In Section 10.6, we showed that a smoothed estimate of the power spectrum can be obtaine...

In Section 10.6, we showed that a smoothed estimate of the power spectrum can be obtained by windowing an estimate of the autocorrelation sequence. It was stated (see Eq. (10.109)) that the variance of the smoothed spectrum estimate is

where F, the variance ratio or variance reduction factor, is

As discussed in Section 10.6, Q is the length of the sequence x[n] and (2M−1) is the length of the symmetric window w_c[m] that is applied to the autocorrelation estimate. Thus, if Q is fixed, the variance of the smoothed spectrum estimate can be reduced by adjusting the shape and duration of the window applied to the correlation function.

In this problem we will show that F decreases as the window length decreases, but we also know from the previous discussion of windows in Chapter 7 that the width of the main lobe of W_c(e^jω) increases with decreasing window length, so that the ability to resolve two adjacent frequency components is reduced as the window width decreases. Thus, there is a trade-off between variance reduction and resolution. We will study this trade-off for the following commonly used windows:

(α = β = 0.5 for the Hanning window, and α = 0.54 and β = 0.46 for the Hamming window.)

(a) Find the Fourier transform of each of the foregoing windows; i.e., compute W_R(e^jω), W_B(e^jω), and W_H (e^jω). Sketch each of these Fourier transforms as functions of ω.

(b) For each of the windows, show that the entries in the following table are approximately true when M >>1: