# Signals, Systems, & Transforms, Global Edition

Free delivery worldwide

Available. Dispatched from the UK in 1 business day

When will my order arrive?

## Description

For sophomore/junior-level signals and systems courses in Electrical and Computer Engineering departments.

This text provides a clear, comprehensive presentation of both the theory and applications in signals, systems, and transforms. It presents the mathematical background of signals and systems, including the Fourier transform, the Fourier series, the Laplace transform, the discrete-time and the discrete Fourier transforms, and the z-transform. The text integrates MATLAB examples into the presentation of signal and system theory and applications.

show more

This text provides a clear, comprehensive presentation of both the theory and applications in signals, systems, and transforms. It presents the mathematical background of signals and systems, including the Fourier transform, the Fourier series, the Laplace transform, the discrete-time and the discrete Fourier transforms, and the z-transform. The text integrates MATLAB examples into the presentation of signal and system theory and applications.

show more

## Product details

- Paperback | 816 pages
- 191 x 233 x 41mm | 1,426g
- 17 Apr 2014
- Pearson Education Limited
- Harlow, United Kingdom
- English
- 5th edition
- 1292015284
- 9781292015286
- 2,592,041

## Table of 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 23

2.1 Transformations of Continuous-Time Signals 24

Time Transformations, 24

Amplitude Transformations, 30

2.2 Signal Characteristics 32

Even and Odd Signals, 32

Periodic Signals, 34

2.3 Common Signals in Engineering 39

2.4 Singularity Functions 45

Unit Step Function, 45

Unit Impulse Function, 49

2.5 Mathematical Functions for Signals 54

2.6 Continuous-Time Systems 59

Interconnecting Systems, 61

Feedback System, 64

2.7 Properties of Continuous-Time Systems 65

Stability, 69

Linearity, 74

Summary 76

Problems 78

3 Continuous-Time Linear Time-Invariant Systems 90

3.1 Impulse Representation of Continuous-Time Signals 91

3.2 Convolution for Continuous-Time LTI Systems 92

3.3 Properties of Convolution 105

3.4 Properties of Continuous-Time LTI Systems 108

Memoryless Systems, 109

Invertibility, 109

Causality, 110

Stability, 111

Unit Step Response, 112

3.5 Differential-Equation Models 113

Solution of Differential Equations, 115

General Case, 117

Relation to Physical Systems, 119

3.6 Terms in the Natural Response 120

Stability, 121

3.7 System Response for Complex-Exponential Inputs 124

Linearity, 124

Complex Inputs for LTI Systems, 125

Impulse Response, 129

3.8 Block Diagrams 130

Direct Form I, 134

Direct Form II, 134

nth-Order Realizations, 134

Practical Considerations, 136

Summary 139

Problems 149

4 Fourier Series 154

4.1 Approximating Periodic Functions 155

Periodic Functions, 155

Approximating Periodic Functions, 156

4.2 Fourier Series 160

Fourier Series, 161

Fourier Coefficients, 162

4.3 Fourier Series and Frequency Spectra 165

Frequency Spectra, 166

4.4 Properties of Fourier Series 175

4.5 System Analysis 178

4.6 Fourier Series Transformations 185

Amplitude Transformations, 186

Time Transformations, 188

Summary 190

Problems 191

5 The Fourier Transform 201

5.1 Definition of the Fourier Transform 201

5.2 Properties of the Fourier Transform 210

Linearity, 211

Time Scaling, 212

Time Shifting, 214

Time Reversal, 215

Time Transformation, 216

Duality, 218

Convolution, 220

Frequency Shifting, 221

Time Integration, 224

Time Differentiation, 226

Frequency Differentiation, 231

Symmetry, 232

Summary, 233

5.3 Fourier Transforms of Time Functions 233

DC Level, 233

Unit Step Function, 233

Switched Cosine, 234

Pulsed Cosine, 234

Exponential Pulse, 236

Fourier Transforms of Periodic Functions, 236

Summary, 241

5.4 Application of the Fourier Transform 241

