Specialized resource providing detailed coverage of recent advances in theory and applications of sparse arrays Sparse Arrays for Radar, Sonar, and Communications discusses various design approaches of sparse arrays, including those seeking to increase the corresponding one-dimensional and two-dimensional virtual array apertures, as well as others that configure the arrays based on solutions of constrained minimization problems. The latter includes statistical bounds and signal-to-interference and noise ratio; in this respect, the book utilizes the recent strides made in convex optimizations and machine learning for sparse array configurability in both fixed and dynamic environments. Similar ideas are presented for sparse array-waveform design. The book also discusses the role of sparse arrays in improving target detection and resolution in radar, improving channel capacity in massive MIMO, and improving underwater target localization in sonar. It covers different sparse array topologies, and provides various approaches that deliver the optimum and semi-optimum sparse array transceivers. . Edited by a world-leading expert in Radar and Signal Processing and contributed to by world-class researchers in their respective fields, Sparse Arrays for Radar, Sonar, and Communications covers topics including: Utilizing sparse arrays in emerging technologies and showing their offerings in various sensing and communications applications Applying sparse arrays to different environments and obtain superior performances over conventional uniform arrays Solving the localization, beamforming, and direction-finding problems using non-uniform array structures for narrowband and wideband signals Designing sparse array structures for both stationary and moving platforms that produce physical and synthesized array apertures. Using deep neural networks that learn the underlying complex nonlinear model and output the sparse array configuration using representations of the input data spatio-temporal observations Solving for optimum sparse array configurations and beamforming coefficients in sensing using iterative convex optimization methods Providing complete coverage of the recent considerable progress in sparse array design and configurations, Sparse Arrays for Radar, Sonar, and Communications is an essential resource on the subject for graduate students and engineers pursuing research and applications in the broad areas of active/passive sensing and communications.

Table of Contents:
About the Editor xvii List of Contributors xviii Preface xxiii 1 Sparse Arrays: Fundamentals 1 Palghat P. Vaidyanathan and Pranav Kulkarni 1.1 Introduction 1 1.2 Basics of Array Processing 2 1.2.1 Expression for the Array Output 2 1.2.2 Sampling the Array Outputs 4 1.2.3 Covariance of the Array Output 4 1.2.4 The MUSIC Algorithm 5 1.2.5 Invertibility of the Array Manifold 6 1.2.6 Beamforming 7 1.3 What Are Sparse Arrays? 