Optimum Array Processing

Transcription

1 Optimum Array Processing Part IV of Detection, Estimation, and Modulation Theory Harry L. Van Trees WILEY- INTERSCIENCE A JOHN WILEY & SONS, INC., PUBLICATION

2 Preface xix 1 Introduction Array Processing Applications Radar Radio Astronomy Sonar Communications Direction Finding Seismology Tomography Array Processing Literature Organization of the Book Interactive Study 14 2 Arrays and Spatial Filters Introduction Frequency-wavenumber Response and Beam Patterns Uniform Linear Arrays Uniformly Weighted Linear Arrays Beam Pattern Parameters Array Steering Array Performance Measures Directivity Array Gain vs. Spatially White Noise (A w ) Sensitivity and the Tolerance Factor Summary Linear Apertures 71 vii

3 Summary Broadband Arrays Summary Problems Planar Arrays and Apertures Rectangular Arrays Uniform Rectangular Arrays Array Manifold Vector Separable Spectral Weightings D z-transforms Least Squares Synthesis Circularly Symmetric Weighting and Windows Wavenumber Sampling and 2-D DFT Transformations from One Dimension to Two Dimensions Null Steering Related Topics Circular Arrays Continuous Circular Arrays (Ring Apertures) Circular Arrays Phase Mode Excitation Beamformers Circular Apertures Separable Weightings Taylor Synthesis for Circular Apertures Sampling the Continuous Distribution Difference Beams Summary Hexagonal Arrays Introduction Beam Pattern Design Hexagonal Grid to Rectangular Grid Transformation Summary Nonplanar Arrays Cylindrical Arrays Spherical Arrays Summary Problems 322

4 x 5 Characterization of Space-time Processes Introduction Snapshot Models Frequency-domain Snapshot Models Narrowband Time-domain Snapshot Models Summary Space-time Random Processes Second-moment Characterization Gaussian Space-time Processes Plane Waves Propagating in Three Dimensions D and 2-D Projections Arrays and Apertures Arrays Apertures Orthogonal Expansions Plane-wave Signals Spatially Spread Signals Frequency-spread Signals Closely Spaced Signals Beamspace Processors Subspaces for Spatially Spread Signals Parametric Wavenumber Models Rational Transfer Function Models Model Relationships Observation Noise Summary Summary Problems Optimum Waveform Estimation Introduction Optimum Beamformers Minimum Variance Distortionless Response (MVDR) Beamformers Minimum Mean-Square Error (MMSE) Estimators Maximum Signal-to-Noise Ratio (SNR) Minimum Power Distortionless Response (MPDR) Beamformers Summary Discrete Interference 452

5 xi Single Plane-wave Interfering Signal Multiple Plane-wave Interferers Summary: Discrete Interference Spatially Spread Interference Physical Noise Models ARMA Models Multiple Plane-wave Signals MVDR Beamformer MMSE Processors Mismatched MVDR and MPDR Beamformers Introduction DOA Mismatch Array Perturbations Diagonal Loading Summary LCMV and LCMP Beamformers Typical Constraints Optimum LCMV and LCMP Beamformers Generalized Sidelobe Cancellers Performance of LCMV and LCMP Beamformers Quiescent Pattern (QP) Constraints Covariance Augmentation Summary Eigenvector Beamformers Principal-component (PC) Beamformers Cross-spectral Eigenspace Beamformers Dominant-mode Rejection Beamformers Summary Beamspace Beamformers Beamspace MPDR Beamspace LCMP Summary: Beamspace Optimum Processors Quadratically Constrained Beamformers Soft-constraint Beamformers Beamforming for Correlated Signal and Interferences Introduction MPDR Beamformer: Correlated Signals and Interference MMSE Beamformer: Correlated Signals and Interference Spatial Smoothing and Forward-Backward Averaging Summary 620

6 xii 6.13 Broadband Beamformers Introduction DFT Beamformers Finite impulse response (FIR) Beamformers Summary: Broadband Processing Summary Problems Adaptive Beamformers Introduction Estimation of Spatial Spectral Matrices Sample Spectral Matrices Asymptotic Behavior Forward-Backward Averaging Structured Spectral Matrix Estimation Parametric Spatial Spectral Matrix Estimation Singular Value Decomposition Summary Sample Matrix Inversion (SMI) SINR ami Behavior: MVDR and MPDR LCMV and LCMP Beamformers Fixed Diagonal Loading Toeplitz Estimators Summary Recursive Least Squares (RLS) Least Squares Formulation Recursive Implementation Recursive Implementation of LSE Beamformer Generalized Sidelobe Canceller Quadratically Constrained RLS Conjugate Symmetric Beamformers Summary Efficient Recursive Implementation Algorithms Introduction QR Decomposition (QRD) Gradient Algorithms Introduction Steepest Descent: MMSE Beamformers Steepest Decent: LCMP Beamformer Summary 805

