LINEAR SYSTEMS AND SIGNALS

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LINEAR SYSTEMS AND SIGNALS

THE OXFORD SERIES IN ELECTRICAL AND COMPUTER ENGINEERING

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LINEAR SYSTEMS AND SIGNALS

THIRD EDITION

B. P. Lathi and R. A. Green

New York Oxford OXFORD UNIVERSITY PRESS 2018

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Library of Congress Cataloging-in-Publication Data Names: Lathi, B. P. (Bhagwandas Pannalal), author. | Green, R. A. (Roger A.), author. Title: Linear systems and signals / B.P. Lathi and R.A. Green. Description: Third Edition. | New York : Oxford University Press, [2018] | Series: The Oxford Series in Electrical and Computer Engineering Identifiers: LCCN 2017034962 | ISBN 9780190200176 (hardcover : acid-free paper) Subjects: LCSH: Signal processing–Mathematics. | System analysis. | Linear time invariant systems. | Digital filters (Mathematics) Classification: LCC TK5102.5 L298 2017 | DDC 621.382/2–dc23 LC record available athttps://lccn.loc.gov/2017034962

ISBN 978–0–19–020017–6

Printing number: 9 8 7 6 5 4 3 2 1

Printed by R.R. Donnelly in the United States of America

CONTENTS

PREFACE xv

B BACKGROUND

• B.1 Complex Numbers 1

  • B.1-1 A Historical Note 1
  • B.1-2 Algebra of Complex Numbers 5

• B.2 Sinusoids 16

  • B.2-1 Addition of Sinusoids 18
  • B.2-2 Sinusoids in Terms of Exponentials 20

• B.3 Sketching Signals 20

  • B.3-1 Monotonic Exponentials 20
  • B.3-2 The Exponentially Varying Sinusoid 22

• B.4 Cramer’s Rule 23

• B.5 Partial Fraction Expansion 25

  • B.5-1 Method of Clearing Fractions 26
  • B.5-2 The Heaviside “Cover-Up” Method 27
  • B.5-3 Repeated Factors of Q(x) 31
  • B.5-4 A Combination of Heaviside “Cover-Up” and Clearing Fractions 32
  • B.5-5 Improper F(x) with m = n 34
  • B.5-6 Modified Partial Fractions 35

• B.6 Vectors and Matrices 36

  • B.6-1 Some Definitions and Properties 37
  • B.6-2 Matrix Algebra 38

• B.7 MATLAB: Elementary Operations 42

  • B.7-1 MATLAB Overview 42
  • B.7-2 Calculator Operations 43
  • B.7-3 Vector Operations 45
  • B.7-4 Simple Plotting 46
  • B.7-5 Element-by-Element Operations 48
  • B.7-6 Matrix Operations 49
  • B.7-7 Partial Fraction Expansions 53

• B.8 Appendix: Useful Mathematical Formulas 54

  • B.8-1 Some Useful Constants 54

• B.8-2 Complex Numbers 54

• B.8-3 Sums 54

• B.8-4 Taylor and Maclaurin Series 55

• B.8-5 Power Series 55

• B.8-6 Trigonometric Identities 55

• B.8-7 Common Derivative Formulas 56

• B.8-8 Indefinite Integrals 57

• B.8-9 L’Hôpital’s Rule 58

• B.8-10 Solution of Quadratic and Cubic Equations 58

• References 58 Problems 59

1 SIGNALS AND SYSTEMS

• 1.1 Size of a Signal 64

  • 1.1-1 Signal Energy 65
  • 1.1-2 Signal Power 65

• 1.2 Some Useful Signal Operations 71

  • 1.2-1 Time Shifting 71
  • 1.2-2 Time Scaling 73
  • 1.2-3 Time Reversal 76
  • 1.2-4 Combined Operations 77

• 1.3 Classification of Signals 78

  • 1.3-1 Continuous-Time and Discrete-Time Signals 78
  • 1.3-2 Analog and Digital Signals 78
  • 1.3-3 Periodic and Aperiodic Signals 79
  • 1.3-4 Energy and Power Signals 82
  • 1.3-5 Deterministic and Random Signals 82

• 1.4 Some Useful Signal Models 82

  • 1.4-1 The Unit Step Function u(t) 83
  • 1.4-2 The Unit Impulse Function δ(t) 86
  • 1.4-3 The Exponential Function est 89

• 1.5 Even and Odd Functions 92

  • 1.5-1 Some Properties of Even and Odd Functions 92
  • 1.5-2 Even and Odd Components of a Signal 93

• 1.6 Systems 95

• 1.7 Classification of Systems 97

  • 1.7-1 Linear and Nonlinear Systems 97
  • 1.7-2 Time-Invariant and Time-Varying Systems 102
  • 1.7-3 Instantaneous and Dynamic Systems 103
  • 1.7-4 Causal and Noncausal Systems 104
  • 1.7-5 Continuous-Time and Discrete-Time Systems 107
  • 1.7-6 Analog and Digital Systems 109
  • 1.7-7 Invertible and Noninvertible Systems 109
  • 1.7-8 Stable and Unstable Systems 110

• 1.8 System Model: Input–Output Description 111

  • 1.8-1 Electrical Systems 111
  • 1.8-2 Mechanical Systems 114
  • 1.8-3 Electromechanical Systems 118

• 1.9 Internal and External Descriptions of a System 119

• 1.10 Internal Description: The State-Space Description 121

• 1.11 MATLAB: Working with Functions 126

  • 1.11-1 Anonymous Functions 126
  • 1.11-2 Relational Operators and the Unit Step Function 128
  • 1.11-3 Visualizing Operations on the Independent Variable 130
  • 1.11-4 Numerical Integration and Estimating Signal Energy 131

• 1.12 Summary 133

References 135 Problems 136

2 TIME-DOMAIN ANALYSIS OF CONTINUOUS-TIME SYSTEMS

• 2.1 Introduction 150

• 2.2 System Response to Internal Conditions: The Zero-Input Response 151 2.2-1 Some Insights into the Zero-Input Behavior of a System 161

