Analog and Digital Signals and Systems

R.K. Rao Yarlagadda

Analog and Digital Signalsand Systems

1 3

R.K. Rao YarlagaddaSchool of Electrical & ComputerEngineering

Oklahoma State UniversityStillwater OK 74078-6028202 Engineering SouthUSArao.yarlagadda@okstate.edu

ISBN 978-1-4419-0033-3 e-ISBN 978-1-4419-0034-0DOI 10.1007/978-1-4419-0034-0Springer New York Dordrecht Heidelberg London

Library of Congress Control Number: 2009929744

# Springer ScienceBusiness Media, LLC 2010All rights reserved. This work may not be translated or copied in whole or in part without the writtenpermission of the publisher (Springer ScienceBusiness Media, LLC, 233 Spring Street, New York,NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use inconnection with any form of information storage and retrieval, electronic adaptation, computersoftware, or by similar or dissimilar methodology now known or hereafter developed is forbidden.The use in this publication of trade names, trademarks, service marks, and similar terms, even if theyare not identified as such, is not to be taken as an expression of opinion as to whether or not they aresubject to proprietary rights.

Printed on acid-free paper

Springer is part of Springer ScienceBusiness Media (www.springer.com)

This book is dedicated to my wifeMarceil, children, Tammy Bardwell, Ryan Yarlagaddaand Travis Yarlagadda and their families

Note to Instructors

The solutions manual can be located on the books webpage http://www/

springer.com/engineering/cirucits%26systems/bok/978-1-4419-0033-3

vii

Preface

This book presents a systematic, comprehensive treatment of analog and discrete

signal analysis and synthesis and an introduction to analog communication

theory. This evolved from my 40 years of teaching at Oklahoma State University

(OSU). It is based on three courses, Signal Analysis (a second semester junior

level course), Active Filters (a first semester senior level course), and Digital

signal processing (a second semester senior level course). I have taught these

courses a number of times using this material along with existing texts. The

references for the books and journals (over 160 references) are listed in the

bibliography section. At the undergraduate level, most signal analysis courses

do not require probability theory. Only, a very small portion of this topic is

included here.

I emphasized the basics in the book with simple mathematics and the sophis-

tication is minimal. Theorem-proof type of material is not emphasized. The book

uses the following model:

1. Learn basics

2. Check the work using bench marks

3. Use software to see if the results are accurate

The book provides detailed examples (over 400) with applications. A three-

number system is used consisting of chapter number section number

example or problem number, thus allowing the student to quickly identify

the related material in the appropriate section of the book. The book

includes well over 400 homework problems. Problem numbers are identified

using the above three-number system. Hints are provided wherever addi-

tional details may be needed and may not have been given in the main part

of the text. A detailed solution manual will be available from the publisher

for the instructors.

Summary of the Chapters

This book starts with an introductory chapter that includes most of the basic

material that a junior in electrical engineering had in the beginning classes. For

those who have forgotten, or have not seen the material recently, it gives enough

ix

background to follow the text. The topics in this chapter include singularity

functions, periodic functions, and others. Chapter 2 deals with convolution

and correlation of periodic and aperiodic functions. Chapter 3 deals with

approximating a function by using a set of basis functions, referred to as the

generalized Fourier series expansion. From these concepts, the three basic

Fourier series expansions are derived. The discussion includes detailed dis-

cussion on the operational properties of the Fourier series and their

convergence.

Chapter 4 deals with Fourier transform theory derived from the Fourier

series. Fourier series and transforms are the bases to this text. Considerable

material in the book is based on these topics. Chapter 5 deals with the relatives

of the Fourier transforms, including Laplace, cosine and sine, Hartley and

Hilbert transforms.

Chapter 6 deals with basic systems analysis that includes linear time-

invariant systems, stability concepts, impulse response, transfer functions,

linear and nonlinear systems, and very simple filter circuits and concepts.

Chapter 7 starts with the Bode plots and later deals with approximations

using classical analog Butterworth, Chebyshev, and Bessel filter functions.

Design techniques, based on both amplitude and phase based, are discussed.

Last part of this chapter deals with analysis and synthesis of active filter

circuits. Examples of basic low-pass, high-pass, band-pass, band elimina-

tion, and delay line filters are included.

Chapter 8 builds a bridge to go from the continuous-time to discrete-time

analysis by starting with sampling theory and the Fourier transform of the

ideally sampled signals. Bulk of this chapter deals with discrete basis func-

tions, discrete-time Fourier series, discrete-time Fourier transform (DTFT),

and the discrete Fourier transform (DFT). Chapter 9 deals with fast

implementations of the DFT, discrete convolution, and correlation. Second

part of the chapter deals the z-transforms and their use in the design of

discrete-data systems. Digital filter designs based on impulse invariance and

bilinear transformations are presented. The chapter ends with digital filter

realizations.

Chapter 10 presents an introduction to analog communication theory,

which includes basic material on analog modulation, such as AM and FM,

demodulation, and multiplexing. Pulse modulation methods are

introduced.

Appendix A reviews the basics on matrices; Appendix B gives a brief intro-

duction onMATLAB; and Appendix C gives a list of useful formulae. The book

concludes with a list of references and Author and Subject indexes.

Suggested Course Content

Instructor is the final judge of what topics will best suit his or her class and in

what depth. The suggestions given below are intended to serve as a guide only.

The book permits flexibility in teaching analysis, synthesis of continuous-time

and discrete-time systems, analog filters, digital signal processing, and an intro-

duction to analog communications. The following table gives suggestions for

courses.

x Preface

Topical Title Related topics in chapters

One semester (Fundamentals of analogsignals and systems ) Chapters 14, 6

One semester Systems and analog filters Chapters 4, 5*, 6, 7

One semester (Introduction to digitalsignal processing ) Chapters 4*, 6*, 8, 9

Two semesters (Signals and an introduction toanalog communications ) Chapters 14, 5*, 6, 8*, 10*Partial coverage

Preface xi

Acknowledgements

The process of writing this book has taken me several years. I am indebted to all

the students who have studied with me and taken classes fromme. Education is a

two-way street. The teachers learn from the students, as well as the students learn

from the teachers. Writing a book is a learning process.

Dr. Jack Cartinhour went through the material in the early stages of the text

and helped me in completing the solution manual. His suggestions made the text

better. I am deeply indebted to him. Dr. George Scheets used an earlier version of

this book in his signal analysis and communications theory class. Dr. Martin

Hagan has reviewed a chapter. Their comments were incorporated into the

manuscript. Beau Lacefield did most of the artwork in the manuscript. Vijay

Venkataraman and Wen Fung Leong have gone through some of the chapters

and their suggestions have been incorporated. In addition, Vijay and Wen have

provided some of the MATLAB programs and artwork. I appreciated Vijays

help in formatting the final version of the manuscript.

An old adage of the uncertainty principle is, no matter how many times the

author goes through the text, mistakes will remain. I sincerely appreciate all the

support provided by Springer. Thanks to Alex Greene. He believed in me to

complete this project. I appreciated the patience and support of Katie Chen.

Thanks to Shanty Jaganathan and her associates of Integra-India. They have

been helpful and gracious in the editorial process.

Dr. Keith Teague, Head, School of Electrical and Computer Engineering at

Oklahoma State University has been very supportive of this project and I

appreciated his encouragement.

Finally, the time spent on this book is the time taken away from my wife

Marceil, children Tammy, Ryan and Travis and my grandchildren. Without my

familys understanding, I could not have completed this book.

Oklahoma, USA R.K. Rao Yarlagadda

xiii

Contents

1 Basic Concepts in Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Introduction to the Book and Signals . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Different Ways of Looking at a Signal . . . . . . . . . . . . . . 1

1.1.2 Continuous-Time and Discrete-Time Signals . . . . . . . . . 3

1.1.3 Analog Versus Digital Signal Processing . . . . . . . . . . . . 5

1.1.4 Examples of Simple Functions . . . . . . . . . . . . . . . . . . . . 6

1.2 Useful Signal Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2.1 Time Shifting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2.2 Time Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2.3 Time Reversal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2.4 Amplitude Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2.5 Simple Symmetries: Even and Odd Functions . . . . . . . . 9

1.2.6 Products of Even and Odd Functions . . . . . . . . . . . . . . . 9

1.2.7 Signum (or sgn) Function . . . . . . . . . . . . . . . . . . . . . . . . 10

1.2.8 Sinc and Sinc2 Functions. . . . . . . . . . . . . . . . . . . . . . . . . 10

1.2.9 Sine Integral Function. . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.3 Derivatives and Integrals of Functions . . . . . . . . . . . . . . . . . . . 11

1.3.1 Integrals of Functions with Symmetries . . . . . . . . . . . . . 12

1.3.2 Useful Functions from Unit Step Function . . . . . . . . . . 12

1.3.3 Leibnizs Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.3.4 Interchange of a Derivative and an Integral . . . . . . . . . . 13

