Adaptive Filtering. Squares. Alexander D. Poularikas. Fundamentals of. Least Mean. with MATLABR. University of Alabama, Huntsville, AL.
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1 Adaptive Filtering Fundamentals of Least Mean Squares with MATLABR Alexander D. Poularikas University of Alabama, Huntsville, AL CRC Press Taylor & Francis Croup Boca Raton London New York CRC Press is an imprint of the Taylor & Francis Croup, an informa business
2 Contents Preface Author Abbreviations MATLAB Functions X1 xiii xv xvii Chapter 1 Vectors Introduction Multiplication by a Constant and Addition and Subtraction Multiplication by a Constant Addition and Subtraction Unit Coordinate Vectors Inner Product Distance between Two Vectors Mean Value of a Vector Direction Cosines The Projection of a Vector Linear Transformations Linear Independence, Vector Spaces, and Basis Vectors Orthogonal Basis Vectors 13 Problems 13 Hints-Suggestions-Solutions 14 Chapter 2 Matrices Introduction General Types of Matrices Diagonal, Identity, and Scalar Matrices Upper and Lower Triangular Matrices Symmetric and Exchange Matrices Toeplitz Matrix Hankel and Hermitian Matrix Operations Determinant of a Matrix Definition and Expansion of a Matrix Trace of a Matrix Inverse of a Matrix Linear Equations Square Matrices (n x ri) Rectangular Matrices (n < m) Rectangular Matrices (m < n) 27
3 yj Contents Quadratic and Hermitian Forms Eigenvalues and Eigenvectors Eigenvectors Properties of Eigenvalues and Eigenvectors 33 Problems 36 Hints-Suggestions-Solutions 37 Chapter 3 Processing of Discrete Deterministic Signals: Discrete Systems Discrete-Time Signals Time-Domain Representation of Basic Continuous and Discrete Signals Transform-Domain Representation of Discrete Signals Discrete-Time Fourier Transform The Discrete FT Properties of DFT The z-transform Discrete-Time Systems Linearity and Shift Invariant Causality Stability Transform-Domain Representation 57 Problems 60 Hints-Suggestions-Solutions 61 Chapter 4 Discrete-Time Random Processes Discrete Random Signals, Probability Distributions, and Averages of Random Variables Stationary and Ergodic Processes Averages of RV Mean Value Correlation Covariance Stationary Processes Autocorrelation Matrix Purely Random Process (White Noise) Random Walk Special Random Signals and pdf's White Noise Gaussian Distribution (Normal Distribution) Exponential Distribution Lognormal Distribution Chi-Square Distribution Wiener-Khinchin Relations Filtering Random Processes 83
4 Contents vii 4.6 Special Types of Random Processes Autoregressive Process Nonparametric Spectra Estimation Periodogram Correlogram Computation of Periodogram and Correlogram Using FFT General Remarks on the Periodogram Windowed Periodogram Proposed Book Modified Method for Better Frequency Resolution Using Transformation of the rv's Blackman-Tukey Method Bartlett Periodogram The Welch Method Proposed Modified Welch Methods Modified Method Using Different Types of Overlapping Modified Welch Method Using Transformation of rv's Ill Problems 113 Hints-Solutions-Suggestions 114 Chapter 5 The Wiener Filter Introduction The LS Technique Linear LS LS Formulation Statistical Properties of LSEs The LS Approach Orthogonality Principle Corollary Projection Operator LS Finite Impulse Response Filter The Mean-Square Error The FIR Wiener Filter The Wiener Solution Orthogonality Condition Normalized Performance Equation Canonical Form of the Error-Performance Surface Wiener Filtering Examples Minimum MSE Optimum Filter (w ) Linear Prediction 161 Problems 162
