## 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.

## Table of Contents

**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 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 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 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 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

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