivity of the TIR to the target structure and orientation with respect to the LOS, it is of paramount importance to appropriately define the TIR uncertainty sets to account for estimation/prediction errors. Following this guideline, the spherical uncertainty set is introduced for the TIR as a norm ball centered around an a-priori known TIR, say t 0 , with uncertainty radius

Robust transmit code and receive filter design 139 Algorithm 3: Algorithm for code synthesis Require: S  , w , L, and I . Ensure: A randomized approximate solution s for P. − − 1: Define d, d , D, D according to (5.11)−(5.13). 2: Draw s¯  ∈ CN from the complex normal distribution  random    vectors − − NC 0, D S + I − D D D− ,  = 1, . . . , L; ˜  = w w† and compute s , with s (i) = d(i)ej arg (¯s (i)) i = 1, . . . , N , and 3: Let W †¯  s (T θ ,W )s , β = min † ¯  θ ∈I s  cv (W )s for all  = 1, . . . , L; 4: Pick the maximum value over {β1 , . . . , βL }, say βi , and select si as optimized solution; 5: Output s = si .

Algorithm 4: : Algorithm for waveform-filter pair design Require: σv2 , rc (n, n ), I , γ , s0 , ζ , K, and L. Ensure: A randomized approximate solution (s , w ) for P;   1: Solve problem P2 finding optimized solution (S ,W ), through Algorithm 1;    2: Use W and S , to find a randomized filter w through Algorithm 2;  3: Use S and w  , to find a randomized code s through Algorithm 3; 4: Output the pair (s , w  ).

r. The larger r, the higher uncertainty about the TIR. In other words, the spherical uncertainty constraint on t can be cast as  t − t 0 2 ≤ r 2 . This setup allows one to model higher uncertainty about the TIR considering larger r values. Figure 5.3 conceptually depicts an illustration of this constraint in real three dimensions. To proceed further, let us define  † 2 w T s   SINR(s, w, t) =  2  2  , E w† Cs + E w† v



We avoid trivial solution t = 0 for the robust design problem, choosing r ≤ t 0 .

140

Radar waveform design based on optimization theory

r

t t0

||t – t0||2 ≤ r2

Figure 5.3 An illustration of the spherical uncertainty constraint

for any arbitrary TIR t, where T is obtained from t according to (5.2). Hence, the corresponding worst case optimization problem, employing the spherical uncertainty constraint, can be expressed as ⎧ max min SINR(s, w, t) ⎪ ⎪ ⎨ s,w t s.t. t − t 0 2 ≤ r 2 ˜ . P ⎪ s2 = 1 ⎪ ⎩ |s(n)|2 ≤ γ , n = 0, . . . , N − 1 Again, according to the structure of the feasible set of the above problem, it is evident ˜ is non-convex. Using the idea developed for a finite uncertainty set in the that P ˜ previous section, the focus of the current section is on a modified version of P, replacing the spherical uncertainty set with a discrete set composed of samples picked up from the sphere. Additionally, it is shown that it is adequate to sample just the surface of the sphere (under the reasonable assumption that the null TIR does not belong to the assumed uncertainty region). Indeed, given a fixed pair (s, w), the objective function SINR(s, w, t) can be expressed as a positive semidefinite quadratic function of t. Specifically, SINR(s, w, t) =

  1 † 2 † |w T s| ∝ |w† S t t|2 = t † S t ww† S t t, g

 −1 where g is the denominator that is fixed, and S t = Nn=0 s(n)J n , with J n being the M × Q-dimensional shift matrix defined as in (5.3). This fact along with the fact that t = 0 does not belong to the admissible TIR, and t − t 0 2 ≤ r 2 is a closed set ˜ lays on the spherical boundary. Hence, it is implies that the optimal solution of P reasonable to restrict the sampling over the sphere boundary instead of the whole spherical uncertainty set. Let us indicate with J the set of random samples taken

Robust transmit code and receive filter design 141 from the boundary of t − t 0 2 ≤ r 2 (say the number of samples R). Hence, the new optimization problem can be derived as ⎧ min SINR(s, w, t) ⎪ ⎨max s,w t∈J  ˜ P . s.t. s2 = 1 ⎪ ⎩ 2 |s(n)| ≤ γ , n = 0, . . . , N − 1 ˜ ˜  and P share the same structure. Hence, the solution for P It can be shown that P can be obtained with the same method developed for P. ˜ can be considered as a benchmark for our An unconstrained version of P approach: ⎧ ⎪ min SINR(s, w, t) ⎨max s,w t Pb s.t. t − t 0 2 ≤ r 2 . ⎪ ⎩ s2 ≤ 1

(5.14)

It is worth noting that problem Pb corresponds to the optimization problem addressed in [18], with the convex constraint t − t 0 2 ≤ r 2 as the TIR vector uncertainty set. ˜  and Pb is addressed The performance comparison of the solutions obtained via P in the next section.

5.6 Numerical results In this section, the performance of the proposed method for different scenarios is assessed, based on the available knowledge about TIR in terms of the uncertainty set description. Additionally, the impact of the PAR constraint on the SINR behavior is studied. In this respect, we consider an S-band radar with operating frequency fc = 3.05 GHz, and rectangular sub-pulse duration of T = 6.67 ns (which gives a range resolution of 1 m). In order to model the TIR of an extended target, radar backscattering data is simulated from three airplanes exploiting the simulation software described in [49,50]. The physical dimensions of the simulated targets are summarized in Table 5.1. Considering the maximum physical extent of the simulated targets and the radar range resolution, the TIR’s support interval length is set to Q = 17 (with the possibility of having zero coefficients at the end of the support interval). Let us now describe the statistical characteristics of the noise and clutter environment. As to the signal-independent interference, a discrete white noise is considered with variance σv2 = 1. With reference to the signal-dependent interference, a homogeneous clutter environment is assumed with an exponentially shaped autocorrelation function:   rc n, n = σc2 ρ −|n−n | ,

(n, n ) ∈ {−N + 1, . . . , M − 1}2 ,

(5.15)

142

Radar waveform design based on optimization theory Table 5.1 Target parameters Target name Target A Target B Target C

Length (m)

Width (m)

19.43 15.76 16.72

13.05 7.15 13.91

where σc2 = 5, and the one-lag coefficient is set to ρ = 0.75 (the values are arbitrarily chosen). For the transmit signal, a code length of N = 20 is assumed, and a linear frequency modulated sequence is employed as the initial code s0 , namely, 1 2 s0 (n) = √ ejπ (n /2N ) , N

n = 0, . . . , N − 1,

(5.16)

which corresponds to a chirp signal that shares a chirp rate (fs2 /2N ), with fs being the underlying sampling frequency. As to the randomization step, the number of random samples is set to K = L = 100. Moreover, for the stop criterion of the proposed methods, ζ = 10−5 is considered. Finally, the CVX toolbox is exploited to find the solution of the involved convex optimization problems [51]. In the following subsections, the effects of 1. 2. 3.

different TAA uncertainty sets for a given target (example 1), different target types characterized by different TAA uncertainty sets (example 2), and spherical uncertainty set for a given TIR (example 3)

are assessed. In the first two examples the uncertainty set Iφφ12 is defined as (see Figure 5.4) Iφφ12 = {kθ |k ∈ Z, kθ ∈ [φ1 , φ2 ] } ,

(5.17)

where θ is the TAA sampling step (in our simulations set to 0.1◦ ). In correspondence of each TAA kθ , within the uncertainty set, the actual TIR is computed resorting to the toolbox in [49,50]. As to the third example, a collection of samples is considered, drawn from the boundary of a radius r sphere centered in the nominal TIR, to account for the lack of knowledge about the actual TIR. To study the performance of our design method, the focus is on two design approaches, representative of two different amounts of knowledge available about the TIR, ●

●

Nominal design, where the exact knowledge about the TIR is assumed and only the nominal TIR is exploited to design the optimized waveform-filter pair. In ◦ examples 1 and 2 the nominal design is performed setting I = I00◦ (let 0◦ be the nominal TAA), while in example 3 is obtained setting r = 0. Robust design, where an uncertainty set for the TIR is considered at the design stage. Hence, a worst case (over the foreseen uncertainty set) optimized waveform-filter pair is provided.

Robust transmit code and receive filter design 143 f2

Los

Extended target of interest

f1

Radar system

Figure 5.4 Pictorial representation of the TAA uncertainty As performance benchmarks, the unconstrained and constant modulus signals performance is considered, defined in Remark 5.1 as two extremes of the developed design method. Finally, it is worth pointing out that, as observed running the procedure with several random starting points, in almost all the steps of the proposed algorithm, the code-filter matrices have rank-one property. This means that usually the code-filter vectors are obtained directly at the output of the iterative procedure without the need of the randomization steps. As a result, in the simulation results reported in this section usually it does not present a synthesis loss.

5.6.1 TAA uncertainty set size analysis In this subsection, the effect of TAA uncertainty set size is assessed, for a given target, by making a comparison between the achieved SINR of the nominal design and that of the robust design method with respect to different TAA uncertainty set sizes. Here, the target is modeled as Target A. Besides, the single-input–single-output version of the unconstrained design approach in [18] is employed, with the known TIR assumption, as a benchmark for our analysis. Moreover, four TAA uncertainty sets are considered each with different intervals, so as to evaluate the effect of the uncertainty set size on the achieved SINR and the robustness to model mismatches of the proposed framework. (m)  versus the iteration index m is shown, In Figure 5.5, the objective function SINR considering different design methods. Note that, for the proposed method, the relaxed (m)  instead of the actual SINR, which is not known before the objective function SINR randomization step, is studied, while for the method of [18], the plot corresponds to the nominal design and the actual SINR values are reported. Figure inspection highlights that the unconstrained nominal design exhibits exactly the same performance as the unconstrained design approach of [18]. In this respect, note that the unconstrained version of our problem (as discussed in Remark 5.1) presents the same feasible set and objective function as that in [18]. As discussed earlier, it is worth observing that, in this specific case study, the obtained pairs (S (m) ,W (m) ) are rank-one, in all the iterations. Consequently, the relaxed objective coincides with the actual SINR at each iteration, leading to the same results for the two different approaches. Moreover, in

144

Radar waveform design based on optimization theory

14

Nominal design

12 S 0.1° –0.1° SINR (m) (dB)

10 S 0.2° –0.2°

8

S 5° –5°

6

4 10° S –10° 2 0

5

10

15

20

25

30

35

40

45

50

Iteration (m) (m)

 versus the number of iterations for Target A: method in [18] Figure 5.5 SINR (◦), nominal design (red), and robust design (green); the robust design 0.1◦ 0.2◦ 5◦ is performed over four different uncertainty sets: I−0.1 ◦ , I−0.2◦ , I−5◦ , ◦ 10 and I−10 ◦ . The unconstrained and constant modulus case studies in each scenario are shown with solid line and dashed line, respectively. nominal design, the constant modulus experiences an SINR loss less than or equal to 0.3 dB with respect to the technique of [18]. It is evident that, increasing the parameter γ toward 1, leads to larger and larger feasible sets that can result in higher and higher SINR values. 0.1◦ 0.2◦ 5◦ As to the robust design, we focus on four uncertainty sets: I−0.1 ◦ , I−0.2◦ , I−5◦ , ◦ 10  and I−10 ◦ . As shown in the plots, larger uncertainty sets lead to less worst case SINR for the robust design. This reflects the fact that, assuming two uncertainty sets I1 and I2 , where I1 ⊆ I2 , then min SINRθ (s, w) ≤ min SINRθ (s, w).

θ ∈I2

θ ∈I1

Hence, the solution to the problem with I2 as uncertainty set has a worst case SINR value lower than or equal to that of the same problem with I1 as uncertainty set. It is also worth mentioning that, for the considered scenario, the constant modulus design exhibits almost the same performance as the unconstrained one. Beyond the analysis of the worst case SINR value, it is also of great importance to assess the performance of the synthesized robust waveform-filter pair over all

Robust transmit code and receive filter design 145 the TAA entries of the uncertainty set. In this respect, in Figure 5.6, the achieved SINR versus the TAAs belonging to the uncertainty set assumed at the design stage is reported. For comparison purposes, also the behavior of the nominal design is included. Observe that the nominal design assumes no uncertainties about the TAA ◦ and just optimizes the SINR in the matched TAA (i.e., over the set I00◦ ), while the robust design performs a worst case SINR optimization over the specified uncertainty 2.5◦ 5◦ 10◦ set. The comparison is performed over the sets: I−2.5 ◦ , I−5◦ , and I−10◦ . In-line with previous considerations, the figures highlight the fact that the larger the TAA uncertainty set, the lower worst case SINR values. Furthermore, the curves show that although the robust technique obtains lower SINR values as compared with the matched SINR of the nominal design, it exhibits higher and flatter SINR values over the entire TAA uncertainty set. Basically, it is exactly the expected behavior for the robust approach idea. In fact, the considered worst case optimization is aimed at maximizing the minimum value of the SINR over all the points in the uncertainty set, whereas according to the nominal design, the goal is to maximize the SINR value in correspondence of the nominal TAA, without any control over the possible target responses.

5.6.2 TAA uncertainty set for different target types In this subsection, the performance of the developed method is assessed for two different target types: B and C. Our aim is to show how the results change from one 5◦ target to another. In Figure 5.7, the achieved SINR versus TAA is plotted for I−5 ◦. As to Target B, Figure 5.7(a) reveals that although the nominal design outperforms the robust design at the nominal TAA with a gain of 10 dB, it experiences severe 5◦ reductions over I−5 ◦ , with losses up to 20 dB as compared with the robust design. On the other hand, the robust design shares a stable SINR over the uncertainty set 5◦ I−5 ◦ . In Figure 5.7(b), the same comparison is made for Target C. The results confirm that the robust design is able to provide a stable SINR behavior over the uncertainty set. Considering the plots in Figures 5.6(b) and 5.7(a) and (b), it can be observed that the SINR exhibits a flat behavior for the three extended target models, which reveals the effectiveness of the proposed design method to deal with different target characteristics.

5.6.3 Spherical uncertainty set In this subsection, a comparison between the worst case SINR achieved by problems ˜  and Pb is performed, as a function of the uncertainty radius r, considering Target P A model; the nominal TIR corresponds to the TAA 0◦ . In the following, two case studies are examined: ● ●

unconstrained, constant modulus.

Moreover, the focus of the study is on the range 10−4 ≤ r ≤ 1 for the uncertainty radius (log10 (r) ∈ {−4, −4 + 1/4, −4 + 2/4, . . . , 0}). For each radius r, R = 100 points are randomly picked up from the uncertainty sphere to construct J . As to

146

Radar waveform design based on optimization theory 15 10

SINR (dB)

5 0 −5 −10 −15

Method in [18] Unconstrained nominal Constant modulus nominal Unconstrained robust Constant modulus robust

−20 −25 −2.5

−2

−1.5

−1

−0.5

(a)

0

0.5

1

1.5

2

2.5

4

5

TAA (°) 15 10

SINR (dB)

5 0 −5 −10 −15 Method in [18] Unconstrained nominal Constant modulus nominal Unconstrained robust Constant modulus robust

−20 −25 −5

−4

−3

−2

−1

(b)

0 TAA (°)

1

2

3

15 10

SINR (dB)

5 0 −5 −10 −15

Method in [18] Unconstrained nominal Constant modulus nominal Unconstrained robust Constant modulus robust

−20 −25 −10

(c)

−8

−6

−4

−2

0

2

4

6

8

10

TAA (°) ◦

2.5 Figure 5.6 SINR versus TAA, for different uncertainty regions: (a) I−2.5 ◦ , (b) 10◦ 5◦ I−5◦ , (c) I−10◦

Robust transmit code and receive filter design 147 15

10

SINR (dB)

5

0

−5

−10

−15 Method in [18] Unconstrained nominal Constant modulus nominal Unconstrained robust Constant modulus robust

−20

−25 −5

−4

−3

−2

−1

(a)

0

1

2

3

4

5

2

3

4

5

TAA (°)

20

10

SINR (dB)

0

−10

−20 Method in [18] Unconstrained nominal Constant modulus nominal Unconstrained robust Constant modulus robust

−30

−40 −5

(b)

−4

−3

−2

−1

0

1

TAA (°) ◦

5 Figure 5.7 SINR versus TAA over I−5 ◦ , for (a) Target B and (b) Target C

starting point involved in the considered procedures, (5.16) is used for r = 10−4 , whereas for r > 10−4 the optimized code obtained for 10[ log10 (r)−0.25] is used, i.e., that obtained in the previous step. Interestingly, as shown in Figure 5.8, the optimized SINR achieved by the unconstrained design has a behavior similar to that of Pb .

148

Radar waveform design based on optimization theory 3

2

Achieved SINR (dB)

1

0

−1

−2

−3

Method in [18] Unconstrained robust Constant modulus robust

−4 10−4

10−3

10−2 r

10−1

100

Figure 5.8 Achieved SINR versus the uncertainty radius, r, for the method in [18], the unconstrained robust and constant modulus robust case studies

Moreover, the constant modulus case enjoys SINR values experiencing a loss smaller than 0.3 dB with respect to the SINR of the unconstrained design. This clearly suggests to exploit constant modulus waveforms that trade off practical advantages of constant modulus with a small SINR loss.

5.7 Conclusions In this chapter, the problem of joint transmit code and receive filter design has been considered to optimize the achieved SINR of extended targets in the presence of signal-dependent interference. Due to uncertainties about the TIR, a worst case optimization framework has been proposed over two different TIR uncertainty sets. The former contains TIR’s samples corresponding to some TAAs, whereas the latter is equivalent to bounding the actual TIR within a spherical set. Additionally, a PAR constraint has been imposed on the transmitted waveform so as to ensure its practical implementation. The obtained non-convex optimization problem has been divided into two sequential relaxed SDP problems. Their corresponding solutions have then been exploited to synthesize the code-filter pair via randomization methods.

Robust transmit code and receive filter design 149 At the analysis stage, two design techniques have been considered: the nominal design where the exact knowledge about the TIR has been assumed and the robust design in which knowledge imperfections about the TIR have been accounted for. The obtained results have shown the robustness of the proposed design method, in some case studies, which guarantees a quite stable SINR behavior corresponding to TIR uncertainties. Moreover, with a comparison between the unconstrained and constant modulus designs, it has been highlighted that the constant modulus signal shares a negligible performance loss with respect to the unconstrained one. Possible future research tracks might account for realistic datasets to better examine the performance of the developed method. Moreover, an extension of the developed framework to the radars with polarimetric channels could also be of interest. Additionally, there are still open issues in current HRR, like the target range migration, which can make uncertainties about the TIR and could be subject to the future activities.

Appendix A Proof of Lemma 5.1 Let us start with the numerator of (5.5) and rewrite it as   † w T θ s2 = w†T θ ss†T † w = w† (T θ , S)w θ ¯ θ ,W )s. = s†T θ ww†T θ s = s† (T †

As to the denominator of (5.5), considering the code s as a fixed parameter, it can be rewritten as

E[|w† Cs|2 ] + E[|w† v|2 ] = w† E[Css† C † ]w + w† E[v† v]w = w† E[CSC † ]w + σv2 w† w = w†  cv (S)w.

Additionally, considering the energy constraint s2 = 1, the denominator of (5.5) can be finally recast as

E[|w† Cs|2 ] + E[|w† v|2 ] = s† E[C † ww† C]s + w† E[v† v]ws2 = s† E[C †W C]s + σv2 tr (W )s† s ¯ cv (W )s. = s† 

150

Radar waveform design based on optimization theory

Appendix B Proof of Proposition 5.1 (m)

 } is a monotonic increasing sequence, i.e., We first prove that the sequence {SINR (m) (m+1)  ≤ SINR  SINR . In fact  SINR

(m)

 tr (T θ , S (m) )W (m)  ≤ v(Pw(m+1) ) = min θ ∈I tr  cv (S (m) )W (m)  ¯ θ ,W (m+1) )S (m) tr (T  ≤ v(Ps(m+1) ) = min ¯ cv (W (m+1) )S (m) θ ∈I tr   = SINR

(m+1)

.

Second, it is shown that the objective function is bounded above, that is,  SINR

(m)

= min θ ∈I

≤ min θ ∈I

= min θ ∈I

≤ min θ ∈I

≤ min θ ∈I

=

  tr ¯ T θ ,W (m) S (m)   ¯ cv W (m) S (m) tr    † tr T θW (m)T θ S (m)   σv2 tr W (m) tr S (m)   † tr T θ S (m)T θW (m)  σv2 tr W (m)    † tr T θ S (m)T θ tr W (m)  σv2 tr W (m)   †  tr S (m) tr T θ T θ σv2

N min t θ 2 , σv2 θ ∈I

(B.1)

(B.2)

(B.3)

(B.4) (B.5)

where for the first inequality, it is exploited (5.10) to claim   ¯ cv W (m)  σv2 tr W (m) I ,   which leads to (B.2) since tr S (m) = 1. Moreover, based on [52], given A, B  0, 0 ≤ tr (AB) ≤ tr (A)tr (B),   † from which (B.3) and (B.4) follow. Finally, the trace tr T θ T θ is simply the summation over the squared norms of the columns of T θ , which according to the structure

Robust transmit code and receive filter design 151   † of T θ can be expressed as tr T θ T θ = N t θ 2 . Hence, the upper-bound on the  sequence SINR  SINR

(m)

(m)

≤

is

N min t θ 2 . σv2 θ ∈I

(B.6)

Due to the finite energy assumption for the TIR, inequality (B.6) along with the mono(m)  } guarantees the convergence of the objective tonic increasing property of {SINR function sequence.

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Haykin S. Cognitive Radar: A Way of the Future. IEEE Signal Processing Magazine. 2006;23(1):30–40. Dai FZ, Liu HW, Wang PH, et al. Adaptive Waveform Design for Range-Spread Target Tracking. Electronics Letters. 2010;46(11):793–794. Guerci JR, Pillai SU. Theory and Application of Optimum Transmit-Receive Radar. In: Record of the IEEE 2000 International Radar Conference, Alexandria, VA, USA; 2000. p. 705–710. Garren D, Osborn M, Odom A, et al. Enhanced Target Detection and Identification via Optimised Radar Transmission Pulse Shape. IET Radar, Sonar & Navigation. 2001;148(3):130–138. Kim HS, Goodman N, Lee C, et al. Improved Waveform Design for Radar Target Classification. Electronics Letters. 2017;53(13):879–881. Zhu Z, Kay S, Raghavan RS. Information-Theoretic Optimal Radar Waveform Design. IEEE Signal Processing Letters. 2017;24(3):274–278. Chen P, Qi C, Wu L, et al. Estimation of Extended Targets Based on Compressed Sensing in Cognitive Radar System. IEEE Transactions on Vehicular Technology. 2016;66(2):941–951. Cui G, Fu Y, Yu X, et al. Robust Transmitter–Receiver Design for Extended Target in Signal-Dependent Interference. Signal Processing. 2018;147: 60–67. Liu Y, Liao G, Xu J, et al. Adaptive OFDM Integrated Radar and Communications Waveform Design Based on Information Theory. IEEE Communications Letters. 2017;21(10):2174–2177. Luo ZQ, Ma WK, So AC, et al. Semidefinite Relaxation of Quadratic Optimization Problems. IEEE Signal Processing Magazine. 2010;27(3):20–34. De Maio A, Huang Y, Piezzo M, et al. Design of Optimized Radar Codes With a Peak to Average Power Ratio Constraint. IEEE Transactions on Signal Processing. 2011;59(6):2683–2697. Carrara WG, Goodman RS, Majewski RM, et al. Spotlight Synthetic Aperture Radar: Signal Processing Algorithms. Boston, MA: Artech House; 1995. Keller JB. Geometrical Theory of Diffraction. Journal of the Optical Society of America. 1962;52(2):116–130. Bachman C. Some Recent Developments in RCS Measurement Techniques. Proceedings of the IEEE. 1965;53(8):962–972. Jacobs SP, O’Sullivan JA. Automatic Target Recognition Using Sequences of High Resolution Radar Range-Profiles. IEEE Transactions on Aerospace and Electronic Systems. 2000;36(2):364–381. Cohen MN. Variability of Ultrahigh-Range-Resolution Radar Profiles and Some Implications for Target Recognition. In: Aerospace Sensing. International Society for Optics and Photonics; 1992. p. 256–266. Ben-Tal A, Nemirovski A. Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications. vol. 2. Philadelphia, PA: SIAM; 2001. Boyd SP, Vandenberghe L. Convex Optimization. Cambridge: Cambridge University Press; 2004.

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Charnes A, Cooper WW. Programming with Linear Fractional Functionals. Naval Research Logistics Quarterly. 1962;9:181–186. Gorshkov SA, Leschenko SP, Orlenko VM, et al. Radar Target Backscattering Simulation Software and User’s Manual. Norwood, MA: Artech House; 2002. Shirman JD. Computer Simulation of Aerial Target Radar Scattering, Recognition, Detection, and Tracking. Norwood, MA: Artech House; 2002. Grant M, Boyd S, Ye Y. CVX: MATLAB Software for Disciplined Convex Programming; 2008. Horn RA, Johnson CR. Matrix Analysis. NewYork, NY: Cambridge University Press; 2012.

[49] [50] [51] [52]

Chapter 6

Optimizing radar transceiver for Doppler processing via non-convex programming Augusto Aubry1 , Mohammad Mahdi Naghsh2 , Ehsan Raei3 , Mohammad Alaee-Kerahroodi3 , and Bhavani Shankar Mysore3

This chapter addresses the performance improvement of a radar system in typical scenarios involving moving targets in the presence of clutter. The transmitted waveform and the receiver architecture form the key elements of the radar system. Hence, the goal is the joint design of the transmitter–receiver pair for Doppler processing application via structured non-convex programming optimization. Specifically, a bank of filters is considered at the receiver end to deal with the unknown Doppler shift of the target and the focus is on the joint design of the transmit waveform and the filter bank. To this end, the worst case signal-to-interference plus noise ratio (SINR) at the output of the filter bank is selected as the figure of merit to optimize. Besides, some suitable constraints are forced on the sought waveform. To handle the resulting nonconvex max–min optimization problem, an appropriate reformulation is provided that paves the way to the design of an innovative iterative procedure, exhibiting a monotonic improvement of the worst case SINR. With regards to algorithm features, each iteration involves both a convex and a generalized fractional programming (GFP) problem that can be globally solved via generalized Dinkelbach’s procedure with a polynomial computational complexity. This chapter first illustrates the system model introducing the transmit waveform, receiver processing, and statistical models for the signal-dependent and signalindependent interference. Then, the joint robust design of the radar signal and the receive filter bank is formulated. Subsequently, to solve the optimization problem, an iterative procedure aimed at producing high-quality waveforms and filters is presented. A comprehensive numerical analysis to investigate and illustrate the

1 Dipartimento di Ingegneria Elettrica e delleTecnologie dell’Informazione, Università degli Studi di Napoli “Federico II,” Napoli, Italy 2 Department of Electrical and Computer Engineering, Isfahan University of Technology, Isfahan, Iran 3 Interdisciplinary Centre for Security, Reliability and Trust (SnT), University of Luxembourg, Luxembourg City, Luxembourg

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effectiveness of the proposed method in various conditions is provided. Finally, interesting research avenues are highlighted in the conclusion.

6.1 Introduction In active radar systems, surveillance performance is highly tied up to the probing signal and the receiver structure. Not surprising, many studies have been directed during the last five decades toward their joint design [1–9] aimed at improving system performance in terms of target detection, identification, and classification capabilities. Nonetheless, many problems still remain unsolved representing current challenges for radar scientists. On the transmitter side, waveform diversity is now recognized as the key instrument for radar performance optimization. Depending on the operational environment, two main research directions have been originated. The first assumes signal-independent interference and relates to radar scenarios where the main contribution to the competing disturbance is represented by system noise, and/or intentional interference (jammers), and/or unintentional emissions from telecommunication apparatus, and/or terrain scattering due to signals from other radar platforms (hot clutter) [6,7,10–19]. The second design methodology accounts for signal-dependent interference due to possible reflections of the signal transmitted by the radar of interest from objects of no tactical importance located in the surveillance area [1–5,8,9,20–22]. Most of the aforementioned techniques focus on either a stationary or a moving target with a-priori known Doppler frequency (refer to [12,23,24] and references therein). While this assumption is reasonable during the confirmation of a detection process or for a target already tracked, it is no longer the case in standard radar tasks of search/estimation. To address this commonplace situation, very few works have modeled the target Doppler shift [25–27] as an unknown parameter during the waveform synthesis [7,28,29]. In particular, in [7], a waveform design algorithm maximizing the worst case detection performance (across Doppler) in the presence of signal-independent disturbance has been proposed and assessed. In [28,29], the worst case SINR has been optimized over the transmit signal and receive structure assuming signal-dependent interference. While in [28], a single receive filter has been employed, in [29] a battery of filters at the receiver side has been implicitly considered. Finally, in [29], a signal design technique optimizing the average SINR over the possible target Doppler shifts has also been devised. Therein, the effects of range-ambiguous clutter have not been considered; besides, the provided solutions may suffer the so-called synthesis loss resulting from the relaxation technique exploited to handle the design optimization problems. In this landscape, this chapter addresses the shortcomings of prevailing works, including targets with unknown Doppler shifts, consideration of range-ambiguous signal-dependent disturbance, and an efficient optimization methodology. It deals with joint design of the transmit signal and the receive filter bank where the worst

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case SINR at the output of the filter bank is optimized to ensure target detectability regardless of its actual Doppler shift.

6.2 Radar system operation 6.2.1 Transmit waveform We consider a monostatic radar system, wherein each transmission burst consists of N pulses, with the ith pulse obtained as s(i)p(t). Here, p(t) can be a conventional radar signal, such as linear frequency modulation waveform, and s(i) is the slowtime modulation sequence. Let s = [s(1), s(2), . . . , s(N )]T ∈ CN denote the vector containing all the slow-time modulation values; this vector is henceforth referred to as transmit radar code since it confers additional features to the transmitted signal. The length of the code is determined by the coherent processing interval (CPI) as well as the pulse repetition interval (PRI) employed by the surveillance system. The transmitted waveform is reflected by targets (e.g., pedestrians in an automotive scenario) and other scatterers (e.g., road and trees). The receiver collects the overall backscattering from the environment, i.e., the superposition of these individual reflections. Hence, the received signal is down-converted to baseband, correlated with the pulse p(t) (pulse-matched filtering operation or pulse compression), and then sampled. Due to Doppler mismatches, the performed correlation may not achieve, in general, ideal signal-to-noise ratio (SNR) gains. Thus, it is assumed in the following that p(t) is Doppler tolerant, in the sense that the pulse compression gain is robust to the actual Doppler shift. The location of the correlation peak provides information about the range of the target. The range estimation process mainly depends on the adopted probing p(t) that is not assumed a degree of freedom in the present chapter. Hence, this chapter focuses on the processing of the slow-time data collected from a specific range–azimuth cell, assumed, without the loss of generality, to be the (0,0) range–azimuth bin.

6.2.2 Receiver processing and signal model Let v(i) be the sample at the output of the matched filter (associated with the range of interest) resulting from the transmission of the kth pulse. Then, the N -dimensional column vector v = [v(1), v(2), . . . , v(N )]T ∈ CN collects the observations from the range–azimuth cell under test within a CPI. Following the derivations in [8], this observation vector takes the form v = αT s  p(νdT ) + c + n ,

(6.1)

with αT a complex parameter accounting for to the effects of channel propagation and target backscattering, νdT = TI fdT the unknown normalized target Doppler frequency with TI the PRI, fdT in (Hz) the actual target Doppler frequency, p(νdT ) = [1, e j2πνdT , . . . , e j2π(N −1)νdT ]T ∈ CN is the temporal steering vector containing

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Radar waveform design based on optimization theory Filter tuned to n1dT w1 Filter tuned to n2dT w2

y1

y2

|.|2

|.|2 Processing logic

v

(e.g., compare the maximum with a threshold)

Filter tuned to nKdT wK

yK

H1 (v^dT) H0

|.|2

Figure 6.1 Block diagram of the filter bank processing

information about Doppler, c the N -dimensional column vector containing the filtered signal-dependent interfering echo samples (clutter), and n the N -dimensional column vector of the filtered signal-independent interference and noise samples. Conventional approaches for the joint transmitter–receiver design, [8,28], employ usually one receive filter, assuming either a known or an unknown target Doppler frequency. The former is meaningful for a confirmation process and allows the use of the matched SINR as the figure of merit. The latter models the lack of a-priori information and entails a robust system synthesis (i.e., worst case optimization) accounting for all the possible Doppler shifts. In this chapter, more degrees of freedom are provided at the receiver side (enabling the use of multiple filters) and focus on the joint design of the transmit code and the entire filter bank (see Figure 6.1). Specifically, it is assumed that v is processed through the array of filters w1 , w2 , . . . , wK , each tuned to a specific Doppler frequency νdi T , i = 1, 2, . . . , K, where each wi is the N -dimensional complex column vector denoting the coefficients of the ith filter. The output corresponding to the ith filter is then given by †

yi = wi v,

i ∈ {1, 2, . . . , N }.

(6.2)

The samples yi are subsequently used to establish the presence of targets exploiting Doppler signatures embedded in the received signal. Figure 6.1 also depicts the traditional strategy that compares the maximum among the filters’ outputs with a suitable threshold to declare the presence of the target. To ensure target detectability regardless of its actual Doppler frequency, the worst case SINR at the output of the filter array, corresponding to the minimum SINR among the available branches, is considered as figure of merit. Thus, in the following, the statistical characterization of clutter and noise components is considered.

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6.2.3 Clutter and signal independent disturbance characterization The clutter vector c arises due to reflections of the transmitted signal from uncorrelated scatterers in various range–azimuth bins; since these reflections superpose over the air, the resulting clutter vector c can be expressed as [8,30,31] c=

N c −1 

L−1 

r=0

k=0

  α(r,k) J r s  p(νd(r,k) ) ,

(6.3)

where Nc − 1 is the number of range rings that interfere with the range–azimuth bin of interest and L is the number of discrete azimuth sectors. Further, α(r,k) and νd(r,k) , respectively, denote the echo complex amplitude and the normalized Doppler frequency of the scatterer in the range–azimuth bin. Since the signal strength from scatterers located far away from the target range–azimuth cell drops rapidly, only Nc ≤ N ambiguous range rings and L discrete azimuth sectors are considered. Furthermore, N -shift matrices, each of dimension N × N , are defined as  1 if l − m = r l, m ∈ {1, 2, . . . , N }, r ∈ {0, 1, . . . , N − 1}. J r (l, m) = 0 if l − m  = r These shift matrices satisfy J −r = J Tr . Being the phase of the complex random variable α(r,k) reasonably  modeled as uniform over [0, 2π], it follows that α(r,k) is zero mean, i.e., E α(r,k) = 0. Furthermore, the mean interfering power associated with the clutter  patch located at the (r, k)th range–azimuth bin is denoted as 2 σ(r,k) = E |α(r,k) |2 . Similarly, the Doppler shift of the clutter patch (r, k), denoted by ν , is modeled as a random variable distributed uniformly in the interval  d(r,k)  ν¯ d(r,k) − ((r,k) /2), ν¯ d(r,k) + ((r,k) /2) [8]. As a result, being the different scatterers uncorrelated, the following statistical characterization of the clutter (6.3) is obtained: E [c] = 0 L−1 c −1    N 2 c (s) = E cc† = σ(r,k) J r (s, (r, k))J †r . r=0

(6.4)

k=0 ν¯ d

(r,k) diag (s)† where ν¯d (l, m) is the In the previous equation, (s, (r, k)) = diag (s)(r,k) covariance matrix of p(νd(r,k) ) and takes the following form for uniformly distributed Doppler shifts: ⎧ ⎨ 1 if l = m ν¯ d(r,k)  , (6.5) (r,k) (l, m) = j2π ν¯ d(r,k) (l−m) sin [π(r,k) (l−m)] ⎩e if l  = m [π(r,k) (l−m)]

with (l, m) ∈ {1, . . . , N }2 . In the aforementioned uniform Doppler distribution, the ν¯ d

(r,k) (l, m) in (6.5) can be appropriately modified to deal with arbitrary expression for (r,k) statistical distribution of the Doppler shifts for the scatterers. According to (6.4), the second-order statistical characterization of the interfer2 ence requires the knowledge of the mean scatterer power σ(r,k) , as well as the Doppler parameters ν¯ d(r,k) and (r,k) , for any clutter patch (r, k). This information can be obtained

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from a cognitive paradigm. Specifically, for a point-like interfering scatterer, it is assumed that the prediction of its parameters is performed using a dynamic environmental database, including a geographical information system, such as the National Land Cover Data, meteorological data, previous scans, and some clutter models. As to the statistical characterization of the signal-independent disturbance vector (accounting for thermal noise, jamming, and hot clutter), it is modeled as azero-mean colored noise with a covariance matrix R  0, i.e., E [n] = 0 and E nn† = R.

6.2.4 Performance metric for Doppler processing The SINR associated with the ith branch, denoted as SINRi , i = {1, 2, . . . , K}, determines the reliability to detect a target whose Doppler shift ν i using yi . It is formally defined as  † |αT |2 |wi s  p(νdi T ) |2 SINRi = † , i = 1, 2, . . . , K, wi  = 0, (6.6) † wi  c (s) wi + wi Rwi where |αT |2 |w†i (s  p(νdi T ))|2 denotes the useful energy at the output of the ith filter (due to a matched target, i.e., the one sharing the nominal Doppler frequency νdi T ), while w†i Rwi and w†i  c (s) wi represent the signal-independent disturbance and the clutter energies at the output of the ith filter, respectively. Hence, in this chapter, it is considered as figure of merit SINRB = min SINRi . i=1,...,K

(6.7)

The metric in (6.7) corresponds to the minimum SINR among the available branches, each tuned to a specific Doppler shift within the assumed discrete set. Such an approach guarantees target detectability independently of its actual Doppler frequency and grants (as it will be shown later) performance improvements as compared with some conventional approaches.

6.3 Problem formulation and design issues This section deals with the joint design of the radar code and the corresponding receive filter bank optimizing the worst case SINR over the unknown target Doppler frequency, under some practical constraints on the shape of the transmit code.

6.3.1 Constraints and optimization problem With the optimization variables and the objective function already defined, before formalizing the joint design problem of interest, it is worth introducing some constraints of practical relevance to force on the sought probing signal. In this respect, a natural design requirement concerns the amount of available transmit power that can be expressed, without loss of generality, as the energy constraint s2 = 1. In addition, the sought code is constrained to be similar to a preselected radar sequence s0 so as to control some relevant feature of the produced waveform. Several reasons motivate this constraint; in fact, an unconstrained optimization can lead to signals

Optimizing radar transceiver for Doppler processing

161

with (i) significant modulus variations impacting hardware implementation, (ii) poor Doppler resolution, (iii) high peak sidelobe levels, and (iv) an undesired ambiguity function response. These drawbacks can be partially circumvented by forcing the solution to be close to a known code s0 (s0 2 = 1) that exhibits desirable properties like constant modulus and reasonable peak sidelobe level. From a mathematical point of view, such similarity constraint is forced requiring that [32] s − s0 2 ≤ δ , where the parameter 0 ≤ δ ≤ 2 controls the size of the trust hypervolume. Based on the aforementioned discussions, the joint design of the radar code and the receive Doppler bank can be formulated as the following constrained optimization problem:  ⎧ † |αT |2 |wi s  p(νdi T ) |2 ⎪ ⎪ ⎨ max min s,w1 ,...,wK i=1,...,K w †  (s) w + w † Rw i i i c i . (6.8) P 2 ⎪ s.t. s = 1 ⎪ ⎩ s − s0 2 ≤ δ Problem P is a non-convex optimization problem, since the objective function is a non-convex and s2 = 1 defines a non-convex set. Besides, the technique developed in [8] cannot be exploited to tackle it. Indeed, although the semidefinite programming relaxation of P with respect to the optimization variable s (see [8,28]) still leads to a quasi-concave problem, it may not be tight due to the large amount of the involved trace constraints. It is therefore necessary to introduce a new optimization technique to tackle this challenging problem. In this respect, a key role is played by the following proposition. Proposition 6.1. Problem P is solvable∗ and, for any 0 ≤ δ < 2, can be equivalently recast† as   ⎧  † ⎪ |αT |Re wi s  p(νdi T ) ⎪ ⎪ ⎪ max min  ⎪ ⎪ s,w1 ,...,wK i=1,...,K  ⎪  ⎪ † 2 w ⎪ + Rs w  (s) ⎪ c i i ⎪ ⎪ ⎪ ⎨ s2 ≤ 1 P1 s.t. , (6.9)  ⎪ † ⎪ ⎪ Re s0 s ≥ δ1 ⎪ ⎪ ⎪ ⎪   ⎪  ⎪ † i ⎪ Re w s  p(ν ) ≥ 0, ⎪ i dT ⎪ ⎪ ⎩ i = 1, 2, . . . , K where δ1 = 1 − (δ/2) > 0. Indeed, given an optimal solution to P, it is possible to construct an optimal solution to P1 , and vice versa. Proof. See Appendix A. ∗

“Solvable” means that the problem is feasible, bounded above, and the optimal value is attained (see [33, p. 13]). † The case δ = 2 is not of great practical relevance, since it refers to the absence of the similarity constraint. In this specific case, the optimal value can be approximated through (6.9), with arbitrary precision, just setting δ = 2 − (1/n) with n large enough.

