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 discrete-time Fourier transform 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, ?) is defined by Eq. (10.18).

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

i.e., for fixed n, c[n, m] is the a periodic autocorrelation of the sequence x[n + r]w[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?

where

(c) What condition must the window w[r] satisfy so that Eq. (P10.39-1) can be used to compute c[n, m] for fixed m and ??

(d) Suppose that?

Find the impulse response h_m[r] for computing the mth autocorrelation lag value, and find the corresponding system function H_m(z). From the system function, draw the block diagram of a causal system for computing the mth autocorrelation lag value c[n, m] for ??

(e) Repeat Part (d) for

IX In. k)feihmdn. c[n. m] = 2 x[n + m]w[m]e im. (10.18) X[n. 2) = Part A c[n. m] = 2 x[n +r]w[r]x[m+n+r]w[m +r]: Part B c[n. m] = x[r]x[r m]hmn-r]. %3D Part C hm[r] = w[-r]w[-(m+r)]. (P10.39-1) Part D Sa'. r20. |0. r < 0. w[-r] = (P10.39-2) Part E ra'. r> 0. w[-r] = 0. r < (0.