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adaptive filter theory 5th.pdf
Cover
Title Page
Copyright Page
Contents
Preface
Acknowledgments
Background and Preview
1. The Filtering Problem
2. Linear Optimum Filters
3. Adaptive Filters
4. Linear Filter Structures
5. Approaches to the Development of Linear Adaptive Filters
6. Adaptive Beamforming
7. Four Classes of Applications
8. Historical Notes
Chapter 1 Stochastic Processes and Models
1.1 Partial Characterization of a Discrete-Time Stochastic Process
1.2 Mean Ergodic Theorem
1.3 Correlation Matrix
1.4 Correlation Matrix of Sine Wave Plus Noise
1.5 Stochastic Models
1.6 Wold Decomposition
1.7 Asymptotic Stationarity of an Autoregressive Process
1.8 Yule–Walker Equations
1.9 Computer Experiment: Autoregressive Process of Order Two
1.10 Selecting the Model Order
1.11 Complex Gaussian Processes
1.12 Power Spectral Density
1.13 Properties of Power Spectral Density
1.14 Transmission of a Stationary Process Through a Linear Filter
1.15 Cramér Spectral Representation for a Stationary Process
1.16 Power Spectrum Estimation
1.17 Other Statistical Characteristics of a Stochastic Process
1.18 Polyspectra
1.19 Spectral-Correlation Density
1.20 Summary and Discussion
Problems
Chapter 2 Wiener Filters
2.1 Linear Optimum Filtering: Statement of the Problem
2.2 Principle of Orthogonality
2.3 Minimum Mean-Square Error
2.4 Wiener–Hopf Equations
2.5 Error-Performance Surface
2.6 Multiple Linear Regression Model
2.7 Example
2.8 Linearly Constrained Minimum-Variance Filter
2.9 Generalized Sidelobe Cancellers
2.10 Summary and Discussion
Problems
Chapter 3 Linear Prediction
3.1 Forward Linear Prediction
3.2 Backward Linear Prediction
3.3 Levinson–Durbin Algorithm
3.4 Properties of Prediction-Error Filters
3.5 Schur–Cohn Test
3.6 Autoregressive Modeling of a Stationary Stochastic Process
3.7 Cholesky Factorization
3.8 Lattice Predictors
3.9 All-Pole, All-Pass Lattice Filter
3.10 Joint-Process Estimation
3.11 Predictive Modeling of Speech
3.12 Summary and Discussion
Problems
Chapter 4 Method of Steepest Descent
4.1 Basic Idea of the Steepest-Descent Algorithm
4.2 The Steepest-Descent Algorithm Applied to the Wiener Filter
4.3 Stability of the Steepest-Descent Algorithm
4.4 Example
4.5 The Steepest-Descent Algorithm Viewed as a Deterministic Search Method
4.6 Virtue and Limitation of the Steepest-Descent Algorithm
4.7 Summary and Discussion
Problems
Chapter 5 Method of Stochastic Gradient Descent
5.1 Principles of Stochastic Gradient Descent
5.2 Application 1: Least-Mean-Square (LMS) Algorithm
5.3 Application 2: Gradient-Adaptive Lattice Filtering Algorithm
5.4 Other Applications of Stochastic Gradient Descent
5.5 Summary and Discussion
Problems
Chapter 6 The Least-Mean-Square (LMS) Algorithm
6.1 Signal-Flow Graph
6.2 Optimality Considerations
6.3 Applications
6.4 Statistical Learning Theory
6.5 Transient Behavior and Convergence Considerations
6.6 Efficiency
6.7 Computer Experiment on Adaptive Prediction
6.8 Computer Experiment on Adaptive Equalization
6.9 Computer Experiment on a Minimum-Variance Distortionless-Response Beamformer
6.10 Summary and Discussion
Problems
Chapter 7 Normalized Least-Mean-Square (LMS) Algorithm and Its Generalization
7.1 Normalized LMS Algorithm: The Solution to a Constrained Optimization Problem
