SIGNALS, SYSTEMS, AND TRANSFORMS

FOURTH EDITION

Charles L. Phillips - John M. Parr - Eve A. Riskin

A clear, comprehensive presentation of the theory and applications of signals, systems, and transforms. presents the mathematical backgroung of signals and systems, including the Fourier transform, the Fourier series, the Laplace transform, the discrte-time and discrete Fourier transforms, and the z-transforms. Organization permits great flexibility in course emphasis. MATLAB® examples are integrated throughout the book. The advanced features of the student version of MATLAB are integrated into the examples and problems.

The interactive Web site at

www.ee.washington.edu/class/SST_textbook/textbook.html has numerous animated demonstrations and interactive examples.

More than 350 homework problems and over 150 examples. Answers to selected problems enable students to gain instant feedback of their understanding of new concepts.

Significant Changes in the Fourth Edition

1. Concepts presented more clearly.
2. More concise introduction to convolution in Chapter 3.
3. Expanded presentation of the Discrete Fourier Transform in Chapter 12.
4. Revised problem sets for most chapters. Problems arranged so sets of problems relating to a common concept are presented together. The answer to at least one problem from each set is provided in an appendix.
5. Biographical information about selected pioneers in the area of signal and system analysis has been added in appropriate sections of the text.

Contents

Preface                                                                                                                                                                    xvii

1          Introduction                                                                                                                                              1

1.1        Modeling   1

1.2        Continuous-Time Physical Systems   4

Electric Circuits,   4

Operational Amplifier Circuits,   6

Simple Pendulum,   9

DC Power Supplies,   10

Analogous Systems,   12

1.3        Samplers and Discrete-Time Physical Systems   14

Analog-to-Digital Converter,   14

Numerical Integration,   16

Picture in a Picture,   17

Compact Disks,   18

Sampling in Telephone Systems,   19

Data-Acquisition System,   21

1.4        Matlab and Simulink   22

2          Continuous-Time Signals and Systems                                                                                   24

2.1        Transformations of Continuous-Time Signals   25

Time Transformations,   25

Amplitude Transformations,   31

2.2        Signal Characteristics   33

Even and Odd Signals,    33

Periodic Signals,   35

2.3        Common Signals in Engineering    40

2.4        Singularity Functions   46

Unit Step Function,   46

Unit Impulse Function,   50

2.5        Mathematical Functions for Signals   55

2.6        Continuous-Time Systems   60

Interconnecting Systems,   62

Feedback System,   64

2.7        Properties of Continuous-Time Systems   66

Stability   70

Linearity   75

Summary   77

Problems   79

3          Continuous-Time Linear Time-Invariant Systems                                                            89

3.1        Impulse Representation of Continuous-Time Signals   90

3.2        Convolution for Continuous-Time LTI Systems   93

3.3        Properties of Convolution   105

3.4        Properties of Continuous-Time LTI Systems   109

Memoryless Systems,   110

Invertibility,   110

Causality,   111

Stability,   112

Unit Step Response,   113

3.5        Differential-Equation Models   114

Solution of Differential Equations,   116

General Case,   118

Relation to Physical Systems,   120

3.6        Terms in the Natural Response   121

Stability,   122

3.7        System Response for Complex-Exponential Inputs   125

Linearity,    125

Complex Inputs for LTI Systems,   126

Impulse Response,   130

3.8        Block Diagrams   131

Direct Form I,   135

Direct Form II,   135

nth-Order Realizations,    135

Practical Considerations,   137

Summary   139

Problems   141

4          Fourier Series                                                                                                                                        152

4.1        Approximating Periodic Functions   153

Periodic Functions,   153

Approximating Periodic Functions,   154

4.2        Fourier Series   158

Fourier Series,   159

Fourier Coefficients,   160

4.3        Fourier Series and Frequency Spectra   163

Frequency Spectra,   164

4.4        Properties of Fourier Series   173

4.5        System Analysis   176

4.6        Fourier Series Transformations   183

Amplitude Transformations,   184

Time Transformations,   186

Summary   188

Problems   189

5          The Fourier Transform                                                                                                                   199