5 viii Contents Additional Problems 164 Hints-Solutions-Suggestions 164 Additional Problems 166 Chapter 6 Eigenvalues of Rx: Properties of the Error Surface The Eigenvalues of the Correlation Matrix Karhunen-Loeve Transformation Geometrical Properties of the Error Surface 174 Problems 178 Hints-Solutions-Suggestions 178 Chapter 7 Newton's and Steepest Descent Methods One-Dimensional Gradient Search Method Gradient Search Algorithm Newton's Method in Gradient Search Steepest Descent Algorithm Steepest Descent Algorithm Applied to Wiener Filter Stability (Convergence) of the Algorithm Transient Behavior of MSE Learning Curve Newton's Method Solution of the Vector Difference Equation 194 Problems 197 Edition Problems 197 Hints-Solutions-Suggestions 198 Additional Problems 200 Chapter 8 The Least Mean- Square Algorithm Introduction The LMS Algorithm Examples Using the LMS Algorithm Performance Analysis of the LMS Algorithm Learning Curve The Coefficient-Error or Weighted-Error Correlation Matrix Excess MSE and Misadjustment Stability The LMS and Steepest Descent Methods Complex Representation of the LMS Algorithm 228 Problems 231 Hints-Solutions-Suggestions 232 Chapter 9 Variants of Least Mean-Square Algorithm The Normalized Least Mean-Square Algorithm Power Normalized LMS 244
6 Contents ix 9.3 Self-Correcting LMS Filter The Sign-Error LMS Algorithm The NLMS Sign-Error Algorithm The Sign-Regressor LMS Algorithm Self-Correcting Sign-Regressor LMS Algorithm The Normalized Sign-Regressor LMS Algorithm The Sign-Sign LMS Algorithm The Normalized Sign-Sign LMS Algorithm Variable Step-Size LMS The Leaky LMS Algorithm The Linearly Constrained LMS Algorithm The Least Mean Fourth Algorithm The Least Mean Mixed Norm LMS Algorithm Short-Length Signal of the LMS Algorithm The Transform Domain LMS Algorithm Convergence The Error Normalized Step-Size LMS Algorithm The Robust Variable Step-Size LMS Algorithm The Modified LMS Algorithm Momentum LMS The Block LMS Algorithm The Complex LMS Algorithm The Affine LMS Algorithm The Complex Affine LMS Algorithm 290 Problems 291 Hints-Solutions-Suggestions 293 Appendix 1: Suggestions and Explanations for MATLAB Use 301 Al. 1 Suggestions and Explanations for MATLAB Use 301 Al.1.1 Creating a Directory 301 Al.1.2 Help 301 A Save and Load 302 A MATLAB as Calculator 302 A1.1.5 Variable Names 302 Al.1.6 Complex Numbers 302 Al.1.7 Array Indexing 302 A1.1.8 Extracting and Inserting Numbers in Arrays 303 Al.1.9 Vectorization 303 A Windowing 304 A Matrices 304 A Producing a Periodic Function 305 Al.1.13 Script Files 305 A Functions 305 A Complex Expressions 306 A Axes 306 Al D Graphics 306
7 x Contents Al D Plots 308 Al Mesh-Type Figures 308 A1.2 General Purpose Commands 309 A1.2.1 Managing Commands and Functions 309 A Managing Variables and Workplace 309 A Operators and Special Characters 309 A Control Flow 310 A 1.3 Elementary Matrices and Matrix Manipulation 311 Al.3.1 Elementary Matrices and Arrays 311 A1.3.2 Matrix Manipulation 311 A1.4 Elementary Mathematical Functions 312 Al.4.1 Elementary Functions 312 A1.5 Numerical Linear Algebra 313 Al.5.1 Matrix Analysis 313 A1.6 Data Analysis 313 Al.6.1 Basic Operations 313 Al.6.2 Filtering and Convolution 313 Al.6.3 Fourier Transforms 314 A 1.7 2D Plotting 314 Al.7.1 2D Plots 314 Appendix 2: Matrix Analysis 317 A2.1 Definitions 317 A2.2 Special Matrices 319 A2.3 Matrix Operation and Formulas 322 A2.4 Eigendecomposition of Matrices 325 A2.5 Matrix Expectations 326 A2.6 Differentiation of a Scalar Function with respect to a Vector 327 Appendix 3: Mathematical Formulas 329 A3.1 Trigonometric Identities 329 A3.2 Orthogonality 330 A3.3 Summation of Trigonometric Forms 331 A3.4 Summation Formulas 331 A3.4.1 Finite Summation Formulas 331 A3.4.2 Infinite Summation Formulas 331 A3.5 Series Expansions 332 A3.6 Logarithms 332 A3.7 Some Definite Integrals 332 Appendix 4: Lagrange Multiplier Method 335 Bibliography 337 Index 339
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