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Fourier and Laplace Transforms

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This 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.

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2. Fourier and Laplace TransformsThis book presents in a unified manner the fundamentals of both continuous anddiscrete versions of the Fourier and Laplace transforms. These transforms play animportant role in the analysis of all kinds of physical phenomena. As a link betweenthe various applications of these transforms the authors use the theory of signals andsystems, 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 Laplacetransform, and finally the discrete versions of these transforms, in particular the Dis-creteFourier Transform together with its fast implementation, and the z-transform.Each part closes with a separate chapter on the applications of the specific transformto signals, systems, and differential equations. The book includes a preliminary partwhich develops the relevant concepts in signal and systems theory and also containsa review of mathematical prerequisites.This textbook is designed for self-study. It includes many worked examples, to-getherwith more than 450 exercises, and will be of great value to undergraduatesand graduate students in applied mathematics, electrical engineering, physics andcomputer science. 3. Fourier and LaplaceTransformsR. 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, TokyoCambridge University PressThe Edinburgh Building, Cambridge CB2 8RU, UKPublished in the United States of America by Cambridge University Press, New Yorkwww.cambridge.orgInformation on this title: www.cambridge.org/9780521806893 Cambridge University Press 2003This publication is in copyright. Subject to statutory exception and to theprovision of relevant collective licensing agreements, no reproduction of any partmay take place without the written permission of Cambridge University Press.First published in print formatISBN-13 978-0-511-67312-2ISBN-13 978-0-521-80689-3ISBN-13 978-0-521-53441-32003eBook (EBL)HardbackPaperbackCambridge University Press has no responsibility for the persistence or accuracyof 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 page ixIntroduction 1Part 1 Applications and foundations1 Signals and systems 71.1 Signals and systems 81.2 Classification of signals 111.3 Classification of systems 162 Mathematical prerequisites 272.1 Complex numbers, polynomials and rational functions 282.2 Partial fraction expansions 352.3 Complex-valued functions 392.4 Sequences and series 452.5 Power series 51Part 2 Fourier series3 Fourier series: definition and properties 603.1 Trigonometric polynomials and series 613.2 Definition of Fourier series 653.3 The spectrum of periodic functions 713.4 Fourier series for some standard functions 723.5 Properties of Fourier series 763.6 Fourier cosine and Fourier sine series 804 The fundamental theorem of Fourier series 864.1 Bessels inequality and RiemannLebesgue lemma 864.2 The fundamental theorem 894.3 Further properties of Fourier series 954.4 The sine integral and Gibbs phenomenon 105v 6. vi Contents5 Applications of Fourier series 1135.1 Linear time-invariant systems with periodic input 1145.2 Partial differential equations 122Part 3 Fourier integrals and distributions6 Fourier integrals: definition and properties 1386.1 An intuitive derivation 1386.2 The Fourier transform 1406.3 Some standard Fourier transforms 1446.4 Properties of the Fourier transform 1496.5 Rapidly decreasing functions 1566.6 Convolution 1587 The fundamental theorem of the Fourier integral 1647.1 The fundamental theorem 1657.2 Consequences of the fundamental theorem 1727.3 Poissons summation formula 1818 Distributions 1888.1 The problem of the delta function 1898.2 Definition and examples of distributions 1928.3 Derivatives of distributions 1978.4 Multiplication and scaling of distributions 2039 The Fourier transform of distributions 2089.1 The Fourier transform of distributions: definitionand examples 2099.2 Properties of the Fourier transform 2179.3 Convolution 22110 Applications of the Fourier integral 22910.1 The impulse response 23010.2 The frequency response 23410.3 Causal stable systems and differential equations 23910.4 Boundary and initial value problems for partialdifferential equations 243Part 4 Laplace transforms11 Complex functions 25311.1 Definition and examples 25311.2 Continuity 256 7. Contents vii11.3 Differentiability 25911.4 The CauchyRiemann equations 26312 The Laplace transform: definition and properties 26712.1 Definition and existence of the Laplace transform 26812.2 Linearity, shifting and scaling 27512.3 Differentiation and integration 28013 Further properties, distributions, and the fundamentaltheorem 28813.1 Convolution 28913.2 Initial and final value theorems 29113.3 Periodic functions 29413.4 Laplace transform of distributions 29713.5 The inverse Laplace transform 30314 Applications of the Laplace transform 31014.1 Linear systems 31114.2 Linear differential equations with constant coefficients 32314.3 Systems of linear differential equations with constantcoefficients 32714.4 Partial differential equations 330Part 5 Discrete transforms15 Sampling of continuous-time signals 34015.1 Discrete-time signals and sampling 34015.2 Reconstruction of continuous-time signals 34415.3 The sampling theorem 34715.4 The aliasing problem 35116 The discrete Fourier transform 35616.1 Introduction and definition of the discreteFourier transform 35616.2 Fundamental theorem of the discrete Fourier transform 36216.3 Properties of the discrete Fourier transform 36416.4 Cyclical convolution 36817 The Fast Fourier Transform 37517.1 The DFT as an operation on matrices 37617.2 The N-point DFT with N = 2m 38017.3 Applications 38318 The z-transform 39118.1 Definition and convergence of the z-transform 39218.2 Properties of the z-transform 396 8. viii Contents18.3 The inverse z-transform of rational functions 40018.4 Convolution 40418.5 Fourier transform of non-periodic discrete-time signals 40719 Applications of discrete transforms 41219.1 The impulse response 41319.2 The transfer function and the frequency response 41919.3 LTD-systems described by difference equations 424Literature 429Tables of transforms and properties 432Index 444 9. PrefaceThis book arose from the development of a course on Fourier and Laplacetransforms for the Open University of the Netherlands. Originally it was theintention to get a suitable course by revising part of the book Analysis andnumerical analysis, part 3 in the series Mathematics for higher educationby R. van Asselt et al. (in Dutch). However, the revision turned out to be sothorough that in fact a completely new book was created. We are gratefulthat Educaboek was willing to publish the original Dutch edition of the bookbesides 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 thosewho 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 Fourierand Laplace transforms as well as to those who appreciate a mathematicallysound treatment of the theory.We assume that the reader has a mathematical background comparableto an undergraduate student in one of the technical sciences. In particularwe assume a basic understanding and skill in differential and integral cal-culus.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 anumber of coherent chapters covering a specific part of the field of Fourierand Laplace transforms. In each chapter we accurately state all the learningobjectives, so that the reader will knowwhat we expect from him or her whenstudying that particular chapter. Besides this, we start each chapter with anintroduction and we close each chapter with a summary and a selftest. Theselftest consists of a series of exercises that readers can use to test their ownknowledge and skills. For selected exercises, answers and extensive hintswill be available on the CUP website.Sections contain such items as definitions, theorems, examples, and so on.These are clearly marked in the left margin, often with a number attached tothem. In the remainder of the text we then refer to these numbered items.ix 10. x PrefaceFor almost all theorems proofs are given following the heading Proof. Theend of a proof is indicated by a right-aligned black square: . In some casesit may be wise to skip the proof of a theorem in a first reading, in order notto lose the main line of argument. The proof can be studied later on.Examples are sometimes included in the running text, but often they arepresented separately. In the latter case they are again clearly marked in theleft 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 maynot 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 con-cernselements such as sections, parts of sections, or exercises which areconsiderably more difficult than the rest of the text. In those parts we godeeper into the material or we present more detailed background material.The book is written in such a way that