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Observe that P1 is still a non-convex optimization problem because the objective function is a non-convex function and the feasible set is non-convex. Nonetheless, differently from P, the equivalent formulation provided by Proposition 6.1 shares a convex feasible set and a quasi-concave objective function with respect to either s or (w1 , w2 , . . . , wK ) allowing the global optimization over each block of variables, as thoroughly explained in the following subsections. Specifically, the technique adopted to obtain an optimized solution is based on a iterative optimization procedure that sequentially optimizes    † |αT |Re wi s  p(νdi T ) g(s, w1 , w2 , . . . , wK ) = min  (6.10) i=1,...,K † † 2 wi  c (s) wi + wi Rwi s  (n−1) (n−1) , an admissible radar code Starting from an array of receive filters w1 , . . . , wK (n−1)

(n−1)

s(n) maximizing g(s, w1 , . . . , wK ) at step n is searched. Whenever s(n) is found, the focus is on the filter bank  (n) (n) w1 , . . . , wK  (n) (n) that maximizes g(s(n) , w1 , w2 , . . . , wK ), and so on. Thus, w1 , . . . , wK is used as the starting point  at step n + 1. To trigger the procedure, the optimal receive filter (0) (0) w1 , . . . , wK , to an admissible code s(0) , is considered. The algorithm can be initialized with s0 .  (n) (n) From an analytical point of view, s(n) and w1 , . . . , wK are optimal solutions to the optimization problems Ps (n) and Pw (n) , respectively, defined as ⎧  (n−1) (n−1) (n−1) ⎪ g s, w , w , . . . , w max ⎪ 1 2 K ⎪ ⎪ s ⎪ ⎪ 2 ⎪ s.t. s ≤ 1 ⎪ ⎪ ⎨  (n) † Ps , Re s0 s ≥ δ1 ⎪ ⎪  ⎪   ⎪ (n) † ⎪ ⎪ Re wi s  p(νdi T ) ≥ 0, ⎪ ⎪ ⎪ ⎩ i = 1, 2, . . . , K and

⎧ ⎪ max ⎪ ⎨w1 ,...,wK (n) Pw s.t. ⎪ ⎪ ⎩

  g s(n) , w1 , w2 , . . . , wK    Re w†i s(n)  p(νdi T ) ≥ 0, i = 1, 2, . . . , K

.

(6.11)

(6.12)

The proposed procedure exhibits some interesting properties summarized in the following proposition.

Optimizing radar transceiver for Doppler processing

163

Proposition 6.2. Assume that problems Pw (n) and Ps (n) are solvable.‡   (n) (n) Let s(n) , w1 , . . . , wK be a sequence of points obtained through the pro(n) posed sequential optimization  procedure; let also SINR be the worst SINR value (n) (n) (n) corresponding to the point s , w1 , . . . , wK at the nth iteration. Then ● ● ●

the sequence SINR(n) is a monotonic increasing sequence; the sequence SINR(n) converges SINR ;  to a finite value  (n) (n) (n) starting from the sequence s , w1 , . . . , wK , it is possible to construct a   (m) (m) subsequence s˜(m) , w ˜1 ,...,w ˜K that converges to a feasible point   ˜ 1, . . . , w ˜ K s˜ , w

●

  of P1 and such that the worst SINR evaluated in s˜ , w˜ 1 , . . . , w˜ K is equal to SINR ;     (n) (n) any cluster point s˜ , w˜ 1 , . . . , w ˜ K to s(n) , w1 , . . . , wK , ˜ i  = 0, i = 1, . . . , K, is a stationary point for P1 . with§ w

Proof. See Appendix B. Proposition 6.2 highlights that the devised algorithm monotonically converges to the limit value associated with the employed starting point and that any cluster point of practical interest (i.e., with w˜ i  = 0, i = 1, . . . , K) is a stationary point. The former ensures the algorithm stability (in terms of convergence), while the latter shows the quality of the obtained solution, since it satisfies necessary optimality conditions. Let us observe that, from a practical point of view, the proposed optimization procedure requires a condition to stop the iterations. There are several ways to impose it. In particular, it is possible to consider a maximum number of iterations, a minimum difference between two consecutive values of the partially optimized SINR (i.e., forcing an iteration gain constraint), or mixing them. The next subsections will study the optimization problems Pw (n) and Ps (n) required to implement the proposed sequential optimization algorithm.

6.3.2 Filter bank optimization: solution to problem Pw (n) This the properties of Pw (n) and finds a closed form optimal solution  section studies (n) (n) w1 , . . . , wK for any fixed s(n) .

In the following subsections, we develop optimization procedures to globally solve both Pw (n) and Ps (n) . Without loss of generality, we may assume filters with unit energy. By doing so, we ensure the existence of cluster points with w˜ i  = 0, i = 1, . . . , K. ‡ §

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Radar waveform design based on optimization theory

Lemma 6.1. To find an optimal solution to Pw (n) , it suffices to solve problems P i , i = 1, . . . , K, given by    min w†  c s(n) w + w† Rws(n) 2 i w    P , (6.13) s.t. Re w† s(n)  p(νdi T ) = 1   i.e., given optimal solutions w i to P i , i = 1, . . . , K, w 1 , w 2 , . . . , w K is an optimal solution to Pw (n) . Proof. See Appendix C. Using Lemma 6.1 and [34], an optimal solution to Pw (n) is given by    −1  (n) s  p(νdi T )  c s(n) + Rs(n) 2 =    ,    (n)  2 (n) (n) 2 −1/2 i s  p(νdT )    c s + Rs  

(n) wi

i = 1, . . . , K, which highlights the influence of s(n) as well as the nominal Doppler frequencies on the optimized responses of the filters.

6.3.3 Radar code optimization: solution to problem Ps (n) In this subsection, the main properties of problem Ps (n) are analyzed. The solvability of the problem is considered followed by an algorithm that finds an optimal solution. To this end, some results from the GFP theory [35,36] are used; these are summarized here in the form of a lemma. Lemma 6.2. Let X ⊆ CN be a convex compact set, {fi (x)}Ii=1 be nonnegative concave functions over X , and {gi (x)}Ii=1 positive convex functions over X . Hence, the GFP problem ⎧ fi (x) ⎨ max min x PGFP (6.14) i=1,...,I gi (x) ⎩s.t. x ∈ X is solvable and an optimal solution can be obtained through Algorithm 1. Algorithm 1 is characterized by a linear convergence rate [35] and only requires the solution of a convex problem; this can be obtained in polynomial time using many convex programming algorithms [37,38]. Additionally, the objective function of PGFP monotonically converges to the optimal value of PGFP and the exit condition Fλ = 0, in practice, is replaced by Fλ ≤ ς, with ς a prescribed accuracy.  

The convergence of Algorithm 1 to an optimal solution of PGFP holds even under some milder technical conditions [35,36].

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Algorithm 1: Generalized Dinkelbach’s algorithm Require: X ⊆ CN , { fi (x)}Ii=1 , and {gi (x)}Ii=1 . Ensure: A solution x to PGFP . 1: set n = 0, λn = 0. 2: repeat   find x n = arg max min fi (x) − λn gi (x) ; x∈X 1≤i≤I   let Fλ = min fi (x n ) − λn gi (x n ) ;

3: 4:

1≤i≤I

n = n + 1; fi (x n ) 6: λn = min ; 1≤i≤I gi (x ) n 7: until Fλ = 0 8: output x = x n . 5:

It should be observed that problem Ps (n) belongs to the class of GFP problems. In fact, it can be recast as

Ps (n)

⎧ ⎪ ⎪ ⎪ ⎪ max ⎪ ⎪ s ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨s.t. ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

  (n−1) †  s  p(νdi T ) |αT |Re wi min 

i=1,...,K

(n−1)

s† (wi

)s

s2 ≤ 1  † Re s0 s ≥ δ1   (n) †  Re wi s  p(νdi T ) ≥ 0,

,

(6.15)

i = 1, 2, . . . , K

where  (w) =

N L−1 c −1 

∗  ν¯ d(r,k) 2 σ(r,k) diag (J −r w) (r,k) diag (J −r w∗)

r=0 k=0

+I w† Rw. As a consequence, since the feasible set of (6.15) is a convex compact set and the functions in the objective comply with the assumptions of Lemma 6.2 (s† (w)s > 0 for any w  = 0 and δ1 > 0), we may apply Algorithm 1 to get an optimal solution. Interestingly, the optimization problem involved in each step of Algorithm 1 can be

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Radar waveform design based on optimization theory

Algorithm 2: Algorithm for transmit–receive system design       Require: σ(r,k) , ν¯ d(r,k) , (r,k) , R, νdTi , s0 , δ.   Ensure: A solution s , w 1 , w 2 , . . . , w K to P. 1: set n = 0, s(n) = s0 ,   −1 (n)  s  p(νdi T )  c s(n) + Rs(n) 2 =    , −1/2 (n)   2 s  p(νdi T )    c s(n) + Rs(n) 2 

(n) wi

i = 1, . . . , K,



2 (n) (n) (n) and SINR = g(s , w1 , w2 , . . . , wK ) ; 2: repeat 3: n = n + 1;  (n−1) , i = 1, . . . , K; 4: construct the matrices  wi (n)

5: 6: 7:

(n)

solve problem Ps (n) finding an optimal radar code s(n) , through the use of Algorithm 1;   construct the matrix  c s(n) ; solve problem Pw (n) finding an optimal receive filter bank    −1 (n) s  p(νdi T )  c s(n) + Rs(n) 2 =    , −1/2 (n)   2 s  p(νdi T )    c s(n) + Rs(n) 2 

(n) wi

8:

let SINR

(n)



i = 1, . . . , K;

2 (n) (n) (n) = g(s , w1 , w2 , . . . , wK ) ; (n)

until |SINR(n) − SINR(n−1) | ≤ ζ (n) 10: output s = s(n) and w i = w i , i = 1, . . . , K. 9:

recast as a second-order cone programming (SOCP) problem¶ exploiting its epigraph form [38].

6.3.4 Transmit–receive system design: optimization procedure In this subsection, the devised iterative optimization procedure is summarized as Algorithm 2. with the radar code s(0) that leads to an optimal  The recursion is started (0) (0) (0) filter bank w1 , w2 , . . . , wK . A natural choice is s(0) = s0 .

Notice that the superlinear convergent algorithm developed in [36] can also be used to tackle Ps (n) . In fact, the denominator functions in Ps (n) are strictly convex functions. However, each iteration of the aforementioned procedure may require a higher computational load than Algorithm 1. The interested reader might consult [35] and [36] for further details on the complexity/convergence rate of several related methods used to solve GFP problems. ¶

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The computational complexity of Algorithm 2 depends on the number of outer iterations N as well as the complexity involved in each iteration. In fact, the overall complexity is linear with respect to N , while, in each iteration, the computation of   (n)  −1  c s + Rs(n) 2 and the implementation of Algorithm 1 need to be considered. The former is in the order of O(KN 3 ) [39]. The latter is linear with respect to the number of inner iterations, while each iteration corresponds to the complexity required to solve an SOCP that is O(N 3.5 log (1/η)) (see [33, p. 249]), where η is a prescribed accuracy. Finally, it is important to remark that the devised procedure is able to improve the performance of any feasible transmit–receive pair, due to its monotonically increasing property.

6.4 Performance analysis This section provides representative numerical results to illustrate the effectiveness of the described method. A coherent stationary radar transmitting a burst of N = 20 pulses and |αT |2 = 10 dB is considered. Further, the generalized Barker code is assumed as reference sequence s0 in the considered similarity constraint. This is a constant modulus sequence exhibiting good correlation properties. The convex optimization problems are solved via the CVX toolbox unless specified otherwise [37]. The termination condition inAlgorithm 1 employs ς = 10−6 , whereasAlgorithm 2 uses ζ = 10−3 . In the following, GFP refers to the proposed algorithm and other simulation parameters are detailed next.

Clutter parameters The simulations assume Nc = 3, leading to two interfering (ambiguous) range rings, and L = 100 azimuth sectors for generating the clutter returns. The interfering signals backscattered from these diverse azimuth sectors are weighted according to the azimuth beam-pattern of a typical linear array [8]. An uniformly dis2 tributed clutter with σ(r,k) = σ 2 = 30 dB is assumed for all the pairs (r, k). In addition, the Doppler shifts of the clutter scatterers are uniformly distributed over the interval c = [ν¯ d − (/2), ν¯ d + (/2)] = [−0.065, 0.065], for any range– azimuth bin. These clutter parameters reasonably represent a homogeneous ground clutter [40]. Throughout the simulations, unless explicitly stated otherwise, the filter bank design is handled assuming νdi T ∈ / c , i = 1, 2, . . . , K. In fact, it is well known that Doppler processing techniques are not effective to detect targets whose Doppler shifts are very close to the clutter-dominated Doppler region. Instead, a different processing strategy, i.e., clutter mapping, is usually employed to deal with such Doppler bins [30].

Signal independent interference parameters An autoregressive model with parameter ρ is chosen to model the signal-independent interference, i.e., R(l, m) = ρ |l−m| , (l, m) ∈ {1, 2, . . . , N }2 . Unless explicitly stated otherwise, a white interference model is considered in the ensuing simulations by letting ρ = 0.

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6.4.1 Monotonicity of the proposed method and the impact of similarity constraint Figure 6.2 illustrates the behavior of SINRB , the objective of P in (6.8), as a function of the number of outer iterations, for different values of the similarity parameter δ. The normalized target Doppler shift is assumed to be in the interval T = [−0.5, −0.34] ∪ [0.34, 0.5]; this is representative of an aerial target with cruise speed of 300 m/s, assuming an L band radar with 1/TI = 4 kHz. The corresponding discrete Doppler frequencies, νdi T , i = 1, 2, . . . , K, are obtained by sampling T uniformly with a step size 0.025. Their exact values are reported in the second column of Table 6.1. As expected, the devised method monotonically improves the lowest SINR at the output of the filter bank. Moreover, the optimized worst case SINR value improves with increasing δ, since the feasible set of the optimization problem becomes larger. Finally, the results highlight the capability of the devised procedure to ensure an enhanced worst case (over unknown Doppler shift) target detectability. In fact, significant performance gains (up to 7 dB) with respect to only receive adaptation, whose (0) optimal SINR is SINRB , can be observed.

10

SINRB (dB)

8 6 d = 0.1 d = 0.2 d = 0.5

4 2

10

20

30

40

50

60

70

Iteration

Figure 6.2 An illustration of the monotonically increasing property of the proposed method

Table 6.1 Center frequencies associated with Figures 6.2 and 6.3 for step size value equal to 0.025 Normalized target Doppler shift interval T

Normalized center frequencies {νdi T }Ki=1

Figures 6.2 and 6.3: T = [−0.5, −0.34] ∪ [0.34, 0.5]

−0.490, −0.465, −0.440, −0.415, −0.390, −0.365, −0.340, 0.340, 0.365, 0.390, 0.415, 0.440, 0.465, 0.490

Optimizing radar transceiver for Doppler processing 12

r=0 r = 0.3

10 SINRi (dB)

169

8 6 4 2 0 –0.5

–0.4

–0.3

–0.2

–0.1

0

0.1

0.2

0.3

0.4

0.5

Normalized Doppler shift

Figure 6.3

SINRi values associated with the different filters of the bank, T is [−0.5, −0.34] ∪ [0.34, 0.5] with colored (ρ = 0.3) as well as white interference (ρ = 0)

6.4.2 Impact of colored interference The optimized value of the worst case SINR depends, among other, on the correlation between signal-independent interference samples, ρ. This subsection addresses the impact of ρ on the performance. The SINR values associated with different filters from the filter bank, i.e., {SINRi }, are illustrated in Figure 6.3 for ρ ∈ {0, 0.3} and T = [−0.5, −0.34] ∪ [0.34, 0.5]. The discrete Doppler frequencies, νdi T , i = 1, 2, . . . , K, are obtained by sampling the corresponding T uniformly with a step size 0.025. Table 6.1 denotes these center frequencies νdi T , i = 1, 2, . . . , K. In the presence of colored interference, it can be observed that the worst case value of the SINR is enhanced. This is not surprising since higher interference correlation leads to lower interference energy in corresponding Doppler bins of interest. This observation is also compatible with the behavior of the upper-bound, SINRUB , in (6.16) with respect to ρ. v(P) ≤

|αT |2 = SINRUB , λmin (R) + min λmin ( c (s))

(6.16)

s,s=1

where, indeed, for the considered structure of R, a larger ρ yields a smaller λmin (R) leading to a larger value for SINRUB . It also observed that the SINR values at the output of the filter bank show a flat behavior; this is an additional confirmation of system robustness against Doppler shifts.

6.4.3 Effect of target Doppler shift interval To ascertain the impact of the target Doppler shift uncertainty set, a variable T = [−1/2, −1/2 + D, 1/2 − D, 1/2], D > 0 is assumed as (normalized) target Doppler

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Radar waveform design based on optimization theory

SINRB (dB)

10

0 Initial SINR Optimized SINR

–10

–20

0

0.1

0.2

0.3

0.4

0.5

D

Figure 6.4 SINRB versus D (T = [−0.5, −0.5 + D] ∪ [0.5 − D, 0.5]): dotted line (with circle marker) for the worst case SINR at step zero and dark line (with square marker) for the worst case SINR at convergence

interval with a varying D. Figure 6.4 plots SINRB versus D assuming a uniform sampling of T with a step size 0.025. The worst case SINR value at convergence as well as the one at step zero (i.e., with only receive adaptation) are reported. The curves highlight that higher the D, lower is the worst case SINR due to both the larger target uncertainty levels as well as the increased vicinity of the possible target Doppler shifts and clutter Doppler region. The figure further confirms the effectiveness of the devised algorithm to significantly improve the performance of Doppler processing applications. Indeed, the optimized transmit–receive pair exhibits an SINRB value higher than the architecture employing only receive adaptation as long as the target Doppler shift lies outside the normalized Doppler bins {−0.05, 0, 0.05} that are impacted by clutter returns.

6.4.4 Impact of receive filter bank size The developed design methodology relies on the receiver structure equipped with a finite number of branches each tuned to a prespecified (target) Doppler. The effects of the number of filters K (in the bank) on the system performance are now discussed. To this end, define the quantity SINRi (ν), 1 ≤ i ≤ K, as the SINR at the output of the ith filter due to a target with Doppler frequency ν ∈ [−0.5, 0.5], i.e., †

SINRi (ν) =

|αT |2 |wi (s  p(ν)) |2 †

†

wi  c (s) wi + wi Rwi

, i = 1, 2, . . . , K.

Figure 6.5(a) shows SINRi (ν) versus ν, for i = 1, 2, 3 (corresponding to the positive center frequencies), assuming K = 6 and T = [−0.5, −0.34] ∪ [0.34, 0.5]. Only the values for positive Doppler shifts are plotted. The figure also depicts SINR(ν)  max1≤i≤K SINRi (ν), namely, the SINR value associated with the optimized filter

0

10

20

30

5

10

0.36

0.4

0.42

0.44

15

25

30

35

The number of filters (K)

20

Normalized Doppler shift n

0.38

SINR3(n) SINR(n)

SINR2(n)

SINR1(n)

40

0.46

45

0.48

50

0.5

(d)

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

(b)

0

0.34

10 9.9 9.8

8 0.34

9

10

2

0.36

0.36

4

0.4

0.4

0.42

0.42

0.44

0.44

6

8

| aT | 2

10

12

Normalized Doppler shift n

0.38

0.38

14

16

0.46

0.46

18

K=6 K=10 K=20 K ∞

0.48

K=20

0.48

K=8

20

0.5

0.5

Figure 6.5 Effect of K on the system performance. Only the positive values of T are shown. (a) SINRi (ν) versus ν for the filter bank with K = 6. (b) SINR(ν) versus ν for the system with K = 8 and K = 20. (c) Straddle loss (defined in (6.17)) versus K. (d) Pd,worst versus |αT |2 for different values of K. In part (d), we consider a non-fluctuating target (Swerling 0) embedded in Gaussian disturbance.

(c)

–5

0

5

10

–10 0.34

(a)

dB

Loss (dB)

dB dB Pd,worst

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Radar waveform design based on optimization theory

bank. It can be observed that mismatches∗∗ between the nominal and the actual target Doppler shift lead to an SINR loss. This is due to the finite number (K) of the filters involved in the bank; this is usually referred to as “straddle loss” [30]. In Figure 6.5(b), the behavior of SINR(ν) versus ν is illustrated for K = 8 and K = 20. As expected, the larger K the smaller the straddle loss. Figure 6.5(c) provides a quantitative analysis of the straddle loss L that is analytically defined as L

minν∈T SINR(ν) minν∈T max1≤i≤K SINRi (ν) = . SINRB min1≤i≤K SINRi

(6.17)

The plot therein shows that L monotonically decreases as K increases; for K ≥ N , the loss is negligible but still nonzero even for K = 50.

Influence of straddle loss on detection To assess the impact of the straddle loss on the radar detection capability , the detection probability of the equivalent coherent detector sharing an SNR equal to the worst case SINR of the devised filter bank is considered. Assuming a non-fluctuating target embedded in Gaussian interference, the mentioned detection probability, denoted as Pd,worst , is referred to as a [12,29] √  2SINR , −2 log (Pfa ) , (6.18) Pd,worst = Q where SINR = minν∈T SINR(ν), Pfa represents the probability of false alarm, and Q(., .) is the generalized Marcum-Q function [41]. Figure 6.5(d) shows Pd,worst versus the target strength |αT |2 in the ideal case (K → ∞) as well as for some finite values of K, assuming Pfa = 10−4 . The curve highlights that the use of a small number of filters in the filter bank, e.g., K = 6, considerably affects the worst case detection probability. On the other hand, using K ≥ 20 leads to negligible performance degradation. Finally, for comparison purposes, the responses (in terms of achieved SINRs) of the filters w˜ i = s0  p(νi ), i = 1, 2, 3, corresponding to a standard Doppler filter bank, are shown in Figure 6.6. In particular, considering the same scenario as in Figure 6.5(a), the following quantity is plotted:  i (ν) = SINR

†

|αT |2 |w˜ i (s0  p(ν)) |2 †

†

w˜ i  c (s0 ) w˜ i + w˜ i Rw˜ i

, i = 1, 2, 3.

The results reveal that the new method significantly outperforms the standard Doppler filter bank.

6.4.5 Impact of sequence length on performance In this subsection, we consider the performance of the proposed method for different sequence lengths. In particular, we consider Golomb sequences of different lengths as initial waveforms and evaluate the performance of the proposed method. ∗∗ In this case, the maximum SINR is reasonably achieved at the output of either the filter associated with the nominal Doppler frequency closest to the actual target Doppler shift or the one immediately adjacent to it.

Optimizing radar transceiver for Doppler processing

173

0

dB

–20 –40 SINR1 (n) SINR2 (n) SINR3 (n)

–60 –80 0.34

0.36

0.38

0.4

0.42

0.44

0.46

0.48

0.5

Normalized Doppler shift n

Figure 6.6

 i (ν) versus ν, for the standard Doppler filter bank with K = 6 SINR

10

SINRB (dB)

8 6 4

N=32 N=20 N=12 N=10

2 0

0

20

40

60

80

100

120

Number of iterations

Figure 6.7

Convergence of the proposed method with different sequence lengths (N = {10, 12, 20, 32} and Golomb sequences as initial waveform)

Figure 6.7 shows the convergence behavior of the proposed algorithm for several sequence lengths. As can be seen in all of the cases, the associated minimum SINR increases monotonically and converges to the optimum value. In addition, Figure 6.8 illustrates the resulting minimum SINR versus various sequence lengths. Observe that the minimum SINR almost meets the upper bound (which is 10 dB in this case) for the lengths longer than 15.

6.4.6 Performance comparison In this chapter, a method based on GFP is proposed to maximize the minimum SINR through the filters in the filter bank. In this regard, the problem leads to solving a max–min optimization problem. In this subsection, we compare the performance of the proposed algorithm with a method that is mentioned in [42]. Authors in [42]

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Radar waveform design based on optimization theory

Optimum min SINR (dB)

10 9.8 9.6 9.4 9.2 9 10

15

20

25

30

35

Sequence length

Figure 6.8

Min SINR achieved by generalized fractional programming (GFP) for different sequence lengths

use majorization–minimization (MM) method to tackle the similar problem. To this end, we consider the target Doppler interval T = [0.34, 0.5], the number of filters K = 10, and a white noise (ρ = 0) plus a colored interference with ρ = 0.2 as the signal-independent interference. Also, MOSEK solver in CVX is employed to tackle subproblems of MM method as addressed in [42] and the same threshold has been applied to terminate both methods. Figure 6.9 shows the convergence behavior of the proposed method and that of [42]. Monotonic convergence can be seen for two methods; however, the proposed method converges to a larger value addressing its effectiveness in terms of achievable SINR. Also, in this example, the proposed method converges in much fewer iterations. Figure 6.10 illustrates the filter bank response of the two methods associated with Figure 6.9. As expected, the filter bank of GFP method provides better SINR(ν) when compared to MM method.

6.5 Conclusions This chapter discussed a robust joint optimization of the transmit signal and receive filter bank in the presence of signal-dependent interference. The Doppler shift of the moving target is modeled as being unknown and the worst case SINR at the output of the filter bank is the figure of merit. A similarity constraint has been imposed on the transmit code to control properties of the optimized code and the energy constraint is included as well. The design objective is formulated as a non-convex optimization problem. Thanks to suitable reformulations of the considered non-convex design problem, an efficient sequential optimization procedure of the transmit signal and the receive filter bank designs has been devised. This method monotonically improves the worst case SINR

Optimizing radar transceiver for Doppler processing

175

8

SINRB (dB)

6 4 2 GFP method MM method

0

0

500

1,000

1,500

2,000

2,500

3,000

Iteration

Figure 6.9

Convergence behavior of GFP and MM methods. The signal-independent interference is white noise with ρ = 0 plus a colored noise with ρ = 0.2

8

GFP method MM method

dB

7.5

7

6.5 0.34

0.36

0.38

0.4

0.42

0.44

0.46

0.48

0.5

Normalized Doppler shift n

Figure 6.10

Filter bank response of GFP and MM methods. The signal-independent interference is white noise with ρ = 0 plus a colored noise with ρ = 0.2

while converging to a stationary point. For a fixed transmit code, the resulting optimization problem is separable with respect to individual filters and each optimal filter is given in closed-form. For a fixed number of filters in the filter bank, the problem belongs to a special class of max–min fractional problems that can be globally solved via Dinkelbach-type algorithms. Thus, the problem involves an outer loop solving for the filters and the code, while optimizing code in each iteration requires Dinkelbach procedure. The computational complexity of the proposed method is linear with the number of outer iterations, and Dinkelbach’s procedure in each iteration involves the solution of convex SOCP problems.

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Radar waveform design based on optimization theory

Several numerical examples have been provided to assess the effectiveness of the proposed method in various scenarios. The results have shown the capability of the procedure to ensure target detectability regardless of its actual Doppler shifts, with significant SINR gains (up to 7 dB), with respect to the traditional approach adapting the receiver side only. Possible future research tracks might concern the analysis of the new technique on real radar data as well as the extension of the proposed framework to a space-time adaptive processing scenario, where also the direction-of-arrival is subject to some uncertainty. Finally, it might be of interest to robustify the developed method to account for possible inaccuracies about clutter/interference statistics and to consider constant modulus or peak-to-average ratio constrained transmitted signals.

Appendix A Proof of Proposition 6.1 Proof. The proof is organized in the following two steps: P1.

we show that P is solvable and shares the same optimal solutions and optimal value of ⎧  † |αT |2 |wi sp(νdi T ) |2 ⎪ ⎪ ⎪ max min † ⎪ † ⎪ 2 ⎪ ⎪s,w1 ,...,wK i=1,...,K wi  c (s) wi +wi Rwi s ⎪ ⎪ ⎨s.t. s2 ≤ 1 , P2  ⎪ † ⎪ ⎪ Re s s ≥ δ 1 ⎪ 0 ⎪ ⎪ ⎪  ⎪ ⎪ † ⎩ Re s s ≥ δ1 s 0

P2. P1.

where δ1 = 1 − (δ/2); we prove that P2 is equivalent to P1 . Let us prove that P is solvable. To this end, notice that solving P is equivalent to solving ⎧  † |αT |2 |wi s  p(νdi T ) |2 ⎪ ⎪ ⎪ max min ⎪ ⎪ s,w1 ,...,wK i=1,...,K w †  (s) w + w † Rw ⎪ ⎪ i i i c i ⎨ s2 = 1 . P¯ s.t. ⎪ ⎪ ⎪ 2 ⎪ wi  = 1, ı = 1, 2, . . . , K ⎪ ⎪ ⎪ ⎩ s − s0 2 ≤ δ In fact, the two problems share the same objective function. Additionally, any feasible solution to P¯ is a feasible solution to P; moreover, for any feasible solution (s, w1 , . . . , wK ) to P, (s, (w1 /w1 ), . . . , (wK /wK )) is a feasible solution to P¯ achieving the same objective value as (s, w1 , . . . , wK ). Now, observe that the objective function of P¯ is a continuous function (the minimum of continuous functions is still a continuous function), and the

Optimizing radar transceiver for Doppler processing

177

constraint set defines a compact set. Therefore, from Weierstrass theorem [43], there exists a feasible point (s , w 1 , . . . , w K ) for P¯ such that   |αT |2 |w i † s  p(νdi T ) |2 ¯ v(P) = min † . i=1,...,K w  (s ) w + w † Rw c i i i i Hence, both P¯ and P are solvable. Next, we focus on the equivalence between P and P2 . Since s − s0 2 = † 2 − 2Re s0 s , P shares the same optimal solutions and the same optimal value of

P¯ 2

P2.

⎧ ⎪ ⎪ ⎪ max ⎪ ⎪ s,w1 ,...,wK ⎪ ⎪ ⎪ ⎪ ⎪ ⎨s.t. ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

min

 † |αT |2 |wi sp(νdi T ) |2

i=1,...,K w †  i c

†

(s) wi +wi Rwi s2

s2 = 1  † Re s0 s ≥ δ1 ,  † Re s0 s ≥ δ1 s

.

Let us now show that the non-convex constraint s2 = 1 in P¯ 2 can be relaxed into the convex constraint s2 ≤ 1, namely, P¯ 2 is equivalent to P2 . Clearly, v(P¯ 2 ) ≤ v(P2 ), because the feasible set of P¯ 2 is contained in the feasible set of P2 . Moreover, given a feasible point sˆ to P2 , s1 = (ˆs/ˆs) is feasible to P¯ 2 . In fact, i. s1 2 = 1 since sˆ  = 0;   † † ii. Re s0 s1 ≥ δ1 = δ1 s1 , due to (1/ˆs)Re s0 sˆ ≥ δ1 ;  † iii. Re s0 s1 ≥ (δ1 /ˆs) ≥ δ1 , because (1/ˆs) ≥ 1. Owing to the scale invariance of the objective function of P2 with respect to s, the proof of step 1 is completed. P2 shares the same optimal solutions as ⎧ ⎪ ⎪ ⎪ max ⎪ ⎪ s,w1 ,...,wK ⎪ ⎪ ⎪ ⎪ ⎪ ⎨s.t. ¯ P1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ and v(P¯ 1 ) =

min 

i=1,...,K

 † |αT ||wi sp(νdi T ) | †

†

wi  c (s)wi +wi Rwis2 s2 ≤ 1 ,  † Re s0 s ≥ δ1  † Re s0 s ≥ δ1 s   v(P2 ) = v(P). More precisely,

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Radar waveform design based on optimization theory i.

ii.

two optimization problems sharing the same feasibility set and whose objective functions are related through a monotonic increasing one-to-one mapping have the same optimal set of solutions; for any x1 , x2 , . . . , xK ≥ 0,  √ √ √  min (x1 , x2 , . . . , xK ) = min x1 , x2 , . . . , xK .

Now, let (s , w 1 , . . . , w K ) be an optimal solution to P¯ 1 , then      s , w 1 exp jarg w 1 † s  p(νd1T ) , . . . ,     , . . . , w K exp jarg w K † s  p(νdKT ) is another optimal solution to P¯ 1 that is also feasible to P1 (with objective function value v(P¯ 1 )); hence, v(P1 ) ≥ v(P¯ 1 ). Furthermore, any feasible point (s, w1 , . . . , wK ) to P1 is also feasible to P¯ 1 , and the objective function of P1 in (s, w1 , . . . , wK ) is lower than or equal to the objective function of P¯ 1 , evaluated at the same point. This implies that v(P1 ) ≤ v(P¯ 1 ). As a consequence, v(P1 ) = v(P¯ 1 ), and any optimal solution to P1 is also optimal to P¯ 1 . In addition, given an optimal solution to P¯ 1 , we can construct an optimal solution to P1 . This completes the proof of step 2. Note: The following constraints †

Re(s0 s) ≥ δ †

Re(s0 s) ≥ δs s ≤ 1, is equivalent to †

Re(s0 s) ≥ δ s ≤ 1

Appendix B Proof of Proposition 6.2 Proof. The proof of the first three items of the proposition mainly follows from [8,   2 (n) (n) Proposition 2.1] and the fact that SINR(n) = g s(n) , w1 , . . . , wK . Let us now show that any cluster point to the generated is a stationary point for our  sequence   (n ) (n ) optimization problem. To this end, let s˜ (n ) , w˜ 1 , . . . , w˜ K be a subsequence

Optimizing radar transceiver for Doppler processing 179     (n) (n) of s(n) , w1 , . . . , wK converging to the cluster point s˜ , w˜ 1 , . . . , w˜ K , we will prove that g(s, w˜ 1 , w˜ 2 , . . . , w˜ K ) ≤ g(˜s , w˜ 1 , w˜ 2 , . . . , w˜ K ), for any feasible s,

g(˜s , w1 , w2 , . . . , wK ) ≤ g(˜s , w˜ 1 , w˜ 2 , . . . , w˜ K ), for any feasible (w1 , w2 ,. . . ,wK ). Let s¯ be an optimal solution to  † ⎧  ⎪ |αT |Re w˜ i s  p(νdi T ) ⎪ ⎪ ⎪ max min  ⎪ ⎪ i=1,...,K ⎪ s ⎪ † † 2 ⎪ w ˜ w ˜  + w ˜ R w ˜ s (s) ⎪ c i i i i ⎪ ⎪ ⎨ s.t. s2 ≤ 1 † ⎪ ⎪ Re s0 s ≥ δ1 ⎪ ⎪  ⎪ ⎪ † ⎪ ⎪ Re s s ≥ δ1 s ⎪ ⎪  0† ⎪  ⎪ ⎩ Re w˜ i s  p(νdi T ) ≥ 0, i = 1, 2, . . . , K

.

Then, the following inequality chain holds (n )

(n )

(n )

g(¯s , w˜ 1 , w˜ 2 , . . . , w˜ K ) ≤  (n ) (n ) (n ) g(˜s(n +1) , w˜ 1 , w˜ 2 , . . . , w˜ K ) ≤  (n +1) (n +1) (n +1) g(˜s(n +1) , w˜ 1 , w˜ 2 , . . . , w˜ K ) ≤  ({n+1} ) ({n+1} ) ({n+1} ) g(˜s({n+1} ) , w˜ 1 , w˜ 2 , . . . , w˜ K ), 

where the first inequality is straightforward if s¯ is a feasible point to Ps (n +1) , (n ) (n ) (n ) otherwise it follows from the fact that in such case g(¯s , w˜ 1 , w˜ 2 , . . . , w˜ K ) < 0,     (n ) (n ) (n ) while g(˜s(n +1) , w˜ 1 , w˜ 2 , . . . , w˜ K ) ≥ 0. The second inequality is due to the optimal (n +1) (n +1) (n +1) , w˜ 2 , . . . , w˜ K ) to Pw (n +1) . Finally, the last inequality ity of the point (w˜ 1 (n) (n) (n) stems from the fact that g(˜s(n) , w˜ 1 , w˜ 2 , . . . , w˜ K ) is an increasing sequence of values and n + 1 ≤ {n + 1} . Therefore, due to the continuity of the function g( · ), as n → ∞, we have g(¯s , w˜ 1 , w˜ 2 , . . . , w˜ K ) ≤ g(˜s , w˜ 1 , w˜ 2 , . . . , w˜ K ).   Similarly, let w¯ 1 , w¯ 2 , . . . , w¯ K be an optimal solution to   ⎧  † ⎪ |αT |Re wi s˜  p(νdi T ) ⎪ ⎪ ⎪ ⎨ max min  w1 ,...,wK i=1,...,K   † † . wi  c s˜ wi + wi Rwi ˜s 2 ⎪ ⎪   ⎪  ⎪ ⎩s.t. Re w† s˜  p(ν i ) ≥ 0, i = 1, 2, . . . , K i

dT

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Radar waveform design based on optimization theory Hence, 



(n )

(n )

(n )

g(˜s(n ) , w¯ 1 , w¯ 2 ,. . . ,w¯ K ) ≤ g(˜s(n ) , w˜ 1 , w˜ 2 ,. . . ,w˜ K ), 

where the inequality is straightforward if (w¯ 1 , w¯ 2 , . . . , w¯ K ) is feasible to Pw (n ) ,  otherwise, as before, it is due to the fact that in this case g(˜s(n ) , w¯ 1 , w¯ 2 , . . . , w¯ K ) < 0,     (n ) (n ) (n ) while g(˜s(n ) , w˜ 1 , w˜ 2 , . . . , w˜ K ) ≥ 0. Thus, as n → ∞, due to the continuity of the function g(·), we have g(˜s , w¯ 1 , w¯ 2 , . . . , w¯ K ) ≤ g(˜s , w˜ 1 , w˜ 2 , . . . , w˜ K ), and the proof is completed.

Appendix C Proof of Lemma 6.1 Proof. The objective function and the constraint functions of Pw (n) are separable functions of the variables wi , i = 1, . . . , K. This implies that an optimal solution to Pw (n) is obtained solving the disjoint optimization problems, i = 1, 2, . . . , K, ⎧    ⎪ |α |Re w† s(n)  p(νdi T ) ⎪ ⎨max  T     w . P¯ i w†  c s(n) w + w† Rws(n) 2 ⎪    ⎪ ⎩s.t. i † (n) Re w s  p(νdT ) ≥ 0, Observe that P¯ i is equivalent to P˜ i , i = 1, 2, . . . , K, defined as ⎧    ⎪ |αT |Re w† s(n)  p(νdi T ) ⎪ ⎨max      w . P˜ i w†  c s(n) w + w† Rws(n) 2 ⎪  (n)  ⎪ ⎩s.t. i † w s  p(ν ) = 1 dT

In fact, v(P¯ i ) ≥ v(P˜ i ), i = 1, . . . , K, since a constraint has been added optimal solution to P¯ i . Then, to the problem. Moreover, let  wi be    an      † (n)  † w¯ i = wi /wi s  p(νdi T )  exp j arg wi s(n)  p(νdi T ) is an optimal solution i i i i ˜ ¯ ˜ ¯ to P feasible to P ; hence, v(P ) = v(P ), i = 1, . . . , K, because the two problems share the same objective function. It can be checked that P˜ i is equivalent to P˜ i1 , i = 1, 2, . . . , K,    min w†  c s(n) w + w† Rws(n) 2 i ˜ w   P1 , s.t. w† s(n)  p(νdi T ) = 1 i.e., v(P˜ i ) = 1/(v(P˜ i1 )), and if w is an optimal solution of P˜ i then it is also an optimal solution of P˜ i1 and vice versa. Finally, P˜ i1 is equivalent to P i ; in fact v(P i ) ≤ v(P˜ i1 ), because a constraint has been let w be an optimal solution to problem P i , then   Moreover,  added.     i i  † (n) † (n) w1 = w /w s p(νdT )  exp jarg w s p(νdT ) is an optimal solution to

Optimizing radar transceiver for Doppler processing 181    2  †   † P˜ i1 and v(P˜ i1 ) = v(P i )/w s(n)  p(νdi T )  . Since |w s(n)  p(νdi T ) | ≥ 1, it follows that w 1 = w . From the previous chain of equivalences, the proof is completed.

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Wang G, Lu Y. Designing single/multiple sparse frequency waveforms with sidelobe constraint. IET Radar, Sonar and Navigation. 2011;5(1):32–38. Patton LK, Frost SW, Rigling BD. Efficient design of radar waveforms for optimised detection in coloured noise. IET Radar, Sonar and Navigation. 2012;6(1):21–29. Pillai SU, Li KY, Himed B. Constant envelope signals with prescribed discrete Fourier transform magnitude. In: IEEE International Radar Conference. Kansas City, MO: IEEE; 2011. p. 470–473. Patton LK, Rigling BD. Phase retrieval for radar waveform optimization. IEEE Transactions on Aerospace and Electronic Systems. 2012;48(4):3287–3302. Patton L, Bryant CA, Himed B. Radar-centric design of waveforms with disjoint spectral support. In: IEEE International Radar Conference. Atlanta, GA: IEEE; 2012. p. 269–274. Chen C, Vaidyanathan PP. MIMO radar waveform optimization with prior information of the extended target and clutter. IEEE Transactions on Signal Processing. 2009;57(9):3533–3544. Friedlander B. Waveform design for MIMO radars. IEEE Transactions on Aerospace and Electronic Systems. 2007;43(3):1227–1238. Aubry A, De Maio A, Piezzo M, et al. Cognitive design of the receive filter and transmitted phase code in reverberating environment. IET Radar, Sonar and Navigation. 2012;6(9):822–833. Naghsh MM, Modarres-Hashemi M, ShahbazPanahi S, et al. Unified optimization framework for multi-static radar code design using information-theoretic criteria. IEEE Transactions on Signal Processing. 2013;61(21):5401–5416. Naghsh MM, Modarres-Hashemi M, Kerahroodi MA, et al. An information theoretic approach to robust constrained code design for MIMO radars. IEEE Transactions on Signal Processing. 2017;65(14):3647–3661. Hsiao J. On the optimization of MTI clutter rejection. IEEE Transactions on Aerospace and Electronic Systems. 1974;10(5):622–629. D’addio E, Farina A, Studer F. Performance comparison of optimum and conventional MTI and Doppler processors. IEEE Transactions on Aerospace and Electronic Systems. 1984;20(6):707–715. Aubry A, De Maio A, Huang Y, et al. Robust design of radar Doppler filters. IEEE Transactions on Signal Processing. 2016;64(22):5848–5860. Naghsh MM, Soltanalian M, Stoica P, et al. A Doppler robust design of transmit sequence and receive filter in the presence of signal-dependent interference. IEEE Transactions on Signal Processing. 2014;62(4):772–785. Naghsh MM, Soltanalian M, Stoica P, et al. Radar code design for detection of moving targets. IEEE Transactions on Aerospace and Electronic Systems. 2014;50(4):2762–2778. Richards MA, Scheer JA, Holm WA. Principles of Modern Radar: Basic Principles. Stevenage: SciTech Publisher; 2010. Ward J. Space-time adaptive processing for airborne radar. In: Technical Report 1015. Lexington, MA: MIT Lincoln Laboratory; 1994.