1.3.5 Interchange of Integrals. . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.4 Singularity Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.4.1 Unit Impulse as the Limit of a Sequence. . . . . . . . . . . . . 15

1.4.2 Step Function and the Impulse Function . . . . . . . . . . . . 16

1.4.3 Functions of Generalized Functions . . . . . . . . . . . . . . . . 17

1.4.4 Functions of Impulse Functions . . . . . . . . . . . . . . . . . . . 18

1.4.5 Functions of Step Functions . . . . . . . . . . . . . . . . . . . . . . 19

1.5 Signal Classification Based on Integrals . . . . . . . . . . . . . . . . . . 19

1.5.1 Effects of Operations on Signals . . . . . . . . . . . . . . . . . . . 21

1.5.2 Periodic Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.5.3 Sum of Two Periodic Functions . . . . . . . . . . . . . . . . . . . 23

1.6 Complex Numbers, Periodic, and Symmetric Periodic

Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

1.6.1 Complex Numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

xv

1.6.2 Complex Periodic Functions . . . . . . . . . . . . . . . . . . . . . . 271.6.3 Functions of Periodic Functions . . . . . . . . . . . . . . . . . . . 271.6.4 Periodic Functions with Additional Symmetries. . . . . . . 28

1.7 Examples of Probability Density Functions and their

Moments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

1.8 Generation of Periodic Functions from Aperiodic

Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

1.9 Decibel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

1.10 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2 Convolution and Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.1.1 Scalar Product and Norm . . . . . . . . . . . . . . . . . . . . . . . . 40

2.2 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.2.1 Properties of the Convolution Integral . . . . . . . . . . . . . . 41

2.2.2 Existence of the Convolution Integral. . . . . . . . . . . . . . . 44

2.3 Interesting Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.4 Convolution and Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

2.4.1 Repeated Convolution and the Central Limit

Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

2.4.2 Deconvolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

2.5 Convolution Involving Periodic and Aperiodic Functions . . . . 54

2.5.1 Convolution of a Periodic Function with an

Aperiodic Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

2.5.2 Convolution of Two Periodic Functions. . . . . . . . . . . . . 55

2.6 Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

2.6.1 Basic Properties of Cross-Correlation Functions . . . . . . 57

2.6.2 Cross-Correlation and Convolution . . . . . . . . . . . . . . . . 57

2.6.3 Bounds on the Cross-Correlation Functions . . . . . . . . . 58

2.6.4 Quantitative Measures of Cross-Correlation . . . . . . . . . 59

2.7 Autocorrelation Functions of Energy Signals . . . . . . . . . . . . . . 63

2.8 Cross- and Autocorrelation of Periodic Functions . . . . . . . . . . 65

2.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3 Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.2 Orthogonal Basis Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

3.2.1 GramSchmidt Orthogonalization . . . . . . . . . . . . . . . . . 74

3.3 Approximation Measures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

3.3.1 Computation of c[k] Based on Partials . . . . . . . . . . . . . . 77

3.3.2 Computation of c[k] Using the Method of Perfect

Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

3.3.3 Parsevals Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

3.4 Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

3.4.1 Complex Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . 80

3.4.2 Trigonometric Fourier Series . . . . . . . . . . . . . . . . . . . . . 83

3.4.3 Complex F-series and the Trigonometric F-series

Coefficients-Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

xvi Contents

3.4.4 Harmonic Form of Trigonometric Fourier Series. . . . . 833.4.5 Parsevals Theorem Revisited . . . . . . . . . . . . . . . . . . . . 843.4.6 Advantages and Disadvantages of the Three Forms

of Fourier Series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

3.5 Fourier Series of Functions with Simple Symmetries. . . . . . . . 853.5.1 Simplification of the Fourier Series Coefficient Integral . 86

3.6 Operational Properties of Fourier Series . . . . . . . . . . . . . . . . . 873.6.1 Principle of Superposition . . . . . . . . . . . . . . . . . . . . . . . 873.6.2 Time Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 873.6.3 Time and Frequency Scaling . . . . . . . . . . . . . . . . . . . . . 883.6.4 Fourier Series Using Derivatives . . . . . . . . . . . . . . . . . . 893.6.5 Bounds and Rates of Fourier Series Convergence

by the Derivative Method . . . . . . . . . . . . . . . . . . . . . . 913.6.6 Integral of a Function and Its Fourier Series . . . . . . . . 933.6.7 Modulation in Time. . . . . . . . . . . . . . . . . . . . . . . . . . . . 933.6.8 Multiplication in Time. . . . . . . . . . . . . . . . . . . . . . . . . . 943.6.9 Frequency Modulation . . . . . . . . . . . . . . . . . . . . . . . . . 953.6.10 Central Ordinate Theorems. . . . . . . . . . . . . . . . . . . . . . 953.6.11 Plancherels Relation (or Theorem). . . . . . . . . . . . . . . . 953.6.12 Power Spectral Analysis . . . . . . . . . . . . . . . . . . . . . . . . 95

3.7 Convergence of the Fourier Series and the Gibbs

Phenomenon. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 963.7.1 Fouriers Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 963.7.2 Gibbs Phenomenon . . . . . . . . . . . . . . . . . . . . . . . . . . . . 973.7.3 Spectral Window Smoothing. . . . . . . . . . . . . . . . . . . . . 99

3.8 Fourier Series Expansion of Periodic Functions with

Special Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1003.8.1 Half-Wave Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . 1003.8.2 Quarter-Wave Symmetry. . . . . . . . . . . . . . . . . . . . . . . 1023.8.3 Even Quarter-Wave Symmetry . . . . . . . . . . . . . . . . . . 1023.8.4 Odd Quarter-Wave Symmetry. . . . . . . . . . . . . . . . . . . 1023.8.5 Hidden Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

3.9 Half-Range Series Expansions . . . . . . . . . . . . . . . . . . . . . . . . . 103

3.10 Fourier Series Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

3.11 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

4 Fourier Transform Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

4.2 Fourier Series to Fourier Integral . . . . . . . . . . . . . . . . . . . . . . . 109

4.2.1 Amplitude and Phase Spectra . . . . . . . . . . . . . . . . . . . . . 112

4.2.2 Bandwidth-Simplistic Ideas . . . . . . . . . . . . . . . . . . . . . . . 114

4.3 Fourier Transform Theorems, Part 1. . . . . . . . . . . . . . . . . . . . . 114

4.3.1 Rayleighs Energy Theorem . . . . . . . . . . . . . . . . . . . . . . 114

4.3.2 Superposition Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 115

4.3.3 Time Delay Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

4.3.4 Scale Change Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 116

4.3.5 Symmetry or Duality Theorem . . . . . . . . . . . . . . . . . . . . 118

4.3.6 Fourier Central Ordinate Theorems . . . . . . . . . . . . . . . . 119

Contents xvii

4.4 Fourier Transform Theorems, Part 2 . . . . . . . . . . . . . . . . . . . . 1194.4.1 Frequency Translation Theorem. . . . . . . . . . . . . . . . . 1204.4.2 Modulation Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 1204.4.3 Fourier Transforms of Periodic and Some Special

Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1214.4.4 Time Differentiation Theorem . . . . . . . . . . . . . . . . . . 1244.4.5 Times-t Property: Frequency Differentiation

Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1264.4.6 Initial Value Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 1284.4.7 Integration Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 128

4.5 Convolution and Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . 1294.5.1 Convolution in Time . . . . . . . . . . . . . . . . . . . . . . . . . . 1294.5.2 Proof of the Integration Theorem . . . . . . . . . . . . . . . . 1324.5.3 Multiplication Theorem (Convolution in

Frequency) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1334.5.4 Energy Spectral Density . . . . . . . . . . . . . . . . . . . . . . . 135

4.6 Autocorrelation and Cross-Correlation . . . . . . . . . . . . . . . . . . 1364.6.1 Power Spectral Density . . . . . . . . . . . . . . . . . . . . . . . . 138

4.7 Bandwidth of a Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1394.7.1 Measures Based on Areas of the Time and Frequency

Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1394.7.2 Measures Based on Moments . . . . . . . . . . . . . . . . . . . 1404.7.3 Uncertainty Principle in Fourier Analysis. . . . . . . . . . 141

4.8 Moments and the Fourier Transform . . . . . . . . . . . . . . . . . . . 143

4.9 Bounds on the Fourier Transform . . . . . . . . . . . . . . . . . . . . . . 144

4.10 Poissons Summation Formula . . . . . . . . . . . . . . . . . . . . . . . . 145

4.11 Interesting Examples and a Short Fourier Transform Table . . 1454.11.1 Raised-Cosine Pulse Function. . . . . . . . . . . . . . . . . . . 146

