Fourier and Laplace transforms

Cover......Page 1
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Contents......Page 7
Preface......Page 11
Introduction......Page 13
Part 1 Applications and foundations......Page 17
INTRODUCTION......Page 19
1.1 Signals and systems......Page 20
1.2.1 Continuous-time and discrete-time signals......Page 23
1.2.2 Periodic signals......Page 24
1.2.3 Power and energy signals......Page 26
1.3 Classification of systems......Page 28
1.3.2 Linear time-invariant systems......Page 29
1.3.4 Real systems......Page 32
1.3.5 Causal systems......Page 33
1.3.7 Systems described by difference equations......Page 34
SUMMARY......Page 35
SELFTEST......Page 36
INTRODUCTION......Page 39
2.1.1 Elementary properties of complex numbers......Page 40
2.1.2 Zeros of polynomials......Page 44
2.2 Partial fraction expansions......Page 47
2.3 Complex-valued functions......Page 51
EXERCISES......Page 56
2.4.1 Basic properties......Page 57
2.4.2 Absolute convergence and convergence tests......Page 59
2.4.3 Series of functions......Page 61
2.5 Power series......Page 63
EXERCISES......Page 66
SELFTEST......Page 67
INTRODUCTION TO PART 2......Page 69
INTRODUCTION......Page 72
3.1 Trigonometric polynomials and series......Page 73
3.2 Definition of Fourier series......Page 77
3.2.1 Fourier series......Page 78
3.2.2 Complex Fourier series......Page 80
EXERCISES......Page 82
EXERCISES......Page 83
3.4.1 The periodic block function......Page 84
3.4.2 The periodic triangle function......Page 86
EXERCISES......Page 87
3.5.1 Linearity......Page 88
3.5.2 Conjugation......Page 89
3.5.3 Shift in time......Page 90
3.5.4 Time reversal......Page 91
3.6 Fourier cosine and Fourier sine series......Page 92
EXERCISES......Page 94
SELFTEST......Page 95
4.1 Bessel’s inequality and Riemann–Lebesgue lemma......Page 98
4.2 The fundamental theorem......Page 101
EXERCISES......Page 106
4.3 Further properties of Fourier series......Page 107
4.3.1 Product and convolution......Page 108
4.3.3 Integration......Page 111
4.3.4 Differentiation......Page 113
EXERCISES......Page 114
4.4 The sine integral and Gibbs’ phenomenon......Page 117
4.4.1 The sine integral......Page 118
4.4.2 Gibbs' phenomenon......Page 119
SUMMARY......Page 121
SELFTEST......Page 122
INTRODUCTION......Page 125
5.1 Linear time-invariant systems with periodic input......Page 126
5.1.1 Systems described by differential equations......Page 127
5.2 Partial differential equations......Page 134
5.2.1 The heat equation......Page 135
5.2.2 The wave equation......Page 139
EXERCISES......Page 142
SUMMARY......Page 143
SELFTEST......Page 144
INTRODUCTION TO PART 3......Page 147
6.1 An intuitive derivation......Page 150
6.2 The Fourier transform......Page 152
EXERCISE......Page 155
6.3.1 The block function......Page 156
6.3.2 The triangle function......Page 157
6.3.3 The function e-a| t |......Page 158
6.3.4 The Gauss function......Page 159
EXERCISES......Page 160
6.4.1 Linearity......Page 161
6.4.4 Shift in the frequency domain......Page 162
6.4.6 Even and odd functions......Page 163
6.4.7 Selfduality......Page 164
6.4.8 Differentiation in the time domain......Page 165
6.4.9 Differentiation in the frequency domain......Page 166
EXERCISES......Page 167
6.5 Rapidly decreasing functions......Page 168
6.6 Convolution......Page 170
SUMMARY......Page 173
SELFTEST......Page 174
INTRODUCTION......Page 176
7.1 The fundamental theorem......Page 177
EXERCISES......Page 183
7.2.1 Uniqueness......Page 184
7.2.2 Fourier pairs......Page 185
7.2.3 Definite integrals......Page 188
7.2.4 Convolution in the frequency domain......Page 189
7.2.5 Parseval’s identities......Page 190
EXERCISES......Page 191
7.3 Poisson's summation formula......Page 193
SUMMARY......Page 196
SELFTEST......Page 197
INTRODUCTION......Page 200
8.1 The problem of the delta function......Page 201
8.2.1 Definition of distributions......Page 204
8.2.2 The delta function......Page 205
8.2.3 Functions as distributions......Page 206
8.3 Derivatives of distributions......Page 209
EXERCISES......Page 214
8.4 Multiplication and scaling of distributions......Page 215
EXERCISES......Page 217
SELFTEST......Page 218
INTRODUCTION......Page 220
9.1.1 Definition of the Fourier transform of distributions......Page 221
9.1.2 Examples of Fourier transforms of distributions......Page 222
9.1.3 The comb distribution and its spectrum......Page 226
EXERCISES......Page 228
9.2.1 Shift in time and frequency domains......Page 229
