Signals and Systems - Elsevier · 2013-12-20 · Signals and Systems Using MATLABR Luis F. Chaparro Department of Electrical and Computer Engineering University of Pittsburgh AMSTERDAM

Signals and SystemsUsing MATLAB R

Luis F. ChaparroDepartment of Electrical and Computer Engineering

University of Pittsburgh

AMSTERDAM • BOSTON • HEIDELBERG • LONDONNEW YORK • OXFORD • PARIS • SAN DIEGO

SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYOAcademic Press is an imprint of Elsevier

Academic Press is an imprint of Elsevier30 Corporate Drive, Suite 400, Burlington, MA 01803, USAElsevier, The Boulevard, Langford Lane, Kidlington, Oxford, OX5 1GB, UK

Copyright c© 2011 Elsevier Inc. All rights reserved.

No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical,including photocopying, recording, or any information storage and retrieval system, without permission in writing fromthe publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and ourarrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can befound at our website: www.elsevier.com/permissions.

This book and the individual contributions contained in it are protected under copyright by the Publisher (other thanas may be noted herein). MATLAB R© is a trademark of The MathWorks, Inc. and is used with permission. The MathWorksdoes not warrant the accuracy of the text or exercises in this book. This books use or discussion of MATLAB R© softwareor related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogicalapproach or particular use of the MATLAB R© software.

NoticesKnowledge and best practice in this field are constantly changing. As new research and experience broaden ourunderstanding, changes in research methods, professional practices, or medical treatment may become necessary.

Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using anyinformation, methods, compounds, or experiments described herein. In using such information or methods theyshould be mindful of their own safety and the safety of others, including parties for whom they have aprofessional responsibility.

To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liabilityfor any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise,or from any use or operation of any methods, products, instructions, or ideas contained in the material herein.

Library of Congress Cataloging-in-Publication DataChaparro, Luis F.

Signals and systems using MATLAB R© / Luis F. Chaparro.p. cm.

ISBN 978-0-12-374716-71. Signal processing–Digital techniques. 2. System analysis. 3. MATLAB. I. Title.

TK5102.9.C472 2010621.382’2–dc22

2010023436

British Library Cataloguing-in-Publication DataA catalogue record for this book is available from the British Library.

For information on all Academic Press publicationsvisit our Web site at www.elsevierdirect.com

Printed in the United States of America10 11 12 13 9 8 7 6 5 4 3 2 1

To my family, with much love.

Contents

PREFACE..................................................................................................................... xi

ACKNOWLEDGMENTS ................................................................................................ xvi

Part 1 Introduction 1

CHAPTER 0 From the Ground Up! ............................................................................. 3

0.1 Signals and Systems and Digital Technologies........................................ 3

0.2 Examples of Signal Processing Applications ........................................... 5

0.2.1 Compact-Disc Player ................................................................ 50.2.2 Software-Defined Radio and Cognitive Radio............................... 60.2.3 Computer-Controlled Systems ................................................... 8

0.3 Analog or Discrete? ............................................................................. 9

0.3.1 Continuous-Time and Discrete-Time Representations .................. 100.3.2 Derivatives and Finite Differences ............................................. 120.3.3 Integrals and Summations......................................................... 130.3.4 Differential and Difference Equations ......................................... 16

0.4 Complex or Real? ................................................................................ 20

0.4.1 Complex Numbers and Vectors.................................................. 200.4.2 Functions of a Complex Variable................................................ 230.4.3 Phasors and Sinusoidal Steady State .......................................... 240.4.4 Phasor Connection ................................................................... 26

0.5 Soft Introduction to MATLAB ............................................................... 29

0.5.1 Numerical Computations .......................................................... 300.5.2 Symbolic Computations ............................................................ 43Problems............................................................................................ 53

Part 2 Theory and Application of Continuous-TimeSignals and Systems 63

CHAPTER 1 Continuous-Time Signals ......................................................................... 65

1.1 Introduction ....................................................................................... 65

1.2 Classification of Time-Dependent Signals............................................... 66

iv

Contents v

1.3 Continuous-Time Signals ..................................................................... 67

1.3.1 Basic Signal Operations—Time Shifting and Reversal ................... 711.3.2 Even and Odd Signals .............................................................. 751.3.3 Periodic and Aperiodic Signals .................................................. 771.3.4 Finite-Energy and Finite Power Signals ...................................... 79

