Fundamentals of Signals and systems

FUNDAMENTALS OF

SIGNALS AND SYSTEMS

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FUNDAMENTALS OF

SIGNALS AND SYSTEMS

BENOIT BOULET

CHARLES RIVER MEDIA

Boston, Massachusetts

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Library of Congress Cataloging-in-Publication Data

Boulet, Benoit, 1967-

Fundamentals of signals and systems / Benoit Boulet.— 1st ed.

p. cm.

Includes index.

ISBN 1-58450-381-5 (hardcover with cd-rom : alk. paper)

1. Signal processing. 2. Signal generators. 3. Electric filters. 4. Signal detection. 5. System analysis.

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eISBN: 1-58450-660-1

Acknowledgments xiii

Preface xv

1 Elementary Continuous-Time and Discrete-Time Signals and Systems 1

Systems in Engineering 2

Functions of Time as Signals 2

Transformations of the Time Variable 4

Periodic Signals 8

Exponential Signals 9

Periodic Complex Exponential and Sinusoidal Signals 17

Finite-Energy and Finite-Power Signals 21

Even and Odd Signals 23

Discrete-Time Impulse and Step Signals 25

Generalized Functions 26

System Models and Basic Properties 34

Summary 42

To Probe Further 43

Exercises 43

2 Linear Time-Invariant Systems 53

Discrete-Time LTI Systems: The Convolution Sum 54

Continuous-Time LTI Systems: The Convolution Integral 67

Properties of Linear Time-Invariant Systems 74

Summary 81

To Probe Further 81

Exercises 81

3 Differential and Difference LTI Systems 91

Causal LTI Systems Described by Differential Equations 92

Causal LTI Systems Described by Difference Equations 96

Contents

v

Impulse Response of a Differential LTI System 101

Impulse Response of a Difference LTI System 109

Characteristic Polynomials and Stability of Differential and

Difference Systems 112

Time Constant and Natural Frequency of a First-Order LTI

Differential System 116

Eigenfunctions of LTI Difference and Differential Systems 117

Summary 118

To Probe Further 119

Exercises 119

4 Fourier Series Representation of Periodic Continuous-Time Signals 131

Linear Combinations of Harmonically Related Complex Exponentials 132

Determination of the Fourier Series Representation of a

Continuous-Time Periodic Signal 134

Graph of the Fourier Series Coefficients: The Line Spectrum 137

Properties of Continuous-Time Fourier Series 139

Fourier Series of a Periodic Rectangular Wave 141

Optimality and Convergence of the Fourier Series 144

Existence of a Fourier Series Representation 146

Gibbs Phenomenon 147

Fourier Series of a Periodic Train of Impulses 148

Parseval Theorem 150

Power Spectrum 151

Total Harmonic Distortion 153

Steady-State Response of an LTI System to a Periodic Signal 155

Summary 157

To Probe Further 157

Exercises 158

5 The Continuous-Time Fourier Transform 175

Fourier Transform as the Limit of a Fourier Series 176

Properties of the Fourier Transform 180

Examples of Fourier Transforms 184

The Inverse Fourier Transform 188

Duality 191

Convergence of the Fourier Transform 192

The Convolution Property in the Analysis of LTI Systems 192

vi Contents

Fourier Transforms of Periodic Signals 199

Filtering 202

Summary 210

To Probe Further 211

Exercises 211

6 The Laplace Transform 223

Definition of the Two-Sided Laplace Transform 224

Inverse Laplace Transform 226

Convergence of the Two-Sided Laplace Transform 234

Poles and Zeros of Rational Laplace Transforms 235

Properties of the Two-Sided Laplace Transform 236

Analysis and Characterization of LTI Systems Using the

Laplace Transform 241

Definition of the Unilateral Laplace Transform 243

Properties of the Unilateral Laplace Transform 244

Summary 247

To Probe Further 248

Exercises 248

7 Application of the Laplace Transform to LTI Differential Systems 259

The Transfer Function of an LTI Differential System 260

Block Diagram Realizations of LTI Differential Systems 264

Analysis of LTI Differential Systems with Initial Conditions Using

the Unilateral Laplace Transform 272

Transient and Steady-State Responses of LTI Differential Systems 274

Summary 276

To Probe Further 276

Exercises 277

8 Time and Frequency Analysis of BIBO Stable,

Continuous-Time LTI Systems 285

Relation of Poles and Zeros of the Transfer Function to the

Frequency Response 286

Bode Plots 290

Frequency Response of First-Order Lag, Lead, and Second-Order

Lead-Lag Systems 296

Contents vii