9.2.2 Differentiation in time and frequency domains......Page 230
9.2.3 Reciprocity......Page 231
9.3 Convolution......Page 233
9.3.1 Intuitive derivation of the convolution of distributions......Page 234
9.3.2 Mathematical treatment of the convolution of distributions......Page 235
SUMMARY......Page 238
SELFTEST......Page 239
INTRODUCTION......Page 241
10.1 The impulse response......Page 242
EXERCISES......Page 245
10.2 The frequency response......Page 246
EXERCISES......Page 249
10.3 Causal stable systems and differential equations......Page 251
EXERCISES......Page 254
10.4 Boundary and initial value problems for partial differential equations......Page 255
SUMMARY......Page 257
SELFTEST......Page 258
INTRODUCTION TO PART 4......Page 261
11.1 Definition and examples......Page 265
11.2 Continuity......Page 268
EXERCISES......Page 270
11.3 Differentiability......Page 271
EXERCISES......Page 274
11.4 The Cauchy–Riemann equations......Page 275
SELFTEST......Page 277
INTRODUCTION......Page 279
12.1 Definition and existence of the Laplace transform......Page 280
12.2.1 Linearity......Page 287
12.2.2 Shift in the time domain......Page 288
12.2.3 Shift in the s-domain......Page 289
12.2.4 Scaling......Page 290
EXERCISES......Page 291
12.3.1 Differentiation in the time domain......Page 292
12.3.2 Differentiation in the s-domain......Page 294
EXERCISES......Page 296
SUMMARY......Page 297
SELFTEST......Page 298
INTRODUCTION......Page 300
13.1 Convolution......Page 301
EXERCISES......Page 302
13.2 Initial and final value theorems......Page 303
13.3 Periodic functions......Page 306
EXERCISES......Page 308
13.4.1 Intuitive derivation......Page 309
13.4.2 Mathematical treatment......Page 312
EXERCISES......Page 314
13.5 The inverse Laplace transform......Page 315
SUMMARY......Page 319
SELFTEST......Page 320
INTRODUCTION......Page 322
14.1.1 The transfer function......Page 323
14.1.2 The method of Laplace transforming......Page 324
14.1.3 Systems described by differential equations......Page 326
14.1.4 Stability......Page 330
14.1.5 The harmonic oscillator......Page 331
EXERCISES......Page 333
14.2 Linear differential equations with constant coefficients......Page 335
EXERCISES......Page 338
14.3 Systems of linear differential equations with constant coefficients......Page 339
EXERCISES......Page 341
14.4 Partial differential equations......Page 342
EXERCISES......Page 344
SUMMARY......Page 345
SELFTEST......Page 346
INTRODUCTION TO PART 5......Page 349
15.1 Discrete-time signals and sampling......Page 352
15.2 Reconstruction of continuous-time signals......Page 356
15.3 The sampling theorem......Page 359
EXERCISES......Page 362
15.4 The aliasing problem......Page 363
SUMMARY......Page 364
SELFTEST......Page 365
16.1 Introduction and definition of the discrete Fourier transform......Page 368
16.1.1 Trapezoidal rule for periodic functions......Page 369
16.1.2 An approximation of the Fourier coefficients......Page 370
16.1.3 Definition of the discrete Fourier transform......Page 371
EXERCISES......Page 373
16.2 Fundamental theorem of the discrete Fourier transform......Page 374
16.3.2 Reciprocity......Page 376
16.3.4 Conjugation......Page 377
16.3.5 Shift in the n-domain......Page 378
16.4 Cyclical convolution......Page 380
SUMMARY......Page 383
SELFTEST......Page 384
INTRODUCTION......Page 387
17.1 The DFT as an operation on matrices......Page 388
17.2 The N-point DFT with N = 2m......Page 392
17.3.1 Calculation of Fourier integrals......Page 395
17.3.2 Fast convolution......Page 398
EXERCISES......Page 399
SELFTEST......Page 400
INTRODUCTION......Page 403
18.1 Definition and convergence of the z-transform......Page 404
18.2.1 Linearity......Page 408
18.2.4 Shift in the n-domain......Page 409
18.2.6 Differentiation in the z-domain......Page 410
EXERCISES......Page 411
18.3 The inverse z-transform of rational functions......Page 412
18.4 Convolution......Page 416
18.5 Fourier transform of non-periodic discrete-time signals......Page 419
SUMMARY......Page 421
SELFTEST......Page 422
INTRODUCTION......Page 424
19.1 The impulse response......Page 425
EXERCISES......Page 430
19.2 The transfer function and the frequency response......Page 431
EXERCISES......Page 435
19.3 LTD-systems described by difference equations......Page 436
SUMMARY......Page 439
SELFTEST......Page 440
Literature......Page 441
Tables of transforms and properties......Page 444
Index......Page 456