7.2 Stability of the Normalized LMS Algorithm
7.3 Step-Size Control for Acoustic Echo Cancellation
7.4 Geometric Considerations Pertaining to the Convergence Process for Real-Valued Data
7.5 Affine Projection Adaptive Filters
7.6 Summary and Discussion
Problems
Chapter 8 Block-Adaptive Filters
8.1 Block-Adaptive Filters: Basic Ideas
8.2 Fast Block LMS Algorithm
8.3 Unconstrained Frequency-Domain Adaptive Filters
8.4 Self-Orthogonalizing Adaptive Filters
8.5 Computer Experiment on Adaptive Equalization
8.6 Subband Adaptive Filters
8.7 Summary and Discussion
Problems
Chapter 9 Method of Least-Squares
9.1 Statement of the Linear Least-Squares Estimation Problem
9.2 Data Windowing
9.3 Principle of Orthogonality Revisited
9.4 Minimum Sum of Error Squares
9.5 Normal Equations and Linear Least-Squares Filters
9.6 Time-Average Correlation Matrix Φ
9.7 Reformulation of the Normal Equations in Terms of Data Matrices
9.8 Properties of Least-Squares Estimates
9.9 Minimum-Variance Distortionless Response (MVDR) Spectrum Estimation
9.10 Regularized MVDR Beamforming
9.11 Singular-Value Decomposition
9.12 Pseudoinverse
9.13 Interpretation of Singular Values and Singular Vectors
9.14 Minimum-Norm Solution to the Linear Least-Squares Problem
9.15 Normalized LMS Algorithm Viewed as the Minimum-Norm Solution to an Underdetermined Least-Squares Estimation Problem
9.16 Summary and Discussion
Problems
Chapter 10 The Recursive Least-Squares (RLS) Algorithm
10.1 Some Preliminaries
10.2 The Matrix Inversion Lemma
10.3 The Exponentially Weighted RLS Algorithm
10.4 Selection of the Regularization Parameter
10.5 Updated Recursion for the Sum of Weighted Error Squares
10.6 Example: Single-Weight Adaptive Noise Canceller
10.7 Statistical Learning Theory
10.8 Efficiency
10.9 Computer Experiment on Adaptive Equalization
10.10 Summary and Discussion
Problems
Chapter 11 Robustness
11.1 Robustness, Adaptation, and Disturbances
11.2 Robustness: Preliminary Considerations Rooted in H[Sup(∞)] Optimization
11.3 Robustness of the LMS Algorithm
11.4 Robustness of the RLS Algorithm
11.5 Comparative Evaluations of the LMS and RLS Algorithms from the Perspective of Robustness
11.6 Risk-Sensitive Optimality
11.7 Trade-Offs Between Robustness and Efficiency
11.8 Summary and Discussion
Problems
Chapter 12 Finite-Precision Effects
12.1 Quantization Errors
12.2 Least-Mean-Square (LMS) Algorithm
12.3 Recursive Least-Squares (RLS) Algorithm
12.4 Summary and Discussion
Problems
Chapter 13 Adaptation in Nonstationary Environments
13.1 Causes and Consequences of Nonstationarity
13.2 The System Identification Problem
13.3 Degree of Nonstationarity
13.4 Criteria for Tracking Assessment
13.5 Tracking Performance of the LMS Algorithm
13.6 Tracking Performance of the RLS Algorithm
13.7 Comparison of the Tracking Performance of LMS and RLS Algorithms
13.8 Tuning of Adaptation Parameters
13.9 Incremental Delta-Bar-Delta (IDBD) Algorithm
13.10 Autostep Method
13.11 Computer Experiment: Mixture of Stationary and Nonstationary Environmental Data
13.12 Summary and Discussion
Problems
Chapter 14 Kalman Filters
14.1 Recursive Minimum Mean-Square Estimation for Scalar Random Variables
14.2 Statement of the Kalman Filtering Problem
14.3 The Innovations Process
14.4 Estimation of the State Using the Innovations Process
14.5 Filtering
14.6 Initial Conditions
