Fourier And Laplace Transforms

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 Discrete 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, 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. 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 Introduction page ix 1 Part 1 Applications and foundations 1 Signals and systems 1.1 Signals and systems 1.2 Classification of signals 1.3 Classification of systems 7 8 11 16 2 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 series 27 28 35 39 45 51 Part 2 Fourier series 3 Fourier series: definition and properties 3.1 Trigonometric polynomials and series 3.2 Definition of Fourier series 3.3 The spectrum of periodic functions 3.4 Fourier series for some standard functions 3.5 Pro