• 2.3 The Unit Impulse Response h(t) 163

• 2.4 System Response to External Input: The Zero-State Response 168

  • 2.4-1 The Convolution Integral 170
  • 2.4-2 Graphical Understanding of Convolution Operation 178
  • 2.4-3 Interconnected Systems 190
  • 2.4-4 A Very Special Function for LTIC Systems:
    • The Everlasting Exponential est 193
  • 2.4-5 Total Response 195

• 2.5 System Stability 196

  • 2.5-1 External (BIBO) Stability 196
  • 2.5-2 Internal (Asymptotic) Stability 198
  • 2.5-3 Relationship Between BIBO and Asymptotic Stability 199

• 2.6 Intuitive Insights into System Behavior 203

  • 2.6-1 Dependence of System Behavior on Characteristic Modes 203
  • 2.6-2 Response Time of a System: The System Time Constant 205
  • 2.6-3 Time Constant and Rise Time of a System 206
  • 2.6-4 Time Constant and Filtering 207
  • 2.6-5 Time Constant and Pulse Dispersion (Spreading) 209
  • 2.6-6 Time Constant and Rate of Information Transmission 209
  • 2.6-7 The Resonance Phenomenon 210

• 2.7 MATLAB: M-Files 212

  • 2.7-1 Script M-Files 213
  • 2.7-2 Function M-Files 214

• 2.7-3 For-Loops 215

• 2.7-4 Graphical Understanding of Convolution 217

• 2.8 Appendix: Determining the Impulse Response 220

• 2.9 Summary 221

References 223 Problems 223

3 TIME-DOMAIN ANALYSIS OF DISCRETE-TIME SYSTEMS

• 3.1 Introduction 237
  • 3.1-1 Size of a Discrete-Time Signal 238
• 3.2 Useful Signal Operations 240
• 3.3 Some Useful Discrete-Time Signal Models 245
  • 3.3-1 Discrete-Time Impulse Function δ[n] 245
  • 3.3-2 Discrete-Time Unit Step Function u[n] 246
  • 3.3-3 Discrete-Time Exponential γ ⁿ 247
  • 3.3-4 Discrete-Time Sinusoid cos(n+θ ) 251
  • 3.3-5 Discrete-Time Complex Exponential ejⁿ 252
• 3.4 Examples of Discrete-Time Systems 253
  • 3.4-1 Classification of Discrete-Time Systems 262
• 3.5 Discrete-Time System Equations 265

3.5-1 Recursive (Iterative) Solution of Difference Equation 266

• 3.6 System Response to Internal Conditions: The Zero-Input Response 270
• 3.7 The Unit Impulse Response h[n] 277
  • 3.7-1 The Closed-Form Solution of h[n] 278
• 3.8 System Response to External Input: The Zero-State Response 280
  • 3.8-1 Graphical Procedure for the Convolution Sum 288
  • 3.8-2 Interconnected Systems 294
  • 3.8-3 Total Response 297
• 3.9 System Stability 298
  • 3.9-1 External (BIBO) Stability 298
  • 3.9-2 Internal (Asymptotic) Stability 299
  • 3.9-3 Relationship Between BIBO and Asymptotic Stability 301
• 3.10 Intuitive Insights into System Behavior 305
• 3.11 MATLAB: Discrete-Time Signals and Systems 306
  • 3.11-1 Discrete-Time Functions and Stem Plots 306
  • 3.11-2 System Responses Through Filtering 308
  • 3.11-3 A Custom Filter Function 310
  • 3.11-4 Discrete-Time Convolution 311
• 3.12 Appendix: Impulse Response for a Special Case 313
• 3.13 Summary 313

Problems 314

4 CONTINUOUS-TIME SYSTEM ANALYSIS USING THE LAPLACE TRANSFORM

• 4.1 The Laplace Transform 330

  • 4.1-1 Finding the Inverse Transform 338

• 4.2 Some Properties of the Laplace Transform 349

  • 4.2-1 Time Shifting 349
  • 4.2-2 Frequency Shifting 353
  • 4.2-3 The Time-Differentiation Property 354
  • 4.2-4 The Time-Integration Property 356
  • 4.2-5 The Scaling Property 357
  • 4.2-6 Time Convolution and Frequency Convolution 357

• 4.3 Solution of Differential and Integro-Differential Equations 360

  • 4.3-1 Comments on Initial Conditions at 0⁻ and at 0⁺ 363
  • 4.3-2 Zero-State Response 366
  • 4.3-3 Stability 371
  • 4.3-4 Inverse Systems 373

• 4.4 Analysis of Electrical Networks: The Transformed Network 373

  • 4.4-1 Analysis of Active Circuits 382

• 4.5 Block Diagrams 386

• 4.6 System Realization 388

  • 4.6-1 Direct Form I Realization 389
  • 4.6-2 Direct Form II Realization 390
  • 4.6-3 Cascade and Parallel Realizations 393
  • 4.6-4 Transposed Realization 396
  • 4.6-5 Using Operational Amplifiers for System Realization 399

• 4.7 Application to Feedback and Controls 404

  • 4.7-1 Analysis of a Simple Control System 406

• 4.8 Frequency Response of an LTIC System 412

  • 4.8-1 Steady-State Response to Causal Sinusoidal Inputs 418

• 4.9 Bode Plots 419

  • 4.9-1 Constant Ka1a2/b1b³ 422
  • 4.9-2 Pole (or Zero) at the Origin 422
  • 4.9-3 First-Order Pole (or Zero) 424
  • 4.9-4 Second-Order Pole (or Zero) 426
  • 4.9-5 The Transfer Function from the Frequency Response 435

• 4.10 Filter Design by Placement of Poles and Zeros of H(s) 436

  • 4.10-1 Dependence of Frequency Response on Poles and Zeros of H(s) 436
  • 4.10-2 Lowpass Filters 439
  • 4.10-3 Bandpass Filters 441
  • 4.10-4 Notch (Bandstop) Filters 441
  • 4.10-5 Practical Filters and Their Specifications 444

