DIGITAL
`
`SIGNAL
`
`PROCESSING
`
`Alan V. Oppenheim
`
`Ronald W. Schafer
`
`Department of Electricc_zl Engineering
`Massachusetts Irzst1'tm‘e of Technology
`
`Bell Telephone Laboratories
`Murray Hill, New Jersey
`
`|NC., Englewood Cliffs, New Jersey.
`PRENTICE-HALL,
`R9-1"L345-2_1027-0001
`Realtek345—2 Ex. 1027
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`RTL345-2_1027-0001
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`

`
`Library of Congress Cataloging in Publication Data
`OPPENHEIM, ALAN V.
`Digital Signal Processing.
`
`Includes bibliographical references.
`1. Signal theory (Telecommunication) 2. Digital electronics. 1. SCHAFER, RONALD W.
`joint author. II. Title.
`TK5]02.5.0245
`621.3819'58’2
`ISBN 0-13-214635-5
`
`74-17280
`
`© 1975 by Alan V. Oppenheim
`and Bell Telephone Laboratories, Inc.
`
`All rights reserved. No part of this book
`may be reproduced in any form or by any means
`without permission in writing from the publisher.
`
`10987654321
`
`RTL345-2_1027-0002
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`

`
`11.3.1 Definition of the Periodogram
`
`As an estimate of the power density spectrum let us consider the Fourier
`transform of the biased autocorrelation estimate cm(m). That is,
`N-1
`
`IN(w)= Z cm(rn)e’j‘”'"
`m=—(]V—1)
`
`(11.24)
`
`Since the Fourier transform of the real finite-length sequence x(n), 0 g n 3
`N — l, is
`
`N—1
`
`X(ej“’) : Z x(n)eT"“’”
`7120
`
`it can be shown that (see Problem 1 of this chapter).
`
`1
`‘co
`1N(w)= E ]X(e’ )|2
`
`(11.25)
`
`The spectrum estimate IN(w) is often called the periodogram.
`As before,
`it is of interest to determine the bias and variance of the
`
`periodogram as an estimate of the power spectrum. The expected value of
`IN(co) is
`
`E[IN(w)] = NS E[c,,,,(m)]e-W"
`m=—(N~1)
`
`(11.26)
`
`Since we have shown that for a zero mean process
`
`Ei:Ca:a:(rn):i = ]%V'"—” mm),
`
`lml < N
`
`then
`
`E[I1\7(w)l = NS
`
`m=—(N—1)
`
`(N—_L1') ¢,,.(m)e""‘°"‘
`
`N
`
`(11.27)
`
`Thus because of the finite limits of summation and the factor (N — |m[)/N,
`E[IN((u)] is not equal to the Fourier transform of :;S,,,.(rn), and therefore the
`periodogram is a biased estimate of the power spectrum, P,,,,(w).
`Alternatively, consider the Fourier transform of the estimate c;,,(m); i.e.,
`
`