5.1        Definition of the Fourier Transform   199

5.2        Properties of the Fourier Transform   208

Linearity,    208

Time Scaling,   210

Time Shifting,   213

Time Transformation,   214

Duality,   215

Convolution,   218

Frequency Shifting,   219

Time Differentiation,   221

Time Integration,   226

Frequency Differentiation,   229

Summary,   229

5.3        Fourier Transforms of Time Functions   230

DC Level,   230

Unit Step Function,   230

Switched Cosine,   231

Pulsed Cosine,   231

Exponential Pulse,   233

Fourier Transforms of Periodic Functions,   233

Summary,   239

5.4        Sampling Continuous-Time Signals   239

Impulse Sampling,   240

Shannon’s Sampling Theorem,   242

Practical Sampling,   244

5.5        Application of the Fourier Transform   244

Frequency Response of Linear Systems,   244

Frequency Spectra of Signals,   253

Summary,   256

5.6        Energy and Power Density Spectra   256

Energy Density Spectrum,   256

Power Density Spectrum,    259

Power and Energy Transmission,   262

Summary,   264

Summary   265

Problems   267

6          Applications of the Fourier Transform                                                                            275

6.1        Ideal Filters   275

6.2        Real Filters   282

RC Low-Pass Filter,   283

Butterworth Filter,   285

Chebyschev and Elliptic Filters,   291

Bandpass Filters,   295

Summary,   296

6.3        Bandwidth Relationships   296

6.4        Reconstruction of signals from sample data   300

Interpolating Function,   302

Digital-to-analog Conversion,   304

6.5        Sinusoidal Amplitude Modulation   307

Frequency-Division Multiplexing,   316

6.6        Pulse-Amplitude Modulation   318

Time-Division Multiplexing,   320

Flat-Top PAM,   322

Summary   325

Problems   326

7          The Laplace Transform                                                                                                                 337

7.1        Definitions of Laplace Transforms   338

7.2        Examples   341

7.3        Laplace Transforms of Functions   346

7.4        Laplace Transform Properties   350

Real Shifting,   351

Differentiation,   355

Integration,   357

7.5        Additional Properties   358

Multiplication by t,   358

Initial Value,   359

Final Value,   360

Time Transformation,   361

7.6        Response of LTI Systems   364

Initial Conditions,   364

Transfer Functions,   365

Convolution,   370

Transforms with Complex Poles,   372

Functions with Repeated Poles,   375

7.7        LTI Systems Characteristics   376

Causality,   376

Stability,   377

Invertibility,   379

Frequency Response,    380

7.8        Bilateral Laplace Transform   382

Region of Convergence,   384

Bilateral Transform from Unilateral Tables,   386

Inverse Bilateral Laplace Transform,   388

7.9        Relationship of the Laplace Transform to the Fourier Transform  390

Summary   391

Problems   392

8          State Variables for Continuous-Time Systems                                                             400

8.1        State-Variable Modeling   401

8.2        Simulation Diagrams   405

8.3        Solution of State Equations   410

Laplace-Transform Solution,   411

Convolution Solution,   416

Infinite Series Solution,   417

8.4        Properties of the State Transition Matrix   420

8.5        Transfer Functions   422

Stability,   424

8.6        Similarity Transformations   426

Transformations,   426

Properties,   432

Summary   434

Problems   436

9          Discrete-Time Signals and Systems                                                                                        445

9.1        Discrete-Time Signals and Systems   447

Unit Step and Unit Impulse Functions,   449

Equivalent Operations,   451

9.2        Transformations of Discrete-Time Signals   452

Time Transformations,   453

Amplitude Transformations,   458

9.3        Characteristics of Discrete-Time Signals   461

Even and Odd Signals,   461

Signals Periodic in n,   464

Signals Periodic in W   467

9.4        Common Discrete-Time Signals   468

9.5        Discrete-Time Systems    474

Interconnecting Systems,   475

9.6        Properties of Discrete-Time Systems   477

Systems with Memory,   477

Invertibility,   478

Inverse of a System,   479

Causality,   479

Stability,   480

Time Invariance,   480

Linearity,    481

Summary   483

Problems   485

10        Discrete-Time Linear Time-Invariant Systems                                                                 493