4.12 Tables of Fourier Transforms Properties and Pairs. . . . . . . . . 147

4.13 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

5 Relatives of Fourier Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

5.2 Fourier Cosine and Sine Transforms . . . . . . . . . . . . . . . . . . . . . 156

5.3 Hartley Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

5.4 Laplace Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

5.4.1 Region of Convergence (ROC) . . . . . . . . . . . . . . . . . . . . 163

5.4.2 Inverse Transform of Two-Sided Laplace Transform. . . 164

5.4.3 Region of Convergence (ROC) of Rational

Functions Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . 165

5.5 Basic Two-Sided Laplace Transform Theorems . . . . . . . . . . . . 165

5.5.1 Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

5.5.2 Time Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

5.5.3 Shift in s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

5.5.4 Time Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

5.5.5 Time Reversal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

5.5.6 Differentiation in Time . . . . . . . . . . . . . . . . . . . . . . . . . . 166

5.5.7 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

5.5.8 Convolution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

xviii Contents

5.6 One-Sided Laplace Transform . . . . . . . . . . . . . . . . . . . . . . . . . 1665.6.1 Properties of the One-Sided Laplace Transform. . . . . 1675.6.2 Comments on the Properties (or Theorems)

of Laplace Transforms . . . . . . . . . . . . . . . . . . . . . . . . 167

5.7 Rational Transform Functions and Inverse Laplace

Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1745.7.1 Rational Functions, Poles, and Zeros . . . . . . . . . . . . . 1755.7.2 Return to the Initial and Final Value Theorems

and Their Use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

5.8 Solutions of Constant Coefficient Differential Equations

Using Laplace Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1785.8.1 Inverse Laplace Transforms . . . . . . . . . . . . . . . . . . . . 1795.8.2 Partial Fraction Expansions . . . . . . . . . . . . . . . . . . . . 179

5.9 Relationship Between Laplace Transforms and Other

Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1835.9.1 Laplace Transforms and Fourier Transforms . . . . . . . 1845.9.2 Hartley Transforms and Laplace Transforms . . . . . . . 185

5.10 Hilbert Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1865.10.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1865.10.2 Hilbert Transform of Signals with Non-overlapping

Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1885.10.3 Analytic Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

5.11 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

6 Systems and Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

6.2 Linear Systems, an Introduction . . . . . . . . . . . . . . . . . . . . . . . . 193

6.3 Ideal Two-Terminal Circuit Components and Kirchhoffs

Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

6.3.1 Two-Terminal Component Equations . . . . . . . . . . . . . . 195

6.3.2 Kirchhoffs Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

6.4 Time-Invariant and Time-Varying Systems . . . . . . . . . . . . . . . . 198

6.5 Impulse Response. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

6.5.1 Eigenfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

6.5.2 Bounded-Input/Bounded-Output (BIBO) Stability . . . . 202

6.5.3 RouthHurwitz Criterion (RH criterion) . . . . . . . . . . . 203

6.5.4 Eigenfunctions in the Fourier Domain . . . . . . . . . . . . . . 206

6.6 Step Response. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

6.7 Distortionless Transmission . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

6.7.1 Group Delay and Phase Delay . . . . . . . . . . . . . . . . . . . . 213

6.8 System Bandwidth Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

6.8.1 Bandwidth Measures Using the Impulse Response

ht and Its Transform Hj! . . . . . . . . . . . . . . . . . . . . . 2166.8.2 Half-Power or 3 dB Bandwidth. . . . . . . . . . . . . . . . . . . . 217

6.8.3 Equivalent Bandwidth or Noise Bandwidth . . . . . . . . . . 217

6.8.4 Root Mean-Squared (RMS) Bandwidth . . . . . . . . . . . . . 218

6.9 Nonlinear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219

6.9.1 Distortion Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

Contents xix

6.9.2 Output Fourier-Transform of a Nonlinear System. . . 2206.9.3 Linearization of Nonlinear System Functions . . . . . . 221

6.10 Ideal Filters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2216.10.1 Low-Pass, High-Pass, Band-Pass, and

Band-Elimination Filters . . . . . . . . . . . . . . . . . . . . . . . 222

6.11 Real and Imaginary Parts of the Fourier Transform

of a Causal Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2276.11.1 Relationship Between Real and Imaginary Parts

of the Fourier Transform of a Causal Function

Using Hilbert Transform . . . . . . . . . . . . . . . . . . . . . . . 2286.11.2 Amplitude Spectrum Hj!j j to a Minimum Phase

Function Hs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2296.12 More on Filters: Source and Load Impedances . . . . . . . . . . . . 229

6.12.1 Simple Low-Pass Filters . . . . . . . . . . . . . . . . . . . . . . . 2316.12.2 Simple High-Pass Filters . . . . . . . . . . . . . . . . . . . . . . . 2316.12.3 Simple Band-Pass Filters . . . . . . . . . . . . . . . . . . . . . . . 2336.12.4 Simple Band-Elimination or Band-Reject

or Notch Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2356.12.5 Maximum Power Transfer. . . . . . . . . . . . . . . . . . . . . . 2386.12.6 A Simple Delay Line Circuit . . . . . . . . . . . . . . . . . . . . 239

6.13 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

7 Approximations and Filter Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

7.2 Bode Plots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

7.2.1 Gain and Phase Margins . . . . . . . . . . . . . . . . . . . . . . . . . 252

7.3 Classical Analog Filter Functions . . . . . . . . . . . . . . . . . . . . . . . 254

7.3.1 Amplitude-Based Design. . . . . . . . . . . . . . . . . . . . . . . . . 254

7.3.2 Butterworth Approximations . . . . . . . . . . . . . . . . . . . . . 255

7.3.3 Chebyshev (Tschebyscheff) Approximations . . . . . . . . . 257

7.4 Phase-Based Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262

7.4.1 Maximally Flat Delay Approximation . . . . . . . . . . . . . . 263

7.4.2 Group Delay of Bessel Functions . . . . . . . . . . . . . . . . . . 264

7.5 Frequency Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266

7.5.1 Normalized Low-Pass to High-Pass Transformation . . . 266

7.5.2 Normalized Low-Pass to Band-Pass Transformation. . . 268

7.5.3 Normalized Low-Pass to Band-Elimination

Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268

7.5.4 Conversions of Specifications from Low-Pass,

High-Pass, Band-Pass, and Band Elimination Filters

to Normalized Low-Pass Filters . . . . . . . . . . . . . . . . . . . 270

7.6 Multi-terminal Components. . . . . . . . . . . . . . . . . . . . . . . . . . . . 273

7.6.1 Two-Port Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 273

7.6.2 Circuit Analysis Involving Multi-terminal Components

and Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277

7.6.3 Controlled Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278

7.7 Active Filter Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279

7.7.1 Operational Amplifiers, an Introduction . . . . . . . . . . . . 279

xx Contents

7.7.2 Inverting Operational Amplifier Circuits . . . . . . . . . . . 2807.7.3 Non-inverting Operational Amplifier Circuits . . . . . . . 2827.7.4 Simple Second-Order Low-Pass and All-Pass Circuits. .. 284

7.8 Gain Constant Adjustment . . . . . . . . . . . . . . . . . . . . . . . . . . . 285

7.9 Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2877.9.1 Amplitude (or Magnitude) Scaling, RLC Circuits . . . . 2877.9.2 Frequency Scaling, RLC Circuits . . . . . . . . . . . . . . . . . 2887.9.3 Amplitude and Frequency Scaling in Active Filters . . . 2887.9.4 Delay Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290

7.10 RCCR Transformations: Low-Pass to High-Pass Circuits . . 292

7.11 Band-Pass, Band-Elimination and Biquad Filters . . . . . . . . . . 294

7.12 Sensitivities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298

7.13 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301

8 Discrete-Time Signals and Their Fourier Transforms . . . . . . . . . . . . . 311

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311

8.2 Sampling of a Signal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312

8.2.1 Ideal Sampling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312

8.2.2 Uniform Low-Pass Sampling or the Nyquist

Low-Pass Sampling Theorem . . . . . . . . . . . . . . . . . . . . . 314

8.2.3 Interpolation Formula and the Generalized Fourier

Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317

8.2.4 Problems Associated with Sampling Below

the Nyquist Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319

8.2.5 Flat Top Sampling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322

8.2.6 Uniform Band-Pass Sampling Theorem . . . . . . . . . . . . . 324

8.2.7 Equivalent continuous-time and discrete-time

systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325

8.3 Basic Discrete-Time (DT) Signals . . . . . . . . . . . . . . . . . . . . . . . 325

8.3.1 Operations on a Discrete Signal . . . . . . . . . . . . . . . . . . . 327

8.3.2 Discrete-Time Convolution and Correlation . . . . . . . . . 329

8.3.3 Finite duration, right-sided, left-sided, two-sided,

and causal sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330

8.3.4 Discrete-Time Energy and Power Signals . . . . . . . . . . . . 330

8.4 Discrete-Time Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . 332

8.4.1 Periodic Convolution of Two Sequences with the

Same Period. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334

8.4.2 Parsevals Identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334

8.5 Discrete-Time Fourier Transforms . . . . . . . . . . . . . . . . . . . . . . 335

8.5.1 Discrete-Time Fourier Transforms (DTFTs) . . . . . . . . . 335

8.5.2 Discrete-Time Fourier Transforms of Real Signals

with Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336

8.6 Properties of the Discrete-Time Fourier Transforms. . . . . . . . . 339

8.6.1 Periodic Nature of the Discrete-Time Fourier

Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339

8.6.2 Superposition or Linearity . . . . . . . . . . . . . . . . . . . . . . . 340

8.6.3 Time Shift or Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341

8.6.4 Modulation or Frequency Shifting . . . . . . . . . . . . . . . . . 341

Contents xxi

8.6.5 Time Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3418.6.6 Differentiation in Frequency . . . . . . . . . . . . . . . . . . . 3428.6.7 Differencing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3428.6.8 Summation or Accumulation . . . . . . . . . . . . . . . . . . 3448.6.9 Convolution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3448.6.10 Multiplication in Time. . . . . . . . . . . . . . . . . . . . . . . . 3458.6.11 Parsevals Identities . . . . . . . . . . . . . . . . . . . . . . . . . . 3468.6.12 Central Ordinate Theorems . . . . . . . . . . . . . . . . . . . . 3468.6.13 Simple Digital Encryption . . . . . . . . . . . . . . . . . . . . . 346

