Statistical and Adaptive Signal Processing

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1 r Statistical and Adaptive Signal Processing Spectral Estimation, Signal Modeling, Adaptive Filtering and Array Processing Dimitris G. Manolakis Massachusetts Institute of Technology Lincoln Laboratory Vinay K. Ingle Northeastern University Stephen M. Kogon Massachusetts Institute of Technology Lincoln Laboratory ARTECH HOUSE BOSTON LONDON artechhouse.com

2 CONTENTS Preface 1 Introduction XVll 1.1 Random Signals Spectral Estimation Signal Modeling Rational or Pole-Zero Models / Fractional Pole-Zero Models and Fractal Models 1.4 Adaptive Filtering Applications of Adaptive Filters / Features of Adaptive Filters 1.5 Array Processing Spatial Filtering or Beamforming /1.5.2 Adaptive Interference Mitigation in Radar Systems /1.5.3 Adaptive Sidelobe Canceler 1.6 Organization of the Book 29 2 Fundamentals of Discrete- Time Signal Processing Discrete-Time Signals Continuous-Time, Discrete- Time, and Digital Signals / Mathematical Description of Signals / Real-World Signals 2.2 Transform-Domain Representation of Deterministic Signals Fourier Transforms and Fourier Series / Sampling of Continuous-Time Signals / The Discrete Fourier Transform / The Z-Transform / Representation of Narrowband Signals Discrete-Time Systems Analysis of Linear, Time-Invariant Systems / Response to Periodic Inputs / Correlation Analysis and Spectral Density 2.4 Minimum-Phase and System Invertibility System Invertibility and Minimum-Phase Systems / All-Pass Systems / Minimum-Phase and All-Pass Decomposition / Spectral Factorization 2.5 Lattice Filter Realizations All-Zero Ixittice Structures / All-Pole Lattice Structures 2.6 Summary 70 Problems 70 3 Random Variables, Vectors, and Sequences Random Variables Distribution and Density Functions / Statistical Averages / Some Useful Random Variables 3.2 Random Vectors Definitions and Second-Order Moments / Linear Transformations of Random Vectors / Normal Random Vectors / Sums of Independent Random Variables 3.3 Discrete-Time Stochastic Processes Description Using Probability Functions / Second-Order Statistical Description / Stationarity /

3 3.3.4 Ergodicity / Random Signal Variability / Frequency-Domain Description of Stationary Processes 3.4 Linear Systems with Stationary Random Inputs Time-Domain Analysis / Frequency-Domain Analysis / Random Signal Memory / General Correlation Matrices / Correlation Matrices from Random Processes 3.5 Whitening and Innovations Representation Transformations Using Eigen-decomposition / Transformations Using Triangular Decomposition / The Discrete Karhunen- Loeve Transform 3.6 Principles of Estimation Theory Properties of Estimators / Estimation of Mean / Estimation of Variance 3.7 Summary 142 Problems Linear Signal Models Introduction Linear Nonparametric Signal Models / Parametric Pole-Zero Signal Models / Mixed Processes and the Wold Decomposition 4.2 All-Pole Models Model Properties / All-Pole Modeling and Linear Prediction / Autoregressive Models / Lower-Order Models 4.3 All-Zero Models Model Properties / Moving-Average Models / Lower-Order Models 4.4 Pole-Zero Models Model Properties / Autoregressive Moving-Average Models / The First-Order Pole-Zero Model 1: PZ(1,1) / Summary and Dualities 4.5 Models with Poles on the Unit Circle Cepstrum of Pole-Zero Models Pole-Zero Models / All-Pole Models / All-Zero Models 4.7 Summary 189 Problems Nonparametric Power Spectrum Estimation Spectral Analysis of Deterministic Signals Effect of Signal Sampling / Windowing, Periodic Extension, and Extrapolation / Effect of Spectrum Sampling / Effects of Windowing: Leakage and Loss of Resolution / Summary 5.2 Estimation of the Autocorrelation of Stationary Random Signals Estimation of the Power Spectrum of Stationary Random Signals Power Spectrum Estimation Using the Periodogram / Power Spectrum Estimation by- Smoothing a Single Periodogram The Blackman-Tukey Method / Power Spectrum Estimation by Averaging Multiple Periodograms The Welch- Bartlett Method / Some Practical Considerations and Examples