1.4 Representation Using Basic Signals....................................................... 85

1.4.1 Complex Exponentials .............................................................. 851.4.2 Unit-Step, Unit-Impulse, and Ramp Signals ................................. 881.4.3 Special Signals—the Sampling Signal and the Sinc ....................... 1001.4.4 Basic Signal Operations—Time Scaling, Frequency Shifting,

and Windowing ....................................................................... 1021.4.5 Generic Representation of Signals.............................................. 105

1.5 What Have We Accomplished? Where Do We Go from Here?.................... 106

Problems............................................................................................ 108

CHAPTER 2 Continuous-Time Systems ....................................................................... 117

2.1 Introduction ....................................................................................... 117

2.2 System Concept .................................................................................. 118

2.2.1 System Classification................................................................ 1182.3 LTI Continuous-Time Systems .............................................................. 119

2.3.1 Linearity ................................................................................. 1202.3.2 Time Invariance....................................................................... 1252.3.3 Representation of Systems by Differential Equations.................... 1302.3.4 Application of Superposition and Time Invariance ....................... 1352.3.5 Convolution Integral................................................................. 1362.3.6 Causality ................................................................................ 1432.3.7 Graphical Computation of Convolution Integral ........................... 1452.3.8 Interconnection of Systems—Block Diagrams .............................. 1472.3.9 Bounded-Input Bounded-Output Stability ................................... 153


Problems............................................................................................ 157

CHAPTER 3 The Laplace Transform ............................................................................ 165

3.1 Introduction ....................................................................................... 165

3.2 The Two-Sided Laplace Transform ........................................................ 166

3.2.1 Eigenfunctions of LTI Systems................................................... 1673.2.2 Poles and Zeros and Region of Convergence ............................... 172

3.3 The One-Sided Laplace Transform ........................................................ 176

3.3.1 Linearity ................................................................................. 1853.3.2 Differentiation ......................................................................... 1883.3.3 Integration .............................................................................. 1933.3.4 Time Shifting........................................................................... 1943.3.5 Convolution Integral................................................................. 196

vi Contents

3.4 Inverse Laplace Transform ................................................................... 197

3.4.1 Inverse of One-Sided Laplace Transforms ................................... 1973.4.2 Inverse of Functions Containing e−ρs Terms ................................ 2093.4.3 Inverse of Two-Sided Laplace Transforms................................... 212

3.5 Analysis of LTI Systems ....................................................................... 214

3.5.1 LTI Systems Represented by Ordinary Differential Equations ........ 2143.5.2 Computation of the Convolution Integral .................................... 221


Problems............................................................................................ 226

CHAPTER 4 Frequency Analysis: The Fourier Series .................................................. 237

4.1 Introduction ....................................................................................... 237

4.2 Eigenfunctions Revisited ..................................................................... 238

4.3 Complex Exponential Fourier Series ...................................................... 245

4.4 Line Spectra ....................................................................................... 248

4.4.1 Parseval’s Theorem—Power Distribution over Frequency ............. 2484.4.2 Symmetry of Line Spectra ......................................................... 250

4.5 Trigonometric Fourier Series ................................................................ 251

4.6 Fourier Coefficients from Laplace.......................................................... 255

4.7 Convergence of the Fourier Series......................................................... 265

4.8 Time and Frequency Shifting................................................................ 270

4.9 Response of LTI Systems to Periodic Signals........................................... 273

4.9.1 Sinusoidal Steady State............................................................. 2744.9.2 Filtering of Periodic Signals ....................................................... 276

4.10 Other Properties of the Fourier Series .................................................... 279

4.10.1 Reflection and Even and Odd Periodic Signals ............................. 2794.10.2 Linearity of Fourier Series—Addition of Periodic Signals ............... 2824.10.3 Multiplication of Periodic Signals ............................................... 2844.10.4 Derivatives and Integrals of Periodic Signals ............................... 285


Problems............................................................................................ 290

CHAPTER 5 Frequency Analysis: The Fourier Transform ........................................... 299

5.1 Introduction ....................................................................................... 299

5.2 From the Fourier Series to the Fourier Transform .................................... 300

5.3 Existence of the Fourier Transform ....................................................... 302

5.4 Fourier Transforms from the Laplace Transform ..................................... 302

5.5 Linearity, Inverse Proportionality, and Duality ........................................ 304

5.5.1 Linearity ................................................................................. 3045.5.2 Inverse Proportionality of Time and Frequency............................ 3055.5.3 Duality ................................................................................... 310