• 4.11 The Bilateral Laplace Transform 445

• 4.11-1 Properties of the Bilateral Laplace Transform 451

• 4.11-2 Using the Bilateral Transform for Linear System Analysis 452

• 4.12 MATLAB: Continuous-Time Filters 455

  • 4.12-1 Frequency Response and Polynomial Evaluation 456
  • 4.12-2 Butterworth Filters and the Find Command 459
  • 4.12-3 Using Cascaded Second-Order Sections for Butterworth Filter Realization 461
  • 4.12-4 Chebyshev Filters 463

• 4.13 Summary 466

References 468 Problems 468

5 DISCRETE-TIME SYSTEM ANALYSIS USING THE z-TRANSFORM

• 5.1 The z-Transform 488
  • 5.1-1 Inverse Transform by Partial Fraction Expansion and Tables 495
  • 5.1-2 Inverse z-Transform by Power Series Expansion 499
• 5.2 Some Properties of the z-Transform 501
  • 5.2-1 Time-Shifting Properties 501
  • 5.2-2 z-Domain Scaling Property (Multiplication by γ ⁿ) 505
  • 5.2-3 z-Domain Differentiation Property (Multiplication by n) 506
  • 5.2-4 Time-Reversal Property 506
  • 5.2-5 Convolution Property 507
• 5.3 z-Transform Solution of Linear Difference Equations 510
  • 5.3-1 Zero-State Response of LTID Systems: The Transfer Function 514
  • 5.3-2 Stability 518
  • 5.3-3 Inverse Systems 519
• 5.4 System Realization 519
• 5.5 Frequency Response of Discrete-Time Systems 526
  • 5.5-1 The Periodic Nature of Frequency Response 532
  • 5.5-2 Aliasing and Sampling Rate 536
• 5.6 Frequency Response from Pole-Zero Locations 538
• 5.7 Digital Processing of Analog Signals 547
• 5.8 The Bilateral z-Transform 554
  • 5.8-1 Properties of the Bilateral z-Transform 559
  • 5.8-2 Using the Bilateral z-Transform for Analysis of LTID Systems 560
• 5.9 Connecting the Laplace and z-Transforms 563
• 5.10 MATLAB: Discrete-Time IIR Filters 565
  • 5.10-1 Frequency Response and Pole-Zero Plots 566
  • 5.10-2 Transformation Basics 567
  • 5.10-3 Transformation by First-Order Backward Difference 568
  • 5.10-4 Bilinear Transformation 569
  • 5.10-5 Bilinear Transformation with Prewarping 570
  • 5.10-6 Example: Butterworth Filter Transformation 571

5.10-7 Problems Finding Polynomial Roots 572

5.10-8 Using Cascaded Second-Order Sections to Improve Design 572

5.11 Summary 574

References 575 Problems 575

6 CONTINUOUS-TIME SIGNAL ANALYSIS: THE FOURIER SERIES

• 6.1 Periodic Signal Representation by Trigonometric Fourier Series 593
  • 6.1-1 The Fourier Spectrum 598
  • 6.1-2 The Effect of Symmetry 607
  • 6.1-3 Determining the Fundamental Frequency and Period 609
• 6.2 Existence and Convergence of the Fourier Series 612
  • 6.2-1 Convergence of a Series 613
  • 6.2-2 The Role of Amplitude and Phase Spectra in Waveshaping 615
• 6.3 Exponential Fourier Series 621
  • 6.3-1 Exponential Fourier Spectra 624
  • 6.3-2 Parseval’s Theorem 632
  • 6.3-3 Properties of the Fourier Series 635
• 6.4 LTIC System Response to Periodic Inputs 637
• 6.5 Generalized Fourier Series: Signals as Vectors 641
  • 6.5-1 Component of a Vector 642
  • 6.5-2 Signal Comparison and Component of a Signal 643
  • 6.5-3 Extension to Complex Signals 645
  • 6.5-4 Signal Representation by an Orthogonal Signal Set 647
• 6.6 Numerical Computation of Dn 659
• 6.7 MATLAB: Fourier Series Applications 661
  • 6.7-1 Periodic Functions and the Gibbs Phenomenon 661
  • 6.7-2 Optimization and Phase Spectra 664
• 6.8 Summary 667 References 668 Problems 669

7 CONTINUOUS-TIME SIGNAL ANALYSIS: THE FOURIER TRANSFORM

• 7.1 Aperiodic Signal Representation by the Fourier Integral 680 7.1-1 Physical Appreciation of the Fourier Transform 687
• 7.2 Transforms of Some Useful Functions 689
  • 7.2-1 Connection Between the Fourier and Laplace Transforms 700
• 7.3 Some Properties of the Fourier Transform 701
• 7.4 Signal Transmission Through LTIC Systems 721
  • 7.4-1 Signal Distortion During Transmission 723
  • 7.4-2 Bandpass Systems and Group Delay 726

xii Contents

• 7.5 Ideal and Practical Filters 730
• 7.6 Signal Energy 733
• 7.7 Application to Communications: Amplitude Modulation 736
  • 7.7-1 Double-Sideband, Suppressed-Carrier (DSB-SC) Modulation 737
  • 7.7-2 Amplitude Modulation (AM) 742
  • 7.7-3 Single-Sideband Modulation (SSB) 746
  • 7.7-4 Frequency-Division Multiplexing 749
• 7.8 Data Truncation: Window Functions 749
  • 7.8-1 Using Windows in Filter Design 755
• 7.9 MATLAB: Fourier Transform Topics 755
  • 7.9-1 The Sinc Function and the Scaling Property 757
  • 7.9-2 Parseval’s Theorem and Essential Bandwidth 758
  • 7.9-3 Spectral Sampling 759
  • 7.9-4 Kaiser Window Functions 760
• 7.10 Summary 762

