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ISBN 10: 0471093904
ISBN 13: 978-0471093909
Author: Harry Van Trees
1 Introduction
1.1 Array Processing
1.2 Applications
1.2.1 Radar
1.2.2 Radio Astronomy
1.2.3 Sonar
1.2.4 Communications
1.2.5 Direction Finding
1.2.6 Seismology
1.2.7 Tomography
1.2.8 Array Processing Literature
1.3 Organization of the Book
1.4 Interactive Study
2 Arrays and Spatial Filters
2.1 Introduction
2.2 Frequency-wavenumber Response and Beam Patterns
2.3 Uniform Linear Arrays
2.4 Uniformly Weighted Linear Arrays
2.4.1 Beam Pattern Parameters
2.5 Array Steering
2.6 Array Performance Measures
2.6.1 Directivity
2.6.2 Array Gain vs. Spatially White Noise (A[sub(w)])
2.6.3 Sensitivity and the Tolerance Factor
2.6.4 Summary
2.7 Linear Apertures
2.7.1 Frequency-wavenumber Response
2.7.2 Aperture Sampling
2.8 Non-isotropic Element Patterns
2.9 Summary
2.10 Problems
3 Synthesis of Linear Arrays and Apertures
3.1 Spectral Weighting
3.2 Array Polynomials and the z-Transform
3.2.1 z-Transform
3.2.2 Real Array Weights
3.2.3 Properties of the Beam Pattern Near a Zero
3.3 Pattern Sampling in Wavenumber Space
3.3.1 Continuous Aperture
3.3.2 Linear Arrays
3.3.3 Discrete Fourier Transform
3.3.4 Norms
3.3.5 Summary
3.4 Minimum Beamwidth for Specified Sidelobe Level
3.4.1 Introduction
3.4.2 Dolph-Chebychev Arrays
3.4.3 Taylor Distribution
3.4.4 Villeneuve n Distribution
3.5 Least Squares Error Pattern Synthesis
3.6 Minimax Design
3.6.1 Alternation Theorem
3.6.2 Parks-McClellan-Rabiner Algorithm
3.6.3 Summary
3.7 Null Steering
3.7.1 Null Constraints
3.7.2 Least Squares Error Pattern Synthesis with Nulls
3.8 Asymmetric Beams
3.9 Spatially Non-uniform Linear Arrays
3.9.1 Introduction
3.9.2 Minimum Redundancy Arrays
3.9.3 Beam Pattern Design Algorithm
3.10 Beamspace Processing
3.10.1 Full-dimension Beamspace
3.10.2 Reduced-dimension Beamspace
3.10.3 Multiple Beam Antennas
3.10.4 Summary
3.11 Broadband Arrays
3.12 Summary
3.13 Problems
4 Planar Arrays and Apertures
4.1 Rectangular Arrays
4.1.1 Uniform Rectangular Arrays
4.1.2 Array Manifold Vector
4.1.3 Separable Spectral Weightings
4.1.4 2-D z-Transforms
4.1.5 Least Squares Synthesis
4.1.6 Circularly Symmetric Weighting and Windows
4.1.7 Wavenumber Sampling and 2-D DFT
4.1.8 Transformations from One Dimension to Two Dimensions
4.1.9 Null Steering
4.1.10 Related Topics
4.2 Circular Arrays
4.2.1 Continuous Circular Arrays (Ring Apertures)
4.2.2 Circular Arrays
4.2.3 Phase Mode Excitation Beamformers
4.3 Circular Apertures
4.3.1 Separable Weightings
4.3.2 Taylor Synthesis for Circular Apertures
4.3.4 Difference Beams
4.3.5 Summary
4.4 Hexagonal Arrays
4.4.1 Introduction
4.4.2 Beam Pattern Design
4.4.3 Hexagonal Grid to Rectangular Grid Transformation
4.4.4 Summary
4.5 Nonplanar Arrays
4.5.1 Cylindrical Arrays
4.5.2 Spherical Arrays
4.6 Summary
4.7 Problems
