Wideband Beamforming

Dr. Wei Liu, University of Sheffield, UKDr Liu received his B.Sc. in Space Physics (1996) and L.L.B in Intellectual Property Law (1997) from Peking University, China; M.Phil. in Filter Banks and Wavelets from the Department of Electrical and Electronic Engineering, University of Hong Kong (2001); PhD in Digital Beamforming from the Communications Research Group, School of Electronics and Computer Science, University of Southampton (2003). He then worked as a postdoc in the same group and later in the Communications and Signal Processing Group, Department of Electrical and Electronic Engineering, Imperial College London, where he moved to the area of blind source separation. In September 2005, Dr Liu joined the Communications Research Group, Department of Electronic and Electrical Engineering, University of Sheffield, as a lecturer.Dr. Stephan Weiss, University of Strathclyde, ScotlandDr Weiss received the degree of Diplom-Ingenieur in Electronic & Electrical Engineering from the University of Erlangen-Nürnberg in 1995, and the PhD in the same subject from the University of Strathclyde in 1998. Interrupted by a leave in 1996/97 as a visiting scholar at the Signal and Image Processing Institute, University of Southern California, he was a research student in the Signal Processing Division, University of Strathclyde. He held a visiting lectureship in the EEE Department at Strathclyde in 1998/99, and joined the Communications Research Group of the Electronics and Computer Science Department at the University of Southampton as a lecturer in 1999. He was promoted to a senior lecturership at University of Strathclyde in 2003.

• About the Series Editors viiPreface xiii1 Introduction 11.1 Array Signal Processing 11.2 Narrowband Beamforming 41.3 Wideband Beamforming 71.4 Wideband Beam Steering 111.4.1 Beam Steering for Narrowband Arrays 121.4.2 Beam Steering for Wideband Arrays 131.4.3 A Unified Interpretation 171.5 Summary 182 Adaptive Wideband Beamforming 192.1 Reference Signal-Based Beamformer 192.1.1 Least Mean Square Algorithm 202.1.2 Normalized Least Mean Square Algorithm 222.1.3 Recursive Least Squares Algorithm 232.1.4 Comparison of Computational Complexities 242.1.5 Frequency-Domain and Subband Adaptive Algorithms 262.1.6 Simulations 262.2 Linearly Constrained Minimum Variance Beamforming 282.2.1 A Simple Formulation of Constraints 292.2.2 Optimum Solution to the LCMV Problem 302.2.3 Frost’s Algorithm for LCMV Beamforming 312.2.4 Simulations 312.3 Constraints Design for LCMV Beamforming 332.3.1 Eigenvector Constraint Design 332.3.2 Design Example 352.3.3 Application to Wideband DOA Estimation 362.4 Generalized Sidelobe Canceller 382.4.1 GSC Structure 382.4.2 GSC with Tapped Delay-Lines 422.4.3 Blocking Matrix Design 462.4.4 Simulations 482.5 Other Minimum Variance Beamformers 482.5.1 Soft Constrained Minimum Variance Beamformer 492.5.2 Correlation Constrained Minimum Variance Beamformer 512.6 Robust Adaptive Beamforming 522.6.1 Spatially Extended Constraints 522.6.2 Norm-Restrained Approaches 572.7 Summary 603 Subband Adaptive Beamforming 613.1 Fundamentals of Filter Banks 613.1.1 Basic Multirate Operations 623.1.2 Perfect Reconstruction Condition for Filter Banks 663.1.3 Oversampled Modulated Filter Banks 683.2 Subband Adaptive Filtering 703.3 General Subband Adaptive Beamforming 743.3.1 Reference Signal Based Beamformer 753.3.2 Generalized Sidelobe Canceller 763.3.3 Reconstruction of the Fullband Beamformer 793.3.4 Simulations 793.4 Subband Adaptive GSC 823.4.1 Structure 823.4.2 Analysis of the Computational Complexity 