References 763 Problems 764

8 SAMPLING: THE BRIDGE FROM CONTINUOUS TO DISCRETE

• 8.1 The Sampling Theorem 776 8.1-1 Practical Sampling 781
• 8.2 Signal Reconstruction 785
  • 8.2-1 Practical Difficulties in Signal Reconstruction 788
  • 8.2-2 Some Applications of the Sampling Theorem 796
• 8.3 Analog-to-Digital (A/D) Conversion 799
• 8.4 Dual of Time Sampling: Spectral Sampling 802
• 8.5 Numerical Computation of the Fourier Transform: The Discrete Fourier Transform 805
  • 8.5-1 Some Properties of the DFT 818
  • 8.5-2 Some Applications of the DFT 820
• 8.6 The Fast Fourier Transform (FFT) 824
• 8.7 MATLAB: The Discrete Fourier Transform 827
  • 8.7-1 Computing the Discrete Fourier Transform 827
  • 8.7-2 Improving the Picture with Zero Padding 829
  • 8.7-3 Quantization 831
• 8.8 Summary 834

References 835 Problems 835

9 FOURIER ANALYSIS OF DISCRETE-TIME SIGNALS

• 9.1 Discrete-Time Fourier Series (DTFS) 845
  • 9.1-1 Periodic Signal Representation by Discrete-Time Fourier Series 846
  • 9.1-2 Fourier Spectra of a Periodic Signal x[n] 848
• 9.2 Aperiodic Signal Representation

by Fourier Integral 855

• 9.2-1 Nature of Fourier Spectra 858
• 9.2-2 Connection Between the DTFT and the z-Transform 866
• 9.3 Properties of the DTFT 867
• 9.4 LTI Discrete-Time System Analysis by DTFT 878
  • 9.4-1 Distortionless Transmission 880
  • 9.4-2 Ideal and Practical Filters 882
• 9.5 DTFT Connection with the CTFT 883
  • 9.5-1 Use of DFT and FFT for Numerical Computation of the DTFT 885
• 9.6 Generalization of the DTFT to the z-Transform 886
• 9.7 MATLAB: Working with the DTFS and the DTFT 889
  • 9.7-1 Computing the Discrete-Time Fourier Series 889
  • 9.7-2 Measuring Code Performance 891
  • 9.7-3 FIR Filter Design by Frequency Sampling 892
• 9.8 Summary 898

Reference 898 Problems 899

10 STATE-SPACE ANALYSIS

• 10.1 Mathematical Preliminaries 909
  • 10.1-1 Derivatives and Integrals of a Matrix 909
  • 10.1-2 The Characteristic Equation of a Matrix:
    • The Cayley–Hamilton Theorem 910
  • 10.1-3 Computation of an Exponential and a Power of a Matrix 912
• 10.2 Introduction to State Space 913
• 10.3 A Systematic Procedure to Determine State Equations 916
  • 10.3-1 Electrical Circuits 916
  • 10.3-2 State Equations from a Transfer Function 919
• 10.4 Solution of State Equations 926
  • 10.4-1 Laplace Transform Solution of State Equations 927
  • 10.4-2 Time-Domain Solution of State Equations 933
• 10.5 Linear Transformation of a State Vector 939
  • 10.5-1 Diagonalization of Matrix A 943
• 10.6 Controllability and Observability 947
  • 10.6-1 Inadequacy of the Transfer Function Description of a System 953

10.7 State-Space Analysis of Discrete-Time Systems 953

10.7-1 Solution in State Space 955

10.7-2 The z-Transform Solution 959

10.8 MATLAB: Toolboxes and State-Space Analysis 961

10.8-1 z-Transform Solutions to Discrete-Time, State-Space Systems 961

10.8-2 Transfer Functions from State-Space Representations 964

10.8-3 Controllability and Observability of Discrete-Time Systems 965

10.8-4 Matrix Exponentiation and the Matrix Exponential 968

10.9 Summary 969

References 970 Problems 970

INDEX 975

PREFACE

This book, Linear Systems and Signals, presents a comprehensive treatment of signals and linear systems at an introductory level. Following our preferred style, it emphasizes a physical appreciation of concepts through heuristic reasoning and the use of metaphors, analogies, and creative explanations. Such an approach is much different from a purely deductive technique that uses mere mathematical manipulation of symbols. There is a temptation to treat engineering subjects as a branch of applied mathematics. Such an approach is a perfect match to the public image of engineering as a dry and dull discipline. It ignores the physical meaning behind various derivations and deprives students of intuitive grasp and the enjoyable experience of logical uncovering of the subject matter. In this book, we use mathematics not so much to prove axiomatic theory as to support and enhance physical and intuitive understanding. Wherever possible, theoretical results are interpreted heuristically and are enhanced by carefully chosen examples and analogies.

This third edition, which closely follows the organization of the second edition, has been refined in many ways. Discussions are streamlined, adding or trimming material as needed. Equation, example, and section labeling is simplified and improved. Computer examples are fully updated to reflect the most current version of MATLAB. Hundreds of added problems provide new opportunities to learn and understand topics. We have taken special care to improve the text without the topic creep and bloat that commonly occurs with each new edition of a text.

NOTABLE FEATURES

The notable features of this book include the following.

• 1. Intuitive and heuristic understanding of the concepts and physical meaning of mathematical results are emphasized throughout. Such an approach not only leads to deeper appreciation and easier comprehension of the concepts, but also makes learning enjoyable for students.
• 2. Often, students lack an adequate background in basic material such as complex numbers, sinusoids, hand-sketching of functions, Cramer’s rule, partial fraction expansion, and matrix algebra. We include a background chapter that addresses these basic and pervasive topics in electrical engineering. Response by students has been unanimously enthusiastic.
• 3. There are hundreds of worked examples in addition to drills (usually with answers) for students to test their understanding. Additionally, there are over 900 end-of-chapter problems of varying difficulty.
• 4. Modern electrical engineering practice requires the use of computer calculation and simulation, most often using the software package MATLAB. Thus, we integrate

MATLAB into many of the worked examples throughout the book. Additionally, each chapter concludes with a section devoted to learning and using MATLAB in the context and support of book topics. Problem sets also contain numerous computer problems.

• 5. The discrete-time and continuous-time systems may be treated in sequence, or they may be integrated by using a parallel approach.
• 6. The summary at the end of each chapter proves helpful to students in summing up essential developments in the chapter.
• 7. There are several historical notes to enhance students’ interest in the subject. This information introduces students to the historical background that influenced the development of electrical engineering.