5 Characterization of Space-time Processes
5.1 Introduction
5.2 Snapshot Models
5.2.1 Frequency-domain Snapshot Models
5.2.2 Narrowband Time-domain Snapshot Models
5.2.3 Summary
5.3 Space-time Random Processes
5.3.1 Second-moment Characterization
5.3.2 Gaussian Space-time Processes
5.3.3 Plane Waves Propagating in Three Dimensions
5.3.4 1-D and 2-D Projections
5.4 Arrays and Apertures
5.4.1 Arrays
5.4.2 Apertures
5.5 Orthogonal Expansions
5.5.1 Plane-wave Signals
5.5.2 Spatially Spread Signals
5.5.3 Frequency-spread Signals
5.5.4 Closely Spaced Signals
5.5.5 Beamspace Processors
5.5.6 Subspaces for Spatially Spread Signals
5.6 Parametric Wavenumber Models
5.6.1 Rational Transfer Function Models
5.6.2 Model Relationships
5.6.3 Observation Noise
5.6.4 Summary
5.7 Summary
5.8 Problems
6 Optimum Waveform Estimation
6.1 Introduction
6.2 Optimum Beamformers
6.2.1 Minimum Variance Distortionless Response (MVDR) Beamformers
6.2.2 Minimum Mean-Square Error (MMSE) Estimators
6.2.3 Maximum Signal-to-Noise Ratio (SNR)
6.2.4 Minimum Power Distortionless Response (MPDR) Beamformers
6.2.5 Summary
6.3 Discrete Interference
6.3.1 Single Plane-wave Interfering Signal
6.3.2 Multiple Plane-wave Interferers
6.3.3 Summary: Discrete Interference
6.4 Spatially Spread Interference
6.4.1 Physical Noise Models
6.4.2 ARMA Models
6.5 Multiple Plane-wave Signals
6.5.1 MVDR Beamformer
6.5.2 MMSE Processors
6.6 Mismatched MVDR and MPDR Beamformers
6.6.1 Introduction
6.6.2 DOA Mismatch
6.6.3 Array Perturbations
6.6.4 Diagonal Loading
6.6.5 Summary
6.7 LCMV and LCMP Beamformers
6.7.1 Typical Constraints
6.7.2 Optimum LCMV and LCMP Beamformers
6.7.3 Generalized Sidelobe Cancellers
6.7.4 Performance of LCMV and LCMP Beamformers
6.7.5 Quiescent Pattern (QP) Constraints
6.7.6 Covariance Augmentation
6.7.7 Summary
6.8 Eigenvector Beamformers
6.8.1 Principal-component (PC) Beamformers
6.8.2 Cross-spectral Eigenspace Beamformers
6.8.3 Dominant-mode Rejection Beamformers
6.8.4 Summary
6.9 Beamspace Beamformers
6.9.1 Beamspace MPDR
6.9.2 Beamspace LCMP
6.9.3 Summary: Beamspace Optimum Processors
6.10 Quadratically Constrained Beamformers
6.11 Soft-constraint Beamformers
6.12 Beamforming for Correlated Signal and Interferences
6.12.1 Introduction
6.12.2 MPDR Beamformer: Correlated Signals and Interference
6.12.3 MMSE Beamformer: Correlated Signals and Interference
6.12.4 Spatial Smoothing and Forward-Backward Averaging
6.12.5 Summary
6.13 Broadband Beamformers
6.13.1 Introduction
6.13.2 DFT Beamformers
6.13.3 Finite impulse response (FIR) Beamformers
6.13.4 Summary: Broadband Processing
6.14 Summary
6.15 Problems
7 Adaptive Beamformers
7.1 Introduction
7.2 Estimation of Spatial Spectral Matrices
7.2.1 Sample Spectral Matrices
7.2.2 Asymptotic Behavior
7.2.3 Forward-Backward Averaging
7.2.4 Structured Spectral Matrix Estimation
7.2.5 Parametric Spatial Spectral Matrix Estimation
