Fourier and Laplace Transforms, 2003

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Fourier and Laplace Transforms
R. J. Beerends, H. G. ter Morsche, J. C. van den Berg and E. M. van de Vrie

Translated from Dutch by R. J. Beerends

CAMBRIDGE UNIVERSITY PRESS, 2003

Preface page ix
Introduction 1

Part 1 Applications and foundations
1 Signals and systems 7
1.1 Signals and systems 8
1.2 Classification of signals 11
1.3 Classification of systems 16
2 Mathematical prerequisites 27
2.1 Complex numbers, polynomials and rational functions 28
2.2 Partial fraction expansions 35
2.3 Complex-valued functions 39
2.4 Sequences and series 45
2.5 Power series 51

Part 2 Fourier series
3 Fourier series: definition and properties 60
3.1 Trigonometric polynomials and series 61
3.2 Definition of Fourier series 65
3.3 The spectrum of periodic functions 71
3.4 Fourier series for some standard functions 72
3.5 Properties of Fourier series 76
3.6 Fourier cosine and Fourier sine series 80
4 The fundamental theorem of Fourier series 86
4.1 Bessel’s inequality and Riemann–Lebesgue lemma 86
4.2 The fundamental theorem 89
4.3 Further properties of Fourier series 95
4.4 The sine integral and Gibbs’ phenomenon 105
5 Applications of Fourier series 113
5.1 Linear time-invariant systems with periodic input 114
5.2 Partial differential equations 122

Part 3 Fourier integrals and distributions
6 Fourier integrals: definition and properties 138
6.1 An intuitive derivation 138
6.2 The Fourier transform 140
6.3 Some standard Fourier transforms 144
6.4 Properties of the Fourier transform 149
6.5 Rapidly decreasing functions 156
6.6 Convolution 158
7 The fundamental theorem of the Fourier integral 164
7.1 The fundamental theorem 165
7.2 Consequences of the fundamental theorem 172
7.3 Poisson’s summation formula∗ 181
8 Distributions 188
8.1 The problem of the delta function 189
8.2 Definition and examples of distributions 192
8.3 Derivatives of distributions 197
8.4 Multiplication and scaling of distributions 203
9 The Fourier transform of distributions 208
9.1 The Fourier transform of distributions: definitionand examples 209
9.2 Properties of the Fourier transform 217
9.3 Convolution 221
10 Applications of the Fourier integral 229
10.1 The impulse response 230
10.2 The frequency response 234
10.3 Causal stable systems and differential equations 239
10.4 Boundary and initial value problems for partialdifferential equations 243

Part 4 Laplace transforms
11 Complex functions 253
11.1 Definition and examples 253
11.2 Continuity 256
11.3 Differentiability 259
11.4 The Cauchy–Riemann equations∗ 263
12 The Laplace transform: definition and properties 267
12.1 Definition and existence of the Laplace transform 268
12.2 Linearity, shifting and scaling 275
12.3 Differentiation and integration 280
13 Further properties, distributions, and the fundamentaltheorem 288
13.1 Convolution 289
13.2 Initial and final value theorems 291
13.3 Periodic functions 294
13.4 Laplace transform of distributions 297
13.5 The inverse Laplace transform 303
14 Applications of the Laplace transform 310
14.1 Linear systems 311
14.2 Linear differential equations with constant coefficients 323
14.3 Systems of linear differential equations with constantcoefficients 327
14.4 Partial differential equations 330

Part 5 Discrete transforms
15 Sampling of continuous-time signals 340
15.1 Discrete-time signals and sampling 340
15.2 Reconstruction of continuous-time signals 344
15.3 The sampling theorem 347
15.4 The aliasing problem∗ 351
16 The discrete Fourier transform 356
16.1 Introduction and definition of the discreteFourier transform 356
16.2 Fundamental theorem of the discrete Fourier transform 362
16.3 Properties of the discrete Fourier transform 364
16.4 Cyclical convolution 368
17 The Fast Fourier Transform 375
17.1 The DFT as an operation on matrices 376
17.2 The N-point DFT with N = 2m 380
17.3 Applications 383
18 The z-transform 391
18.1 Definition and convergence of the z-transform 392
18.2 Properties of the z-transform 396
18.3 The inverse z-transform of rational functions 400
18.4 Convolution 404
18.5 Fourier transform of non-periodic discrete-time signals 407
19 Applications of discrete transforms 412
19.1 The impulse response 413
19.2 The transfer function and the frequency response 419
19.3 LTD-systems described by difference equations 424

Literature 429
Tables of transforms and properties 432
Index 444

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