ORGANIZATION

The book may be conceived as divided into five parts:

• 1. Introduction (Chs. B and 1).
• 2. Time-domain analysis of linear time-invariant (LTI) systems (Chs. 2 and 3).
• 3. Frequency-domain (transform) analysis of LTI systems (Chs. 4 and 5).
• 4. Signal analysis (Chs. 6, 7, 8, and 9).
• 5. State-space analysis of LTI systems (Ch. 10).

The organization of the book permits much flexibility in teaching the continuous-time and discrete-time concepts. The natural sequence of chapters is meant to integrate continuous-time and discrete-time analysis. It is also possible to use a sequential approach in which all the continuous-time analysis is covered first (Chs. 1, 2, 4, 6, 7, and 8), followed by discrete-time analysis (Chs. 3, 5, and 9).

SUGGESTIONS FOR USING THIS BOOK

The book can be readily tailored for a variety of courses spanning 30 to 45 lecture hours. Most of the material in the first eight chapters can be covered at a brisk pace in about 45 hours. The book can also be used for a 30-lecture-hour course by covering only analog material (Chs. 1, 2, 4, 6, 7, and possibly selected topics in Ch. 8). Alternately, one can also select Chs. 1 to 5 for courses purely devoted to systems analysis or transform techniques. To treat continuous- and discrete-time systems by using an integrated (or parallel) approach, the appropriate sequence of chapters is 1, 2, 3, 4, 5, 6, 7, and 8. For a sequential approach, where the continuous-time analysis is followed by discrete-time analysis, the proper chapter sequence is 1, 2, 4, 6, 7, 8, 3, 5, and possibly 9 (depending on the time available).

MATLAB

MATLAB is a sophisticated language that serves as a powerful tool to better understand engineering topics, including control theory, filter design, and, of course, linear systems and signals. MATLAB’s flexible programming structure promotes rapid development and analysis. Outstanding visualization capabilities provide unique insight into system behavior and signal character.

As with any language, learning MATLAB is incremental and requires practice. This book provides two levels of exposure to MATLAB. First, MATLAB is integrated into many examples throughout the text to reinforce concepts and perform various computations. These examples utilize standard MATLAB functions as well as functions from the control system, signal-processing, and symbolic math toolboxes. MATLAB has many more toolboxes available, but these three are commonly available in most engineering departments.

A second and deeper level of exposure to MATLAB is achieved by concluding each chapter with a separate MATLAB section. Taken together, these eleven sections provide a self-contained introduction to the MATLAB environment that allows even novice users to quickly gain MATLAB proficiency and competence. These sessions provide detailed instruction on how to use MATLAB to solve problems in linear systems and signals. Except for the very last chapter, special care has been taken to avoid the use of toolbox functions in the MATLAB sessions. Rather, readers are shown the process of developing their own code. In this way, those readers without toolbox access are not at a disadvantage. All of this book’s MATLAB code is available for download at the OUP companion website www.oup.com/us/lathi.

CREDITS AND ACKNOWLEDGMENTS

The portraits of Gauss, Laplace, Heaviside, Fourier, and Michelson have been reprinted courtesy of the Smithsonian Institution Libraries. The likenesses of Cardano and Gibbs have been reprinted courtesy of the Library of Congress. The engraving of Napoleon has been reprinted courtesy of Bettmann/Corbis. The many fine cartoons throughout the text are the work of Joseph Coniglio, a former student of Dr. Lathi.

Many individuals have helped us in the preparation of this book, as well as its earlier editions. We are grateful to each and every one for helpful suggestions and comments. Book writing is an obsessively time-consuming activity, which causes much hardship for an author’s family. We both are grateful to our families for their enormous but invisible sacrifices.

B. P. Lathi R. A. Green

BACKGROUND

The topics discussed in this chapter are not entirely new to students taking this course. You have already studied many of these topics in earlier courses or are expected to know them from your previous training. Even so, this background material deserves a review because it is so pervasive in the area of signals and systems. Investing a little time in such a review will pay big dividends later. Furthermore, this material is useful not only for this course but also for several courses that follow. It will also be helpful later, as reference material in your professional career.

B.1 COMPLEX NUMBERS

Complex numbers are an extension of ordinary numbers and are an integral part of the modern number system. Complex numbers, particularly imaginary numbers, sometimes seem mysterious and unreal. This feeling of unreality derives from their unfamiliarity and novelty rather than their supposed nonexistence! Mathematicians blundered in calling these numbers “imaginary,” for the term immediately prejudices perception. Had these numbers been called by some other name, they would have become demystified long ago, just as irrational numbers or negative numbers were. Many futile attempts have been made to ascribe some physical meaning to imaginary numbers. However, this effort is needless. In mathematics we assign symbols and operations any meaning we wish as long as internal consistency is maintained. The history of mathematics is full of entities that were unfamiliar and held in abhorrence until familiarity made them acceptable. This fact will become clear from the following historical note.

B.1-1 A Historical Note

Among early people the number system consisted only of natural numbers (positive integers) needed to express the number of children, cattle, and quivers of arrows. These people had no need for fractions. Whoever heard of two and one-half children or three and one-fourth cows!

However, with the advent of agriculture, people needed to measure continuously varying quantities, such as the length of a field and the weight of a quantity of butter. The number system, therefore, was extended to include fractions. The ancient Egyptians and Babylonians knew how

2 CHAPTER B BACKGROUND

to handle fractions, but Pythagoras discovered that some numbers (like the diagonal of a unit square) could not be expressed as a whole number or a fraction. Pythagoras, a number mystic, who regarded numbers as the essence and principle of all things in the universe, was so appalled at his discovery that he swore his followers to secrecy and imposed a death penalty for divulging this secret [1]. These numbers, however, were included in the number system by the time of Descartes, and they are now known as irrational numbers.