Frequency Response of Second-Order Systems 300

Step Response of Stable LTI Systems 307

Ideal Delay Systems 315

Group Delay 316

Non-Minimum Phase and All-Pass Systems 316

Summary 319

To Probe Further 319

Exercises 319

9 Application of Laplace Transform Techniques to

Electric Circuit Analysis 329

Review of Nodal Analysis and Mesh Analysis of Circuits 330

Transform Circuit Diagrams: Transient and Steady-State Analysis 334

Operational Amplifier Circuits 340

Summary 344

To Probe Further 344

Exercises 344

10 State Models of Continuous-Time LTI Systems 351

State Models of Continuous-Time LTI Differential Systems 352

Zero-State Response and Zero-Input Response of a

Continuous-Time State-Space System 361

Laplace-Transform Solution for Continuous-Time State-Space Systems 367

State Trajectories and the Phase Plane 370

Block Diagram Representation of Continuous-Time State-Space Systems 372

Summary 373

To Probe Further 373

Exercises 373

11 Application of Transform Techniques to LTI Feedback

Control Systems 381

Introduction to LTI Feedback Control Systems 382

Closed-Loop Stability and the Root Locus 394

The Nyquist Stability Criterion 404

Stability Robustness: Gain and Phase Margins 409

Summary 413

To Probe Further 413

Exercises 413

viii Contents

12 Discrete-Time Fourier Series and Fourier Transform 425

Response of Discrete-Time LTI Systems to Complex Exponentials 426

Fourier Series Representation of Discrete-Time Periodic Signals 426

Properties of the Discrete-Time Fourier Series 430

Discrete-Time Fourier Transform 435

Properties of the Discrete-Time Fourier Transform 439

DTFT of Periodic Signals and Step Signals 445

Duality 449

Summary 450

To Probe Further 450

Exercises 450

13 The z-Transform 459

Development of the Two-Sided z-Transform 460

ROC of the z-Transform 464

Properties of the Two-Sided z-Transform 465

The Inverse z-Transform 468

Analysis and Characterization of DLTI Systems Using the z-Transform 474

The Unilateral z-Transform 483

Summary 486

To Probe Further 487

Exercises 487

14 Time and Frequency Analysis of Discrete-Time Signals and Systems 497

Geometric Evaluation of the DTFT From the Pole-Zero Plot 498

Frequency Analysis of First-Order and Second-Order Systems 504

Ideal Discrete-Time Filters 510

Infinite Impulse Response and Finite Impulse Response Filters 519

Summary 531

To Probe Further 531

Exercises 532

15 Sampling Systems 541

Sampling of Continuous-Time Signals 542

Signal Reconstruction 546

Discrete-Time Processing of Continuous-Time Signals 552

Sampling of Discrete-Time Signals 557

Contents ix

Summary 564

To Probe Further 564

Exercises 564

16 Introduction to Communication Systems 577

Complex Exponential and Sinusoidal Amplitude Modulation 578

Demodulation of Sinusoidal AM 581

Single-Sideband Amplitude Modulation 587

Modulation of a Pulse-Train Carrier 591

Pulse-Amplitude Modulation 592

Time-Division Multiplexing 595

Frequency-Division Multiplexing 597

Angle Modulation 599

Summary 604

To Probe Further 605

Exercises 605

17 System Discretization and Discrete-Time LTI State-Space Models 617

Controllable Canonical Form 618

Observable Canonical Form 621

Zero-State and Zero-Input Response of a Discrete-Time

State-Space System 622

z-Transform Solution of Discrete-Time State-Space Systems 625

Discretization of Continuous-Time Systems 628

Summary 636

To Probe Further 637

Exercises 637

Appendix A: Using MATLAB 645

Appendix B: Mathematical Notation and Useful Formulas 647

Appendix C: About the CD-ROM 649

Appendix D: Tables of Transforms 651

Index 665

x Contents

List of Lectures

Lecture 1: Signal Models 1

Lecture 2: Some Useful Signals 12

Lecture 3: Generalized Functions and Input-Output System Models 26

Lecture 4: Basic System Properties 38

Lecture 5: LTI systems: Convolution Sum 53

Lecture 6: Convolution Sum and Convolution Integral 62

Lecture 7: Convolution Integral 69

Lecture 8: Properties of LTI Systems 74

Lecture 9: Definition of Differential and Difference Systems 91

Lecture 10: Impulse Response of a Differential System 101

Lecture 11: Impulse Response of a Difference System; Characteristic Polynomial

and Stability 109

Lecture 12: Definition and Properties of the Fourier Series 131

Lecture 13: Convergence of the Fourier Series 141

Lecture 14: Parseval Theorem, Power Spectrum, Response of LTI System to Periodic Input 148

