《SIGNALS & SYSTEMS SECOND EDITION》

1 SIGNALS AND SYSTEMS1

1.0 Introduction1

1.1 Continuous-Time and Discrete-Time Signals1

1.1.1 Examples and Mathematical Representation1

1.1.2 Signal Energy and Power5

1.2 Transformations of the Independent Variable7

1.2.1 Examples of Transformations of the Independent Variable8

1.2.2 Periodic Signals11

1.2.3 Even and Odd Signals13

1.3 Exponential and Sinusoidal Signals14

1.3.1 Continuous-Time Complex Exponential and Sinusoidal Signals15

1.3.2 Discrete-Time Complex Exponential and Sinusoidal Signals21

1.3.3 Periodicity Properties of Discrete-Time Complex Exponentials25

1.4 The Unit Impulse and Unit Step Functions30

1.4.1 The Discrete-Time Unit Impulse and Unit Step Sequences30

1.4.2 The Continuous-Time Unit Step and Unit Impulse Functions32

1.5 Continuous-Time and Discrete-Time Systems38

1.5.1 Simple Examples of Systems39

1.5.2 Interconnections of Systems41

1.6 Basic System Properties44

1.6.1 Systems with and without Memory44

1.6.2 Invertibility and Inverse Systems45

1.6.3 Causality46

1.6.4 Stability48

1.6.5 Time Invariance50

1.6.6 Linearity53

1.7 Summary56

Problems57

2 LINEAR TIME-INVARIANT SYSTEMS74

2.0 Introduction74

2.1 Discrete-Time LTI Systems: The Convolution Sum75

2.1.1 The Representation of Discrete-Time Signals in Terms of Impulses75

2.1.2 The Discrete-Time Unit Impulse Response and the Convolution-Sum Representation of LTI Systems77

2.2 Continuous-Time LTI Systems: The Convolution Integral90

2.2.1 The Representation of Continuous-Time Signals in Terms of Impulses90

2.2.2 The Continuous-Time Unit Impulse Response and the Convolution Integral Representation of LTI Systems94

2.3 Properties of Linear Time-Invariant Systems103

2.3.1 The Commutative Property104

2.3.2 The Distributive Property104

2.3.3 The Associative Property107

2.3.4 LTI Systems with and without Memory108

2.3.5 Invertibility of LTI Systems109

2.3.6 Causality for LTI Systems112

2.3.7 Stability for LTI Systems113

2.3.8 The Unit Step Response of an LTI System115

2.4 Causal LTI Systems Described by Differential and Difference Equations116

2.4.1 Linear Constant-Coefficient Differential Equations117

2.4.2 Linear Constant-Coefficient Difference Equations121

2.4.3 Block Diagram Representations of First-Order Systems Described by Differential and Difference Equations124

2.5 Singularity Functions127

2.5.1 The Unit Impulse as an Idealized Short Pulse128

2.5.2 Defining the Unit Impulse through Convolution131

2.5.3 Unit Doublets and Other Singularity Functions132

2.6 Summary137

Problems137

3 FOURIER SERIES REPRESENTATION OF PERIODIC SIGNALS177

3.0 Introduction177

3.1 A Historical Perspective178

3.2 The Response of LTI Systems to Complex Exponentials182

3.3 Fourier Series Representation of Continuous-Time Periodic Signals186

3.3.1 Linear Combinations of Harmonically Related Complex Exponentials186

3.3.2 Determination of the Fourier Series Representation of a Continuous-Time Periodic Signal190

3.4 Convergence of the Fourier Series195

3.5 Properties of Continuous-Time Fourier Series202

3.5.1 Linearity202

3.5.2 Time Shifting202

3.5.3 Time Reversal203

3.5.4 Time Scaling204

3.5.5 Multiplication204

3.5.6 Conjugation and Conjugate Symmetry204

3.5.7 Parseval's Relation for Continuous-Time Periodic Signals205

3.5.8 Summary of Properties of the Continuous-Time Fourier Series205

