A Practical Approach to Signals and Systems

Preface xiii

Abbreviations xv

1 Introduction 1

1.1 The Organization of this Book 1

2 Discrete Signals 5

2.1 Classification of Signals 5

2.1.1 Continuous, Discrete and Digital Signals 5

2.1.2 Periodic and Aperiodic Signals 7

2.1.3 Energy and Power Signals 7

2.1.4 Even- and Odd-symmetric Signals 8

2.1.5 Causal and Noncausal Signals 10

2.1.6 Deterministic and Random Signals 10

2.2 Basic Signals 11

2.2.1 Unit-impulse Signal 11

2.2.2 Unit-step Signal 12

2.2.3 Unit-ramp Signal 13

2.2.4 Sinusoids and Exponentials 13

2.3 Signal Operations 20

2.3.1 Time Shifting 21

2.3.2 Time Reversal 21

2.3.3 Time Scaling 22

2.4 Summary 23

Further Reading 23

Exercises 23

3 Continuous Signals 29

3.1 Classification of Signals 29

3.1.1 Continuous Signals 29

3.1.2 Periodic and Aperiodic Signals 30

3.1.3 Energy and Power Signals 31

3.1.4 Even- and Odd-symmetric Signals 31

3.1.5 Causal and Noncausal Signals 33

3.2 Basic Signals 33

3.2.1 Unit-step Signal 33

3.2.2 Unit-impulse Signal 34

3.2.3 Unit-ramp Signal 42

3.2.4 Sinusoids 43

3.3 Signal Operations 45

3.3.1 Time Shifting 45

3.3.2 Time Reversal 46

3.3.3 Time Scaling 47

3.4 Summary 48

Further Reading 48

Exercises 48

4 Time-domain Analysis of Discrete Systems 53

4.1 Difference Equation Model 53

4.1.1 System Response 55

4.1.2 Impulse Response 58

4.1.3 Characterization of Systems by their Responses to Impulse and Unit-step Signals 60

4.2 Classification of Systems 61

4.2.1 Linear and Nonlinear Systems 61

4.2.2 Time-invariant and Time-varying Systems 62

4.2.3 Causal and Noncausal Systems 63

4.2.4 Instantaneous and Dynamic Systems 64

4.2.5 Inverse Systems 64

4.2.6 Continuous and Discrete Systems 64

4.3 Convolution–Summation Model 64

4.3.1 Properties of Convolution–Summation 67

4.3.2 The Difference Equation and Convolution–Summation 68

4.3.3 Response to Complex Exponential Input 69

4.4 System Stability 71

4.5 Realization of Discrete Systems 72

4.5.1 Decomposition of Higher-order Systems 73

4.5.2 Feedback Systems 74

4.6 Summary 74

Further Reading 75

Exercises 75

5 Time-domain Analysis of Continuous Systems 79

5.1 Classification of Systems 80

5.1.1 Linear and Nonlinear Systems 80

5.1.2 Time-invariant and Time-varying Systems 81

5.1.3 Causal and Noncausal Systems 82

5.1.4 Instantaneous and Dynamic Systems 83

5.1.5 Lumped-parameter and Distributed-parameter Systems 83

5.1.6 Inverse Systems 83

5.2 Differential Equation Model 83

5.3 Convolution-integral Model 85

5.3.1 Properties of the Convolution-integral 87

5.4 System Response 88

5.4.1 Impulse Response 88

5.4.2 Response to Unit-step Input 89

5.4.3 Characterization of Systems by their Responses to Impulse and Unit-step Signals 91

5.4.4 Response to Complex Exponential Input 92

5.5 System Stability 93

5.6 Realization of Continuous Systems 94

5.6.1 Decomposition of Higher-order Systems 94

5.6.2 Feedback Systems 95

5.7 Summary 96

Further Reading 97

Exercises 97

6 The Discrete Fourier Transform 101

6.1 The Time-domain and the Frequency-domain 101

6.2 Fourier Analysis 102

6.2.1 Versions of Fourier Analysis 104

6.3 The Discrete Fourier Transform 104

6.3.1 The Approximation of Arbitrary Waveforms with a Finite Number of Samples 104

