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2. Fourier and Laplace TransformsThis book presents in a unied 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 nally the discrete versions of these transforms, in particular the Discrete Fourier Transform together with its fast implementation, and the z-transform. Each part closes with a separate chapter on the applications of the specic 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, together 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. 3. Fourier and Laplace Transforms R. J. Beerends, H. G. ter Morsche, J. C. van den Berg and E. M. van de VrieTranslated from Dutch byR. J. Beerends 4. CAMBRIDGE UNIVERSITY PRESSCambridge, New York, Melbourne, Madrid, Cape Town, Singapore, So 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 Information on this title: 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-13978-0-511-67312-2eBook (EBL)ISBN-13978-0-521-80689-3HardbackISBN-13978-0-521-53441-3PaperbackCambridge 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. 5. ContentsPreface Introductionpage ix 1Part 1 Applications and foundations 1 Signals and systems 1.1 Signals and systems 1.2 Classication of signals 1.3 Classication of systems7 8 11 162 Mathematical prerequisites 2.1 Complex numbers, polynomials and rational functions 2.2 Partial fraction expansions 2.3 Complex-valued functions 2.4 Sequences and series 2.5 Power series27 28 35 39 45 51Part 2 Fourier series 3 Fourier series: denition and properties 3.1 Trigonometric polynomials and series 3.2 Denition of Fourier series 3.3 The spectrum of periodic functions 3.4 Fourier series for some standard functions 3.5 Properties of Fourier series 3.6 Fourier cosine and Fourier sine series 4 The fundamental theorem of Fourier series 4.1 Bessels inequality and RiemannLebesgue lemma 4.2 The fundamental theorem 4.3 Further properties of Fourier series 4.4 The sine integral and Gibbs phenomenon60 61 65 71 72 76 80 86 86 89 95 105 v 6. viContents5 Applications of Fourier series 5.1 Linear time-invariant systems with periodic input 5.2 Partial differential equations113 114 122Part 3 Fourier integrals and distributions 6 Fourier integrals: denition and properties 6.1 An intuitive derivation 6.2 The Fourier transform 6.3 Some standard Fourier transforms 6.4 Properties of the Fourier transform 6.5 Rapidly decreasing functions 6.6 Convolution138 138 140 144 149 156 1587 The fundamental theorem of the Fourier integral 7.1 The fundamental theorem 7.2 Consequences of the fundamental theorem 7.3 Poissons summation formula164 165 172 1818 Distributions 8.1 The problem of the delta function 8.2 Denition and examples of distributions 8.3 Derivatives of distributions 8.4 Multiplication and scaling of distributions188 189 192 197 2039 The Fourier transform of distributions 9.1 The Fourier transform of distributions: denition and examples 9.2 Properties of the Fourier transform 9.3 Convolution20810 Applications of the Fourier integral 10.1 The impulse response 10.2 The frequency response 10.3 Causal stable systems and differential equations 10.4 Boundary and initial value problems for partial differential equations209 217 221 229 230 234 239 243Part 4 Laplace transforms 11 Complex functions 11.1 Denition and examples 11.2 Continuity253 253 256 7. Contents11.3 Differentiability 11.4 The CauchyRiemann equationsvii259 26312 The Laplace transform: denition and properties 12.1 Denition and existence of the Laplace transform 12.2 Linearity, shifting and scaling 12.3 Differentiation and integration267 268 275 28013 Further properties, distributions, and the fundamental theorem 13.1 Convolution 13.2 Initial and nal value theorems 13.3 Periodic functions 13.4 Laplace transform of distributions 13.5 The inverse Laplace transform288 289 291 294 297 30314 Applications of the Laplace transform 14.1 Linear systems 14.2 Linear differential equations with constant coefcients 14.3 Systems of linear differential equations with constant coefcients 14.4 Partial differential equations310 311 323 327 330Part 5 Discrete transforms 15 Sampling of continuous-time signals 15.1 Discrete-time signals and sampling 15.2 Reconstruction of continuous-time signals 15.3 The sampling theorem 15.4 The aliasing problem340 340 344 347 35116 The discrete Fourier transform 16.1 Introduction and denition of the discrete Fourier transform 16.2 Fundamental theorem of the discrete Fourier transform 16.3 Properties of the discrete Fourier transform 16.4 Cyclical convolution356 356 362 364 36817 The Fast Fourier Transform 17.1 The DFT as an operation on matrices 17.2 The N -point DFT with N = 2m 17.3 Applications375 376 380 38318 The z-transform 18.1 Denition and convergence of the z-transform 18.2 Properties of the z-transform391 392 396 8. viiiContents18.3 The inverse z-transform of rational functions 18.4 Convolution 18.5 Fourier transform of non-periodic discrete-time signals400 404 40719 Applications of discrete transforms 19.1 The impulse response 19.2 The transfer function and the frequency response 19.3 LTD-systems described by difference equations412 413 419 424Literature Tables of transforms and properties Index429 432 444 9. PrefaceThis book arose from the development of a course on Fourier and Laplace transforms for the Open University of the Netherlands. Originally it was the intention to get a suitable course by revising part of the book Analysis and numerical analysis, part 3 in the series Mathematics for higher education by R. van Asselt et al. (in Dutch). However, the revision turned out to be so thorough that in fact a completely new book was created. We are grateful that Educaboek was willing to publish the original Dutch edition of the book besides the existing series. In writing this book, the authors were led by a twofold objective: - the didactical structure should be such that the book is suitable for those who want to learn this material through self-study or distance teaching, without damaging its usefulness for classroom use; - the material should be of interest to those who want to apply the Fourier and Laplace transforms as well as to those who appreciate a mathematically sound treatment of the theory. We assume that the reader has a mathematical background comparable to an undergraduate student in one of the technical sciences. In particular we assume a basic understanding and skill in differential and integral calculus. Some familiarity with complex numbers and series is also presumed, although chapter 2 provides an opportunity to refresh this subject. The material in this book is subdivided into parts. Each part consists of a number of coherent chapters covering a specic part of the eld of Fourier and Laplace transforms. In each chapter we accurately state all the learning objectives, so that the reader will know what we expect from him or her when studying that particular chapter. Besides this, we start each chapter with an introduction and we close each chapter with a summary and a selftest. The selftest consists of a series of exercises that readers can use to test their own knowledge and skills. For selected exercises, answers and extensive hints will be available on the CUP website. Sections contain such items as denitions, theorems, examples, and so on. These are clearly marked in the left margin, often with a number attached to them. In the remainder of the text we then refer to these numbered items. ix 10. xPrefaceFor almost all theorems proofs are given following the heading Proof. The end of a proof is indicated by a right-aligned black square: . In some cases it may be wise to skip the proof of a theorem in a rst reading, in order not to lose the main line of argument. The proof can be studied later on. Examples are sometimes included in the running text, but often they are presented separately. In the latter case they are again clearly marked in the left margin (with possibly a number, if this is needed as a reference later on). The end of an example is indicated by a right-aligned black triangle: . Mathematical formulas that are displayed on a separate line may or may not be numbered. Only formulas referred to later on in the text have a number (right-aligned and in brackets). Some parts of the book have been marked with an asterisk: . This concerns elements such as sections, parts of sections, or exercises which are considerably more difcult than the rest of the text. In those parts we go deeper into the material or we