3.5.9 Examples205

3.6 Fourier Series Representation of Discrete-Time Periodic Signals211

3.6.1 Linear Combinations of Harmonically Related Complex Exponentials211

3.6.2 Determination of the Fourier Series Representation of a Periodic Signal212

3.7 Properties of Discrete-Time Fourier Series221

3.7.1 Multiplication222

3.7.2 First Difference222

3.7.3 Parseval's Relation for Discrete-Time Periodic Signals223

3.7.4 Examples223

3.8 Fourier Series and LTI Systems226

3.9 Filtering231

3.9.1 Frequency-Shaping Filters232

3.9.2 Frequency-Selective Filters236

3.10 Examples of Continuous-Time Filters Described by Differential Equations239

3.10.1 A Simple RC Lowpass Filter239

3.10.2 A Simple RC Highpass Filter241

3.11 Examples of Discrete-Time Filters Described by Difference Equations244

3.11.1 First-Order Recursive Discrete-Time Filters244

3.11.2 Nonrecursive Discrete-Time Filters245

3.12 Summary249

Problems250

4 THE CONTINUOUS-TIME FOURIER TRANSFORM284

4.0 Introduction284

4.1 Representation of Aperiodic Signals: The Continuous-Time Fourier Transform285

4.1.1 Development of the Fourier Transform Representation of an Aperiodic Signal285

4.1.2 Convergence of Fourier Transforms289

4.1.3 Examples of Continuous-Time Fourier Transforms290

4.2 The Fourier Transform for Periodic Signals296

4.3 Properties of the Continuous-Time Fourier Transform300

4.3.1 Linearity301

4.3.2 Time Shifting301

4.3.3 Conjugation and Conjugate Symmetry303

4.3.4 Differentiation and Integration306

4.3.5 Time and Frequency Scaling308

4.3.6 Duality309

4.3.7 Parseval's Relation312

4.4 The Convolution Property314

4.4.1 Examples317

4.5 The Multiplication Property322

4.5.1 Frequency-Selective Filtering with Variable Center Frequency325

4.6 Tables of Fourier Properties and of Basic Fourier Transform Pairs328

4.7 Systems Characterized by Linear Constant-Coefficient Differential Equations330

4.8 Summary333

Problems334

5 THE DISCRETE-TIME FOURIER TRANSFORM358

5.0 Introduction358

5.1 Representation of Aperiodic Signals: The Discrete-Time Fourier Transform359

5.1.1 Development of the Discrete-Time Fourier Transform359

5.1.2 Examples of Discrete-Time Fourier Transforms362

5.1.3 Convergence Issues Associated with the Discrete-Time Fourier Transform366

5.2 The Fourier Transform for Periodic Signals367

5.3 Properties of the Discrete-Time Fourier Transform372

5.3.1 Periodicity of the Discrete-Time Fourier Transform373

5.3.2 Linearity of the Fourier Transform373

5.3.3 Time Shifting and Frequency Shifting373

5.3.4 Conjugation and Conjugate Symmetry375

5.3.5 Differencing and Accumulation375

5.3.6 Time Reversal376

5.3.7 Time Expansion377

5.3.8 Differentiation in Frequency380

5.3.9 Parseval's Relation380

5.4 The Convolution Property382

5.4.1 Examples383

5.5 The Multiplication Property388

5.6 Tables of Fourier Transform Properties and Basic Fourier Transform Pairs390

5.7 Duality390

5.7.1 Duality in the Discrete-Time Fourier Series391

5.7.2 Duality between the Discrete-Time Fourier Transform and the Continuous-Time Fourier Series395

5.8 Systems Characterized by Linear Constant-Coefficient Difference Equations396

5.9 Summary399

Problems400

6 TIME AND FREQUENCY CHARACTERIZATION OF SIGNALS AND SYSTEMS423

6.0 Introduction423

6.1 The Magnitude-Phase Representation of the Fourier Transform423