Contents vii

5.6 Spectral Representation ....................................................................... 313

5.6.1 Signal Modulation .................................................................... 3135.6.2 Fourier Transform of Periodic Signals ......................................... 3175.6.3 Parseval’s Energy Conservation................................................. 3205.6.4 Symmetry of Spectral Representations........................................ 322

5.7 Convolution and Filtering..................................................................... 327

5.7.1 Basics of Filtering .................................................................... 3295.7.2 Ideal Filters ............................................................................. 3325.7.3 Frequency Response from Poles and Zeros.................................. 3375.7.4 Spectrum Analyzer................................................................... 341

5.8 Additional Properties .......................................................................... 344

5.8.1 Time Shifting .......................................................................... 3445.8.2 Differentiation and Integration .................................................. 346

5.9 What Have We Accomplished? What Is Next? ....................................... 350

Problems............................................................................................ 350

CHAPTER 6 Application to Control and Communications ........................................... 359

6.1 Introduction ....................................................................................... 359

6.2 System Connections and Block Diagrams ............................................... 360

6.3 Application to Classic Control............................................................... 363

6.3.1 Stability and Stabilization ......................................................... 3696.3.2 Transient Analysis of First- and Second-Order Control Systems ..... 371

6.4 Application to Communications ............................................................ 377

6.4.1 AM with Suppressed Carrier ..................................................... 3796.4.2 Commercial AM....................................................................... 3806.4.3 AM Single Sideband ................................................................. 3826.4.4 Quadrature AM and Frequency-Division Multiplexing .................. 3836.4.5 Angle Modulation .................................................................... 385

6.5 Analog Filtering.................................................................................. 390

6.5.1 Filtering Basics........................................................................ 3906.5.2 Butterworth Low-Pass Filter Design........................................... 3936.5.3 Chebyshev Low-Pass Filter Design ............................................ 3966.5.4 Frequency Transformations ...................................................... 4026.5.5 Filter Design with MATLAB ...................................................... 405

6.6 What Have We Accomplished? What Is Next? ........................................ 409

Problems............................................................................................ 409

Part 3 Theory and Application of Discrete-TimeSignals and Systems 417

CHAPTER 7 Sampling Theory ...................................................................................... 419

7.1 Introduction ....................................................................................... 419

viii Contents

7.2 Uniform Sampling ............................................................................... 420

7.2.1 Pulse Amplitude Modulation ..................................................... 4207.2.2 Ideal Impulse Sampling ............................................................ 4217.2.3 Reconstruction of the Original Continuous-Time Signal ................ 4287.2.4 Signal Reconstruction from Sinc Interpolation.............................. 4327.2.5 Sampling Simulation with MATLAB ........................................... 433

7.3 The Nyquist-Shannon Sampling Theorem .............................................. 437

7.3.1 Sampling of Modulated Signals .................................................. 4387.4 Practical Aspects of Sampling............................................................... 439

7.4.1 Sample-and-Hold Sampling ....................................................... 4397.4.2 Quantization and Coding .......................................................... 4417.4.3 Sampling, Quantizing, and Coding with MATLAB........................ 444


Problems............................................................................................ 447

CHAPTER 8 Discrete-Time Signals and Systems ......................................................... 451

8.1 Introduction ..................................................................................... 451

8.2 Discrete-Time Signals .......................................................................... 452

8.2.1 Periodic and Aperiodic Signals .................................................. 4548.2.2 Finite-Energy and Finite-Power Discrete-Time Signals ................. 4588.2.3 Even and Odd Signals .............................................................. 4618.2.4 Basic Discrete-Time Signals ...................................................... 465

8.3 Discrete-Time Systems ........................................................................ 478

8.3.1 Recursive and Nonrecursive Discrete-Time Systems..................... 4818.3.2 Discrete-Time Systems Represented by Difference

Equations ............................................................................... 4868.3.3 The Convolution Sum ............................................................... 4878.3.4 Linear and Nonlinear Filtering with MATLAB.............................. 4948.3.5 Causality and Stability of Discrete-Time Systems ......................... 497


Problems............................................................................................ 502

CHAPTER 9 The Z-Transform ...................................................................................... 511

9.1 Introduction ....................................................................................... 511

9.2 Laplace Transform of Sampled Signals................................................... 512

9.3 Two-Sided Z-Transform ....................................................................... 515

9.3.1 Region of Convergence............................................................. 5169.4 One-Sided Z-Transform........................................................................ 521

9.4.1 Computing the Z-Transform with Symbolic MATLAB ................... 5229.4.2 Signal Behavior and Poles ......................................................... 5229.4.3 Convolution Sum and Transfer Function ..................................... 526

