In Section 10.3 we defined the time-dependent Fourier transform of the signal x[m] so th...

In Section 10.3 we defined the time-dependent Fourier transform of the signal x[m] so that, for fixed n, it is equivalent to the regular DTFT of the sequence x[n+m]w[m], where w[m] is a window sequence. It is also useful to define a time-dependent autocorrelation function for the sequence x[n] such that, for fixed n, its regular Fourier transform is the magnitude squared of the time-dependent Fourier transform. Specifically, the time-dependent autocorrelation function is defined as

where X[n, λ) is defined by Eq. (10.18).

(a) Show that if x[n] is real

i.e., for fixed n, c[n,m] is the aperiodic autocorrelation of the sequence x[n + r]w[r], −∞ < r < ∞.

b) Show that the time-dependent autocorrelation function is an even function of m for n fixed, and use this fact to obtain the equivalent expression

(c) What condition must the window w[r] satisfy so that Eq. (P10.45-1) can be used to compute c[n,m] for fixed m and −∞ < n < ∞by causal operations?

(d) Suppose that

Find the impulse response hm[r] for computing the mth autocorrelation lag value, and find the corresponding system function Hm(z). From the system function, draw the block diagram of a causal system for computing the mth autocorrelation lag value c[n,m] for −∞ < n < ∞for the window of Eq. (P10.45-2).

(e) Repeat part (d) for