`
`11.3 The Periodogram as an Estimate of the Power Spectrum 543
`
`Again, because of the finite limits of summation, this is a biased estimate of
`Pm(cu), even though cm(m) is an unbiased estimate of <,{>m(m).
`We can interpret Eqs.
`(11.27) and (11.29) as Fourier transforms of
`windowed autocorrelation sequences. In the case of Eq. (11.27) the window
`is the triangular window
`
`N — [ml
`T ’
`
`wB(m) : {
`
`0,
`
`N
`
`< N
`
`‘mi
`
`otherwise
`
`(11-30)
`
`In Chapter 5 we called this the Bartlett window. For Eq. (11.29) the window
`is rectangular; i.e.,
`
`wR<n> = {
`
`1,
`0,
`
`[ml < N
`otherwise
`
`(1131)
`
`Using the concepts introduced in Chapter 5 we can see that Eqs. (11.27)
`and (11.29) can be interpreted in the frequency domain as the convolutions
`
`and
`
`where
`
`and
`
`EuN(w>J : 2% _”Pmm(6)WB(ej(m_9)) d6
`
`E[PN(w)] = 31; _flPxw(6)WR(ej(w_9)) «I6
`
`WB(e
`
`M, _1 sin [a)N/2] 2
`
`_ Ni sin [co/2] )
`
`WR(e5w) :
`sin [cu/2]
`
`(11.32)
`
`(11.33)
`
`(“'34)
`
`are the Fourier transforms of the Bartlett and rectangular windows, respec-
`tively.
`
`11.3.2 Variance of the Periodogram
`
`To obtain an expression for the Variance of the periodogram, it is con-
`venient to first assume that the sequence 2:07), 0 g 11 g N — 1, is a sample
`of a real, white, zero-mean process with Gaussian probability density
`functions. The periodogram IN(co) can be expressed as
`
`IN((”) = 1%]
`
`|X(e"“’)l2
`
`1 N—1 N—1
`= — Z Z x(l)x(111)ej“’"‘e"“’l
`Nl=U m=0
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`To evaluate the covariance of IN(a>) at two frequencies cal and C02 we first
`consider
`
`E [11v(w1)11v(w2)] =
`
`1 N—1 N—1 N—1 N—1
`2
`Z Z 2 2E[x(k)x(l)x(,n)x(n)]ei[a)1(k—l)+w2(m—n)]
`N k=0 [=0 m=0 n=0
`
`(11.36)
`
`To obtain a useful result, we must simplify Eq. (11.36). In general, it is not
`possible to obtain a very simple result even when x(n) is white, because
`E[x(n)x(n -1- m)] = a,2,(3(m) does not guarantee a simple expression for
`E[x(/c)x(l)x(m)x(n)] for all combinations of k, I, m, and 11. However, i11 the
`case of a white Gaussian process, it can be shown [7] that
`
`E[x(Ic)x(I)x(m)x(n)] = E[x(l<)x(l)]E[x(m)x(n)]
`
`~— E[x(k)x(m)]E[x(l)x(n)]
`
`T E[X(k)«‘<(")lE[X(l)X(m)]
`
`Therefore,
`
`E[x(k)x(l)x(m)x(n)] =
`
`a,
`
`k:landm=n
`ork=mandl=n
`ork=nandl=m
`
`0,
`
`otherwise
`
`(11.37)
`
`For other than Gaussian joint density functions, the result will not neces-
`sarily be so simple. However, our objective is to give a result that will lend
`insight into the problems of spectrum estimation rather than to give a general
`formula with wide validity which would be difiicult to interpret. Thus, if
`we substitute Eq. (11.37) into Eq. (11.36), we obtain
`
`0,4‘
`2
`N—1N~1 _(
`E[I1\’((’)1)I1\’((’)2)]: “ii” +20 Zoe” '"‘"
`7n= 11:
`
`H
`
`)
`
`N~1N—1 _(
`+2 2 e’
`771:0 'n=0
`
`H
`
`)1
`‘°1“”2J
`
`OI‘
`
`EiI1\’(w1)I1V(w2)l : a:{1 +
`
`(11.38)
`
`(If the signal is non-Gaussian, Eq. (11.38) contains additional terms which are
`proportional to 1/N [4, 8].) The covariance of the periodogram is
`
`COV [[.\’((’)1)s [.\'(")2)l = El[.\'((')1)I1\*(“32)l — ElI1\'(")1)lE[IN(C°2)]
`
`‘(l 1.39)
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`11.3 The Periodogram as an Estimate of the Power Spectrum 545
`
`Since E[IN(w1)] = E[IN(co2)] = 0,2,, from Eqs. (11.38) and (11.39) we obtain
`
`_.mwmwmH
`
`cov1Ino»o.Ina»a1~<n{(3;;;;E;;j;7;5Efi)
`mmm—mwm2
`+ (N sin [(w1 — (112)/2])i
`
`0””
`
`From Eq. (11.40) we can draw a number of interesting conclusions about
`the periodogram. The variance of the estimate of the spectrum at a particular
`frequency to : col = (02 is
`
`var [IN(cu)] = cov [IN(w), IN(a))] = a:{1 +
`
`S111 co
`
`(11.41)
`
`the variance of IN(w) does not approach zero as N approaches
`Clearly,
`infinity. Thus the periodogram is not a consistent estimate. In fact,
`var [IN(a))] is of the order of 05% no matter how N is chosen.
`We also see from Eq. (11.40) that for frequencies col = 27rk/N and (112 =
`2171/N, where k and I are integers,
`
`mwmmammm=¢“
`
`sin [7T(k —— 1)]
`
`2
`
`N sin [7T(k —— l)/N]) + (N sin [7r(k — 1)/N])i
`
`sin [7T(/C — 1)]
`
`2
`
`(11.42)
`
`which is equal to zero for k gré 1. Thus values of the periodogram spaced in
`frequency by integer multiples of 271-/N are uncorrelated. As N increases,
`these uncorrelated frequency samples with zero covariance come closer
`together. It is reasonable to expect that a good estimate of the power spec-
`trum should approach a constant as N increases since we have assumed that
`the signal was white. A consequence of the fact that the variance of the
`periodogram approaches a non-zero constant and that the spacing between
`spectral samples with zero covariance decreases as N increases is that as the
`record length becomes longer, the rapidity of the fluctuations in the periodo-
`gram increases. This behavior is illustrated in Fig. 11.3, where the periodo-
`gram is plotted for record lengths ofN = 14, 51, 135, and 452 samples.
`
`11.3.3 General Variance Expressions
`
`All the previous discussion was for the case of estimating the spectrum
`of white noise. If we consider data that are nonwhite but Gaussian,
`the
`
`analysis is considerably more difficult. In evaluating the covariance between
`spectrum samples in this more general case, it is useful to take a heuristic
`approach and develop an approximate expression. The approach that we
`shall take is heuristic; a more rigorous derivation is given by Jenkins and
`Watts [5]. With the application of some approximations to their result,
`the approximate results derived here can be obtained. The basis for the
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