6.2 The Magnitude-Phase Representation of the Frequency Response of LTI Systems427

6.2.1 Linear and Nonlinear Phase428

6.2.2 Group Delay430

6.2.3 Log-Magnitude and Bode Plots436

6.3 Time-Domain Properties of Ideal Frequency-Selective Filters439

6.4 Time-Domain and Frequency-Domain Aspects of Nonideal Filters444

6.5 First-Order and Second-Order Continuous-Time Systems448

6.5.1 First-Order Continuous-Time Systems448

6.5.2 Second-Order Continuous-Time Systems451

6.5.3 Bode Plots for Rational Frequency Responses456

6.6 First-Order and Second-Order Discrete-Time Systems461

6.6.1 First-Order Discrete-Time Systems461

6.6.2 Second-Order Discrete-Time Systems465

6.7 Examples of Time- and Frequency-Domain Analysis of Systems472

6.7.1 Analysis of an Automobile Suspension System473

6.7.2 Examples of Discrete-Time Nonrecursive Filters476

6.8 Summary482

Problems483

7 SAMPLING514

7.0 Introduction514

7.1 Representation of a Continuous-Time Signal by Its Samples: The Sampling Theorem515

7.1.1 Impulse-Train Sampling516

7.1.2 Sampling with a Zero-Order Hold520

7.2 Reconstruction of a Signal from Its Samples Using Interpolation522

7.3 The Effect of Undersampling: Aliasing527

7.4 Discrete-Time Processing of Continuous-Time Signals534

7.4.1 Digital Differentiator541

7.4.2 Half-Sample Delay543

7.5 Sampling of Discrete-Time Signals545

7.5.1 Impulse-Train Sampling545

7.5.2 Discrete-Time Decimation and Interpolation549

7.6 Summary555

Problems556

8 COMMUNICATION SYSTEMS582

8.0 Introduction582

8.1 Complex Exponential and Sinusoidal Amplitude Modulation583

8.1.1 Amplitude Modulation with a Complex Exponential Carrier583

8.1.2 Amplitude Modulation with a Sinusoidal Carrier585

8.2 Demodulation for Sinusoidal AM587

8.2.1 Synchronous Demodulation587

8.2.2 Asynchronous Demodulation590

8.3 Frequency-Division Multiplexing594

8.4 Single-Sideband Sinusoidal Amplitude Modulation597

8.5 Amplitude Modulation with a Pulse-Train Carrier601

8.5.1 Modulation of a Pulse-Train Carrier601

8.5.2 Time-Division Multiplexing604

8.6 Pulse-Amplitude Modulation604

8.6.1 Pulse-Amplitude Modulated Signals604

8.6.2 Intersymbol Interference in PAM Systems607

8.6.3 Digital Pulse-Amplitude and Pulse-Code Modulation610

8.7 Sinusoidal Frequency Modulation611

8.7.1 Narrowband Frequency Modulation613

8.7.2 Wideband Frequency Modulation615

8.7.3 Periodic Square-Wave Modulating Signal617

8.8 Discrete-Time Modulation619

8.8.1 Discrete-Time Sinusoidal Amplitude Modulation619

8.8.2 Discrete-Time Transmodulation623

8.9 Summary623

Problems625

9 THE LAPLACE TRANSFORM654

9.0 Introduction654

9.1 The Laplace Transform655

9.2 The Region of Convergence for Laplace Transforms662

9.3 The Inverse Laplace Transform670

9.4 Geometric Evaluation of the Fourier Transform from the Pole-Zero Plot674

9.4.1 First-Order Systems676

9.4.2 Second-Order Systems677

9.4.3 All-Pass Systems681

9.5 Properties of the Laplace Transform682

9.5.1 Linearity of the Laplace Transform683

9.5.2 Time Shifting684

9.5.3 Shifting in the 5-Domain685

9.5.4 Time Scaling685

9.5.5 Conjugation687

9.5.6 Convolution Property687

9.5.7 Differentiation in the Time Domain688

9.5.8 Differentiation in the s-Domain688

9.5.9 Integration in the Time Domain690

9.5.10 The Initial-and Final-Value Theorems690

9.5.11 Table of Properties691

