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Adaptive Filter Theory



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Prentice Hall
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Adaptive Filter Theory, 5e, is ideal for courses in Adaptive Filters.

Haykin examines both the mathematical theory behind various linear adaptive filters and the elements of supervised multilayer perceptrons. In its fifth edition, this highly successful book has been updated and refined to stay current with the field and develop concepts in as unified and accessible a manner as possible.

Author Biography

Simon Haykin received his B.Sc. (First-class Honours), Ph.D., and D.Sc., all in Electrical Engineering from the University of Birmingham, England. He is a Fellow of the Royal Society of Canada, and a Fellow of the Institute of Electrical and Electronics Engineers. He is the recipient of the Henry Booker Gold Medal from URSI, 2002, the Honorary Degree of Doctor of Technical Sciences from ETH Zentrum, Zurich, Switzerland, 1999, and many other medals and prizes.

He is a pioneer in adaptive signal-processing with emphasis on applications in radar and communications, an area of research which has occupied much of his professional life.

Table of Contents




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 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



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



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



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 as a Deterministic Search Method

4.6            Virtue and Limitation of the Steepest-Descent Algorithm

4.7            Summary and Discussion



Chapter 5            Method of Stochastic Gradient Descent

5.1            Principles of Stochastic Gradient Descent

5.2            Application: Least-Mean-Square (LMS) Algorithm

5.3            Gradient-Adaptive Lattice Filtering Algorithm

5.4            Other Applications of Stochastic Gradient Descent

5.5            Summary and Discussion



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 Minimum-Variance Distortionless-Response Beamformer

6.10            Summary and Discussion



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



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



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<ALT1> Revisited</ALT>

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 Least-Mean-Square (LMS) Algorithm Viewed as the Minimum-Norm Solution to an Underdetermined Least-Squares Estimation Problem

9.16            Summary and Discussion



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            Update 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



Chapter 11            Robustness

11.1        Robustness, Adaptation, and Disturbances

11.2        Robustness: Preliminary Considerations Rooted in H 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



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



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 the 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



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            Kalman Filter as the Unifying Basis for RLS Algorithms

14.9            Variants of the Kalman Filter

14.10            Summary and Discussion



Chapter 15            Square-Root Adaptive Filters

15.1            Square-Root Kalman Filters

15.2            Building Square-Root Adaptive Filters on Their Kalman Filter Counterparts

15.3            QRD-RLS Algorithm

15.4            Adaptive Beamforming

15.5            Inverse QRD-RLS Algorithm

15.6            Finite-Precision Effects

15.7            Summary



Chapter 16            Order-Recursive Adaptive Filters

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



Chapter 17            Blind Deconvolution

17.1            Overview of the 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 Factor of Signal Source

17.9            Summary and Discussion




1.            Robusness, Efficiency, and Complexity

2.            Kernel-Based Nonlinear Adaptive Filtering


Appendix A            Theory of Complex Variables

A.1            Cauchy—Reimann 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            Computation of Derivatives in the Complex Domain

B.1            Differentiability and Analyticity

B.2            Wirtinger Derivatives

B.3            Matrix and Vector Derivatives

B.4            Newton Updates


Appendix C            Method of Lagrange Multipliers

C.1            Optimization Involving a Single Equality Constraint

C.2            Optimization Involving Multiple Equality Constraints

C.3            Optimization 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 Φ −1(n)




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