823.4.3 Reconstruction of the Fullband Beamformer 833.4.4 Simulations 833.5 Temporally/Spatially Subband-Selective Beamforming 843.5.1 Partially Adaptive GSC 853.5.2 Temporally/Spatially Subband-Selective Blocking Matrix 873.5.3 Temporally/Spatially Subband-Selective Transformation Matrix 953.5.4 Application to Subband Adaptive GSC 983.5.5 Extension to the General Subband Adaptive Beamforming Structure 1003.5.6 Simulations 1033.6 Frequency-Domain Adaptive Beamforming 1053.6.1 Frequency-Domain Formulation 1063.6.2 Constrained Frequency-Domain Adaptive Algorithm 1083.6.3 Frequency-Domain GSC 1093.6.4 Simulations 1113.7 Transform-Domain Adaptive Beamforming 1123.7.1 Transform-Domain GSC 1133.7.2 Subband-Selective Transform-Domain GSC 1153.7.3 Simulations 1153.8 Summary 1184 Design of Fixed Wideband Beamformers 1194.1 Iterative Optimization 1194.1.1 Traditional Methods 1194.1.2 Convex Optimization 1204.2 The Least Squares Approach 1264.2.1 Standard Formulation 1264.2.2 Constrained Least Squares 1284.3 The Eigenfilter Approach 1314.3.1 Standard Approach 1324.3.2 Maximum Energy 1374.3.3 Total Least Squares 1394.4 Summary 1425 Frequency Invariant Beamforming 1435.1 Introduction 1435.2 Design Based on Multi-Dimensional Inverse Fourier Transform 1445.2.1 Continuous Sensor and Signals 1445.2.2 Discrete Sensors and Signals 1515.2.3 Design Examples 1555.2.4 Further Generalization to the FIB Design 1635.3 Subband Design of Frequency Invariant Beamformers 1675.3.1 First Implementation 1695.3.2 Second Implementation–Scaled Aperture 1735.3.3 Design Examples 1755.4 Frequency Invariant Beamforming for Circular Arrays 1765.4.1 Phase Mode Processing 1775.4.2 FIB Design 1815.4.3 Design Example 1815.5 Direct Optimization for Frequency Invariant Beamforming 1825.5.1 Convex Optimization 1825.5.2 Least Squares 1855.5.3 Eigenfilter 1865.6 Beamspace Adaptive Wideband Beamforming 1885.6.1 Structure 1885.6.2 Analysis of the Beamspace Adaptive Method 1905.6.3 Design of Independent FIBs 1925.6.4 Simulations 1935.7 Summary 1976 Blind Wideband Beamforming 1996.1 Blind Source Separation 1996.1.1 Introduction 1996.1.2 A Blind Source Extraction Example 2016.2 Blind Wideband Beamforming 2046.3 Blind Beamforming Based on Frequency Invariant Transformation 2066.3.1 Structure 2076.3.2 The Algorithm 2086.3.3 Simulations 2086.4 Summary 2117 Wideband Beamforming with Sensor Delay-Lines 2137.1 Sensor Delay-Line Based Structures 2137.1.1 Introduction 2137.1.2 Wideband Response of the SDL-Based Structure 2177.2 Frequency Invariant Beamforming 2187.2.1 2-D Arrays 2207.2.2 3-D Arrays 2247.3 Adaptive Beamforming 2287.3.1 Reference Signal Based Beamformer 2297.3.2 Linearly Constrained Minimum Variance Beamformer 2307.3.3 Discussions 2327.3.4 Simulations 2337.4 Beamspace Adaptive Beamforming 2357.4.1 Structure 2357.4.2 Simulations 2367.5 Summary 2388 Wideband Beamforming for Multipath Signals 2398.1 The Wideband Multipath Problem 2408.2 Approach Based on a Narrowband Beamformer 2418.2.1 Structure 2418.2.2 Simulations 2438.3 Approach Based on Blind Source Separation 2468.3.1 Structure 2468.3.2 Simulations 2478.4 MIMO System 2498.4.1 Evolution to a MIMO System 2508.4.2 MIMO Beamforming and Equalization 2528.5 Summary 254Appendix A: Matrix Approximation 255Appendix B: Differentiation with Respect to a Vector 259Appendix C: Genetic Algorithm 261C. 1 The Principle 261C.1. 1 Chromosome Representation 261C.1. 2 Parent Selection 262C.1. 3 Genetic Operation 262C.1. 4 Fitness Evaluation 263C.1. 5 Initialization 263C.1. 6 Termination 263C. 2 Design Example in Section 3.5.2 264Bibliography 267Index 283