1 Introduction to Signals and Systems 1

1.1 Introduction 1

1.2 Introduction to Signal Manipulation 3

1.3 A Few Useful Signals 9

1.4 The Sinusoidal Signal 17

1.5 Phase Change vs. Time Shift 21

1.6 Useful Hints and Help with MATLAB 25

1.7 Conclusions 26

2.1 Introduction 30

2.2 Periodic Signals 31

2.3 Odd and Even Signals 38

2.4 Energy and Power Signals 47

2.5 Complex Signals 52

2.6 Discrete Time Signals 56

2.7 Digital Signals 58

2.8 Random Signals 58

2.9 Useful Hints and Help with MATLAB 60

2.10 Conclusions 61

3.1 Introduction 66

3.2 Definition of a Linear System 67

3.3 Linear System Response Function h(t) 73

3.4 Convolution 73

3.5 Determining h(t) in an Unknown System 88

3.6 Causality 91

3.7 Combined Systems 92

3.8 Convolution and Random Numbers 94

3.9 Useful Hints and Help with MATLAB 96

3.10 Chapter Summary 97

3.11 Conclusions 97

4.1 Introduction 101

4.2 Expressing Signals by Components 102

4.3 Part One—Orthogonal Signals 106

4.4 Orthogonality 107

4.5 Part Two—The Fourier Series 118

4.6 Computing Fourier Series Components 121

4.7 Fundamental Frequency Component 123

4.8 Practical Harmonics 126

4.9 Odd and Even Square Waves 128

4.10 Gibb’s Phenomenon 131

4.11 Setting Up the Fourier Series Calculation 132

4.12 Some Common Fourier Series 136

4.13 Part Three—The Complex Fourier Series 137

4.14 The Complex Fourier Series 138

4.15 Complex Fourier Series Components 143

4.16 Properties of the Complex Fourier Series 151

4.17 Analysis of a DC Power Supply 152

4.18 The Fourier Series with MATLAB 158

4.19 Conclusions 165

5.1 Introduction 171

5.2 Properties of the Fourier Transform 178

5.3 The Rectangle Signal 181

5.4 The Sinc Function 182

5.5 Signal Manipulations: Time and Frequency 189

5.6 Fourier Transform Pairs 198

5.7 Rapid Changes vs. High Frequencies 200

5.8 Conclusions 203

6.1 Introduction 206

6.2 Convolution: Time and Frequency 206

6.3 Transfer Function of a Linear System 210

6.4 Energy in Signals: Parseval’s Theorem for the Fourier Transform 213

6.5 Data Smoothing and the Frequency Domain 215

6.6 Ideal Filters 216

6.7 A Real Lowpass Filter 220

6.8 The Modulation Theorem 224

6.9 Periodic Signals and the Fourier Transform 230

6.10 The Analog Spectrum Analyzer 233

6.11 Conclusions 235

7.1 Introduction 241

7.2 The Laplace Transform 241

7.3 Exploring the s-Domain 243

7.4 Visualizing the Laplace Transform 251

7.5 Properties of the Laplace Transform 267

7.6 Differential Equations 267

7.7 Laplace Transform Pairs 270

7.8 Circuit Analysis with the Laplace Transform 272

7.9 State Variable Analysis 285

7.10 Conclusions 295

8.1 Introduction 301

8.2 Discrete Time vs. Continuous Time Signals 301

8.3 A Discrete Time Signal 303

8.3.1 A Periodic Discrete Time Signal 303

8.4 Data Collection and Sampling Rate 304

8.5 Introduction to Digital Filtering 319

8.6 Illustrative Examples 328

8.7 Discrete Time Filtering with MATLAB 338

8.8 Conclusions 340

9.1 Introduction 344

9.2 The z-Transform 344

9.3 Calculating the z-Transform 348

9.4 A Discrete Time Laplace Transform 356

9.5 Properties of the z-Transform 358

9.6 z-Transform Pairs 359

9.7 Transfer Function of a Discrete Linear System 359

9.8 MATLAB Analysis with the z-Transform 360

9.9 Digital Filtering—FIR Filter 366

9.10 Digital Filtering—IIR Filter 373

9.11 Conclusions 378

10.1 Introduction 381

10.2 Amplitude Modulation 385

10.3 Suppressed Carrier Transmission 394

10.4 Superheterodyne Receiver 398

10.5 Digital Communications 402

10.6 Phase Shift Keying 407

10.7 Conclusions 409