Suppose that an estimate of the power spectrum of a signal is obtained by the method of avetraging periodograms, as discussed in Section 10.6.3 This is, the spectrum estimate is?

where the K periodograms I_r(?) are computed from L-point segments of the signal using Eqs. (10.67) and (10.68). We define an estimate of the autocorrelation function as the invese Fourier transform of I(?); i.e.,

where L is the length of the segments, U is a normalizing factor given by Eq. (10.64), and c_ww[m], give by Eq. (10.60), is the aperiodic autocorrelation function of the window that is applied to the signal segments.

(b) In the application of periodogram averaging, we normally use an FFT algorithm to compute I(?) at N equally spaced frequencies; i.e.,

I[k] = I(2?k/N),? ? ? ? ??k = 0, 1,. . . , N ? 1,

Where N ? L. Suppose that we compute an estimate of the autocorrelation funtion by computing the inverse DFT of I[k], as in?

Obtain an expression for ?{?_p[m]},

(c) How should N be chosen so that?

?{?_p[m]} = ?{?[m]}, m = 0, 1,?. , L ? 1 ?

Transcribed Image Text:

K-1 i(w) = 1,(w). %3D (w)em do. [m] = 2л 0 sn< L-1. x,(n] = x[r R+ n]w{n]. (10.67) %3D 1, (w) = X, (e"). LU (10.68) Part a Eto [m}} Cww/m/0.[m). LU %3D L-1 Cww [m] = w[n]w[n+ m}. (10.60) 2-ti L-1 ΙΣo, (10.64) n-0 Part b I[k]e(2/Nkm ở „\m] = m = 0. 1..... N- 1. %3D %3D k=0