Signals and Systems

, by Oppenheim, Alan V.; Willsky, Alan S.; Hamid, with S.

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ISBN: 9780138147570 | 0138147574
Cover: Hardcover
Copyright: 8/6/1996

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This authoritative book, highly regarded for its intellectual quality and contributions provides a solid foundation and life-long reference for anyone studying the most important methods of modern signal and system analysis.The major changes of the revision are reorganization of chapter material and the addition of a much wider range of difficulties.

PREFACE

XVII

(8)

ACKNOWLEDGMENTS

XXV

(2)

FOREWORD

XXVII

1 SIGNALS AND SYSTEMS

(73)

1.0 Introduction

(1)

1.1 Continuous-Time and Discrete-Time Signals

(6)

1.1.1 Examples and Mathematical Representation

(4)

1.1.2 Signal Energy and Power

(2)

1.2 Transformations of the Independent Variable

(7)

1.2.1 Examples of Transformations of the Independent Variable

(3)

1.2.2 Periodic Signals

(2)

1.2.3 Even and Odd Signals

(1)

1.3 Exponential and Sinusoidal Signals

(16)

1.3.1 Continuous-Time Complex Exponential and Sinusoidal Signals

(6)

1.3.2 Discrete-Time Complex Exponential and Sinusoidal Signals

(4)

1.3.3 Periodicity Properties of Discrete-Time Complex Exponentials

(5)

1.4 The Unit Impulse and Unit Step Functions

(8)

1.4.1 The Discrete-Time Unit Impulse and Unit Step Sequences

(2)

1.4.2 The Continuous-Time Unit Step and Unit Impulse Functions

(6)

1.5 Continuous-Time and Discrete-Time Systems

(6)

1.5.1 Simple Examples of Systems

(2)

1.5.2 Interconnections of Systems

(3)

1.6 Basic System Properties

(12)

1.6.1 Systems with and without Memory

(1)

1.6.2 Invertibility and Inverse Systems

(1)

1.6.3 Causality

(2)

1.6.4 Stability

(2)

1.6.5 Time Invariance

(3)

1.6.6 Linearity

(3)

1.7 Summary

(1)

Problems

(17)

2 LINEAR TIME-INVARIANT SYSTEMS

(103)

2.0 Introduction

(1)

2.1 Discrete-Time LTI Systems: The Convolution Sum

(15)

2.1.1 The Representation of Discrete-Time Signals in Terms of Impulses

(2)

2.1.2 The Discrete-Time Unit Impulse Response and the Convolution-Sum Representation of LTI Systems

(13)

2.2 Continuous-Time LTI Systems: The Convolution Integral

(13)

2.2.1 The Representation of Continuous-Time Signals in Terms of Impulses

(4)

2.2.2 The Continuous-Time Unit Impulse Response and the Convolution Integral Representation of LTI Systems

(9)

2.3 Properties of Linear Time-Invariant Systems

103

(13)

2.3.1 The Commutative Property

104

(1)

2.3.2 The Distributive Property

104

(3)

2.3.3 The Associative Property

107

(1)

2.3.4 LTI Systems with and without Memory

108

(1)

2.3.5 Invertibility of LTI Systems

109

(3)

2.3.6 Causality for LTI Systems

112

(1)

2.3.7 Stability for LTI Systems

113

(2)

2.3.8 The Unit Step Response of an LTI System

115

(1)

2.4 Causal LTI Systems Described by Differential and Difference Equations

116

(11)

2.4.1 Linear Constant-Coefficient Differential Equations

117

(4)

2.4.2 Linear Constant-Coefficient Difference Equations

121

(3)

2.4.3 Block Diagram Representations of First-Order Systems Described by Differential and Difference Equations

124

(3)

2.5 Singularity Functions

127

(10)

2.5.1 The Unit Impulse as an Idealized Short Pulse

128

(3)

2.5.2 Defining the Unit Impulse through Convolution

131

(1)

2.5.3 Unit Doublets and Other Singularity Functions

132

(5)

2.6 Summary

137

(1)