Contents ix

9.4.4 Interconnection of Discrete-Time Systems................................... 5379.4.5 Initial and Final Value Properties ............................................... 539

9.5 One-Sided Z-Transform Inverse ............................................................ 542

9.5.1 Long-Division Method .............................................................. 5429.5.2 Partial Fraction Expansion ........................................................ 5449.5.3 Inverse Z-Transform with MATLAB............................................ 5479.5.4 Solution of Difference Equations ................................................ 5509.5.5 Inverse of Two-Sided Z-Transforms............................................ 561


Problems............................................................................................ 564

CHAPTER 10 Fourier Analysis of Discrete-Time Signals and Systems........................... 571

10.1 Introduction ....................................................................................... 571

10.2 Discrete-Time Fourier Transform .......................................................... 572

10.2.1 Sampling, Z-Transform, Eigenfunctions, and the DTFT ................. 57310.2.2 Duality in Time and Frequency .................................................. 57510.2.3 Computation of the DTFT Using MATLAB .................................. 57710.2.4 Time and Frequency Supports ................................................... 58010.2.5 Parseval’s Energy Result........................................................... 58510.2.6 Time and Frequency Shifts........................................................ 58710.2.7 Symmetry ............................................................................... 58910.2.8 Convolution Sum ..................................................................... 595

10.3 Fourier Series of Discrete-Time Periodic Signals...................................... 596

10.3.1 Complex Exponential Discrete Fourier Series .............................. 59910.3.2 Connection with the Z-Transform .............................................. 60110.3.3 DTFT of Periodic Signals ........................................................... 60210.3.4 Response of LTI Systems to Periodic Signals ............................... 60410.3.5 Circular Shifting and Periodic Convolution .................................. 607

10.4 Discrete Fourier Transform .................................................................. 614

10.4.1 DFT of Periodic Discrete-Time Signals ........................................ 61410.4.2 DFT of Aperiodic Discrete-Time Signals ...................................... 61610.4.3 Computation of the DFT via the FFT .......................................... 61710.4.4 Linear and Circular Convolution Sums ........................................ 622


Problems............................................................................................ 629

CHAPTER 11 Introduction to the Design of Discrete Filters .......................................... 639

11.1 Introduction ....................................................................................... 639

11.2 Frequency-Selective Discrete Filters...................................................... 641

11.2.1 Linear Phase ........................................................................... 64111.2.2 IIR and FIR Discrete Filters ....................................................... 643

x Contents

11.3 Filter Specifications ............................................................................. 648

11.3.1 Frequency-Domain Specifications .............................................. 64811.3.2 Time-Domain Specifications ...................................................... 652

11.4 IIR Filter Design.................................................................................. 653

11.4.1 Transformation Design of IIR Discrete Filters .............................. 65411.4.2 Design of Butterworth Low-Pass Discrete Filters ......................... 65811.4.3 Design of Chebyshev Low-Pass Discrete Filters ........................... 66611.4.4 Rational Frequency Transformations .......................................... 67211.4.5 General IIR Filter Design with MATLAB ..................................... 677

11.5 FIR Filter Design ................................................................................. 679

11.5.1 Window Design Method ........................................................... 68111.5.2 Window Functions ................................................................... 683

11.6 Realization of Discrete Filters ............................................................... 689

11.6.1 Realization of IIR Filters ............................................................ 69011.6.2 Realization of FIR Filters ........................................................... 699


Problems............................................................................................ 701

CHAPTER 12 Applications of Discrete-Time Signals and Systems................................. 709

12.1 Introduction ....................................................................................... 709

12.2 Application to Digital Signal Processing................................................. 710

12.2.1 Fast Fourier Transform ............................................................. 71112.2.2 Computation of the Inverse DFT ................................................ 71512.2.3 General Approach of FFT Algorithms ......................................... 716

12.3 Application to Sampled-Data and Digital Control Systems ........................ 722

12.3.1 Open-Loop Sampled-Data System .............................................. 72412.3.2 Closed-Loop Sampled-Data System ............................................ 726

12.4 Application to Digital Communications .................................................. 729

12.4.1 Pulse Code Modulation ............................................................. 73012.4.2 Time-Division Multiplexing ....................................................... 73312.4.3 Spread Spectrum and Orthogonal Frequency-Division

Multiplexing............................................................................ 73512.5 What Have We Accomplished? Where Do We Go from Here?.................... 742