8.7 Tables of Discrete-Time Fourier Transform (DTFT)

Properties and Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347

8.8 Discrete-Time Fourier-transforms from Samples of the

Continuous-Time Fourier-Transforms . . . . . . . . . . . . . . . . . . 348

8.9 Discrete Fourier Transforms (DFTs) . . . . . . . . . . . . . . . . . . . . 3508.9.1 Matrix Representations of the DFT and

the IDFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3528.9.2 Requirements for Direct Computation of

the DFT . . . . . . . . . . . . . . . . . . . . . . . . . . . 353

8.10 Discrete Fourier Transform Properties . . . . . . . . . . . . . . . . . . 3548.10.1 DFTs and IDFTs of Real Sequences. . . . . . . . . . . . . 3548.10.2 Linearity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3548.10.3 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3558.10.4 Time Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3558.10.5 Frequency Shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3568.10.6 Even Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3568.10.7 Odd Sequences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3568.10.8 Discrete-Time Convolution Theorem . . . . . . . . . . . . 3578.10.9 Discrete-Frequency Convolution Theorem. . . . . . . . 3588.10.10 Discrete-Time Correlation Theorem . . . . . . . . . . . . . 3598.10.11 Parsevals Identity or Theorem . . . . . . . . . . . . . . . . . 3598.10.12 Zero Padding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3598.10.13 Signal Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . 3608.10.14 Decimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361

8.11 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361

9 Discrete Data Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367

9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367

9.2 Computation of Discrete Fourier Transforms (DFTs) . . . . . . . 368

9.2.1 Symbolic Diagrams in Discrete-Time

Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368

9.2.2 Fast Fourier Transforms (FFTs). . . . . . . . . . . . . . . . . . . 369

9.3 DFT (FFT) Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372

9.3.1 Hidden Periodicity in a Signal. . . . . . . . . . . . . . . . . . . . . 372

9.3.2 Convolution of Time-Limited Sequences . . . . . . . . . . . . 374

9.3.3 Correlation of Discrete Signals . . . . . . . . . . . . . . . . . . . . 377

9.3.4 Discrete Deconvolution. . . . . . . . . . . . . . . . . . . . . . . . . . 378

9.4 z-Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380

9.4.1 Region of Convergence (ROC) . . . . . . . . . . . . . . . . . . . . 381

xxii Contents

9.4.2 z-Transform and the Discrete-Time Fourier

Transform (DTFT) . . . . . . . . . . . . . . . . . . . . . . . . . . . 384

9.5 Properties of the z-Transform . . . . . . . . . . . . . . . . . . . . . . . . . 3849.5.1 Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3849.5.2 Time-Shifted Sequences. . . . . . . . . . . . . . . . . . . . . . . . 3859.5.3 Time Reversal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3859.5.4 Multiplication by an Exponential . . . . . . . . . . . . . . . . 3859.5.5 Multiplication by n . . . . . . . . . . . . . . . . . . . . . . . . . . . 3869.5.6 Difference and Accumulation . . . . . . . . . . . . . . . . . . . 3869.5.7 Convolution Theorem and the z-Transform . . . . . . . . 3869.5.8 Correlation Theorem and the z-Transform. . . . . . . . . 3879.5.9 Initial Value Theorem in the Discrete Domain . . . . . . 3889.5.10 Final Value Theorem in the Discrete Domain . . . . . . 388

9.6 Tables of z-Transform Properties and Pairs. . . . . . . . . . . . . . . 389

9.7 Inverse z-Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3909.7.1 Inversion Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3909.7.2 Use of Transform Tables (Partial Fraction

Expansion Method) . . . . . . . . . . . . . . . . . . . . . . . . . . . 3919.7.3 Inverse z-Transforms by Power Series Expansion. . . . 394

9.8 The Unilateral or the One-Sided z-Transform . . . . . . . . . . . . . 3959.8.1 Time-Shifting Property . . . . . . . . . . . . . . . . . . . . . . . . 395

9.9 Discrete-Data Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3979.9.1 Discrete-Time Transfer Functions. . . . . . . . . . . . . . . . 4009.9.2 SchurCohn Stability Test. . . . . . . . . . . . . . . . . . . . . . 4019.9.3 Bilinear Transformations. . . . . . . . . . . . . . . . . . . . . . . 401

9.10 Designs by the Time and Frequency Domain Criteria. . . . . . . 4039.10.1 Impulse Invariance Method by Using the Time

Domain Criterion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4059.10.2 Bilinear Transformation Method by Using the

Frequency Domain Criterion. . . . . . . . . . . . . . . . . . . . 407

9.11 Finite Impulse Response (FIR) Filter Design . . . . . . . . . . . . . 4109.11.1 Low-Pass FIR Filter Design . . . . . . . . . . . . . . . . . . . . 4119.11.2 High-Pass, Band-Pass, and Band-Elimination

FIR Filter Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4139.11.3 Windows in Fourier Design. . . . . . . . . . . . . . . . . . . . . . . . . .1416

9.12 Digital Filter Realizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4199.12.1 Cascade Form of Realization . . . . . . . . . . . . . . . . . . . 4229.12.2 Parallel Form of Realization . . . . . . . . . . . . . . . . . . . . 4229.12.3 All-Pass Filter Realization. . . . . . . . . . . . . . . . . . . . . . 4239.12.4 Digital Filter Transposed Structures . . . . . . . . . . . . . . 4239.12.5 FIR Filter Realizations . . . . . . . . . . . . . . . . . . . . . . . . 423

9.13 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425

10 Analog Modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429

10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429

10.2 Limiters and Mixers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43110.2.1 Mixers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432

10.3 Linear Modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432

Contents xxiii

10.3.1 Double-Sideband (DSB) Modulation . . . . . . . . . . . 43210.3.2 Demodulation of DSB Signals. . . . . . . . . . . . . . . . . 433

10.4 Frequency Multipliers and Dividers. . . . . . . . . . . . . . . . . . . . 435

10.5 Amplitude Modulation (AM). . . . . . . . . . . . . . . . . . . . . . . . . 43710.5.1 Percentage Modulation . . . . . . . . . . . . . . . . . . . . . . 43810.5.2 Bandwidth Requirements . . . . . . . . . . . . . . . . . . . . 43810.5.3 Power and Efficiency of an Amplitude

Modulated Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . 43910.5.4 Average Power Contained in an AM Signal . . . . . . 440

10.6 Generation of AM Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . 44110.6.1 Square-Law Modulators . . . . . . . . . . . . . . . . . . . . . 44110.6.2 Switching Modulators . . . . . . . . . . . . . . . . . . . . . . . 44110.6.3 Balanced Modulators. . . . . . . . . . . . . . . . . . . . . . . . 442

10.7 Demodulation of AM Signals . . . . . . . . . . . . . . . . . . . . . . . . 44310.7.1 Rectifier Detector. . . . . . . . . . . . . . . . . . . . . . . . . . . 44310.7.2 Coherent or a Synchronous Detector . . . . . . . . . . . 44310.7.3 Square-Law Detector. . . . . . . . . . . . . . . . . . . . . . . . 44410.7.4 Envelope Detector . . . . . . . . . . . . . . . . . . . . . . . . . . 444

10.8 Asymmetric Sideband Signals . . . . . . . . . . . . . . . . . . . . . . . . 44610.8.1 Single-Sideband Signals . . . . . . . . . . . . . . . . . . . . . . 44610.8.2 Vestigial Sideband Modulated Signals . . . . . . . . . . 44710.8.3 Demodulation of SSB and VSB Signals . . . . . . . . . 44810.8.4 Non-coherent Demodulation of SSB. . . . . . . . . . . . 44910.8.5 Phase-Shift Modulators and Demodulators . . . . . . 449

10.9 Frequency Translation and Mixing . . . . . . . . . . . . . . . . . . . . 450

10.10 Superheterodyne AM Receiver. . . . . . . . . . . . . . . . . . . . . . . . 453

10.11 Angle Modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45510.11.1 Narrowband (NB) Angle Modulation. . . . . . . . . . . 45810.11.2 Generation of Angle Modulated Signals . . . . . . . . . 459