Frequency Response of Linear Systems, 241

Frequency Spectra of Signals, 250

Summary, 252

5.5 Energy and Power Density Spectra 253

Energy Density Spectrum, 253

Power Density Spectrum, 256

Power and Energy Transmission, 258

Summary, 260

Summary 262

Problems 263

6 Applications of the Fourier Transform 272

6.1 I deal Filters 272

6.2 Real Filters 279

RC Low-Pass Filter, 280

Butterworth Filter, 282

Bandpass Filters, 288

Active Filters, 289

Summary, 291

6.3 Bandwidth Relationships 291

6.4 Sampling Continuous-Time Signals 295

Impulse Sampling, 296

Shannon's Sampling Theorem, 299

Practical Sampling, 299

6.5 Reconstruction of Signals from Sample Data 300

Interpolating Function, 302

Digital-to-Analog Conversion, 304

Quantization Error, 306

6.6 Sinusoidal Amplitude Modulation 308

Frequency-Division Multiplexing, 317

6.7 Pulse-Amplitude Modulation 319

Time-Division Multiplexing, 321

Flat-Top PAM, 323

Summary 326

Problems 326

7 The Laplace Transform 336

7.1 Definitions of Laplace Transforms 337

7.2 Examples 340

7.3 Laplace Transforms of Functions 345

7.4 Laplace Transform Properties 349

Real Shifting, 350

Differentiation, 354

Integration, 356

7.5 Additional Properties 357

Multiplication by t, 357

Initial Value, 358

Final Value, 359

Time Transformation, 360

7.6 Response of LTI Systems 363

Initial Conditions, 363

Transfer Functions, 364

Convolution, 369

Transforms with Complex Poles, 371

Functions with Repeated Poles, 374

7.7 LTI Systems Characteristics 375

Causality, 375

Stability, 376

Invertibility, 378

Frequency Response, 379

Step Response, 380

7.8 Bilateral Laplace Transform 382

Region of Convergence, 384

Bilateral Transform from Unilateral Tables, 386

Inverse Bilateral Laplace Transform, 389

7.9 Relationship of the Laplace Transform to the Fourier Transform 391

Summary 392

Problems 393

8 State Variables for Continuous-Time Systems 401

8.1 State-Variable Modeling 402

8.2 Simulation Diagrams 406

8.3 Solution of State Equations 412

Laplace-Transform Solution, 412

Convolution Solution, 417

Infinite Series Solution, 418

8.4 Properties of the State-Transition Matrix 421

8.5 Transfer Functions 423

Stability, 425

8.6 Similarity Transformations 427

Transformations, 427

Properties, 433

Summary 435

Problems 437

9 Discrete-Time Signals and Systems 446

9.1 Discrete-Time Signals and Systems 448

Unit Step and Unit Impulse Functions, 450

Equivalent Operations, 452

9.2 Transformations of Discrete-Time Signals 453

Time Transformations, 454

Amplitude Transformations, 459

9.3 Characteristics of Discrete-Time Signals 462

Even and Odd Signals, 462

Signals Periodic in n, 465

Signals Periodic in , 468

9.4 Common Discrete-Time Signals 469

9.5 Discrete-Time Systems 475

Interconnecting Systems, 476

9.6 Properties of Discrete-Time Systems 478

Systems with Memory, 478

Invertibility, 479

Inverse of a System, 480

Causality, 480

Stability, 481

Time Invariance, 481

Linearity, 482

Summary 484

Problems 486

10 Discrete-Time Linear Time-Invariant Systems 495

10.1 Impulse Representation of Discrete-Time Signals 496

10.2 Convolution for Discrete-Time Systems 497

Properties of Convolution, 506

10.3 Properties of Discrete-Time LTI Systems 509

Memory, 510

Invertibility, 510

Causality, 510

Stability, 511

Unit Step Response, 513

10.4 Difference-Equation Models 514

Difference-Equation Models, 514

Classical Method, 516

Solution by Iteration, 521

10.5 Terms in the Natural Response 522

Stability, 523

10.6 Block Diagrams 525

Two Standard Forms, 527

10.7 System Response for Complex-Exponential Inputs 531

Linearity, 532

Complex Inputs for LTI Systems, 532

Stability, 537

Sampled Signals, 537

Impulse Response, 537

Summary 539

Problems 540

11 The z-Transform 552

11.1 Definitions of z-Transforms 552

11.2 Examples 555

Two z-Transforms, 555

Digital-Filter Example, 558

11.3 z-Transforms of Functions 560