7.2.6 Singular Value Decomposition
7.2.7 Summary
7.3 Sample Matrix Inversion (SMI)
7.3.1 SINR[sub(smi)] Behavior: MVDR and MPDR
7.3.2 LCMV and LCMP Beamformers
7.3.3 Fixed Diagonal Loading
7.3.4 Toeplitz Estimators
7.3.5 Summary
7.4 Recursive Least Squares (RLS)
7.4.1 Least Squares Formulation
7.4.2 Recursive Implementation
7.4.3 Recursive Implementation of LSE Beamformer
7.4.4 Generalized Sidelobe Canceller
7.4.5 Quadratically Constrained RLS
7.4.6 Conjugate Symmetric Beamformers
7.4.7 Summary
7.5 Efficient Recursive Implementation Algorithms
7.5.1 Introduction
7.5.2 QR Decomposition (QRD)
7.6 Gradient Algorithms
7.6.1 Introduction
7.6.2 Steepest Descent: MMSE Beamformers
7.6.3 Steepest Decent: LCMP Beamformer
7.6.4 Summary
7.7 LMS Algorithms
7.7.1 Derivation of the LMS Algorithms
7.7.2 Performance of the LMS Algorithms
7.7.3 LMS Algorithm Behavior
7.7.4 Quadratic Constraints
7.7.5 Summary: LMS algorithms
7.8 Detection of Signal Subspace Dimension
7.8.1 Detection Algorithms
7.8.2 Eigenvector Detection Tests
7.9 Eigenspace and DMR Beamformers
7.9.1 Performance of SMI Eigenspace Beamformers
7.9.2 Eigenspace and DMR Beamformers: Detection of Subspace Dimension
7.9.3 Subspace tracking
7.9.4 Summary
7.10 Beamspace Beamformers
7.10.1 Beamspace SMI
7.10.2 Beamspace RLS
7.10.3 Beamspace LMS
7.10.4 Summary: Adaptive Beamspace Processing
7.11 Broadband Beamformers
7.11.1 SMI Implementation
7.11.2 LMS Implementation
7.11.3 GSC: Multichannel Lattice Filters
7.11.4 Summary
7.12 Summary
7.13 Problems
8 Parameter Estimation I: Maximum Likelihood
8.1 Introduction
8.2 Maximum Likelihood and Maximum a posteriori Estimators
8.2.1 Maximum Likelihood (ML) Estimator
8.2.2 Maximum a posteriori (MAP) Estimator
8.2.3 Cramér-Rao Bounds
8.3 Parameter Estimation Model
8.3.1 Multiple Plane Waves
8.3.2 Model Perturbations
8.3.3 Parametric Spatially Spread Signals
8.3.4 Summary
8.4 Cramér-Rao Bounds
8.4.1 Gaussian Model: Unknown Signal Spectrum
8.4.2 Gaussian Model: Uncorrelated Signals with Unknown Power
8.4.3 Gaussian Model: Known Signal Spectrum
8.4.4 Nonrandom (Conditional) Signal Model
8.4.5 Known Signal Waveforms
8.4.6 Summary
8.5 Maximum Likelihood Estimation
8.5.1 Maximum Likelihood Estimation
8.5.2 Conditional Maximum Likelihood Estimators
8.5.3 Weighted Subspace Fitting
8.5.4 Asymptotic Performance
8.5.5 Wideband Signals
8.5.6 Summary
8.6 Computational Algorithms
8.6.1 Optimization Techniques
8.6.2 Alternating Maximization Algorithms
8.6.3 Expectation Maximization Algorithm
8.6.4 Summary
8.7 Polynomial Parameterization
8.7.1 Polynomial Parameterization
8.7.2 Iterative Quadratic Maximum Likelihood (IQML)
8.7.3 Polynomial WSF (MODE)
8.7.4 Summary
8.8 Detection of Number of Signals
8.9 Spatially Spread Signals
8.9.1 Parameterized S(θ, φ)
8.9.2 Spatial ARMA Process
8.9.3 Summary
8.10 Beamspace algorithms
8.10.1 Introduction
8.10.2 Beamspace Matrices
