Adaptive Filtering

Fundamentals of Least Mean Squares with MATLAB®

Alexander D. Poularikas

• Publisher: CRC Press
• Year: 2017
• Print ISBN:
  9781138417915
• eBook ISBN:
  9781351831024
• Topic: Technologie et ingénierie
• Subtopic: Sciences générales de l'informatique

Adaptive filters are used in many diverse applications, appearing in everything from military instruments to cellphones and home appliances. Adaptive Filtering: Fundamentals of Least Mean Squares with MATLAB® covers the core concepts of this important field, focusing on a vital part of the statistical signal processing area—the least mean square (LMS) adaptive filter.

This largely self-contained text:

• Discusses random variables, stochastic processes, vectors, matrices, determinants, discrete random signals, and probability distributions
• Explains how to find the eigenvalues and eigenvectors of a matrix and the properties of the error surfaces
• Explores the Wiener filter and its practical uses, details the steepest descent method, and develops the Newton's algorithm
• Addresses the basics of the LMS adaptive filter algorithm, considers LMS adaptive filter variants, and provides numerous examples
• Delivers a concise introduction to MATLAB®, supplying problems, computer experiments, and more than 110 functions and script files

Featuring robust appendices complete with mathematical tables and formulas, Adaptive Filtering: Fundamentals of Least Mean Squares with MATLAB® clearly describes the key principles of adaptive filtering and effectively demonstrates how to apply them to solve real-world problems.

1

Vectors

1.1 INTRODUCTION

In this text we present vectors with boldfaced lowercase Latin letters, in columnar format. Matrices are presented with boldfaced capital Latin letters and numbers with lowercase Latin letters. For example, for three different types of three element vectors, we write

$\mathit{x}=\left[\begin{array}{l}1\\ 2.3\\ -2\end{array}\right];\mathit{y}=\left[\begin{array}{l}{y}_1\\ y_2\\ y_3\end{array}\right];\mathit{z}=\left[\begin{array}{l}2.33\\ 4+j5\\ -j\end{array}\right]$

(1.1)

The elements y₁, y₂, and y₃ are constants. For any one of the above vectors, we could also write its equivalent form, for example, $x^T=\left[\begin{array}{ccc}1& 2.3& -2\end{array}\right]$ or x^T=(1, 2.3, −2), where the letter T stands for the word transpose. The MATLAB^® column vector format is written in the command window as follows:

>>x = [1 2.3 −2]’;%column vector, accent indicates transpose;

>>x = [1; 2.3; −2];%column vector;

>>x = [1 2.3 −2];%row vector;() are not used in MATLAB for

>> %vector manipulations;

We can consider a column vector as a special matrix with m rows, in this case 3, and n columns, in this case 1.

1.1.1 MULTIPLICATION BY A CONSTANT AND ADDITION AND SUBTRACTION

1.1.1.1 Multiplication by a Constant

Using two vectors from (1.1), we write $a_1{x}^T=[...$

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Table of Contents
• Preface
• Author
• Abbreviations
• MATLAB® Functions
• Chapter 1 Vectors
• Chapter 2 Matrices
• Chapter 3 Processing of Discrete Deterministic Signals: Discrete Systems
• Chapter 4 Discrete-Time Random Processes
• Chapter 5 The Wiener Filter
• Chapter 6 Eigenvalues of Rx: Properties of the Error Surface
• Chapter 7 Newton’s and Steepest Descent Methods
• Chapter 8 The Least Mean-Square Algorithm
• Chapter 9 Variants of Least Mean-Square Algorithm
• Appendix 1: Suggestions and Explanations for MATLAB Use
• Appendix 2: Matrix Analysis
• Appendix 3: Mathematical Formulas
• Appendix 4: Lagrange Multiplier Method
• Bibliography
• Index