4 5.4 Joint Signal Analysis Estimation of Cross-Power Spectrum / Estimation of Frequency Response Functions 5.5 Multitaper Power Spectrum Estimation Estimation of Auto Power Spectrum / Estimation of Cross Power Spectrum 5.6 Summary 254 Problems Optimum Linear Filters Optimum Signal Estimation Linear Mean Square Error Estimation Error Performance Surface / Derivation of the Linear MMSE Estimator / Principal- Component Analysis of the Optimum Linear Estimator / Geometric Interpretations and the Principle of Orthogonality / Summary and Further Properties 6.3 Solution of the Normal Equations Optimum Finite Impulse Response Filters Design and Properties / Optimum FIR Filters for Stationary Processes / Frequency-Domain Interpretations 6.5 Linear Prediction Linear Signal Estimation / Forward Linear Prediction / Backward Linear Prediction / Stationary Processes / Properties 6.6 Optimum Infinite Impulse Response Filters Noncausal IIR Filters / Causal IIR Filters / Filtering of Additive Noise / Linear Prediction Using the Infinite Past Whitening 6.7 Inverse Filtering and Deconvolution Channel Equalization in Data Transmission Systems Nyquist's Criterion for Zero ISI / Equivalent Discrete-Time Channel Model / Linear Equalizers / Zero-Forcing Equalizers / Minimum MSE Equalizers 6.9 Matched Filters and Eigenfilters Deterministic Signal in Noise / Random Signal in Noise 6.10 Summary 325 Problems Algorithms and Structures for Optimum Linear Filters Fundamentals of Order- Recursive Algorithms Matrix Partitioning and Optimum Nesting / Inversion of Partitioned Hermitian Matrices / Levinson Recursion for the Optimum Estimator / Order- Recursive Computation of the LDL H Decomposition / Order- Recursive Computation of the Optimum Estimate 7.2 Interpretations of Algorithmic Quantities Innovations and Backward Prediction / Partial Correlation / Order Decomposition of the Optimum Estimate / Gram-Schmidt Orthogonalization 7.3 Order-Recursive Algorithms for Optimum FIR Filters Order-Recursive Computation of the Optimum Filter / 7.3.2

5 Lattice-Ladder Structure / Simplifications for Stationary Stochastic Processes / Algorithms Based on the UDU H Decomposition 7.4 Algorithms of Levinson and Levinson-Durbin Lattice Structures for Optimum FIR Filters and Predictors Lattice-Ladder Structures / Some Properties and Interpretations / Parameter Conversions 7.6 Algorithm of Schtir Direct Schur Algorithm / Implementation Considerations / Inverse Schu'r Algorithm 7.7 Triangularization and Inversion of Toeplitz Matrices LDL H Decomposition of Inverse of a Toeplitz Matrix / LDL H Decomposition of a Toeplitz Matrix / Inversion of Real Toeplitz Matrices 7.8 Kalman Filter Algorithm Preliminary Development / Development of Kalman Filter 7.9 Summary 387 Problems Linear Least-Squares Signal Estimation Signal Estimation and Linear Prediction / Combined Forward and Backward Linear Prediction (FBLP) / Narrowband Interference Cancelation 8.5 LS Computations Using the Normal Equations Linear LSE Estimation / LSE FIR Filtering and Prediction 8.6 LS Computations Using Orthogonalization Techniques Householder Reflections / The Givens Rotations / Gram-Schmidt Orthogonalization 8.7 LS Computations Using the Singular Value Decomposition Singular Value Decomposition / Solution of the LS Problem / Rank-Deficient LS Problems 8.8 Summary 438 Problems Signal Modeling and Parametric Spectral Estimation Least-Squares Filtering and Prediction The Principle of Least Squares Linear Least-Squares Error Estimation Derivation of the Normal Equations / Statistical Properties of Least-Squares Estimators 8.3 Least-Squares FIR Filters The Modeling Process: Theory and Practice Estimation of All-Pole Models Direct Structures / Lattice Structures / Maximum Entropy Method / Excitations with Line Spectra 9.3 Estimation of Pole-Zero Models 462 9J.7 Known Excitation / Unknown Excitation / 9.3.3