10.12 Spectrum of an Angle Modulated Signal . . . . . . . . . . . . . . . . 46010.12.1 Properties of Bessel Functions . . . . . . . . . . . . . . . . . 46110.12.2 Power Content in an Angle Modulated Signal . . . . 463

10.13 Demodulation of Angle Modulated Signals. . . . . . . . . . . . . . 46510.13.1 Frequency Discriminators . . . . . . . . . . . . . . . . . . . . 46510.13.2 Delay Lines as Differentiators . . . . . . . . . . . . . . . . . 467

10.14 FM Receivers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46810.14.1 Distortions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46810.14.2 Pre-emphasis and De-emphasis . . . . . . . . . . . . . . . . 46910.14.3 Distortions Caused by Multipath Effect . . . . . . . . . 470

10.15 Frequency-Division Multiplexing (FDM) . . . . . . . . . . . . . . . 47110.15.1 Quadrature Amplitude Modulation (QAM)

or Quadrature Multiplexing (QM). . . . . . . . . . . . . . 47210.15.2 FM Stereo Multiplexing and the FM Radio . . . . . . 473

10.16 Pulse Modulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47410.16.1 Pulse Amplitude Modulation (PAM) . . . . . . . . . . . 47510.16.2 Problems with Pulse Modulations . . . . . . . . . . . . . . 47510.16.3 Time-Division Multiplexing (TDM) . . . . . . . . . . . . 477

10.17 Pulse Code Modulation (PCM) . . . . . . . . . . . . . . . . . . . . . . . 47810.17.1 Quantization Process . . . . . . . . . . . . . . . . . . . . . . . . 47810.17.2 More on Coding. . . . . . . . . . . . . . . . . . . . . . . . . . . . 480

xxiv Contents

10.17.3 Tradeoffs Between Channel Bandwidth and

Signal-to-Quantization Noise Ratio . . . . . . . . . . . . 48110.17.4 Digital Carrier Modulation . . . . . . . . . . . . . . . . . . . 482

10.18 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484

Appendix A: Matrix Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489

A.1 Matrix Notations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489

A.2 Elements of Matrix Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . 490

A.2.1 Vector Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491

A.3 Solutions of Matrix Equations . . . . . . . . . . . . . . . . . . . . . . . . . 492

A.3.1 Determinants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492

A.3.2 Cramers Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493

A.3.3 Rank of a Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494

A.4 Inverses of Matrices and Their Use in Determining

the Solutions of a Set of Equations . . . . . . . . . . . . . . . . . . . . . . 495

A.5 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . . . 496

A.6 Singular Value Decomposition (SVD) . . . . . . . . . . . . . . . . . . . 500

A.7 Generalized Inverses of Matrices . . . . . . . . . . . . . . . . . . . . . . . 501

A.8 Over- and Underdetermined System of Equations . . . . . . . . . . 502

A.8.1 Least-Squares Solutions of Overdetermined

System of Equations (m > n) . . . . . . . . . . . . . . . . . . . . 502A.8.2 Least-Squares Solution of Underdetermined

System of Equations (m n) . . . . . . . . . . . . . . . . . . . . 504A.9 Numerical-Based Interpolations: Polynomial and Lagrange

Interpolations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505A.9.1 Polynomial Approximations . . . . . . . . . . . . . . . . . . . . . . . . . .505A.9.2 Lagrange Interpolation Formula . . . . . . . . . . . . . . . . . . . . . .506

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506

Appendix B: MATLAB1 for Digital Signal Processing . . . . . . . . . . . . . . . 509

B.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509

B.2 Signal Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509

B.3 Signal Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511

B.4 Fast Fourier Transforms (FFTs) . . . . . . . . . . . . . . . . . . . . . . . 511

B.5 Convolution of Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513

B.6 Differentiation Using Numerical Methods . . . . . . . . . . . . . . . 515

B.7 Fourier Series Computation . . . . . . . . . . . . . . . . . . . . . . . . . . 515

B.8 Roots of Polynomials, Partial Fraction Expansions,

PoleZero Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517B.8.1 Partial Fraction Expansions . . . . . . . . . . . . . . . . . . . . 518

B.9 Bode Plots, Impulse and Step Responses . . . . . . . . . . . . . . . . 518

B.9.1 Bode Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 518

B.9.2 Impulse and Step Responses . . . . . . . . . . . . . . . . . . . . 518

B.10 Frequency Responses of Digital Filter Transfer

Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 520

B.11 Introduction to the Construction of Simple MATLAB

Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 520

B.12 Additional MATLAB Code. . . . . . . . . . . . . . . . . . . . . . . . . . . 521

Contents xxv

Appendix C: Mathematical Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523

C.1 Trigonometric Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523

C.2 Logarithms, Exponents and Complex Numbers . . . . . . . . . . . . 523

C.3 Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524

C.4 Indefinite Integrals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524

C.5 Definite Integrals and Useful Identities. . . . . . . . . . . . . . . . . . . 525

C.6 Summation Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525

C.7 Series Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526

C.8 Special Constants and Factorials . . . . . . . . . . . . . . . . . . . . . . . 526

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527

Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 531

Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535

xxvi Contents

List of Tables

Table 1.4.1 Properties of the impulse function. . . . . . . . . . . . . . . . . . . 18

Table 1.9.1 Sound Power (loudness) Comparison . . . . . . . . . . . . . . . . 33

Table 1.9.2 Power ratios and their corresponding values in dB. . . . . . 33

Table 2.4.1 Properties of aperiodic convolution . . . . . . . . . . . . . . . . . 53

Table 2.6.1 Example 2.6.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

Table 3.4.1 Summary of the three Fourier series representations . . . . 84

Table 3.10.1 Symmetries of real periodic functions and their

Fourier-series coefficients . . . . . . . . . . . . . . . . . . . . . . . . . 105

Table 3.10.2 Periodic functions and their Trigonometric Fourier Series . . 105

Table 4.12.1 Fourier transform properties. . . . . . . . . . . . . . . . . . . . . . . 148

Table 4.12.2 Fourier Transform Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . 149

Table 5.6.1 One-sided Laplace transform properties . . . . . . . . . . . . . . 168

Table 5.6.2 One-sided Laplace tranform pairs . . . . . . . . . . . . . . . . . . . 175

Table 5.8.1 Typical rational replace transforms and their inverses . . . 182

Table 5.9.1 One sided Laplace transforms and Fourier transforms. . . 185

Table 5.10.1 Hilbert transform pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

Table 7.1.1 Formula for computing sensitivities . . . . . . . . . . . . . . . . . 244

Table 7.4.1 Normalized frequencies, ! !0. Time delay and a losstable giving the normalized frequency ! at which the

zero frequency delay and loss values deviate by specified

amounts for Bessel filter functions . . . . . . . . . . . . . . . . . . 265

Table 7.5.1 Frequency transformations . . . . . . . . . . . . . . . . . . . . . . . . 269

Table 7.7.1 Guidelines for passive components . . . . . . . . . . . . . . . . . . 283

Table 8.1.1 Fourier representations of discrete-time and

continuous-time signals . . . . . . . . . . . . . . . . . . . . . . . . . . . 312

Table 8.2.1 Common interpolation functions . . . . . . . . . . . . . . . . . . . 319

Table 8.2.2 Spectral occupancy of Xj! n!s; ! 2f;n 0;1;2;3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325

Table 8.3.1 Properties of discrete convolution . . . . . . . . . . . . . . . . . . . 329

Table 8.7.1 Discrete-time Fourier transform (DTFT) properties . . . . 347

Table 8.7.2 Discrete-time Fourier transform (DTFT) pairs. . . . . . . . . 348

Table 8.10.1 Discrete Fourier transform (DFT) properties . . . . . . . . . . 36

Table 9.1.1 Discrete-time and continuous-time signals and their

transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367

xxvii

1

Table 9.2.1 Properties of the functionWN ej2=N. . . . . . . . . . . . . 369Table 9.6.1 Z-transform properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 389

Table 9.6.2 Z-transform pairs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390

Table 9.11.1 Ideal low-pass filter FIR coefficients with c =4 . . . . . 412Table 9.11.2 FIR Filter Coefficients for the Four Basic Filters. . . . . . . 415

Table 10.9.1 Inputs and outputs of the system in Fig. 10.9.1 . . . . . . . . 453

Table 10.12.1 Bessel function values . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463

Table 10.17.1 Quantization values and codes corresponding to

Fig. 10.17.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479

Table 10.17.2 Binary representation of quantized values . . . . . . . . . . . . 480

Table 10.17.3 Normal binary and Gray code representations

for N8.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481Table B.7.1 Amplitudes and phase angles of the harmonic Fourier

series coefficients (Example B.7.1). . . . . . . . . . . . . . . . . . . 516

xxviii List of Tables

Chapter 1

Basic Concepts in Signals

1.1 Introduction to the Book and Signals

The primary goal of this book is to introduce the

reader on the basic principles of signals and to

provide tools thereby to deal with the analysis of

analog and digital signals, either obtained natu-

rally or by sampling analog signals, study the

concepts of various transforming techniques, fil-

tering analog and digital signals, and finally

introduce the concepts of communicating analog

signals using simple modulation techniques. The

basic material in this book can be found in several

books. See references at the end of the book.