APPENDIX Useful Formulas ....................................................................................... 743

BIBLIOGRAPHY ........................................................................................................... 746

INDEX ......................................................................................................................... 749

Preface

In this book I have only made up a bunchof other men’s flowers, providing of my own

only the string that ties them together.M. de Montaigne (1533–1592)

French essayist

Although it is hardly possible to keep up with advances in technology, it is reassuring to know that in scienceand engineering, development and innovation are possible through a solid understanding of basic principles.The theory of signals and systems is one of those fundamentals, and it will be the foundation of much researchand development in engineering for years to come. Not only engineers will need to know about signals andsystems—to some degree everybody will. The pervasiveness of computers, cell phones, digital recording, anddigital communications will require it.

Learning as well as teaching signals and systems is complicated by the combination of mathematical abstractionand concrete engineering applications. Mathematical sophistication and maturity in engineering are needed.Thus, a course in signals and systems needs to be designed to nurture the students’ interest in applications,but also to make them appreciate the significance of the mathematical tools. In writing this textbook, as inteaching this material for many years, the author has found it practical to follow Einstein’s recommendationthat “Everything should be made as simple as possible, but not simpler,” and Melzak’s [47] dictum that “It isdownright sinful to teach the abstract before the concrete.” The aim of this textbook is to serve the students’needs in learning signals and systems theory as well as to facilitate the teaching of the material for faculty byproposing an approach that the author has found effective in his own teaching.

We consider the use of MATLAB, an essential tool in the practice of engineering, of great significance in the learn-ing process. It not only helps to illustrate the theoretical results but makes students aware of the computationalissues that engineers face in implementing them. Some familiarity with MATLAB is beneficial but not required.

LEVELThe material in this textbook is intended for courses in signals and systems at the junior level in electrical andcomputer engineering, but it could also be used in teaching this material to mechanical engineering and bioengi-neering students and it might be of interest to students in applied mathematics. The “student-friendly” natureof the text also makes it useful to practicing engineers interested in learning or reviewing the basic principles ofsignals and systems on their own. The material is organized so that students not only get a solid understand-ing of the theory—through analytic examples as well as software examples using MATLAB—and learn aboutapplications, but also develop confidence and proficiency in the material by working on problems.

xi

xii Preface

The organization of the material in the book follows the assumption that the student has been exposed to thetheory of linear circuits, differential equations, and linear algebra, and that this material will be followed bycourses in control, communications, or digital signal processing. The content is guided by the goal of nurturingthe interest of students in applications, and of assisting them in becoming more sophisticated mathematically.In teaching signals and systems, the author has found that students typically lack basic skills in manipulatingcomplex variables, in understanding differential equations, and are not yet comfortable with basic concepts incalculus. Introducing discrete-time signals and systems makes students face new concepts that were not exploredin their calculus courses, such as summations, finite differences, and difference equations. This text attempts tofill the gap and nurture interest in the mathematical tools.

APPROACHIn writing this text, we have taken the following approach:

1. The material is divided into three parts: introduction, theory and applications of continuous-time signalsand systems, and theory and applications of discrete-time signals and systems. To help students under-stand the connection between continuous- and discrete-time signals and systems, the connection betweeninfinitesimal and finite calculus is made in the introduction part, together with a motivation as to why com-plex numbers and functions are used in the study of signals and systems. The treatment of continuous- anddiscrete-time signals and systems is then done separately in the next two parts; combining them is found tobe confusing to students. Likewise, the author believes it is important for students to understand the connec-tions and relevance of each of the transformations used in the analysis of signals and systems so that thesetransformations are seen as a progression rather than as disconnected methods. Thus, the author advocatesthe presentation of the Laplace analysis followed by the Fourier analysis, and the Z-transform followed by thediscrete Fourier, and capping each of these topics with applications to communications, control, and filter-ing. The mathematical abstraction and the applications become more sophisticated as the material unfolds,taking advantage as needed of the background on circuits that students have.

2. An overview of the topics to be discussed in the book and how each connects with some basic mathematicalconcepts—needed in the rest of the book—is given in Chapter 0 (analogous to the ground floor of a build-ing). The emphasis is in relating summations, differences, difference equations, and sequence of numberswith the calculus concepts that the students are familiar with, and in doing so providing a new interpreta-tion to integrals, derivatives, differential equations, and functions of time. This chapter also links the theoryof complex numbers and functions to vectors and to phasors learned in circuit theory. Because we stronglybelieve that the material in this chapter should be covered before beginning the discussion of signals andsystems, it is not relegated to an appendix but placed at the front of the book where it cannot be ignored. Asoft introduction to MATLAB is also provided in this chapter.

