Fourier and Laplace transforms

This textbook presents in a unified manner the fundamentals of both continuous and discrete versions of the Fourier and Laplace transforms. These transforms play an important role in the analysis of all kinds of physical phenomena. As a link between the various applications of these transforms the authors use the theory of signals and systems, as well as the theory of ordinary and partial differential equations. The book is divided into four major parts: periodic functions and Fourier series, non-periodic functions and the Fourier integral, switched-on signals and the Laplace transform, and finally the discrete versions of these transforms, in particular the Discrete Fourier Transform together with its fast implementation, and the z-transform. This textbook is designed for self-study. It includes many worked examples, together with more than 120 exercises, and will be of great value to undergraduates and graduate students in applied mathematics, electrical engineering, physics and computer science. 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 This Textbook Describes In Detail The Various Fourier And Laplace Transforms That Are Used To Analyze Problems In Mathematics, The Natural Sciences And Engineering. These Transforms Decompose Complicated Signals Into Elementary Signals, And Are Widely Used Across The Spectrum Of Science And Engineering. Applications Include Electrical And Mechanical Networks, Heat Conduction And Filters. In Contrast With Other Books, Continuous And Discrete Transforms Are Given Equal Coverage. Fourier and Laplace transforms provide a technique to solve differential equations which frequently occur when translating a physical problem into a mathematical model.