Sinusoids, 561

11.4 z-Transform Properties 565

Real Shifting, 565

Initial and Final Values, 568

11.5 Additional Properties 570

Time Scaling, 570

Convolution in Time, 572

11.6 L TI System Applications 573

Transfer Functions, 573

Inverse z-Transform, 575

Complex Poles, 578

Causality, 580

Stability, 581

Invertibility, 584

Frequency Response, 585

11.7 Bilateral z-Transform 588

Bilateral Transforms, 592

Regions of Convergence, 594

Inverse Bilateral Transforms, 595

Summary 598

Problems 599

12 Fourier Transforms of Discrete-Time Signals 609

12.1 Discrete-Time Fourier Transform 610

z-Transform, 612

12.2 Properties of the Discrete-Time Fourier Transform 617

Periodicity, 618

Linearity, 619

Time Shift, 619

Frequency Shift, 620

Symmetry, 620

Time Reversal, 621

Convolution in Time, 621

Convolution in Frequency, 622

Multiplication by n, 623

Parseval's Theorem, 623

12.3 Discrete-Time Fourier Transform of Periodic Sequences 624

12.4 Discrete Fourier Transform 630

Shorthand Notation for the DFT, 632

Frequency Resolution of the DFT, 632

Validity of the DFT, 634

Summary, 638

12.5 Fast Fourier Transform 638

Decomposition-in-Time Fast Fourier Transform Algorithm, 638

Decomposition-in-Frequency Fast Fourier Transform, 643

Summary, 646

12.6 Applications of the Discrete Fourier Transform 646

Calculation of Fourier Transforms, 646

Convolution, 654

Filtering, 663

Correlation, 671

Energy Spectral Density Estimation, 677

Summary, 678

12.7 The Discrete Cosine Transform, 678

Summary 683

Problems 684

13 State Variables for Discrete-Time Systems 692

13.1 State-Variable Modeling 693

13.2 Simulation Diagrams 697

13.3 Solution of State Equations 703

Recursive Solution, 703

z-Transform Solution, 705

13.4 Properties of the State Transition Matrix 710

13.5 Transfer Functions 712

Stability, 714

13.6 Similarity Transformations 715

Properties, 719

Summary 720

Problems 721

Appendices 718

A. Integrals and Trigonometric Identities 730

Integrals, 730

Trigonometric Identities, 731

B. Leibnitz's and L'Hopital's Rules 732

Leibnitz's Rule, 732

L'Hopital's Rule, 733

C. Summation Formulas for Geometric Series 734

D. Complex Numbers and Euler's Relation 735

Complex-Number Arithmetic, 736

Euler's Relation, 739

Conversion Between Forms, 740

E. Solution of Differential Equations 742

Complementary Function, 742

Particular Solution, 743

General Solution, 744

Repeated Roots, 744

F. Partial-Fraction Expansions 746

G. Review of Matrices 749

Algebra of Matrices, 753

Other Relationships, 754

H. Answers to Selected Problems 756

I. Signals and Systems References 770

Index

show more

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 23

2.1 Transformations of Continuous-Time Signals 24

Time Transformations, 24

Amplitude Transformations, 30

2.2 Signal Characteristics 32

Even and Odd Signals, 32

Periodic Signals, 34

2.3 Common Signals in Engineering 39

2.4 Singularity Functions 45

Unit Step Function, 45

Unit Impulse Function, 49

2.5 Mathematical Functions for Signals 54

2.6 Continuous-Time Systems 59

Interconnecting Systems, 61

Feedback System, 64

2.7 Properties of Continuous-Time Systems 65

Stability, 69

Linearity, 74

Summary 76

Problems 78

3 Continuous-Time Linear Time-Invariant Systems 90

3.1 Impulse Representation of Continuous-Time Signals 91

3.2 Convolution for Continuous-Time LTI Systems 92

3.3 Properties of Convolution 105

3.4 Properties of Continuous-Time LTI Systems 108

Memoryless Systems, 109

Invertibility, 109

Causality, 110

Stability, 111

Unit Step Response, 112

3.5 Differential-Equation Models 113

Solution of Differential Equations, 115

General Case, 117

Relation to Physical Systems, 119

3.6 Terms in the Natural Response 120

Stability, 121

3.7 System Response for Complex-Exponential Inputs 124

Linearity, 124

Complex Inputs for LTI Systems, 125

Impulse Response, 129

