Second Edition
`
`550 Paulo Singapore Sydney Tokyo Toronto
`
`Ronald N. Bracewell
`Lewis M. Terman Professor of Electrical Engineering
`Stanford University
`
`The Fourier Transform
`and Its Applications
`
`McGraw-llill Book Company
`New York St. Louis San Francisco Auckland Bogoté Diisseldorf
`Johannesburg London Madrid Mexico Montreal New Delhi
`Panama Paris
`
`1
`
`Micro Motion 1041
`
`
`
`1
`
`Micro Motion 1041
`
`
`
`550 Paulo Singapore Sydney Tokyo Toronto
`
`Ronald N. Bracewell
`Lewis M. Terman Professor of Electrical Engineering
`Stanford University
`
`The Fourier Transform
`and Its Applications
`
`McGraw-llill Book Company
`New York St. Louis San Francisco Auckland Bogoté Diisseldorf
`Johannesburg London Madrid Mexico Montreal New Delhi
`Panama Paris
`
`1
`
`Micro Motion 1041
`
`
`
`1
`
`Micro Motion 1041
`
`
`
`
`
`The Fourier Transform and Its Applications
`
`Copyright © 1978, 1965 by McGraw-Hill, Inc. All rights reserved.
`Printed in the United States of America. No part of this publication
`may be reproduced, stored in a retrieval system, or transmitted, in any
`form or by any means, electronic, mechanical, photocopying, recording, or
`otherwise, without the prior written permission of the publisher.
`
`7890 DODO 83
`
`This book was set in Scotch Roman by Bi—Comp, Incorporated.
`The editors were Julienne V. Brown and Michael Gardner;
`the production supervisor was Dennis J. Conroy.
`The drawings were done by J & R Services, Inc.
`R. R. Donnelley & Sons Company was printer and binder.
`
`Library of Congress Cataloging in Publication Data
`Bracewell, Ronald Newbold, date
`The Fourier transform and its applications.
`
`(McGraw-Hill electrical and electronic engineering series)
`Includes index.
`9. Transformations
`1. Fourier transformations.
`(Mathematics)
`3. Harmonic analysis.
`I. Title.
`QA403.5.B7
`1978
`515'.723
`77-13376
`ISBN 0-07-007013-X
`
`
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`Groundwork
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`7
`
`The, customary formulas exhibiting the reversibility of the Fourier
`
`transformation are
`'
`
`
`F(8) = f; f(:c)e—””‘ d2:
`f(x) = f: F(s)e"2mds.W
`In this form, two successive transfermations are made to yield the original
`function. The second transformation, however, is not exactly the same as
`the first, and where it is necessary to distinguish between these two sorts of
`Fourier transform, we shall say that F(s) is the minus-i transform of f(x)
`.
`.
`and that f(:r) is the plus-z transform of F (8).
`Writing the two successive transformations as a repeated integral, we
`obtain the usual statement of Fourier’s integral theorem:
`
`f(xj = [:0 [ f: f(x)e—i21rzs dx] 8mm d3.
`The conditions under which this is true are given in the next section, but it
`must be stated at once that where f(:v) is discontinuous the left-hand side
`should be replaced by afar +) + f(:v —)], that is, by the mean of the
`unequal limits of flat) as x is approached from above and below.
`The factor 21r appearing in the transform formulas may be lumped with
`s to yield the following version (system 2):
`
`F(s) = f.” f(x)e“'” dx
`f(x) = i _: F(s)ei”ds.
`
`And for the sake of symmetry, authors occasionally write (system 3):
`
`—'i:l:a
`
`I
`”
`dx
`m) _ waflx)‘;
`1
`a
`‘
`f(x) = W [W F(s)e1 ‘ds.
`
`f(s). Various advantages and disadvantages are found in both notations.
`
`All three versions are in common use, but here we shall keep the 211' in the
`exponent (system 1).
`If f(:r) and F(s) are a transform pair in system 1,
`then f(.z‘) and F(3/210 are a transform pair in system 2, andf[x/ (21rfl and
`F[s/(21r)f] are a transform pair in system 3. An example of a transform
`pair in each of the three systems follows.
`
`System 1
`
`KI)
`e~1nrz
`
`F6?)
`e—n”
`
`System 2
`
`f(16)
`e—wz’
`
`F(s)
`e—s2/41r
`
`System 3
`
`KI)
`e—h’
`
`F(S)
`e—h’
`
`An excellent notation which may be used as an alternative to F (s) is
`
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`358
`
`THE FOURIER TRANSFORM AND ITS APPLICATIONS
`
`utt)
`
` —o.25
`
`0.25
`
`t
`
`Fig. 18.2 A function of the continuous variable t and one way of representing it
`by eight sample values.
`
`It will be noticed
`In what follows, f(-r) forms the point of departure.
`that no provision is made for cases where there is no starting point, as
`with a function such as exp (—t”). This is in keeping with the practical
`character of the discrete transform, which does not contemplate data
`trains dating back to the indefinitely remote past. A second feature to
`note is that the finishing point must occur after a finite time. However,
`it need not come at -r = 5 as in Table 18.2; one might choose to let -r run
`on to 15 and assign values of zero to the extra samples. This is a con-
`scious choice that must always be made.
`It may be important; for
`example, Table 18.2 does not convey the information given in the equa-
`tion preceding it——that following the half-period cosine,
`the voltage
`remains zero. The table remains silent on that point, and if it is impor-
`tant, the necessary number of zeros would need to be appended.
`By definition, f(-r) possesses a discrete Fourier transform F (u) given by
`N.—i
`.
`
`F(u) = N—l E f(T)e—i2r(le)1-.
`r=0
`
`(1)
`
`The quantity u/N is analogous to frequency measured in cycles per sam-
`pling interval. The correspondence of symbols may be summarized as
`follows:
`
`Time
`
`Frequency
`
`Continuous case
`Discrete case
`
`t
`-r
`
`f
`u/N
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`
`The symbol v has been chosen in the discrete case, instead of f, to
`emphasize that the frequency integer 1} is related to frequency but is not
`the same as frequency f. For example, if the sampling interval is 1 second
`and there are eight samples (N = 8), then the component of frequency f
`will be found at u = 8f; conversely, the frequency represented by a fre-
`quency integer u = 1 will be @— hertz.
`
`
`
`5
`
`
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