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Chapter 7

Radar waveform design via the majorization–minimization framework Linlong Wu1 and Daniel P. Palomar1

7.1 Introduction A multiple-input–multiple-output (MIMO) radar transmits several probing waveforms simultaneously from its transmit antennas. It can improve the interference rejection capability and parameter identifiability as well as provide flexibility for transmit beam pattern design [1]. In addition, a cognitive approach for radar systems, which capitalizes on the information obtained from the surrounding environment or the prior knowledge stored in the platform, was proposed [2]. The significance of MIMO radar and the cognitive approach has recently motivated active research into the waveform design. A well-designed waveform can allow for more accurate detection and estimation. However, we are facing several challenges when designing radar waveform: 1.

2.

Radar system is an application with extreme emphasis on real time, which requires a fast implementation of waveform design. Moreover, for a MIMO radar, the simultaneously transmitted waveforms are independent from each other, which requires more computation cost compared with the traditional radars. In addition, if the radar system runs in a cognitive manner, the adaptability is achieved by redesigning the waveform before the next transmission, which means a short time window for the design phase. With the consideration of hardware configuration and application scenarios, some waveform constraints are necessarily incorporated. However, these constraints probably raise the design difficulty. For example, in [3], closed-form solutions can be found when only the finite-energy constraint is considered. If other constraints are considered, then it is not as easy as the former case.

Thus, for radar waveform design, an algorithmic framework is desired that it is not only computationally efficient but also flexible so that various waveform constraints can be handled. One type of the promising solutions is the derived algorithms based on the MM method. The MM method refers to the majorization–minimization or 1 Department of Electronic and Computer Engineering, Hong Kong University of Science and Technology (HKUST), Hong Kong, China

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the minorization–maximization method, which is a general algorithmic framework to solve the original problem via iteratively solving a sequence of simpler problems. The idea of MM dates back to the 1970s [4] and started taking shape as a general algorithm framework around 2000 [5–7]. Many existing algorithms can be interpreted from the MM’s perspective, and it has already been successfully applied to a large number of problems and applications. Recently, some works have applied the MM method successfully on radar waveform design problems [8–12]. The results presented in these papers have shown that the algorithms based on the MM method are competitive alternatives compared with the benchmark algorithms for those design problems. This chapter presents, in a self-contained manner, the MM framework and its applications to two radar waveform design problems. Through these design problems, several key issues of applying MM and its properties are illustrated. We hope this selfcontained chapter can introduce you to this useful algorithmic framework on solving waveform design problems. The rest of the chapter is organized as follows. In Section 7.2, we give a relatively complete description of the MM method. In Section 7.3, we apply the MM method to the joint design of transmit waveform and receive filter. In Section 7.4, we apply the extension of the vanilla MM method to the robust joint design problem. Finally, conclusions are given in Section 7.5.

7.2 Preliminaries: the MM method The MM framework is a powerful optimization scheme, especially when the problem is hard to tackle directly. The idea behind the MM algorithm is to convert the original problem into a sequence of simpler problems to be solved until convergence. The key to using the MM method is to construct a simple majorized or minorized problem that can be solved efficiently. This section serves as a concise description of the MM method, which includes the general procedure of MM, convergence analysis, acceleration schemes, and the extension to the maximin case. Interested readers may refer to [13], and references therein for more details.

7.2.1 The vanilla MM method Consider a general optimization problem: min

f (x)

subject to

x ∈X,

x

(7.1)

where X is a nonempty closed set and f : X → R1 is a continuous function. Notice that the convexity of the feasible set X and the objective function f is not assumed. Initialized at x (0) ∈ X , the MM method generates a sequence of feasible points () {x }∈N according to the following update rule:   (7.2) x (+1) ∈ argmin u x, x () , x∈X

Radar waveform design via the majorization–minimization framework

187

Algorithm 1: Majorization–minimization (MM) Require: Initial point x (0) Ensure: Converged point x 1:  = 0 2: repeat   3: Construct a majorizer u x, x ()  4: x (+1) ∈ argmin u x, x () x∈X

←+1 6: until convergence 5:

  where x () is the point generated by the MM method at the th iteration, and u x, x () is a majorizing function (majorizer) of f (x) at x () , which should satisfy at least the following two properties: u (x, y) ≥ f (x) , for ∀x, y ∈ X

(7.3)

u (y, y) = f (y) , for ∀y ∈ X . (7.4)   () In other words, the majorizer u x, x should be a global upper bound of f (x) and coincide with f (x) at x () . Therefore, the algorithm based on the MM method for problem (7.1) is summarized as Algorithm 1. One interesting and useful property of MM-based   methods is the monotonicity of the generated sequence of objective values f x () :         f x (+1) ≤ u x (+1) , x () ≤ u x () , x () = f x () , (7.5) where the first inequality follows from (7.3), the second one follows from (7.2), and (+1) the last equality follows is not chosen as the  from (7.4). Furthermore, even if x () minimizer of u x, x , the monotonicity can still be guaranteed as long as it improves    the function u x (+1) , x () ≤ u x () , x () . The key to applying the MM method lies in the careful construction of the majorizer. On the one hand, the majorizer should be tight to the original function for a fast convergence speed in the sense of the number of iterations. On the other hand, it should be simple enough so that the majorized problem per iteration can be solved at low computational cost. Thus, there is clearly a trade-off between the complexity of each iteration and the number of iterations, which should be considered when constructing the majorizer.

7.2.2 Convergence analysis

      Given the monotonicity of f x () , it is obvious that f x () is guaranteed to converge as long as f (x) is bounded below  over the feasible set. In the following, we will focus on the convergence of x () . The convexity of the constraint set X decides which type of stationary points the MM method converges to. Suppose the

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constraint set X is convex. The majorizer must satisfy the following conditions to guarantee the convergence to a d(irectional) stationary point [14]: u (x, y) ≥ f (x) , for ∀x, y ∈ X

(7.6)

u (y, y) = f (y) , for ∀y ∈ X

(7.7)

u (y, y; d) = f  (y; d) , for ∀d with y + d ∈ X

(7.8)

u (x, y) is continuous on (x, y) ,

(7.9)

where f  (y; d) is the directional derivative defined as f  (y; d) = limλ→0 inf ((f (y+ λd) − f (y))/λ). In [14], it is proved that the limit point x (∞) satisfies   f  x (∞) ; d ≥ 0, for ∀d with x (∞) + d ∈ X . (7.10) Thus, x (∞) is called d(irectional) stationary point. Suppose the constraint set X is nonconvex. The condition (7.8) needs to be modified to guarantee the convergence: u (y, y; d) = f  (y; d) , for ∀d ∈ TX (y) ,

(7.11)

where u and f are defined on the whole R1 space and TX (y) is the Boulingand tangent cone of X at y. Following this modification, we can prove that the limit point x (∞) satisfies     f  x (∞) ; d ≥ 0, for ∀d ∈ TX x (∞) . (7.12) Thus, x (∞) is called the B(oulingand) stationary point [15,16].∗

7.2.3 Acceleration schemes In some cases, if the majorizer is ill-constructed in the sense that it yields a closed-form solution but at the cost of slow convergence speed, some acceleration technique has to be adopted as a remedy. Luckily, the MM algorithm can be implemented with some acceleration techniques without loss of convergence.† Different types of accelerators have been proposed in the literature based on line search [18], inexact descent [19], and fixed-point equations [20,21]. Among them, the accelerator named SQUAREM (squared iterative method) was proposed by Varadhan and Roland [20] and can be easily implemented as an “off-the-shelf ” accelerator for the MM algorithm.

∗

Following the notations and problem settings of [14], the proof mostly follows that of [14, Theorem 1] with only “z + d ∈ X ” replaced by d ∈ TX (z), which is based on [17, Proposition 4.7.1]. † The proof of the convergence of the accelerated MM mostly follows that of [14, Theorem 1] with slight modifications on one equation: (following the notations and problem settings of [14]) u (x, xrj ) ≥  u (MM (xrj ) , xrj ) ≥ f (MM (xrj )) ≥ f xrj +1 ≥ f (xrj+1 ) ≥ u (xrj+1 , xrj+1 ), where MM (·) is the MM algorithm mapping and xrj +1 is the next iteration point found by the acceleration technique. Thus, every limit point of the generated sequence by the accelerated MM algorithm is a stationary point of the solving problem.

Radar waveform design via the majorization–minimization framework

189

Algorithm 2: MM accelerated via SQUAREM Require: Initial point x (0) Ensure: Converged point x 1:  = 0 2: repeat 3: x0 = x () 4: x1 = FMM (x0 ) 5: x2 = FMM (x1 ) 6: r = x1 − x0 7: v = (x2 − x1 ) − r 8: α = − r2 / v2 9: x = x1 − 2αr + α 2 v 10: x (+1) = FMM (x) 11: ←+1 12: until convergence

Let FMM (·) denote the fixed-point iteration map of theMM algorithm. Then, the MM updating scheme can be expressed as x (+1) ∈ FMM x () . Define F (x) = FMM (x) − x, a Newton step update of finding a zero of G is given by   −1  ()  x (+1) = x () − ∇F x () F x , (7.13) where ∇F is the Jacobian of F , which will be further approximated via the secant method. Combining this approximate update rule with the Cauchy–Barzilai–Borwein method [52], we obtain the SQUAREM given in Algorithm 2.

7.2.4 Extension to the maximin case We have illustrated the MM framework by considering a minimization problem. For a maximization problem, the above results can be directly applied with simple modifications. Specifically, the update rule (7.2) should be replaced by   (7.14) x (+1) ∈ argmax u x, x () , 

where u x, x

 ()

x∈X

is the minorizing function (minorizer) of f (x) at x () with

u (x, y) ≤ f (x) , for ∀x, y ∈ X

(7.15)

u (y, y) = f (y) , for ∀y ∈ X .

(7.16)

Now, we consider a maximin optimization problem: max x

min fi (x)

i=1,...,I

subject to x ∈ X ,

(7.17)

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where the objective takes the form of F (x) = mini=1,...,I fi (x) (the fi ’s are assumed differentiable). One minorizer of F (x) is given by U (x, y) = min ui (x, y) ,

(7.18)

i=1,...,I

with each ui being a tight lower bound of fi , satisfying: ∀i, ui (x, y) ≤ fi (x) , for ∀x, y ∈ X

(7.19)

ui (y, y) = fi (y) , for ∀y ∈ X

(7.20)

∇ui (y, y) = ∇fi (y) , for ∀y ∈ X

(7.21)

ui (x, y) is continuous on (x, y) .

(7.22)

Now, we check whether U (x, y) satisfies the conditions for stationarity convergence: Checking (7.16): ∀y ∈ X , U (y, y) = min ui (y, y) = min fi (y) = F (y) . i=1,...,I

i=1,...,I

(7.23)

Checking (7.15): ∀x, y ∈ X , ui (x, y) ≤ fi (x) ⇒

min ui (x, y) ≤ min fi (x) ⇒U (x, y) ≤ F (x) .

i=1,...,I

i=1,...,I

(7.24)

Checking (7.8): According to [22, Theorem 9.16], given d, the directional derivative of U in (7.14) can be expressed as U  (y, y; d) = max {ξ , d : ξ ∈ ∂U (y, y)} ,

(7.25)

where ∂U (y, y) = conv({∇ui (y, y) : U (y, y) = ui (y, y)}) and conv(A ) is the convex hull of the set A . We also derive the directional derivative of F: F  (y; d) = max {ξ , d : ξ ∈ ∂F (y)} ,

(7.26)

where ∂F (y) = conv ({∇fi (y) : F (y) = fi (y)}). From U (y, y) = F (y) and ui (y, y) = fi (y), we obtain {i|U (y, y) = ui (y, y)} = {i|F (y) = fi (y)}. Besides, fi satisfies ∇ui (y, y) = ∇fi (y), ∀i, so ∂U (y, y) = ∂F (y). If X is convex, U  (y, y; d) = F  (y; d), ∀d with y + d ∈ X . If X is nonconvex, U  (y, y; d) = F  (y; d), ∀d ∈ TX (y). Checking: (7.9): Obvious. Thus, the piecewise function U (x, y) in (7.18) satisfies the four conditions of convergence and the limit point of (7.14), i.e., x (∞) , shall be the stationary point: ● ●

When X is a convex set, d-stationarity is achieved. When X is a nonconvex set, B-stationarity is achieved.

7.3 Joint design of transmit waveform and receive filter In this section, we consider the joint design of transmit waveform and receive filter in the presence of signal-dependent interference for a colocated MIMO radar. For the joint design problem, a popular approach is to adopt the alternating optimization

Radar waveform design via the majorization–minimization framework

191

scheme and then resort to semidefinite relaxation (SDR) and rank-one reconstruction [23–30]. Recently, [11] has applied the MM method successfully on this problem, on which the content of this section is based. Interested readers may refer to [13], and references therein for more details. The rest of this section is organized as follows. We first introduce the system model and formulate the problem. Then, we derive the general algorithm within the MM framework and consider several constraints on the problem. Finally, we analyze the numerical performance of the proposed algorithm for each case and compare our methods with the corresponding benchmarks.

7.3.1 System model and problem formulation Consider a colocated MIMO radar system with Nt transmit antennas and Nr receive antennas. Each transmit antenna emits a waveform sm (n) , m = 1, . . . , Nt , n = 1, . . . , N , with N being the number of samples of each transmitted pulse. Let s (n) ∈ CNt ×1 represent the nth sample of the Nt waveforms. Suppose the target of interest is located at the range-angle position (r0 , θ0 ) with r0 = 0. Thus, the signals at the receive antennas can be represented by x (n) = αar (θ0 ) at (θ0 )T s (n) ej2π (n−1)ν0 + d (n) + v (n) ,

(7.27)

where ● ● ●

  α accounts for the target radar cross section with E |α|2 = σ02 . ν0 is the Doppler frequency of the target. at (θ0 ) ∈ CNt ×1 denotes the steering vector and ar (θ ) ∈ CNr ×1 is the propagation vector. Both the transmit and receive antennas are assumed to be uniform linear arrays (ULAs) with half-wavelength separation so that T 1  at (θ) = √ 1, e−jπ 1 sin θ , . . . , e−jπ(Nt −1) sin θ , Nt

●

(7.28)

and similarly for ar (θ ). d (n) accounts for the superposition of K signal-dependent uncorrelated point-like interferers. Specifically, the kth interferer is located at the range-angle position (rk , θk ), where rk ∈ {0, 1, . . . , N }, θk ∈ {0, 1, . . . , L} × (2π /(L + 1)) with L being the number of discrete azimuth sectors. The received interfering vector d (n) can be expressed as follows: d (n) =

K 

αk ar (θk ) at (θk )T s (n − rk ) ej2π(n−1)νk ,

(7.29)

k=1

●

  where αk is the complex amplitude of the kth interferer with E |αk |2 = σk2 and νk is its Doppler frequency.   v (n) denotes the additive Gaussian noise with v (n) ∼ N 0, σv2 I .

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  T T T  Let x = x (1)T . . . x (N )T , s = s (1)T . . . s (N )T , and v = v (1)T . . . v (N )T . Expression (7.27) can be represented as x = αA (r0 , θ0 ) s +

K 

αk A (rk , θk ) s + v,

(7.30)

k=1

where A (rk , θk ) is given by    A (rk , θk ) = Diag (p (νk )) ⊗ ar (θk ) at (θk )T Jrk , where Jrk is the shift matrix given by 1, m − n = Nt × rk Jrk (m, n) = (m, n) ∈ {1, . . . , Nt N }2 , 0, m − n  = Nt × rk and

 T p (νk ) = 1, ej2πνk , . . . , ej2π (N −1)νk

(7.31)

(7.32)

(7.33)

with νk being the Doppler frequency for k = 0, 1, . . . , K. Hereafter, the Doppler frequencies {νk }Kk=0 are assumed to be zero without loss of generality (i.e., both the target and the interferers are assumed to be slowly moving or stay still), and A (rk , θk ) will be denoted by Ak for simplicity of notation. Suppose linear finite impulse response receive filters w are deployed. Then, the output of the filter is given by r = w† x = αw† A0 s + w†

K 

αk Ak s + w† v.

(7.34)

k=1

Based on the model, the output SINR is given by

2

σ02 w† A0 s



SINR = . K 2 † A† w + σ 2 w † w w† σ A ss k v k k=1 k

(7.35)

Thus, the design problem can be formulated from the perspective of maximizing the SINR as follows:

†

w A0 s 2 max s, w (7.36) w† (s) w + w† w subject to s ∈ S ,   where (s) = Kk=1 qk Ak ss† A†k with qk = σk2 /σv2 > 0, S = s| s2 = 1, s ∈ Sc , with Sc ∈ CNNt denoting a nonempty but not necessarily convex set.

7.3.2 MM-based method for joint design with multiple constraints We first derive an algorithmic framework for the joint design problem based on the MM method. Then, we will consider four waveform constraints for the derived algorithmic framework. Finally, we will summarize the proposed algorithms with analysis of their computational complexity.

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193

7.3.2.1 Majorized iteration method for joint design For a given s, problem (7.36) with respect to w can be equivalently reformulated into min

w† [ (s) + I ] w

subject to

w† A0 s = 1,

w

(7.37)

with a closed-form solution given by w =

[ (s) + I ]−1 A0 s s† A0 [ (s) + I ]−1 A0 s †

.

(7.38)

Substituting (7.38) into (7.36), and after some algebraic manipulations, problem (7.36) reduces to 

† s† A0 [ (s) + I ]−1 A0 s max s (7.39) subject to s ∈ S . Defining s = ss† , problem (7.39) is equivalent to 

† − s† A0 [ (s) + I ]−1 A0 s min s,S

(7.40)

subject to s = ss† , s ∈ S , where (s) = Kk=1 qk Ak sA†k . Lemma 7.1. Denote the objective function of problem (7.40) by f (s, S). Then, f (s, S) is a concave function of s and s jointly, and a majorizer of f (s, S) is  † (7.41) u1 (s, S; s , s ) = −2Re z s + 2Tr (p s) − 2Tr (p s ) − f (s , s ) , where z = A†0 [ (s ) + I ]−1 A0 s , p = † Qk = A0 [ (s ) + I ]−1 Ak .

K k=1

 † qk Qk s Qk , and

Proof. See Appendix A. Ignoring the constant terms and undoing the change of variables s = ss† in the function (7.40), the first majorized problem is then given by  † s† p s − Re z s min s (7.42) subject to s ∈ S . Lemma 7.2. Let L be an n × n Hermitian matrix and M be another n × n Hermitian n † matrix such that M  L. Then  x0 ∈† C , the quadratic function x Lx is  † for any point † majorized by x Mx + 2Re x (L − M ) x0 + x0 (M − L) x0 at x0 [31].

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Radar waveform design based on optimization theory

By using the above lemma, a majorizer of the objective function of problem (7.42) can be constructed as follows:   u2 (s, S )=λu (P ) s† s + 2Re s† (P − λu (P ) I ) s  (7.43) † † + s (λu (P ) I − P ) s − Re z s , where λu (P ) is an upper bound of the eigenvalues of the positive semidefinite matrix P , which could be simply chosen as Tr (P ) considering the computation cost. Please note that the tightness of the upper bound λu (P ) affects the performance of the convergence speed. Due to s† s = 1, we have the second majorized problem:  † min Re v s s (7.44) subject to s ∈ S , where v = 2 (P − λu (P ) I ) s − z .

(7.45)

7.3.2.2 Four waveform constraint cases We consider four constraints on the problem: the constant modulus, the similarity, the peak-to-average power ratio (PAR), and the spectrum compatibility. For each constraint case, we will derive the closed-form solution or algorithms for the corresponding majorized problems.

Constant modulus constraint Note that in practice, due to the limitations of hardware components (such as the maximum signal amplitude clip of A/D converters and power amplifiers), it is usually desirable to transmit constant modulus waveforms. Problem (7.44) becomes  † min Re v s s (7.46) 1 subject to |sn | = √ , for i = 1, . . . , NNt , NNt which has the closed-form solution ejarg(v ) s=−√ , NNt

(7.47)

where ejarg(·) is an element-wise operation that extracts the phase.

Similarity constraint We will consider the similarity constraint for the first and second majorized problems. If the finite energy constraint is reduced to the constant modulus constraint, the

Radar waveform design via the majorization–minimization framework

195

similarity constraint can be recast as in [32], and the first majorized problem (7.42) becomes  † s† P s − Re z s min s (7.48) 1 subject to |sn | = √ , arg (sn ) ∈ [γn , γn + δ] , NNt   2 with sref being the reference where γn = arg (sref (n)) −arccos 1 − NN t  /2  sequence and δ = 2arccos 1 − NNt  2 /2 with  being the similarity parameter defined by s − sref ∞ ≤ . The block coordinate descent method (BCD) [33, Chapter 2] can be employed for problem (7.48). Assuming all the elements of s, except sn , are fixed, the problem of sn is given by min

Re(a∗n sn )

subject to

|sn | = √

sn

1 , NNt

arg (sn ) ∈ [γn , γn + δ] ,

(7.49)

t where an = 2 NN i=1,i =n si Pn,i − zn with Pn,i being the (n, i)th entry of P , and zn is the nth element of z. The closed-form solution to problem (7.49) is already shown in [30] and rewritten as follows: ⎧ ejan   √ arg (an ) ∈ γn + 2δ + (2k − 2) π , γn + (2k − 1) π ⎪ NNt ⎪ ⎨ jarg(an ) (7.50) sn = −e√NNt arg (an ) ∈ [γn + (2k − 1) π , γn + δ + (2k − 1) π ] ⎪ ⎪ ⎩ e√j(an +δ) otherwise, NNt

where ∃k ∈ Z, for n = 1, 2, . . . , NNt . For the second majorized problem (7.44), we have  † Re v s min s

subject to

|sn | = √

1 , NNt

(7.51) arg (sn ) ∈ [γn , γn + δ] ,

which has a closed-form solution similar to (7.50).

PAR constraint The PAR constraint is expressed as   max |sn |2 1≤n≤NNt ≤ , PAR = (1/NNt ) s2

(7.52)

where  is the parameter controlling the acceptable level of PAR with 1 ≤  ≤ NNt . Considering the PAR constraint and letting γ = (/NNt ), the problem becomes  † Re v s min s (7.53) √ subject to s2 = 1, |sn | ≤ γ ,

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Radar waveform design based on optimization theory

Lemma 7.3. Without loss of generality, we assume that |v1 | ≥ |v2 | ≥ · · · ≥ |vN | and the number of nonzero elements of v is m. Then, the solution to problem (7.53) is

where

s = PS (v) ,

(7.54)

 √ PS (·) = − 1R≥0 (1 − mγ ) γ um  ejarg(·)   √ − 1R εslc : μ+1 = 2μ ; else: μ+1 = μ 5: ← +1 (I,II,III): |f (s ) − f (s−1 )| ≤ εobj 6: until |f (s ) − f (s−1 )| ≤ εobj & t ≤ εslc (IV) : 7:

w=

[ (s )+I ]−1 a(θ )s

s a(θ)† [ (s )+I ]−1 a(θ )s †

7.3.2.3 Summary of algorithm and complexity analysis To summarize, the description of the algorithm with respect to the above constraints is given in Algorithm 3. Hereafter, we use MIA-CMC, MIA-CMSC, MIA-PC, and

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199

MIA-SCCG to denote the proposed Majorized Iterative Algorithm with the Constant Modulus Constraint, Constant Modulus and Similarity Constraint, PAR Constraint and Spectrum Compatibility Constraint for Global design, respectively. MIA-CMSC can be subdivided into MIA-CMSC1 (MIA-CMSC based on the first majorization) and MIA-CMSC2 (MIA-CMSC based on the second majorization). We now analyze the computational complexity of the MIA-based algorithms. The overall complexity of each MIA-XXX method is linear with respect to the number of iterations. For the convenience of analysis, we focus on the deterministic cost on a per-iteration basis, which comes from the following  three sources: z , P , and s+1 . The computational cost of z is O (Nr N )3 because of the inversion operation. With [  (s) + I ]−1 already computed, the computational cost  k of Q is O (Nr N ) · (Nt N )2 , and, thus, the computational cost of P should be     O (Nr N ) · (Nt N )2 + O (Nt N )3 . To sum up, z and P contributes the total amount   3 of complexity O N · (max {Nr , Nt })3 , neglecting the lower order terms. The update of s+1 varies case by case: for MIA-CMC, MIA-CMSC2, and MIAPAR, the computation cost mainly comes from the computation of λu (P ) and v . The as Tr update of λu (P ) is simply chosen  (P ), so the computation cost is O (Nt N ). The  computational cost of v is O (Nt N )2 . For MIA-CMSC1, there are NNt subproblems due to the BCD scheme, and the computational cost of each subproblem  is O (N  tN ) because of the update of an . Thus, the total computational cost is O (Nt N )2 . For MIA-SCCG, MOSEK will reformulate problem (7.65) into the epigraph form by introducing one more slack variable. Then, there is one linear constraint and three second-order cone (SOC) constraints. The complexity of solving the  computational  problem should be upper bounded by O (Nt N )3.5 , the same order as SOC programing (SOCP).

7.3.3 Numerical experiments Assume that both the transmitter and receiver are ULAs with half-wavelength separation, Nt = 8, Nr = 8, and N = 20. A target is located at the range-angle position (0, 15◦ ) with power |α0 |2 = 20 dB, and three fixed interferers are located at the range-angle positions (0, −50◦ ), (1, −10◦ ) , and (2, 40◦ ), respectively. The power for each interferer is |αi |2 = 20 dB, for i = 1, 2, 3. The noise variance is σν2 = 0 dB. The orthogonal linear frequency modulation (LFM) waveforms are set as the initial and also the reference waveforms for MIA-CMC, MIA-CMSC, and MIA-PC. Denote the space-time sequence matrix of the LFM waveform by S0 . The (k, n)th entry of S0 is given by S0 (k, n) = √

1 exp {j2π (n − 1) (k + n − 1) /N } , NNt

(7.66)

where k = 1, . . . , Nt and n = 1, . . . , N . The initial sequence S0 ∈ CNNt ×1 is obtained by stacking the columns of s0 . The initial filter w0 is obtained according to (7.38) by using s0 . Unless otherwise specified, all the parameters are the same in the numerical experiments. In the following experiment, we also implement the accelerated version of the corresponding MIA-type method, denoted by MIA-XXX-Accelerated.

200

Radar waveform design based on optimization theory 20 19 18 20

SINR (dB)

17 16

19.8

15

19.6

14

19.4 0

13

50

12

MIA-CMC (proposed) MIA-CMC-accelerated (proposed) SOA1-CMC SOA2-CMC

11 10

100

0

200

400

600

800 1,000 1,200 CPU time (s)

1,400

1,600

1,800

2,000

Figure 7.1 Convergence plot: SINR versus CPU time for the constant modulus case

SQUAREM [20], as an off-the-shelf acceleration scheme, is deployed as the acceleration scheme. All experiments were carried out on a Windows laptop with a 2.60 GHz i7-5600U CPU and 8 GB RAM.

7.3.3.1 Joint design with the constant modulus constraint The benchmark methods for the constant modulus constraint are SOA1-CMC (sequential optimization algorithm 1 with constant modulus constraint) and SOA2-CMC (sequential optimization algorithm 2 with constant modulus constraint) [23]. Both of them employ the SDR and randomization to solve the rank-one constrained SDP problems iteratively. In Figure 7.1, we can see clearly that an obvious advantage of the MIA-type methods is the guarantee of monotonicity, compared with the fluctuation of the SINR achieved by the SOA methods. Besides, both MIA-CMC and MIA-CMC-Accelerated are much faster than the SOA-type methods in terms of the CPU time.

7.3.3.2 Joint design with the similarity constraint The benchmarks are SOA1-CMSC (sequential optimization algorithm 1 with constant modulus and similarity constraints), SOA2-CMSC (sequential optimization algorithm 2 with constant modulus and similarity constraints) in [23], and Algorithm 2 in [40] named ALT-DB. SOA1-CMSC and SOA2-CMSC are similar to SOA1-CMC and

Radar waveform design via the majorization–minimization framework 15

201

14.88 14.86

14.5

14.84 14

14.82 14.8

SINR (dB)

13.5

14.78 0

13

2

4

6

12.5 12 11.5

MIA-CMSC1 (proposed) MIA-CMSC2 (proposed) SOA1-CMSC SOA2-CMSC ALT-DB

11 10.5

0

50

100

150 CPU time (s)

200

250

300

Figure 7.2 Convergence plot: SINR versus √ CPU time for the similarity case. Similarity parameter:  = 1/ NNt

SOA2-CMC, respectively. ALT-DB deploys the alternating scheme, and for the problem with respect to s, the BCD method is used within the Dinkelbach framework. From Figure 7.2, we can see that the MIA-type methods and ALT-DB are better than the SOA-type methods in terms of both CPU time and converged SINR. Further, MIA-CMSC2 is slightly better than ALT-DB in terms of the achieved SINR with almost the same performance on CPU time.

7.3.3.3 Joint design with the PAR constraint The SOA1-type method is modified and named SOA1-PC to be a benchmark for the PAR case. The other benchmark is the method proposed in [41] named ALT-SDP here. ALT-SDP can be decomposed into two steps: SDPs of w and s are solved alternately in the first step, and w and s are synthesized by randomization in the second step. In the experiment about computation efficiency, we only consider the first step of ALT-SDP, where the SDP of w is replaced by the closed-form solution of w. Note that the SINR achieved after the second step may be smaller than that obtained in the first step, and the running time should include the time for the synthesis stage. Figure 7.3 shows that the converged SINR achieved by MIA-PC is about 3 dB higher than both SOA1-PC and ALT-SDP. In addition, MIA-PC converges much faster in terms of CPU time.

202

Radar waveform design based on optimization theory 20 MIA-PC SOA1-PC ALT-SDP

19 18

SINR (dB)

17 16 15 14 13 12 11 10

0

50

100

150

200 250 CPU time (s)

300

350

400

Figure 7.3 Convergence plot: SINR versus CPU time for the PAR case. PAR parameter: γ = 0.5

7.3.3.4 Joint design with the spectrum compatibility constraint The experiment settings as follows: N = 40, Nt = 2, and Nr = 4. The first fre 1 are 1 quency interval is flower , fupper = [0.2, 0.3], together with the spectral compatibility matrix R and the maximum allowed interference E1 . The second frequency interval 1  2  2 is flower = [0.75, 0.85], together with the spectral compatibility matrix R2 and , fupper the maximum allowed interference E2 . For global design, the spectral compatibility † t matrix is R = Nk=1 Uk Rk Uk , and the total allowed interference is EI . The benchmark is the method proposed in [42] named ARCO here. Figure 7.4 shows the convergence curves and the PSD comparison of the designed waveforms. Our method is faster than the benchmark although both methods converge to almost the same SINR. Two deep nulls of the PSD designed by MIA-SCCG are much deeper than the counterparts designed by the benchmark.

7.4 Robust joint design for the worst-case SINR maximization In the previous section, we formulate problem as a maximization of SINR. Further, the maximization of the minimum of SINR is also of interest in radar signal processing, where the maximin metric aims at ensuring the worst-case performance guarantee. In radar target detection, many works of waveform design are based on

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203

20

19.5

SINR (dB)

19

18.5

18

17.5

17

16.5

MIA-SCCG (proposed) ARCO

0

5

10

15

20 25 CPU time (s)

30

35

40

20

Average PSD (dB)

0

–20

–40

–60

–80 MIA-SCCG (proposed) ARCO

–100

0

0.2

0.6 0.4 Normalized frequency

0.8

1

Figure 7.4 Convergence plot (left) and designed average PSD (right) for spectrum compatibility case. Parameter settings: EI = |λmax (r1 ) + λmin (r1 )| × 10−4 + |λmax (r2 ) + λmin (r2 )| × 10−4

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either known Doppler shifts [3,43,44] or signal-independent interference [37,45,46]. In practice, Doppler shifts are often unknown, especially when the detection process has just been launched and the target has not yet been tracked. The assumption of signal-independent interference fails to take into account possible reflections of transmitting signals from other objects (hence signal-dependent interference). One pioneering work combining these two considerations is [26], which proposed a novel algorithm, DESIDE, to conduct the maximin optimization. The DESIDE algorithm is cyclic and SDR-based. The more recent work [27] improved the design of [26] by incorporating a filter bank. Apart from the problems with unknown Doppler, there exists a similar problem in a colocated MIMO radar system [23,47–49]. Both signaldependent interference and some uncertainty in the target angle are considered. For the introduced works above, the optimization problems take basically the same form as follows: max

min SINRi

s (or si ), wi

i=1,2,...,I

(7.67)

subject to s (or si ) ∈ S ,

where s (or si ) ∈ CN denotes the transmitting sequence, wi ∈ CM represents the receiving filter, and S models the constraint set. In this section, we focus on solving the above robust design problem with consideration of uncertain parameters such as Doppler frequency. The content of this section is based on [12]. Interested readers may refer to [12], and references therein for more details. The rest of this section is organized as follows. We first specify the problem formulation and then propose the MM-based algorithmic framework for solving the maximin problem, which is followed by the numerical simulations.

7.4.1 Problem formulation We denote the length of the transmitting sequence(s) and receiving filters as N and M , respectively. Recall that s is the transmitting sequence and wi is the ith receiving filter of the filter bank. The SINR is expressed as (the derivation is similar to (7.38) in Section 7.3.2.1)

SINRi =



† 2 αi wi Hi s

†

†

wi  I (s) wi + wi RWi

.

(7.68)

with  I (s) =



†

βj Mj ss† Mj ,

(7.69)

j

where αi > 0 is the parameter representing the path gain (or loss), Hi ∈ CM ×N represents the channel matrix,  I (s) (cf. [23,29,50]) is the interference covariance matrix with βj > 0, and the matrix Mj ∈ CM ×N is an application-dependent constant matrix, which will be specified later.

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205

Therefore, the problem of interest is given by max

2



† αi wi Hi s

min

†

†

i=1,2,...,I wi  I (s)wi +wi Rwi

s,wi

subject to s ∈ S , # !

" where S = s ∈ CN s2 = 1, s∞ ≤ (ρ/N ) with ρ being the parameter of the PAR constraint and I is the number of the filters of the filter bank.

7.4.2 MM-based method for robust joint design We are now prepared to present the MM algorithmic framework for solving the formulated problem. The MM method is naturally split into two stages: the minorization and the maximization, which will be illustrated in two subsections.

7.4.2.1 Minorizer construction It is not hard to show that given s, the optimal solution for wi is (up to a positive scaling factor) wi =

( I (s) + R)−1 Hi s s† Hi ( I (s) + R)−1 Hi s †

.

(7.70)

Then the original problem (7.67) is reduced to min SINRi (s)

max s∈CN

i=1,2,...,I

subject to s ∈ S ,

(7.71)

where SINRi (s) = αi s† Hi ( I (s) + R)−1 Hi s. †

(7.72)

We already know from Section 7.2 that finding a minorizing function for mini=1,2,...,I SINRi (s) boils down to finding one for each SINRi (s). Thus, we can focus on the expression of SINRi (s) only. In the following, we are going to find a tight lower bound for SINRi (s) at the current iteration value s(n) . We do a change of variable: let G be  I (s), and then SINRi = SINRi (s, G) = αi s† Hi† (G + R)−1 Hi s. We can see that SINRi is a matrix fractional function and proves to be jointly convex in (s, G), as can be seen from [51, Example 3.4]. In that sense, a simple first-order Taylor expansion with respect to (s, G) at (s0 , G0 ) gives us a tight lower bound: $ % 

SINRi (s, G) ≥ SINRi (s0 , G0 ) + 2Re b†i (s − s0 ) − Tr ai ai† · (G − G0 ) , (7.73) where bi = αi Hi (G0 + R)−1 Hi s0 , √ ai = αi (G0 + R)−1 Hi s0 , †

(7.74) (7.75)

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Radar waveform design based on optimization theory †

and −ai ai is the gradient with respect to G. Now, we undo the change of variable G =  I (s) and let s0 be s(n) , the transmitting code at the nth iteration:    SINRi (s,  I (s)) ≥ SINRi s(n) ,  I s(n) $   (7.76) %   † †  + 2Re bi s − s(n) − Tr ai ai ·  I (s) −  I s(n) , where bi and ai should also be adjusted:   −1 † bi = αi Hi  I s(n) + R Hi s(n) and ai =

−1 √   (n)  αi  I s + R Hi s(n) .

(7.77)

(7.78)

Lemma 7.4 provides a minorizer of SINRi (s,  I (s)). Lemma 7.4. A minorizing function of SINRi (s) at s = s(n) is given as $  & &2     % † SINRi s, S (n) = SINRi s(n) + 2Re ci s − s(n) − λu,i &s − s(n) &2 ,

(7.79)

where ci = bi − Ai s(n) ,  † † βj Mj ai ai Mj  0, Ai =

(7.80) (7.81)

j

and λu,i = λmax (Ai ) > 0.

(7.82)

Proof. See Appendix B. Then, the minorizing function for mini=1,2,...,I SINRi (s) is   min SINRi s, S (n) . i=1,2,...,I

(7.83)

According to the framework of the MM method, at every iteration we just need to solve the following problem:   max min SINRi s, S (n) i=1,2,...,I s∈CN (7.84) subject to s ∈ S .

7.4.2.2 Maximization solution pursuit &

& Since s2 = 1 and &s(n) &2 = 1, we can now rewrite (7.84) as $ † % max min di + 2Re ci + λu,i s(n) s , s∈S

i=1,2,...,I

(7.85)

where

% $   † di = SINRi s(n) − 2Re ci s(n) − 2λu,i .

(7.86)

Radar waveform design via the majorization–minimization framework

207

The discrete minimum in (7.85) can be equivalently rewritten as follows: $ † % min di + 2Re ci + λu,i s(n) s i=1,2,...,I

(a)

=min

p∈P

I 

 $ † % pi di + 2Re ci + λu,i s(n) s

i=1

(b)

=min pT d + 2Re p∈P

(7.87)

$  † % C + s(l) λTu p s ,

  where (a) P = p ∈ RI |1T p = 1, p ≥ 0 is a simplex and (b) d = [d1 , d2 , · · · , dI ]T ,  T C = [c1 , c2 , . . . , cI ], and λu = λu,1 , λu,2 , · · · , λu,I . Thus, (7.85) is equivalent to $  † % max min 2Re C + s(l) λTu p s + pT d. (7.88) s∈S p∈P

Lemma 7.5. In problem (7.88), a saddle point exists and it can be obtained from solving the relaxed problem where the nonconvex constraint set S is relaxed to ( ) ' ρ N . (7.89) Srelaxed = s ∈ C | s2 ≤ 1, s∞ ≤ N Proof. See Appendix C. Now we look into the relaxed problem: $ † % max min di + 2Re ci + λu,i s(l) s , s∈Srelaxed i=1,2,...,I

(7.90)

or, equivalently, max min 2Re

s∈Srelaxed p∈P

$

 † % C + s(l) λTu p s + pT d.

(7.91)

Problem (7.91) is derived from (7.88) by changing S to Srelaxed and problem (7.90) is a reformulation of (7.91) into the discrete minimum format. We are going to propose two approaches for solving the relaxed problem. The first approach: If we focus on (7.90), we can solve it via an off-the-shelf solver directly. To accelerate the convergence speed of the MM algorithm, we adopt the following technique‡ : at any iteration, say the nth iteration, we utilize the optimal

‡

The convergence result also holds for the accelerated MM algorithm and the proof mostly follows that of [14, Theorem 1] with slight modifications on one equation: the notations and problem settings of  (following  [14]) u (x, xrj ) ≥ u (MM (xrj ) , xrj ) ≥ f (MM (xrj )) ≥ f xrj +1 ≥ f (xrj+1 ) = u (xrj+1 , xrj+1 ), where MM () is the MM algorithm mapping and xrj +1 is the next iteration point found by the acceleration technique. In this case, subsequence stationarity convergence is maintained.

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Algorithm 4: Accelerated solver-based MM algorithm Require: feasible s(0) , n = 0; 1: repeat 2: Compute d, C, and λu (cf. (7.86), (7.80), and (7.82), respectively); 3: Solve (7.90) via some off-the-shelf solver and get its optimal solution sˆ (n) ; 4: Apply acceleration technique (7.92) for step size β; s(n) +β sˆ (n) −s(n) 5: s(n+1) = s(n) +β (sˆ(n) −s(n) ) ; ( )2 6: n = n + 1; 7: until convergence

s, denoted by sˆ (n) , to provide an ascent direction, sˆ (n) − s(n) , and do the line search as [52]: choose α (> 1) ; β = 1; s(n) +αβ sˆ (n) −s(n) stemp = s(n) +αβ (sˆ(n) −s(n) ) ; ( & &  )2    " while min SINRi stemp ≥ min SINRi sˆ (n) and &stemp &∞ ≤ Nρ i=1,2,...,I

β = αβ;

i=1,2,...,I

(7.92)

s(n) +αβ sˆ (n) −s(n) stemp = s(n) +αβ (sˆ(n) −s(n) ) ; ( )2 end

The first algorithm is summarized in Algorithm 4. The second approach: Now we focus on (7.91). The objective function in (7.91) is bilinear in s and p; Srelaxed and P are both nonempty compact convex sets. Following the results of [53, Corollary 37.6.2 and Lemma 36.2], a saddle point exists and we can swap maximin to be minimax without affecting the solutions: $  † % min max 2Re C + s(n) λTu p s + pT d, (7.93) p∈P s∈Srelaxed

which can be compactly rewritten as min h (p) ,

(7.94)

  h (p) = max 2Re (Bp)† s + pT d

(7.95)

p∈P

where

s∈Srelaxed

and b = C + s(n) λTu . In particular,   √ √ ● when ρ = 1, Srelaxed = s ∈ CN | s∞ ≤ 1/N and h (p) = 2 1/N Bp1 + pT d;   ● when ρ = N , Srelaxed = s ∈ CN | s2 ≤ 1 and h (p) = 2 Bp2 + pT d.