Problems

137

(40)

3 FOURIER SERIES REPRESENTATION OF PERIODIC SIGNALS

177

(107)

3.0 Introduction

177

(1)

3.1 A Historical Perspective

178

(4)

3.2 The Response of LTI Systems to Complex Exponentials

182

(4)

3.3 Fourier Series Representation of Continuous-Time Periodic Signals

186

(9)

3.3.1 Linear Combinations of Harmonically Related Complex Exponentials

186

(4)

3.3.2 Determination of the Fourier Series Representation of a Continuous-Time Periodic Signal

190

(5)

3.4 Convergence of the Fourier Series

195

(7)

3.5 Properties of Continuous-Time Fourier Series

202

(9)

3.5.1 Linearity

202

(1)

3.5.2 Time Shifting

202

(1)

3.5.3 Time Reversal

203

(1)

3.5.4 Time Scaling

204

(1)

3.5.5 Multiplication

204

(1)

3.5.6 Conjugation and Conjugate Symmetry

204

(1)

3.5.7 Parseval's Relation for Continuous-Time Periodic Signals

205

(1)

3.5.8 Summary of Properties of the Continuous-Time Fourier Series

205

(1)

3.5.9 Examples

205

(6)

3.6 Fourier Series Representation of Discrete-Time Periodic Signals

211

(10)

3.6.1 Linear Combinations of Harmonically Related Complex Exponentials

211

(1)

3.6.2 Determination of the Fourier Series Representation of a Periodic Signal

212

(9)

3.7 Properties of Discrete-Time Fourier Series

221

(5)

3.7.1 Multiplication

222

(1)

3.7.2 First Difference

222

(1)

3.7.3 Parseval's Relation for Discrete-Time Periodic Signals

223

(1)

3.7.4 Examples

223

(3)

3.8 Fourier Series and LTI Systems

226

(5)

3.9 Filtering

231

(8)

3.9.1 Frequency-Shaping Filters

232

(4)

3.9.2 Frequency-Selective Filters

236

(3)

3.10 Examples of Continuous-Time Filters Described by Differential Equations

239

(5)

3.10.1 A Simple RC Lowpass Filter

239

(2)

3.10.2 A Simple RC Highpass Filter

241

(3)

3.11 Examples of Discrete-Time Filters Described by Difference Equations

244

(5)

3.11.1 First-Order Recursive Discrete-Time Filters

244

(1)

3.11.2 Nonrecursive Discrete-Time Filters

245

(4)

3.12 Summary

249

(1)

Problems

250

(34)

4 THE CONTINUOUS-TIME FOURIER TRANSFORM

284

(74)

4.0 Introduction

284

(1)

4.1 Representation of Aperiodic Signals: The Continuous-Time Fourier Transform

285

(11)

4.1.1 Development of the Fourier Transform Representation of an Aperiodic Signal

285

(4)

4.1.2 Convergence of Fourier Transforms

289

(1)

4.1.3 Examples of Continuous-Time Fourier Transforms

290

(6)

4.2 The Fourier Transform for Periodic Signals

296

(4)

4.3 Properties of the Continuous-Time Fourier Transform

300

(14)

4.3.1 Linearity

301

(1)

4.3.2 Time Shifting

301

(2)

4.3.3 Conjugation and Conjugate Symmetry

303

(3)

4.3.4 Differentiation and Integration

306

(2)

4.3.5 Time and Frequency Scaling

308

(1)

4.3.6 Duality

309

(3)

4.3.7 Parseval's Relation

312

(2)

4.4 The Convolution Property

314

(8)

4.4.1 Examples

317

(5)

4.5 The Multiplication Property

322

(6)

4.5.1 Frequency-Selective Filtering with Variable Center Frequency

325

(3)

4.6 Tables of Fourier Properties and of Basic Fourier Transform Pairs

328

(2)

4.7 Systems Characterized by Linear Constant-Coefficient Differential Equations

330

(3)

4.8 Summary

333

(1)

Problems

334

(24)

