Suppose that an estimate of the power spectrum of a signal is obtained by the method of...

Suppose that an estimate of the power spectrum of a signal is obtained by the method of averaging periodograms, as discussed in Section 10.5.3. That is, the spectrum estimate is

where the K periodograms I_r(ω) are computed from L-point segments of the signal using Eqs. (10.82) and (10.83). We define an estimate of the autocorrelation function as the inverse Fourier transform of i.e.,

(a) Show that

where L is the length of the segments, U is a normalizing factor given by Eq. (10.79), and c_ww[m], given by Eq. (10.75), 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 at N equally spaced frequencies; i.e.,

where N ≥ L. Suppose that we compute an estimate of the autocorrelation function by computing the inverse DFT of as in

Obtain an expression for

(c) How should N be chosen so that