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Algorithm 5: MDA algorithm Require: feasible p(0) , m = 0; 1: repeat   2: Get subgradient: h(m) ∈ ∂h p(m) ; p(m) exp −γ h(m) 3: p(m+1) = 1T p(m) exp( −γm h(m)) ; ( ( m )) 4: m = m + 1; 5: until convergence

We solve (7.94) via the mirror descent algorithm (MDA) [54]. Since P is the unit simplex, we omit the derivation details and present the MDA for problem (7.94) in Algorithm 5. Now we are only left with computing h(m) in line 2 ofAlgorithm 5. The subgradient (m) h is given as   h(m) = 2Re B† x (m) + d, (7.96) $ % † where x (m) = arg maxx∈Srelaxed Re Bp(m) x . In particular, 

−1   √ ● when ρ = 1, x (m) = 1/N Bp(m)  Bp(m) (|·|−1 operation is imposed element wisely); &   & ● when ρ = N , x (m) = Bp(m) / &Bp(m) &2 ; ● when 1 < ρ < N , x (m) follows the closed-form solution in [34, Algorithm 2]. The phases of x (m) are aligned with those of Bp(m) . Denote the number of nonzero elements of Bp(m) as K (≤

N ), and the set containing all the corresponding indexes as K . The solution of x (m) is as follows: – if Kρ/N ≤ " 1, the solution is ρ ∀n ∈ K ,

(m)

x = * N (7.97) n 1−Kρ/N ∀n ∈ /K; N −K – if Kρ/N > 1, the solution is

(m)  (m) √ρ/N

x = η Bp

, (7.98) & 0 & √ & & ρ/N b where η satisfies &[η|Bp(m) |]0 & = 1 (|·| and [·]a denote the element-wise 2 absolute value and the &element-wise projection onto [a, b], respectively). & √ & ρ/N & (m) Observing that g(η) = &[η|Bp |]0 & is a strictly increasing function 2  √    (m)  



on 0, ( ρ/N )/ minn∈K , there is a unique η satisfying Bp n g (η) = 1. The second algorithm is finally summarized in Algorithm 6.

7.4.2.3 Computational complexity Now, we discuss the computational complexity of Algorithm 4 and 6. The only difference between the two algorithms is the way they solve the subproblem (7.84). We

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Algorithm 6: Accelerated MDA-based MM algorithm Require: feasible s(0) , n = 0; 1: repeat 2: Compute d, C, and λu (cf. (7.86), (7.80), and (7.82), respectively); 3: Solve (7.94) via MDA (Algorithm 5) for p and sˆ (n) ; 4: Apply acceleration technique (7.92) for step size β; s(n) +β sˆ (n) −s(n) 5: s(n+1) = s(n) +β (sˆ(n) −s(n) ) ; ( )2 6: n = n + 1; 7: until convergence

analyze the computational complexity on a per-iteration basis or, more precisely, on a per-outer-iteration basis. For analytical convenience, we focus on the deterministic cost only. The deterministic computational cost mainly comes from two sources: (1) computing d, C, and λu and (2) solving the simple convex problem (7.90) or (7.91). We assume M and N are of the same order (Hi , Mj ∈ CM ×N ). First, we look into the computation of d, C, and λu (cf. (7.86), (7.80), and (7.82), respectively). The most costly operation in computing one element of d ∈ RI and      −1 one column of C∈ CN ×I needs O N 3 because of  I s(n) + r , so the overall complexity is O IN 3 . Recall that λu,i = λmax (ai ), where ai  0. The computation of λu,i can be replaced by Tr (ai ) because, first, this change does not violate any of the inequalities in the algorithm design and, second, computing Tr (ai ) is very efficient, only O (N ). So, the overall cost is O (IN  ). To this moment, the first source contributes a total amount of complexity O IN 3 , neglecting the lower order terms. Next, we move on to the simple convex problem. An off-the-shelf solver, e.g., MOSEK [39], will reformulate the problem into the epigraph form with one more slack variable. Thus, we have I linear constraints. The 2 - and ∞ -norm constraints√can be rewritten as SOC constraints: √ (1) 2 : s2 ≤ 1 and (2) ∞ : ∀n, |sn | ≤ ρ/N =⇒ [Re [sn ] , Im [sn ]]2 ≤ ρ/N , hence a total of N + 1 SOC constraints. To sum up, there are I linear constraints and N + 1 SOC constraints, so the computational of solving the simple convex problem should be upper  complexity  bounded by O N 3.5 , the same order as SOCP. If we take a closer look at those SOC constraints, we find that they are of very small size (only two variables) and much simpler than those in [27]: no Hadamard product, no matrix decomposition, and no affine transformation. The resulting SOCP is quite sparse, and modern conic solvers such as MOSEK can exploit the sparsity of the problem very efficiently.  That  is why the practical complexity is far below the worst-case complexity O N 3.5 . The periteration complexity of MDA is elaborated as follows. MDA consists of two main steps in each iteration: (1) computation of subgradient H (m) : this step involves matrix multiplications Bp(m) and b† x (m) , of complexity O (NI ) (b ∈ CN ×I , p(m) ∈ RI , and x (m) ∈ CN ); (2) update of p(m) to p(m+1) : this step involves element-wise exponent, Hadamard product, and summation, of complexity O (I ). Therefore, the per-iteration complexity of MDA is O (NI ).

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7.4.2.4 Robust design with the uncertainty of Doppler Following the setting in [27], we set M = N (the filter has the same length as the sequence); the channel matrix is given as    (7.99) Hi = Diag p νdi T ,   j2πν T where p (ν) = 1, e , . . . , ej2π (N −1)ν is the Doppler steering vector and νdi T is the   ith sampled normalized target Doppler frequency, falling within νdT ,lower , νdT ,upper . The interference covariance matrix  I (s) is specifically expressed as + , N L−1 c −1   ν¯ d  I (s) = σ(n2 c ,l) Jnc (n(nc c,l),l)  ss† Jn†c , (7.100) nc =0 l=0

where Nc (< N ) is the number of range rings, L is the number of azimuth sectors, the range-azimuth bin is denoted as (nc , l), the bin of interest is (0, 0) (where we receive signals), σ(n2 c ,l) is the mean interfering power associated with the clutter patch located at the range-azimuth bin (nc , l) whose (normalized) Doppler shift νd(nc ,l) is uniformly distributed over the interval   ν¯ d(nc ,l) − (nc ,l) /2, ν¯ d(nc ,l) + (nc ,l) /2 , Jnc is a Toeplitz matrix with the nc th subdiagonal ν¯ d   entries being 1 and 0 elsewhere, and (n(nc c,l),l) is the covariance matrix of p νd(nc ,l) , given as ν¯ d     (n(nc c,l),l) (m, n) = exp j2π ν¯ d(nc ,l) (m − n) × sinc (nc ,l) (m − n) (7.101) with sinc (x) = sin (π x) / (πx). In this case, the expression of ai is specified as ,T + N L−1 c −1   ν¯ d(n ,l) † 2 † c σ(nc ,l) Jnc ai ai Jnc  (nc ,l) . (7.102) ai = nc =0 l=0

7.4.3 Numerical experiments We present numerical results with respect to Doppler robust design. All experiments were performed on a PC with a 3.20-GHz i5-4570 CPU and 8-GB RAM. The off-theshelf solver is specified as (1) MOSEK [39] built in the CVX toolbox [55], shortly denoted as CVX, and/or (2) the Fusion MATLAB® API in MOSEK, shortly denoted as MOSEK. The proposed algorithms are terminated when the improvement between iterations is smaller than a threshold (by default 10−6 ) or the number of iterations reaches a predetermined maximum (by default 500).

7.4.3.1 Experiment settings The transmitting sequence length is N = 20. We assume Nc = 2 interfering range rings and L = 100 azimuth sectors. A homogeneous ground clutter is adopted: ∀ (nc , l), a uniformly distributed clutter is assumed with σ(n2 c ,l) = σ 2 = 1000 and the Doppler shift of the clutter scatterer νd(nc ,l) is uniformly distributed over c =   ν¯ d(nc ,l) − (nc ,l) /2, ν¯ d(nc ,l) + (nc ,l) /2 = (−0.065, 0.065). As for the target, αi = α = 10 dB, ∀i. The background noise covariance matrix r is I (white noise). The filter bank is designed by assuming νdi T ∈ / c , ∀i, i.e., the uncertainty interval of the

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8 MM: solver (MOSEK), ρ = N MM: MDA, ρ = N MM: solver (CVX), ρ = N MM: solver (MOSEK), ρ = N/4 MM: MDA, ρ = N/4 MM: solver (CVX), ρ = N/4 MM: solver (MOSEK), ρ = 1 MM: MDA, ρ = 1 MM: solver (CVX), ρ = 1

7.5

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Figure 7.5 Convergence plot: worst-case SINR versus CPU time, N = 20

  target Doppler frequency T = νdT ,lower , νdT ,upper does not overlap with c . We set T = [0.34, 0.5]. The number of filters is I = 10. For the PAR constraint threshold, we set ρ = 1, N /4, and N for performance comparison. Unless otherwise specified, all the values of the parameters are unchanged in the numerical experiments.

7.4.3.2 Monotonic property of the proposed algorithms We implement both solver-based MM and MDA-based MM; both algorithms initiate from a known sequence: the generalized Barker code (with unit energy) [56]. In Figure 7.5, we show the monotonic property of the proposed algorithms. The worstcase SINR (i.e., the objective function value) monotonically increases with the time, until it becomes saturated at a certain level. When we increase the parameter ρ, the optimized worst-case SINR also increases because the constraint set becomes more and more relaxed. We may notice that the two algorithms need different numbers of iterations and time to converge, and they may not converge to exactly the same solution. In the current settings, when ρ = N /4 and N , MDA-based MM reaches a slightly higher optimized value, while solver-based MM converges slightly faster, especially in the case of MOSEK; when ρ = 1, solver-based MM using MOSEK directly reaches the highest optimized value and converges the fastest.

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213

7.4.3.3 Robust versus non-robust design We adopt the MDA-based MM as the proposed method, which initiates from the generalized Barker code (with unit energy). In the non-robust design, only the nominal target Doppler frequency is considered. Here we set the nominal value to be the center of the uncertainty interval, i.e., νˆ dT = (0.34 + 0.5) /2 = 0.42. The radar detection performance is measured by SINR (ν), which is defined as

2

†

αi wi H (ν) s

SINR (ν) = max † , (7.103) i=1,2,...,I w  (s) w + w † rw I i i i i  T with H (ν) = Diag 1, ej2πν , . . . , ej2π(N −1)ν . The variable s is derived from optimization; once s is known, the optimal {wi } is also known (cf. (7.70)). The reason for using SINR (ν) is related to the detection mechanism of the filter bank: once the received signal is passed through the filter bank, we pick the largest SINR to compare with a predetermined threshold for detection; when the target Doppler is actually ν (still falling within the uncertainty interval), the largest SINR for threshold comparison is thus expressed as SINR (ν) and in the performance evaluation, we want SINR (ν) to be as large as possible. In Figure 7.6, we compare between robust and non-robust designs under different PAR levels and noise. Under both white noise and colored noise, the robust design has a much smaller scale of fluctuation than its non-robust counterpart. Although the non-robust design achieves slightly higher SINR in a small neighborhood around the nominal value, its worst-case performance across the interval can be arbitrarily bad. Moreover, when imposing different levels of the PAR constraint, we see no significant change of SINR (ν) in the robust design different from the non-robust case.

7.4.3.4 Comparison with existing methods We compare the proposed two algorithms with the existing DESIDE [26] and the SOCP-based algorithm in [27]. To enable fair comparison, only the energy constraint is enforced, i.e., ρ = N , and all four methods initiate from the same initial sequence s(0) . In Figure 7.7, we plot SINR (ν) in the uncertainty interval for the four methods. Our proposed methods achieve a worst-case SINR (the smallest value across the uncertainty interval, i.e., minν∈T SINR (ν)) of around 9.8 dB, while both benchmark algorithms achieve around 8.2 dB.

7.5 Conclusion In this chapter, we have first given a complete description of the MM optimization method, followed by its application to two radar waveform design problems. The first problem is the joint design of transmit waveform and receive filter for a colocated MIMO radar, which is formulated as maximizing the SINR subject to multiple waveform constraints. We have derived the MIA based on the MM method, which can handle multiple waveform constraints, namely, the constant modulus, the similarity,

(a)

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Figure 7.6 Robust designs versus non-robust designs under different PAR levels and noise: (a) White noise (R = I ): ρ = N (left), ρ = N /4 (middle), and ρ = 1 (right); N = 20 and (b) colored noise (R(m, n) = 0.4|m−n| ): ρ = N (left), ρ = N /4 (middle), and ρ = 1 (right); N = 20

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Figure 7.7 SINR (ν) versus Doppler shift ν for four methods: MM: MDA, MM: solver (proposed methods) and SOCP, DESIDE (benchmark methods)

the PAR, and the spectrum compatibility. Numerical experiments show the good performance of MIA under the above four constraints and emphasize its efficiency in terms of the achieved SINR and the CPU time. The second problem is the robust joint design with the uncertainty of Doppler frequency. It is an extension of the first problem in the sense of mathematical expression, which is formulated as a of the minimum of several SINR’s. Based on the maximin extension of MM, an MM-based algorithm for solving the robust design problem has been proposed. Within the proposed algorithm, we have provided two solving approaches for the minorized problem at each iteration. Numerical simulations have shown that the proposed MM algorithms, both solverbased and MDA-based, achieve higher objective values as well as a faster convergence speed compared with the benchmarks, and the achieved robust SINR is higher than the benchmark. The results for the two problems have shown that the algorithms based on the MM method are competitive alternatives compared with the benchmark algorithms for radar waveform design.

Appendix A Proof of Lemma 7.1 Proof. Denote the objective function of (7.40) by f (s, S) and define g (x, Y ) = x † Y −1 x, where Y  0. The function g (x, Y ) = x † Y −1 x is jointly convex in x and Y

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† [51]. Let x = A0 s and Y = I + Kk=1 qk Ak SAk . Both are affine transformations. Thus, f (s, S) is jointly concave of s and S. Since f (s, S) is jointly concave in s and S, the first-order approximation of f (s, S), denoted by u1 (s, S; s , S ), is a majorizer of f (s, S) at the point (s , S ), which is given by u1 (s, S; s , S )

  = f (s , S ) + Ds f |s (s − s ) + Ds∗ f |s∗ s∗ − s∗ , + + ,

T

T   ∂f ∂f ∗ ∗ ∗ − S + Tr S +Tr | | − S (S )   ∂S S ∂S ∗ S 

† † = f (s , S ) − s A0 [ (S ) + I ]−1 A0 (s − s ) 

T   † s∗ − s∗ −sT A0 [ (S ) + I ]−1 A0 

 k † K k − S +Tr q Q S Q (S )     k=1 k , +

T   k †  K k ∗ ∗ S − S S Q +Tr k=1 qk Q 

† † = −f (s , S ) − 2Re s A0 [ (S ) + I ]−1 A0 s 

 k † K k +2Tr q S Q Q   S  k=1 k 

  K k † k −2Tr Q q S Q   S .  k=1 k

(A.1)

Appendix B Proof of Lemma 7.4 Proof. According to  I (s) from (7.69), one term in (7.76) can be rewritten as    †   − Tr ai ai† ·  I (s) −  I s(n) (B.1) = −s† Ai s + s(n) Ai s(n) , where ai follows (7.81). Thus, $   †   % † SINRi (s) ≥SINRi s(n) + 2Re bi s − s(n) − s† Ai s + s(n) Ai s(n) . (B.2)  †    We minorize SINRi (s) by applying s† Ai s ≤ s(n) Ai s(n) + 2Re s(n)H Ai s − s(n) + & &2 λmax (Ai ) &s − s(n) &2 : $  & &2   % SINRi (s) ≥SINRi s(n) + 2Re ci† s − s(n) − λu,i &s − s(n) &2 , (B.3) where ci and λu,i are defined in (7.80) and (7.82), respectively.

Appendix C Proof of Lemma 7.5 Proof. If we relax S to be Srelaxed , then the problem becomes (same as (7.91)) $  † % max min 2Re C + s(l) λTu p s + pT d. (C.1) s∈Srelaxed p∈P

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The objective function is concave–convex in s and p, and Srelaxed and P are both nonempty compact convex sets. Following the results of [53, Corollary 37.6.2 and Lemma 36.2], a saddle point exists for the relaxed problem. The saddle point of the  2 < 1. relaxed problem, denoted by (s , p ), must satisfy s 2 = 1. Suppose

 s √   There always exist some element of s , denoted by sj , such that sj < ρ/N . If * 2 √ √ ρ/N × N = ρ ≥ 1, causing contradiction. Then we reset not, then s 2 ≥   the phase of sj to be aligned with the jth element of C + s(l) λTu p and increase its modulus by a small amount without violating feasibility. The objective can be pushed  up from theside of s, causing   contradiction with the saddle point nature of s . The jth (l) T element of C + s λu p has been assumed to be nonzero for simplicity. In case it becomes zero, the optimal solution of sj may be nonunique (and thus the saddle point is nonunique), but we can always find one on the boundary by properly increasing the modulus of the currently obtained sj if necessary. Since the saddle point (or at least one saddle point) of the relaxed problem naturally satisfies s ∈ S and p ∈ P, there must exist a saddle point for problem (7.88), and the saddle point can be obtained from solving the relaxed problem.

Acknowledgment This work was supported by the Hong Kong RGC Theme-based Research Scheme (TRS) Grant T21-602/15R.

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Naghsh MM, Soltanalian M, Stoica P, et al. Radar code design for detection of moving targets. IEEE Transactions on Aerospace and Electronic Systems. 2014;50(4):2762–2778. Naghsh MM, Soltanalian M, Stoica P, et al. A Doppler robust design of transmit sequence and receive filter in the presence of signal-dependent interference. IEEE Transactions on Signal Processing. 2014;62(4):772–785. Aubry A, De Maio A, Naghsh MM. Optimizing radar waveform and Doppler filter bank via generalized fractional programming. IEEE Journal of Selected Topics in Signal Processing. 2015;9(8):1387–1399. Stoica P, He H, Li J. Optimization of the receive filter and transmit sequence for active sensing. IEEE Transactions on Signal Processing. 2012;60(4):1730– 1740. Aubry A, De Maio A, Farina A, et al. Knowledge-aided (potentially cognitive) transmit signal and receive filter design in signal-dependent clutter. IEEE Transactions on Aerospace and Electronic Systems. 2013;49(1):93–117. Yu X, Cui G, Kong L, et al. Space-time transmit code and receive filter design for colocated MIMO radar. In: Proceedings of the IEEE Radar Conference (RadarConf); 2016. p. 1–6. Song J, Babu P, Palomar DP. Optimization methods for designing sequences with low autocorrelation sidelobes. IEEE Transactions on Signal Processing. 2015;63(15):3998–4009. De Maio A, De Nicola S, Huang Y, et al. Design of phase codes for radar performance optimization with a similarity constraint. IEEE Transactions on Signal Processing. 2009;57(2):610–621. Bertsekas DP. Nonlinear programming. Journal of the Operational Research Society. 1997;48(3):334–334. Tropp J, Dhillon IS, Heath RW, et al. Designing structured tight frames via an alternating projection method. IEEE Transactions on Information Theory. 2005;51(1):188–209. Scutari G, Palomar DP, Barbarossa S. Cognitive MIMO radio. IEEE Signal Processing Magazine. 2008;25(6):46–59. Scutari G, Palomar DP. MIMO cognitive radio: a game theoretical approach. IEEE Transactions on Signal Processing. 2010;58(2):761–780. Aubry A, De Maio A, Piezzo M, et al. Radar waveform design in a spectrally crowded environment via nonconvex quadratic optimization. IEEE Transactions on Aerospace and Electronic Systems. 2014;50(2):1138– 1152. Mehanna O, Huang K, Gopalakrishnan B, et al. Feasible point pursuit and successive approximation of non-convex QCQPs. IEEE Signal Processing Letters. 2015;22(7):804–808. The MOSEK optimization toolbox for MATLAB manual. Version 7.1 (Revision 63). Denmark: MOSEK ApS. 2015; [Online] Available: http://docs. mosek.com/7.1/toolbox/index.html. Cui G, Yu X, Carotenuto V, et al. Space-time transmit code and receive filter design for colocated MIMO radar. IEEE Transactions on Signal Processing. 2017;65(5):1116–1129.

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Karbasi SM, Aubry A, De Maio A, et al. Robust transmit code and receive filter design for extended targets in clutter. IEEE Transactions on Signal Processing. 2015;63(8):1965–1976. Aubry A, De Maio A, Piezzo M, et al. Cognitive radar waveform design for spectral coexistence in signal-dependent interference. In: Proceedings of the IEEE Radar Conference (RadarConf). IEEE; 2014. p. 474–478. Stoica P, Li J, Xie Y. On probing signal design for MIMO radar. IEEE Transactions on Signal Processing. 2007;55(8):4151–4161. Friedlander B. Waveform design for MIMO radars. IEEE Transactions on Aerospace and Electronic Systems. 2007;43(3):1227–1238. De Maio A, Huang Y, Piezzo M, et al. Design of optimized radar codes with a peak to average power ratio constraint. IEEE Transactions on Signal Processing. 2011;59(6):2683–2697. De Maio A, Huang Y, Piezzo M. A Doppler robust max-min approach to radar code design. IEEE Transactions on Signal Processing. 2010;58(9):4943–4947. Li J, Stoica P. MIMO radar with colocated antennas. IEEE Signal Processing Magazine. 2007;24(5):106–114. Xu H, Blum RS, Wang J, et al. Colocated MIMO radar waveform design for transmit beampattern formation. IEEE Transactions on Aerospace and Electronic Systems. 2015;51(2):1558–1568. Karbasi SM, Aubry A, Carotenuto V, et al. Knowledge-based design of space-time transmit code and receive filter for a multiple-input-multipleoutput radar in signal-dependent interference. IET Radar, Sonar & Navigation. 2015;9(8):1124–1135. Soltanalian M, Tang B, Li J, et al. Joint design of the receive filter and transmit sequence for active sensing. IEEE Signal Processing Letters. 2013;20(5): 423–426. Boyd S, Vandenberghe L. Convex Optimization. Cambridge: Cambridge University Press; 2004. Lipp T, Boyd S. Variations and extension of the convex–concave procedure. Optimization and Engineering. 2016;17(2):263–287. Rockafellar RT. Convex Analysis. Princeton, NJ: Princeton University Press; 1970. Beck A, Teboulle M. Mirror descent and nonlinear projected subgradient methods for convex optimization. Operations Research Letters. 2003;31(3): 167–175. Grant M, Boyd S. CVX: Matlab Software for Disciplined Convex Programming, version 2.1. 2014. Available http://cvxr.com/cvx. Golomb S, Scholtz R. Generalized barker sequences. IEEE Transactions on Information theory. 1965;11(4):533–537.

[42]

[43] [44] [45]

[46] [47] [48]

[49]

[50]

[51] [52] [53] [54]

[55] [56]

Chapter 8

Lagrange programming neural network for radar waveform design Junli Liang1 , Yang Jing1 , Hing Cheung So2 , Chi Sing Leung2 , Jian Li3 , and Alfonso Farina4

In this chapter, Lagrange programming neural network (LPNN) that is a constrained optimization solver is exploited to design radar probing waveforms under constraints due to unit-modulus, spectral requirement and/or ambiguity function (AF). In the LPNN, there are two types of neurons to compute the optimum solution, namely, variable and Lagrangian neurons, which are responsible for finding a minimum point of the cost function as well as the solution at an equilibrium point and leading the dynamic trajectory into the feasible region. The local stability conditions of the dynamic neuron model are also analyzed. Simulation results show that the LPNN-based approach is a competitive alternative for waveform design.

8.1 Introduction Probing waveforms play an important role in active sensing systems [1–8], such as multiple-input multiple-output (MIMO) radar, sonar and communications. The design of these waveforms is crucial to accurately determine useful information of the targets and the propagation medium via emitting them toward an area of interest and analyzing the received reflections from the scene [9]. For example, a land-based surveillance radar system can send electromagnetic sequences to the environment and then estimate the range of a target and its radial speed component by measuring the round-trip time delay and Doppler shift, respectively, from the received signal [10–18]. In probing waveform design, constraints, including constant modulus [8,19–25], correlation properties [9–18], AF characteristics [10–12,23–26] and/or spectral containment restrictions [27–34], should be considered to improve sensing performances. These factors are elaborated as follows. 1

School of Electronics and Information, Northwestern Polytechnical University, Xi’an, China Department of Electrical Engineering, City University of Hong Kong, Hong Kong, China 3 Department of Electrical and Computer Engineering, University of Florida, Gainesville, FL, USA 4 Selex ES (Retired), Roma, Italy 2

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First, in many real-world applications, high-power amplifiers are used to maximize the transmitted power in the saturation region. It is desirable that the probing waveforms are constrained to be constant modulus to avoid distortions by the highpower amplifiers [9]. Without loss of generality, the constant modulus constraint is often assumed as unit-modulus. Next, an emitted probing waveform should maximize the signal-to-noise ratio when complemented by a matched filter at the receiver side while significantly weakening signals from adjacent range bins [9]. Therefore, a well-designed waveform set should possess both low autocorrelation (AC) sidelobes and low cross-correlation (CC) properties in the sense that the correlations among different echoes and the same echo with different time delays are minimized to improve the target-detection performance [10–12,23–25]. Many existing waveform design schemes, such as the cyclic algorithm-new (CAN) [12] and monotonic integrated sidelobe level algorithm (MISL) [23–25], aim at producing aperiodic unimodular probing sequences with low AC sidelobes by minimizing the integrated sidelobe level (ISL) metric. Furthermore, for moving targets, the Doppler effect needs to be considered in the sequence design. The AF has been introduced to describe the response after matched filtering. A good Doppler sensitive sequence can be obtained by minimizing the sidelobes of the corresponding AF [9,26,33–41]. Note that when the Doppler shift is ignored, the AF reduces to the AC function (ACF). Recently, based on the ISL metric, the time–frequency CAN (TFCA) method [9], an extension of CAN, is developed to design unimodular waveforms with low-sidelobe AF in a local time delay and Doppler shift region (see Ch. 6 of [9] for details). The efficient gradient method (EGM) [39] and efficient coordinate descent algorithm [42] are proposed to design sequences with low peak sidelobe levels (PSLs). The radio frequency electromagnetic spectrum has been used for a wide range of applications, including communications, radio and television broadcasting, navigation and radar [18,43]. Nearly all services have a need for greater bandwidth. Due to the increasing demand of defense and civilian applications, the limited radio frequency spectrum becomes more and more crowded [43]. A major impact of spectrum crowding is that many users will be forced to coexist with one another within a finite spectrum allocation, thereby increasing the likelihood of mutual interference. Therefore, radio frequency sensing should coexist with other applications: (i) the sequences in different applications of active sensing system should satisfy spectral containment restrictions and occupy different frequency bands and (ii) even if common frequency bands are used for different applications, the mutual interferences should be kept lower than acceptable levels [43]. Based on the fact that the AC and power spectral density (PSD) form a Fourier transform pair, the periodic CAN (PeCAN) is developed to design unimodular waveforms with flat spectrum and impulse ACF [9,12,13]. To adapt to the crowded radio frequency spectrum, PeCAN is modified as the SHAPE algorithm [27,28] for spectrally constrained waveform design. Similar to the former, the latter introduces auxiliary phase (AP) variables and then determines the sequences and AP variables alternately. The works in [29] and its enhanced version [30] allow designing suitable spectral shapes via time-domain constraints on the radiated energy instead of unit-modulus. Another design technique

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is based on stochastic optimization, and a representative example is particle swarm optimization [31]. In this chapter, we design radar probing waveforms fulfilling unimodular property, spectral containment restriction and good AC at the same time via the LPNN [44–46]. The rest of this chapter is organized as follows. An introduction of LPNN is given in Section 8.2. In Section 8.3, LPNN is exploited to design radar probing waveforms under the constraints of unit-modulus, spectral requirement and/or AF. In Section 8.4, based on the alternating direction method of multipliers (ADMM) framework [47], the hybrid LPNN-ADMM method is developed. Numerical examples are given in Section 8.5. Finally, conclusions are drawn in Section 8.6.

8.2 Basics of LPNN In this section, we present the basic framework of LPNN and then analyze its stability according to [22,26,44–46].

8.2.1 Problem statement Consider the following nonlinear optimization problem: min f (x) s.t. hm (x) = 0, m = 1, . . . , M , x

(8.1)

with x = [x(1), . . . , x(N )]T ∈ RN ×1 , and functions f : RN ×1 → R and hm : RN ×1 → R, m = 1, . . . , M . The Lagrangian function of (8.1) is constructed as L(x, λ) = f (x) + λT h(x),

(8.2)

where λ = [λ(1), . . . , λ(M )]T and h(x) = [h1 (x), . . . , hM (x)]T . Denote (x  , λ ) as a local minimum of (8.2). Then x  can be computed by solving the following equations: x L(x  , λ ) = 0N ×1 , λ L(x  , λ ) = h(x  ) = 0M ×1 ,

(8.3)

where the gradient vector of function (·) with respect to x is x (·) = [(∂(·)/∂x1 ), . . . , (∂(·)/∂xN )]T , and 0m×n represents the m × n zero matrix, and we may drop its dimensions for brevity when they are not required. Note that together with λ  = 0M ×1 , (8.3) is the Karush–Kuhn–Tucker (KKT) condition, which is necessary for a local optimum of problem (8.1). Since the problems are sometimes ill-conditioned and mostly high dimensional and/or nonlinear, the traditional method that solves (8.3) directly does not work well [44]. Zhang and Constantinides [44] have proposed the LPNN to provide an approximate solution to (8.3) based on the neural network techniques and the well-known Lagrange multiplier method [48].

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8.2.2 Lagrange programming neural network [44] Basically, the LPNN is an analog neural network approach for solving constrained optimization problems based on the method of Lagrange multipliers. The mechanism of the LPNN is presented as follows.

8.2.2.1 Principle of LPNN For the problem in (8.1), the Lagrangian function is defined in (8.2), the variable neurons hold the variables in x being optimized, while the Lagrangian neurons hold the Lagrange multiplier vector λ for the M constraints: hm (x) = 0, m = 1, . . . , M . Define the transient behavior of the LPNN as dx dλ = −x L(x, λ), = λ L(x, λ), dt dt

(8.4)

with the variable neurons x, Lagrangian neurons λ and time variable t. The states of the variable neurons x and Lagrangian neurons λ at time (t + 1) are obtained via performing the following integration procedure:  du u(t+1) = u(t) + dt, (8.5) dt where u denotes the neuron. Obviously, one can determine x (t+1) and λ(t+1) according to (8.4) and (8.5) easily. Noting that an integration can be defined as summing infinite rectangles with widths approaching zero, we approximate u(t+1) of (8.5) by accumulating (du/dt) to u(t) , i.e.,  ∞  du  (t+1) (t) =u + ρ , (8.6) u dt u=u(s) s=0 where ρ > 0 is the step size. The transient behavior of the LPNN has been studied via the Runge–Kutta method [44]. The LPNN first sets up a Lagrangian function and defines two kinds of neurons, namely, the variable neurons and the Lagrangian neurons, then it calculates the transient behaviors simultaneously and approximates variables at time t + 1 by accumulating the transient behaviors at time t and uses the neural network to seek for an approximate solution to the original problem.

8.2.2.2 Theoretical aspects Definition 8.1. (x , λ ) is an equilibrium point of (8.4) if it satisfies the constraints (dx/dt)|x=x ,λ=λ = 0N ×1 and (dλ/dt)|x=x ,λ=λ = 0M ×1 [44]. Definition 8.2. x‡ is a regular point if it satisfies the constraints hm (x‡ ) = 0, m = 1, . . . , M , and the gradient vectors x hm |x=x‡ , m = 1, . . . , M , are linearly independent of one another [44].

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225

For the network, we have the following two lemmas: Lemma 8.1. The Lagrangian function L(x, λ0 ) with λ0 being constant decreases, and the Lagrangian function L(x0 , λ) with x0 being constant increases [44]. Proof. See [44]. For stable network, there exists an equilibrium point, namely, (x  , λ ), that satisfies (dx/dt)|x=x ,λ=λ = 0N ×1 and (dλ/dt)|x=x ,λ=λ = 0M ×1 [44]. Lemma 8.2. The equilibrium point (x , λ ) satisfies (8.3) and thus provides a solution to (8.1) [44]. Proof. See [44].

⎡

⎢ The Hessian matrix of (·) is defined as 2xx (·) = ⎢ ⎣  2xx L(x, λ)x=x0 ,λ=λ0 as 2xx L(x 0 , λ0 ).

∂ 2 (·) ∂x1 ∂x1

··· .. .

∂ 2 (·) ∂x1 ∂xN

∂ 2 (·) ∂xN ∂x1

···

∂ 2 (·) ∂xN ∂xN

.. .

.. .

⎤ ⎥ ⎥. Denote ⎦

Theorem 8.1. [44]: For an equilibrium point (x , λ ) of (8.4), assume that x is a regular point of (8.1) and the Hessian matrix 2xx L(x, λ ) of the Lagrangian function L(x, λ ) satisfies 2xx L(x , λ )  0(N +M )×(N +M ) . Then (x , λ ) is an asymptotically stable point of the neural network. Proof. Let 2xx L(x  , λ ) h(x  ) , h(x  ) = [x h1 , . . . , x hM ]. h(x  )T 0M ×M

 G =

(8.7)

The linearized neural dynamics around the equilibrium point are

 dx dt dλ dt

= −G



x − x λ − λ



  2xx L(x  , λ ) h(x  ) x − x  =− . (h(x  ))T 0M ×M λ − λ

(8.8)

 z  = 0(M +N )×1 be the corresponding Denote β as an eigenvalue of G , and let w eigenvector where z ∈ CN ×1 and w ∈ CM ×1 . From the eigenvector definition, we have 

      † † z z = Re β z w Re [z w ]G w w 

†

†



= Re(β)(z2 + w2 ).

(8.9)

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According to the definitions of G  and h(x  ) in (8.7), (z † (h(x  ))T w)† = w† h(x  )z holds. And we obtain      z = Re z † 2xx L(x  , λ )z + z † h(x  )w − w† h(x  )† z Re [z † w† ]G  w   (8.10) = Re z † 2xx L(x  , λ )z . Equating (8.9) and (8.10) yields   Re z † 2xx L(x  , λ )z = Re (β) (z2 + w2 ).

(8.11)

Assuming that G  is positive definite, we have   Re z † 2xx L(x  , λ )z > 0,

z  = 0N ×1 .

(8.12)

For z  = 0N ×1 , (8.11) implies Re (β) > 0. Now we consider z = 0N ×1 :    z z 0N ×1 =β =β . G w w w 

(8.13)

Also, from the definition of G  in (8.7), if z is a zero vector, then we obtain G

  z h(x  )w = . w 0M ×1

(8.14)

From (8.13)–(8.14), z = 0N ×1 implies h(x  )w = 0M ×1 .

(8.15)

Since x  is a regular point of (8.1), i.e., the column vectors of h(x  ) have been shown to be linearly independent of one  another, it follows that w is a zero vector. This z contradicts our assumption of  = 0(M +N )×1 . Consequently, we get Re(β) > 0. w Thus, the real part of each eigenvalue in G  is strictly positive

[44].  This means that each equilibrium point is an asymptotically stable point of

dx dt dλ dt

[49]. The proof is

complete. Theorem 8.1 indicates that the network using (8.4) as its transient behavior will always converge to the equilibrium point (x  , λ ) of L(x, λ) with arbitrary initial point within the attraction domain of (x  , λ ), if x  is regular and 2xx L(x  , λ )  0 holds. Combining Lemma 8.1 and Theorem 8.1, it can be seen that along the trajectory of the neural network, the Lagrangian function is always decreasing with x and increasing with λ, until the network reaches an equilibrium [44].

Lagrange programming neural network for radar waveform design

dx dt

∫

x Variable neurons

dλ dt

∫

λ Lagrangian neurons

227

Figure 8.1 LPNN procedure To improve the convexity, the common practice is to add an augmented term into the objective function. The constrained optimization problem stated in (8.1) becomes min f (x) + x

C0 h(x)2 s.t. h(x) = 0M ×1 , 2

(8.16)

with C0 > 0, the augmented term (C0 /2)h(x)2 , and the Lagrangian function Lc (x, λ) = f (x) + (C0 /2)h(x)2 + λT h(x), which is the augmented Lagrangian function of the objective function of (8.1). Note that for a local minimum point (x  , λ ), we have 2xx Lc (x  , λ ) = 2xx L(x  , λ ) + C0

M 

x hm (x  )x hTm (x  ).

(8.17)

m=1

For any Hermitian matrices A and B, A  (  )0 and A  (  )B mean that A and A − B are positive (semi-)definite. Since x hm (x  )x hTm (x  )  0, 2xx Lc (x  , λ )  0(N +M )×(N +M ) holds if C0 is chosen sufficiently large. This means that the introduction of the augmented term (C0 /2)h(x)2 convexifies the original problem prior to employing the LPNN model. This is theoretically valid for relaxing the condition for stability and is also useful in accelerating the convergence of the neural networks. In particular, a large value of C0 forces the state to approach the feasible region more quickly.

8.2.2.3 Procedure of LPNN The procedure of LPNN is illustrated in Figure 8.1. We utilize the neural dynamics to govern the state transition of the neurons. After the neurons settle down at one of equilibrium points, the solution is obtained by measuring the neuron outputs at this stable equilibrium point. The LPNN dynamics try to fulfill (8.3), namely, the first-order necessary conditions of optimality in the state space. The variable and Lagrangian neurons are responsible for finding a minimum point of the cost function as well as the solution at an equilibrium point, and leading the dynamic trajectory into the feasible region. The dynamic dx/dt is employed to seek a solution of x with the minimum objective value. The dynamic dλ/dt is used to govern the variables

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Radar waveform design based on optimization theory

such that they satisfy the constraints. An equilibrium point of (8.4) satisfies the KKT condition of (8.1). According to Theorem 8.1, LPNN will converge to the equilibrium point (x  , λ ) with arbitrary initial point (x (0) , λ(0) ), and then outputs (x (t) , λ(t) ) upon convergence.

8.3 LPNN for waveform design with spectral constraints To adapt to the increasingly crowded radio frequency spectrum and coexist with other applications rather than acting as interferences to each other, we need to control the spectral shape of waveform x = [x1 , . . . , xN ]T ∈ CN ×1 . The LPNN approach is developed to generate unimodular sequences with spectral constraints.

8.3.1 Problem statement Taking the discrete Fourier transform (DFT) of x yields X (n) =

N −1 

x(q)e−j(2π /N )qn , n = 0, . . . , N − 1,

(8.18)

q=0

where {X (n)} are the N uniformly spaced frequency samples of the spectrum and we are going to specify their magnitudes. In the following, we first study the design problem with a flat spectrum requirement, followed by its generalization to an arbitrary spectral shape.

8.3.1.1 Flat spectrum waveform design According to Parseval’s theorem and the requirement of the unit-modulus, we have  −1 −1 (1/N ) Nn=0 |X (n)|2 = Nn=0 |x(n)|2 = N . As PSD and AC form a Fourier transform pair, a constant PSD for all frequencies, or known as flat spectrum, corresponds to an impulse ACF [9,12,13]. That is, the elements in x are uncorrelated or E[x(n)x∗ (m)] = 0 for m  = n. In this case, the waveform design can be formulated as the following optimization problem: min x

N −1 

(|X (n)|2 − N )2 s.t. |x(n)| = 1, n = 0, . . . , N − 1.

(8.19)

n=0

Define the DFT vector f n ∈ C1×N whose kth element is fn (k) = e−j(2π/N )kn ,

n = 0, . . . , N − 1,

k = 0, . . . , N − 1.

(8.20)

Employing (8.20), (8.18) can be written as X (n) = f n x.

(8.21)

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229

Furthermore, we denote en ∈ R1×N as a selection vector with its nth element being 1 and 0 otherwise. With the use of (8.21) and en , (8.19) can be represented in a vector form as min

N −1 

x

(x † f †n f n x − N )2 s.t. x † e†n en x = 1, n = 0, . . . , N − 1.

(8.22)

n=0

The solution to (8.22) will produce x with unit-modulus and flat spectrum. It is worth mentioning that the PeCAN approach [9,12,13,38] solves (8.22) via transforming the objective function and introducing auxiliary variables.

8.3.1.2 Generalized spectrally constrained waveform design Apart from flat spectrum, there is a need for the magnitude spectrum of x to satisfy other arbitrary shapes. Let Sn be the desired power spectrum point, up to a scalar, on the frequency grid 2πn/N for n = 0, . . . , N − 1. The waveform design problem can be easily formulated by modifying (8.22) as min

N −1 

x,α

(x † f †n f n x − αSn )2 s.t. x † e†n en x = 1, n = 0, . . . , N − 1,

(8.23)

n=0

where α > 0 is a real-valued scalar. Similar to the strategy for emphasizing/deemphasizing different measurements in the weighted least squares approach, we further generalize (8.23) as N −1 

min x,α

Wn (x † f †n f n x − αSn )2 s.t. x † e†n en x = 1, n = 0, . . . , N − 1, (8.24)

n=0

where Wn ≥ 0, n = 0, . . . , N − 1, denotes the weight at each frequency grid to tradeoff between the passband and the stopband properties. Note that when α is fixed to 1, together with Sn = N and Wn = 1 for n = 0, . . . , N − 1, (8.24) reduces to (8.22). This means that the flat spectrum waveform design problem is just a special case of (8.24). As a result, we only present the algorithm for solving (8.24) in next subsection.

8.3.2 Algorithm development In this subsection, the solver based on the LPNN is first developed. Then, we analyze the local stability conditions of the dynamic neuron model with variable neurons and Lagrangian neurons. Equation (8.24) is rewritten as real-valued form: min x,α ˆ

N −1 

T

Wn (xˆ T F Tn F n xˆ − αSn )2 s.t. xˆ T Eˆ n Eˆ n xˆ = 1, n = 0, . . . , N − 1,

(8.25)

n=0

where F n =

T T f¯ n −f˜ n

T

, xˆ = [x¯ x˜ ] , Eˆ n = T

T T f˜ n f¯ n Im(x), f¯ n = Re( f n ) and f˜ n = Im( f n ).