3.8 Block Diagrams 130

Direct Form I, 134

Direct Form II, 134

nth-Order Realizations, 134

Practical Considerations, 136

Summary 139

Problems 149

4 Fourier Series 154

4.1 Approximating Periodic Functions 155

Periodic Functions, 155

Approximating Periodic Functions, 156

4.2 Fourier Series 160

Fourier Series, 161

Fourier Coefficients, 162

4.3 Fourier Series and Frequency Spectra 165

Frequency Spectra, 166

4.4 Properties of Fourier Series 175

4.5 System Analysis 178

4.6 Fourier Series Transformations 185

Amplitude Transformations, 186

Time Transformations, 188

Summary 190

Problems 191

5 The Fourier Transform 201

5.1 Definition of the Fourier Transform 201

5.2 Properties of the Fourier Transform 210

Linearity, 211

Time Scaling, 212

Time Shifting, 214

Time Reversal, 215

Time Transformation, 216

Duality, 218

Convolution, 220

Frequency Shifting, 221

Time Integration, 224

Time Differentiation, 226

Frequency Differentiation, 231

Symmetry, 232

Summary, 233

5.3 Fourier Transforms of Time Functions 233

DC Level, 233

Unit Step Function, 233

Switched Cosine, 234

Pulsed Cosine, 234

Exponential Pulse, 236

Fourier Transforms of Periodic Functions, 236

Summary, 241

5.4 Application of the Fourier Transform 241

Frequency Response of Linear Systems, 241

Frequency Spectra of Signals, 250

Summary, 252

5.5 Energy and Power Density Spectra 253

Energy Density Spectrum, 253

Power Density Spectrum, 256

Power and Energy Transmission, 258

Summary, 260

Summary 262

Problems 263

6 Applications of the Fourier Transform 272

6.1 I deal Filters 272

6.2 Real Filters 279

RC Low-Pass Filter, 280

Butterworth Filter, 282

Bandpass Filters, 288

Active Filters, 289

Summary, 291

6.3 Bandwidth Relationships 291

6.4 Sampling Continuous-Time Signals 295

Impulse Sampling, 296

Shannon's Sampling Theorem, 299

Practical Sampling, 299

6.5 Reconstruction of Signals from Sample Data 300

Interpolating Function, 302

Digital-to-Analog Conversion, 304

Quantization Error, 306

6.6 Sinusoidal Amplitude Modulation 308

Frequency-Division Multiplexing, 317

6.7 Pulse-Amplitude Modulation 319

Time-Division Multiplexing, 321

Flat-Top PAM, 323

Summary 326

Problems 326

7 The Laplace Transform 336

7.1 Definitions of Laplace Transforms 337

7.2 Examples 340

7.3 Laplace Transforms of Functions 345

7.4 Laplace Transform Properties 349

Real Shifting, 350

Differentiation, 354

Integration, 356

7.5 Additional Properties 357

Multiplication by t, 357

Initial Value, 358

Final Value, 359

Time Transformation, 360

7.6 Response of LTI Systems 363

Initial Conditions, 363

Transfer Functions, 364

Convolution, 369

Transforms with Complex Poles, 371

Functions with Repeated Poles, 374

7.7 LTI Systems Characteristics 375

Causality, 375

Stability, 376

Invertibility, 378

Frequency Response, 379

Step Response, 380

7.8 Bilateral Laplace Transform 382

Region of Convergence, 384

Bilateral Transform from Unilateral Tables, 386

Inverse Bilateral Laplace Transform, 389

7.9 Relationship of the Laplace Transform to the Fourier Transform 391

Summary 392

Problems 393

8 State Variables for Continuous-Time Systems 401

8.1 State-Variable Modeling 402

8.2 Simulation Diagrams 406

8.3 Solution of State Equations 412

Laplace-Transform Solution, 412

Convolution Solution, 417

Infinite Series Solution, 418

8.4 Properties of the State-Transition Matrix 421

8.5 Transfer Functions 423

Stability, 425

8.6 Similarity Transformations 427

Transformations, 427

Properties, 433

Summary 435

Problems 437

9 Discrete-Time Signals and Systems 446

9.1 Discrete-Time Signals and Systems 448

Unit Step and Unit Impulse Functions, 450

Equivalent Operations, 452

9.2 Transformations of Discrete-Time Signals 453

Time Transformations, 454

Amplitude Transformations, 459

9.3 Characteristics of Discrete-Time Signals 462

Even and Odd Signals, 462

Signals Periodic in n, 465

