(a) Consider the system of Figure P10.24-1 with input sampling period T = 10−4...

(a) Consider the system of Figure P10.24-1 with input sampling period

T = 10⁻⁴, and

What is the smallest nonzero value of N such that X_w[k] is nonzero at exactly one value of k?

(b) Suppose now that N = 32, the input signal is and the sampling period T is chosen such that no aliasing occurs during the sampling process. Figures P10.24-2 and P10.24-3 show the magnitude of the sequence X_w[k] for k = 0, . . . , 31 for the following two different choices of w[n]:

Indicate which figure corresponds to which choice of w[n]. State your reasoning clearly.

(c) For the input signal and system parameters of part (b), we would like to estimate the value of _0 from Figure P10.24-3 when the sampling period is T = 10⁻⁴. Assuming that the sequence and that the sampling period is sufficient to ensure that no aliasing occurs during sampling, estimate the value of . Is your estimate exact? If it is not, what is the

maximum possible error of your frequency estimate?

(d) Suppose you were provided with the exact values of the 32-point DFT X_w[k] for the window choices w₁[n] and w₂[n]. Briefly describe a procedure to obtain a precise estimate of .