3. A great deal of effort has been put into making the text “student friendly.” To make sure that the student doesnot miss some of the important issues presented in a section, we have inserted well-thought-out remarks—we want to minimize the common misunderstandings we have observed from our students in the past.Plenty of analytic examples with different levels of complexity are given to illustrate issues. Each chapterhas a set of examples in MATLAB, illustrating topics presented in the text or special issues that the studentshould know. The MATLAB code is given so that students can learn by example from it. To help studentsfollow the mathematical derivations, we provide extra steps whenever necessary and do not skip steps thatare necessary in the understanding of a derivation. Summaries of important issues are boxed and conceptsand terms are emphasized to help students grasp the main points and terminology.

4. Without any doubt, learning the material in signals and systems requires working analytical as well as com-putational problems. It is important to provide problems of different levels of complexity to exercise notonly basic problem-solving skills, but to achieve a level of proficiency and mathematical sophistication.The problems at the end of the chapter are of different types, some to be done analytically, others using

Preface xiii

MATLAB, and some both. The repetitive type of problem was avoided. Some of the problems explore issuesnot covered in the text but related to it. The MATLAB problems were designed so that a better understandingof the theoretical concepts is attained by the student working them out.

5. We feel two additional features would be beneficial to students. One is the inclusion of quotations andfootnotes to present interesting ideas or historical comments, and the other is the inclusion of sidebars thatattempt to teach historical or technical information that students should be aware of. The theory of signalsand systems clearly connects with mathematics and a great number of mathematicians have contributed toit. Likewise, there is a large number of engineers who have contributed significantly to the development andapplication of signals and systems. All of them need to be recognized for their contributions, and we shouldlearn from their experiences.

6. Finally, other features are: (1) the design of the index of the book so that it can be used by students to finddefinitions, symbols, and MATLAB functions used in the text; and (2) a list of references to the material.

CONTENTThe core of the material is presented in the second and third part of the book. The second part of the bookcovers the basics of continuous-time signals and systems and illustrates their application. Because the conceptsof signals and systems are relatively new to students, we provide an extensive and complete presentation of thesetopics in Chapters 1 and 2. The presentation in Chapter 1 goes from a very general characterization of signalsto very specific classes that will be used in the rest of the book. One of the aims is to familiarize students withcontinuous-time as well as discrete-time signals so as to avoid confusion in their processing later on—a commondifficulty encountered by students. Chapter 1 initiates the representation of signals in terms of basic signals thatwill be easily processed later with the transform methods. Chapter 2 introduces the general concept of systems,in particular continuous-time systems. The concepts of linearity, time invariance, causality, and stability areintroduced in this chapter, trying as much as possible to use the students’ background in circuit theory. Usinglinearity and time invariance, the computation of the output of a continuous-time system using the convolutionintegral is introduced and illustrated with relatively simple examples. More complex examples are treated withthe Laplace transform in the following chapter.

Chapter 3 covers the basics of the Laplace transform and its application in the analysis of continuous-timesignals and systems. It introduces the student to the concept of poles and zeros, damping and frequency, andtheir connection with the signal as a function of time. This chapter emphasizes the solution of differentialequations representing linear time-invariant (LTI) systems, paying special attention to transient solutions dueto their importance in control, as well as to steady-state solutions due to their importance in filtering and incommunications. The convolution integral is dealt with in time and using the Laplace transform to emphasizethe operational power of the transform. The important concept of transfer function for LTI systems and thesignificance of its poles and zeros are studied in detail. Different approaches are considered in computing theinverse Laplace transform, including MATLAB methods.

Fourier analysis of continuous-time signals and systems is covered in detail in Chapters 4 and 5. The Fourierseries analysis of periodic signals, covered in Chapter 4, is extended to the analysis of aperiodic signals resultingin the Fourier transform of Chapter 5. The Fourier transform is useful in representing both periodic and aperi-odic signals. Special attention is given to the connection of these methods with the Laplace transform so that,whenever possible, known Laplace transforms can be used to compute the Fourier series coefficients and theFourier transform—thus avoiding integration but using the concept of the region of convergence. The conceptof frequency, the response of the system (connected to the location of poles and zeros of the transfer function),and the steady-state response are emphasized in these chapters.