Until recently, negative numbers were not a part of the number system. The concept of negative numbers must have appeared absurd to early man. However, the medieval Hindus had a clear understanding of the significance of positive and negative numbers [2, 3]. They were also the first to recognize the existence of absolute negative quantities [4]. The works of Bhaskar (1114–1185) on arithmetic (L¯ilavat ¯ ¯i) and algebra (B¯ijaganit) not only use the decimal system but also give rules for dealing with negative quantities. Bhaskar recognized that positive numbers have two square roots [5]. Much later, in Europe, the men who developed the banking system that arose in Florence and Venice during the late Renaissance (fifteenth century) are credited with introducing a crude form of negative numbers. The seemingly absurd subtraction of 7 from 5 seemed reasonable when bankers began to allow their clients to draw seven gold ducats while their deposit stood at five. All that was necessary for this purpose was to write the difference, 2, on the debit side of a ledger [6].

Thus, the number system was once again broadened (generalized) to include negative numbers. The acceptance of negative numbers made it possible to solve equations such as x+5=0, which had no solution before. Yet for equations such as x² + 1 = 0, leading to x² = −1, the solution could not be found in the real number system. It was therefore necessary to define a completely new kind of number with its square equal to −1. During the time of Descartes and Newton, imaginary (or complex) numbers came to be accepted as part of the number system, but they were still regarded as algebraic fiction. The Swiss mathematician Leonhard Euler introduced the notation ⁱ (for imaginary) around 1777 to represent √−1. Electrical engineers use the notation j instead of i to avoid confusion with the notation i often used for electrical current. Thus,

$$j^2 = -1 \qquad \text{and} \qquad \sqrt{-1} = \pm j$$

This notation allows us to determine the square root of any negative number. For example,

$$\sqrt{-4} = \sqrt{4} \times \sqrt{-1} = \pm 2j$$

When imaginary numbers are included in the number system, the resulting numbers are called complex numbers.

ORIGINS OF COMPLEX NUMBERS

Ironically (and contrary to popular belief), it was not the solution of a quadratic equation, such as x² + 1 = 0, but a cubic equation with real roots that made imaginary numbers plausible and acceptable to early mathematicians. They could dismiss √−1 as pure nonsense when it appeared as a solution to x² + 1 = 0 because this equation has no real solution. But in 1545, Gerolamo Cardano of Milan published Ars Magna (The Great Art), the most important algebraic work of the Renaissance. In this book, he gave a method of solving a general cubic equation in which a root of a negative number appeared in an intermediate step. According to his method, the solution to a third-order equation†

$$x^3 + ax + b = 0$$

is given by

$$x = \sqrt[3]{-\frac{b}{2} + \sqrt{\frac{b^2}{4} + \frac{a^3}{27}}} + \sqrt[3]{-\frac{b}{2} - \sqrt{\frac{b^2}{4} + \frac{a^3}{27}}}$$

For example, to find a solution of x³ + 6x − 20 = 0, we substitute a = 6,b = −20 in the foregoing equation to obtain

$$x = \sqrt[3]{10 + \sqrt{108}} + \sqrt[3]{10 - \sqrt{108}} = \sqrt[3]{20.392} - \sqrt[3]{0.392} = 2$$

We can readily verify that 2 is indeed a solution of x³ + 6x − 20 = 0. But when Cardano tried to solve the equation x³ −15x −4 = 0 by this formula, his solution was

$$x = \sqrt[3]{2 + \sqrt{-121}} + \sqrt[3]{2 - \sqrt{-121}}$$

What was Cardano to make of this equation in the year 1545? In those days, negative numbers were themselves suspect, and a square root of a negative number was doubly preposterous! Today, we know that

$$(2 \pm j)^3 = 2 \pm j11 = 2 \pm \sqrt{-121}$$

Therefore, Cardano’s formula gives

$$x = (2+j) + (2-j) = 4$$

We can readily verify that x = 4 is indeed a solution of x³ − 15x − 4 = 0. Cardano tried to explain halfheartedly the presence of √−121 but ultimately dismissed the whole enterprise as being “as subtle as it is useless.” A generation later, however, Raphael Bombelli (1526–1573), after examining Cardano’s results, proposed acceptance of imaginary numbers as a necessary vehicle that would transport the mathematician from the real cubic equation to its real solution. In other words, although we begin and end with real numbers, we seem compelled to move into an unfamiliar world of imaginaries to complete our journey. To mathematicians of the day, this proposal seemed incredibly strange [7]. Yet they could not dismiss the idea of imaginary numbers so easily because this concept yielded the real solution of an equation. It took two more centuries for the full importance of complex numbers to become evident in the works of Euler, Gauss, and Cauchy. Still, Bombelli deserves credit for recognizing that such numbers have a role to play in algebra [7].

^† This equation is known as the depressed cubic equation. A general cubic equation

^{y3 +}py^{2 +}qy+^{r = 0}

can always be reduced to a depressed cubic form by substituting y = x − (p/3). Therefore, any general cubic equation can be solved if we know the solution to the depressed cubic. The depressed cubic was independently solved, first by Scipione del Ferro (1465–1526) and then by Niccolo Fontana (1499–1557). The latter is better known in the history of mathematics as Tartaglia (“Stammerer”). Cardano learned the secret of the depressed cubic solution from Tartaglia. He then showed that by using the substitution y = x−(p/3), a general cubic is reduced to a depressed cubic.

4 CHAPTER B BACKGROUND

In 1799 the German mathematician Karl Friedrich Gauss, at the ripe age of 22, proved the fundamental theorem of algebra, namely that every algebraic equation in one unknown has a root in the form of a complex number. He showed that every equation of the nth order has exactly n solutions (roots), no more and no less. Gauss was also one of the first to give a coherent account of complex numbers and to interpret them as points in a complex plane. It is he who introduced the term complex numbers and paved the way for their general and systematic use. The number system was once again broadened or generalized to include imaginary numbers. Ordinary (or real) numbers became a special case of generalized (or complex) numbers.

The utility of complex numbers can be understood readily by an analogy with two neighboring countries X and Y, as illustrated in Fig. B.1. If we want to travel from City a to City b (both in

Gerolamo Cardano Karl Friedrich Gauss

Figure B.1 Use of complex numbers can reduce the work.

