# Signals and Systems Primer with MATLAB

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## Book Description

Signals and Systems Primer with MATLAB® equally emphasizes the fundamentals of both analog and digital signals and systems. To ensure insight into the basic concepts and methods, the text presents a variety of examples that illustrate a wide range of applications, from microelectromechanical to worldwide communication systems. It also provides MATLAB functions and procedures for practice and verification of these concepts.

Taking a pedagogical approach, the author builds a solid foundation in signal processing as well as analog and digital systems. The book first introduces orthogonal signals, linear and time-invariant continuous-time systems, discrete-type systems, periodic signals represented by Fourier series, Gibbs's phenomenon, and the sampling theorem. After chapters on various transforms, the book discusses analog filter design, both finite and infinite impulse response digital filters, and the fundamentals of random digital signal processing, including the nonparametric spectral estimation. The final chapter presents different types of filtering and their uses for random digital signal processing, specifically, the use of Wiener filtering and least mean squares filtering.

Balancing the study of signals with system modeling and interactions, this text will help readers accurately develop mathematical representations of systems.

## Table of Contents

SIGNALS AND THEIR FUNCTIONAL REPRESENTATION

Some applications involving signals

Fundamental representation of simple time signals

Signal conditioning and manipulation

Representation of signals

Appendix 1: Elementary matrix algebra

Appendix 2: Complex numbers

Appendix 1 Problems

Appendix 2 Problems

LINEAR CONTINUOUS-TIME SYSTEMS

Properties of systems

Modeling simple continuous systems

Solutions of first-order systems

Evaluation of integration constants: initial conditions

Block diagram representation

Convolution and correlation of continuous-time signals

Impulse response

DISCRETE SYSTEMS

Discrete systems and equations

Digital simulation of analog systems

Digital simulation of higher-order differential equations

Convolution of discrete-time signals

Appendix 1: Method of variation of parameters

Appendix 2: Euler's approximation for differential equations

PERIODIC CONTINUOUS SIGNALS AND THEIR SPECTRUMS

Complex functions

Fourier series of continuous functions

Features of periodic continuous functions

Linear systems with periodic inputs

NONPERIODIC SIGNALS AND THEIR FOURIER TRANSFORM

Direct and inverse Fourier transform

Properties of Fourier transforms

Some special Fourier transform pairs

Effects of truncation and Gibbs' phenomenon

Linear time-invariant filters

Appendix

SAMPLING OF CONTINUOUS SIGNALS

Fundamentals of sampling

The sampling theorem

DISCRETE-TIME TRANSFORMS

Discrete-time Fourier transform (DTFT)

Summary of DTFT properties

DTFT of finite time sequences

Frequency response of linear time-invariant (LTI) discrete systems

The discrete Fourier transform (DFT)

Summary of the DFT properties

Multirate digital signal processing

Appendix 1: Proofs of the DTFT properties

Appendix 2: Proofs of DFT properties

Appendix 3: Fast Fourier transform (FFT)

LAPLACE TRANSFORM

One-sided Laplace transform

Summary of the Laplace transform properties

Systems analysis: transfer functions of LTI systems

Inverse Laplace transform (ILT)

Problem solving with Laplace transform

Frequency response of LTI systems

Pole location and the stability of LTI systems

Feedback for linear systems

Bode plots

Appendix: Proofs of Laplace transform properties

THE Z-TRANSFORM, DIFFERENCE EQUATIONS, AND DISCRETE SYSTEMS

The z-transform

Convergence of the z-transform

Properties of the z-transform

z-Transform pairs

Inverse z-transform

Transfer function

Frequency response of first-order discrete systems

Frequency response of higher-order digital systems

z-Transform solution of first-order difference equations

Higher-order difference equations

Appendix: Proofs of the z-transform properties

ANALOG FILTER DESIGN

General aspects of filters

Butterworth filter

Chebyshev low-pass filter

Phase characteristics

Frequency transformations

Analog filter design using MATLAB functions

FINITE IMPULSE RESPONSE (FIR) FILTERS

Properties of FIR filters

FIR filters using the Fourier series approach

FIR filters using windows

Prescribed filter specifications using a Kaiser window

MATLAB FIR filter design

INFINITE IMPULSE RESPONSE (IIR) FILTERS

The impulse-invariant method approximation in the time domain

Bilinear transformation

Frequency transformation for digital filters

Recursive versus nonrecursive design

RANDOM VARIABLES, SEQUENCES, AND POWER SPECTRA DENSITIES

Random signals and distributions

Averages

Stationary processes

Special random signals and probability density functions

Wiener-Kintchin relations

Filtering random processes

Nonparametric spectra estimation

LEAST SQUARE SYSTEM DESIGN, WIENER FILTER, AND THE LMS FILTER

The least-squares technique

The mean square error

Wiener filtering examples

The least mean square (LMS) algorithm

Examples using the LMS algorithm

APPENDIX A: MATHEMATICAL FORMULAS

Trigonometric identities

Orthigonality

Summation of trigonometric forms

Summation formulas

Series expansions

Logarithms

Some definite integrals

APPENDIX B: SUGGESTIONS AND EXPLANATIONS FOR MATLAB USE

Creating a directory

Help

Save and load

MATLAB as calculator

Variable names

Complex numbers

Array indexing

Extracting and inserting numbers in arrays

Vectorization

Matrices

Produce a periodic function

Script files

Functions

Subplots

Figures

Changing the scales of the axes of a figure

Writing Greek letters

Subscripts and superscripts

Lines in plots

INDEX

Each chapter features Important Definitions and Concepts as well as Problems.

## Reviews

"The book is written with a high pedagogical mastership; the style of exposition is clear and attractive, the typographical presentation is excellent . . . Much valuable information is contained in the book at a moderate mathematical level . . . we think the present volume is an excellent book on SS and can be a serious candidate for a reference book in presenting the SS domain."

– Dumitru Stanomir,in Zentralblatt Math, 2009