7 1.4 How Sparse Arrays Identify �(N 2) Sources 9 1.4.1 The Difference Coarray 10 1.4.2 The Weight Function and the Estimation of R[l] 11 1.4.3 Central ULA 11 1.4.3.1 Degrees of Freedom 12 1.4.4 How Coarrays Arise in Other Contexts 13 1.5 Identifying DOAs from Correlations 13 1.5.1 Factorization of the Matrix R 14 1.5.2 Proof of Theorem 1.1 15 1.6 Coarray MUSIC 15 1.6.1 Unique Identifiability 16 1.6.2 Estimating the Signal Powers 16 1.6.3 Subtleties Which Arise in Practice 17 1.6.4 Spatial Smoothing 17 1.6.4.1 Steps in the Computation of Coarray-MUSIC for Sparse Arrays 19 1.7 Examples of Sparse Arrays 19 1.7.1 Nested Arrays 19 1.7.2 Coprime Arrays 20 1.7.2.1 Coarray of the Coprime Array 22 1.8 Examples of Optimal Sparse Arrays 23 1.8.1 Minimum Redundancy Arrays 24 1.8.2 Minimum Hole Arrays 25 1.9 Coprime DFT Beamformers 26 1.9.1 Definition of a Set of N 1 N 2 Product Filters 26 1.9.2 Realization of the Set of N 1 N 2 Beamformers 29 1.9.3 Summary: Coprime DFT Beamformer 30 1.10 Directions for Further Reading 31 1.10.1 Sparse Reconstruction Methods for DOAs 31 1.10.2 Cramér–Rao Bounds for Sparse Arrays 32 1.10.2.1 CRB Versus MSE for Coarray Methods 34 1.10.3 Direct MUSIC on Sparse Arrays 34 1.10.4 Further Developments on Sparse Array Geometry 35 Acknowledgment 36 References 36 2 Sparse Array Interpolation for Direction-of-Arrival Estimation 41 Chengwei Zhou, Yujie Gu, Yimin D. Zhang, and Zhiguo Shi 2.1 Introduction 41 2.2 Virtual Array Interpolation for Gridless DOA Estimation 43 2.2.1 Discontiguous Coarray Model 43 2.2.2 Virtual Array Interpolation and Its Atomic Norm 44 2.2.2.1 Array Interpolation for Virtual ULA 45 2.2.2.2 Atomic Norm of Multiple Virtual Measurements 45 2.2.2.3 Properties of Virtual Domain Atomic Norm 47 2.2.3 Toeplitz Matrix Reconstruction for DOA Estimation with Interpolated Virtual Array 50 2.2.4 Coarray Cramér–Rao Bound 53 2.2.5 Simulation Results 54 2.2.5.1 Comparison of Resolution 55 2.2.5.2 Comparison of DOFs 57 2.2.5.3 Comparison of Estimation Accuracy 57 2.2.5.4 Comparison of Computational Complexity 61 2.3 Physical Array Interpolation for Off-grid DOA Estimation 62 2.3.1 Physical Array Interpolation and Signal Model 62 2.3.2 Covariance Matrix Recovery for Off-grid DOA Estimation 64 2.3.3 Push the Limit of Achievable Degrees-of-Freedom 65 2.3.4 Simulation Results 66 2.4 Prospective Research Directions 67 2.4.1 Interpolation-Aware Sparse Array Design 67 2.4.2 Multi-dimensional Sparse Array Interpolation 69 2.4.3 Sparse Array Interpolation in Tensor Signal Processing 69 Acknowledgments 70 References 70 3 Wideband and Multi-frequency Sparse Array Processing 75 Fauzia Ahmad, Peter Gerstoft, and Wei Liu 3.1 Introduction 75 3.2 Wideband DOA Estimation 76 3.2.1 Wideband Array Model 76 3.2.2 Sparsity-Based DOA Estimation at a Single Frequency 78 3.2.3 Wideband DOA Estimation Based on Group Sparsity 80 3.2.4 Simulation Results 81 3.3 Multi-frequency DOA Estimation 83 3.3.1 Multi-frequency Signal Model 83 3.3.2 DOA Estimation Under Proportional Spectra 85 3.3.3 DOA Estimation Under Nonproportional Spectra 86 3.3.4 Simulation Results 86 3.4 Wideband SBL for Beamforming 89 3.4.1 SBL for Beamforming at a Single Frequency 90 3.4.2 Wideband SBL for Beamforming 92 3.4.3 Experimental Results 93 3.5 Suggested Further Reading 96 3.6 Conclusion 97 References 98 4 Sparse Arrays in Sample Starved Regimes: Algorithms and Performance Analysis 103 Piya Pal and Heng Qiao 4.1 Introduction 103 4.2 Background on Correlation-Aware Sparse Support Recovery with Sparse Arrays 104 4.2.1 Fundamental Limits: Is |�| 2 > M Achievable? 