Country X), the shortest route is through Country Y, although the journey begins and ends in Country X. We may, if we desire, perform this journey by an alternate route that lies exclusively in X, but this alternate route is longer. In mathematics we have a similar situation with real numbers (Country X) and complex numbers (Country Y). Most real-world problems start with real numbers, and the final results must also be in real numbers. But the derivation of results is considerably simplified by using complex numbers as an intermediary. It is also possible to solve any real-world problem by an alternate method, using real numbers exclusively, but such procedures would increase the work needlessly.

B.1-2 Algebra of Complex Numbers

A complex number (a,b) or a + jb can be represented graphically by a point whose Cartesian coordinates are (a,b) in a complex plane (Fig. B.2). Let us denote this complex number by z so that

$$z = a + jb \tag{B.1}$$

This representation is the Cartesian (or rectangular) form of complex number z. The numbers a and b (the abscissa and the ordinate) of z are the real part and the imaginary part, respectively, of z. They are also expressed as

$$Re z = a \qquad \text{and} \qquad Im z = b$$

Note that in this plane all real numbers lie on the horizontal axis, and all imaginary numbers lie on the vertical axis.

Complex numbers may also be expressed in terms of polar coordinates. If (r, θ ) are the polar coordinates of a point z = a+jb (see Fig. B.2), then

$$a = r \cos \theta$$

and $b = r \sin \theta$

Consequently,

$$z = a + jb = r\cos\theta + jr\sin\theta = r(\cos\theta + j\sin\theta)$$

(B.2)

Euler’s formula states that

$$e^{j\theta} = \cos\theta + j\sin\theta \tag{B.3}$$

To prove Euler’s formula, we use a Maclaurin series to expand ej^θ , cos θ, and sin θ:

$$e^{j\theta} = 1 + j\theta + \frac{(j\theta)^2}{2!} + \frac{(j\theta)^3}{3!} + \frac{(j\theta)^4}{4!} + \frac{(j\theta)^5}{5!} + \frac{(j\theta)^6}{6!} + \cdots$$

\n

$$= 1 + j\theta - \frac{\theta^2}{2!} - j\frac{\theta^3}{3!} + \frac{\theta^4}{4!} + j\frac{\theta^5}{5!} - \frac{\theta^6}{6!} - \cdots$$

\n

$$\cos \theta = 1 - \frac{\theta^2}{2!} + \frac{\theta^4}{4!} - \frac{\theta^6}{6!} + \frac{\theta^8}{8!} + \cdots$$

\n

$$\sin \theta = \theta - \frac{\theta^3}{3!} + \frac{\theta^5}{5!} - \frac{\theta^7}{7!} + \cdots$$

Clearly, it follows that ej^θ = cos θ +jsin θ. Using Eq. (B.3) in Eq. (B.2) yields

$$z = re^{i\theta} \tag{B.4}$$

This representation is the polar form of complex number z.

Summarizing, a complex number can be expressed in rectangular form a + jb or polar form rej^θ with

$$a = r \cos \theta$$

\n

$$b = r \sin \theta$$

and

$$r = \sqrt{a^2 + b^2}$$

\n

$$\theta = \tan^{-1} \left(\frac{b}{a}\right)$$

(B.5)

Observe that r is the distance of the point z from the origin. For this reason, r is also called the magnitude (or absolute value) of z and is denoted by |z|. Similarly, θ is called the angle of z and is denoted by z. Therefore, we can also write polar form of Eq. (B.4) as

$$z = |z|e^{j\angle z}$$

where $|z| = r$ and $\angle z = \theta$

Using polar form, we see that the reciprocal of a complex number is given by

$$\frac{1}{z} = \frac{1}{re^{j\theta}} = \frac{1}{r}e^{-j\theta} = \frac{1}{|z|}e^{-j\sqrt{z}}$$

CONJUGATE OF A COMPLEX NUMBER

We define z∗, the conjugate of z = a+jb, as

$$z^* = a - jb = re^{-j\theta} = |z|e^{-j\angle z}$$

(B.6)

The graphical representations of a number z and its conjugate z^∗ are depicted in Fig. B.2. Observe that z^∗ is a mirror image of z about the horizontal axis. To find the conjugate of any number, we need only replace j with −j in that number (which is the same as changing the sign of its angle).

The sum of a complex number and its conjugate is a real number equal to twice the real part of the number:

$$z + z^* = (a + jb) + (a - jb) = 2a = 2 \operatorname{Re} z$$

Thus, we see that the real part of complex number z can be computed as

$$\text{Re}\,z = \frac{z + z^*}{2} \tag{B.7}$$

Similarly, the imaginary part of complex number z can be computed as

$$\operatorname{Im} z = \frac{z - z^*}{2j} \tag{B.8}$$

The product of a complex number z and its conjugate is a real number |z| 2, the square of the magnitude of the number:

$$zz^* = |z|e^{j\angle z}|z|e^{-j\angle z} = |z|^2$$

(B.9)

UNDERSTANDING SOME USEFUL IDENTITIES

In a complex plane, rej^θ represents a point at a distance r from the origin and at an angle θ with the horizontal axis, as shown in Fig. B.3a. For example, the number −1 is at a unit distance from the origin and has an angle π or −π (more generally, π plus any integer multiple of 2π), as seen from Fig. B.3b. Therefore,

$$-1 = e^{j(\pi + 2\pi n)} \qquad n \text{ integer}$$

The number 1, on the other hand, is also at a unit distance from the origin, but has an angle 0 (more generally, 0 plus any integer multiple of 2π). Therefore,

$$1 = e^{j2\pi n} \qquad n \text{ integer}$$

(B.10)

The number j is at a unit distance from the origin and its angle is ^{π 2} (more generally, ^{π 2} plus any integer multiple of 2π), as seen from Fig. B.3b. Therefore,

$$j = e^{j(\frac{\pi}{2} + 2\pi n)} \qquad n \text{ integer}$$

Similarly,

$$-j = e^{j(-\frac{\pi}{2} + 2\pi n)} \qquad n \text{ integer}$$

Notice that the angle of any complex number is only known within an integer multiple of 2π.