A signal is a pattern of some kind used to convey

a message. Examples include smoke signals, a set of

flags, traffic lights, speech, image, seismic signals,

and many others. Smoke signals were used for con-

veying information that goes back before recorded

history. Greeks and Romans used light beacons in

the pre-Christian era. England employed a long

chain of beacons to warn that Spanish Armada is

approaching in the late sixteenth century. Around

this time, the word signal came into use perceptible

by sight, hearing, etc., conveying information. The

present day signaling started with the invention of

the Morse code in 1838. Since then, a variety of

signals have been studied. These include the follow-

ing inventions: Facsimile by Alexander Bain in

1843; telephone by Alexander Bell in 1876; wireless

telegraph system by Gugliemo Marconi in 1897;

transmission of speech signals via radio by Reginald

Fessenden in 1905, invention and demonstration of

television, the birth of television by Vladimir Zwor-

ykin in the 1920 s, and many others. In addition, the

development of radar and television systems during

World War II, proposition of satellite communica-

tion systems, demonstration of a laser in 1955, and

the research and developments of many signal pro-

cessing techniques and their use in communication

systems. Since the early stages of communications,

research has exploded into several areas connected

directly, or indirectly, to signal analysis and com-

munications. Signal analysis has taken a signifi-

cant role in medicine, for example, monitoring

the heart beat, blood pressure and temperature of

a patient, and vital signs of patients. Others include

the study of weather phenomenon, the geological

formations below the surface and deep in the

ground and under the ocean floors for oil and gas

exploration, mapping the underground surface

using seismometers, and others. Researchers have

concluded that computers are powerful and neces-

sary that they need to be an integral part of any

communication system, thus generating significant

research in digital signal processing, development

of Internet, research on HDTV, mobile and cellu-

lar telephone systems, and others. Defense indus-

try has been one of the major organizations in

advancing research in signal processing, coding,

and transmission of data. Several research areas

have surfaced in signals that include processing of

speech, image, radar, seismic, medical, and other

signals.

1.1.1 Different Ways of Lookingat a Signal

Consider a signal xt, a function representing aphysical quantity, such as voltage, current, pres-

sure, or any other variable with respect to a second

variable t, such as time. The terms of interest are the

time t and the signal xt. One of the main topics of

R.K.R. Yarlagadda, Analog and Digital Signals and Systems, DOI 10.1007/978-1-4419-0034-0_1, Springer ScienceBusiness Media, LLC 2010

1

this book is the analysis of signals. Websters dic-

tionary defines the analysis as

1. Separation of a thing into the parts or elements

of which it is composed.

2. An examination of a thing to determine its parts

or elements.

3. A statement showing the results of such an

examination.

There are other definitions. In the following the

three parts are considered using simple examples.

Consider the sinusoidal function and its expansion

using Eulers formula:

xt A0 coso0t y0

A02ejy0

ejo0t A0

2ejy0

ejo0t

ReA0ejo0tejy0:

(1:1:1)

In (1.1.1) A0 is assumed to be positive and real and

A0ejy0 is a complex number carrying the amplitude

and phase angle of the sinusoidal function and is by

definition the phasor representation of the given

sinusoidal function. Some authors refer to this as

phasor transform of the sinusoidal signal, as it trans-

forms the time domain sinusoidal function to the

complex frequency domain. A brief discussion on

complex numbers is included later in Section 1.6.

This signal can be described in another domain, i.e.,

such as the frequency domain. The amplitude is

A0=2 and the phase angles of y0 correspondingto the frequencies f0 o0=2p Hz. In reality,only positive frequencies are available, but Eulers

formula in (1.1.1) dictates that both the positive and

negative frequencies need to be identified as illu-

strated in Fig. 1.1.1a. This description is the two-

sided amplitude and phase line spectra of xt.Amplitudes are always positive and are located at

f o0=2p f0 Hz, symmetrically locatedaround the zero frequency, i.e., with even symme-

try. The phase spectrum consists of two angles

y y0 corresponding to the positive and negativefrequencies, respectively, with odd symmetry. Since

t is real, we can pictorially describe it by one- ortwo-sided amplitude and phase line spectra as shown

in Fig. 1.1.1a,b,c,d. The following example illus-

trates the three steps.

Example 1.1.1 Express the following function in

terms of a sum of cosine functions:

xt A0 A1 coso1t y1

A2 coso2t A3 sino3t y3;Ai > 0:(1:1:2)

Solution: Using trigonometric relations to express

each term in (1.1.2) in the form of Ai cosoit yiresults in

xt A0 cos0t 180

A1 coso1t y1 A2 coso2t 180

A3 coso3t y3 90:

(1:1:3)

In the first and the third terms either 1808 or 1808could be used, as the end result is the same. The

two-sided line spectra of the function in (1.1.2) are

shown in Fig. 1.1.2. How would one get the func-

tions of the type shown in (1.1.3) for an arbitrary

Fig. 1.1.1 xt A0 coso0t y0. (a) Two-sided amplitudespectrum, (b) two-sided phase spectrum, (c) one-sided ampli-tude spectrum, and (d) one-sided phase spectrum

2 1 Basic Concepts in Signals

function? The sign and cosine functions are the

building blocks of the Fourier series in Chapter 3

and later the Fourier transforms in Chapter 4. The

function xt has four frequencies:

f10; f2; f3; f4 with amplitudes A0;A1;A2;A3and phases 180o; y1;180o; y3 90o:

Figure 1.1.2 illustrates pictorially the discrete loca-

tions of the frequencies, their amplitudes, and

phases. The signal in (1.1.2) can be described by

using the time domain function or in terms of fre-

quencies. In the figures, o0 2pf 0s in radians persecond could have been used rather than f 0s in Hz.&

1.1.2 Continuous-Time and Discrete-TimeSignals

A signal xt is a continuous-time signal if t is acontinuous variable. It can take on any value in

the continuous interval a; b. Continuous-time sig-nal is an analog signal. If a function yn is defined atdiscrete times, then it is a discrete-time signal, where

n takes integer values. In Chapter 8 discrete-time

signals will be studied by sampling the continuous

signals at equal sampling intervals of ts seconds and

write xnts;where n an integer. This is expressed by

xn xnts: (1:1:4)

Example(s) 1.1.2 In this example several specific

examples of interest are considered. In the first one,

part of the time signal illustrating a male voice of

speech in the sentence . . .Show the rich lady out is

shown in Fig. 1.1.3. The speech signal is sampled at

8000 samples per second. There are three portions

of the speech /. . ./, /sh/, /o/ shown in the figure.

The first part of the signal does not have any speech

in it and the small amplitudes of the signal represent

the noise in the tape recorder and/or in the room

where the speech was recorded. It represents a ran-

dom signal and can be described only by statistical

means. Random signal analysis is not discussed in

any detail in this book, as it requires knowledge of

probability theory. The second part represents the

phoneme sh that does not show any observable

pattern. It is a time signal for a very short time and

has finite energy. Power and energy signals are stu-

died in Section 1.5. Third part of the figure repre-

sents the vowel o, showing a structure of (almost)

periodic pulses for a short time. In this book, aper-

iodic or non-periodic signals with finite energy and

periodic signals with finite average power will be

studied. One goal is to come up with a model for

each portion of a signal that can be transmitted and

reconstructed at the receiver.

Next three examples are from food industry.

Small businesses are sprouting that use signal pro-

cessing. For example, when we go to a grocery store

we may like to buy a watermelon. It may not always

be possible to judge the ripeness of the watermelon

Fig. 1.1.3 Speech . . .sho in . . .show ._male 2000 Samples @8000 samples per second. Printed with the permission fromHassan et al. (1994)

Fig. 1.1.2 (a) Two-sided amplitude spectra and (b) two-sided phase spectra

1.1 Introduction to the Book and Signals 3

by outward characteristics such as external color,

stem conditions, or just the way it looks. A sure way

of looking at the quality is to cut the watermelon

open and taste it before we buy it. This implies we

break it first, which is destructive testing. Instead,

we can use our grandmothers procedure in select-

ing a watermelon. She uses her knuckles to send a

signal into the watermelon. From the audio

response of the watermelon she decides whether it

is good or not based on her prior experience.We can

simulate this by putting the watermelon on a stand,

use a small hammer like device, give a slight tap on

the watermelon, and record the response. A simplis-

tic model of this is shown in Fig. 1.1.4. The

responses can be categorized by studying the out-

puts of tasty watermelons. For an interesting

research work on this topic, see Stone et al (1996).

Image processing can be used to check for

burned crusts, topping amount distribution, such

as the location of pepperoni pizza slices, and others.