5 THE DISCRETE-TIME FOURIER TRANSFORM

358

(65)

5.0 Introduction

358

(1)

5.1 Representation of Aperiodic Signals: The Discrete-Time Fourier Transform

359

(8)

5.1.1 Development of the Discrete-Time Fourier Transform

359

(3)

5.1.2 Examples of Discrete-Time Fourier Transforms

362

(4)

5.1.3 Convergence Issues Associated with the Discrete-Time Fourier Transform

366

(1)

5.2 The Fourier Transform for Periodic Signals

367

(5)

5.3 Properties of the Discrete-Time Fourier Transform

372

(10)

5.3.1 Periodicity of the Discrete-Time Fourier Transform

373

(1)

5.3.2 Linearity of the Fourier Transform

373

(1)

5.3.3 Time Shifting and Frequency Shifting

373

(2)

5.3.4 Conjugation and Conjugate Symmetry

375

(1)

5.3.5 Differencing and Accumulation

375

(1)

5.3.6 Time Reversal

376

(1)

5.3.7 Time Expansion

377

(3)

5.3.8 Differentiation in Frequency

380

(1)

5.3.9 Parseval's Relation

380

(2)

5.4 The Convolution Property

382

(6)

5.4.1 Examples

383

(5)

5.5 The Multiplication Property

388

(2)

5.6 Tables of Fourier Transform Properties and Basic Fourier Transform Pairs

390

(1)

5.7 Duality

390

(6)

5.7.1 Duality in the Discrete-Time Fourier Series

391

(4)

5.7.2 Duality between the Discrete-Time Fourier Transform and the Continuous-Time Fourier Series

395

(1)

5.8 Systems Characterized by Linear Constant-Coefficient Difference Equations

396

(3)

5.9 Summary

399

(1)

Problems

400

(23)

6 TIME AND FREQUENCY CHARACTERIZATION OF SIGNALS AND SYSTEMS

423

(91)

6.0 Introduction

423

(1)

6.1 The Magnitude-Phase Representation of the Fourier Transform

423

(4)

6.2 The Magnitude-Phase Representation of the Frequency Response of LTI Systems

427

(12)

6.2.1 Linear and Nonlinear Phase

428

(2)

6.2.2 Group Delay

430

(6)

6.2.3 Log-Magnitude and Bode Plots

436

(3)

6.3 Time-Domain Properties of Ideal Frequency-Selective Filters

439

(5)

6.4 Time-Domain and Frequency-Domain Aspects of Nonideal Filters

444

(4)

6.5 First-Order and Second-Order Continuous-Time Systems

448

(13)

6.5.1 First-Order Continuous-Time Systems

448

(3)

6.5.2 Second-Order Continuous-Time Systems

451

(5)

6.5.3 Bode Plots for Rational Frequency Responses

456

(5)

6.6 First-Order and Second-Order Discrete-Time Systems

461

(11)

6.6.1 First-Order Discrete-Time Systems

461

(4)

6.6.2 Second-Order Discrete-Time Systems

465

(7)

6.7 Examples of Time- and Frequency-Domain Analysis of Systems

472

(10)

6.7.1 Analysis of an Automobile Suspension System

473

(3)

6.7.2 Examples of Discrete-Time Nonrecursive Filters

476

(6)

6.8 Summary

482

(1)

Problems

483

(31)

7 SAMPLING

514

(68)

7.0 Introduction

514

(1)

7.1 Representation of a Continuous-Time Signal by Its Samples: The Sampling Theorem

515

(7)

7.1.1 Impulse-Train Sampling

516

(4)

7.1.2 Sampling with a Zero-Order Hold

520

(2)

7.2 Reconstruction of a Signal from Its Samples Using Interpolation

522

(5)

7.3 The Effect of Undersampling: Aliasing

527

(7)

7.4 Discrete-Time Processing of Continuous-Time Signals

534

(11)

7.4.1 Digital Differentiator

541

(2)

7.4.2 Half-Sample Delay

543

(2)

7.5 Sampling of Discrete-Time Signals

545

(10)