This discussion shows the usefulness of the graphic picture of rej^θ . This picture is also helpful in several other applications. For example, to determine the limit of e(α+jω)^t as t → ∞, we note that

ejω*^t*

$$e^{(\alpha+j\omega)t} = e^{\alpha t} e^{j\omega t}$$

\n

$$\lim_{\rho \to 0} \frac{e^{j\theta}}{\log \rho} = \lim_{\rho \to 0} \frac{e^{j\theta}}{\log \rho} = \lim_{\rho \to 0} \frac{1}{\sqrt{\frac{\pi}{2}}} e^{j\theta}$$

\n

$$\lim_{\rho \to 0} \frac{1}{\sqrt{\frac{\pi}{2}}} e^{j\theta} = \lim_{\rho \to 0} \frac{1}{\sqrt{\frac{\pi}{2}}} e^{j\theta}$$

\n

\lim_{\rho \to 0} \frac{1}{\sqrt{\frac{\pi}{2}}} e^{j\theta} = \lim_{\rho \to 0} \frac{1}{\sqrt{\frac{\pi}{2}}} e^{j\theta} = \lim_{\rho \to 0} \frac{1}{\sqrt{\frac{\pi}{2}}} e^{j\theta} = \lim_{\rho \to 0} \frac{1}{\sqrt{\frac{\pi}{2}}} e^{j\theta} = \lim_{\rho \to 0} \frac{1}{\sqrt{\frac{\pi}{2}}} e^{j\theta} = \lim_{\rho \to 0} \frac{1}{\sqrt{\frac{\pi}{2}}} e^{j\theta} = \lim_{\rho \to 0} \frac{1}{\sqrt{\frac{\pi}{2}}} e^{j\theta} = \lim_{\rho \to 0} \frac{1}{\sqrt{\frac{\pi}{2}}} e^{j\theta} = \lim_{\rho \to 0} \frac{1}{\sqrt{\frac{\pi}{2}}} e^{j\theta} = \lim_{\rho \to 0} \frac{1}{\sqrt{\frac{\pi}{2}}} e^{j\theta} = \lim_{\rho \to 0} \frac{1}{\sqrt{\frac{\pi}{2}}} e^{j\theta} = \lim_{\rho \to 0} \frac{1}{\sqrt{\frac{\pi}{2}}} e^{j\theta} = \lim_{\rho \to 0} \frac{1}{\sqrt{\frac{\pi}{2}}} e^{j\theta} = \lim_{\rho \to 0} \frac{1}{\sqrt{\frac{\pi}{2}}} e^{j\theta} = \lim_{\rho \to 0} \frac{1}{\sqrt{\frac{\pi}{2}}} e^{j\theta} = \lim_{\rho \to 0} \frac{1}{\sqrt{\frac{\pi}{2}}} e^{j\theta} = \lim_{\rho \to 0} \frac{1}{\sqrt{\frac{\pi}{2}}} e^{j\theta} = \lim_{\rho \to 0} \frac{1}{\sqrt{\frac{\pi}{2}}} e^{j\theta} = \lim_{\rho \to 0} \frac{

Figure B.3 Understanding some useful identities in terms of rej^θ .

8 CHAPTER B BACKGROUND

Now the magnitude of ejω*^t* is unity regardless of the value of ω or t because ejω*^t* = rej^θ with r = 1. Therefore, eα*^t* determines the behavior of e(α+jω)^t as t → ∞ and

$$\lim_{t \to \infty} e^{(\alpha + j\omega)t} = \lim_{t \to \infty} e^{\alpha t} e^{j\omega t} = \begin{cases} 0 & \alpha < 0 \\ \infty & \alpha > 0 \end{cases}$$

In future discussions, you will find it very useful to remember rej^θ as a number at a distance r from the origin and at an angle θ with the horizontal axis of the complex plane.

A WARNING ABOUT COMPUTING ANGLES WITH CALCULATORS

From the Cartesian form a + jb, we can readily compute the polar form rej^θ [see Eq. (B.5)]. Calculators provide ready conversion of rectangular into polar and vice versa. However, if a calculator computes an angle of a complex number by using an inverse tangent function θ = tan−1(b/a), proper attention must be paid to the quadrant in which the number is located. For instance, θ corresponding to the number −2 − j3 is tan−¹(−3/−2). This result is not the same as tan−¹(3/2). The former is −123.7◦, whereas the latter is 56.3◦. A calculator cannot make this distinction and can give a correct answer only for angles in the first and fourth quadrants.† A calculator will read tan−¹(−3/−2) as tan−¹(3/2), which is clearly wrong. When you are computing inverse trigonometric functions, if the angle appears in the second or third quadrant, the answer of the calculator is off by 180◦. The correct answer is obtained by adding or subtracting 180◦ to the value found with the calculator (either adding or subtracting yields the correct answer). For this reason, it is advisable to draw the point in the complex plane and determine the quadrant in which the point lies. This issue will be clarified by the following examples.

EXAMPLE B.1 Cartesian to Polar Form

Express the following numbers in polar form: (a) 2+j3, (b) −2+j1, (c) −2−j3, and (d) 1−j3.

(a)

$$|z| = \sqrt{2^2 + 3^2} = \sqrt{13}$$

$\angle z = \tan^{-1}(\frac{3}{2}) = 56.3^{\circ}$

In this case the number is in the first quadrant, and a calculator will give the correct value of 56.3◦. Therefore (see Fig. B.4a), we can write

$$2 + j3 = \sqrt{13} e^{j56.3^{\circ}}$$

(b)

$$|z| = \sqrt{(-2)^2 + 1^2} = \sqrt{5}$$

$\angle z = \tan^{-1}(\frac{1}{-2}) = 153.4^{\circ}$

In this case the angle is in the second quadrant (see Fig. B.4b), and therefore the answer given by the calculator, tan−¹(1/−2) = −26.6◦, is off by 180◦. The correct answer is

^† Calculators with two-argument inverse tangent functions will correctly compute angles.

(−26.6±180)◦ = 153.4◦ or −206.6◦. Both values are correct because they represent the same angle. It is a common practice to choose an angle whose numerical value is less than 180◦. Such a value is called the principal value of the angle, which in this case is 153.4◦. Therefore,

$$-2 + j1 = \sqrt{5}e^{j153.4^{\circ}}$$