9.6 Some Laplace Transform Pairs692

9.7 Analysis and Characterization of LTI Systems Using the Laplace Transform693

9.7.1 Causality693

9.7.2 Stability695

9.7.3 LTI Systems Characterized by Linear Constant-Coefficient Differential Equations698

9.7.4 Examples Relating System Behavior to the System Function701

9.7.5 Butterworth Filters703

9.8 System Function Algebra and Block Diagram Representations706

9.8.1 System Functions for Interconnections of LTI Systems707

9.8.2 Block Diagram Representations for Causal LTI Systems Described by Differential Equations and Rational System Functions708

9.9 The Unilateral Laplace Transform714

9.9.1 Examples of Unilateral Laplace Transforms714

9.9.2 Properties of the Unilateral Laplace Transform716

9.9.3 Solving Differential Equations Using the Unilateral Laplace Transform719

9.10 Summary720

Problems721

10 THE Z-TRANSFORM741

10.0 Introduction741

10.1 The z-Transform741

10.2 The Region of Convergence for the z-Transform748

10.3 The Inverse z-Transform757

10.4 Geometric Evaluation of the Fourier Transform from the Pole-Zero Plot763

10.4.1 First-Order Systems763

10.4.2 Second-Order Systems765

10.5 Properties of the z-Transform767

10.5.1 Linearity767

10.5.2 Time Shifting767

10.5.3 Scaling in the z-Domain768

10.5.4 Time Reversal769

10.5.5 Time Expansion769

10.5.6 Conjugation770

10.5.7 The Convolution Property770

10.5.8 Differentiation in the z-Domain772

10.5.9 The Initial-Value Theorem773

10.5.10 Summary of Properties774

10.6 Some Common z-Transform Pairs774

10.7 Analysis and Characterization of LTI Systems Using z-Transforms774

10.7.1 Causality776

10.7.2 Stability777

10.7.3 LTI Systems Characterized by Linear Constant-Coefficient Difference Equations779

10.7.4 Examples Relating System Behavior to the System Function781

10.8 System Function Algebra and Block Diagram Representations783

10.8.1 System Functions for Interconnections of LTI Systems784

10.8.2 Block Diagram Representations for Causal LTI Systems Described by Difference Equations and Rational System Functions784

10.9 The Unilateral z-Transform789

10.9.1 Examples of Unilateral z-Transforms and Inverse Transforms790

10.9.2 Properties of the Unilateral z-Transform792

10.9.3 Solving Difference Equations Using the Unilateral z-Transform795

10.10 Summary796

Problems797

11 LINEAR FEEDBACK SYSTEMS816

11.0 Introduction816

11.1 Linear Feedback Systems819

11.2 Some Applications and Consequences of Feedback820

11.2.1 Inverse System Design820

11.2.2 Compensation for Nonideal Elements821

11.2.3 Stabilization of Unstable Systems823

11.2.4 Sampled-Data Feedback Systems826

11.2.5 Tracking Systems828

11.2.6 Destabilization Caused by Feedback830

11.3 Root-Locus Analysis of Linear Feedback Systems832

11.3.1 An Introductory Example833

11.3.2 Equation for the Closed-Loop Poles834

11.3.3 The End Points of the Root Locus: The Closed-Loop Poles for K = 0 and |K| = +∞836

11.3.4 The Angle Criterion836

11.3.5 Properties of the Root Locus841

11.4 The Nyquist Stability Criterion846

11.4.1 The Encirclement Property847

11.4.2 The Nyquist Criterion for Continuous-Time LTI Feedback Systems850

11.4.3 The Nyquist Criterion for Discrete-Time LTI Feedback Systems856

11.5 Gain and Phase Margins858

11.6 Summary866

Problems867

APPENDIX PARTIAL-FRACTION EXPANSION909

BIBLIOGRAPHY921

ANSWERS931

INDEX941