The ordering of the presentation of the Laplace and the Fourier transformations (similar to the Z-transformand the Fourier representation of discrete-time signals) is significant for learning and teaching of the material.

xiv Preface

Our approach of presenting first the Laplace transform and then the Fourier series and Fourier transform isjustified by several reasons. For one, students coming into a signals and systems course have been familiarizedwith the Laplace transform in their previous circuits or differential equations courses, and will continue usingit in control courses. So expertise in this topic is important and the learned material will stay with them longer.Another is that a common difficulty students have in applying the Fourier series and the Fourier transform isconnected with the required integration. The Laplace transform can be used not only to sidestep the integrationbut to provide a more comprehensive understanding of the frequency representation. By asking students toconsider the two-sided Laplace transform and the significance of its region of convergence, they will appreciatebetter the Fourier representation as a special case of Laplace’s in many cases. More importantly, these transformscan be seen as a continuum rather than as different transforms. It also makes theoretical sense to deal withthe Laplace representation of systems first to justify the existence of the steady-state solution considered in theFourier representations, which would not exist unless stability of the system is guaranteed, and stability can onlybe tested using the Laplace transform. The paradigm of interest is the connection of transient and steady-stateresponses that must be understood by students before they can understand the connections between Fourier andLaplace analyses.

Chapter 6 presents applications of the Laplace and the Fourier transforms to control, communications, and fil-tering. The intent of the chapter is to motivate interest in these areas. The chapter illustrates the significance ofthe concepts of transfer function, response of systems, and stability in control, and of modulation in communi-cations. An introduction to analog filtering is provided. Analytic as well as MATLAB examples illustrate differentapplications to control, communications, and filter design.

Using the sampling theory as a bridge, the third part of the book covers the theory and illustrates the applicationof discrete-time signals and systems. Chapter 7 presents the theory of sampling: the conditions under which thesignal does not lose information in the sampling process and the recovery of the analog signal from the sampledsignal. Once the basic concepts are given, the analog-to-digital and digital-to-analog converters are consideredto provide a practical understanding of the conversion of analog-to-digital and digital-to-analog signals.

Discrete-time signals and systems are discussed in Chapter 8, while Chapter 9 introduces the Z-transform.Although the treatment of discrete-time signals and systems in Chapter 8 mirrors that of continuous-time sig-nals and systems, special emphasis is given in this chapter to issues that are different in the two domains. Issuessuch as the discrete nature of the time, the periodicity of the discrete frequency, the possible lack of periodicityof discrete sinusoids, etc. are considered. Chapter 9 provides the basic theory of the Z-transform and how itrelates to the Laplace transform. The material in this chapter bears similarity to the one on the Laplace trans-form in terms of operational solution of difference equations, transfer function, and the significance of poles andzeros.

Chapter 10 presents the Fourier analysis of discrete signals and systems. Given the accumulated experience ofthe students with continuous-time signals and systems, we build the discrete-time Fourier transform (DTFT) onthe Z-transform and consider special cases where the Z-transform cannot be used. The discrete Fourier transform(DFT) is obtained from the Fourier series of discrete-time signals and sampling in frequency. The DFT will beof great significance in digital signal processing. The computation of the DFT of periodic and aperiodic discrete-time signals using the fast Fourier transform (FFT) is illustrated. The FFT is an efficient algorithm for computingthe DFT, and some of the basics of this algorithm are discussed in Chapter 12.

Chapter 11 introduces students to discrete filtering, thus extending the analog filtering in Chapter 6. In thischapter we show how to use the theory of analog filters to design recursive discrete low-pass filters. Frequencytransformations are then presented to show how to obtain different types of filters from low-pass prototypefilters. The design of finite-impulse filters using the window method is considered next. Finally, the implementa-tion of recursive and nonrecursive filters is shown using some basic techniques. By using MATLAB for the designof recursive and nonrecursive discrete filters, it is expected that students will be motivated to pursue on theirown the use of more sophisticated filter designs.

Preface xv

Finally, Chapter 12 explores topics of interest in digital communications, computer control, and digital signalprocessing. The aim of this chapter is to provide a brief presentation of topics that students could pursue afterthe basic courses in signals and systems.

TEACHING USING THIS TEXTThe material in this text is intended for a two-term sequence in signals and systems: one on continuous-timesignals and systems, followed by a term in discrete-time signals and systems with a lab component using MAT-LAB. These two courses would cover most of the chapters in the text with various degrees of depth, dependingon the emphasis the faculty would like to give to the course. As indicated, Chapter 0 was written as a necessaryintroduction to the rest of the material, but does not need to be covered in great detail—students can refer to it asneeded. Chapters 6 and 11 need to be considered together if the emphasis on applications is in filter design. Thecontrol, communications, and digital signal processing material in Chapters 6 and 12 can be used to motivatestudents toward those areas.

