DIGITAL
`SIGNAL
`PROCESSING
`
`Alan V. Oppenheim
`Department of Electrical Engineering
`Massachusetts Institute'of Technology
`
`Ronald W. Schafer
`Bell Telephone Laboratories
`Murray Hill, New Jersey
`
`'"
`
`PRENTICE-HALL, INC., Englewood Cliffs, New Jersey.
`
`,.~
`
`RTL345-1_1029-0001
`
`

`
`Library of COllgress Cataloglllg III Publlcatloll Data
`
`OPPENHEIM, ALAN V.
`Digital Signal Processing.
`
`Includes bibliographical references.
`1. Signal theory (Telecommunication) 2 . Digital electronics. I. SCHAFER, RONALD W.
`joint author. II. Title.
`TK5102.5.0245
`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.
`
`10 9 8 7 6 5 4 3 2 1
`
`Printed in the United States of America
`
`PRENTICE-HALL INTERNATIONAL, INC., LOlldon
`PRENTICE-HALL OF AUSTRALIA, PTy. LTD., Sydlley
`PRENTICE-HALL OF CANADA, LTD., Toronto
`PRENTICE-HALL OF INDIA PRIVATE LIMITED, New Delhi
`PRENTICE-HALL OF JAPAN, INC., Tokyo
`
`l
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`RTL345-1_1029-0002
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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 c",,,,(m). That is,
`
`IN(w) = L c",.,(m)e- iwm
`
`N-I
`
`m=-( N-I )
`
`(11.24)
`
`Since the Fourier transform of the real finite-length sequence x(n), 0 :c::; n :c::;
`N - 1, is
`
`X(e iW
`
`N-I
`
`) = L x(n)e- iwn
`
`n=O
`it can be shown that (see Problem 1 of this chapter) .
`
`(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(w ) is
`
`E[IN(w)] = L E[c",,,,(m)]e- iwm
`
`N-I
`
`m=-(N-I)
`
`(11.26)
`
`Since we have shown that for a zero mean process
`
`E[c",,,,(m)] =
`
`N-Iml
`cP",.,(m) ,
`N
`
`Iml <N
`
`then
`
`(11.27)
`
`Iml)/N,
`Thus because of the finite limits of summation and the factor (N -
`E[IN(w)] is not equal to the Fourier transform of cP",x(m) , and therefore the
`periodogram is a biased estimate of the power spectrum, P",x(w).
`Alternatively, consider the Fourier transform of the estimate c~", (m); i.e.,
`
`PN(W) = L <.,(m)e-iwm
`
`N-I
`
`m=-(N-l)
`
`The expected value of PN(W) is
`
`E[P,y(W)] = L
`
`N-l
`
`E[c~xCm)]e-iW'"
`m=-(1\'-I)
`
`N-l L
`
`cPxx(m) e- iwm
`
`711=-(1\'- 1)
`
`(11.28)
`
`(11.29)
`
`

`
`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
`P",,,,(w) , even though c",,,,(m) is an unbiased estimate of 1>",,,,(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
`
`wB(m) =
`
`(
`
`N-Iml
`N
`'
`
`0,
`
`Iml < N
`
`otherwise
`
`(11.30)
`
`In Chapter 5 we called this the Bartlett window. For Eq. (11.29) the window
`is rectangular ; i.e.,
`
`Iml <N
`otherwise
`
`(11.31)
`
`Using the concepts introduced in Chapter 5 we can see that Eqs. (I 1.27)
`and (I 1.29) can be interpreted in the frequency domain as the convolutions
`
`and
`
`where
`
`and
`
`1 (Sin [WNI2])2
`iw
`W (e ) -
`-
`- N
`sin [wI2]
`B
`
`W (eiw ) _ s _in---=--[ w---,-(2_N_-_1..:..:.) 1--,,2]
`sin [wI2 ]
`R
`-
`
`(11.32)
`
`(11.33)
`
`(11.34)
`
`(11.35)
`
`are the Fourier transforms of the Bartlett and rectangular windows, respec(cid:173)
`tively.
