This page intentionally left blank Fourier and Laplace Transforms This book 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 Dis- crete Fourier Transform together with its fast implementation, and the z-transform. Each part closes with a separate chapter on the applications of the specific transform to signals, systems, and differential equations. The book includes a preliminary part which develops the relevant concepts in signal and systems theory and also contains a review of mathematical prerequisites. This textbook is designed for self-study. It includes many worked examples, to- gether with more than 450 exercises, and will be of great value to undergraduates and graduate students in applied mathematics, electrical engineering, physics and computer science. 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 Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Dubai, Tokyo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521806893 © Cambridge University Press 2003 This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2003 ISBN-13 978-0-511-67312-2 eBook (EBL) ISBN-13 978-0-521-80689-3 Hardback ISBN-13 978-0-521-53441-3 Paperback Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. Contents 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 v vi Contents 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: definition and 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 partial differential equations 243 Part 4 Laplace transforms 11 Complex functions 253 11.1 Definition and examples 253 11.2 Continuity 256 Contents vii 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 fundamental theorem 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 constant coefficients 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 discrete Fourier 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 m 17.2 The N-point DFT with N = 2 380 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 viii Contents 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