T T

en

0N ×N

 0N ×N , x¯ = Re(x), x˜ = en

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Radar waveform design based on optimization theory

Note that although the term xˆ T F Tn F n xˆ in (8.25) is a quadratic function of x, ˆ its square makes the objective function to be highly nonlinear. Moreover, the unimodular T constraints xˆ T Eˆ n Eˆ n xˆ = 1 for n = 0, . . . , N − 1, are not convex. According to the augmented Lagrangian, we introduce an augmented term (C0 /2) 2 N −1  T T ˆ Eˆ n Eˆ n xˆ − 1 , C0 > 0 into (8.25), yielding n=0 x min x,α ˆ

N −1 

Wn (xˆ

n=0

T

F Tn F n xˆ

N −1 2 C0   T ˆ T ˆ − αSn ) + xˆ E n E n xˆ − 1 2 n=0 2

T

s.t. xˆ T Eˆ n Eˆ n xˆ = 1, n = 0, . . . , N − 1,

(8.26)

where the variable neurons hold the variables [xˆ T , α] being optimized, while the Lagrangian neurons hold the Lagrange multiplier vector λ = [λ0 , . . . , λN −1 ]T for the N constraints corresponding to the unit-modulus property. If C0 is chosen sufficiently large, the local convexity condition holds under fairly mild conditions [50]. This is theoretically valid for relaxing the condition for stability and is also useful in accelerating the convergence of the neural networks. Especially, a large value of C0 forces the state to approach the feasible region more quickly. Based on (8.26), the Lagrangian function is set up as   L x, ˆ α, λ =

N −1 

Wn (xˆ T F Tn F n xˆ − αSn )2 +

n=0

+

N −1 

N −1 2 C0   T ˆ T ˆ xˆ E n E n xˆ − 1 2 n=0

T λn (xˆ T Eˆ n Eˆ n xˆ − 1) .

(8.27)

n=0

Applying (8.3), the transient behaviors of the neurons are computed as   ˆ α, λ ∂L x, d xˆ =− dt ∂ xˆ N −1  4Wn (xˆ T F Tn F n xˆ − αSn )F Tn F n xˆ =−

(8.28)

n=0

−

N −1 

N −1    T T T 2C0 xˆ T Eˆ n Eˆ n xˆ − 1 Eˆ n Eˆ n xˆ − 2λn Eˆ n Eˆ n x, ˆ

n=0

n=0

  N −1  ∂L x, ˆ α, λ dα 2Wn Sn (xˆ T F Tn F n xˆ − αSn ), =− = dt ∂α n=0   ∂L x, ˆ α, λ dλn T = = xˆ T Eˆ n Eˆ n xˆ − 1, n = 0, . . . , N − 1. dt ∂ λn Then we update x, ˆ α and λ according to (8.6).

(8.29)

(8.30)

Lagrange programming neural network for radar waveform design

231

Note that −∂L /∂ xˆ and −∂L /∂ α require F n x, ˆ and we apply fast Fourier transform (FFT) of x to simplify the realization. It is worth pointing out that in the analog implementation, we use capacitors, amplifiers and resistances to realize the integration processes [51–53].

8.3.3 LPNN stability analysis An important theoretical aspect of the LPNN is to ensure that all equilibrium points correspond to a stable system [44]. Denote an equilibrium point by (xˆ  , α  , λ ). According to Theorem 8.1, two conditions are needed to ensure local stability: (i) (xˆ  , α  ) is a regular point of (8.26) and (ii) A , the Hessian matrix of the Lagrangian function with respect to (xˆ  , α  ), is positive definite.

8.3.3.1 Regular point T From (8.26), the gradient vectors of the constraints xˆ T Eˆ n Eˆ n xˆ − 1 = 0, n =  0, . . . , N − 1 at the equilibrium point (xˆ , α  ) are [01×n Re x (n) 01×(N −1) Im x (n) 01×(N −n) ]T , n = 0, . . . , N − 1. We employ these gradient vectors to construct a (2N + 1) × N matrix where there are two N × N diagonal sub-matrices with nonzero elements Re(x (n)) and/or Im(x (n)). This matrix has the form of

       ⎤ diag Re x (0) , Re x (1) , . . . , Re x (N − 1) ⎢      ⎥     ⎥ V =⎢ ⎣diag Im x (0) , Im x (1) , . . . , Im x (N − 1) ⎦ 01×N ⎡

(8.31)

At the equilibrium point, we have from (8.30)    2    2 Re x (n) + Im x (n) = 1,

(8.32)

and thus each column of (8.31) is unitary. In addition, since the nonzero element positions of these columns are different from each other, this implies that the gradient vectors are linearly independent of one another. Thus, (xˆ  , α  ) is a regular point of (8.26).

8.3.3.2 Positive definiteness of A

We now show that for a sufficiently large value of C0 , A is positive definite under fairly mild conditions [50]. Note that ⎡ A =

∂2L ˆ xˆ T ⎣ ∂ x∂

∂2L ∂ α∂ xˆ T

∂2L ∂ x∂ ˆ α ∂2L ∂ α2

⎤ ⎦

, 

x= ˆ xˆ ,α=α  ,λ=λ

(8.33)

232

Radar waveform design based on optimization theory

where  ∂ 2 L  ∂ x∂ ˆ xˆ T 



x= ˆ xˆ ,α=α  ,λ=λ

=

N −1 

8Wn F Tn F n xˆ  (xˆ  )T F Tn F n

n=0

+

N −1 

4Wn ((xˆ  )T F Tn F n xˆ  − α  Sn )F Tn F n

n=0

+

N −1 

T T 4C0 Eˆ n Eˆ n xˆ  (xˆ  )T Eˆ n Eˆ n

n=0

+

N −1 

N −1    T T T 2C0 (xˆ  )T Eˆ n Eˆ n xˆ  − 1 Eˆ n Eˆ n + 2λn Eˆ n Eˆ n ,

n=0

 ∂ 2 L  ∂ x∂ ˆ α  ∂ 2 L  ∂ α2 

 x= ˆ xˆ  ,α=α  ,λ=λ



x= ˆ xˆ ,α=α  ,λ=λ

= =

 ∂ 2 L  ∂ α∂ xˆ 

N −1 

n=0

(8.34)

T x= ˆ xˆ  ,α=α  ,λ=λ

=

N −1 

−4Wn Sn F Tn F n xˆ  , (8.35)

n=0

8Wn Sn2 ,

(8.36)

n=0

T    ∂ 2 L  ∂ 2 L  = ∂ x∂ ˆ λT x= ∂ λ∂ xˆ T x= ˆ xˆ  ,α=α  ,λ=λ ˆ xˆ  ,α=α  ,λ=λ  T  T = 2 Eˆ 0 Eˆ 0 xˆ  · · · Eˆ N −1 Eˆ N −1 xˆ  ,  T   ∂ 2 L  ∂ 2 L  = = 01×N . ∂ λ∂ α x= ∂ α∂ λT x= ˆ xˆ  ,α=α  ,λ=λ ˆ xˆ  ,α=α  ,λ=λ

(8.37) (8.38)

 T T  T ˆ ) Eˆ n Eˆ n xˆ  − 1 Eˆ n Eˆ n is equal to zero, since at an n=0 2C0 (x  −1 T T T equilibrium point (xˆ  )T Eˆ n Eˆ n xˆ  = 1 for all n. Also, 4C0 Nn=0 Eˆ n Eˆ n xˆ  (xˆ  )T Eˆ n Eˆ n is  −1 T T positive definite or positive semi-definite in (8.34). Let θ = 4 Nn=0 Eˆ n Eˆ n xˆ  (xˆ  )T Eˆ n Eˆ n .

In (8.34), the term

N −1

We have

⎡ ⎤ xˆ   (xˆ  )T α  (λ )T A ⎣α  ⎦ = C0 (xˆ  )T θ xˆ  + ξ , λ

(8.39)

where ξ is a scalar, which is independent of C0 . Hence, if xˆ  is not orthogonal to the space spanned by θ , then (xˆ  )T θ xˆ  is positive when C0 is chosen large enough [50]. Thus, A is positive definite for a sufficiently large C0 . Then, according to Sections 8.3.3.1 and 8.3.3.2, and Theorem 8.1, the equilibrium point is an asymptotically stable point of (8.28)–(8.30).

Lagrange programming neural network for radar waveform design

233

8.4 LPNN for designing waveform with low PSL Different from Section 8.3 that solves the problem with equality constraints, this section deals with the problem of inequality constraints via providing the hybrid LPNN-ADMM method. We consider generating unimodular sequences with low PSL in local time delay and Doppler shift regions to deal with moving targets.

8.4.1 Problem statement For sequence x = [x(1), . . . , x(N )]T , its ACF is defined as rk = N a discrete ∗ n=k+1 x(n)x (n − k), k ∈ , whereas the discrete AF is given by [9] rl,k =

N 

x(n)x∗ (n − k)e−j2π l(n−k)/N = x †Al,k x,

(8.40)

n=k+1

where = {−N + 1, . . . , N − 1}, l, k ∈ denote the Doppler frequency shift bin and time delay bin indices, and Al,k = J k Dl with Dl = diag(d l ), d l = [1, ej2π l/N , . . . ,  0k×k 0 , J −k = J Tk for k ∈ {0, . . . , N − 1}, and the ej2π l(N −1)/N ]† , J k = k×(N −k) I N −k 0(N −k)×k identity matrix I m ∈ Rm×m . Obviously, rk = r0,k holds. Note that Al,k , l, k ∈ are generally not Hermitian. With (8.40), the problem of minimizing PSL is formulated as the following optimization problem with fourth-order polynomial constraints: min η x,η

s.t. ωl,k |x †Al,k x|2 ≤ η, l, k ∈ , |x(n)| = 1, n = 1, . . . , N ,

(8.41)

where ωl,k denotes a selection function. If the (l, k)th grid lies in the sidelobe region of interest , we set ωl,k = 1; otherwise, ωl,k = 0. In terms of the phase angle θn of x(n) that satisfies the constraint |x(n)| = 1, we have | cos (θn ) + j sin (θn )| = 1, n = 1, . . . , N . Thus, we replace the constraint |x(n)| = 1 by h(x, ¯ θ ) = x¯ − φ = 02N ×1 ,

(8.42) 

 Re(x) cos (θ ) , φ= and θ = [θ(1) · · · θ(N )]T . Im(x) sin (θ ) Then, (8.41) is rewritten in the real-valued form as where real-valued vectors x¯ =

min η x,η,θ ¯

¯ θ ) = 02N ×1 , ¯ 2 + (x¯ T B¯ l,k x) ¯ 2 ) ≤ η, l, k ∈ , h(x, s.t. ωl,k ((x¯ T A¯ l,k x)

(8.43)

234

Radar waveform design based on optimization theory   IN ¯ Re(Al,k ) −Im(Al,k ) 0 where A¯ l,k = A . Introducing realand B¯ l,k = N ×N −I N 0N ×N l,k Im(Al,k ) Re(Al,k ) valued variables fl,k and gl,k with the forms of fl,k = x¯ T A¯ l,k x¯ and gl,k = x¯ T B¯ l,k x, ¯ for l, k ∈ , (8.43) can be expressed as min

x,θ,η,f ¯ ,g

η

2 s.t. ωl,k ( fl,k2 + gl,k ) ≤ η,

fl,k = x¯ T A¯ l,k x, ¯

gl,k = x¯ T B¯ l,k x, ¯

l, k ∈ ,

h(x, ¯ θ ) = 02N ×1 ,

(8.44)

where f and g denote the vectors with fl,k and gl,k as their (l + N )(2N − 1) + (k + N )th elements, respectively. Ignoring the constraints with respect to variables θ and η of problem (8.44), we have ¯ min η s.t. fl,k = x¯ T A¯ l,k x, x,f ¯ ,g

gl,k = x¯ T B¯ l,k x, ¯

l, k ∈ .

(8.45)

We construct the augmented Lagrangian function of (8.45) as    T ¯   T ¯ 2   x¯ Al,k x¯ x¯ Al,k x¯  C1  ωl,k α Tl,k yl,k − T ωl,k yl,k − T , + L =η+ 2 l,k x¯ B¯ l,k x¯ x¯ B¯ l,k x¯  l,k  fl,k αf ,l,k where yl,k = , α l,k = , with the real-valued variables αf ,l,k and αg,l,k gl,k αg,l,k being the Lagrange multipliers for the constraints fl,k = x¯ T A¯ l,k x¯ and gl,k = x¯ T B¯ l,k x, ¯ respectively, l, k ∈ , and C1 > 0 is the step size in the augmented Lagrangian function [47]. Note that if C1 is large enough, fl,k and gl,k will be close to x¯ T A¯ l,k x¯ and x¯ T B¯ l,k x, ¯ respectively. Then (8.44) can be transformed as 

2 ¯ θ ) = 02N ×1 . min L s.t. ωl,k ( fl,k2 + gl,k ) ≤ η, l, k ∈ , h(x,

x,θ,η,f ¯ ,g

(8.46)

We will design unimodular sequences with low PSL of local AF via solving (8.46). Problem (8.41) includes both equality and inequality constrains. Notice that the constraints h(x, ¯ θ ) = 02N ×1 of x¯ are not related to fl,k and gl,k , and the inequality 2 constraints ωl,k ( fl,k2 + gl,k ) ≤ η are independent of the variables x¯ and θ. Therefore, we apply the alternating optimization idea and the decomposition–coordination procedure in the ADMM framework [47] to solve (8.46) via the following iterative steps.

8.4.2 Algorithm description Step 1: Update x¯ and θ via LPNN update step (i) This step determines x¯ (i+1) and θ (i+1) with the obtained η(i) , f (i) , g (i) , α f and α (i) g in the ith iteration, where α f and α g denote the matrices with αf ,l,k and αg,l,k as their (l, k)th elements, respectively. After removing the constant terms dependent on

Lagrange programming neural network for radar waveform design

235

¯ with the fourth-order η(i) , f (i) , g (i) , α f and α (i) g , (8.46) becomes a problem involving x polynomial-like objective function and unimodular constraints h(x, ¯ θ ) = 02N ×1 , i.e., (i)

(i) ¯ θ | η(i) , f (i) , g (i) , α f , α (i) ¯ θ) = 02N ×1 , min L (x, g ) s.t. h(x, x,θ ¯

(8.47)

where   (i) L x, ¯ θ | η(i) , f (i) , g (i) , α f , α (i) g =η + (i)



α (i) ωl,k (α l,k )

l,k

T





(i) yl,k

x¯ T A¯ l,k x¯ − T x¯ B¯ l,k x¯



  T ¯ 2  (i) x¯ Al,k x¯  C1    ωl,k yl,k − T . + 2 l,k x¯ B¯ l,k x¯ 

Here, we apply the LPNN to solve such a subproblem that is called “LPNN update step.” In this step, we first construct another Lagrangian function in the LPNN framework (see [44] for details): T ¯ θ | η(i) , f (i) , g (i) , α f , α (i) Lx¯0 = L (x, g )+β h+ (i)

C0 T h h, 2

(8.48)

where β = [β(1), . . . , β(2N )]T contains the Lagrangian neurons, x¯ and θ are variable neurons and C0 is the penalty parameter for the penalty term hT h [44]. It is pointed out that the penalty parameter C0 is used to improve the convexity and convergence performance of the optimization problem [44]. From Appendix A.1, it is seen that the convexity of (8.48) is not guaranteed. Next, we will construct a convex function to replace (8.48). Note that once the constraints h(x, ¯ θ ) = 02N ×1 are satisfied, x¯ 2 (n) − cos2 (θ(n)) = 0, x¯ 2 (N + n) − sin2 (θ(n)) = 0, n = 1, . . . , N and therefore Lx¯0 = Lx¯ holds, where the convex function Lx¯ is T ¯ θ | η(i) , f (i) , g (i) , α f , α (i) Lx¯ = L (x, g )+β h+ (i)

+

C0 T h h 2

N  qC0   2 (¯x (n) − cos2 (θ (n))) + (¯x2 (N + n) − sin2 (θ(n))) , (8.49) 2 n=1

with a large enough C0 (see details about the convexity property of (8.49) in Appendix A.3.3). Therefore, with a given parameter q > 0, we replace (8.48) by (8.49). Thus, we construct the LPNN by defining the transient behaviors d x/dt, ¯ dθ/dt and dβ/dt of the variable neurons x, ¯ θ and Lagrangian neurons β by applying (8.3): d x¯ = −∇x¯ Lx¯ = −(Z 1 + C1 Z 2 + qC0 )x¯ − (β + C0 h), dt

(8.50)

  dθ = −∇θ Lx¯ = D2 + C0 D4 −(D1 + C0 D3 ) φ, (8.51) dt with the nth element (βN +n + C0 x¯ (N + n)) cos θ (n) − (βn + C0 x¯ (n)) sin θ (n), n = 1, . . . , N , dβ = ∇β Lx¯ = h, dt

(8.52)

236

Radar waveform design based on optimization theory

where Z1 = −



ωl,k [(C1 fl,k + αf ,l,k )(A¯ l,k + A¯ l,k ) + (C1 gl,k + αg,l,k )(B¯ l,k + B¯ l,k )], T

T

l,k

Z2 =



ωl,k [x¯ T A¯ l,k x( ¯ A¯ l,k + A¯ l,k ) + x¯ T B¯ l,k x( ¯ B¯ l,k + B¯ l,k )], T

T

l,k

D1 = diag(β(1), . . . , β(N )), D2 = diag(β(N + 1), . . . , β(2N )), D3 = diag(¯x(1), . . . , x¯ (N )), D4 = diag(¯x(N + 1), . . . , x¯ (2N )).

(8.53)

Then the states of the variable neurons x, ¯ θ and Lagrangian neurons β are obtained according to (8.6). Step 2: Update η, f and g (i) In this step, we update η(i+1) , f (i+1) and g (i+1) with given x¯ (i+1) , α f and α (i) g via solving the following optimization problem: 2 min L (η, f , g | x¯ (i+1) , α f , α (i) g ) s.t. ωl,k y l,k  ≤ η, (i)

(8.54)

η,f ,g

where L (η, f , g | x¯

(i+1)

(i) , α f , α (i) g )

= η+



 T α (i) ωl,k (α l,k )

l,k

+ C21

 l,k

   ωl,k  yl,k 

 (x¯ (i+1) )T A¯ l,k x¯ (i+1) yl,k − (x¯ (i+1) )T B¯ l,k x¯ (i+1)  2

(x¯ (i+1) )T A¯ l,k x¯ (i+1)  −  . (x¯ (i+1) )T B¯ l,k x¯ (i+1) 

It is clear that (8.54) is a quadratic convex problem of yl,k with inequality constraints. Removing the constant term, (8.54) reduces to min Ly s.t. ωl,k yl,k 2 ≤ η, l, k ∈ , η ∈ [η0 , ηP ],

(8.55)

η,f ,g

where the objective function is Ly = η + (C1 /2)



ωl,k yl,k − y˜ l,k 2 , and the ele-

l,k

ments f˜l,k and g˜ l,k of the vector y˜ l,k = [f˜l,k g˜ l,k ]T are given by (i)

f˜l,k = (x¯ (i+1) )T A¯ l,k x¯ (i+1) −

αf ,l,k C1

(i)

, g˜ l,k = (x¯ (i+1) )T B¯ l,k x¯ (i+1) −

αg,l,k C1

.

Apparently, η is coupled with fl,k , gl,k due to the constraints ωl,k yl,k 2 ≤ η, l, k ∈

(i+1)  f (i+1) . Once η is provided, the optimal yl,k = l,k(i+1) is given by gl,k ⎧ ⎨y˜ l,k , ωl,k ˜yl,k 2 ≤ η, (i+1) √ (8.56) yl,k = ⎩ η y˜ l,k , ωl,k ˜yl,k 2 > η. ˜y  l,k

Lagrange programming neural network for radar waveform design

237

Substituting (8.56) into (8.55), we have an optimization problem of η:    C1  C1  √ ωl,k Sl,k η − C1 ωl,k Sl,k ˜yl,k  η + ωl,k Sl,k ˜yl,k 2 , min 1 + η 2 l,k 2 l,k l,k (8.57) 

0, ωl,k ˜yl,k 2 ≤ η . 1, ωl,k ˜yl,k 2 > η According to the constraints of (8.41), there exists a specified range [η0 , ηP ] ⊆ [0, N 2 ] for η, and thus we divide [η0 , ηP ] into P subregions. Assume the ascend(i) ing order of the obtained yl,k 2 as {η0 , . . . , ηP }. Then, when η ∈ [ηp−1 , ηp ], p ∈ √ {1, . . . , P}, (8.57) can be expressed as a quadratic function of η: √ (8.58) min√η Lη ( η),  √ √ √ ap = 1 + (C1 /2) l,k ωl,k Sl,k , bp = where Lη ( η) = ap ( η)2 + bp η+ cp , −C1 l,k ωl,k Sl,k ˜yl,k , cp = (C1 /2) l,k ωl,k Sl,k ˜yl,k 2 . The solution to (8.58) is given by ⎧ −bp −b ( )2 , ( 2app )2 ∈ [ηp−1 , ηp ], ⎪ ⎪ ⎨ 2ap −b ( 2app )2 < ηp−1 , (8.59) η˜ p = ηp−1 , ⎪ ⎪ −bp 2 ⎩η , ( ) >η , where Sl,k =

p

p

2ap

and the objective function of (8.55) can be computed by substituting η˜ p into the objective function of (8.58) (see Appendix A.2). Thus, the optimal value of η(i+1) is obtained by comparing the local minimum: ! (8.60) η(i+1) = arg min{Lη ( η˜ p ), p = 1, . . . , P}. η˜ p

(i+1)

Finally, substituting η(i+1) into (8.56) yields fl,k

(i+1)

and gl,k .

Step 3: Updating Lagrange multipliers (i) With given x¯ (i+1) , θ (i+1) , η(i+1) , f (i+1) , g (i+1) , α f and α (i) g , the Lagrange multipliers (i+1) (i+1) αf ,l,k and αg,l,k are updated as [47]

(i+1)

αf ,l,k

(i+1)

αg,l,k



=

(i)

αf ,l,k (i)

αg,l,k



 + C1

(i+1) yl,k

(x¯ (i+1) )T A¯ l,k x¯ (i+1) − (x¯ (i+1) )T B¯ l,k x¯ (i+1)

Steps 1–3 are repeated until convergence.

8.4.3 LPNN stability analysis This subsection will study the stability of the LPNN.

 .

(8.61)

238

Radar waveform design based on optimization theory

8.4.3.1 Regular point The gradients of h(x, ¯ θ) = [h1 , . . . , h2N ]T with respect to the arbitrary point (x, ¯ θ ) are ⎡

en

⎤

⎥ ⎢0 ⎢ (n−1)×1 ⎥ ∇hn = ⎢ ⎥ ∈ R3N ×1 , n = 1, . . . , N ; ⎣sin (θ (n))⎦ 0(N −n)×1 ⎤ ⎡ en ⎥ ⎢ 0 (n−N −1)×1 ⎥ ⎢ ∇hn = ⎢ ⎥ ∈ R3N ×1 , n = N + 1, . . . , 2N , ⎣− cos (θ (n − N ))⎦

(8.62)

(8.63)

0(2N −n)×1 where the elements of vector en ∈ R2N ×1 are 0 except the nth one which is 1. Obviously, the first 2N row vectors of ∇h(x, ¯ θ ) form an identity matrix, and thus ∇hn , n = 1, . . . , 2N are linearly independent. Therefore, the point {x, ¯ θ} is a regular point of the constraints h(x, ¯ θ ) = 02N ×1 .

8.4.3.2 Positive definiteness of Hessian matrix According to (8.50) and (8.51), we have 1. 2.

∂ 2 Lx¯ ∂θ 2 (n)

= (β(n) + C0 x¯ (n)) cos θ (n) + (βN +n + C0 x¯ (N + n)) sin θn ,

2L x¯ C0 sin θ(n) and ∂ x¯ (N∂+n)∂θ(n) = −C0 cos θ(n) for n = 1, . . . , N ; 2 ∂ Lx¯ ∂ 2 Lx¯ ∂ 2 Lx¯ = 0 and ∂ x¯ (n1 )∂θ(n2 ) = ∂ x¯ (N +n = 0 for n1 , n2 = ∂θ (n1 )∂θ (n2 ) 1 )∂θ(n2 ) n2 .

∂ 2 Lx¯ ∂ x¯ (n)∂θ (n)

=

1, . . . , N and n1  =

The Hessian matrix of (8.49) is

H=

2 ∇x¯2x¯ Lx¯ ∇xθ ¯ Lx¯

∇θ2x¯ Lx¯ ∇θ2θ Lx¯

 ,

(8.64)

where ∇x¯2x¯ Lx¯ = Z 0 + qC0 I 2N , Z 0 = Z 1 + C1 Z 2 + C1 Z 3 + C0 I 2N ,  T T T T Z3 = ωl,k [(A¯ l,k + A¯ l,k )x¯ x¯ T (A¯ l,k + A¯ l,k ) + (B¯ l,k + B¯ l,k )x¯ x¯ T (B¯ l,k + B¯ l,k )], l,k

∇θ2θ Lx¯ = (D1 + C0 D3 )D6 + (D2 + C0 D4 )D5 , ⎡ ∂ 2 Lx¯ ∂ 2 Lx¯ ∂ 2 Lx¯ · · · ∂ x¯ (1)∂θ(N ∂ x¯ (1)∂θ(1) ∂ x¯ (1)∂θ(2) ) ⎢ .. .. . . 2 ⎢ .. .. ∇xθ . . ¯ Lx¯ = ⎣ ∂ 2 Lx¯ ∂ 2 Lx¯ ∂ x¯ (2N )∂θ(1) ∂ x¯ (2N )∂θ(2)

···

∂ 2 Lx¯ ∂ x¯ (2N )∂θ(N )

⎤ ⎥ ⎥ ⎦

Lagrange programming neural network for radar waveform design  diag (sin (θ )) D5 ∈ R2N ×N , = C0 −diag (cos (θ )) −D6 T  2 , ∇θ2x¯ Lx¯ = ∇xθ ¯ Lx¯

239



= C0

(8.65)

D5 = diag (sin (θ )) , D6 = diag (cos (θ)), and Z 1 , Z 2 , D1 , D2 , D3 and D4 have been defined in (8.53). ∇θ2θ Lx¯ is a diagonal matrix with the nth diagonal element (β(n) + C0 x¯ (n)) cos θ (n) + (β(N + n) + C0 x¯ (N + n)) sin θn . Let the Hessian matrix of (8.49) at the equilibrium point (x¯  , θ  , β  ) be H  = 2 H |x= ¯ x¯  ,θ =θ  ,β=β  . Define W 0 = ∇x¯ x¯ Lx¯ |x= ¯ x¯  ,θ =θ  ,β=β  . Denote W n as the (2N + n)thorder principal minor determinant of H  , denote U n as the nth-order principal minor determinant of ∇θ2θ Lx¯ |x= ¯ x¯  ,θ =θ  ,β=β  , and denote V n as the matrix consisting of the 2 first n column vectors of ∇xθ ¯ x¯  ,θ =θ  ,β=β  , n = 1, . . . , N . Then we have ¯ Lx¯ |x= Wn =



W0 Vn V Tn U n

=

W0

02N ×n

V Tn U n − V Tn W −1 0 Vn



 I 2N W −1 0 Vn , 0n×2N In

and |W n | = |W 0 ||U n − V Tn W −1 0 V n |, n = 1, . . . , N . The equilibrium point satisfies ∇β Lx¯ |x= ¯ x¯  ,θ =θ  ,β=β  = 02N ×1 . Then accord= 0 ¯ ing to (8.52), we have h|x=    2N ×1 , namely, x(n)| ¯ x¯ ,θ =θ ,β=β x= ¯ x¯  ,θ =θ  ,β=β  = cos (θ(n))|x= and x(N ¯ + n)| = sin (θ (n))|        ¯ x¯ ,θ =θ ,β=β x= ¯ x¯ ,θ =θ ,β=β x= ¯ x¯ ,θ =θ  ,β=β  , n = 1, . . . , N . Therefore, the ith diagonal element of the diagonal matrix U n is (βn cos (θ  (n)) + β  (N + n) sin (θ  (n))) + C0 , i = 1, . . . , n, n = 1, . . . , N . If C0 → ∞, both W 0 → (1 + q)C0 I 2N and U n → C0 I n are positive definite matrices according to (8.54) and (8.65). Note that we have V Tn V n = C02 I n , C0 > 0 and q > 0. Then for arbitrary n = 1, . . . , N ,   I 2N V n  = qn (1 + q)2N −n C02N +n > 0 |W n | → |(1 + q)C0 I 2N |C0 I n − V Tn (1 + q)C0 holds, when C0 → ∞. Therefore, H  is positive definite. Then if C0 is sufficiently large, the equilibrium point (x¯  , θ  , β  ) is an asymptotically stable point of (8.49) according to Sections 8.3.3.1 and 8.3.3.2 and Theorem 8.1. Thus, the LPNN can provide local solutions to (8.49) and the proposed method provides a local optimal solution to (8.44). The adaptive selection scheme of C0 is shown in Appendix A.3. The waveform can be designed off-line in practical applications. Similar to [9,12,21], the solutions with small objective values of (8.41) are utilized in practice.

8.4.4 Summary of proposed algorithm The LPNN update step for solving (8.47) is summarized in Algorithm 1. The proposed method for solving (8.46) is summarized in Algorithm 2. Denote S and M as the iteration number of Algorithm 1 and the number of grids in the region of interest (i.e., M = l,k ωl,k ), respectively. Then the computational complexity of updating x in Algorithm 2 is O(N 2 MS). The updates of η(i+1) , f (i+1) and

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Radar waveform design based on optimization theory

Algorithm 1: LPNN update step Initialization: Known parameters ω, C1 , x¯ (i) , f (i)√, g (i) , α (i) and q. 2−q+

q2 +4

. Obtain m according to m ≥ 2q [t] (i) t = 0, x¯ = x¯ , and initialization of θ [t] and β [t] .

While: Obtain C0 according to (A.11)-(A.13); Compute ddtx¯ = −(Z 1 + C1 Z 2 + qC0 )x¯ [t] − (β [t] + C0 h) using (8.50); Compute dθ = [D2 + C0 D4 − (D1 + C0 D3 )] φ [t] using (8.51); dt dβ Compute dt = h using (8.52); and dβ to x¯ [t] , θ [t] Compute x¯ [t+1] , θ [t+1] and β [t+1] by accumulating ddtx¯ , dθ dt dt [t] and β respectively; t = t + 1. End while until convergence. Output: x¯ (i+1) = x¯ [t] .

Algorithm 2: Hybrid LPNN-ADMM method Initialization: N , i = 0, ω, C1 and the initial values for x¯ (0) , f (0) , g (0) , α (0) . While: Update x¯ (i+1) using Algorithm 1; Obtain η(i+1) , f (i+1) and g (i+1) using (8.54); Update α (i+1) according to (8.61); i = i + 1. End while until convergence. Obtain waveform: x(n) = x¯ (i) (n) + j x¯ (i) (N + n), n = 1, . . . , N . Output: x.

g (i+1) have complexities of O(M ), O(N 2 M ) and O(N 2 M ), respectively. Obviously, the complexity O(N 2 MS) is involved in every iteration of the proposed method.

8.5 Numerical examples To evaluate the performance of the proposed method, we present several sequence design examples in this section.

Lagrange programming neural network for radar waveform design

241

30

Spectrum (dB)

25 20 15 10 5 0

0

0.4 0.6 Frequency

0.2

0.8

1

Figure 8.2 Spectrum of random phase sequence in Experiment 1

Squared modulus of ACF for RP (dB)

0 −5 −10 −15 −20 −25 −30 −35 −40

−100

−50

0 Delay bin

50

100

Figure 8.3 ACF of random phase sequence in Experiment 1

8.5.1 Experiment 1: Flat spectrum waveform design As LPNN is an analog solution, we employ discretization in our digital implementation: ⎧ ⎪ x(Iter ˆ + 1) = x(Iter) ˆ + ρ ddtxˆ ⎪ ⎨ , (8.66) α(Iter + 1) = α(Iter) + ρ dα dt ⎪ ⎪ ⎩ λ(Iter + 1) = λ(Iter) + ρ dλ dt

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Radar waveform design based on optimization theory 100

80

60

40

20 0 −50

−40

−30

−20

−10

0

Squared modulus of CCF for RP (dB)

Figure 8.4 CCF histogram of random phase sequence in Experiment 1 21.1 PeCAN Proposed

Spectrum (dB)

21.09

21.08

21.07

21.06

21.05

0

0.2

0.4

0.6 Frequency

0.8

1

Figure 8.5 Spectrum of designed sequence in Experiment 1

where dλ/dt = [(dλ0 /dt) · · · (dλN −1 /dt)]T , Iter means the iteration number and ρ stands for the step size. In our study, we set ρ = 0.001, C0 = 20, and the maximum number of iterations is Iter=105 while the values of x(0), ˆ α(0) and λ(0) are initialized randomly. In this experiment, we consider designing a 128-element unimodular sequence with flat spectrum. Two random phase (RP) sequences are used for initialization. The spectrum of one RP sequence is shown in Figure 8.2, and we see that it has a huge dynamic range from 0.45 to 28.72 dB. As a result, this spectrum deviates far from

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243

Squared modulus of ACF (dB)

0 PeCAN Proposed

−20 −40 −60 −80 −100 −120 −140

−100

−50

0 Delay bin

50

100

Figure 8.6 ACF of designed sequence in Experiment 1 100

80

60

40

20

0 −50

−40

−30 −20 −10 Squared modulus of CCF (dB)

0

Figure 8.7 CCF histogram of designed sequence using proposed algorithm in Experiment 1

the flat spectrum, and thus ACF of the RP sequence is not quite impulse-like enough as shown in Figure 8.3. Figure 8.4 shows the CC functions (CCFs) of the two RP sequences. Note that CCF is also considered as a performance metric because in applications such as MIMO radar, multistatic sonar and communication systems [9,28], a set of sequences that has good CC properties is desired to achieve an increased virtual aperture, which in turn improves detection performance and enhance resolution. For the unimodular waveform design problem that is NP-hard, there are many local solutions. Employing different initializations (or RP sequences) will result in different

244

Radar waveform design based on optimization theory 100

80

60

40

20

0 −50

−40

−30 −20 −10 Squared modulus of CCF (dB)

0

Figure 8.8 CCF histogram of designed sequence using PeCAN in Experiment 1 Table 8.1 Performance comparison in Experiment 1 Method

RP

PeCAN

Proposed

PSL (ACF, dB) PSL (CCF, dB) Max (spectrum, dB) Min (spectrum, dB) −3 dB main lobe width (delay bin)

−13.89 −13.68 28.72 0.45 0.166

−79.35 −13.80 21.08 21.07 0.063

−104.99 −14.72 21.07 21.07 0.050

local solutions and this idea has been exploited for generating multiple sequences in MIMO applications [9,12,13,27,28]. Using the two RP sequences as initializations, we run the proposed method and PeCAN [9] to design the flat spectrum sequences. Both sequences designed by PeCAN and the proposed method are unimodular. Figure 8.5 plots the spectrum of the designed sequences according to the two methods, and flat spectra are observed. Therefore, its ACF corresponds to an impulse, as shown in Figure 8.6. Figure 8.7 shows the CCF of the two sequences obtained by the proposed algorithm. For comparison purposes, Figures 8.5 and 8.6 plot the spectrum and ACF of the sequences obtained by PeCAN, respectively, and Figure 8.8 plots the corresponding CCF. Comparing the curves in Figures 8.5 (spectrum) and 8.6 (ACF), 8.7 and 8.8 (CCF), we find that the flat spectrum is helpful to improve the correlation property, especially in regard to the AC performance.∗ To further contrast the performance of these two methods, we provide the PSLs of the ACF and CCF in Table 8.1. In addition, to assess the flatness of the spectrum, ∗

The level of ACFs is an analytic result, potentially difficult to reach in practical systems.

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120

Weight (dB)

100 80 60 40 20 0 0

20

40

60 80 100 Weight index

120

140

160

Figure 8.9 Weight assignment in Experiment 2

we extract the maximum and minimum of the corresponding spectrum to evaluate their dynamic ranges for the given −3 dB main lobe width. From Table 8.1, it is seen that (i) the correlation properties (especially the AC) of the PeCAN and the proposed method are better than RP; (ii) the more flat the spectrum of the sequence is or the smaller the dynamic range of the spectrum is, the better the AC property is and (iii) the proposed method has the narrowest −3 dB main lobe width.

8.5.2 Experiment 2: Spectrally constrained waveform design for radar Now we consider a radar waveform design example [54], where the overlayed and foreseen telecommunication systems are required to spectrally coexist with it. The radar sampling rate is 810 kHz and the pulse duration is 200 μs, which implies that the length of the sequence is N = 162. The corresponding stopbands for the radar are [0.0000, 0.0617], [0.0988, 0.2469], [0.2593, 0.2840], [0.3086, 0.3827], [0.4074, 0.4938], [0.5185, 0.5556] and [0.9383, 1.0000], and the rest are the passbands available for the radar. The weights used in the proposed algorithm are shown in Figure 8.9, and the expected spectrum masks are provided in Figure 8.10, where a large weight of 106 is used to ensure the small response for the narrow stopbands, whereas a smaller weight of 103 is assigned to the wide stopbands. In addition, 105 is assigned to the extremely narrow passband of [0.2469, 0.2593]. Similar to Experiment 1, two RP sequences are used for initialization. The magnitudes of the sequences obtained by SHAPE and the proposed method are all ones. Figures 8.11–8.13 plot their spectra, ACF and CCF. For comparison purposes,

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Radar waveform design based on optimization theory 5

Spectrum (dB)

0 –5 –10 –15 –20 –25

0

0.1

0.2

0.3

0.4 0.5 0.6 Frequency

0.7

0.8

0.9

1

Figure 8.10 Spectrum mask in Experiment 2

Power spectrum (dB)

0

–10

SHAPE Proposed

–20

–30 0

0.2

0.4

0.6

0.8

1

Normalized frequency

Figure 8.11 Spectrum of designed sequence in Experiment 2

the spectrum of the sequence designed by the SHAPE method [27,28] is given in Figure 8.11. In addition, Figures 8.14 and 8.15 show the corresponding ACF and CCF. It is observed that for this design application, the SHAPE method cannot ensure sufficiently low stopbands.

Lagrange programming neural network for radar waveform design

Squared modulus of ACF (dB)

0

−10

−20

−30

−40

−50

−100

−50

0 Delay bin

50

100

Figure 8.12 ACF of designed sequence in Experiment 2 100 80 60 40 20 0 −50

−40

−30 −20 −10 Squared modulus of CCF (dB)

0

Figure 8.13 CCF of designed sequence in Experiment 2 0 Squared modulus of ACF (dB)

−5 −10 −15 −20 −25 −30 −35 −40 −45 −50 −150

−100

−50

0 Delay bin

50

100

150

Figure 8.14 ACF of designed sequence using SHAPE in Experiment 2

247

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Radar waveform design based on optimization theory 100 80 60 40 20 0 −50

−40

−30 −20 −10 Squared modulus of CCF (dB)

0

Figure 8.15 CCF of designed sequence using SHAPE in Experiment 2

Table 8.2 Performance comparison in Experiment 2 Design method

SHAPE

Proposed

PSL of ACF (dB) PSL of CCF (dB) PVS of spectrum (dB) −3 dB main lobe width (delay bin)

−5.85 −11.75 −13.42 1.020

−12.70 −10.28 −20.45 0.968

To further assess the performance of the two methods, we provide the PSLs of the ACF and CCF, and −3 dB main lobe width in Table 8.2. Moreover, we extract the Peak Values of the corresponding stopbands (PVS). From Table 8.2, we see that (i) the CC property of the SHAPE method is slightly better than the proposed method; (ii) the AC property of the proposed method is better than SHAPE and (iii) the proposed method can attain lower stopband peak and narrower main lobe width than SHAPE. Therefore, the proposed method is a competitive alternative for waveform design with unit-modulus and spectral shape constraints. We provide the main complexity comparison using the discretized realization based on (8.66). For each iteration, the proposed method needs to compute F n x, ˆ corresponding to 0.5N log2 N multiplications assuming that FFT is applied. Moreover, the updates of xˆ , α and λ in (8.66) require N (7 + 4N ) multiplications. On the other hand, two FFTs are involved in both PeCAN and SHAPE, which correspond to N log2 N multiplications. Based on the digital implementation of LPNN, our computational complexity is higher than those of the PeCAN and SHAPE algorithms.

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−15

Normalized PSL (dB)

−20 −25 −30 −35 −40 −45

0

2

4 Number of iterations

6

8 × 104

Figure 8.16 PSL versus iteration number of proposed algorithm in Experiment 3

8.5.3 Experiment 3: Region of interest around main lobe In this experiment, we investigate the ability of the proposed method to design the sequence with the region of interest around the main lobe and the convergence property of the proposed algorithm. We consider designing a sequence x of length N = 128 with low PSL in the Doppler bin and time delay region of interest = {−4, . . . , 4}l × {−10, . . . , 10}k . Here, we initialize θ with RPs in the range [0, 2π]. For each iteration in"one LPNN update step, we adaptively select C0 with q = 10−2 and m = (2 − q + q2 + 4)/(2q) (see details in Appendix A.3). And C1 = 1 is assigned. The algorithm is terminated when x¯ (i+1) − x¯ (i)  ≤ 10−6 , x¯ (i)  or when the iteration number reaches 8 × 104 . For each iteration, we compute the corresponding normalized PSL with unit of dB, i.e., # r  $  l,k  PSL = 20 log10 max   , {l, k} ∈ , N and plot them in Figure 8.16. We can see that the proposed algorithm converges after 2 × 104 iterations, which implies that the proposed adaptive scheme in the LPNN update step performs well to contribute to the convergence of the LPNN-ADMM method. For comparison purposes, we run the TFCA [9] and EGM [39] and plot the AFs of the proposed EGM and TFCA methods in Figures 8.17–8.19. It is found that (i) the proposed method has a more flat sidelobe region within the region of interest and

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Radar waveform design based on optimization theory

Normalized AF (dB)

0 −10 −20 −30 −40 −50 −60 4

0

−4

−10

Doppler shift

0

10

Time delay

Figure 8.17 Normalized AF of designed sequence in Experiment 3

0

Normalized AF (dB)

−10 −20 −30 −40 −50 −60 4

0 Doppler shift

−4

−10

0

10

Time delay

Figure 8.18 Normalized AF of designed sequence using EGM in Experiment 3

(ii) the PSL of the proposed algorithm is −40.28 dB, which is much smaller than −33.79 dB (EGM) and −19.94 dB (TFCA).