Lecture 15: Definition and Properties of the Continuous-Time Fourier Transform 175

Lecture 16: Examples of Fourier Transforms, Inverse Fourier Transform 184

Lecture 17: Convergence of the Fourier Transform, Convolution Property and

LTI Systems 192

Lecture 18: LTI Systems, Fourier Transform of Periodic Signals 197

Lecture 19: Filtering 202

Lecture 20: Definition of the Laplace Transform 223

Lecture 21: Properties of the Laplace Transform, Transfer Function of an LTI System 236

Lecture 22: Definition and Properties of the Unilateral Laplace Transform 243

Lecture 23: LTI Differential Systems and Rational Transfer Functions 259

Lecture 24: Analysis of LTI Differential Systems with Block Diagrams 264

Lecture 25: Response of LTI Differential Systems with Initial Conditions 272

Lecture 26: Impulse Response of a Differential System 285

Lecture 27: The Bode Plot 290

Lecture 28: Frequency Responses of Lead, Lag, and Lead-Lag Systems 296

Lecture 29: Frequency Response of Second-Order Systems 300

Lecture 30: The Step Response 307

Lecture 31: Review of Nodal Analysis and Mesh Analysis of Circuits 329

Lecture 32: Transform Circuit Diagrams, Op-Amp Circuits 334

Lecture 33: State Models of Continuous-Time LTI Systems 351

Lecture 34: Zero-State Response and Zero-Input Response 361

Lecture 35: Laplace Transform Solution of State-Space Systems 367

Lecture 36: Introduction to LTI Feedback Control Systems 381

Lecture 37: Sensitivity Function and Transmission 387

Lecture 38: Closed-Loop Stability Analysis 394

Lecture 39: Stability Analysis Using the Root Locus 400

Lecture 40: They Nyquist Stability Criterion 404

Lecture 41: Gain and Phase Margins 409

Lecture 42: Definition of the Discrete-Time Fourier Series 425

Lecture 43: Properties of the Discrete-Time Fourier Series 430

Lecture 44: Definition of the Discrete-Time Fourier Transform 435

Contents xi

Lecture 45: Properties of the Discrete-Time Fourier Transform 439

Lecture 46: DTFT of Periodic and Step Signals, Duality 444

Lecture 47: Definition and Convergence of the z-Transform 459

Lecture 48: Properties of the z-Transform 465

Lecture 49: The Inverse z-Transform 468

Lecture 50: Transfer Function Characterization of DLTI Systems 474

Lecture 51: LTI Difference Systems and Rational Transfer Functions 478

Lecture 52: The Unilateral z-Transform 483

Lecture 53: Relationship Between the DTFT and the z-Transform 497

Lecture 54: Frequency Analysis of First-Order and Second-Order Systems 504

Lecture 55: Ideal Discrete-Time Filters 509

Lecture 56: IIR and FIR Filters 519

Lecture 57: FIR Filter Design by Windowing 524

Lecture 58: Sampling 541

Lecture 59: Signal Reconstruction and Aliasing 546

Lecture 60: Discrete-Time Processing of Continuous-Time Signals 552

Lecture 61: Equivalence to Continuous-Time Filtering; Sampling of

Discrete-Time Signals 556

Lecture 62: Decimation, Upsampling and Interpolation 558

Lecture 63: Amplitude Modulation and Synchronous Demodulation 577

Lecture 64: Asynchronous Demodulation 583

Lecture 65: Single Sideband Amplitude Modulation 586

Lecture 66: Pulse-Train and Pulse Amplitude Modulation 591

Lecture 67: Frequency-Division and Time-Division Multiplexing; Angle Modulation 595

Lecture 68: State Models of LTI Difference Systems 617

Lecture 69: Zero-State and Zero-Input Responses of Discrete-Time State Models 622

Lecture 70: Discretization of Continuous-Time LTI Systems 628

xii Contents

I

wish to acknowledge the contribution of Dr. Maier L. Blostein, emeritus pro￾fessor in the Department of Electrical and Computer Engineering at McGill

University. Our discussions over the past few years have led us to the current

course syllabi for Signals & Systems I and II, essentially forming the table of con￾tents of this textbook.

I would like to thank the many students whom, over the years, have reported

mistakes and suggested useful revisions to my Signals & Systems I and II course

notes.

The interesting and useful applets on the companion CD-ROM were pro￾grammed by the following students: Rafic El-Fakir (Bode plot applet) and Gul Pil

Joo (Fourier series and convolution applets). I thank them for their excellent work

and for letting me use their programs.

Acknowledgments

xiii

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