105 4.2.2 Role of Difference Sets 106 4.3 Universal Recovery Guarantees for OOSA: The Role of Non-negativity 108 4.3.1 Why Positivity Alone Suffices 108 4.3.2 Stable Recovery in the Regime |�| > M with Correlation Estimates: Preliminaries 110 4.3.3 Universal Upper Bounds on Error with Non-negative Constraint When |�| > m 110 4.3.4 Stability Guarantees for Generic Correlation-Matching Techniques 112 4.4 Support Recovery with High Probability: How Many Snapshots Suffice? 113 4.4.1 Characterizing the Snapshot Requirement for Support Recovery with High Probability 113 4.4.2 Tightness of the Upper Bound 115 4.4.3 Numerical Experiments 116 4.4.3.1 Power Estimation Error and the Universal Upper Bound 116 4.4.3.2 Comparison of Support Recovery as a function of L and s 117 4.4.3.3 Comparison with Vector Approximate Message Passing 117 4.4.3.4 Phase Transition 118 4.4.3.5 Achievability of Upper Bound 119 4.4.3.6 Performance of “Correlation-Aware” Algorithms for MMV Models 120 4.5 Single-Snapshot Virtual Array Interpolation: Deterministic Guarantees 120 4.5.1 Matrix Completion with Nested Array 121 4.5.2 Guaranteed Single Snapshot Interpolation with Nested Matrix Completion 122 4.5.3 Numerical Examples 123 4.6 Concluding Remarks and Future Directions 124 References 124 5 Sparse Sensor Arrays for Two-dimensional Direction-of-arrival Estimation 131 Ali H. Muqaibel and Saleh A. Alawsh 5.1 Introduction 131 5.2 Two-Dimensional DOA Estimation Essentials 132 5.2.1 2D System Model 132 5.2.2 Terminology of 2D Arrays 134 5.2.3 Coarrays in 2D 134 5.3 Sparse Array Geometries for 2D-DOA Estimation 136 5.3.1 Parallel Arrays 138 5.3.1.1 Parallel Coprime Array (PCA) 138 5.3.1.2 Three Parallel Coprime Array (TPCA) 138 5.3.1.3 Parallel Nested Array (PNA) 140 5.3.1.4 Coprime-displaced Three Parallel Nested Arrays (CDTPNA) 140 5.3.1.5 Other Parallel Arrays 140 5.3.1.6 Parallel Arrays with Motion 141 5.3.2 Nonparallel Linear Arrays 143 5.3.2.1 L-Shaped Array 143 5.3.2.2 Cross-shaped Array 145 5.3.2.3 Generalized L-shaped Array with Odd–Even Locations (GLA-OEL) 145 5.3.2.4 Synthetic Augmented Cross Array (SACA) 145 5.3.2.5 V-shaped Array 146 5.3.2.6 Billboard Array 146 5.3.2.7 Open Box Array (OBA) 146 5.3.2.8 T-shaped Array (TSA) 146 5.3.3 Interleaved Rectangular Arrays 147 5.3.3.1 Coprime Planar Array (CPA) 147 5.3.3.2 Unfolded Coprime Planar Array (UCPA) 149 5.3.3.3 Symmetric Displaced Coprime Planar Array (SDCPA) 149 5.3.3.4 Nested Planar Array (NPA) 151 5.3.3.5 Nested Coprime Planar Array (NCPA) 151 5.3.3.6 Planar Arrays with Motion 152 5.3.4 Conformal Arrays 153 5.3.5 Other 2D Arrays 155 5.3.5.1 Half Open Box Array-2 (HOBA-2) 155 5.3.5.2 Hourglass Array 156 5.3.5.3 Thermos Array 156 5.3.5.4 Concentric Rectangular Array (CcRA) 156 5.3.5.5 Extended Sparse Convolutional Array (ESCA) 157 5.3.5.6 Half H Array (HHA) and Ladder Array (LAA) 157 5.4 Comparative Evaluation 159 5.4.1 DOF and Number of Sensors 159 5.4.2 Aperture Size and Mutual Coupling 166 5.5 Summary 171 References 171 6 Sparse Array Design for Direction Finding Using Deep Learning 181 Kumar Vijay Mishra, Ahmet M. Elbir, and Koichi Ichige 6.1 Introduction 181 6.1.1 Prior Art and Historical Notes 181 6.1.2 Learning-Based