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PART ONE 1

Chapter 1．Introduction 1

1-1．Fourier Analysis 1

1-2．The Laplace Transform 2

1-3．Linear Systems 4

1-4．Singularity Functions 5

1-5．The Fourier Transform in Probability Theory 6

Chapter 2．Basic Theorems and Examples 7

2-1．The Fourier Integral 7

2-2．Special Forms of the Fourier Integral 10

2-3．Simple Theorems 14

2-4．Examples 18

2-5．The Convolution Theorem 25

2-6．On the Proof of the Fourier-integral Theorem 29

Chapter 3．Singularity Functions and Line Spectra 35

3-1．Basic Examples 36

3-2．Fourier Series 42

3-3．Poisson＇s Sum Formula 47

3-4．Periodic-frequency Spectra 49

3-5．Sampling Theorem 50

Chapter 4．Numerical Techniques and Uncertainty Principle 53

4-1．Evaluation of the Fourier Transform 53

4-2．Evaluation of the Inversion Integral 56

4-3．Approximate Evaluation ofthe Convolution Integral 59

4-4．Duration of a Signal and Uncertainty Principle 62

4-5．Generalization of the Uncertainty Principle 67

Problems 75

Solutions 77

PART TWO 81

Chapter 5．Linear Systems 81

5-1．Definitions 81

5-2．The System Function 86

5-3．Evaluation of the Step Response 89

Chapter 6．Low-pass Filters 94

6-1．Definitions 94

6-2．Amplitude Distortion 97

6-3．Causal Systems with Linear Phase 106

6-4．Phase Distortion 108

6-5．Summary 119

Chapter 7．Bandpass Filters 120

7-1．Symmetrical Systems 120

7-2．Modulated Input 127

7-3．Unsymmetrical Systems 131

7-4．Modulated Input 133

7-5．Group，Phase，and Signal-front Delay 134

7-6．Group，PHase，and Signal-front Velocity 136

7-7．The Principle of Stationary Phase 139

Chapter 8．Spectrum Analyzers 144

8-1．Simultaneous Spectral Analysis 145

8-2．Sequential Spectral Analysis 150

8-3．Periodic Signals 153

Problems 161

Solutions 165

PART THREE 169

Chapter 9．The Laplace Transform 169

9-1．The Unilateral Laplace Transform 169

9-2．Relationship between the Fourier Integral of a Causal Function and the Unilateral Laplace Transform 172

9-3．The Inversion Formula 175

9-4．Evaluation of f(t) 176

9-5．Initial-value Theorem 186

9-6．The Bilateral Laplace Transform 187

Chapter 10．Integral Theorems 192

10-1．Integral Theorems 192

10-2．Relationship between R(ω)and X(ω) 195

10-3．Minimum-phase-shift Functinos 204

10-4．Energy of a Signal 212

10-5．Causality Conditions 213

Problems 218

Solutions 220

PART FOUR 223

Chapter 11．Positive Functions and Limit Theorems 223

11-1．The Density Function 223

11-2．Repeated Convolution 226

11-3．The Central-limit Theorem 227

11-4．Error Correction 233

Chapter 12．Generalized Harmonic Analysis，Correlation，and Power Spectra 240

12-1．Introduction 240

12-2．Finite Energy Signals 241

12-3．Finite Power Signals 245

12-4．Functions with Arbitrary Power Spctra 254

12-5．Generalized Harmonic Analysis 259

Problems 265

Solutions 266

APPENDIXES 269

Appendix Ⅰ．The Impulse Function as Distribution 269

Ⅰ-1．Definitions 270

Ⅰ-2．Generalized Limits 277

Ⅰ-3．Physical Concepts as Distributions 281

Appendix Ⅱ．Analytic Functions 283

Ⅱ-1．Definitions 283

Ⅱ-2．Integration 290

Ⅱ-3．Calculus of Residues 296

Ⅱ-4．Saddle-point Method of Integration 302

Ⅱ-5．Positive Real Functions 307

Index 313