Dr. Moeness G. Amin, Villanova University, USA. Since 1985, Dr. Amin has been with the Faculty of the Department of Electrical and Computer Engineering, Villanova University, PA, USA, where he became the Director of the Center for Advanced Communications, College of Engineering, in 2002. He has more than 900 journal and conference publications in signal processing theory and applications, covering the areas of wireless communications, radar, sonar, satellite navigations, ultrasound, and RFID.

About the Editor xvii List of Contributors xviii Preface xxiii 1 Sparse Arrays: Fundamentals 1Palghat 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 41Chengwei 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 75Fauzia 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 103Piya Pal and Heng Qiao 4.1 Introduction 103 4.2 Background on Correlation-Aware Sparse Support Recovery with Sparse Arrays 104 > 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 > M with Correlation Estimates: Preliminaries 110 > 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 131Ali 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 181Kumar 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 215Syed 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 251Geert 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 ¿ Parameter 261 8.3.1.1 Optimality Criteria 261 8.3.1.2 Sparse Sampler Design 263 8.3.2 Unknown ¿ 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 273Robin Rajamäki and Visa Koivunen 9.1 Introduction 273 9.1.1 Goals, Scope, and...