Van Trees H.L. Detection, Estimation, and Modulation Theory, Part IV: Optimum Array Processing

John Wiley, 2002. — 1470 p.This book is the fourth in a set of four volumes.
Array processing has played an important role in many diverse application areas. Most modern radar and sonar systems rely on antenna arrays or hydrophone arrays as an essential component of the system. Many communication systems utilize phased arrays or multiple beam antennas to achieve their performance objectives. Seismic arrays are widely used for oil exploration and detection of underground nuclear tests. Various medical diagnosis and treatment techniques exploit arrays. Radio astronomy utilizes very large antenna arrays to achieve resolution goals. It appears that the third generation of wireless systems will utilize adaptive array processing to achieve the desired system capacity. We discuss various applications in Chapter.
The four issues of interest are.
A Array configuration.
B Spatial and temporal characteristics of the signal.
C Spatial and temporal characteristics of the interference.
D Objective of the array processing.Introduction
Array Processing
Applications
Radar.
Radio Astronomy.
Sonar.
Communications.
Direction Finding.
Seismology.
Tomography.
Array Processing Literature.
Organization of the Book
Interactive StudyArrays 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_ω).
Sensitivity and the Tolerance Factor.Linear Apertures
Frequency-wavenumber Response.
Aperture Sampling.
Non-isotropic Element Patterns

Synthesis of Linear Arrays and Apertures
Spectral Weighting
Array Polynomials and the z-Transform
z-Transform.
Real Array Weights.
Properties of the Beam Pattern Near a Zero.
Pattern Sampling in Wavenumber Space
Continuous Aperture.
Linear Arrays.
Discrete Fourier Transform.
Norms.Minimum Beam-width for Specified Sidelobe LevelDolph-Chebychev Arrays.
Taylor Distribution.
Villeneuve n Distribution.Least Squares Error Pattern Synthesis
Minimax Design
Alternation Theorem.
Parks-McClellan-Rabiner Algorithm.Null Steering
Null Constraints.
Least Squares Error Pattern Synthesis with Nulls.
Asymmetric Beams
Spatially Non-uniform Linear ArraysMinimum Redundancy Arrays.
Beam Pattern Design Algorithm.
Beamspace Processing
Full-dimension Beamspace.
Reduced-dimension Beamspace.
Multiple Beam Antennas.Broadband Arrays