`
`11.3.2 Variance of the Periodogram
`
`To obtain an expression for the variance of the periodogram, it is con(cid:173)
`
`venient to first assume that the sequence x(n), ° :::;; n :::;; N - 1, is a sample
`
`of a real , white, zero-mean process with Gaussian probability density
`functi.ons. The periodogram I N(w) can be expressed as
`
`I N(w) = ~ IX(e iW)12
`1 N-IN-l
`= - I I x(l)x(m)e'W"'e-'W!
`N I~O m~O
`
`. .
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`544
`
`Power Spectrum Estimation
`
`To evaluate the covariance of I N ( w) at two frequencies WI and W 2 we first
`consider
`
`E[IN(w 1)IN(W 2) ] =
`
`1 N-I N -I N -I N-I
`--;; L L L L E [x(k)x(l)x(m)x(n)]e'[W 1(k-O+w 2(m-n)]
`N k= O 1=0 m=O n = O
`
`.
`
`(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 + m)] = a;o(m) does not guarantee a simple expression for
`E [x(k)x(l)x(m)x(n)] for all combinations of k, I, m, and n. However, in the
`case of a white Gaussian process , it can be shown [7] that
`
`E[x(k)x( l)x(m)x(n)] = E[x(k)x(l)]E[x(m)x(n)]
`+ E[x(k)x(m)]E[x(l)x(n)]
`+ E[x(k)x(n)]E[x(l)x(m)]
`
`Therefore,
`
`E [x(k) x( l)x(m)x(n) ] =
`
`a!,
`
`(
`
`0,
`
`k = 1 and m = n
`or k = m and 1 = n
`or k = It and I = m
`otherwise
`
`(11.37)
`
`For other than Gaussian joint density functions, the result will not neces(cid:173)
`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
`formul a with wide validity which would be difficult to interpret. Thus, if
`we substitute Eq. (11.37) into Eq. (11.36), we obtain
`
`or
`
`(11.38)
`
`(If the signal is non-Gaussian, Eq . (11.38) contains additional terms which are
`proportional to l iN [4, 8].) The covariance of the periodogram is
`
`cov [IN(W I ), IN(W 2)] = E [IN (w 1)IN (w 2)] - E [IN (w l )]E[IN (w 2)]
`~(l1.39)
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`
`11.3 The Periodogram as an Estimate of the Power Spectrum
`
`545
`
`---'.'-'----''------------'=----'-----''-
`
`Since E[IN(w l )] = E[IN(w 2 )] = a;, from Eqs. (11.38) and (11.39) we obtain
`4{( sin [(WI + W 2)N/2])2
`cov [IN(w I ), IN(W 2)] = ax
`N sm [(WI + w2)/2]
`+ ( sin .[(wl
`- W 2)N/2])2}
`N sm [(WI - w 2)/2]
`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 W = WI = W 2 is
`(Sin [WN])2}
`4{
`var [IN(w)] = cov [IN(W), IN(w)] = ax 1 +
`.
`Nsmw
`Clearly, the variance of IN(W) do es not approach zero as N approaches
`infinity. Thus the periodogram is not a consistent estimate. In fact,
`var [IN(w)] is of the order of a! no matter how N is chosen.
`We also see from Eq. (11.40) that for frequencies WI = 27Tk/N and W 2 =
`27Tl/N, where k and 1 are integers,
`( sin [7T(k -
`)2
`4{(
`sin [7T(k + l)]
`cov [IN(w I ), IN(w 2)] = ax N sin [7T(k + l)/N] + N sin [7T(k -
`
`(11.40)
`
`(11.41)
`
`l)] )2}
`l)/N]
`
`(11.42)
`which is equal to zero for k *- l. T~us values of the periodogram spaced in
`frequency by integer multiples of 27T/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(cid:173)
`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(cid:173)
`gram increases. This behavior is illustrated in Fig. 11.3 , where the periodo(cid:173)
`gram is plotted for record lengths of N = 14, 51 , l35, 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 derivatiori 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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