Signals Periodic in , 468

9.4 Common Discrete-Time Signals 469

9.5 Discrete-Time Systems 475

Interconnecting Systems, 476

9.6 Properties of Discrete-Time Systems 478

Systems with Memory, 478

Invertibility, 479

Inverse of a System, 480

Causality, 480

Stability, 481

Time Invariance, 481

Linearity, 482

Summary 484

Problems 486

10 Discrete-Time Linear Time-Invariant Systems 495

10.1 Impulse Representation of Discrete-Time Signals 496

10.2 Convolution for Discrete-Time Systems 497

Properties of Convolution, 506

10.3 Properties of Discrete-Time LTI Systems 509

Memory, 510

Invertibility, 510

Causality, 510

Stability, 511

Unit Step Response, 513

10.4 Difference-Equation Models 514

Difference-Equation Models, 514

Classical Method, 516

Solution by Iteration, 521

10.5 Terms in the Natural Response 522

Stability, 523

10.6 Block Diagrams 525

Two Standard Forms, 527

10.7 System Response for Complex-Exponential Inputs 531

Linearity, 532

Complex Inputs for LTI Systems, 532

Stability, 537

Sampled Signals, 537

Impulse Response, 537

Summary 539

Problems 540

11 The z-Transform 552

11.1 Definitions of z-Transforms 552

11.2 Examples 555

Two z-Transforms, 555

Digital-Filter Example, 558

11.3 z-Transforms of Functions 560

Sinusoids, 561

11.4 z-Transform Properties 565

Real Shifting, 565

Initial and Final Values, 568

11.5 Additional Properties 570

Time Scaling, 570

Convolution in Time, 572

11.6 L TI System Applications 573

Transfer Functions, 573

Inverse z-Transform, 575

Complex Poles, 578

Causality, 580

Stability, 581

Invertibility, 584

Frequency Response, 585

11.7 Bilateral z-Transform 588

Bilateral Transforms, 592

Regions of Convergence, 594

Inverse Bilateral Transforms, 595

Summary 598

Problems 599

12 Fourier Transforms of Discrete-Time Signals 609

12.1 Discrete-Time Fourier Transform 610

z-Transform, 612

12.2 Properties of the Discrete-Time Fourier Transform 617

Periodicity, 618

Linearity, 619

Time Shift, 619

Frequency Shift, 620

Symmetry, 620

Time Reversal, 621

Convolution in Time, 621

Convolution in Frequency, 622

Multiplication by n, 623

Parseval's Theorem, 623

12.3 Discrete-Time Fourier Transform of Periodic Sequences 624

12.4 Discrete Fourier Transform 630

Shorthand Notation for the DFT, 632

Frequency Resolution of the DFT, 632

Validity of the DFT, 634

Summary, 638

12.5 Fast Fourier Transform 638

Decomposition-in-Time Fast Fourier Transform Algorithm, 638

Decomposition-in-Frequency Fast Fourier Transform, 643

Summary, 646

12.6 Applications of the Discrete Fourier Transform 646

Calculation of Fourier Transforms, 646

Convolution, 654

Filtering, 663

Correlation, 671

Energy Spectral Density Estimation, 677

Summary, 678

12.7 The Discrete Cosine Transform, 678

Summary 683

Problems 684

13 State Variables for Discrete-Time Systems 692

13.1 State-Variable Modeling 693

13.2 Simulation Diagrams 697

13.3 Solution of State Equations 703

Recursive Solution, 703

z-Transform Solution, 705

13.4 Properties of the State Transition Matrix 710

13.5 Transfer Functions 712

Stability, 714

13.6 Similarity Transformations 715

Properties, 719

Summary 720

Problems 721

Appendices 718

A. Integrals and Trigonometric Identities 730

Integrals, 730

Trigonometric Identities, 731

B. Leibnitz's and L'Hopital's Rules 732

Leibnitz's Rule, 732

L'Hopital's Rule, 733

C. Summation Formulas for Geometric Series 734

D. Complex Numbers and Euler's Relation 735

Complex-Number Arithmetic, 736

Euler's Relation, 739

Conversion Between Forms, 740

E. Solution of Differential Equations 742

Complementary Function, 742

Particular Solution, 743

General Solution, 744

Repeated Roots, 744

F. Partial-Fraction Expansions 746

G. Review of Matrices 749

Algebra of Matrices, 753

Other Relationships, 754

H. Answers to Selected Problems 756

I. Signals and Systems References 770

Index

show more