For an interesting article on this subject, seeWagner

(1983), which has several applications in the food

industry.

The next two examples are from the surface seis-

mic signal analysis. In the first one, we use a source

in the form of dynamite sticks representing a source,

dig a small hole, and blow them in the hole. The

ground responds to this input and the response is

recorded using a seismometer and a tape recorder.

The analysis of the recorded waveform can provide

information about the underground cavities and

pockets of oil and other important measures.

Geologists drill holes into the ground and a small

slice of the core sample is used to measure the oil

content by looking at the percentage of the area

with dark spots on the slice, which is image

processing.

Another example of interest is measuring the

distance from a ground station to an airplane.

Send a signal with square wave pulses toward the

airplane and when the signal hits it, a return signal is

received at the ground station. A simple model is

shown in Fig. 1.1.5. If we can measure the time

between the time the signal left from the ground

station and the time it returned, identified as T in

the figure, we can determine the distance between

the ground station and the target by the formula

x 3108 m=s

Tsignal round trip time in seconds=2:(1:1:5)

The constant c 3108 m/s is the speed of light.Radar and sonar signal processing are two impor-

tant areas of signal processing applications.

An exciting field of study is the biomedical

area. We are well aware of a healthy heart that

beats periodically, which can be seen from a record

of an electrocardiogram (ECG). The ECG

represents changes in the voltage potential due

(a)

(b)

Fig. 1.1.5 (a) Radar range measurement and (b) transmittedand received filtered signals

Fig. 1.1.4 Watermelon responses to a tap

4 1 Basic Concepts in Signals

to electrochemical processes in the heart cells.

Inferences can be made about the health of the

heart under observation from the ECG. Another

important example is the electroencephalogram

(EEG), which measures the electrical activity in

brain. &

Signal processing is an important area that inter-

ests every engineer. Pattern recognition and classifi-

cation is almost on top of the list. See, for example,

OShaughnessy (1987) and Tou and Gonzalez

(1974). For example, how do we distinguish two

phonemes, one is a vowel and the other one is a

consonant. A rough measure of frequency of a

waveform with zero average value is the number of

zero crossings per unit time. We will study in much

more detail the frequency content in a signal later in

terms of Fourier transforms in Chapter 4. Vowel

sounds have lower frequency content than the con-

sonants. A simple procedure to measure frequency

in a speech segment is by computing the number of

zero crossings in that segment. To differentiate a

vowel from a consonant, set a threshold level for

the frequency content for vowels and consonants

that differentiate between vowels and consonants.

If the frequency content is higher than this thresh-

old, then the phoneme is a consonant. Otherwise, it

is a vowel. If we like to distinguish one vowel from

another wemay needmore than onemeasure. Vocal

tract can be modeled as an acoustic tube with reso-

nances, called formants. Two formant frequencies

can be used to distinguish two vowels, say /u/ and /

a/. SeeProblem 1.1.1. Two formant frequencies may

not be enough to distinguish all the phonemes, espe-

cially if the signal is corrupted by noise.

Consider a simple pattern classification problem

with M prototype patterns z1; z2; . . . ; zM, where ziis a vector representing an ith pattern. For simpli-

city we assume that each pattern can be represented

by a pair of numbers, say zi zi1; zi2 ; i=1,2 . . .Mand classify an arbitrary pattern x x1; x2 torepresent one of the prototype patterns. The Eucli-

dean distance between a pattern x and the ith pro-

totype pattern is defined by

Di x zik k ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix1 zi1 2 x2 zi2 2

q: (1:1:6)

A simple classifier is a minimum distance classifier

that computes the distance from a pattern x of the

signal to be classified to the prototype of each class

and assigns the unknown pattern to the class which it

is closest to. That is, ifDi5Dj; for all i 6 j, then wemake the decision that x belongs to the ith prototype

pattern. Ties are rare and if there are, they are

resolved arbitrarily. In the above discussion two

measures are assumed for each pattern. More mea-

sures give a better separation between classes.

There are several issues that would interest a

biomedical signal processor. These include removal

of any noise present in the signals, such as 60-Hz

interference picked up by the instruments, interference

of the tools or meters that measure a parameter, and

other signals that interfere with the desired signal.

Finding the important facets in a signal, such as the

frequency content, and many others is of interest. &

1.1.3 Analog Versus Digital SignalProcessing

Most signals are analog signals. Analog signal pro-

cessing uses analog circuit elements, such as resistors,

capacitors, inductors, and active components, such

as operational amplifiers and non-linear devices.

Since the inductors are made from magnetic mate-

rial, they have inherent resistance and capacitance.

This brings the quality of the components low. They

tend to be bulky and their effectiveness is reduced. To

alleviate this problem, activeRC networks have been

popular. Analog processing is a natural way to solve

differential equations that describe physical systems,

without having to resort to approximate solutions.

Solutions are obtained in real time. In Chapter 10 we

will see an example of analog encryption of a signal,

wherein the analog speech is scrambled by the use of

modulation techniques.

Digital signal processing makes use of a special

purpose computer, which has three basic elements,

namely adders, multipliers, and memory for sto-

rage. Digital signal processing consists of numerical

computations and there is no guarantee that the

processing can be done in real time. To encrypt a

set of numbers, these need to be converted into

another set of numbers in the digital encryption

scheme, for example. The complete encrypted signal

is needed before it can be decrypted. In addition, if

the input and the output signals are analog, then an

1.1 Introduction to the Book and Signals 5

analog-to-digital converter (A/D), a digital proces-

sor, and a digital-to-analog converter (D/A) are

needed to implement analog processing by digital

means. Special purpose processor with A/D and

D/A converters can be expensive.

Digital approach has distinct advantages over

analog approaches. Digital processor can be used

to implement different versions of a system by chan-

ging the software on the processor. It has flexibility

and repeatability. In the analog case, the system has

to be redesigned every time the specifications are

changed. Design components may not be available

and may have to live with the component values

within some tolerance. Components suffer from

parameter variations due to room temperature,

humidity, supply voltages, and many other aspects,

such as aging, component failure. In a particular

situation, many of the above problems need to be

investigated before a complete decision can be

made. Future appears to be more and more digital.

Many of the digital signal processing filter designs

are based on using analog filter designs. Learning

both analog and digital signal processing is desired.

Deterministic and random signals: Deterministic

signals are specified for any given time. They can be

modeled by a known function of time. There is no

uncertainty with respect to any value at any time.

For example, xt sint is a deterministic signal.A random signal yt xt nt can take ran-dom values at any given time, as there is uncertainty

about the noise signal nt. We can only describesuch signals through statistical means, and the dis-

cussion on this topic will be minimal.

1.1.4 Examples of Simple Functions

To begin the study we need to look into the concept

of expressing a signal in terms of functions that can

be generated in a laboratory. One such function is

the sinusoidal function xt A0 coso0t y0 seenearlier in (1.1.1), where A0;o0; and y0 are someconstants. A digital signal can be defined as a

sequence in the forms

xn an; n 00; n50

; (1:1:7)

fxng f. . . ; 0; 0; 0; 1; a; a2; . . . ; an; . . .g; (1:1:8)

fxng f. . . ; 1#; a; a2; . . . ; an; . . .g or fxng

f1#; a; a2; . . . ; an; . . .g

(1:1:9)

In (1.1.8) reference points are not identified. In

(1.1.9), the arrow below 1 is the 0 index term. The

first term is assumed to be zero index term if there is

no arrow and all the values of the sequence are zero

for n50. We will come back to this in Chapter 8.A signal xt is a real signal if its value at some t

is a real number. A complex signal xt consists oftwo real signals, x1t and x2t such that x(t)= x_1(t) + jx2t; where j

ffiffiffiffiffiffiffiffi1:p

The symbol j (or i) is

used to represent the imaginary part.

Interesting functions: a. P Function: The P func-tion is centered at t0 with a width of t s shown inFig. 1.1.6. It is not defined at t t0 t=2 and issymbolically expressed by

Y t t0t

h i 1; t t0j j5t=20; otherwise

(1:1:10)

xtP t t0t

h i

xt; t t0j j5t=20; otherwise

This P function is a deterministic signal. It is even, asPt Pt andPt0 t Pt t0. Some use thesymbol rectand1=trectt t0=t isarectangularpulse ofwidth t s centered at t t0 with aheight 1=t:

b. L Function: The triangular function shown inFig. 1.1.7 is defined by

Lt t0t

h i 1

jtt0jt ; jt t0j5t

0; otherwise:

((1:1:11)

0t t

Fig. 1.1.6 A P function

6 1 Basic Concepts in Signals

Rectangular function defined in (1.1.10) has a width

of t seconds, whereas the triangular functiondefined in (1.1.11) has a width of 2t s. The symboltri is also used for a triangular function and

trit t0=t describes the function in (1.1.11).

c. Unit step function: It is shown in Fig. 1.1.8 and is

u t 1; t40

0; t50

not defined at t 0 : (1:1:12)

The unit step function at t 0 can be defined expli-citly as 0 or 1 or u0 u0=2=.5

d. Exponential decaying function: A simple such

function is

x t X0et=Tc ; t 0;Tc40

0; otherwise:

((1:1:13)

See Fig. 1.1.9. It has a special significance, as xt isthe solution of a first-order differential equation.