7.5.1 Impulse-Train Sampling

545

(4)

7.5.2 Discrete-Time Decimation and Interpolation

549

(6)

7.6 Summary

555

(1)

Problems

556

(26)

8 COMMUNICATION SYSTEMS

582

(72)

8.0 Introduction

582

(1)

8.1 Complex Exponential and Sinusoidal Amplitude Modulation

583

(4)

8.1.1 Amplitude Modulation with a Complex Exponential Carrier

583

(2)

8.1.2 Amplitude Modulation with a Sinusoidal Carrier

585

(2)

8.2 Demodulation for Sinusoidal AM

587

(7)

8.2.1 Synchronous Demodulation

587

(3)

8.2.2 Asynchronous Demodulation

590

(4)

8.3 Frequency-Division Multiplexing

594

(3)

8.4 Single-Sideband Sinusoidal Amplitude Modulation

597

(4)

8.5 Amplitude Modulation with a Pulse-Train Carrier

601

(3)

8.5.1 Modulation of a Pulse-Train Carrier

601

(3)

8.5.2 Time-Division Multiplexing

604

(1)

8.6 Pulse-Amplitude Modulation

604

(7)

8.6.1 Pulse-Amplitude Modulated Signals

604

(3)

8.6.2 Intersymbol Interference in PAM Systems

607

(3)

8.6.3 Digital Pulse-Amplitude and Pulse-Code Modulation

610

(1)

8.7 Sinusoidal Frequency Modulation

611

(8)

8.7.1 Narrowband Frequency Modulation

613

(2)

8.7.2 Wideband Frequency Modulation

615

(2)

8.7.3 Periodic Square-Wave Modulating Signal

617

(2)

8.8 Discrete-Time Modulation

619

(4)

8.8.1 Discrete-Time Sinusoidal Amplitude Modulation

619

(4)

8.8.2 Discrete-Time Transmodulation

623

(1)

8.9 Summary

623

(2)

Problems

625

(29)

9 THE LAPLACE TRANSFORM

654

(87)

9.0 Introduction

654

(1)

9.1 The Laplace Transform

655

(7)

9.2 The Region of Convergence for Laplace Transforms

662

(8)

9.3 The Inverse Laplace Transform

670

(4)

9.4 Geometric Evaluation of the Fourier Transform from the Pole-Zero Plot

674

(8)

9.4.1 First-Order Systems

676

(1)

9.4.2 Second-Order Systems

677

(4)

9.4.3 All-Pass Systems

681

(1)

9.5 Properties of the Laplace Transform

682

(10)

9.5.1 Linearity of the Laplace Transform

683

(1)

9.5.2 Time Shifting

684

(1)

9.5.3 Shifting in the s-Domain

685

(1)

9.5.4 Time Scaling

685

(2)

9.5.5 Conjugation

687

(1)

9.5.6 Convolution Property

687

(1)

9.5.7 Differentiation in the Time Domain

688

(1)

9.5.8 Differentiation in the s-Domain

688

(2)

9.5.9 Integration in the Time Domain

690

(1)

9.5.10 The Initial- and Final-Value Theorems

690

(1)

9.5.11 Table of Properties

691

(1)

9.6 Some Laplace Transform Pairs

692

(1)

9.7 Analysis and Characterization of LTI Systems Using the Laplace Transform

693

(13)

9.7.1 Causality

693

(2)

9.7.2 Stability

695

(3)

9.7.3 LTI Systems Characterized by Linear Constant-Coefficient Differential Equations

698

(3)

9.7.4 Examples Relating System Behavior to the System Function

701

(2)

9.7.5 Butterworth Filters

703

(3)

9.8 System Function Algebra and Block Diagram Representations

706

(8)

9.8.1 System Functions for Interconnections of LTI Systems

707

(1)

9.8.2 Block Diagram Representations for Causal LTI Systems Described by Differential Equations and Rational System Functions

708

(6)

9.9 The Unilateral Laplace Transform

714

(6)

9.9.1 Examples of Unilateral Laplace Transforms

714

(2)