14.7 Summary of the Kalman Filter
14.8 Optimality Criteria for Kalman Filtering
14.9 Kalman Filter as the Unifying Basis for RLS Algorithms
14.10 Covariance Filtering Algorithm
14.11 Information Filtering Algorithm
14.12 Summary and Discussion
Problems
Chapter 15 Square-Root Adaptive Filtering Algorithms
15.1 Square-Root Kalman Filters
15.2 Building Square-Root Adaptive Filters on the Two Kalman Filter Variants
15.3 QRD-RLS Algorithm
15.4 Adaptive Beamforming
15.5 Inverse QRD-RLS Algorithm
15.6 Finite-Precision Effects
15.7 Summary and Discussion
Problems
Chapter 16 Order-Recursive Adaptive Filtering Algorithm
16.1 Order-Recursive Adaptive Filters Using Least-Squares Estimation: An Overview
16.2 Adaptive Forward Linear Prediction
16.3 Adaptive Backward Linear Prediction
16.4 Conversion Factor
16.5 Least-Squares Lattice (LSL) Predictor
16.6 Angle-Normalized Estimation Errors
16.7 First-Order State-Space Models for Lattice Filtering
16.8 QR-Decomposition-Based Least-Squares Lattice (QRD-LSL) Filters
16.9 Fundamental Properties of the QRD-LSL Filter
16.10 Computer Experiment on Adaptive Equalization
16.11 Recursive (LSL) Filters Using A Posteriori Estimation Errors
16.12 Recursive LSL Filters Using A Priori Estimation Errors with Error Feedback
16.13 Relation Between Recursive LSL and RLS Algorithms
16.14 Finite-Precision Effects
16.15 Summary and Discussion
Problems
Chapter 17 Blind Deconvolution
17.1 Overview of Blind Deconvolution
17.2 Channel Identifiability Using Cyclostationary Statistics
17.3 Subspace Decomposition for Fractionally Spaced Blind Identification
17.4 Bussgang Algorithm for Blind Equalization
17.5 Extension of the Bussgang Algorithm to Complex Baseband Channels
17.6 Special Cases of the Bussgang Algorithm
17.7 Fractionally Spaced Bussgang Equalizers
17.8 Estimation of Unknown Probability Distribution Function of Signal Source
17.9 Summary and Discussion
Problems
Epilogue
1. Robustness, Efficiency, and Complexity
2. Kernel-Based Nonlinear Adaptive Filtering
Appendix A: Theory of Complex Variables
A.1 Cauchy–Riemann Equations
A.2 Cauchy’s Integral Formula
A.3 Laurent’s Series
A.4 Singularities and Residues
A.5 Cauchy’s Residue Theorem
A.6 Principle of the Argument
A.7 Inversion Integral for the z-Transform
A.8 Parseval’s Theorem
Appendix B: Wirtinger Calculus for Computing Complex Gradients
B.1 Wirtinger Calculus: Scalar Gradients
B.2 Generalized Wirtinger Calculus: Gradient Vectors
B.3 Another Approach to Compute Gradient Vectors
B.4 Expressions for the Partial Derivatives əf/əz and əf/əz*
Appendix C: Method of Lagrange Multipliers
C.1 Optimization Involving a Single Equality Constraint
C.2 Optimization Involving Multiple Equality Constraints
C.3 Optimum Beamformer
Appendix D: Estimation Theory
D.1 Likelihood Function
D.2 Cramér–Rao Inequality
D.3 Properties of Maximum-Likelihood Estimators
D.4 Conditional Mean Estimator
Appendix E: Eigenanalysis
E.1 The Eigenvalue Problem
E.2 Properties of Eigenvalues and Eigenvectors
E.3 Low-Rank Modeling
E.4 Eigenfilters
E.5 Eigenvalue Computations
Appendix F: Langevin Equation of Nonequilibrium Thermodynamics
F.1 Brownian Motion
F.2 Langevin Equation
Appendix G: Rotations and Reflections
G.1 Plane Rotations
G.2 Two-Sided Jacobi Algorithm
G.3 Cyclic Jacobi Algorithm
G.4 Householder Transformation
G.5 The QR Algorithm
Appendix H: Complex Wishart Distribution