The constantsX0 and Tc can take different values and

xTcX0

e1 :37 or xTc :37X0: (1:1:14)

xt decreases to about 37% of its initial value in Tcs and is the time constant. It decreases to 2% in four

time constants and X4Tc :018X0: A measureassociated with exponential functions is the half

life Th defined by

xTh 1

2X0; e

Th=Tc

12; Th Tc loge2 ffi :693Tc:

(1:1:15)

e. One-sided and two-sided exponentials: These are

described by

x1t eat; t 00; t50

; a40

x2t 0; t 0eat; t50

a40; x3t eajtj; a40:

(1:1:16)

x1t is the right-sided exponential, x2t is the left-sided exponential, and x3t is the two-sided expo-nential. These are sketched in Fig. 1.1.10. Using the

unit step function, we have x2t x3tut andx1t x3tut.

Fig. 1.1.8 Unit step function

/0( ) ( ), 0

ct Tx t X e u t = >

Fig. 1.1.9 Exponential decaying function

0t t

Fig. 1.1.7 A L function

2 ( )x t 1( )x t 3 ( )x t

(a) (b) (c)

Fig. 1.1.10 Exponentialfunctions (a) x2 (t) = e

at

u(t),a > 0,(b) x1t eatut; a > 0;and (c) x3t ea tj j; a > 0

1.1 Introduction to the Book and Signals 7

1.2 Useful Signal Operations

1.2.1 Time Shifting

Consider an arbitrary signal starting at t 0shown in Fig. 1.2.1a. It can be shifted to the

right as shown in Fig. 1.2.1b. It starts at time

t a > 0, a delayed version of the one inFig. 1.2.1a. Similarly it can be shifted to the left

starting at time a shown in Fig. 1.2.1c. It is anadvanced version of the one in Fig. 1.2.1a. We

now have three functions: xt, xt a; andxt a with a > 0. The delayed and advancedunit step functions are

u t a 1; t4a

0; t5a

;

ut a 1; t4 a0; t5 a

; a40: (1:2:1)

From (1.1.16), the right-sided delayed exponential

decaying function is

x1t t eattut t; a > 0; t > 0: (1:2:2)

1.2.2 Time Scaling

The compression or expansion of a signal in time

is known as time scaling. It is expanded in time if

a5 1 and compressed in time if a > 1 in

f t x at ; a > 0: (1:2:3)

Example 1.2.1 Illustrate the rectangular pulse func-

tions Pt;P 2t ; and P t=2 .

Solution: These are shown in Fig. 1.2.2 and are of

widths 1, (1/2), and 2, respectively. The pulseP 2t is a compressed version and the pulse Pt=2 is anexpanded version of the pulse function P t . &

1.2.3 Time Reversal

If a 1 in (1.2.3), that is, ft xt, then thesignal is time reversed (or folded).

Example 1.2.2 Let x1t eatut: Give its time-reversed signal.

Solution: The time-reversed signal of x1t isx2t eatut. &

1.2.4 Amplitude Shift

The amplitude shift of xt by a constant K isft K xt.

Combined operations: Some of the above signal

operations can be combined into a general form.

The signal yt xat t0 may be described byone of the two ways, namely:

[t] [2t] [t /2]

(a) (b) (c)Fig. 1.2.2 Pulse functions

x(t) x(t a) x(t + a)

(a) (b) (c)

Fig. 1.2.1 (a) x(t), (b) x(t a), (c) x(t + a), a > 0

8 1 Basic Concepts in Signals

1. Time shift of t0 followed by time scaling by a.

2. Time scaling by (a) followed by time shift of t0=a.

These can be visualized by the following:

1: x t shiftt! t t0

! v t x t t0 scale

t! at! y t

v at x at t0 :(1:2:4)

2: x t scalet! at! g t x at

shift

t! t t0=a ! y t

g t t0=a x at t0 :(1:2:5)

Notation can be simplified by writing

yt xat t0=a. Noting that b at t0 is alinear equation in terms of two constants a and t0, it

follows:

y0 xt0 and yt0=a x0: (1:2:6)

These two equations provide checks to verify the

end result of the transformation. Following exam-

ple illustrates some pitfalls in the order of time

shifting and time scaling.

Example 1.2.3 Derive the expression for

yt x3t 2 assuming xt Pt=2.

Solution: Using (1.2.4) with a 3 and t0 2, wehave

vt xt t0 Pt 22

;

yt v3t P 3t 22

P t 2=3

2=3

:

Using (1.2.5), we have

gt xat P 3t2

;

yt gt t0a gt 2

3

P 3t 2=32

P t 2=3

2=3

:

It is a rectangular pulse of unit amplitude centered

at t 2=3 with width (2/3). We can check the

equations in (1.2.6) and y0 x3 0 andyt0=a y3=2 0. &

1.2.5 Simple Symmetries: Even and OddFunctions

Continuous-time even and odd functions satisfy

xt xt xet, an even function, x ( t) =xt x0t, an odd function. (1.2.7)

Examples of even and odd functions are shown in

Fig. 1.2.3. The function coso0t is an even functionand x0t sino0t is an odd function. An arbi-trary real signal, xt; can be expressed in terms ofits even and odd parts by

xt xet x0t; xet

xt xt=2 ; x0t

xt xt=2 :

(1:2:8)

1.2.6 Products of Even and Odd Functions

Let xet and yet be two even functions and x0tand y0t be two odd functions and arbitrary. Somegeneral comments can bemade about their products.

xetyet xetyet; even function: (1:2:9)

xety0t xety0t; odd function: (1:2:10)

x0ty0t 12x0ty0t

x0ty0t; even function:(1:2:11)

xe(t) x0(t)

(a) (b)

Fig. 1.2.3 (a) Even function and (b) odd function

1.2 Useful Signal Operations 9

Note that the functionsP[t],L[t] andP[t]coso0t areeven functions and Pt sino0t is an odd function.The even and odd parts of the exponential pulse

x1t etut are shown in Fig. 1.2.4 and are

x1et 1

2x1t x1t;

x10t 1

2x1t x1t:

(1:2:12)

1.2.7 Signum (or sgn) Function

The signum (or sgn) function is an odd function

shown in Fig. 1.2.5:

sgnt ut ut 2ut 1: (1:2:13)

sgnt lima!0eatut eatut; a > 0: (1:2:14)

It is not defined at t 0 and is chosen as 0.

1.2.8 Sinc and Sinc2 Functions

The sinc and sinc2 functions are defined in terms of

an independent variable l by

sincpl sinplpl

; sinc2pl sin2plpl2

(1:2:15)

Some authors use sincl for sincpl in (1.2.15).Notation in (1.2.15) is common. Sinc pl is inde-terminate at t 0: Using the LHospitals rule,

liml!0

sinplpl

liml!0

d sinpldl

dpldl

liml!0

p cosplp

1:

(1:2:16)

In addition, since sinpl is equal to zero forl n; n an integer, it follows that

sincpl 0; l n; n 6 0 and an integer:(1:2:17a)

Interestingly, the function sincplj j is bounded by1=plj j as sinplj j 1. The side lobes ofsincplj j are larger than the side lobes ofsinc2pl, which follows from the fact that thesquare of a fraction is less than the fraction we

started with. Both the sinc and the sinc2 functions

are even. That is,

sinc pl sincpl and sinc2pl sinc2pl:(1:2:17b)

These functions can be evaluated easily by a calcu-

lator. For the sketch of a sinc function using

MATLAB, see Fig. B.5.2 in Appendix B.

1.2.9 Sine Integral Function

The sine integral function is an odd function defined

by (Spiegel, 1968)

sgn(t)

t

Fig. 1.2.5 Signum function sgnt

x1(t)

x2(t)

x3(t)

(a) (b) (c)

Fig. 1.2.4 (a) x1t, (b) xie(t)= x2 (t) even part of x1(t),and (c) x10t= x3 (t) oddpart of x1t

10 1 Basic Concepts in Signals

Siy Zy

0

sinaa

da: (1:2:18a)

The values of this function can be computed

numerically using the series expression

Siy y11!y3

33!y5

55!y7

77! . . .

(1:2:18b)

Some of its important properties are

Siy Siy;Si0 0;Sip ffi 2:0123;Si1 p=2 : (1:2:18c)

Si function converges fast and only a few terms in

(1.2.18b) are needed for a good approximation.

1.3 Derivatives and Integralsof Functions

It will be assumed that the reader is familiar with some

of the basic properties associated with the derivative

and integral operations. We should caution that

derivatives of discontinuous functions do not exist

in the conventional sense. To handle such cases,

generalized functions are defined in the next section.

The thr