8.5.4 Experiment 4: Region of interest on one side of main lobe We investigate the capability of the proposed method for designing the sequence with the region of interest on one side of the main lobe. In this experiment, we design

Lagrange programming neural network for radar waveform design

251

0

Normalized AF (dB)

−10 −20 −30 −40 −50 −60 4

0

−4

−10

Doppler shift

10

0 Time delay

Figure 8.19 Normalized AF of designed sequence using TFCA in Experiment 3

Normalized AF (dB)

0 −20 −40 −60 −80 −100 15

10 Doppler shift

5

10 Time delay

15

Figure 8.20 Normalized AF of designed sequence in Experiment 4

a sequence with length N = 128, where the region of interest is = {5, . . . , 15}l × {5, . . . , 15}k . We implement EGM and TFCA for comparison. The corresponding normalized AFs of the proposed algorithm, EGM and TFCA are plotted in Figures 8.20–8.22. It is seen that although EGM can yield the flat sidelobe levels, the PSL of the proposed method (−56.84 dB) is smaller than those of EGM (−49.87 dB) and TFCA (−10.63 dB). Clearly, the proposed method can design sequences with lower PSL in a region of interest without main lobe.

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Radar waveform design based on optimization theory

Normalized AF (dB)

0 −20 −40 −60 −80 −100 15

10 Doppler shift

5

10 Time delay

15

Figure 8.21 Normalized AF of designed sequence using EGM in Experiment 4 0

Normalized AF (dB)

−10 −20 −30 −40 −50 −60 15

10 Doppler shift

5

10 Time delay

15

Figure 8.22 Normalized AF of designed sequence using TFCA in Experiment 4

8.5.5 Experiment 5: Low-sidelobe autocorrelation level We consider designing aperiodic sequences for stationary targets. Then the AF metric reduces to the ACF metric. The time delays of interest are = {0}l × {1, . . . , 127}k . For comparison purposes, we also implement the CAN [12], monotonic ISL (MISL) [23], MMPSL [24] and EGM methods. The ACFs of the obtained sequences are plotted in Figure 8.23. The PSLs of the proposed CAN, MISL, MMPSL and EGM methods are −34.18, −26.24, −26.65, −28.27 and −32.50 dB, respectively. Among them, the proposed approach can design sequences with the lowest PSL.

Lagrange programming neural network for radar waveform design

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Normalized autocorrelation function (dB)

0 CAN MISL MMPSL EGM Proposed

–10 –20 –30 –40 –50 –60 –128

–80

–40

0 40 Time delay

80

128

Figure 8.23 Normalized AF (CAN, MISL, MMPSL, EGM and the proposed method) in Experiment 5

8.6 Conclusions We have introduced the LPNN method to design radar probing waveforms under practical constraints, including unit-modulus, spectral requirement and/or AF. The performance of the LPNN-based approach has been evaluated via numerical examples, demonstrating its superiority over a number of state-of-the-art methods. Note that both objective function and constraint function(s) are required to be differentiable in order to compute neurons of LPNN by partial derivatives. However, for some engineering problems, the corresponding objective function and constraint function constructed may not be differentiable. Thus, these problems cannot be solved by using LPNN directly. For example, the ellipse fitting model mina |aT x|p with (0 < p < 1) that has been successfully applied to computer vision, satellite attitude estimation, iris location and other fields is non-differentiable [55,56]. To solve these problems, extending LPNN to apply to non-differentiable problem is one of our follow-up studies. One possible method is to construct a differentiable function to approximate the non-differentiable one.

Appendix A A.1 Positive definiteness of Hessian matrix of (8.48) The Hessian matrix of (8.48) at the equilibrium point (x¯  , θ  , β  ) is  2  ∇x¯2x¯ Lx¯ − qC0 I 2N ∇xθ ¯ Lx¯  , 2 2 ∇θ x¯ Lx¯ ∇θ θ Lx¯ x= ¯ x¯  ,θ =θ  ,β=β 

 H =

(A.1)

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Radar waveform design based on optimization theory

2 2 2 2 where ∇x¯2x¯ L , ∇xθ ¯ Lx¯ , ∇θ x¯ Lx¯ and ∇θ θ L are given in (8.65). Define W 0 = (∇x¯ x¯ Lx¯ − . Similarly, denote W , U and V , n = 1, . . . , N , as those in qC0 I 2N )|x=    n n n ¯ x¯ ,θ=θ ,β=β Section 8.4.3.2. Then we have 

W0 Vn , V Tn V n = I n , Wn = V Tn U n

and |W n | = |W 0 ||U n − V Tn W −1 0 V n |, n = 1, . . . , N . According to (dβ/dt)|x= ¯ x¯  ,θ =θ  ,β=β  = 02N ×1 , we have x¯ n = cos (θ  (n)), x¯ N +n = sin (θ  (n)). Denote un = β  (n) cos (θ  (n)) + β  (N + n) sin (θ  (n)). u1 , . . . , C0 + un ), n = 1, . . . , N , hold. Therefore, we have |W n | ≤ |(C0 + |λ |)I 2N ||U n − V Tn = (C0 + |λ |)2N −n

Then

U n = diag(C0 +

I 2N V n| C0 + |λ |

n % 

 C0 (un + |λ |) + un |λ | ,

i=1

where |λ | denotes the largest modulus of the eigenvalues of (Z 1 + C1 Z 2 + C1 Z 3 )     |x= ¯ x¯  ,θ=θ  ,β=β  . Then, once (u1 + |λ |) < 0 holds, we have u1 |λ | < 0 and |W 1 | ≤ 0. Therefore, H  cannot maintain positive definiteness for all possible equilibrium points.

A.2 Solution to (8.58) In the Cartesian coordinate system, the general function of degree 2 is f (x) = ax2 + bx + c with a, b, c ∈ R, a  = 0. Then, for a > 0, (i) f (x) achieves a minimum when x = −(b/2a) and (ii) f (x) decreases and increases in ( − ∞, −(b/2a)] and √ [ − (b/2a), ∞), respectively. Denote x = η, a = a = 1 + (C /2) p 1 l,k ωl,k Sl,k >   0, b = bp = −C1 l,k ωl,k Sl,k ˜yl,k  < 0, c = cp = (C1 /2) l,k ωl,k Sl,k ˜yl,k 2 . Then (−bp /2ap ) > 0 and for η ∈ [ηp−1 , ηp ], the minimum point (x , f  ) is achieved when ⎧ −b p ⎪ , ⎪ 2ap ⎪ ⎪ ⎨ √ x = ηp−1 , ⎪ ⎪ ⎪ ⎪ ⎩√η , p

  

−bp 2ap −bp 2ap −bp 2ap

  

√ √ ∈ [ ηp−1 , ηp ],


√ ηp .

Therefore, (8.59) is the solution to (8.58).

(A.2)

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255

A.3 Adaptive selection scheme of C 0 The penalty parameter C0 serves as a weight-like parameter to compromise between (i) T the terms L (x, ¯ θ | η(i) , f (i) , g (i) , α f , α (i) g ) and h h. On the one hand, a larger C0 drives the Lagrangian neurons toward the feasible region, namely, satisfying the unimodular constraints h(x, ¯ θ) = 02N ×1 in (8.47). Under fairly mild conditions (see more details in Section 1.4 of [50]), a sufficiently large value of C0 makes the problem convex [22,44]. On the other hand, a smaller C0 belittles the unimodular constraints but comparatively pays more attention to the objective function. Since it is not easy to set a reasonable value for C0 , it is necessary to develop an adaptive selection scheme of C0 to take into account of both the objective function and constraints and to improve the convexity and convergence performance. According to Section 8.4.3.2, it is also difficult to determine C0 to ensure that the Hessian matrix H  defined in (8.64) is positive definite. However, one can first obtain a relaxed penalty parameter by making both Z 0 and ∇θ2θ Lx¯ positive definite. Then, C0 is determined by multiplying the relaxed penalty parameter with a factor greater than 1. The detailed procedure is given next.

A.3.1 On positive definiteness of ∇θ2θ Lx¯ In order to make ∇θ2θ Lx¯ defined in (8.65) positive definite, C0 must satisfy C0 (¯x(n) cos (θ (n)) + x¯ (N + n) sin (θ (n))) + (β(n) cos (θ(n)) + β(N + n) sin (θ (n))) ≥ 0, n = 1, . . . , N . Since h(x, ¯ θ ) = 02N ×1 , holds, and then we have

x¯ (n) cos (θ (n)) + x¯ (N + n) sin (θ(n)) = 1

C0 ≥ −(β(n) cos (θ (n)) + β(N + n) sin (θ (n))), n = 1, . . . , N .

(A.3)

Because − (β(n) cos (θ (n)) + β(N + n) sin (θ(n))) (β 2 (n) + cos (θ (n))2 ) + (β 2 (N + n) + sin (θ(n))2 ) 2 (β 2 (n) + β 2 (N + n)) + 1 , = 2

≤

we relax (A.3) as C0 ≥ γ ,

(A.4)

to ensure ∇θ2θ Lx¯ to be positive definite, where γ = max{(((β 2 (n) + β 2 (N + n)) + 1)/ 2), n = 1, . . . , N }.

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Radar waveform design based on optimization theory

A.3.2 On positive definiteness of Z0 To ensure Z 0 defined in (8.65) to be positive definite, we require C0 I 2N + (Z 1 + C1 Z 2 + C1 Z 3 )  0. If there exist values ai satisfying ai I 2N + Z i  0, i = 1, 2, 3, we can obtain a relaxed value for C0 C˜ 0 = a1 + C1 a2 + C1 a3 .

(A.5)

Furthermore, we compute ai for Z i , i = 1, 2, 3 as follows: 1.

2.

Compute a3 . Obviously, the matrix Z 3 defined in Z 0 is positive semi-definite. Since C1 > 0, we have C1 Z 3  02N ×2N , namely, a3 = 0. Compute a2 .       T T T max min ¯ ¯ ¯ ¯ ¯ ¯ Let λAmin = λ A + A = λ , λ A B + A = λ , λ + B min l,k max l,k min l,k ¯ l,k ¯ l,k l,k l,k , B¯ l,k   Al,k T = λmax B¯ l,k + B¯ l,k . Since and λmax B¯ l,k

min max min ¯ ¯T ¯ ¯T λAmax ¯ l,k I N  Al,k + Al,k  λA¯ l,k I N , λB¯ l,k I N  Bl,k + Bl,k  λB¯ l,k I N ,

we have x¯ T A¯ l,k x( ¯ A¯ l,k + A¯ l,k ) − ul,k al,k I N  0N ×N , x¯ T B¯ l,k x( ¯ B¯ l,k + B¯ l,k ) − vl,k bl,k I N T

T

 0N ×N , (A.6) where al,k = x¯ T A¯ l,k x, ¯ bl,k = x¯ T B¯ l,k x, ¯ ul,k =

⎧ ⎨λmin , al,k ≥ 0, A¯ l,k

⎩λmax , al,k < 0, A¯

vl,k =

l,k

⎧ ⎨λmin , bl,k ≥ 0, B¯ l,k

⎩λmax , bl,k < 0. B¯

(A.7)

l,k

Thus, we draw the conclusion −(ul,k al,k + vl,k bl,k ) ≤

2 2 ul,k + vl,k + a2l,k + b2l,k

2

,

(A.8)

and the second part of Z 0 satisfies C1 a2 I 2N + C1 Z 2  02N ×2N ,  u2 +v2 +a2 +b2 where a2 = ωl,k l,k l,k 2 l,k l,k .

(A.9)

l,k

3.

Compute a1 . Notice that all variables in Z 1 can be obtained while updating x, ¯ θ and β in the LPNN step, so the minimum eigenvalue of Z 1 is known and we denote it as λZ 1 . Therefore, we have a1 = −λZ 1 that satisfies a1 I 2N + Z 1  02N ×2N .

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Thus, we have C˜ 0 = C1



ωl,k

2 2 ul,k + vl,k + a2l,k + b2l,k

2

l,k

− λZ 1 .

(A.10)

Combining (A.4) and (A.10) results in C0 > C¯ 0 ,

(A.11)

where the lower boundary of C0 is C¯ 0 = max{C˜ 0 , γ , 0}.

A.3.3 On positive definiteness of Hessian matrix H of (8.49) Similarly, denote W 0 , W n , U n and V n , n = 1, . . . , N , as those in Section 8.4.3.2. We have the following lemma. Lemma 8.3. The adaptive selected C0 : C0 = (1 + m)C¯ 0 ,

"

(A.12)

where q > 0, m ≥ ((2 − q + q2 + 4)/2q), makes H , i.e., the Hessian matrix H defined in (8.64) at the equilibrium point (¯x , θ  , β  ), positive definite. Proof. Note that C0 selected by (A.11) ensures the positive definiteness of W 0 and U n , n = 1, . . . , N . To ensure the positive definiteness of H  , C0 is determined by multiplying C¯ 0 with a factor greater than 1, i.e., C0 = (1 + m)C¯ 0 > C¯ 0 , m > 0.

(A.13)

Then, we have W 0 = Z 0 + qC0 I 2N  mC¯ 0 + q(1 + m)C¯ 0 I 2N  (1 + q)mC¯ 0 I 2N  02N ×2N , U n  mC¯ 0 I n and V Tn V n = (m + 1)2 C¯ 02 I n , n = 1, . . . , N . Then the following holds    (m + 1)2 C¯ 0  −1 T  ¯ ¯ |W n | = |W 0 ||U n − V n W 0 V n | > |(1 + q)mC0 I 2N | mC0 I n − In , (1 + q)m  " n = 1, . . . , N . Since q > 0 and m ≥ ((2 − q + q2 + 4)/2q) > 0, we have |W n | > 0, n = 1, . . . , N . Therefore, all the principal minor determinants of H  are nonnegative. And thus  H is positive definite. Lemma 8.3 shows that one can apply the scheme in (A.12) to adaptively select the penalty parameter C0 for each LPNN update step to ensure that the Hessian matrix H  at the equilibrium point (x¯  , θ  , β  ) is positive definite.

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Chapter 9

Cognitive local ambiguity function shaping with spectral coexistence and experiments Guolong Cui1 , Jing Yang1 , Xiangxiang Yu1 , and Lingjiang Kong1

9.1 Introduction Ambiguity function (AF) plays an important role in evaluating waveform performance as it is widely used in time–frequency signal processing to analyze the range–Doppler resolutions and mismatch loss. Hence, many techniques in terms of AF shaping have been proposed with reference to various tasks, operating environments, and constraints. Among the metrics of optimizing the AF, integrated sidelobe level (ISL) of range–Doppler plane is widely adopted as a merit to be minimized [1]. Nevertheless, it is impossible to achieve a quite low ISL over the whole AF due to its fixed volume [2]. To this end, by resorting to the prior information provided by cognitive radar [3,4], shaping a reasonable local AF (LAF) through minimizing the weighted ISL (WISL) over range–Doppler bins of interest has received significant attention. In this respect, two categories concerning the study of the LAF optimization have been analyzed. The former focuses on the shaping of a slow-time AF. Maximum block improvement [5], majorization–minimization (MM) [6], and adaptive sequential refinement [7] algorithms are introduced to synthesize slow-time-coded pulses with constant modulus restriction by minimizing the average value of the AF over interfering bins so as to improve the detectability of slow-moving targets. The work accounting for peak-to-average-ratio (PAR) constraint has been also addressed in [6]. The latter emphasizes on the optimization of a fast-time LAF. In [8], an efficient gradient method to optimize the periodic/aperiodic AF is presented to improve the detectability of high-speed targets forcing a constant modulus constraint on the probing signal. In [9], the study focusing on the aperiodic AF over arbitrary range– Doppler bins with a constant modulus constraint is addressed through an accelerated iterative sequential optimization (AISO) algorithm that actually falls into the MM framework [10,11]. Since more and more electromagnetic spectra are crowded, spectral coexistence with surrounding electronic systems becomes pretty important to avoid interfering with each other, which has already been studied in many works [12–28]. In particular, 1 School of Information and Communication Engineering, University of Electronic Science and Technology of China, Chengdu, China

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considering the desired AF and spectral coexistence, the constant modulus waveform design problem by optimizing jointly the autocorrelation and spectral properties is studied in [12–16]. However, these works only focus on the suppression of range sidelobes over the zero-Doppler bin with the constant modulus sequence [29]. In this chapter, we cope with the joint optimization of fast-time LAF and waveform spectrum. A novel objective function with a weighting factor to trade-off the WISL with respect to range–Doppler sidelobes and the spectral stopband energy is proposed. The energy and PAR constraints are forced on the designed signal, which further generalize the problem we developed. To handle the resulting non-convex quadratic problem, a new iterative sequential quadratic optimization (ISQO) algorithm is proposed [30]. In each iteration, it transforms the optimization problem into a tractable quadratic subproblem that is further converted to a linear optimization problem with a closed-form solution by resorting to the first-order Taylor expansion. The proposed algorithm has polynomial time complexity that is linear with the number of iterations and polynomial with the size of the designed waveform and the number of the considered range–Doppler bins. The simulation results highlight the superiority of the proposed algorithm contrast to AISO algorithm in terms of the objective value, computational time, and the capability of prohibiting the sidelobes of a strong return and narrowband spectral interferences from masking weak targets. The experimental results show that the hardware limitation influences the level suppressed in both energy spectral density (ESD) and WISL and confirms the effectiveness of spectral coexistence in prohibiting the sidelobes suppressed in the LAF rising. The rest of the chapter is organized as follows. In Section 9.2, the optimization problem is formulated. In Section 9.3, we provide the ISQO algorithm procedure. In Section 9.4, we evaluate the performance of the devised algorithm. Finally, in Section 9.5, we provide conclusions and future works.

9.2 Problem formulation 9.2.1 Weighted integrated sidelobe level Consider a fast-time transmitted sequence x = [x (1) , x (2) , . . . , x (N )]T ∈ CN with N being the number of coded sub-pulses. The discrete AF value dictated by x is expressed as    2 χ (k, f ) = x† Jk diag pf x ,

(9.1)

where k and f denote the range bin and the normalized Doppler frequency, respectively. The steering vector pf and the shift matrix Jk ∈ CN ×N are, respectively, given by T  pf = e j2π f , . . . , e j2πNf

(9.2)

Cognitive local ambiguity function shaping and



if n − m = k (9.3) else .   = Jk diag pf , the discrete WISL on the LAF interested can be

Jk (n, m) = Letting Mk, f formulated as

265

1, 0,

 2 1  wk, f x† Mk, f x , (9.4) Q (k, f )∈

where  = (k, f )|wk, f ≥ 0 is the region ofinterest to shape the LAF with wk, f being the weight of the (k, f )th bin and Q = (k, f )∈ wk, f . F(x) =

9.2.2 Spectral coexistence Assume that the hth frequency stopband of the cooperative radiators coexisting with the radar occupies over h = [ fh1 , fh2 ], h = 1, . . . , Ns

(9.5)

in which fh1 and fh2 denote the lower and upper bound normalized frequencies, respectively, and Ns is the number of stopband. The energy transmitted on the hth band is computed as fh2 S( f )df = x† RhI x,

(9.6)

fh1

2    −1 xn e−j2π fn  is the ESD, f ∈ [0, 1] denotes the normalized where S( f ) =  Nn=0 frequency, and  fh2 − fh1 if m = n h RI (m, n) = e j2πfh2 (m−n) −e j2πfh1 (m−n) (9.7) else j2π (m−n) with (m, n) ∈ {1, · · · , N }2 . Hence, the total energy in all the stopbands is given by G(x) = x† Rc x,  s ch RhI where ch denotes the weight on the hth band. with Rc = Nh=1

(9.8)

9.2.3 Optimization problem To improve the target detectability as well as ensure the spectral coexistence with other nearby radiators, the optimization problem with an energy constraint and a PAR constraint forced on the probing waveform is formulated as follows: ⎧ W (x) = βF(x) + (1 − β)G(x) ⎪ ⎨ min x (9.9) P0 s.t. x† x = 1 ⎪ ⎩ |x(n)| ≤ γ , n = 1, . . . , N ,

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where β ∈ [0, 1) is a weighted coefficient playing a trade-off  role between the √ LAF optimization and the spectral nulling, and γ ∈ 1/ N , 1 rules the maximum allowable PAR that is defined as [31] PAR =

max |x(n)|2

n=1,...,N

(1/N )x2

= N max |x(n)|2 . n=1,...,N

(9.10)

It is worth noting that P0 can reduce to the √ optimization of LAF with a constant modulus constraint letting β = 1 and γ = 1/ N as discussed in [9]. And P0 can be further transformed to the optimization of aperiodic autocorrelation properties with w(k, f ) > 0 only when f = 0. However, this chapter focuses on the joint optimization of LAF and the spectral coexistence under energy and PAR constraints, which thus is generalized. P0 is a non-convex problem due to the non-convexity of the quadratic objective function and waveform energy constraint. Interestingly, the AISO algorithm (actually involved the MM framework ) in [9] can be extended to handle the non-convex problem in polynomial time. Nevertheless, it relies on the minimization of the surrogate function rather than the origin function and thus probably exhibits a slow convergence speed. Next, a novel iterative algorithm sharing a lower computational time is introduced to offer a high-quality solution to P0 .

9.3 Iterative sequential quadratic optimization algorithm Before proceeding P0 , the WISL F(x), which is a quadratic function, can be further recast as       1  † † wk, f x† Mk, f x x† Mk, f x + x† Mk, f x x† Mk, f x F(x) = 2Q (k, f )∈ ⎡ ⎤    (9.11) 1 † † wk, f Mk, f xx† Mk, f + Mk, f xx† Mk, f ⎦ x = x† ⎣ 2Q (k, f )∈ = x† ϒ(x)x, where ϒ(x) =

  1  † † wk, f Mk, f xx† Mk, f + Mk, f xx† Mk, f . 2Q (k, f )∈

(9.12)

Lemma 9.1. Letting B = Axx†A† for arbitrary A ∈ CN ×N and x ∈ CN , then (1) B is a positive semidefinite matrix (i.e., B  0); (2) Rank (B) ≤ 1, λmax (B) = x†A†Ax, other eigenvalues of B are 0.  2 Proof. Exploiting the fact B = Axx†A† , we have s† Bs = s†Ax ≥ 0, ∀s, thus implying B  0 and its all eigenvalues are nonnegative. Since B is the product of two vectors

Cognitive local ambiguity function shaping

267

and B  0, Rank (B) ≤ 1. Based on this, B only has one unique positive eigenvalue λmax (B) = x†A†Ax when Ax = 0. From the previous discussion, we construct a special matrix ϒ(x) that is the weighted sum of matrices with the same structure of B in Lemma 9.1. Thus, ϒ(x)  0 and satisfies s† ϒ(x)s = x† ϒ(s)x, ∀x, s ∈ CN . To this end, we rewrite P0 into the following equivalent problem: ⎧ −βx† ϒ(x)x − (1 − β)x† Rc x ⎨ max x P1 s.t. x† x = 1, ⎩ |x(n)| ≤ γ , n = 1, . . . , N .

(9.13)

(9.14)

We exploit the previous equivalent high-order polynomial problem and present a new fast iteration procedure to P0 via the following proposition. Proposition 9.1. P1 can be handled by sequentially solving the following approximation problem:   ⎧ u x, x(t) ⎨ max x (9.15) P(t+1) s.t. x† x = 1 ⎩ |x(n)| ≤ γ , n = 1, . . . , N , (t) where x(t) denotes the tth iteration   (t) solution of the proposed iteration algorithm, λ (t) satisfies that λ ≥ βλmax ϒ x and

   

 u x, x(t) = x† λ(t) IN − βϒ x(t) x − (1 − β)x† Rc x(t) .

(9.16)

Proof. The properties of monotonicity and convergence for the presented iteration procedure are analyzed in Appendix A. In this respect, for each iteration, we transform   the original objective function u (x, x) into a quadratic objective function u x, x(t) . Next, we show how to solve P(t+1) . Before proceeding, to save the computational time, λ(t) can be assigned to a constant given by λ(t) = β.

(9.17)

Proof. See Appendix B. It can be observed that P(t+1) is still non-convex. But its objective function is convex. To this end, we can optimize its first-order expansion to find a solution to P(t+1) by the following proposition.

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Proposition 9.2. P(t+1) can be successively solved through a sequential iteration procedure whose i + 1th iteration is required to solve the following problem:   ⎧ ⎪

xˆ † v(i) ⎨ max xˆ Pi+1 s.t. xˆ † xˆ = 1 (9.18) ⎪   ⎩ xˆ (n) ≤ γ , n = 1, . . . , N , where

  

v(i) = 2β IN − ϒ x(t) − (1 − β)Rc xˆ (i)

(9.19)

and xˆ (i) stands for the ith iteration solution of the sequential iteration procedure. Proof. See Appendix C. Without loss of generality, we assume that v(i) (1) ≥ v(i) (2) ≥ · · · ≥ v(i) (N ) and the number of nonzero elements of v(i) is m. The solution to Pi+1 is as follows [32]:   (9.20) xˆ (i+1) = Px v(i) where

   Px (·) = 1R≥ 0 1 − mγ 2 γ um  e j arg(·)    + 1R< 0 1 − mγ 2 min {δ |·| , γ 1}  e j arg(·) ,  1, if x ∈ A , 1A (x) = 0, else ⎡

1 − mγ 2 ,..., ⎢ 1, . . . , 1, um = ⎣    (N − m) γ 2 m   N −m

⎤T 1 − mγ 2 ⎥ (N − m) γ 2 ⎦ , 

(9.21)

(9.22)

(9.23)

and  δ ∈ δ|

N  n=1

"    2 min δ 2 v(i) (n) , γ 2 = 1, δ ∈ 0, 

γ    

(i)   min v (n) , v(i) (n) = 0

#$ . (9.24)

According to the monotonicity requirement in Appendix A, x(t+1) needs to satisfy     (9.25) u x(t−1) , x(t) ≤ u x(t+1) , x(t) . Thus, x(t+1) can be derived not only by one-step iteration with xˆ (i) = x(t−1) but also by multi-iterations with xˆ (i) = x(t) to fulfill (9.25). Hence, based on the aforementioned discussions, the sequential iteration procedure for solving P(t+1) can be summarized in Algorithm 1. In each iteration of Algorithm 1, the computational complexity mainly relies on v taking the order of O(MN 2 ), where M is the number of wk, f = 0.

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269

Algorithm 1: Sequential iteration procedure for optimizing P(t+1)   Require: λ, β, ϒ x(t) , Rc ;     Ensure: A solution x(t+1) satisfies u x(t−1) , x(t) ≤ u x(t+1) , x(t) ; 1: Initialize i = 0 and x ˆ (i) = x(t−1) or x(t) ; (i) 2: Compute v by (9.19); 3: Obtain x ˆ (i+1) according to (9.20); 4: i := i + 1;     5: If u x(t−1) , x(t) ≤ u x ˆ (i) , x(t) , output x(t+1) = xˆ (i) and stop; otherwise, go to Step 2;

Algorithm 2: ISQO without acceleration Require: β, x0 , wk, f and Rc ; Ensure: A suboptimal solution x∗ to P(t+1) ; 1: For t = 0 and x(t) = x0 ; 2: Update x(t+1) by Algorithm 1; 3: t := t + 1; 4: If |x(t) − x(t−1) | ≤ κ, output x∗ = x(t) . Otherwise, return to Step 2.

The ISQO procedure for solving P(t+1) is summarized in Algorithm 2. However, it suffers from a slow convergence speed, mainly because of the objective function of P(t+1) is not optimized directly but surrogated by the first-order Taylor expansion. In order to alleviate the drawback, expectation–maximization (EM) algorithm that can be significantly accelerated via a two-point iteration strategy is adopted [33]. Herein, the monotonicity of the proposed ISQO algorithm will be violated and the updating scheme should be a fixed-point iteration procedure. Thus, a strategy based on backtracking is adopted to enforce the monotonicity of the original objective function of P0 and let    xˆ (i+1) = Px q xˆ (i) , (9.26)   

  where q xˆ (i) = 2β IN − ϒ x(i) − (1 − β)Rc xˆ (i) . The algorithm with EM acceleration is reported in Algorithm 3. Remark 9.1. In Algorithm 3, Step 8 is performed to ensure that the objective value in P0 decreases monotonically. Besides, Step 4 prevents the iteration from endless loop when α = −1 and terminates the algorithm with x∗ = x(t) in such a scenario. To get a stationary point, we might refine the solution obtained via Algorithm 3 by initializing it for MM method [7].

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Algorithm 3: ISQO with acceleration Require: β, x0 , wk, f , and Rc ; Ensure: A suboptimal solution x∗ to P(t+1) ; 1: Set t = 0 and x(t) = x(t−1) = x0 ; 2: x1 = Px (q (x0 )); 3: x2 = Px (q (x1 ));   4: If W (x2 ) > W x(t) , output x∗ = x(t) , break; End 5: r = x1 − x(t−1) , u = x2 − x1 − r; 6: Compute the step  (t)length α = −r/u;  2 7: x(t+1) = P q x  (t+1) x − 2αr  (t)+  α u ; 8: While W x >W x α ← (α − 1)/2;    x(t+1) = Px q x(t) − 2αr + α 2 u ; End 9: If x(t+1) satisfy the stop condition, output x∗ = x(t+1) . Otherwise, t := t + 1 and return to Step 2;

9.4 Numerical results In this section, we present numerical simulations and experiment data to verify the performance of the proposed method.

9.4.1 Simulation results This subsection is devoted to assessing the performance of the proposed algorithm in terms of the achievable objective value W (x), the WISL F(x), and the frequency stopband energy G(x). For comparison purposes, AISO algorithm in [9] is also considered to solve P0 . A random phase-coded unimodular sequence with N = 1, 024 as the initial sequence for both considered algorithms is exploited. We consider different PAR values with the local region  = {(k, f )|k = [2, 70], f ∈ [ − 0.002, 0.002]}. The number of the cooperative radiator is set as Ns = 1 and the corresponding frequency stopband interval is [0.1, 0.2]. The exit condition of running CPU time for each experiment equals to 1,500 s. Finally, the simulations are performed using MATLAB® 2013a version on a standard PC (with a 3.4 GHz Core i7-6700 CPU and 8 GB RAM).

9.4.1.1 Performance for β ∈ [0.1, 1] To illustrate how β affects the performance, we increase β from 0.1 to 1 and depict the trend of the WISL F(x) and the frequency stopband energy G(x)† for AISO and

†

The WISL F(x) and the frequency stopband energy G(x) are, respectively, normalized with respect to those of the initial sequence.

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Figure 9.1 F(x) and G(x) versus β from 0.1 to 1 ISQO algorithms in Figure 9.1 when PAR=1. The result for β less than 0.1 is not shown because it leads to little LAF shaping. The curves show that a larger β gives more weight to the WISL thus resulting in a decreased F(x) at the cost of an increased G(x) for both the algorithms. In particular, F(x) and G(x) vary rapidly with β ∈ [0.9, 1]. Interestingly, ISQO algorithm achieves lower values of F(x) and G(x) in comparison with AISO method. For example, ISQO algorithm achieves G(x) of −70.93 dB when β = 0.1, whereas AISO only attains −58.99 dB. When β = 1, the WISL F(x) can be easily optimized without spectral nulling that leads to high sidelobes, and ISQO achieves F(x) of −240.5 dB, whereas AISO only attains −193.3 dB. Through Figure 9.1, we can reasonably choose β according to practical performance requirements. Without loss of generality, β = 0.5 and β = 1 are, respectively, selected to further illustrate the influence of β in the following simulations.

9.4.1.2 Optimization of both the WISL and the frequency stopband energy for β = 0.5 Figure 9.2 depicts the objective value W (x) versus computational time (in seconds) of AISO and ISQO algorithms for different PAR values. A lower objective value is achieved for a larger PAR for both ISQO and AISO due to allowing more degrees of freedom on the waveform amplitude. The objective values achieved by both the algorithms monotonically decrease as time passes, whereas ISQO algorithm achieves a lower objective value with a faster speed than AISO. This is probably due to the fact that the proposed algorithm directly optimizes the objective function W (x) without the use of surrogate function in the procedure of converting P0 to Pt+1 compared with AISO algorithm.

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Figure 9.2 W (x) versus time for β = 0.5

Figure 9.3 exhibits the corresponding WISL F(x) and the total frequency stopbands energy G(x) against computational time. The results highlight that the proposed algorithm achieves a much lower WISL and stopband energy with a faster convergence speed in comparison with AISO algorithm. Figure 9.4 shows AFs of the initial sequence and the designed sequences obtained through AISO and ISQO algorithms for PAR = 1 at running time 1,500 s when β = 0.5. Interestingly, the AF of the optimized signal by ISQO algorithm achieves a much lower sidelobe value in correspondence of the region  than that of AISO algorithm. Figure 9.5 offers the ESDs of the designed sequences. As expected, the ESDs show the deeper and deeper notches within the region [0.1,0.2] as PAR value increases. In particular, ISQO algorithm is able to attain a deeper notch than AISO. These performance behaviors are consistent with those in Figure 9.3. Finally, it is worth pointing out that the proposed algorithm is capable of shaping reasonably an AF and an ESD to confront interferences with efficiency which will be illustrated further in the aspect of an application in Section 9.4.1.4.

9.4.1.3 Optimization of the WISL only for β = 1 Figure 9.6 exhibits the objective value W (x) against computational time (in seconds) of AISO and ISQO algorithms. Note that ISQO stops before time restriction as Step 4 in Algorithm 3 is executed. As expected, Figure 9.6 again confirms that the higher PAR value, the better performance, and ISQO algorithm achieves a much lower objective value with a faster speed than AISO. Specifically, AISO algorithm achieves

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Figure 9.3 (a) F(x) and (b) G(x) versus time for β = 0.5

the objective value of −193.3 dB taking 1,500 s, while ISQO algorithm spends only around 26.3 s when PAR = 1. In Figure 9.7, AFs of the sequences obtained, respectively, through AISO and ISQO algorithms, and their zero-Doppler profile for PAR = 1 is presented. As expected, the AF of the designed signal by ISQO algorithm achieves a much lower value than that of AISO algorithm, which is consistent with Figure 9.6.

9.4.1.4 Application: detection of multiple high-speed targets We utilize the sequences obtained via the proposed algorithm and AISO to detect two high-speed targets (i.e., strong target 1 and weak target 2) embedded in a narrowband interference emitted by a nearby jammer, as shown in Figure 9.8.

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Radar waveform design based on optimization theory Target 2 Target 1

Cognitive radar

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Jammer

Figure 9.8 The simulation scene of multiple high-speed targets detection Assume a cognitive radar with the working wavelength λ = 0.03 m, the transmitted signal bandwidth B = 5 MHz, and the sampling interval Ts = 5 × 10−8 s. Target 1 with velocity v1 = 1, 200 m/s (the corresponding normalized frequency f1 = 2v1 Ts /λ = 0.004) is located at R1 = 100 km (the corresponding range cell l1 = 2R1 /(cTs ) = 13,333, where c is the speed of light). Target 2 with velocity v2 = 1,500 m/s (the corresponding normalized frequency f2 = 0.005) is located at range R2 = 100.3 km (the corresponding range cell l2 = 13,373). Thus, the nth sample of the received echo y can be expressed as y(n) = a1 x(n − l1 )e j2π f1 n + a2 x(n − l2 )e j2πf2 n + z(n) + p(n)

(9.27)

where z(n) is a zero-mean circularly symmetric complex Gaussian random variable with variance σ 2 , and p(n) = a3 e j2π f3 n is a narrowband interference with a normalized frequency f3 = 0.15. We set SNR1 = 30 dB, SJR1 = −10 dB, SNR2 = 0 dB, and SJR2 = −40 dB where   a2i E |x(n)|2 , i = 1, 2, (9.28a) SNRi = 10 log10 σ2   a2i E |x(n)|2   , i = 1, 2. (9.28b) SJRi = 10 log10 E |p(n)|2 We depict the cross AFs (CAFs) [9] of the initial sequence, the sequence obtained through ISQO with β = 1 and PAR = 1, and the sequences obtained through AISO and ISQO algorithms with β = 0.5 and PAR = 1 in Figure 9.9(a)–(d), respectively. Not only the strong return of the target 1 but also the interference leads to the missing detection in CAF of the initial sequence in Figure 9.9(a). Although the designed sequence without spectral coexistence in Figure 9.9(b) can achieve a lower WISL, the weak target 2 is masked by the interference. Thus, spectral coexistence technique

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plays an important role in confronting the nearby narrowband spectral interference. Considering both the LAF shaping and spectral coexistence, the CAF obtained by ISQO algorithm with β = 0.5 shows the strong target 1 and the weak target 2 more clearly than that of AISO (see Figure 9.9(c) and (d)) as both the spectral stopband energy and the WISL achieved by ISQO are lower as shown in Figures 9.4 and 9.5. The results highlight that the sequence optimized by the proposed algorithm can resist nearby narrowband spectral interferences and detect weak targets efficiently.

9.4.2 Experimental results In order to further assess the effectiveness of the proposed technique, the practical implementation of radar waveforms to shape the LAF and ensure spectral coexistence is studied experimentally. More in detail, we set N = 1, 024,  = {(k, f )|k = [10, 50], f = 0}, and PAR = 1. Two waveforms s1 and s2 are synthesized via MATLAB software and

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Figure 9.11 The test bed of the experiment

then loaded to an arbitrary waveform generator (AWG), where s1 shares a frequency stopband within [0.45, 0.55], while s2 has no stopband. The corresponding autocorrelations and ESDs of two sequences are shown in Figure 9.10(a) and (b), respectively. The test bed comprises an AWG to generate waveforms according to loaded waveform data and an oscilloscope to measure the received signal. Precisely, the AWG is a Tektronix AWG7082C with a sample rate up to 8 GS/s and 10-bit vertical resolution. The oscilloscope is a Tektronix DSA70404B with 4 GHz analog bandwidth and 25 GS/s real-time sample rate. Figure 9.11 shows the connection between antennas and facilities. The signal and interference are generated simultaneously via the AWG and then transmitted via two external antennas. The oscilloscope receives the synthetic signal via an external antenna. Besides, the logical negation of the designed sequence in the AWG is input in the oscilloscope to synchronize.

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The carrier frequency f0 , the sampling rate fs for both AWG and oscilloscope, and the pulse duration T are set as 1.75 GHz, 6.25 GHz and 10 μs, respectively. The amplitude of waveform generated by AWG is set as 1 Vpp. The received waveform is analog-to-digital sampled by the oscilloscope and stored. The collected data are further digitally down-converted and recovered to the baseband complex signal. To reduce the amount of data, the complex signal is decimated and further processed. First, we transmit the designed signals s1 and s2 only. In Figure 9.12, the screenshots of the oscilloscope show the real-time ESDs of the received signals when s1 and s2 are, respectively, transmitted. The notches in Figure 9.12(a) are obvious and the cursor with dashed line marks the notch around 1.7 GHz, while no notch is shown in Figure 9.12(b), which is coincident with the ESDs of the transmitted signals. Further, we handle the received discrete-time baseband-decimated signals associated with transmitting s1 and s2 by MATLAB, which can be denoted as y1 = b1 s1 + z1 and y2 = b2 s2 + z2 , respectively, where b1 (b2 ) is the amplitude of the useful echo s1 (s2 ), zi , i = 1, 2 are noise signals. Figure 9.13(a) and (b) shows matched filtering results and ESDs of y1 and y2 , respectively. Compared with Figure 9.11, it can be seen that the spectral stopband level of y1 and the sidelobe levels in the suppressed area of the matched filtering results for both y1 and y2 increase. This is probably due to the hardware limitations (e.g., device internal noise, quantization errors, and so on). Hence, the stop condition of ISQO algorithm can be set based on the experimental WISL and stopband level limitation. Then we transmit s1 (s2 ) and a narrowband interference p1 (n) = e j2π fp n/fs simultaneously, where fp = 1.7 GHz. Figure 9.14 indicates the real-time ESDs of the received

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Figure 9.12 Screenshots of real-time ESDs on oscilloscope when transmitting (a) s1 and (b) s2

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signals on the screen of oscilloscope when s1 (s2 ) and an interference are transmitted simultaneously. We can see that the interference on the left notch of s1 is much stronger than the designed signals. The received signals are further handled by MATLAB. In particular, the received discrete-time baseband-decimated signals can be denoted as y3 = b3 s1 + c3 p1 + z3 and y4 = b4 s2 + c4 p1 + z4 , respectively, where zi , i = 3, 4 denote noise signals, b3 and b4 are, respectively, the amplitudes of the useful echoes s1 and s2 , c3 and c4 are the amplitudes of the received echo of interference p1 for the two experiments, respectively. Figure 9.15 shows matched filtering results and

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ESDs of y3 and y4 , respectively. Again, we can see that the interference is very evident in ESDs. The sidelobe level in the suppressed range of y4 after matched filtering is much higher than that of y3 as it is impacted by the strong narrowband interference.

9.5 Conclusions This chapter has addressed the shaping of LAF in the presence of narrowband spectral interferences with PAR and energy constraints. We have presented an iterative sequential algorithm to solve the resulting non-convex quadratic optimization problem. In each iteration, we turn the quadratic problem into a quadratic optimization problem and then convert it to a linear optimization problem with a closed-form solution. Numerical simulations have shown the proposed algorithm outperforms AISO in terms of the objective value, the computational time and the capability of prohibiting narrowband spectral interferences and the sidelobes of a strong target from masking weak targets. The experimental results have shown that the effectiveness of spectral coexistence in prohibiting the sidelobes suppressed in the LAF rising. Possible future research tracks might further concern the extension of ISQO to other quadratic problems, such as transmit beampattern design for colocated MIMO radar [34–37]. Moreover, an extension of LAF shaping to dual-use signal design for a novel dual-functional radar-communication architecture [26] is also of interest.

Appendix A Proof of Proposition 9.1 Starting from x(t) and assuming that x(t+1) is an enhanced solution of P(t+1) with respect to x(t) , x(t+1) satisfies     (A.1) u x(t+1) , x(t) ≥ u x(t) , x(t) .

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Radar waveform design based on optimization theory   According to the special structure of ϒ x(t) and Rc ∈ HN , that is,     u x(t+1) , x(t) = u x(t) , x(t+1) .

(A.2)

Then assorting to solve P(t+2) , we can get     u x(t+2) , x(t+1) ≥ u x(t) , x(t+1) .