10.1      Impulse Representation of Discrete-Time Signals   494

10.2      Convolution for Discrete-Time Systems   495

Properties of Convolution,   504

10.3      Properties of Discrete-Time LTI Systems    507

Memory,   508

Invertibility,   508

Causality,   508

Stability,   509

Unit Step Response,   511

10.4      Difference-Equation Models   512

Difference-Equation Models,   512

Classical Method,   514

Solution by Iteration,   519

10.5      Terms in the Natural Response   520

Stability,   521

10.6      Block Diagrams   523

Two Standard Forms,   525

10.7      System Response for Complex-Exponential Inputs   529

Linearity,    530

Complex Inputs for LTI Systems,   530

Stability,   535

Sampled Signals,   535

Impulse Response,   535

Summary   537

Problems   538

11        The z-Transform                                                                                                                                  547

11.1      Definitions of z-Transforms   547

11.2      Examples   550

Two z-Transforms,   550

Digital-Filter Example,   553

11.3      z-Transforms of Functions   555

Sinusoids,   557

11.4      z-Transform Properties   560

Real Shifting,   560

Initial and Final Values,   563

11.5      Additional Properties   565

Time Scaling,   565

Convolution in Time,   567

11.6      LTI System Applications   568

Transfer Functions,   569

Inverse z-Transform,   571

Complex Poles,   574

Causality,   575

Stability,   576

Invertibility,   579

11.7      Bilateral z-Transform   580

Bilateral Transforms,   585

Regions of Convergence,     586

Inverse Bilateral Transforms,   588

Summary   590

Problems   591

12        Fourier Transforms of Discrete-Time Signals                                                              599

12.1      Discrete-Time Fourier Transform   600

z-Transform,   602

12.2      Properties of the Discrete-Time Fourier Transform   605

Periodicity,   605

Linearity,    606

Time Shift,   606

Frequency Shift,   607

Symmetry,   608

Time Reversal,   608

Convolution in Time,   609

Convolution in Frequency,   609

Multiplication by n,   610

Parseval’s Theorem,   610

12.3      Discrete-Time Fourier Transform of Periodic Sequences   611

12.4      Discrete Fourier Transform   617

Shorthand Notation for the DFT,   620

Frequency Resolution of the DFT,   621

Validity of the DFT,   622

Summary,   626

12.5      Fast Fourier Transform   627

Decomposition-in-Time Fast Fourier Transform Algorithm,   627

Decomposition-in-Frequency Fast Fourier Transform,   632

Summary,   635

12.6      Applications of the Discrete Fourier Transform   635

Calculation of Fourier Transforms,   635

Convolution,   646

Filtering,    653

Correlation,   660

Energy Spectral Density Estimation,   666

Summary,   667

12.7      The Discrete Cosine Transform,   667

Summary   669

Problems   671

13        State Variables for Discrete-Time Systems                                                                    677

13.1      State-Variable Modeling   678

13.2      Simulation Diagrams    682

13.3      Solution of State Equations   688

Recursive Solution,   688

z-Transform Solution,    690

13.4      Properties of the State Transition Matrix    695

13.5      Transfer Functions   697

Stability,   699

13.6      Similarity Transformations   700

Properties,   704

Summary   705

Problems   707

Appendices                                                                                                                                                            714

A.        Integrals and Trigonometric Identities   714

Integrals,   714

Trigonometric Identities,   715

B.         Leibnitz’s and L’Hôpital’s Rules   716

Leibnitz’s Rule,   716

L’Hôpital’s Rule,   717

C.         Summation Formulas for Geometric Series   718

D.         Complex Numbers and Euler’s Relation   719

Complex-Number Arithmetic,   720

Euler’s Relation,   723

Conversion Between Forms,   724

E.         Solution of Differential Equations   726

Complementary Function,   726

Particular Solution,   727

General Solution,   728

Repeated Roots,   728

F.         Partial-Fraction Expansions   730

G.         Review of Matrices   733

Algebra of Matrices,   737

Other Relationships,   738

H.         Answers to Selected Problems   740

I.          Signals and Systems References   750

Index                                                                                                                                                                         759