H.1 Definition
H.2 The Chi-Square Distribution as a Special Case
H.3 Properties of the Complex Wishart Distribution
H.4 Expectation of the Inverse Correlation Matrix Ф[Sup(–1)](n)
Glossary
Text Conventions
Abbreviations
Principal Symbols
Bibliography
Suggested Reading
Index
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
Y
Z
AdAptive Filter theory
Fifth edition
Simon haykin
Communications Research Laboratory
McMaster University
Hamilton, Ontario, Canada
Upper Saddle River Boston Columbus San Francisco New York
Indianapolis London Toronto Sydney Singapore Tokyo Montreal
Dubai Madrid Hong Kong Mexico City Munich Paris Amsterdam Cape Town
Vice President and Editorial Director, ECS: Marcia J. Horton
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Editorial Assistant: William Opaluch
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Production Editor: Pavithra Jayapaul, Jouve India
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Executive Marketing Manager: Tim Galligan
Art Editor: Greg Dulles
Art Director: Jayne Conte
Cover Designer: Suzanne Behnke
Cover Image: © Gordon Swanson
Composition/Full-Service Project Management: Jouve India
Copyright © 2014, 2002, 1996, 1991 by Pearson Education, Inc., Upper Saddle River, New Jersey 07458. All rights
reserved. Manufactured in the United States of America. This publication is protected by Copyright and permissions
should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or
transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. To obtain
permission(s) to use materials from this work, please submit a written request to Pearson Higher Education,
Permissions Department, 1 Lake Street, Upper Saddle River, NJ 07458.
The author and publisher of this book have used their best efforts in preparing this book. These efforts include the
development, research, and testing of the theories and programs to determine their effectiveness. The author and
publisher make no warranty of any kind, expressed or implied, with regard to these programs or the documentation
contained in this book. The author and publisher shall not be liable in any event for incidental or consequential
damages in connection with, or arising out of, the furnishing, performance, or use of these programs.
MATLAB is a registered trademark of The Math Works, Inc., 3 Apple Hill Drive, Natick, MA 01760-2098.
Library of Congress Cataloging-in-Publication Data
Simon Haykin
Adaptive filter theory / Simon Haykin, Communications Research Laboratory, McMaster University, Hamilton,
Ontario, Canada.—Fifth edition.
pages cm
ISBN-13: 978-0-13-267145-3
ISBN-10: 0-13-267145-X
1. Adaptive filters. I. Title.
TK7872.F5H368 2012
621.3815'324—dc23
2012025640
10 9 8 7 6 5 4 3 2 1
ISBN 13: 978-0-13-267145-3
ISBN 10:
0-13-267145-X
This book is dedicated to
• The many researchers around the world for their contributions
to the ever-growing literature on adaptive filtering, and
• The many reviewers, new and old, for their useful inputs and
critical comments.
Contents
Preface x
Acknowledgments xvi
Background and Preview 1
The Filtering Problem 1
Linear Optimum Filters 4
Adaptive Filters 4
Linear Filter Structures 6
Approaches to the Development of Linear Adaptive Filters 12
Adaptive Beamforming 13
Four Classes of Applications 17
Historical Notes 20
1.
2.
3.
4.
5.
6.
7.
8.