9.9.2 Properties of the Unilateral Laplace Transform

716

(3)

9.9.3 Solving Differential Equations Using the Unilateral Laplace Transform

719

(1)

9.10 Summary

720

(1)

Problems

721

(20)

10 THE Z-TRANSFORM

741

(75)

10.0 Introduction

741

(1)

10.1 The z-Transform

741

(7)

10.2 The Region of Convergence for the z-Transform

748

(9)

10.3 The Inverse z-Transform

757

(6)

10.4 Geometric Evaluation of the Fourier Transform from the Pole-Zero Plot

763

(4)

10.4.1 First-Order Systems

763

(2)

10.4.2 Second-Order Systems

765

(2)

10.5 Properties of the z-Transform

767

(7)

10.5.1 Linearity

767

(1)

10.5.2 Time Shifting

767

(1)

10.5.3 Scaling in the z-Domain

768

(1)

10.5.4 Time Reversal

769

(1)

10.5.5 Time Expansion

769

(1)

10.5.6 Conjugation

770

(1)

10.5.7 The Convolution Property

770

(2)

10.5.8 Differentiation in the z-Domain

772

(1)

10.5.9 The Initial-Value Theorem

773

(1)

10.5.10 Summary of Properties

774

(1)

10.6 Some Common z-Transform Pairs

774

(1)

10.7 Analysis and Characterization of LTI Systems Using z-Transforms

774

(9)

10.7.1 Causality

776

(1)

10.7.2 Stability

777

(2)

10.7.3 LTI Systems Characterized by Linear Constant-Coefficient Difference Equations

779

(2)

10.7.4 Examples Relating System Behavior to the System Function

781

(2)

10.8 System Function Algebra and Block Diagram Representations

783

(6)

10.8.1 System Functions for Interconnections of LTI Systems

784

(1)

10.8.2 Block Diagram Representations for Causal LTI Systems Described by Difference Equations and Rational System Functions

784

(5)

10.9 The Unilateral z-Transform

789

(7)

10.9.1 Examples of Unilateral z-Transforms and Inverse Transforms

790

(2)

10.9.2 Properties of the Unilateral z-Transform

792

(3)

10.9.3 Solving Difference Equations Using the Unilateral z-Transform

795

(1)

10.10 Summary

796

(1)

Problems

797

(19)

11 LINEAR FEEDBACK SYSTEMS

816

(93)

11.0 Introduction

816

(3)

11.1 Linear Feedback Systems

819

(1)

11.2 Some Applications and Consequences of Feedback

820

(12)

11.2.1 Inverse System Design

820

(1)

11.2.2 Compensation for Nonideal Elements

821

(2)

11.2.3 Stabilization of Unstable Systems

823

(3)

11.2.4 Sampled-Data Feedback Systems

826

(2)

11.2.5 Tracking Systems

828

(2)

11.2.6 Destabilization Caused by Feedback

830

(2)

11.3 Root-Locus Analysis of Linear Feedback Systems

832

(14)

11.3.1 An Introductory Example

833

(1)

11.3.2 Equation for the Closed-Loop Poles

834

(2)

11.3.3 The End Points of the Root Locus: The Closed-Loop Poles for K = 0 and ¦K¦ = +(XXX)

836

(1)

11.3.4 The Angle Criterion

836

(5)

11.3.5 Properties of the Root Locus

841

(5)

11.4 The Nyquist Stability Criterion

846

(12)

11.4.1 The Encirclement Property

847

(3)

11.4.2 The Nyquist Criterion for Continuous-Time LTI Feedback Systems

850

(6)

11.4.3 The Nyquist Criterion for Discrete-Time LTI Feedback Systems

856

(2)

11.5 Gain and Phase Margins

858

(8)

11.6 Summary

866

(1)

Problems

867

(42)

APPENDIX PARTIAL-FRACTION EXPANSION

909

(12)

BIBLIOGRAPHY

921

(10)

ANSWERS

931

(10)

INDEX

941

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Signals and Systems

Summary

Table of Contents

Supplemental Materials