(A.3)

Hence, we have the relationship that     u x(t) , x(t) ≤ u x(t+1) , x(t) =     u x(t) , x(t+1) ≤ u x(t+2) , x(t+1) =     u x(t+1) , x(t+2) ≤ u x(t+3) , x(t+2) = · · ·

(A.4)

Thus, the objective value sequence in P(t+1) monotonically increases with the rise of t. Besides, since P0 has the zero lower bound, P(t+1) has a upper bound. Hence, P(t+1) monotonically increases and converges to a finite value. From the previous discussion, we reasonably assume that limt→∞ x(t) = x(t+1) = ∗ x , which means that the objective value of P(t+1) does not exhibit any jumping behavior. Herein, we have         u x(0) , x(0) ≤ u x(0) , x(1) ≤, . . . , = u x(t) , x(t+1) = u x(∗) , x(∗) . (A.5)  (0)  On the contrary, we get W x ≥ W (x∗ ). Thus an enhance solution x∗ to P0 is achieved.

Appendix B Proof of (9.17) Since ϒ(x) is the sum of several positive semidefinite matrices and ϒ(x)  0, we can get [38]      1  † † λmax (ϒ(x)) ≤ wk, f λmax Mk, f xx† Mk, f + λmax Mk, f xx† Mk, f .(B.1) 2Q (k, f )∈ Besides, Lemma 9.1 indicates,   † † λmax Mk, f xx† Mk, f = x† Mk, f Mk, f x,   † † λmax Mk, f xx† Mk, f = x† Mk, f Mk, f x.

(B.2) (B.3)

Herein, by computing (B.2) and (B.3), we can get †

Mk, f Mk, f = J−k Jk ,

(B.4a)

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Mk, f Mk, f = Jk J−k . When k > 0, Jk J−k =

%

0 0 0 IN −k

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283

(B.6a)

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Remark 9.2. λ(t) influences the speed of decline and convergence of P(t+1) greatly. More specifically, the objective function of P(t+1) in each iteration declines rapidly for λ(t) is close to βλmax (ϒ(x)). Besides, the quadratic term G(x) in the optimization problem is much larger than the quadratic term F(x) when β is very small, which may lead the proposed algorithm in vain. At that situation, we can increase λ(t) properly to handle it.

Appendix C Proof of Proposition 9.2   According to the first-order Taylor expansion of the convex u xˆ , x(t) at xˆ = xˆ (i) , we have [39]       u xˆ , x(t) ≥ u xˆ (i) , x(t) + (ˆx − xˆ (i) )† v(i) .

(C.1)

  Hence, we only need to maximize xˆ † v(i) at the i + 1th iteration of the sequential   iteration procedure (i.e., solve Pi+1 ) to find a solution xˆ (i+1) ensuring xˆ (i+1)† v(i)       ≥ xˆ (i)† v(i) . Based on this, we can further get u xˆ (i+1) , x(t) ≥ u xˆ (i) , x(t) guaranteeing that the objective value in P(t+1) increases monotonically. Besides,  function  it is easy to prove that u xˆ , x(t) shares an upper bound. Thus, the sequential iteration procedure can be used to solve P(t+1) .

Acknowledgment This work was supported in part by the National Natural Science Foundation of China under grants 61771109 and 61871080, by Changjiang Scholars Program, by the 111 project no. B17008, by the Fundamental Research Funds for the Central Universities under grant 2672018ZYGX2018J016.

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Varadhan R, Roland C. Simple and globally convergent methods for accelerating the convergence of any EM algorithm. Scandinavian Journal of Statistics. 2008;35(2):335–353. Yu X, Cui G, Zhang T, Kong L. Constrained transmit beampattern design for colocated MIMO radar. Signal Processing. 2018;144:145–154. Yu X, Cui G, Kong L, et al. Constrained waveform design for colocated MIMO radar with uncertain steering matrices. IEEE Transactions on Aerospace and Electronic Systems. 2019;55(1):356–370. Yu X, Cui G, Yang J, et al. Wideband MIMO radar beampattern shaping with space-frequency nulling. Signal Processing. 2019;160:80–87. Yu X, Cui G, Yang J, et al. Wideband MIMO radar waveform design. IEEE Transactions on Signal Processing. 2019;67(13):3487–3501. Seber GA. A Matrix Handbook for Statisticians. Hoboken, NJ: John Wiley & Sons; 2008. Soltanalian M, Stoica P. Designing unimodular codes via quadratic optimization. IEEE Transactions on Signal Processing. 2014;62(5):1221–1234.

[34] [35]

[36] [37] [38] [39]

Chapter 10

Relative entropy-based waveform design for MIMO radar Bo Tang1 and Jun Tang2

10.1 Introduction Multiple-input–multiple-output (MIMO) radar has received considerable attention in recent years. Two typical configurations of MIMO radar have been extensively discussed (see, e.g., [1–3] and the references therein): one is called statistical MIMO radar (or distributed MIMO radar), i.e., MIMO radar with widely separated antennas. Statistical MIMO radar uses the spatial diversity to enhance the target detection and parameter estimation performance. The other is called colocated MIMO radar or coherent MIMO radar that exploits waveform diversity and can be understood as an extension of traditional phased-array radar systems, showing improved parameter identifiability and superior flexibility in synthesizing transmit beampatterns. For both types of MIMO radar, one of the most interesting problems is how to design the transmitted waveforms properly. Thus, much effort has been devoted to the optimization of MIMO radar waveforms to achieve performance gains. Particularly, we list some addressed waveform optimization problems as follows: ●

●

1 2

Design of quasi-orthogonal waveforms. Orthogonal waveforms facilitate the extraction of target information with matched filtering, from the echoes associated with the multiple propagation paths of a MIMO radar. In [4–6], the authors proposed to minimize the autocorrelation and cross-correlation sidelobes of the waveforms and developed efficient optimization methods to design the quasi-orthogonal waveforms. Synthesis of waveforms to approximate a desired transmit beampattern. Transmitting probing signals to maximize the power around the target of interest can enhance the radar performance (see, e.g., [7,8]). Hence, in [7,9–11], the authors focused on the synthesis of MIMO radar waveforms to match a desired beampattern. Remarkably, many beampattern-matching problems can be formulated as a semidefinite programming (SDP) problem, meaning that we can obtain the globally optimal solution with polynomial complexity.

College of Electronic Engineering, National University of Defense Technology, Hefei, China Department of Electronic Engineering, Tsinghua University, Beijing, China

288 ●

●

Radar waveform design based on optimization theory Design of waveforms maximizing signal-to-interference-plus-noise ratio (SINR). SINR has very close connection with the detection performance of radar systems. Therefore, in [12–15], the authors studied the waveform design method based on maximizing SINR. They proposed cyclic optimization methods to tackle the associated fractional programming problem. The proposed cyclic optimization methods therein monotonically increase the SINR and have guaranteed convergence of SINR. Waveform optimization with information-theoretic criteria (including mutual information, relative entropy, and Bhattacharyya distance). Waveforms synthesized with information-theoretic measures have been shown to exhibit superior performance (see, e.g., [16–20]). In [21], it is shown that by transmitting waveforms that maximize the mutual information between the random target ensemble and the target reflections, performance of target parameter estimation, classification, and identification can be improved. In [16], the authors showed that for a linear-Gaussian channel, the optimal MIMO radar waveforms maximizing the mutual information coincided with those minimizing the minimum mean square error (MMSE). In [17], it is proved that waveforms maximizing the mutual information achieve the Chernoff bound for MIMO radar systems sensing uncorrelated targets (i.e., random targets with their covariance matrices equal to scaled identity matrices).

In this chapter, we design waveforms to improve the detection performance of MIMO radar systems in the presence of disturbances (including signal-dependent clutter and signal-independent colored interference). Given that waveform design based on the maximization of the detection probability (for a given probability of false alarm) is intractable, we use relative entropy associated with the detection problem (i.e., relative entropy between the distributions of the observations under the null and alternative hypotheses) as the design metric. We show that the waveform design problem is non-convex. To tackle the non-convex waveform design problem, we propose a two-stage design algorithm. The two-stage design method involves one stage to optimize a hidden variable (i.e., C S defined in Section 10.3) and a second stage to extract waveforms from the hidden variable. Moreover, in the optimization of the hidden variable, the two-stage algorithm makes use of the minorization–maximization (MM) technique to deal with the non-convex objective, and an iterative trick to deal with the non-convex constraint. Owing to the ascent property of MM-based algorithms, the proposed two-stage design method increases the objective value monotonically and has guaranteed convergence of the objective values. Furthermore, we discuss how to synthesize practical waveforms (including constant-modulus waveforms and waveforms under a similarity constraint) from the hidden variable. However, at every iteration, the two-stage method needs to solve a determinant maximization (MAXDET) problem and its computational complexity is high. Thus, it is highly desirable to devise more efficient methods to reduce this complexity. To this end, we propose a one-stage design algorithm. The one-stage method is also based on MM technique and uses a quadratic function or a linear function (of the waveforms) to minorize the objective function. Since we can obtain a

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closed-form solution for the quadratic/linear programming problem at every iteration, the per-iteration computational complexity of the proposed algorithm is much lower than that of the two-stage algorithm. In addition, we exploit an accelerated scheme, called squared iterative method (SQUAREM), to enhance the convergence rate of the one-stage algorithm. We show that the accelerated one-stage algorithm can be computationally more efficient than the two-stage method. The rest of this chapter is outlined as follows. Section 10.2 establishes the signal model and formulates the waveform design problem. We propose a two-stage deign algorithm in Section 10.3. To reduce the computational complexity of the twostage algorithm, we propose a one-stage design algorithm in Section 10.4. In Section 10.5, we analyze the performance of the proposed algorithms. Finally, we provide concluding remarks in Section 10.6.

10.2 Signal model and problem formulation 10.2.1 Signal model Consider a MIMO radar system with NT transmitting and NR receiving antennas. Let sm (t) denote the emitted signal of the mth transmitter. First, we establish the signal model of target returns. Assume that the targets of interest are moving slowly such that their intra-pulse Doppler shifts are negligible. Under this assumption, the down-converted target return in the nth receiver can be modeled by rnt (t) =

NT 

hˆ m,n sm (t − τm,n ) exp ( − j2πfc τm,n ),

(10.1)

m=1

where hˆ m,n , the strength of which is proportional to the radar cross section of the target, is the amplitude of the target echoes from the mth transmitter to the nth receiver, τm,n is the associated two-way propagation delay, and fc is the carrier frequency of the emitted signals. Assume that the emitted signals are narrowband such that the delay differences between various paths are insignificant within the complex envelope, i.e., sm (t − τm,n ) ≈ sm (t − τ ), m = 1, 2, . . . , NT , n = 1, 2, . . . , NR ,

(10.2)

where τ is a constant. Let sm = [sm (1), sm (2), . . . , sm (L)]T with sm (l) denoting the lth discrete sample of sm (t), where L is the code length (L > NT ), and let S = [s1 , s2 , . . . , sNT ] ∈ CL×NT be the waveform matrix. Define hm,n = hˆ m,n exp ( − j2π fc τm,n ) and hn = [h1,n , h2,n , . . . , hNT ,n ]T ∈ CNT . Then the discrete-time target model corresponding to (10.1) can be written as r tn = Shn , n = 1, 2, . . . , NR .

(10.3)

with r tn = [rnt (1), rnt (2), . . . , rnt (L)]T ∈ CL , n = 1, 2, . . . , NR , and rnt (l) being the lth discrete sample of rnt (t).

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Let r t = [(r t1 )T , (r t2 )T , . . . , (r tNR )T ]T ∈ CNRL be the vector of the collected target returns and h = [hT1 , hT2 , . . . , hTNR ]T ∈ CNTR be the target response vector (associated with the NTR propagation paths), where NTR = NT NR and NRL = NR L, then r t can be written as rt =  Sh,

(10.4)

where  S = (I NR ⊗ S). Next, we establish the clutter model. In radar systems, clutter refers to reflections from unwanted targets, e.g., returns from ground, sea, and cloud. Hence, clutter is a signal-dependent interference. For stationary clutter or clutter with low Doppler frequencies, it can be modeled by rc =  Sc,

(10.5)

where c ∈ CNTR stands for the clutter response vector. Combining the results in (10.4) and (10.5), we establish the following signal model: y = r t + r c + w, NRL ×1

where y ∈ C denotes the received signal, and w ∈ C independent) colored interference.

(10.6) NRL ×1

is the (signal-

10.2.2 Problem formulation Detecting the targets is tantamount to the following binary hypothesis testing problem:  H0 : y = r c + w, (10.7) H1 : y = r t + r c + w. Assume that h, c, and w are mutually independent and circularly symmetric complex Gaussian random processes with zero mean and covariance matrices RH ∈ CNTR ×NTR , RC ∈ CNTR ×NTR , and RW ∈ CNRL ×NRL , respectively. We also assume that RW has a Kronecker structure, i.e., RW = RS ⊗ RT ,

(10.8)

where RS and RT denote the spatial and temporal correlations of the interference, † respectively. Under this assumption, we can observe that E[wn wk ] = RS (n, k)RT . Here, RS (n, k) might account for the antenna pattern and the mutual coupling between the nth and kth receiving antennas [22]. Under the previous assumptions, we can obtain the optimal Neyman–Pearson detector straightforwardly via the estimator-correlator theorem [23]. In addition, closed-form expressions for the probabilities of detection and false alarm associated with the detector are immediate [24]. However, these formulas are too complicated to be used as the waveform design metric. Instead, we resort to relative entropy, which is an information-theoretic metric, to measure the detection performance and design the waveforms. Indeed, the connection between relative entropy and detection performance can be understood using Stein’s lemma [25], which states that, if the probability

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of false alarm Pfa is fixed, the probability of miss Pmiss is exponentially small, with an exponential rate equal to the relative entropy between P0 and P1 , where P0 and P1 denote the distributions of the observations under the two hypotheses, respectively. Therefore, the maximization of the relative entropy allows us to minimize the probability of miss asymptotically (i.e., maximize the probability of detection). To proceed, we write the probability density functions of y under the two hypotheses as follows:   −1  1 † †     P0 (y) = SR exp −y S + R y , (10.9) C W † π NRL det  SRC S + RW   −1  1 † †     exp −y SRHC S + RW P1 (y) = y , (10.10) † S + RW π NRL det  SRHC where RHC = RH + RC denotes the sum of the target and clutter covariance matrices. As a result, the relative entropy between P0 (y) and P1 (y) can be calculated: P0 (y) D(P0 P1 ) = P0 (y) log dy P1 (y) † † = log det( SRHC S + RW ) − log det( S + RW ) SRC † † + tr[( SRHC S + RW )−1 ( SRC S + RW )] − NRL .

(10.11)

Thus, we formulate the waveform design problem based on maximizing relative entropy as follows: † † S + RW ) − log det ( S + RW ) SRHC SRC max log det ( S

S + RW )−1 ( S + RW )] SRC + tr[( SRHC †

s.t. tr(S † S) ≤ et ,

†

(10.12)

where et denotes the total available transmit energy, and we have ignored the constant terms. We can notice that the formulation of the relative-entropy-based waveform design problem requires a priori knowledge of the second-order statistics of the target, the clutter, as well as the colored interference. Thus, we assume that such prior knowledge can be available by using the information collected from previous scans, or by the access to an environmental database, including a geographical information system, National Land Cover Data, and digital terrain elevation data. Such a design methodology is widely used in radar waveform optimizations, and it is extremely useful in a target confirmation/tracking stage when the radar looks over a specific cell. We conclude this section with a remark that the signal model in (10.6) can be applied to several typical types of MIMO radar systems. For a colocated MIMO radar, the target response matrix can be modeled by [3]  H= αt,k a(θt,k )bT (θt,k ), (10.13) k

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where αt,k is the amplitude of the kth target, a(θt,k ) and b(θt,k ) are the associated transmit and receive array steering vectors, respectively, and θt,k is its direction-ofarrival (DOA). As a result, the target response vector is h = vec(H ) = αt,k b(θt,k ) ⊗ a(θt,k ), and the target covariance matrix RH can be written as  2 RH = σt,k (b(θt,k ) ⊗ a(θt,k ))(b(θt,k ) ⊗ a(θt,k ))† ,

(10.14)

(10.15)

k 2 = E[|αt,k |2 ] represents the average strength of the kth target. Similarly, the where σt,k clutter covariance matrix can be modeled as  2 RC = σc,k (b(θc,k ) ⊗ a(θc,k ))(b(θc,k ) ⊗ a(θc,k ))† , (10.16) k

and θc,k stand for the average strength and the DOA of the kth clutter patch, respectively. For a bistatic MIMO radar [26], the target and clutter covariance matrix can be modeled by  2 RH = σt,k (b(ψt,k ) ⊗ a(θt,k ))(b(ψt,k ) ⊗ a(θt,k ))† (10.17) 2 where σc,k

k

and RC =



2 σc,k (b(ψc,k ) ⊗ a(θc,k ))(b(ψc,k ) ⊗ a(θc,k ))† ,

(10.18)

k

where θt,k and ψt,k denote the direction-of-departure (DOD) and DOA of the kth target, respectively, and θc,k and ψc,k denote the DOD and DOA of the kth clutter patch, respectively. For a statistical MIMO radar system with widely separated receive antennas, we can ignore the correlations among the columns of the target response matrix H [2]. Then RH can be written as RH = Blkdiag(RT,1 ; RT,2 ; . . . ; RT,NR ), where RT,k = E[hk h†k ], hk denotes the kth column of H , and Blkdiag([A; B]) denotes the block-diagonal matrix formed by the matrices A and B. If we assume that the correlations of the columns of H are identical (i.e., RT,k = RT , k = 1, 2, . . . , NR ), then RH = I NR ⊗ RT .

10.3 Two-stage algorithm design In this section, we propose a two-stage optimization algorithm to tackle the problem in (10.12). Specifically, in the first stage, we propose a method to optimize C S = S † RT−1 S; we obtain the waveform matrix S from C S in the second stage. Then, we analyze the computational complexity and convergence of the algorithm. In addition, we extend the proposed algorithm to synthesize practical waveforms, including constant-modulus waveforms and similarity-constrained waveforms.

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10.3.1 Synthesis of energy-constrained waveforms Let h : S + → R≥ 0 be the mapping given by h(M ) = log det (I + M ) + tr[(I + M )−1 ], where S + denotes the set of positive semidefinite matrices, respectively. Then, as shown in Appendix A, the objective function in (10.12) can be written as    −1 1/2 −1 1/2 h RH C + NRL − NTR , (10.19) + R R C H S † −1  S =  S = R−1 S RW where C S ⊗ C S . As a result, the optimization problem in (10.12) can be recast as 1/2 −1 −1 1/2 max h(RH (C S + RC ) RH ) C S ,S

s.t. tr(S † S) ≤ et ,

(10.20)

where we have ignored the irrelevant constant terms. Next, we introduce a lemma to characterize the structure of the optimal solution to (10.20). Lemma 10.1. Let CS and RT have the eigen-decompositions CS = V1  ↓1 V†1 and

 RT = VT T↑VT† , respectively. Herein,  ↓1 = diag [σ1,1 , . . . , σ1,NT ] such that σ1,1 ≥

 · · · ≥ σ1,NT ; T↑ = diag [σT,1 , . . . , σT,L ] such that σT,1 ≤ · · · ≤ σT,L . Then, the optimal solution to (10.20) has the following structure: T  1/2 S = V s V1 , †

(10.21)

T being the first NT with  s = diag[σ1,1 σT,1 , . . . , σ1,NT σT,NT ] and the columns of V columns of VT . Proof : See Appendix B. We can observe from (10.21) that the left singular vectors of the optimal waveform matrix should be aligned with the least interfered interference subspace. Furthermore, the optimal waveform matrix can be synthesized from the optimal C S . Note also that using (10.21), the energy constraint in (10.20) can be rewritten as tr(S S) = †

NT 

σT,k σ1,k ≤ et .

(10.22)

k=1

Therefore, the waveform design problem in (10.20) can be recast as 1/2 −1 −1 1/2 max h(RH (C S + RC ) RH ) CS

s.t.

NT 

σT,k σ1,k ≤ et .

(10.23)

k=1

Note that the optimization variable in (10.23) becomes C S . When we find a solution to (10.23) (denoted by C S ), we can obtain S  from C S by (10.21).

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To tackle the optimization problem in (10.23), we apply the matrix inversion lemma [27] and obtain that −1  −1 1/2  1/2 −1 1/2  S − C S R1/2  S + RC =C (10.24) C C (RC C S RC + I NTR ) RC C S . In addition, recall that (see Appendix A)  −1 1/2 −1 −1 1/2 + NRL − NTR + R ) R tr I NTR + RH (C C H S   † † = tr ( SRHC S + RW )−1 ( S + RW ) SRC     † † 1/2 †  1/2 S (SRHC S + RW )−1 S + RW )−1 RW . (10.25) = tr RC  SRC + tr ( SRHC Using the matrix inversion lemma again, we have † 1/2 †  1/2 S (SRHC S + RW )−1 SRC RC  1/2  1/2 1/2  1/2 1/2  1/2 −1 1/2  1/2 = RC C S RC − RC C S RHC (I NTR + RHC C S RHC ) RHC C S RC .

(10.26)

Moreover, using the results of [18, Lemma 4] yields † S + RW )−1 RW ] tr[( SRHC † −1/2 −1/2 = tr[(RW  SRHC S RW + I NRL )−1 ] 1/2  1/2 −1 = tr[(RHC C S RHC + I NTR ) ] + (NRL − NTR ).

Define



S ; M ) = C 1 (C

1/2  1/2 I NTR + RH C S RH − M

1/2  1/2 RH C S RC

1/2  1/2 RC C S RH

1/2  1/2 I NTR + RC C S RC   1/2 1/2  1/2 S R1/2 RC C RC C S RHC C −M  . C 2 (C S ; M ) = 1/2  1/2 1/2  1/2 RHC C I NTR + RHC C S RC S RHC

(10.27)  ,

(10.28)

(10.29)

Combining the results in (10.24)–(10.27), and using the definitions in (10.28) and (10.29), we can rewrite the optimization problem in (10.23) as follows:   1/2  1/2 cn cn −1 C log det (M ) + tr(M ) + tr (R R + I ) max NTR HC S HC 1 2 cn cn C S ,M 1 ,M 2

cn S ; M cn  s.t. C 1 (C 1 )  0, C 2 (C S ; M 2 )  0, cn S = R−1  C S ⊗ C S , C S  0, C S  0, M 1 0, NT M cn σT,k σ1,k ≤ et , 2  0, k=1

(10.30)

cn where M cn 1 0 and M 2  0 are auxiliary matrices, and we have used the Schur complement theorem [27] in the formulation of the first two constraints. Note that the first two terms of the objective function in (10.30) are concave with 1/2  1/2 cn −1 respect to (w.r.t.) M cn 1 and M 2 . However, the convexity of tr[(RHC C S RHC + I NTR ) ]

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S [28] leads to the non-concavity of the third term. Therefore, the waveform w.r.t. C design problem in (10.30) is non-convex. Regarding the energy constraint (i.e., the last constraint in (10.30)), we can notice that {σT,k }k is increasing w.r.t. k, whereas {σ1,k }k is decreasing w.r.t. k for k = 1, . . . , NT , i.e., the two sets of eigenvalues are sorted in different orders. The convex analysis results in [29, Chapter 11] show that the constraint on a weighted sum of the eigenvalues of a positive semidefinite matrix is convex only if the weights and eigenvalues are sorted in the same order. Hence, the constraint set in (10.30) is not convex w.r.t. σ1,k , which leads to another challenge. To tackle the non-convex problem in (10.30), we first employ MM technique to deal with the non-convex objective function (we refer to Appendix C for a brief introduction to MM methods). Then, we propose an iterative procedure to handle the non-convex constraint set. • Handling the non-convex objective function: To tackle the optimization problem in (10.30) with MM technique, we need to 1/2  1/2 −1 find a minorizer for tr[(RHC C S RHC + I NTR ) ]. To this end, we note that, for a general differentiable convex function Q(θ ), it is minorized by its supporting hyperplanes [28], i.e., Q(θ ) ≥ Q(θ (k) ) + ∇Q(θ (k) )T (θ − θ (k) ),

(10.31)

where ∇Q(θ ) is the gradient of Q at θ . Hence, it can be verified that a minorizer 1/2  1/2 −1 (k) for tr[(RHC C S RHC + I NTR ) ] at C S is given by (k)

(k)

1/2 (k) 1/2 −1 (k)  S ; C (k) (k) J1 (C S ) = tr[(RHC C S RHC + I NTR ) ] + tr[A (C S − C S )]

(10.32)

where 1/2 1/2 (k) 1/2 −2 1/2 A(k) = −RHC (RHC C S RHC + I NTR ) RHC 1/2  1/2 is the gradient of tr[(RHC C S RHC

(10.33)

(k) I NTR )−1 ] at C S , (k)

+ and we have used standard derivative properties in the calculation of A [30]. Therefore, we consider the following minorizer as the objective function at the (k + 1)th iteration: cn (k)  log det (M cn 1 ) + tr(M 2 ) + tr[A C S ] + const, 1/2 (k) 1/2 tr[(RHC C S RHC

(10.34)

(k) tr(A(k) C S ).

+ I NTR )−1 ] − Note that the objective where const = function in (10.34) is now concave. Hence, next we handle the non-convex constraint in (10.30). • Handling the non-convex energy constraint: Now we devise an iterative procedure to deal with the non-convex energy con T straint (i.e., the constraint that Nk=1 σT,k σ1,k ≤ et ). To this end, we consider the following optimization problems: Pg : max g(C S ), s.t. CS

NT 

σT,k σ1,k ≤ et ,

(10.35)

k=1

and (i) Pg(i) : max g(C S ), s.t. tr(C S Rˆ T ) ≤ et , CS

(10.36)

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Radar waveform design based on optimization theory

(i) (i) ˆT = ˆ T (Vˆ T(i) )† with  where g(C S ) is a concave function w.r.t. C S , Rˆ T = Vˆ T  diag([σT,1 , . . . , σT,NT ]) being a diagonal matrix of the NT smallest eigenvalues of RT , (i) (i) (i) and Vˆ T is a unitary matrix that satisfies (Vˆ T )†Vˆ T = I NT . Herein, the superscript (i) is associated with the ith iteration of the proposed procedure. The next lemma lays the ground for the derivation of the proposed procedure.

Lemma 10.2. ([31, H.1.h] ) Suppose A and B are n × n matrices in S + . Let † † A = UA  A UA and B = UB  B UB be their eigen-decompositions, respectively, where , αn ]),  B = diag([β1 , . . . , βn ]), α1 ≥ · · · ≥ αn , and β1 ≤ · · · ≤  A = diag([α1 , . . .  βn . Then, tr(AB) ≥ ni=1 αi βi and the lower bound is achieved if and only if UA = UB . According to Lemma 10.2, if Cˇ S is feasible for (10.36), we have NT 

(i) σT,k σˇ 1,k ≤ tr(Cˇ S Rˆ T ) ≤ et ,

(10.37)

k=1

with σˇ 1,1 ≥ · · · ≥ σˇ 1,NT being the eigenvalues of Cˇ S . The previous result means that Cˇ S is also feasible for (10.35), meaning that the feasibility region of (10.36) is smaller than that of (10.35). Therefore, one can infer that v(Pg(i) ) ≤ v(Pg ). (i) Let C˘ S denote the optimal solution of (10.36) with the following eigendecomposition (i) (i) (i) † ˘ 1 (V˘ (i) C˘ S = V˘ 1  1 ) ,

(10.38)

˘ 1 = diag([σ˘ 1,1 , . . . , σ˘ 1,N ]) and {σ˘ 1,k }k is nonincreasing. Set where  T (i)

(i)

(i)

(i)

(i+1) (i) = V˘ 1 , Vˆ T (i+1) Rˆ T

=

(10.39)

(i+1) ˆ T (Vˆ T(i+1) )† , Vˆ T 

and consider the following optimization problem at the (i + 1)th iteration of the proposed procedure: (i+1)

Pg(i+1) : max g(C S ), s.t. tr(C S Rˆ T CS

) ≤ et .

(10.40)

Note that C˘ S is feasible for problem (10.40), since (i)

(i) (i+1) tr(C˘ S Rˆ T ) =

NT 

(i) (i) (i) σ˘ 1,k σT,k ≤ tr(C˘ S Rˆ T ) ≤ et ,

(10.41)

k=1

where the first inequality follows from Lemma 10.2, and the second is due to the (i) feasibility of C˘ S for (10.36). Consequently, it is checked that v(Pg(i) ) ≤ v(Pg(i+1) ) ≤ v(Pg ).

(10.42)

Relative entropy based waveform design for MIMO radar

297

Using the results (10.36)–(10.42), we can formulate an iterative procedure to deal with the non-convex constraint in (10.30). Moreover, the proposed iterative procedure guarantees the monotonic increase of the objective value during the iterations. To sum up, at the (i, k + 1)th iteration, we consider the following optimization problem: ⎧ cn (i,k)  ⎪ max log det (M cn C S] 1 ) + tr(M 2 ) + tr[A ⎪ cn ,M cn ⎪ C ,M S ⎪ 1 2 ⎪ ⎨ cn S ; M cn  s.t. C 1 (C (i,k+1) 1 )  0, C 2 (C S ; M 2 )  0, (10.43) = Ph  S = R−1 ⎪ C ⎪ S ⊗ C S , C S  0, C S  0, ⎪ ⎪ ⎪ ⎩ cn ˆ (i) M cn 1  0, M 2  0, tr(C S RT ) ≤ et , S with C S ). where A(i,k) is available via using (10.33) (by replacing C Note that the constraints in (10.43) are convex and hence, (10.43) is a convex optimization problem. More precisely, this optimization is a determinant maximization (MAXDET) problem [32] (belonging to the class of SDP problems), which can be solved efficiently by interior point methods with polynomial time [28]. Algorithm 1 summarizes the two-stage waveform design algorithm for MIMO radar detection in the presence of clutter. Note that the method has outer iterations (indicated with superscript (i)) to handle the non-convex constraint as well as inner iterations (indicated with superscript (k)) to deal with the non-concave objective. For both inner and outer iterations, we terminate the loops when the increase of the relative entropy associated with the synthesized waveform matrix at the current iteration becomes insignificant. (k)

(i,k)

10.3.2 Convergence and computational complexity analysis We first analyze the convergence of the proposed method summarized in Algorithm 1. Let Nk denote the number of iterations until convergence of the MM algorithm in the ith outer iteration. Note that we have (i,Nk−1 )

v(Ph

(i+1,0)

) = v(Ph

(i+1,1)

) ≤ v(Ph

(i+1,Nk )

) ≤ · · · ≤ v(Ph

),

(10.44)

where the first equality holds because we use the solution obtained in the ith iteration as the initial point for the (i + 1)th iteration. The inequalities are the result of the monotonic ascent property of the MM algorithm and the proposed iterative procedure. Consequently, the relative entropy associated with C S increases at each (outer) iteration (i). This observation along with the fact that the relative entropy is bounded leads to the convergence of the sequence of the objective values. Regarding the computational complexity of the proposed algorithm, it is determined by the number of iterations until convergence and the per-iteration computational complexity. Moreover, the per-iteration computational complexity is mainly due to the overall inner iterations (for solving the convex MAXDET problem). Note 3 that the calculation of A(i,k) requires O(NTR ) operations, and solving the MAXDET 6.5 problem requires O(NTR ) operations. Thus, the computational complexity for every 6.5 3 inner iteration is about O(NTR + NTR ).

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Radar waveform design based on optimization theory

Algorithm 1: Two-stage waveform design method for MIMO radar detection in the presence of clutter Input: RH , RC , RW , et . Output: S. 1 Initialize: 2 i = 0. (i) 3 Initialize Vˆ T with a random unitary matrix. 4 5

6 7 8 9 10 11 12 13 14 15

16 17 18 19 20 21

(i) (i) ˆ T (Vˆ T(i) )† . Rˆ T = Vˆ T  (i,0) Initialize C S with a random positive semidefinite matrix such that (i) (i,0) tr(Rˆ T C S ) ≤ et . repeat k = 1. repeat Compute A(i,k−1) via (10.33). (i,k), Solve the MAXDET problem in (10.43) to obtain C S . (i,k+1) (i,k), Let C S = CS . k = k + 1. until convergence; (i),MM (i,k) C˘ S = CS .

Perform the eigen-decomposition of C˘ S like that in (10.38) with the (i) associated unitary matrix denoted by V˘ 1 . (i),MM

(i+1) (i+1) ˆ TVˆ T(i+1) with Vˆ T(i+1) = V˘ (i) Rˆ T = Vˆ T  1 . (i),MM (i+1,0) ˘ Let C S = CS . i = i + 1. until convergence; (i,0) C S = C S . Obtain S from C S with (10.21).

10.3.3 Extension to the synthesis of constant-modulus waveforms In practical radar systems, constant-modulus waveforms are of particular interest. Constant modulus property avoids unnecessary nonlinear effects in transmitters and allows the radio-frequency amplifier to operate at maximum efficiency. Thus, in this subsection, we extend the proposed two-stage algorithm to design constant-modulus waveforms. Motivated by the idea in [33] and considering the fact that C S = S † RT−1 S, we consider the next minimization problem to synthesize constant-modulus waveforms: −1/2

min RT

S∈C ,Q

S − QK 2F ,

(10.45)

Relative entropy based waveform design for MIMO radar

299

where C denotes the constant-modulus constraint set, Q is a semiunitary matrix (i.e., Q† Q = I NT ), K = (C S )1/2 , and we call the objective value in (10.45), the matching error. Note that for small matching errors, we have C S = S † RT−1 S ≈ C S and hence, the relative entropy of the synthesized constant-modulus waveforms approaches that of the energy-constrained waveforms. Note also that the problem in (10.45) is nonconvex. Herein, we propose a cyclic approach to tackle this problem. The cyclic algorithm tackles the optimization problem in (10.45) iteratively and includes two steps at each iteration: a. b.

−1/2

Given S, solve minQ RT S − QK 2F ; −1/2 Given Q, solve minS∈C RT S − QK 2F . −1/2

†

To solve the minimization problem (10.45) for given S, we let RT SK = U Q  QV Q −1/2 be the singular value decomposition of RT SK where U Q ∈ CL×NT ,  Q ∈ CNT ×NT , ˆ = U QV †Q and V Q ∈ CNT ×NT . Then the optimal solution to problem (a) is given by Q [33]. As to the case of given Q, note that −1/2

RT

S − QK 2F = tr(S † RT−1 S) + tr(C S ) − 2Re(tr(S † QT )), −1/2

where QT = RT as follows:

(10.46)

QK . Thus, the associated optimization problem can be formulated

min tr(S † RT−1 S) − 2Re(tr(S † QT )) S √ s.t. |S(l, n)| = ps , l = 1, . . . , L, n = 1, . . . , NT ,

(10.47)

where ps = et /(NT L). The optimization problem in (10.47) belongs to the class of unimodular quadratic programming (UQP) problems. We can use majorization–minimization methods to tackle such problems efficiently.∗ Specifically, we note that −1 I L )(S − S (k) )) ≤ 0, tr((S − S (k) )† (RT−1 − σT,1

(10.48)

where S (k) denotes the solution of S at the kth iteration. After some algebraic manipulations, we can obtain that −1 −1 I L )S (k) )) + 2σT,1 et tr(S † RT−1 S) ≤ 2Re(tr(S † (RT−1 − σT,1

−tr((S (k) )† RT−1 S (k) ).

(10.49)

It can be verified that the right-hand side of (10.49) is a majorizer of tr(S † RT−1 S). As a result, the majorized problem at the (k + 1)th iteration can be formulated as (k)

max 2Re(tr(S † QT )) S √ s.t. |S(l, n)| = ps , l = 1, . . . , L, n = 1, . . . , NT , ∗

(10.50)

The principle of majorization–minimization is similar to that of MM (see, e.g., Appendix C for an introduction).

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Radar waveform design based on optimization theory

−1 I L )S (k) and we have ignored the constant terms. It is where QT = QT − (RT−1 − σT,1 easy to verify that the solution to (10.50) is √ (k) S(l, n) = ps exp (jarg(QT (l, n))), l = 1, . . . , L, n = 1, . . . , NT . (10.51) (k)

10.3.4 Extension to the synthesis of similarity-constrained waveforms In this subsection, we extend the proposed algorithm to synthesize waveforms with a similarity constraint. Enforcing a similarity constraint on a waveform can control the shape of the ambiguity function of the waveform, partially circumventing the drawbacks, including significant modulus variation, poor range resolution, and/or high peak sidelobe level problems [34]. In this subsection, we consider the following similarity constraint on the waveforms: s − s0 22 ≤ δ,

(10.52)

where s = vec(S), s0 = vec(S 0 ), S 0 denotes the reference waveforms (with some desired properties and the same energy as S), and δ is the user-specified parameter ruling the size of similarity region and 0 ≤ δ ≤ 2et . When the similarity constraint is enforced, we can synthesize the waveforms using the idea in Section 10.3.3. More precisely, the problem stated in part (a) of the previous subsection remains unchanged; for part (b), we can also use MM methods to tackle the associated optimization problem. Specifically, at the (k + 1)-iteration, we consider the following optimization problem for given Q: (k)

max 2Re(s† qT ) S

s.t. s† s = et , s − s0 22 ≤ δ, (k) qT

(10.53)

(k) vec(QT ).

where = Note that the similarity constraint can be equivalently expressed as (2et − δ) . 2 Thus, (10.53) can be recast as Re(s† s0 ) ≥

(10.54)

(k)

max 2Re(s† qT ) S

s.t. s† s = et , Re(s† s0 ) ≥ (2et − δ)/2.

(10.55)

The optimization problem (10.55) can be tackled via the Lagrange multipliers method. More precisely, let the Lagrangian associated with problem (10.55) be (k)

F(s, λ1 , λ2 ) = −2Re(s† qT ) + λ1 (s† s − et ) + λ2 (2et − δ − 2Re(s† s0 )), where λ1 and λ2 ≥ 0 are the Lagrange multipliers associated with the constraints. The minimizer of F(s, λ1 , λ2 ) for fixed λ1 > 0 and λ2 is given by (k)

s =

q T + λ2 s 0 λ1

(10.56)

Relative entropy based waveform design for MIMO radar

301

and the corresponding minimum value becomes (k)

−

(k)

(qT + λ2 s0 )† (qT + λ2 s0 ) − λ1 et + λ2 (2et − δ). λ1

(10.57)

In addition, the optimal λ1 can be obtained considering the constraint s† s = et :  (k) (k) (qT + λ2 s0 )† (qT + λ2 s0 ) . (10.58) λ1 = et Finally, λ2 is available by maximizing the function F(s , λ1 , λ2 ) that leads to the following problem:  (k) (k) max −2 et (qT + λ2 s0 )† (qT + λ2 s0 ) + λ2 (2et − δ) λ2

s.t. λ2 ≥ 0.

(10.59)

Let g(λ2 ) denote the first derivative of the previous objective function (w.r.t. λ2 ). It is straightforward to verify that † (k)

g(λ2 ) = 2et − δ − 

et (2et λ2 + 2Re(s0 qT )) (k) et (qT

+

(k) λ2 s0 )† (qT

(10.60)

+ λ2 s 0 )

and (k)

g  (λ2 ) =

(k)

† (k)

−4[et (qT )† qT − (Re(s0 qT ))2 ] (k)

(k)

[(qT + λ2 s0 )† (qT + λ2 , s0 )]3/2

< 0,

(10.61)

meaning that the optimization problem in (10.59) is concave w.r.t. λ2 , and g(λ2 ) is a monotonically decreasing function of λ2 . Consequently, if g(0) ≤ 0, the optimal solution to the concave problem (10.59) is λ2 = 0. Otherwise, we can obtain λ2 via solving the equation g(λ2 ) = 0. Considering the monotonicity of g(λ2 ), we can find the solution efficiently (e.g., via Newton’s method).

10.4 One-stage algorithm design In this section, we propose a one-stage algorithm to tackle the optimization problem in (10.12). This algorithm is also devised based on MM, and it proposes a quadratic or a linear function of s to minorize the highly nonlinear objective function in (10.12). Due to the simpler minorizer, we can obtain the optimal solution at every iteration more efficiently. We also extend the proposed one-stage algorithm to design waveforms under constant-modulus and similarity constraint, respectively. Moreover, we introduce accelerated schemes to speed up the convergence of the one-stage algorithm. First, we rewrite the first two parts of the objective of (10.12) as † † log det( SRHC S + RW ) − log det( S + RW ) SRC 1/2  1/2 1/2  1/2 = log det(RHC C S RHC + I NTR ) − log det(RC C S RC + I NTR ),

302

Radar waveform design based on optimization theory

where we have used the standard property of matrix determinant that det (I + AB) = det (I + BA). Thus, we can split the objective of (10.12) into three parts: 1/2  1/2 1/2  1/2 f (S) = log det (RHC C + I NTR ) − log det (RC C + I NTR ) SR SR   HC    C  Part A

Part B

+ tr[( SRHC S + RW )−1 ( S + RW )] . SRC    †

†

(10.62)

Part C

Next, we find the minorizer for each part of (10.62).

10.4.1 Minorizing Part A By using the standard property of matrix determinant, we obtain S RHC + I NTR ) = − log det (RHC C S RHC + I NTR )−1 . log det (RHC C 1/2

1/2

1/2

1/2

(10.63)

Noting that with the matrix inversion lemma, we have † 1/2  1/2 1/2 †  1/2 −1 (RHC C S (SRHC S + RW )−1 SRHC . = I NTR − RHC  S RHC + I NTR )

(10.64)

Moreover, using the block matrix inversion lemma [27], we can rewrite the right-hand side of (10.64) as follows: † 1/2 †  1/2 † −1 I NTR − RHC  S (SRHC S + RW )−1 SRHC = (J A B−1 A J A) ,

where J A = [I NTR , 0NTR ×NRL ] and ⎤ ⎡ 1/2 † RHC  I NTR S ⎦. BA = ⎣ † 1/2    SRHC RW + SRHC S

(10.65)

(10.66)

Thus, Part A of the objective can be rewritten as 1/2  1/2 −1 † log det (RHC C S RHC + I NTR ) = log det (J A BA J A ).