Approaches 182 6.2 General Design Procedures 184 6.2.1 Antenna Selection Setups 184 6.2.2 DoA Estimation Setups 185 6.3 Cognitive Sparse Array Design for DoA Estimation 186 6.3.1 Signal Model 186 6.3.2 Antenna Selection via Deep Learning 188 6.3.2.1 Input Data 188 6.3.2.2 Labeling 189 6.3.2.3 Network Architecture 190 6.3.3 Numerical Experiments 191 6.4 TL for Sparse Arrays 194 6.4.1 Knowledge Transfer Across Different Array Geometries 196 6.4.2 Deep Network Realization and Training 197 6.4.3 Performance in Source Domain 197 6.4.4 Performance for TL 198 6.5 Large Planar Sparse Array Design with SA-Assisted dl 200 6.6 DL-Based Sparse Array Design for Hybrid Beamforming 204 6.7 Deep Sparse Arrays for ISAC 206 6.8 Summary 207 Acknowledgments 208 References 208 7 Sparse Array Design for Optimum Beamforming Using Deep Learning 215 Syed A. Hamza, Kyle Juretus, and Moeness G. Amin 7.1 Motivation 215 7.2 Contributions 217 7.3 Problem Formulation 217 7.4 Efficient Generation of Training Data for Optimum Beamforming 219 7.4.1 Sparse Array Design Through the SDR Algorithm 220 7.4.1.1 Modified Re-weighting for Fully Augmentable Hybrid Array 221 7.4.2 Sparse Array Design Through SCA Algorithm 223 7.4.3 SBSA Design 225 7.4.3.1 The Role of Spare Configuration in MaxSINR 225 7.4.4 Summary of Data Generation Approaches 230 7.5 Machine-Learning Methods for Sparse Array Design 232 7.5.1 Generalization 232 7.5.2 Noisy Input–Output Space 233 7.5.3 Input Data Format 233 7.5.4 Obtaining the Full Correlation Matrix 233 7.5.5 Network Architectures 234 7.5.5.1 Dual Network Architecture 234 7.5.5.2 Binary Switching Strategies 235 7.5.5.3 Binary Switching Network Architectures 235 7.6 Simulation Results 236 7.6.1 DNN Simulation Performance 236 7.6.1.1 Data Generation 236 7.6.1.2 Results 237 7.6.2 DNN-Based SBSA Design 238 7.6.3 MLP and CNN Simulation Performance 240 7.6.3.1 Dataset Generation 240 7.6.3.2 Results 241 7.6.4 Comparison of the Network Architectures 243 7.7 Future Directions 244 7.7.1 Multiple Direction of Arrivals 244 7.7.2 Utilizing Limited Snapsots 244 7.7.3 Missing Correlation Data 244 7.7.4 Rapid Dynamic Environments 245 7.8 Conclusions 246 References 246 8 Sensor Placement for Distributed Sensing 251 Geert Leus, Mario Coutino, and Sundeep Prabhakar Chepuri 8.1 Data Model 252 8.1.1 Solution Approaches 253 8.1.2 Running Example 254 8.2 Distributed Estimation 255 8.2.1 Estimation Optimality Criteria 255 8.2.2 Uncorrelated Observations 256 8.2.3 Correlated Observations 258 8.3 Distributed Detection 260 8.3.1 Known θ Parameter 261 8.3.1.1 Optimality Criteria 261 8.3.1.2 Sparse Sampler Design 263 8.3.2 Unknown θ Parameters 268 8.3.2.1 Optimality Criteria 268 8.3.2.2 Sparse Sampler Design 269 8.4 Conclusions 270 References 270 9 Sparse Sensor Arrays for Active Sensing: Models, Configurations, and Applications 273 Robin Rajamäki and Visa Koivunen 9.1 Introduction 273 9.1.1 Goals, Scope, and Organization 274 9.1.2 Notation 275 9.2 Active Sensing Signal Model 275 9.2.1 Physical Array Model 275 9.2.1.1 Angle-Delay-Doppler Model 276 9.2.1.2 Simplified Angle-Only Model 276 9.2.1.3 Waveform Matrix 277 9.2.2 Virtual Array Model 277 9.3 Sparse Array Configurations 279 9.3.1 Categorization of Array Configurations