6.3.2 The DFT and the IDFT 105

6.3.3 DFT of Some Basic Signals 107

6.4 Properties of the Discrete Fourier Transform 110

6.4.1 Linearity 110

6.4.2 Periodicity 110

6.4.3 Circular Shift of a Sequence 110

6.4.4 Circular Shift of a Spectrum 111

6.4.5 Symmetry 111

6.4.6 Circular Convolution of Time-domain Sequences 112

6.4.7 Circular Convolution of Frequency-domain Sequences 113

6.4.8 Parseval’s Theorem 114

6.5 Applications of the Discrete Fourier Transform 114

6.5.1 Computation of the Linear Convolution Using the DFT 114

6.5.2 Interpolation and Decimation 115

6.6 Summary 119

Further Reading 119

Exercises 119

7 Fourier Series 123

7.1 Fourier Series 123

7.1.1 FS as the Limiting Case of the DFT 123

7.1.2 The Compact Trigonometric Form of the FS 125

7.1.3 The Trigonometric Form of the FS 126

7.1.4 Periodicity of the FS 126

7.1.5 Existence of the FS 126

7.1.6 Gibbs Phenomenon 130

7.2 Properties of the Fourier Series 132

7.2.1 Linearity 133

7.2.2 Symmetry 133

7.2.3 Time Shifting 135

7.2.4 Frequency Shifting 135

7.2.5 Convolution in the Time-domain 136

7.2.6 Convolution in the Frequency-domain 137

7.2.7 Duality 138

7.2.8 Time Scaling 138

7.2.9 Time Differentiation 139

7.2.10 Time Integration 140

7.2.11 Parseval’s Theorem 140

7.3 Approximation of the Fourier Series 141

7.3.1 Aliasing Effect 142

7.4 Applications of the Fourier Series 144

7.5 Summary 145

Further Reading 145

Exercises 145

8 The Discrete-time Fourier Transform 151

8.1 The Discrete-time Fourier Transform 151

8.1.1 The DTFT as the Limiting Case of the DFT 151

8.1.2 The Dual Relationship Between the DTFT and the FS 156

8.1.3 The DTFT of a Discrete Periodic Signal 158

8.1.4 Determination of the DFT from the DTFT 158

8.2 Properties of the Discrete-time Fourier Transform 159

8.2.1 Linearity 159

8.2.2 Time Shifting 159

8.2.3 Frequency Shifting 160

8.2.4 Convolution in the Time-domain 161

8.2.5 Convolution in the Frequency-domain 162

8.2.6 Symmetry 163

8.2.7 Time Reversal 164

8.2.8 Time Expansion 164

8.2.9 Frequency-differentiation 166

8.2.10 Difference 166

8.2.11 Summation 167

8.2.12 Parseval’s Theorem and the Energy Transfer Function 168

8.3 Approximation of the Discrete-time Fourier Transform 168

8.3.1 Approximation of the Inverse DTFT by the IDFT 170

8.4 Applications of the Discrete-time Fourier Transform 171

8.4.1 Transfer Function and the System Response 171

8.4.2 Digital Filter Design Using DTFT 174

8.4.3 Digital Differentiator 174

8.4.4 Hilbert Transform 175

8.5 Summary 178

Further Reading 178

Exercises 178

9 The Fourier Transform 183

9.1 The Fourier Transform 183

9.1.1 The FT as a Limiting Case of the DTFT 183

9.1.2 Existence of the FT 185

9.2 Properties of the Fourier Transform 190

9.2.1 Linearity 190

9.2.2 Duality 190

9.2.3 Symmetry 191

9.2.4 Time Shifting 192

9.2.5 Frequency Shifting 192

9.2.6 Convolution in the Time-domain 193

9.2.7 Convolution in the Frequency-domain 194

9.2.8 Conjugation 194

9.2.9 Time Reversal 194

9.2.10 Time Scaling 194

9.2.11 Time-differentiation 195

9.2.12 Time-integration 197

9.2.13 Frequency-differentiation 198

9.2.14 Parseval’s Theorem and the Energy Transfer Function 198

9.3 Fourier Transform of Mixed Classes of Signals 200