Chapter 1 Stochastic Processes and Models 30
Partial Characterization of a Discrete-Time Stochastic Process 30
Yule–Walker Equations 51
Computer Experiment: Autoregressive Process of Order Two 52
Correlation Matrix 34
Correlation Matrix of Sine Wave Plus Noise 39
Stochastic Models 40
1.1
1.2 Mean Ergodic Theorem 32
1.3
1.4
1.5
1.6 Wold Decomposition 46
1.7 Asymptotic Stationarity of an Autoregressive Process 49
1.8
1.9
1.10 Selecting the Model Order 60
1.11 Complex Gaussian Processes 63
1.12 Power Spectral Density 65
1.13 Properties of Power Spectral Density 67
1.14 Transmission of a Stationary Process Through a Linear Filter 69
1.15 Cramér Spectral Representation for a Stationary Process 72
1.16 Power Spectrum Estimation 74
1.17 Other Statistical Characteristics of a Stochastic Process 77
1.18 Polyspectra 78
1.19 Spectral-Correlation Density 81
1.20 Summary and Discussion 84
Problems 85
Chapter 2 Wiener Filters 90
2.1
2.2
Linear Optimum Filtering: Statement of the Problem 90
Principle of Orthogonality 92
iv
Contents v
Error-Performance Surface 100
2.3 Minimum Mean-Square Error 96
2.4 Wiener–Hopf Equations 98
2.5
2.6 Multiple Linear Regression Model 104
2.7
2.8
2.9 Generalized Sidelobe Cancellers 116
2.10 Summary and Discussion 122
Example 106
Linearly Constrained Minimum-Variance Filter 111
Problems 124
Chapter 3 Linear Prediction 132
Forward Linear Prediction 132
Backward Linear Prediction 139
Levinson–Durbin Algorithm 144
Properties of Prediction-Error Filters 153
Schur–Cohn Test 162
3.1
3.2
3.3
3.4
3.5
3.6 Autoregressive Modeling of a Stationary Stochastic Process 164
3.7
3.8
3.9 All-Pole, All-Pass Lattice Filter 175
3.10
3.11 Predictive Modeling of Speech 181
3.12 Summary and Discussion 188
Cholesky Factorization 167
Lattice Predictors 170
Joint-Process Estimation 177
Problems 189
Chapter 4 Method of Steepest Descent 199
Basic Idea of the Steepest-Descent Algorithm 199
The Steepest-Descent Algorithm Applied to the Wiener Filter 200
Stability of the Steepest-Descent Algorithm 204
Example 209
The Steepest-Descent Algorithm Viewed as a Deterministic Search Method 221
4.1
4.2
4.3
4.4
4.5
4.6 Virtue and Limitation of the Steepest-Descent Algorithm 222
4.7
Summary and Discussion 223
Problems 224
Chapter 5 Method of Stochastic Gradient Descent 228
Principles of Stochastic Gradient Descent 228
5.1
5.2 Application 1: Least-Mean-Square (LMS) Algorithm 230
5.3 Application 2: Gradient-Adaptive Lattice Filtering Algorithm 237
5.4 Other Applications of Stochastic Gradient Descent 244
5.5
Summary and Discussion 245
Problems 246
Chapter 6 The Least-Mean-Square (LMS) Algorithm 248
Signal-Flow Graph 248
6.1
6.2 Optimality Considerations 250
6.3 Applications 252
6.4
6.5
6.6
6.7
6.8
Statistical Learning Theory 272
Transient Behavior and Convergence Considerations 283
Efficiency 286
Computer Experiment on Adaptive Prediction 288
Computer Experiment on Adaptive Equalization 293
vi Contents
6.9
Computer Experiment on a Minimum-Variance Distortionless-Response
Beamformer 302
6.10 Summary and Discussion 306
Problems 308
Chapter 7
Normalized Least-Mean-Square (LMS) Algorithm and Its
Generalization 315
7.1 Normalized LMS Algorithm: The Solution to a Constrained Optimization Problem 315
7.2
7.3
7.4 Geometric Considerations Pertaining to the Convergence Process for Real-Valued
Stability of the Normalized LMS Algorithm 319
Step-Size Control for Acoustic Echo Cancellation 322
Data 327
7.5 Affine Projection Adaptive Filters 330
7.6
Summary and Discussion 334
Problems 335
Chapter 8 Block-Adaptive Filters 339
Block-Adaptive Filters: Basic Ideas 340
Fast Block LMS Algorithm 344
8.1
