Wideband Beamforming: Concepts and Techniques

About the Series Editors vii

Preface xiii

1 Introduction 1

1.1 Array Signal Processing 1

1.2 Narrowband Beamforming 4

1.3 Wideband Beamforming 7

1.4 Wideband Beam Steering 11

1.4.1 Beam Steering for Narrowband Arrays 12

1.4.2 Beam Steering for Wideband Arrays 13

1.4.3 A Unified Interpretation 17

1.5 Summary 18

2 Adaptive Wideband Beamforming 19

2.1 Reference Signal-Based Beamformer 19

2.1.1 Least Mean Square Algorithm 20

2.1.2 Normalized Least Mean Square Algorithm 22

2.1.3 Recursive Least Squares Algorithm 23

2.1.4 Comparison of Computational Complexities 24

2.1.5 Frequency-Domain and Subband Adaptive Algorithms 26

2.1.6 Simulations 26

2.2 Linearly Constrained Minimum Variance Beamforming 28

2.2.1 A Simple Formulation of Constraints 29

2.2.2 Optimum Solution to the LCMV Problem 30

2.2.3 Frost’s Algorithm for LCMV Beamforming 31

2.2.4 Simulations 31

2.3 Constraints Design for LCMV Beamforming 33

2.3.1 Eigenvector Constraint Design 33

2.3.2 Design Example 35

2.3.3 Application to Wideband DOA Estimation 36

2.4 Generalized Sidelobe Canceller 38

2.4.1 GSC Structure 38

2.4.2 GSC with Tapped Delay-Lines 42

2.4.3 Blocking Matrix Design 46

2.4.4 Simulations 48

2.5 Other Minimum Variance Beamformers 48

2.5.1 Soft Constrained Minimum Variance Beamformer 49

2.5.2 Correlation Constrained Minimum Variance Beamformer 51

2.6 Robust Adaptive Beamforming 52

2.6.1 Spatially Extended Constraints 52

2.6.2 Norm-Restrained Approaches 57

2.7 Summary 60

3 Subband Adaptive Beamforming 61

3.1 Fundamentals of Filter Banks 61

3.1.1 Basic Multirate Operations 62

3.1.2 Perfect Reconstruction Condition for Filter Banks 66

3.1.3 Oversampled Modulated Filter Banks 68

3.2 Subband Adaptive Filtering 70

3.3 General Subband Adaptive Beamforming 74

3.3.1 Reference Signal Based Beamformer 75

3.3.2 Generalized Sidelobe Canceller 76

3.3.3 Reconstruction of the Fullband Beamformer 79

3.3.4 Simulations 79

3.4 Subband Adaptive GSC 82

3.4.1 Structure 82

3.4.2 Analysis of the Computational Complexity 82

3.4.3 Reconstruction of the Fullband Beamformer 83

3.4.4 Simulations 83

3.5 Temporally/Spatially Subband-Selective Beamforming 84

3.5.1 Partially Adaptive GSC 85

3.5.2 Temporally/Spatially Subband-Selective Blocking Matrix 87

3.5.3 Temporally/Spatially Subband-Selective Transformation Matrix 95

3.5.4 Application to Subband Adaptive GSC 98

3.5.5 Extension to the General Subband Adaptive Beamforming Structure 100

3.5.6 Simulations 103

3.6 Frequency-Domain Adaptive Beamforming 105

3.6.1 Frequency-Domain Formulation 106

3.6.2 Constrained Frequency-Domain Adaptive Algorithm 108

3.6.3 Frequency-Domain GSC 109

3.6.4 Simulations 111

3.7 Transform-Domain Adaptive Beamforming 112

3.7.1 Transform-Domain GSC 113

3.7.2 Subband-Selective Transform-Domain GSC 115

3.7.3 Simulations 115

3.8 Summary 118

4 Design of Fixed Wideband Beamformers 119

4.1 Iterative Optimization 119

4.1.1 Traditional Methods 119

4.1.2 Convex Optimization 120

4.2 The Least Squares Approach 126

4.2.1 Standard Formulation 126

4.2.2 Constrained Least Squares 128

4.3 The Eigenfilter Approach 131

4.3.1 Standard Approach 132

4.3.2 Maximum Energy 137

4.3.3 Total Least Squares 139

4.4 Summary 142

5 Frequency Invariant Beamforming 143

5.1 Introduction 143

5.2 Design Based on Multi-Dimensional Inverse Fourier Transform 144

5.2.1 Continuous Sensor and Signals 144

5.2.2 Discrete Sensors and Signals 151

5.2.3 Design Examples 155

5.2.4 Further Generalization to the FIB Design 163

5.3 Subband Design of Frequency Invariant Beamformers 167

5.3.1 First Implementation 169

5.3.2 Second Implementation–Scaled Aperture 173

5.3.3 Design Examples 175

5.4 Frequency Invariant Beamforming for Circular Arrays 176

5.4.1 Phase Mode Processing 177

5.4.2 FIB Design 181

5.4.3 Design Example 181

5.5 Direct Optimization for Frequency Invariant Beamforming 182

5.5.1 Convex Optimization 182

5.5.2 Least Squares 185

5.5.3 Eigenfilter 186

5.6 Beamspace Adaptive Wideband Beamforming 188

5.6.1 Structure 188

5.6.2 Analysis of the Beamspace Adaptive Method 190

5.6.3 Design of Independent FIBs 192

5.6.4 Simulations 193

5.7 Summary 197

6 Blind Wideband Beamforming 199

6.1 Blind Source Separation 199

6.1.1 Introduction 199

6.1.2 A Blind Source Extraction Example 201

6.2 Blind Wideband Beamforming 204

6.3 Blind Beamforming Based on Frequency Invariant Transformation 206

6.3.1 Structure 207

6.3.2 The Algorithm 208

6.3.3 Simulations 208

6.4 Summary 211

7 Wideband Beamforming with Sensor Delay-Lines 213

7.1 Sensor Delay-Line Based Structures 213

7.1.1 Introduction 213

7.1.2 Wideband Response of the SDL-Based Structure 217

7.2 Frequency Invariant Beamforming 218

7.2.1 2-D Arrays 220

7.2.2 3-D Arrays 224

7.3 Adaptive Beamforming 228

7.3.1 Reference Signal Based Beamformer 229

7.3.2 Linearly Constrained Minimum Variance Beamformer 230

7.3.3 Discussions 232

7.3.4 Simulations 233

7.4 Beamspace Adaptive Beamforming 235

7.4.1 Structure 235

7.4.2 Simulations 236

7.5 Summary 238

8 Wideband Beamforming for Multipath Signals 239

8.1 The Wideband Multipath Problem 240

8.2 Approach Based on a Narrowband Beamformer 241

8.2.1 Structure 241

8.2.2 Simulations 243

8.3 Approach Based on Blind Source Separation 246

8.3.1 Structure 246

8.3.2 Simulations 247

8.4 MIMO System 249

8.4.1 Evolution to a MIMO System 250

8.4.2 MIMO Beamforming and Equalization 252

8.5 Summary 254

Appendix A:Matrix Approximation 255

Appendix B: Differentiation with Respect to a Vector 259

Appendix C: Genetic Algorithm 261

C.1 The Principle 261

C.1.1 Chromosome Representation 261

C.1.2 Parent Selection 262

C.1.3 Genetic Operation 262

C.1.4 Fitness Evaluation 263

C.1.5 Initialization 263

C.1.6 Termination 263

C.2 Design Example in Section 3.5.2 264

Bibliography 267

Index 283