7 xiii 7.7 LMS Algorithms Derivation of the LMS Algorithms Performance of the LMS Algorithms LMS Algorithm Behavior Quadratic Constraints Summary: LMS algorithms Detection of Signal Subspace Dimension Detection Algorithms Eigenvector Detection Tests Eigenspace and DMR Beamformers Performance of SMI Eigenspace Beamformers Eigenspace and DMR Beamformers: Detection of Subspace Dimension Subspace tracking Summary Beamspace Beamformers Beamspace SMI Beamspace RLS Beamspace LMS Summary: Adaptive Beamspace Processing Broadband Beamformers SMI Implementation LMS Implementation GSC: Multichannel Lattice Filters Summary Summary Problems Parameter Estimation I: Maximum Likelihood Introduction Maximum Likelihood and Maximum a posteriori Estimators Maximum Likelihood (ML) Estimator Maximum a posteriori (MAP) Estimator Cramer-Rao Bounds Parameter Estimation Model Multiple Plane Waves Model Perturbations Parametric Spatially Spread Signals Summary Cramer-Rao Bounds 938

8 xiv Gaussian Model: Unknown Signal Spectrum Gaussian Model: Uncorrelated Signals with Unknown Power Gaussian Model: Known Signal Spectrum Nonrandom (Conditional) Signal Model Known Signal Waveforms Summary Maximum Likelihood Estimation Maximum Likelihood Estimation Conditional Maximum Likelihood Estimators Weighted Subspace Fitting Asymptotic Performance Wideband Signals Summary Computational Algorithms Optimization Techniques Alternating Maximization Algorithms Expectation Maximization Algorithm Summary Polynomial Parameterization Polynomial Parameterization Iterative Quadratic Maximum Likelihood (IQML) Polynomial WSF (MODE) Summary Detection of Number of Signals Spatially Spread Signals Parameterized S(0,< ) Spatial ARMA Process Summary Beamspace algorithms Introduction Beamspace Matrices Beamspace Cramer-Rao Bound Beamspace Maximum Likelihood Summary Sensitivity, Robustness, and Calibration Model Perturbations Cramer-Rao Bounds Sensitivity of ML Estimators MAP Joint Estimation 1099

9 XV Self-Calibration Algorithms Summary Summary Major Results Related Topics Algorithm complexity Problems Parameter Estimation II Introduction Quadratic Algorithms Introduction Beamscan Algorithms MVDR (Capon) Algorithm Root Versions of Quadratic Algorithms Performance of MVDR Algorithms Summary Subspace Algorithms Introduction MUSIC Minimum-Norm Algorithm ESPRIT Algorithm Comparison Summary Linear Prediction Asymptotic Performance Error Behavior Resolution of MUSIC and Min-Norm Small Error Behavior of Algorithms Summary Correlated and Coherent Signals Introduction Forward-Backward Spatial Smoothing Summary Beamspace Algorithms Beamspace MUSIC Beamspace Unitary ESPRIT Beamspace Summary Sensitivity and Robustness Planar Arrays 1255

10 XVI Standard Rectangular Arrays Hexagonal Arrays Summary: Planar Arrays Summary Major Results Related Topics Discussion Problems Detection and Other Topics Optimum Detection Classic Binary Detection Matched Subspace Detector Spatially Spread Gaussian Signal Processes Adaptive Detection Related Topics Epilogue Problems 1329 A Matrix Operations 1340 A.l Introduction 1340 A.2 Basic Definitions and Properties 1341 A.2.1 Basic Definitions 1341 A.2.2 Matrix Inverses 1347 A.2.3 Quadratic Forms 1348 A.2.4 Partitioned Matrices 1349 A.2.5 Matrix products 1351 A.2.6 Matrix Inequalities 1356 A.3 Special Vectors and Matrices 1356 A.3.1 Elementary Vectors and Matrices 1356 A.3.2 The vec(a) matrix 1358 A.3.3 Diagonal Matrices 1359 A.3.4 Exchange Matrix and Conjugate Symmetric Vectors A.3.5 Persymmetric and Centrohermitian Matrices 1362 A.3.6 Toeplitz and Hankel Matrices 1364 A.3.7 Circulant Matrices 1365 A.3.8 Triangular Matrices 1366 A.3.9 Unitary and Orthogonal Matrices 1367 A.3.10 Vandermonde Matrices 1368 A.3.11 Projection Matrices 1369

11 xvii A.3.12 Generalized Inverse 1370 A.4 Eigensystems 1372 A.4.1 Eigendecomposition 1372 A.4.2 Special Matrices 1376 A.5 Singular Value Decomposition 1381 A.6 QR Decomposition 1387 A.6.1 Introduction ' 1387 A.6.2 QR Decomposition 1388 A.6.3 Givens Rotation 1390 A.6.4 Householder Transformation 1394 A.7 Derivative Operations 1397 A.7.1 Derivative of Scalar with Respect to Vector 1397 A.7.2 Derivative of Scalar with Respect to Matrix 1399 A.7.3 Derivatives with Respect to Parameter 1401 A.7.4 Complex Gradients 1402 В Array Processing Literature 1407 B.l Journals 1407 B.2 Books 1408 B.3 Duality 1409 С Notation 1414 C.l Conventions 1414 C.2 Acronyms 1415 C.3 Mathematical Symbols 1418 C.4 Symbols 1419 Index 1434