9.3.1 The FT of a Continuous Periodic Signal 200

9.3.2 Determination of the FS from the FT 202

9.3.3 The FT of a Sampled Signal and the Aliasing Effect 203

9.3.4 The FT of a Sampled Aperiodic Signal and the DTFT 206

9.3.5 The FT of a Sampled Periodic Signal and the DFT 207

9.3.6 Approximation of a Continuous Signal from its Sampled Version 209

9.4 Approximation of the Fourier Transform 209

9.5 Applications of the Fourier Transform 211

9.5.1 Transfer Function and System Response 211

9.5.2 Ideal Filters and their Unrealizability 214

9.5.3 Modulation and Demodulation 215

9.6 Summary 219

Further Reading 219

Exercises 219

10 The z-Transform 227

10.1 Fourier Analysis and the z-Transform 227

10.2 The z-Transform 228

10.3 Properties of the z-Transform 232

10.3.1 Linearity 232

10.3.2 Left Shift of a Sequence 233

10.3.3 Right Shift of a sequence 234

10.3.4 Convolution 234

10.3.5 Multiplication by n 235

10.3.6 Multiplication by an 235

10.3.7 Summation 236

10.3.8 Initial Value 236

10.3.9 Final Value 237

10.3.10 Transform of Semiperiodic Functions 237

10.4 The Inverse z-Transform 237

10.4.1 Finding the Inverse z-Transform 238

10.5 Applications of the z-Transform 243

10.5.1 Transfer Function and System Response 243

10.5.2 Characterization of a System by its Poles and Zeros 245

10.5.3 System Stability 247

10.5.4 Realization of Systems 248

10.5.5 Feedback Systems 251

10.6 Summary 253

Further Reading 253

Exercises 253

11 The Laplace Transform 259

11.1 The Laplace Transform 259

11.1.1 Relationship Between the Laplace Transform and the z-Transform 262

11.2 Properties of the Laplace Transform 263

11.2.1 Linearity 263

11.2.2 Time Shifting 264

11.2.3 Frequency Shifting 264

11.2.4 Time-differentiation 265

11.2.5 Integration 267

11.2.6 Time Scaling 268

11.2.7 Convolution in Time 268

11.2.8 Multiplication by t 269

11.2.9 Initial Value 269

11.2.10 Final Value 270

11.2.11 Transform of Semiperiodic Functions 270

11.3 The Inverse Laplace Transform 271

11.4 Applications of the Laplace Transform 272

11.4.1 Transfer Function and System Response 272

11.4.2 Characterization of a System by its Poles and Zeros 273

11.4.3 System Stability 274

11.4.4 Realization of Systems 276

11.4.5 Frequency-domain Representation of Circuits 276

11.4.6 Feedback Systems 279

11.4.7 Analog Filters 282

11.5 Summary 285

Further Reading 285

Exercises 285

12 State-space Analysis of Discrete Systems 293

12.1 The State-space Model 293

12.1.1 Parallel Realization 297

12.1.2 Cascade Realization 299

12.2 Time-domain Solution of the State Equation 300

12.2.1 Iterative Solution 300

12.2.2 Closed-form Solution 301

12.2.3 The Impulse Response 307

12.3 Frequency-domain Solution of the State Equation 308

12.4 Linear Transformation of State Vectors 310

12.5 Summary 312

Further Reading 313

Exercises 313

13 State-space Analysis of Continuous Systems 317

13.1 The State-space Model 317

13.2 Time-domain Solution of the State Equation 322

13.3 Frequency-domain Solution of the State Equation 327

13.4 Linear Transformation of State Vectors 330

13.5 Summary 332

Further Reading 333

Exercises 333

Appendix A: Transform Pairs and Properties 337

Appendix B: Useful Mathematical Formulas 349

Answers to Selected Exercises 355

Index 377