6 i tents Nonlinear Least-Squares Optimization 9.4 Applicatons Spectral Estimation / Speech Modeling 9.5 Minimum-Variance Spectrum Estimation Harmonic Models and Frequency Estimation Techniques Harmonic Model / Pisarenko Harmonic Decomposition / MUSIC Algorithm I Minimum-Norm Method / ESPRIT Algorithm 9.7 Summary 493 Problems Adaptive Filters Typical Applications of Adaptive Filters Echo Cancelation in Communications / Equalization of Data Communications Channels / Linear Predictive Coding / Noise Cancelation 10.2 Principles of Adaptive Filters Features of Adaptive Filters / Optimum versus Adaptive Filters / Stability and Steady-State Performance of Adaptive Filters / Some Practical Considerations 10.3 Method of Steepest Descent Least-Mean-Square Adaptive Filters Derivation / Adaptation in a Stationary SOE / Summary and Design Guidelines / Applications of the IMS Algorithm / Some Practical Considerations 10.5 Recursive Least-Squares Adaptive Filters LS Adaptive Filters / Conventional Recursive Least-Squares Algorithm / Some Practical Considerations / Convergence and Performance Analysis 10.6 RLS Algorithms for Array Processing LS Computations Using the Cholesky and QR Decompositions / Two Useful Lemmas / The QR-RLS Algorithm / Extended QR-RLS Algorithm / The Inverse QR-RLS Algorithm / Implementation of QR-RLS Algorithm Using the Givens Rotations / Implementation of Inverse QR-RLS Algorithm Using the Givens Rotations / Classification of RLS Algorithms for Array Processing 10.7 Fast RLS Algorithms for FIR Filtering Fast Fixed-Order RLS FIR Filters / RLS Lattice- Ladder Filters / RLS Lattice-Ladder Filters Using Error Feedback Updatings / Givens Rotation-Based LS Lattice- Ladder Algorithms / Classification of RLS Algorithms for FIR Filtering 10.8 Tracking Performance of Adaptive Algorithms Approaches for Nonstationary SOE I Preliminaries in Performance Analysis I The LMS Algorithm / The RLS Algorithm with Exponential Forgetting / Comparison of Tracking Performance 10.9 Summary 607 Problems 608

7 11 Array Processing Array Fundamentals Spatial Signals / Modulation-Demodulation / Array Signal Model / The Sensor Array: Spatial Sampling 11.2 Conventional Spatial Filtering: Beamforming Spatial Matched Filter / Tapered Beamforming 11.3 Optimum Array Processing Optimum Beamforming / Eigenanalysis of the Optimum Beamformer / Interference Cancelation Performance / Tapered Optimum Beamforming / The Generalized Sidelobe Canceler 11.4 Performance Considerations for Optimum Beamformers Effect of Signal Mismatch / Effect of Bandwidth 11.5 Adaptive Beamforming Sample Matrix Inversion / Diagonal Loading with the SMI Beamformer / Implementation of the SMI Beamformer / Sample-by- Sample Adaptive Methods 11.6 Other Adaptive Array Processing Methods Linearly Constrained Minimum-Variance Beamformers / Partially Adaptive Arrays / Sidelobe Cancelers Angle Estimation Maximum-Likelihood Angle Estimation / Cramer-Rao Lower Bound on Angle Accuracy / Beamsplitting Algorithms / Model-Based Methods Space-Time Adaptive Processing Summary Problems Further Topics Higher-Order Statistics in Signal Processing /./ Moments, Cumulants. and Polyspectra / Higher- Order Moments and LTI Systems / Higher-Order Moments of Linear Signal Models 12.2 Blind Deconvolution Unsupervised Adaptive Filters Blind Equalizers Blind Equalization / Symbol Rate Blind Equalizers / Constant- Modulus Algorithm 12.4 Fractionally Spaced Equalizers Zero-Forcing Fractionally Spaced Equalizers / MMSE Fractionally Spaced Equalizers / Blind Fractionally Spaced Equalizers 12.5 Fractional Pole-Zero Signal Models Fractional Unit-Pole Model / Fractional Pole- Zero Models: FPZ (p. d. q) / Symmetric a-stable Fractional Pole-Zero Processes 12.6 Self-Similar Random Signal Models 725 /2.6. / Self-Similar Stochastic Processes / Fractional Bmwnian Motion / Fractional Gaussian Noise / Simulation of Fractional Brownian Motions and Fractional Gaussian Noises / Estimation of Long Memory /

8 tents Fractional Levy Stable Motion 12.7 Summary Problems Appendix A Matrix Inversion Lemma Appendix B Gradients and Optimization in Complex Space B.I Gradient B.2 Lagrange Multipliers Appendix C MATLAB Functions Appendix D Useful Results from Matrix Algebra 755 D.I Complex-Valued Vector Space 755 Some Definitions D.2 D.3 D.4 D.5 Matrices 756 D.2.7 Some Definitions / D.2.2 Properties of Square Matrices Determinant of a Square Matrix 760 D.3.1 Properties of the Determinant / D.3.2 Condition Number Unitary Matrices 762 D.4.1 Hermitian Forms after Unitary Transformations / D.4.2 Significant Integral of Quadratic and Hermitian Forms Positive Definite Matrices 764 AnncnH iv F. Minimum Phnsp Test for Polynomials 767 Bibliography Index

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