8.10.3 Beamspace Cramér-Rao Bound
8.10.4 Beamspace Maximum Likelihood
8.10.5 Summary
8.11 Sensitivity, Robustness, and Calibration
8.11.1 Model Perturbations
8.11.2 Cramér-Rao Bounds
8.11.3 Sensitivity of ML Estimators
8.11.4 MAP Joint Estimation
8.11.5 Self-Calibration Algorithms
8.11.6 Summary
8.12 Summary
8.12.1 Major Results
8.12.2 Related Topics
8.12.3 Algorithm complexity
8.13 Problems
9 Parameter Estimation II
9.1 Introduction
9.2 Quadratic Algorithms
9.2.1 Introduction
9.2.2 Beamscan Algorithms
9.2.3 MVDR (Capon) Algorithm
9.2.4 Root Versions of Quadratic Algorithms
9.2.5 Performance of MVDR Algorithms
9.2.6 Summary
9.3 Subspace Algorithms
9.3.1 Introduction
9.3.2 MUSIC
9.3.3 Minimum-Norm Algorithm
9.3.4 ESPRIT
9.3.5 Algorithm Comparison
9.3.6 Summary
9.4 Linear Prediction
9.5 Asymptotic Performance
9.5.1 Error Behavior
9.5.2 Resolution of MUSIC and Min-Norm
9.5.3 Small Error Behavior of Algorithms
9.5.4 Summary
9.6 Correlated and Coherent Signals
9.6.1 Introduction
9.6.2 Forward-Backward Spatial Smoothing
9.6.3 Summary
9.7 Beamspace Algorithms
9.7.1 Beamspace MUSIC
9.7.2 Beamspace Unitary ESPRIT
9.7.3 Beamspace Summary
9.8 Sensitivity and Robustness
9.9 Planar Arrays
9.9.1 Standard Rectangular Arrays
9.9.2 Hexagonal Arrays
9.9.3 Summary: Planar Arrays
9.10 Summary
9.10.1 Major Results
9.10.2 Related Topics
9.10.3 Discussion
9.11 Problems
10 Detection and Other Topics
10.1 Optimum Detection
10.1.1 Classic Binary Detection
10.1.2 Matched Subspace Detector
10.1.3 Spatially Spread Gaussian Signal Processes
10.1.4 Adaptive Detection
10.2 Related Topics
10.3 Epilogue
10.4 Problems
A: Matrix Operations
A.1 Introduction
A.2 Basic Definitions and Properties
A.2.1 Basic Definitions
A.2.2 Matrix Inverses
A.2.3 Quadratic Forms
A.2.4 Partitioned Matrices
A.2.5 Matrix products
A.2.6 Matrix Inequalities
A.3 Special Vectors and Matrices
A.3.1 Elementary Vectors and Matrices
A.3.2 The vec(A) matrix
A.3.3 Diagonal Matrices
A.3.4 Exchange Matrix and Conjugate Symmetric Vectors
A.3.5 Persymmetric and Centrohermitian Matrices
A.3.6 Toeplitz and Hankel Matrices
A.3.7 Circulant Matrices
A.3.8 Triangular Matrices
A.3.9 Unitary and Orthogonal Matrices
A.3.10 Vandermonde Matrices
A.3.11 Projection Matrices
A.3.12 Generalized Inverse
A.4 Eigensystems
A.4.1 Eigendecomposition
A.4.2 Special Matrices
A.5 Singular Value Decomposition
A.6 QR Decomposition
A.6.1 Introduction
A.6.2 QR Decomposition
A.6.3 Givens Rotation
A.6.4 Householder Transformation
A.7 Derivative Operations
A.7.1 Derivative of Scalar with Respect to Vector
A.7.2 Derivative of Scalar with Respect to Matrix
A.7.3 Derivatives with Respect to Parameter
A.7.4 Complex Gradients
B: Array Processing Literature
B.1 Journals
B.2 Books
B.3 Duality
C: Notation
C.1 Conventions
C.2 Acronyms
C.3 Mathematical Symbols
C.4 Symbols
Index
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