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