Based on Overlap Between Tx and Rx Arrays 279 9.3.2 Minimum-Redundancy Array 280 9.3.2.1 Redundancy 281 9.3.2.2 Definition of MRA for Active Sensing 281 9.3.2.3 Known MRAs 282 9.3.3 Symmetric Sparse Array Configurations 282 9.3.3.1 Generic Symmetric Array 283 9.3.3.2 Symmetric Nested Array 284 9.3.3.3 Other Symmetric Arrays 286 9.4 Beamforming 287 9.4.1 Rx, Tx, and Joint Tx–Rx Beamforming 287 9.4.1.1 Receive Beamforming 288 9.4.1.2 Transmit Beamforming 288 9.4.1.3 Joint Transmit–Receive Beamforming 289 9.4.2 Image Addition 290 9.4.2.1 Joint Optimization of Tx and Rx Beamformers 291 9.5 Applications 293 9.5.1 Imaging 293 9.5.2 MIMO Radar 293 9.5.3 Wireless Communications 294 9.6 Conclusions 294 Acknowledgment 294 References 295 10 Sparse MIMO Array Transceiver Design in Dynamic Environment 301 Xiangrong Wang, Weitong Zhai, and Xianghua Wang 10.1 Review of MIMO Arrays and Sparse Arrays 302 10.2 Sparse MIMO Transceiver Design for MaxSINR with Known Environmental Information 307 10.2.1 Problem Formulation 308 10.2.2 Sparse Array Transceiver Design 309 10.2.2.1 Group Sparse Solutions via SCA 309 10.2.2.2 Reweighting Update 311 10.2.3 Simulation 312 10.2.3.1 Example 1 312 10.2.3.2 Example 2 312 10.3 Cognitive-Driven Optimization of Sparse Transceiver for Adaptive Beamforming 314 10.3.1 Full Covariance Construction 315 10.3.2 Optimal Transceiver Design 318 10.3.2.1 Beamforming for MIMO Radar 318 10.3.2.2 Sparse Transceiver Design 318 10.3.2.3 Reweighted l 2,1 -norm 320 10.3.3 Optimized Transceiver Reconfiguration 321 10.3.4 Simulations 321 10.3.4.1 Example 1 321 10.3.4.2 Example 2 322 10.3.4.3 Example 3 322 10.3.4.4 Example 4 323 10.4 Sparse MIMO Transceiver Design for Multi-source DOA Estimation 323 10.4.1 Cramer–Rao Bound of Multi-source DOA Estimation 324 10.4.2 Sparse MIMO Array Transceiver Design in the Metric of CRB 325 10.4.3 Simulations 327 10.5 Conclusion 329 References 329 11 Generalized Structured Sparse Arrays for Fixed and Moving Platforms 335 Guodong Qin and Si Qin 11.1 Introduction 335 11.2 Generalized Coprime Array Configurations 336 11.2.1 Prototype Coprime Array and Difference Coarray 336 11.2.2 Coprime Array with Compressed Inter-element Spacing 337 11.2.3 Coprime Array with Displaced Subarrays 339 11.3 Synthetic Structured Arrays Exploiting Array Motions 342 11.3.1 Array Synthetic Fundamentals 342 11.3.2 The Synthetic Structured Sparse Arrays 344 11.3.2.1 Coprime Array 344 11.3.2.2 Other Sparse Arrays 349 11.4 Structured Arrays Design for Moving Platforms 352 11.5 DOA Estimation Exploiting Array Motions 355 11.6 Other Structured Arrays for Fixed and Moving Platforms 356 11.7 Conclusion 359 References 359 12 Optimization and Learning-Based Methods for Radar Imaging with Sparse and Limited Apertures 363 Ammar Saleem, Alper Güngör, M. Burak Alver, Emre Güven, and Müjdat Çetin 12.1 Introduction 363 12.2 SAR Observation Model 364 12.3 Model-Based Imaging and the Role of Sparsity 367 12.3.1 Overview 367 12.3.2 Feature-Enhanced Sparse SAR Imaging 368 12.3.2.1 Strong Scatterer Enhancement 369 12.3.2.2 Region Enhancement 369 12.3.2.3 Point Target and Region Enhancement 370 12.3.2.4 Transform Domain Enhancement 370 12.3.3 Proximal Algorithms for SAR Imaging 370 12.3.3.1 Alternating