ProblemsPlanar 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.Hexagonal ArraysBeam Pattern Design.
Hexagonal Grid to Rectangular Grid Transformation.Nonplanar Arrays
Cylindrical Arrays.
Spherical Arrays.
Summary
ProblemsCharacterization of Space-time Processes
Introduction
Snapshot Models
Frequency-domain Snapshot Models.
Narrowband Time-domain Snapshot Models.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
ProblemsOptimum 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.Discrete Interference
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 BeamformersDOA Mismatch.
Array Perturbations.
Diagonal Loading.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.Eigenvector Beamformers
Principal-component (PC) Beamformers.
Cross-spectral Eigenspace Beamformers.
Dominant-mode Rejection Beamformers.Beamspace Beamformers
Beamspace MPDR.
Beamspace LCMP.
Summary: Beamspace Optimum Processors.
Quadratically Constrained Beamformers
Soft-constraint Beamformers.
Beamforming for Correlated Signal and InterferencesMPDR Beamformer: Correlated Signals and Interference.
MMSE Beamformer: Correlated Signals and Interference.
Spatial Smoothing and Forward-Backward Averaging.Broadband BeamformersDFT Beamformers.
Finite impulse response (FIR) Beamformers.
Summary: Broadband Processing.
Summary
ProblemsAdaptive 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.Sample Matrix Inversion (SMI)
SINR_smi Behavior: MVDR and MPDR.
LCMV and LCMP Beamformers.
Fixed Diagonal Loading.
Toeplitz Estimators.Recursive Least Squares (RLS)
Least Squares Formulation.
Recursive Implementation.
Recursive Implementation of LSE Beamformer.
Generalized Sidelobe Canceller.
Quadratically Constrained RLS.
Conjugate Symmetric Beamformers.Efficient Recursive Implementation AlgorithmsQR Decomposition (QRD).
Gradient AlgorithmsSteepest Descent: MMSE Beamformers.
Steepest Decent: LCMP Beamformer.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.Beamspace Beamformers
Beamspace SMI.
Beamspace RLS.
Beamspace LMS.
Summary: Adaptive Beamspace Processing.
Broadband Beamformers
SMI Implementation.
LMS Implementation.
GSC: Multichannel Lattice Filters.Summary
ProblemsParameter 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.Cramer-Rao Bounds
Gaussian Model: Unknown Signal Spectrum.
Gaussian Model: Uncorrelated Signals with Unknown Power.
Gaussian Model: Known Signal Spectrum.
Nonrandom (Conditional) Signal Model.
Known Signal Waveforms.Maximum Likelihood Estimation
Maximum Likelihood Estimation.
Conditional Maximum Likelihood Estimators.
Weighted Subspace Fitting.
Asymptotic Performance.
Wideband Signals.Computational Algorithms
Optimization Techniques.
Alternating Maximization Algorithms.
Expectation Maximization Algorithm.Polynomial Parameterization
Polynomial Parameterization.
Iterative Quadratic Maximum Likelihood (IQML).
Polynomial WSF (MODE).Detection of Number of Signals
Spatially Spread Signals
Parameterized S(θ,φ).
Spatial ARMA Process.Beamspace algorithmsBeamspace Matrices.
Beamspace Cramer-Rao Bound.
Beamspace Maximum Likelihood.Sensitivity, Robustness, and Calibration
Model Perturbations.
Cramer-Rao Bounds.
Sensitivity of ML Estimators.
MAP Joint Estimation.
Self-Calibration Algorithms.Major Results.
Related Topics.
Algorithm complexity.Parameter Estimation II
Introduction
Quadratic AlgorithmsBeamscan Algorithms.
MVDR (Capon) Algorithm.
Root Versions of Quadratic Algorithms.
Performance of MVDR Algorithms.Subspace AlgorithmsMUSIC.
Minimum-NormAlgorithm.
ESPRIT.
Algorithm Comparison.Linear Prediction
Asymptotic Performance
Error Behavior.
Resolution of MUSIC and Min-Norm.
Small Error Behavior of Algorithms.Correlated and Coherent SignalsForward-Backward Spatial SmoothingBeamspace Algorithms
Beamspace MUSIC.
Beamspace Unitary ESPRIT.
Beamspace Summary.
Sensitivity and Robustness
Planar Arrays.
Standard Rectangular Arrays.
Hexagonal Arrays.
Summary: Planar Arrays.Major Results.
Related Topics.
Discussion.Detection and Other Topics
Optimum Detection
Classic Binary Detection.
Matched Subspace Detector.
Spatially Spread Gaussian Signal Processes.
Adaptive Detection.
Related Topics
Epilogue.
ProblemsMatrix Operations
Introduction
Basic Definitions and Properties
Basic Definitions.
Matrix Inverses.
Quadratic Forms.
Partitioned Matrices.
Matrix products.
Matrix Inequalities.
Special Vectors and Matrices
Elementary Vectors and Matrices.
The vec(A) matrix.
Diagonal Matrices.
Exchange Matrix and Conjugate Symmetric Vectors.
Persymmetric and Centrohermitian Matrices.
Toeplitz and Hankel Matrices.
Circulant Matrices.
Triangular Matrices.
Unitary and Orthogonal Matrices.
Vandermonde Matrices.
Projection Matrices.
Generalized Inverse.
Eigensystems
Eigendecomposition.
Special Matrices.
Singular Value Decomposition
QR DecompositionQR Decomposition.
Givens Rotation.
Householder Transformation.
Derivative Operations
Derivative of Scalar with Respect to Vector.
Derivative of Scalar with Respect to Matrix.
Derivatives with Respect to Parameter.
Complex Gradients.Array Processing Literature
Journals.Books.Duality.Conventions.
Acronyms.
Mathematical Symbols.
Symbols.