(10.67)

Lemma 10.3. [35] Assume that B ∈ Cn×n is positive definite. Then for any full row rank matrix J ∈ Cm×n (m ≤ n), log det (JB−1 J† ) is convex w.r.t. B. With Lemma 10.3 and noting that J A has a full row rank, we can verify that Part A of the objective is convex w.r.t. BA . In addition, since convex functions are minorized by their supporting hyperplanes [28], we have   (k) −1 † (k) † (k) log det (J A B−1 (10.68) A J A ) ≥ log det (J A (BA ) J A ) + tr G (BA − BA ) , where G (k) = −(BA )−1 J †A (J A (BA )−1 J †A )−1 J A (BA )−1 (k)

(k)

(k)

(10.69)

Relative entropy based waveform design for MIMO radar

303

is the gradient of log det (J A B−1 A J A ) at BA [30]. To further simplify the expression of the minorizer (i.e., the right-hand side of (10.68)), we let G (k) be partitioned as   (k) (k) G 11 G 12 (k) G = , (10.70) (k) (k) (G 12 )† G 22 (k)

†

where G 11 ∈ CNTR ×NTR , G 12 ∈ CNTR ×NRL , and G 22 ∈ CNRL ×NRL . Then tr(G (k) BA ) can be written as      † (k) (k) 1/2 (k) tr(G (k) BA ) = c0 + 2Re tr  SRHC G 12 + tr G 22  SRHC S , (10.71) (k)

(k)

(k)

(k)

(k)

(k)

where c0 = tr(G 11 + RW G 22 ) is a constant term not depending on S. 1/2 1/2 Therefore, a minorizer of log det (RHC C s RHC + I NTR ) is given by      † (k) (k) 1/2 (k) cA + 2Re tr  SRHC G 12 + tr G 22  SRHC S ,     (k) (k) (k) (k) where cA = c0 + log det J A (BA )−1 J †A − tr G (k) BA .

(10.72)

10.4.2 Minorizing Part B It is easy to verify that − log det (I + A) is convex w.r.t. A  0 [28]. Then 1/2  1/2 1/2 (k) 1/2 − log det (RC C S RC + I NTR ) ≥ − log det (RC C S RC + I )

S − C S )], + tr[Q(k) (C (k)

where 1/2 1/2 (k) 1/2 −1 1/2 Q(k) = −RC (RC C S RC + I NTR ) RC

(10.73)

(k) 1/2  1/2 (k)  (k) is the gradient of − log det (RC C S RC + I NTR ) at C S . Thus, cB + tr(Q C S ) is a (k) 1/2  1/2 1/2 (k) 1/2 minorizer of − log det (RC C S RC + I NTR ), where cB = − log det (RC C S RC + (k) I NTR ) − tr(Q(k) C S ).

10.4.3 Minorizing Part C Before finding a minorizer of Part C, we first note that † † S + RW )−1 ( S + RW )] SRC tr[( SRHC † † = NRL − tr[ S ( S + RW )−1 ]. SRH SRHC

(10.74)

Using the matrix inversion lemma again, we obtain 1

1

1

1

† −1 −1 2 2  2 2 † −1 ( SRHC S + RW )−1 = RW SRHC (I + RHC C S RHC S RW . (10.75) − RW )−1 RHC

Hence, the result in (10.74) can be rewritten as † † S + RW )−1 ( S + RW )] SRC tr[( SRHC 1

1

1

1

2 2  2 2  S − C S RHC = NRL − tr[RH (C C S RHC C S )]. (I + RHC )−1 RHC

(10.76)

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Radar waveform design based on optimization theory

According to the block matrix inversion lemma, we notice that the right-hand side of (10.76) can be further written as † † † −1 tr[( SRHC S + RW )−1 ( S + RW )] = NRL − tr[RH (J C B−1 SRC C J C ) ],

where J C = [I NTR , 0NTR ×NTR ], and   S R1/2 S C C HC . BC = 1/2  1/2  1/2 RHC C S I NTR + RHC C S RHC

(10.77)

(10.78)

Lemma 10.4. [35] Assume that B ∈ C2NTR ×2NTR and R ∈ CNTR ×NTR are positive † definite, then −tr[R(JC B−1 JC )−1 ] is convex w.r.t. B. By using Lemma 10.4 and the property of convex functions, we have −1 −1 −1 (k) −tr[RH (J C B−1 C J C ) ] ≥ −tr[RH (J C (BC ) J C ) ] + tr[H (BC − BC )], (k)

†

(k)

†

where H (k) = −(BC )−1 J C X C RH X C J C (BC )−1 (k)

†

(k)

(k)

(k)

† −1 −tr[RH (J C B−1 C J C) ]

is the gradient of partition H (k) as   (k) (k) H H 11 12 , H (k) = (k) (k) (H 12 )† H 22

at

(k) BC ,

(10.79) (k) XC

=

(k) † (J C (BC )−1 J C )−1 .

(10.80)

where H 11 , H 12 , H 22 ∈ CNTR ×NTR . Then (k) S ], tr(H (k) BC ) = c2 + tr[M (k) C (k)

where

(k) c2

=

(k)

(k)

(k) tr(H 22 ) (k)

(10.81)

is a constant term not depending on S and 1/2

(k)

(k)

1/2

1/2

(k)

1/2

M (k) = H 11 + RHC (H 12 )† + H 12 RHC + RHC H 22 RHC . (k) c˜ 2

Next we

(10.82)

S ] is a minorizer of Part C of the objective, where As a result, + tr[M (k) C (k) (k) −1 † −1 (k) = c2 − tr[RH (J C (BC ) J C ) ] − tr(H (k) BC ). (k) c˜ 2

10.4.4 The minorized problem at the (k + 1)th iteration With the results in Sections 10.4.1–10.4.3, the minorized problem at the (k + 1)th iteration can be formulated as        † † −1 (k) 1/2 (k)  SRHC S + tr (M (k) + Q(k) ) S SRHC G 12 + tr G 22  S RW max Re tr  S

s.t. tr(SS † ) ≤ et .

(10.83)

Next we show that we can recast the objective function of (10.83) as a quadratic function of s. By using the identities that tr(AT B) = vecT (A)vec(B) and tr(ABCD) = vecT (D)(A ⊗ C T )vec(BT ) [36], we can rewrite the objective of (10.83) as  s˜ + 2Re(˜s† s˜ †T t ), (k)

(k)

(10.84)

Relative entropy based waveform design for MIMO radar

305

where s˜ = vec( S), (k) −1 (k) = (M (k) + Q(k) )∗ ⊗ RW T + R∗HC ⊗ G 22 ,

(10.85)

 t =

(10.86)

(k)

(k) 1/2 vec((G 12 )† RHC ).

Proposition 1. [35] s˜ = As s, where As = E ⊗ I L , E = [E 1 , E 2 , . . . , E NTR ]T ,

(10.87)

with E i , i = 1, 2, . . . , NTR , denoting an NT × NR elementary matrix whose (ir , ic )th element equals 1 and other elements equal zeros, ir = 1 + mod(i − 1, NT ), and ir = i/NT . Using Proposition 1, we reformulate the optimization problem in (10.83) as Pq(k+1) : max s†T (k) s + 2Re(s† t (k) ), s.t. s† s ≤ et ,

(10.88)

s

(k) (k)As , t (k) = A†s t . where T (k) = A†s T It can be proved that (10.88) is a hidden convex problem [37] and can be solved via the Lagrange multipliers method. Specifically, the associated Lagrangian is given by

L(s, ν) = −s†T (k) s − 2Re(s† t (k) ) + ν(s† s − et ),

(10.89)

where ν is the Lagrange multiplier associated with the energy constraint. The minimizer can be obtained by differentiating (10.89) w.r.t. s and setting the differentiation to zero: s(k+1) = (ν (k+1) I NT L − T (k) )−1 t (k) ,

(10.90)

where ν (k+1) is the solution to the following equation: (t (k) )† (νI NT L − T (k) )−2 t (k) = et .

(10.91)

We summarize the proposed one-stage design method in Algorithm 2.

10.4.5 Convergence and computational complexity analysis First, we analyze the convergence of the one-stage algorithm. Let H (s; s(k) ) denote the minorized function at the (k + 1)th iteration, which is given by H (s; s(k) ) = s†T (k) s + 2Re(s† t (k) ) + const(k) , where const noting that

(k)

=

(k) cA

+

(k) cB

+

(k) c˜ 2

(10.92)

is a constant term depending on s . In addition,

v(Pq(k) ) ≤ v(Pq(k+1) ),

(k)

(10.93)

it is straightforward to verify that s(k+1) satisfies H (s(k) ; s(k) ) ≤ H (s(k+1) ; s(k) ).

(10.94)

As a consequence, we can obtain that D(s(k) ) = H (s(k) ; s(k) ) ≤ H (s(k+1) ; s(k) ) ≤ D(s(k+1) ),

(10.95)

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Radar waveform design based on optimization theory

Algorithm 2: One-stage waveform design method for MIMO radar detection in the presence of clutter

1 2 3 4 5 6 7 8 9 10 11

Input: RH , RC , RW , et . Output: S. Initialize: k = 0, s(0) . repeat Calculate BA with (10.66). Calculate G (k) with (10.69). Calculate Q(k) with (10.73). Calculate BC with (10.78). Calculate H (k) with (10.79). Calculate M (k) with (10.82). (k) −1 (k) = (M (k) + Q(k) )∗ ⊗ RW T + R∗HC ⊗ G 22 ; (k) (k) 1/2  t = vec((G 12 )† RHC ); (k)  As ; T (k) = A†s T

t (k) = A†s t ; 13 Update s(k+1) by (10.90). 14 k = k + 1. 15 until convergence; 12

(k)

where D(s(k) ) denotes the relative entropy associated with s(k) , and the last inequality holds because of the property of minorizers (cf. (C.1)). Regarding the computational complexity of the one-stage design algorithm, it is also linear w.r.t. the number of iterations until convergence and the complexity involved in each iteration. Precisely, at each iteration, the computational complexity for each step is listed in Table 10.1. Therefore, the overall computation at every 3 3 iteration needs O(NTR + (NTR NRL )2 + NRL ) operations. In contrast, the two-stage 6.5 algorithm needs O(NTR ) operations at every iteration. For typical values of NT , NR , and L, the per-iteration computational complexity of the one-stage algorithm will be much lower than that of the two-stage algorithm.

10.4.6 Extension to include other constraints In this subsection, we extend the proposed one-stage design method to synthesize constant-modulus waveforms and similarity-constrained waveforms. When the constant-modulus constraint is imposed, the minorized problem at the (k + 1)th iteration is formulated as max s†T (k) s + 2Re(s† t (k) ) s √ s.t. |s(m)| = ps , m = 1, 2, . . . , NT L,

(10.96)

Relative entropy based waveform design for MIMO radar

307

Table 10.1 The computational complexity at every iteration of Algorithm 2 Calculation

Computational complexity

BA

2 O(NTR NRL )

G (k)

2 3 O(NTR NRL + NRL )

Q BC

3 O(NTR ) 2 3 O(NTR NRL + NTR )

H (k)

3 O(NTR )

M (k)

3 O(NTR )

T

2 2 O(NTR NRL )

(k)

t

(k)

3 O(NTR )

(k)

Update of s(k)

3 O(NRL )

Similar to the method in Section 10.3.3, we can find a minorizer for s†T (k) s, which is given by 2Re(s† (T (k) − λmin (T (k) )I NRL )s(k) ) + 2λmin (T (k) )et − (s(k) )†T (k) s(k) .

(10.97)

Thus, the objective function in (10.96) is minorized by 2Re(s† u(k) ) + 2λmin (T (k) )et − (s(k) )†T (k) s(k) ,

(10.98)

where u = t + (T − λmin (T )I NRL )s . As a result, the associated minorized problem at the (k + 1)th iteration can be further written as √ (10.99) max 2Re(s† u(k) ), s.t. |s(m)| = ps , m = 1, 2, . . . , NT L. (k)

(k)

(k)

(k)

(k)

s

The optimal solution to (10.99) is given by √ s(k+1) (m) = ps exp (j arg (u(k) (m))).

(10.100)

When the similarity constraint is considered, we consider the following minorized problem at the (k + 1)th iteration: max 2Re(s† u(k) ), s.t. s† s = et , s − s0 22 ≤ δ. s

(10.101)

The above problem can be tackled with the method proposed in Section 10.3.4.

10.4.7 Accelerated schemes for the one-stage methods In MM-based algorithms, the tightness of the minorizer affects the convergence rate of the algorithms. Note that the proposed minorizer in the one-stage method is a quadratic/linear function of s. It is not as tight as the minorizer proposed in the two-stage method. We find that, though the computational complexity of the onestage algorithm at every iteration is lower than that of the two-stage algorithm, the

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Radar waveform design based on optimization theory

convergence of the one-stage algorithm might be much slower. Under this situation, the total run-time of the one-stage algorithm might be even longer than the two-stage algorithm. Thus, we need to speed up the convergence of the one-stage algorithm. To this end, we employ the accelerated scheme in [38], i.e., SQUAREM, to accelerate the one-stage algorithm (see also [39] for the applications of the accelerated scheme to a wide range of signal processing problems). Next we outline the main steps of SQUAREM and modify it accordingly to deal with the waveform design problem we encounter. Let F denote the fixed-point mapping during the iterations of the one-stage algorithm (note that the mapping varies with the enforced constraints), i.e., s(k+1) = F(s(k) ).

(10.102)

Motivated by the Cauchy–Barzilai–Borwein method to accelerate the convergence of the classical Cauchy method for solving linear equations, the SQUAREM method iterates as follows: s(k+1) = s(k) − 2α (k) r (k) + (α (k) )2 v(k) ,

(10.103)

where r (k) = F(s(k) ) − s(k) , v(k) = F(F(s(k) )) − F(s(k) ) − r (k) , and α (k) is the step-length in the kth iteration. In [38], the authors proposed three types of step-lengths and the empirical studies therein suggested the use of the following step-length: α (k) = −

r (k)  . v(k) 

(10.104)

However, two issues might arise when applying the update equation in (10.103) to our waveform design problem. On the one hand, s(k+1) might not be feasible; on the other hand, the convergence of the objective values is not ensured. To overcome the first issue, we modify the update equation in (10.103). Let z (k) = s(k) − 2α (k) r (k) + (α (k) )2 v(k) . If only the energy constraint is considered, the update equation can be adapted as follows to ensure the feasibility: s(k+1) (α (k) ) =

√ z (k) et (k) . z 

(10.105)

For the constant-modulus constraint, the update equation can be adapted to s(k+1) (α (k) ) = ps exp (j arg (z (k) )).

(10.106)

For the similarity constraint, we adapt the update equation through solving the following problem (with the solution denoted by s(k+1) (α (k) )): min s − z (k) 22 , s.t. s† s = et , s − s0 22 ≤ δ. s

(10.107)

The solution to (10.107) can be obtained with the method in Section 10.3.4. To keep the ascent property of the MM algorithm and ensure the convergence of the objective value, we use a backtracking strategy to adapt the step-length α (k) , i.e., we let α (k) = (α (k) − 1)/2 [38] until the relative entropy of s(k+1) (α (k) ) is larger than that of s(k) (note that the trick is to repeatedly halve the distance between α (k) and −1,

Relative entropy based waveform design for MIMO radar

309

Algorithm 3: Accelerated schemes for the one-stage design method

1 2 3 4

Input: RH , RC , RW , et . Output: S. Initialize: k = 0, s(0) . repeat r (k) = F(s(k) ) − s(k) ; v(k) = F(F(s(k) )) − F(s(k) ) − r (k) ; r  α (k) = − v (k)  ; (k) (k) z = s − 2α (k) r (k) + (α (k) )2 v(k) ; Compute s(k+1) (α (k) ). repeat α (k) = (α (k) − 1)/2; z (k) = s(k) − 2α (k) r (k) + (α (k) )2 v(k) ; Compute s(k+1) (α (k) ). until the relative entropy of s(k+1) (α (k) ) is larger than that of s(k) ; s(k+1) = s(k+1) (α (k) ). k = k + 1. until convergence; (k)

5 6 7 8 9 10 11 12 13 14 15

and if α (k) = −1, s(k+1) (α (k) ) = F(F(s(k) )) and the SQUAREM update is actually the same as the MM update). We summarize the accelerated algorithm in Algorithm 3.

10.5 Numerical examples In this section, we provide several numerical examples to demonstrate the performance of the proposed algorithms. Due to space limitations, we focus on the performance of a colocated MIMO radar system with uniform linear transmit and receive antenna arrays. The numbers of array elements in the transmitter and in the receiver are NT = 4 and NR = 4, respectively. The inter-element spacings are dT = 2λ and dR = λ/2, respectively (λ stands for the radar wavelength). The transmit code length is L = 10. The DOA of the target is θt = 10◦ , with strength σt2 = 1. The clutter scatterers are uniformly distributed from θk = −40◦ to θk = −35◦ with scattering 2 strength σc,k = 20 dB (for all k). We assume for simplicity that the spatial correlations of the colored interference can be ignored (i.e., RW = RS ⊗ RT , and RS = I NR ). We model RT based on an auto-regressive process of the first order (namely, AR(1)), i.e., RT (i, j) = ρ |i−j| with ρ = 0.4 as the one-lag correlation coefficient. We initialize the proposed algorithm with randomly generated quasi-orthogonal waveforms. In addition, the quasi-orthogonal waveforms are employed as the reference waveforms (i.e., S 0 ) when the similarity constraint is enforced. As to the stopping criterion of

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Radar waveform design based on optimization theory

Table 10.2 Run-time analysis of various algorithms Two-stage algorithm

Ttotal (s) Niter Ttotal (s) Niter

One-stage algorithm

Accelerated one-stage algorithm

C1

C2

C3

C1

C2

C3

C1

C2

C3

172 11 15.7

– – –

– – –

2,134 104,763 0.0204

3,094.6 209,289 0.0148

1,503.2 63,521 0.0237

0.50 14 0.036

1.046 16 0.065

0.11 3 0.038

Ttotal denotes the run-time to reach convergence. Niter denotes the number of overall iterations. C1 , C2 , and C3 represent the energy constraint, the constant-modulus constraint, and the similarity constraint, respectively.

the proposed algorithms, we terminate the algorithm if the relative difference of the objective value is less than  = 10−4 . Finally, all the numerical analysis is performed on a PC with CPU Core i5-6500 3.2 GHz and 8 GB of RAM. First, we analyze the convergence of the proposed algorithms. Figure 10.1(a) plots the relative entropy of the energy-constrained waveforms by the three algorithms (including the two-stage algorithm, the one-stage algorithms with and without acceleration) at every iteration, where et = 1.† Moreover, parts (b) and (c) of Figure 10.1, respectively, show the relative entropy of the constant-modulus waveforms, the similarity-constrained waveforms (with δ = 0.5et ) synthesized by the one-stage algorithms. The associated run-time to reach convergence and the total number of iterations as well as the average time per iteration are given in Table 10.2. The results in Figure 10.1 confirm the monotonically increasing behavior of the proposed algorithms. In addition, as seen from Figure 10.1 and Table 10.2, for this typical parameter set, the per-iteration computational time of the two-stage algorithm is much longer than that of the one-stage algorithm. However, the convergence rate of the one-stage algorithm (without acceleration) is rather slow and the relative entropy at convergence is smaller than the two-stage algorithm. Therefore, although the computational complexity of the one-stage algorithm at each iteration is much lower than that of the two-stage algorithm, the run-time to reach convergence can be much longer. Thanks to the accelerated scheme, the accelerated one-stage algorithm achieves much faster convergence with a slightly higher per-iteration computational complexity than the one-stage algorithm without acceleration. Thus, the run-time of the accelerated one-stage algorithm is significantly less than that of the two-stage algorithm and the one-stage algorithm without acceleration. Next, we analyze the detection performance of the proposed algorithms. Figure 10.2(a) shows the relative entropy of the synthesized constant-modulus waveforms by the proposed algorithms, versus the transmit energy et . Figure 10.2(b) plots the probabilities of detection associated with the curve in Figure 10.2(a) for Pfa = 10−4 . Herein, †

For the two-stage algorithm, Figure 10.1(a) actually plots the relative entropy associated with C S .

2.6

2.6

2.4

2.4

2 1.8 1.6 1.4 1.2 1 100

(a)

311

2.2

2.2

Relative entropy

Relative entropy

Relative entropy based waveform design for MIMO radar

101

2 1.8 1.6 1.4 1.2 1

Two-stage One-stage, without acceleration One-stage, with acceleration

0.8

102

0.6 100

103

104

105

One-stage, without acceleration One-stage, with acceleration

101

(b)

Number of iterations

102

103

104

105

106

Number of iterations

2.2

Relative entropy

2

One-stage, without acceleration One-stage, with acceleration

1.8 1.6 1.4 1.2 1 0.8 100

(c)

101

102 103 Number of iterations

104

105

Figure 10.1 Relative entropy of the devised waveforms versus iteration number (et = 1): (a) energy constraint, (b) constant-modulus constraint, (c) similarity constraint, δ = 0.5et the performance of the energy-constrained waveforms is drawn as a benchmark. Moreover, we do not include the performance of the original one-stage algorithm due to its slow convergence and relatively poor performance (namely, the one-stage algorithm here refers to the accelerated one-stage algorithm). The results show that the energy-constrained waveforms synthesized by the two algorithms have almost identical performance. When designing constant-modulus waveforms, the performance of the one-stage algorithm is slightly better than that of the two-stage algorithm. This can be explained by the fact that, when the constant-modulus constraint is enforced, the two-stage algorithm requires an additional step to synthesize the constrained waveforms from C S . If the matching error is non-negligible, the two-stage approach will degrade the performance of the synthesized waveforms. Figure 10.3 assesses the performance of the devised waveforms when the similarity constraint is enforced, where the similarity parameter is δ = 0.5et . We can observe that the performance of the waveforms devised by the one-stage algorithm outperforms that by the two-stage algorithm. Finally, we examine the robustness of the proposed algorithms w.r.t. the model uncertainties. To this end, we assume that the true target angle is not exactly known

312

Radar waveform design based on optimization theory

5

1 0.9

3

4.5

0.8

4 Probability of detection

Relative entropy

2.5 3.5 3

2 –2

–1

0

1

2

2.5 2 Two-stage, energy One-stage, energy Two-stage, constant-modulus One-stage, constant-modulus

1.5 1 0.5 –10

–5

(a)

0 Transmit energy (dB)

5

Two-stage, energy One-stage, energy Two-stage, constant-modulus One-stage, constant-modulus

0.7 0.6 0.5

0.78 0.76 0.74 0.72 0.7 0.68

0.4 0.3 0.2 0.1 0 –10

10

–1 –5

(b)

–0.5

0 Transmit energy (dB)

0

0.5

5

10

5

1

4.5

0.9

4

0.8 Probability of detection

Relative entropy

Figure 10.2 Performance of the synthesized constant-modulus waveforms: (a) relative entropy versus transmit energy et and (b) detection performance associated with part (a) versus transmit energy et for Pfa = 10−4

3.5 3 2.5 2 1.5 Two-stage, energy One-stage, energy Two-stage, similarity One-stage, similarity

1 0.5 0 –10

(a)

–5

0 Transmit energy (dB)

5

0.7 0.6 0.5 0.4 0.3 Two-stage, energy One-stage, energy Two-stage, similarity One-stage, similarity

0.2 0.1 0 –10

10

(b)

–5

0

5

10

Transmit energy (dB)

Figure 10.3 Performance of the synthesized similarity-constrained waveforms, δ = 0.5et : (a) relative entropy versus transmit energy et and (b) detection performance associated with part (a) versus transmit energy et for Pfa = 10−4

and is subject to some errors but the nominal value (i.e., 10◦ ) is used for waveform design. For simplicity, we compare the synthesized energy-constrained waveforms with the waveforms synthesized by the SINR criterion (see, e.g., [12,14] for cyclic methods on designing such waveforms), and the quasi-orthogonal waveforms.‡ The

‡

Considering that the one-stage algorithm and two-stage algorithm have identical performance when designing energy-constrained waveforms, we only include the result associated with the one-stage algorithm.

4.5

1

4

0.9

3.5

0.8 Probability of detection

Relative entropy

Relative entropy based waveform design for MIMO radar

3 2.5 2 1.5 1 Orthogonal waveforms SINR waveforms Proposed waveforms

0.5 0

(a)

0

5

10

15

Target angle (°)

313

0.7 0.6 0.5 0.4 0.3 0.2

Orthogonal waveforms SINR waveforms Proposed waveforms

0.1 0

20

(b)

0

5

10

15

20

Target angle (°)

Figure 10.4 Robustness w.r.t. model uncertainties: (a) relative entropy of the proposed method, SINR-based algorithm, and orthogonal waveforms versus target angle; (b) detection performance associated with part (a) for Pfa = 0.01. et = 5 relative entropy for the proposed method, SINR-based method, and quasi-orthogonal waveforms versus true target angle are depicted in Figure 10.4(a). Figure 10.4(b) shows the corresponding detection performance for Pfa = 0.01. It is observed that the proposed method is approximately robust w.r.t. the interval of target angle error of |θe | ≤ 5◦ . Moreover, the proposed algorithm is more robust than the SINR-based method. This might be due to the fact that the design of the SINR-based waveforms depends on the designed receiver in the previous iteration, which is not reliable when the target knowledge is imprecise. Note that the quasi-orthogonal waveforms show a robust performance for the whole considered interval for θe but with the cost of poorer performance when accurate target estimations are available (e.g., at the confirmation/tracking stage).

10.6 Concluding remarks In this chapter, we have considered the waveform design problem for MIMO radars in the presence of signal-dependent clutter. We aim at improving the detection performance of the radar system. To this end, we have employed the relative entropy associated with the detection problem as the design metric. We have studied the optimal structure of the energy-constrained waveforms and shown that the optimal left singular vectors of the waveform matrix should be aligned with the least interfered interference subspace. Two algorithms, which are both devised based on the MM techniques, are proposed to tackle the non-convex design problem. We have proved the convergence of the objective values for both algorithms. For the two-stage algorithm, the associated minorizer is tighter to the objective function than that of the one-stage algorithm. Thus, it converges faster than the one-stage algorithm but with a much higher per-iteration computational complexity. Fortunately, for the considered

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Radar waveform design based on optimization theory

constraints in this chapter, we can use SQUAREM to speed up the convergence of the one-stage algorithm, enabling significantly reduced run-time for the one-stage algorithm. Possible future research might include the robust design of multiple waveforms when the prior knowledge of the target and interference is inaccurate. Moreover, the extension of the signal model to the scenarios with interference from multiple range cells is also very important.

Appendix A Proof of (10.19) First, we note that † † log det ( SRHC S + RW ) − log det ( SRC S + RW ) † 1/2 †  1/2 = log det [I NTR + RH  S (SRC S + RW )−1 SRH ].

(A.1)

In addition, † † tr[( SRHC S + RW )−1 ( S + RW )] SRC 1 1 † † † = tr{[I NRL + ( S + RW )− 2  SRH S ( S + RW )− 2 ]−1 } SRC SRC 1

1

† † = tr{[I NTR + RH2  S ( S + RW )−1 SRH2 ]−1 } + NRL − NTR , SRC

(A.2)

where the second equality follows from the result in [18, Lemma 4]. Moreover, by applying the matrix inversion lemma [27], we can obtain that † †  S ( S + RW )−1 S SRC 1/2 1/2 −1 1/2  S − C S R1/2 S RC (RC C = C C + I NTR ) RC C S S )−1 + RC ]−1 , = [(C

(A.3)

−1  where we have used the fact that  S RW S = R−1 S ⊗ C S = C S. Combining the results in (A.1), (A.2), and (A.3), we complete the proof of (10.19). †

Appendix B Proof of Lemma 10.1 Definition 1 ([40]). Let f : S + → R > 0 be a given function. For any n × n positive semidefinite matrices M 1 and M 2 where M 1 M 2 , if f (M 1 ) > f (M 2 ), then we say f (M ) is monotonically increasing with M  0. Lemma 10.5. h(M) = log det (I + M) + tr[(I + M)−1 ] is a monotonically increasing function of M. Proof : Let λ1 (M l ) ≥ · · · ≥ λn (M l ) denote the eigenvalues of M l , l = 1, 2. If M 1 M 2 , then λk (M n [27]. Note that log det (I + 1 ) > λk (M 2 ), k = 1, . . . , M l ) + tr[ (I + M i )−1 ] = nk=1 log (1 + λk (M l )) + nk=1 (1 + λk (M l ))−1 and also

Relative entropy based waveform design for MIMO radar

315

f1 (x) = log (1 + x) + 1/(1 + x) is a monotonically increasing function§ for x > 0. Therefore, log det (I + M 1 ) + tr[ (I + M 1 )−1 ] > log det (I + M 2 ) + tr[ (I + M 2 )−1 ] and hence, h(M ) = log det (I + M ) + tr[(I + M )−1 ] monotonically increases with M . 1/2 −1 −1 1/2 To prove Lemma 10.1, we first show that h(RH (C S + RC ) RH ) monotonically increases with C S . To this end, let C 1 and C 2 be two positive definite −1 1 = R−1  matrices with C 1 C 2 . Define C S ⊗ C 1 and C 2 = RS ⊗ C 2 . It is verified −1 −1 −1 −1 −1 −1 2 . Thus [C 1 + RC ] [C 2 + RC ] . Besides, as shown in Lemma 1 ≺ C that C 1/2 −1 −1 1/2 10.5, h(M ) monotonically increases with M . Therefore, h(RH (C 1 + RC ) RH ) > −1 1/2 −1 1/2 1/2 1/2 S + RC )−1 RH ) is monotonically increasing h(RH (C 2 + RC )−1 RH ) and h(RH (C with C S . To proceed with the proof, we introduce the following lemma: Lemma 10.6 ([18,41]). Let R ∈ Cn×n be a positive semidefinite matrix with the eigen-decomposition R = V ↓V† , where  ↓ = diag([σ1 , . . . , σn ]) and σ1 ≥ · · · ≥ σn . † Assume that  S ∈ Cn×m satisfies  S R S = D, where D = diag ([d1 , . . . , dm ]), d1 ≥ S ∈ Cn×m which satisfies · · · ≥ dm and m < n, then it is always possible to find ! † † † † ! S R! S = α S R S with tr( S S ) = tr(Sˆ Sˆ ) and α ≥ 1, and ! S is given by !1/2 ! 0m×(n−m) ]T , S = V[ s

(B.1)

   "s = αdiag [d1 σ1−1 , . . . , dm σm−1 ] , α = ( m di dU ,i )/( m di σi−1 ), dU ,i is where  i=1 i=1 the ith diagonal element of UR−1 U† and U ∈ Cn×n is a unitary matrix which satisfies S = U[D1/2 0m×(n−m) ]T . R1/2 To employ the result of the above lemma, note that RT−1 = V T (T↑ )−1V T† in which the diagonal elements of (T↑ )−1 are sorted in decreasing order. In addition, −1 S 1 =  ↓1 V 1 C SV 1 = (SV 1 )† RT−1 (SV 1 ) = S 1 RW †

†

(B.2)

is diagonal with decreasing elements. Lemma 10.6 indicates that we can find † † ! S which satisfies tr(! S! S ) = tr(S 1 S †1 ) = tr(SS † ) and ! S = α ↓1 , where ! S RT−1! S= 1/2  V T  s ,  s = αdiag[σ1,1 σT,1 , . . . , σ1,NT σT,NT ], S 1 = SV 1 and α ≥ 1. Moreover, † ˆ †1 = Sˆ †1 RT−1 Sˆ 1 = αV 1  ↓1 V †1 = αC S V 1 Sˆ RT−1 SV

§



Because we have f1 (x) = 1/(1 + x) − 1/(1 + x)2 > 0 for x > 0.

(B.3)

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Radar waveform design based on optimization theory

† −1 ˆ †1 . Note also that h(R1/2 and Sˆ 1 RT−1 Sˆ 1 = αC S  C S = S † RT−1 S with Sˆ 1 = SV H (C S + 1/2 RC )−1 RH ) monotonically increases with C S . Hence, we have † 1/2 †  1/2 S (SRC h(RH  S + RW )−1 SRH ) −1

S + RC )−1 RH ) = h(RH (C 1/2

1/2

S )−1 + RC )−1 RH ) ≤ h(RH ((α C 1/2

1/2

† 1/2 †  1/2 S 1 (S 1 RC S 1 + RW )−1 S 1 RH ), = h(RH 

(B.4)

where  S 1 = I NR ⊗ Sˆ 1 . Consequently, the optimal waveform matrix is given by S  = 1/2 T  s V †1 , namely, its left singular vectors should be the eigenvectors corresponding V to the smallest NT eigenvalues of RT . In other words, we should avoid putting transmit energy in the directions where the interference is severe.

Appendix C An introduction to minorization–maximization MM refer to a class of iterative methods to deal with hard optimization problems. It can be treated as a generalization of the well-known expectation–maximization method. It also has many close connections with other popular optimization methods, including convex–concave procedure, difference of convex functions programming, and proximal algorithms [39]. Assume that we hope to tackle the maximization problem maxθ ∈ g(θ) over the feasibility set . Starting from a feasible solution, typical MM methods include two steps at the (k + 1)th iteration: • Minorization step: find a surrogate function Q(θ ; θ (k) ) to minorize g(θ) at θ (k) , where Q(θ; θ (k) ) is called a minorizer of g(θ), i.e., Q(θ ; θ (k) ) satisfies g(θ) ≥ Q(θ; θ (k) ) for ∀θ ∈ , g(θ (k) ) = Q(θ (k) ; θ (k) ),

(C.1)

where θ represents the value of θ at the kth iteration. • Maximization step: solve the maximization problem (k)

¶

θ (k+1) = arg max Q(θ ; θ (k) ). θ ∈

(C.2)

The MM methods can monotonically increase the objective value during the iterations, since g(θ (k+1) ) ≥ Q(θ (k+1) ; θ (k) ) ≥ Q(θ (k) ; θ (k) ) = g(θ (k) ),

(C.3)

where the first inequality holds because of (C.1) and the second one is the result of the maximization step. Owing to the ascent property and the remarkable numerical stabilities, MM methods have found many applications (see, e.g., [39,42,43] for a comprehensive survey, and [35,44–46] for the application of MM methods in radar waveform design). 

As pointed in [42], the acronym MM does double duty. For minimization problems, “MM” stands for majorization–minimization; while for maximization problems, “MM” stands for minorization– maximization. ¶ The function Q(θ ; θ (k) ) is said to majorize g(θ ) at θ (k) if −Q(θ ; θ (k) ) minorizes −g(θ ) at θ (k) .

Relative entropy based waveform design for MIMO radar

317

Acknowledgment This work was supported in part by the National Natural Science Foundation of China under grant 61671453, in part by the Anhui Provincial Natural Science Foundation under grant 1608085MF123, and in part by the Young Elite Scientist Sponsorship Program under grant 17-JCJQ-QT-041.

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Index

accelerated iterative sequential optimization (AISO) algorithm 263–4, 266, 270–2 accelerated schemes 289, 301, 307–9 acceleration techniques 188–9 adaptive selection scheme 239, 255–7 algebraic methods 2–9 almost perfect autocorrelation sequences (APAS) 8–9 alternating direction method of multipliers (ADMM) framework 223 ambiguity function (AF) 263, 270, 277–81 arbitrary waveform generator (AWG) 278 autocorrelation function (ACF) 108, 113 autocorrelation sidelobes 63 Barker sequences 2–3, 64, 75 beampattern-matching problems 287 binary sequence designs 1–9 CAN (PeCAN) family of algorithms 17 1bCAN 27–8 1bPeCAN 28–30 CANARY algorithm 22–4 cyclic algorithm new (CAN) 18–20, 65, 75, 79, 81 PeCANARY 24–6 periodic cyclic algorithm-new 20–1 iterative twisted approximation (ITROX) algorithm 10–15

block coordinate descent method (BCD) 195 Cauchy–Barzilai–Borwein method 308 Charnes–Cooper transformation 136 clutter impulse response (CIR) 130 code matrix optimization 136–7 code synthesis 137–8 coherent multiple-input–multiple-output (MIMO) radar 287 coherent processing interval (CPI) 157 colocated MIMO radar 287 computational complexity 209–10, 297, 305–6 computation algorithms 10–30 constant modulus 38–9, 148–9, 194, 200, 222 constant modulus constraint 17, 37, 194, 200, 298–300 continuous phase case, iterative algorithm for 40–2 continuous phase code design 69–72 Continuous Phase Method (CPM) algorithm 73–4, 76, 79 convergence analysis 187–8 coordinate-descent (CD) algorithm 14–17, 30–3, 64–7, 69–73, 85 determinant maximization (MAXDET) 288, 297 Dinkelbach-type algorithms 175 discrete Fourier transform (DFT) 18, 72–3, 228 Discrete Phase Method (DPM) algorithm 74–5, 77–9, 81

322

Radar waveform design based on optimization theory

Doppler processing, optimizing radar transceiver for 155–74 Doppler shift 49, 106, 157, 233 Doppler velocity 47 efficient coordinate descent algorithm 222 see also coordinate-descent (CD) algorithm efficient gradient method (EGM) 222 energy-constrained waveforms, synthesis of 293–7 energy spectral density (ESD) 57, 98, 108, 113, 264 enlarged quadratic problem (EQP) 104–5 fast Fourier transform (FFT) 64, 73, 75, 231 feasibility set, derivation of 88–9 filter bank optimization 163–4 filter matrix optimization 135–6 filter synthesis 137 flat spectrum waveform design 241–5 Fourier transform pair 222, 228 Frank sequence 75–7, 79 frequency stopband energy 271–2 Gaussian interference 97 Gaussian noise 128 Gaussian random variable 276 Gaussian random vector 95 generalized fractional programming (GFP) 155 Gold sequences 6–7 Golomb sequences 77 Hermitian matrices 11–12 Hessian matrix, positive definiteness of 238–9, 253–4 high-range resolution (HRR) radars 127, 129 information-theoretic criteria 288 initial codes, impact of 47–9

integrated sidelobe level (ISL) 2, 16, 18, 20, 27, 30, 63–7, 74–6, 79–82, 85, 93, 97, 109, 114, 222, 252, 263 Interference/Similarity (I/S) achievable region 103, 113 iterative algorithm for continuous phase case (IA-CPC) algorithm 38, 40–2, 46–7, 58 iterative algorithm for discrete phase case (IA-DPC) algorithm 42–6, 58 block mode 1 of IA-DPC (IA-DPC-BM1) 47–8 iterative sequential quadratic optimization (ISQO) algorithm 264, 266–73, 277, 279 Karush–Kuhn–Tucker (KKT) condition 223 Lagrange programming neural network (LPNN) 221, 223–33, 227–8, 233–9, 239–5, 245–3 Lagrangian function 223, 230, 234 Lagrangian neurons 230 Legendre sequences 3–5, 9 linear feedback shift register (LFSR) 4 linear frequency modulation (LFM) waveforms 63, 199 low-sidelobe autocorrelation level 252–3 majorization– minimization/minorization– maximization (MM) framework 174, 185, 187–8, 205–11, 288, 316 joint design of transmit waveform and receive filter 190–9 maximin case, extension to 189–90, 199–202 robust joint design 202, 204–13 vanilla MM method 186–7 majorized iteration method for joint design 193–4

Index majorized iterative algorithm (MIA) 198 with constant modulus and similarity constraints (MIA-CMSC) 199 with spectrum compatibility constraint for global design (MIA-SCCG) 199 maximum block improvement (MBI) rule 66 merit factor (MF) 74 minimum peak sidelobe (MPS) sequences 64 mirror descent algorithm (MDA) 209 MOSEK 197, 210–11 m-sequences 4–5 multiple high-speed targets, detection of 273–7 multiple-input–multiple-output (MIMO) radar 185 mutual information (MI) 128 Neyman–Pearson criterion 97 non-convex 295–6 non-deterministic polynomial-time hardness (NP-hard) 38, 64 orthogonal waveforms 287 Pareto optimization framework 65, 82–5 peak sidelobe levels (PSLs) 2, 13–17, 23, 26–7, 29–33, 63–7, 74–9, 85, 94, 97, 109, 114, 222, 233–40, 244, 248–2 peak-to-average power ratio (PAR) constraint 37–8, 129, 195–6, 201, 263 phase-only sequences, computational design of 63–7, 73 power method-like approaches for both continuous and discrete phase cases 45–6 power spectral density (PSD) 222

323

quadratically constrained quadratic program (QCQP) 94, 100–3, 110–11, 115, 119, 197 quantized discrete-phase sequence 75 quasi-orthogonal waveforms 287 radar line-of-sight (LOS) 127 radio environment map (REM) 96 random phase (RP) sequences 242 rank-one decomposition techniques 38 rank-one reconstruction 191 relative entropy-based waveform design for MIMO radar 287–313 scalarization technique 67 second-order cone programing (SOCP) 199, 210 semidefinite programming (SDP) problem 100, 129, 287 semi-definite relaxation (SDR) 38, 191 sequential optimization algorithm 1 with constant modulus and similarity constraints (SOA1-CMSC) 200 sequential optimization algorithm 2 with constant modulus and similarity constraints (SOA2-CMSC) 200 signal-to-interference-plus-noise ratio (SINR) 94–7, 100, 103, 106, 109, 115, 128, 155, 158, 160, 213, 288 signal-to-noise ratio (SNR) 47, 128 similarity constraint 37–9, 47, 50, 53, 58, 97, 103, 109, 114–15, 117–18, 161, 167–8, 174, 194–5, 199–200, 288, 300–301, 307–11 single input single output (SISO) radar systems 65–6 soft-power constraint transmit waveform design technique 107 spectral coexistence 94, 103, 263–266, 276–7, 281

324

Radar waveform design based on optimization theory

spectral compatibility 98, 109–10, 115, 117, 119, 196–7, 202 spectral constraints 94, 113, 228–32 spectrally crowded environments 103–18 spherical uncertainty set 138–41, 145–8 squared iterative method (SQUAREM) 188–9, 200, 289, 308, 314 statistical MIMO radar 287 Stein’s lemma 290 target aspect angle (TAA) uncertainty set 143–5

target impulse response (TIR) 127 time–frequency CAN (TFCA) method 222 uniform linear arrays (ULAs) 191, 199 unimodular quadratic programming (UQP) problems 299 waveform diversity 156 Weierstrass theorem 70 weighted integrated sidelobe level (WISL) 263–5, 271–3