TO THE STUDENTIt is important for you to understand the features of this book, so you can take advantage of them to learn thematerial:

1. Refer as often as necessary to the material in Chapter 0 to review or to learn the mathematical background;to understand the overall structure of the material; or to review or learn MATLAB as it applies to signalprocessing.

2. As you will see, the complexity of the material grows as it develops. The material in part three has beenwritten assuming good understanding of the material in the first two. See also the connection of the materialwith applications in your own areas of interest.

3. To help you learn the material, clear and concise results are emphasized by putting them in boxes. Justi-fication of these results is then given, complemented with remarks regarding issues that need a bit moreclarification, and illustrated with plenty of analytic and computational examples. Important terms areemphasized throughout the text. Tables provide a good summary of properties and formulas.

4. A heading is used in each of the problems at the end of the chapters, indicating how it relates to specifictopics and if it requires to use MATLAB to solve it.

5. One of the objectives of this text is to help you learn MATLAB, as it applies to signal and systems, on yourown. This is done by providing the soft introduction to MATLAB in Chapter 0, and then by showing examplesusing simple code in each of the chapters. You will notice that in the first two parts basic components ofMATLAB (scripts, functions, plotting, etc.) are given in more detail than in part three. It is assumed you arevery proficient by then to supply that on your own.

6. Finally, notice the footnotes, the vignettes, and the historical sidebars that have been included to provide aglance at the background in which the theory and practice of signals and systems have developed.

Acknowledgments

I would like to acknowledge with gratitude the support and efforts of many people who made the writing of this textpossible. First, to my family—my wife Cathy, my children William, Camila, and Juan, and their own families—manythanks for their support and encouragement despite being deprived of my attention. To my academic mentor, ProfessorEliahu I. Jury, a deep sense of gratitude for his teachings and for having inculcated in me the love for a scholarly careerand for the theory and practice of signals and systems. Thanks to Professor William Stanchina, chair of the Departmentof Electrical and Computer Engineering at the University of Pittsburgh, for his encouragement and support that madeit possible to dedicate time to the project. Sincere thanks to Seda Senay and Mircea Lupus, graduate students in mydepartment. Their contribution to the painful editing and proofreading of the manuscript, and the generation of thesolution manual (especially from Ms. Senay) are much appreciated. Equally, thanks to the publisher and its editors, inparticular to Joe Hayton and Steve Merken, for their patience, advising, and help with the publishing issues. Thanksalso to Sarah Binns for her help with the final editing of the manuscript. Equally, I would like to thank Professor JamesRowland from the University of Kansas and the following reviewers for providing significant input and changes tothe manuscript: Dimitrie Popescu, Old Dominion University; Hossein Hakim, Worcester Polytechnic Institute; MarkBudnik, Valparaiso University; Periasamy Rajan, Tennessee Tech University; and Mohamed Zohdy, Oakland University.Thanks to my colleagues Amro El-Jaroudi and Juan Manfredi for their early comments and suggestions.

Lastly, I feel indebted to the many students I have had in my courses in signals and systems over the years I havebeen teaching this material in the Department of Electrical and Computer Engineering at the University of Pittsburgh.Unknown to them, they contributed to my impetus to write a book that I felt would make the teaching of signals andsystems more accessible and fun to future students in and outside the university.

RESOURCES THAT ACCOMPANY THIS BOOKA companion website containing downloadable MATLAB code for the worked examples in the book is available at:

http://booksite.academicpress.com/chaparro

For instructors, a solutions manual and image bank containing electronic versions of figures from the book areavailable by registering at:

www.textbooks.elsevier.com

Also Available for Use with This Book – Elsevier Online Testing

Web-based testing and assessment feature that allows instructors to create online tests and assignments whichautomatically assess student responses and performance, providing them with immediate feedback. Elsevier’sonline testing includes a selection of algorithmic questions, giving instructors the ability to create virtually unlim-ited variations of the same problem. Contact your local sales representative for additional information, or visithttp://booksite.academicpress.com/chaparro/ to view a demo chapter.

xvi

Documents

Signals and Systems - Elsevier · 2013-12-20 · Signals and Systems Using MATLABR Luis F. Chaparro Department of Electrical and Computer Engineering University of Pittsburgh AMSTERDAM