8.2
8.3 Unconstrained Frequency-Domain Adaptive Filters 350
8.4
8.5
8.6
8.7
Self-Orthogonalizing Adaptive Filters 351
Computer Experiment on Adaptive Equalization 361
Subband Adaptive Filters 367
Summary and Discussion 375
Problems 376
Chapter 9 Method of Least-Squares 380
Principle of Orthogonality Revisited 384
Time-Average Correlation Matrix ≥ 391
Statement of the Linear Least-Squares Estimation Problem 380
9.1
9.2 Data Windowing 383
9.3
9.4 Minimum Sum of Error Squares 387
9.5 Normal Equations and Linear Least-Squares Filters 388
9.6
9.7 Reformulation of the Normal Equations in Terms of Data Matrices 393
9.8
9.9 Minimum-Variance Distortionless Response (MVDR) Spectrum Estimation 401
9.10 Regularized MVDR Beamforming 404
9.11 Singular-Value Decomposition 409
9.12 Pseudoinverse 416
9.13
9.14 Minimum-Norm Solution to the Linear Least-Squares Problem 419
9.15 Normalized LMS Algorithm Viewed as the Minimum-Norm Solution to an
Interpretation of Singular Values and Singular Vectors 418
Properties of Least-Squares Estimates 397
Underdetermined Least-Squares Estimation Problem 422
9.16 Summary and Discussion 424
Problems 425
Chapter 10 The Recursive Least-Squares (RLS) Algorithm 431
10.1 Some Preliminaries 431
10.2 The Matrix Inversion Lemma 435
10.3 The Exponentially Weighted RLS Algorithm 436
10.4 Selection of the Regularization Parameter 439
10.5 Updated Recursion for the Sum of Weighted Error Squares 441
10.6 Example: Single-Weight Adaptive Noise Canceller 443
10.7 Statistical Learning Theory 444
10.8 Efficiency 449
10.9 Computer Experiment on Adaptive Equalization 450
10.10 Summary and Discussion 453
Problems 454
Chapter 11 Robustness 456
Contents vii
11.1 Robustness, Adaptation, and Disturbances 456
11.2 Robustness: Preliminary Considerations Rooted in H∞ Optimization 457
11.3 Robustness of the LMS Algorithm 460
11.4 Robustness of the RLS Algorithm 465
11.5 Comparative Evaluations of the LMS and RLS Algorithms from the Perspective of
Robustness 470
11.6 Risk-Sensitive Optimality 470
11.7 Trade-Offs Between Robustness and Efficiency 472
11.8 Summary and Discussion 474
Problems 474
Chapter 12 Finite-Precision Effects 479
12.1 Quantization Errors 480
12.2 Least-Mean-Square (LMS) Algorithm 482
12.3 Recursive Least-Squares (RLS) Algorithm 491
12.4 Summary and Discussion 497
Problems 498
Chapter 13 Adaptation in Nonstationary Environments 500
13.1 Causes and Consequences of Nonstationarity 500
13.2 The System Identification Problem 501
13.3 Degree of Nonstationarity 504
13.4 Criteria for Tracking Assessment 505
13.5 Tracking Performance of the LMS Algorithm 507
13.6 Tracking Performance of the RLS Algorithm 510
13.7 Comparison of the Tracking Performance of LMS and RLS Algorithms 514
13.8 Tuning of Adaptation Parameters 518
13.9
13.10 Autostep Method 526
13.11 Computer Experiment: Mixture of Stationary and Nonstationary Environmental
Incremental Delta-Bar-Delta (IDBD) Algorithm 520
13.12 Summary and Discussion 534
Data 530
Problems 535
Chapter 14 Kalman Filters 540
14.1 Recursive Minimum Mean-Square Estimation for Scalar Random Variables 541
14.2 Statement of the Kalman Filtering Problem 544
14.3 The Innovations Process 547
14.4 Estimation of the State Using the Innovations Process 549
14.5 Filtering 555
14.6
14.7 Summary of the Kalman Filter 558
14.8 Optimality Criteria for Kalman Filtering 559
14.9 Kalman Filter as the Unifying Basis for RLS Algorithms 561
14.10 Covariance Filtering Algorithm 566
14.11 Information Filtering Algorithm 568
14.12 Summary and Discussion 571
Initial Conditions 557
Problems 572
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