Direction Method of Multipliers 371 12.3.3.2 ADMM-Based SAR Reconstruction 372 12.3.3.3 Illustrative Examples 374 12.3.4 Imaging in the Presence of Model Errors 376 12.3.4.1 Sparsity-Driven Autofocus 376 12.3.4.2 Autofocusing with Compressive SAR Imaging Using ADMM 377 12.3.4.3 Illustrative Examples 379 12.4 Learning-Based SAR Imaging 383 12.4.1 Overview 383 12.4.2 Dictionary Learning-Based SAR Image Reconstruction 383 12.4.3 Plug-and-Play Methods for SAR Image Reconstruction 383 12.4.3.1 PnP-CNN-SAR Image Denoiser with ADMM 383 12.4.3.2 Phase Estimation for PnP-CNN-SAR 384 12.4.3.3 Magnitude Estimation for PnP-CNN-SAR 385 12.4.3.4 Auxiliary Update for PnP-SAR 385 12.4.4 Illustrative Examples 388 12.5 Conclusion 389 References 391 13 Sparse Arrays for Sonar 395 Kaushallya Adhikari and Kathleen E. Wage 13.1 Introduction 395 13.2 Active Sonar Processing 397 13.2.1 Review of Uniform Line Arrays 397 13.2.2 Simple Active Sensing Example 398 13.2.3 Echo-Sounding and Mills Cross Array 400 13.3 Passive Sonar Processing 401 13.3.1 Passive ULAs and the Difference Coarray 401 13.3.2 Difference Coarrays of Sparse Arrays 402 13.3.3 Sparse Passive Processing Algorithms 406 13.3.4 Review of the Predominant Processors 409 13.3.4.1 Conventional Beamforming 409 13.3.4.2 Product Processing 409 13.3.4.3 Min Processing 411 13.3.4.4 Augmented Processor 412 13.3.5 Simulation Example 412 13.4 Experimental Sonar Examples 414 13.5 Further Reading on Sparse Sonar 416 Acknowledgements 418 References 418 14 Unconventional Array Architectures for Next Generation Wireless Communications 423 Nicola Anselmi, Sotirios Goudos, Giacomo Oliveri, Lorenzo Poli, Paolo Rocca, Marco Salucci, and Andrea Massa 14.1 Introduction 423 14.2 Sparseness-Promoting Techniques for the Design of Unconventional Architectures 425 14.2.1 Sparse Array Synthesis Through Bayesian Compressive Sensing 425 14.2.1.1 ST-BCS Synthesis Method 426 14.2.1.2 MT-BCS Synthesis Method 427 14.2.2 Dictionary-Based Compressing Sensing Method 429 14.2.3 Total-Variation Regularization Techniques 433 14.3 Co-design of Unconventional Architectures and Radiating Elements 435 14.3.1 5G Base Station Antenna Design Problem 436 14.3.2 Co-design Synthesis Strategy 439 14.4 Capacity-Driven Synthesis of Next Generation Base Station Phased Arrays 443 14.4.1 Modular Array Capacity-Driven Synthesis 443 14.5 Final Remarks and Envisaged Trends 448 Acknowledgments 449 References 450 15 MIMO Communication with Sparse Arrays 455 Ahmed Alkhateeb, Xiang Gao, and Elias Aboutanios 15.1 Introduction 455 15.2 Fully Digital Architectures with Sparse Arrays 456 15.2.1 Architectures 457 15.2.2 Design Criteria and Signal Processing Approaches 458 15.2.3 Simulation Results 461 15.3 Hybrid Analog–Digital Architectures with Sparse Arrays 461 15.3.1 Basic Hybrid Analog–Digital Architectures 462 15.3.1.1 Fully Connected Hybrid Architecture 462 15.3.1.2 Array of Sub-Arrays Architecture 463 15.3.2 Hybrid Architectures with Sparse Arrays 464 15.3.3 Design Criteria and Signal Processing Approaches 465 15.3.3.1 Optimal Hybrid Beamforming Design Via Exhaustive Search 466 15.3.3.2 Hybrid Beamformer Design Via Convex Optimization 467 15.4 Conclusion and Future Directions 471 References 471 Index 477