Problem 6.1

(A) Suppose that the analysis window for the TDFT  Xw [n,ω)of  x[n] is a 16-point signal    w[n] such that w[n] is nonzero for 0 ≤ n ≤ 15. For what positive integer values of L can we definitely say that  Xw [nL,ω) loses information about the signal   x[n] . Justify

your answer.

(B) Let  Xw [n,k]= Xw [2n, 2πk / 8). Here  x[n] = ej ( n−4) and  w[n]= u[n]− u[n − 8] .  Sketch

Xw [6,k] as a function of k. Justify your answer.

Problem 6.2

(A) Throughout this problem, let  x[n] be an arbitrary 200-point signal whose TDFT is given as Xw [n,ω). The analysis window is given by w[n]= u[n]− u[n − 100] and the discrete      TDFT of  x[n] is specified as  Xw [n, k]= Xw [100n, 2πk / 200) for  0 三 k 三199 and zero       otherwise. For what values of n is it guaranteed that  Xw [n, 0]= 0 regardless of the         actual values of the 200-point signal  x[n] ? Justify your answer.

(B) Throughout this problem, let  x[n] be a signal with discrete TDFT  Xw [n, k] given as:

Xw [n, k] = x[2n + p]e−  when 0 ≤ k ≤ 7

(a) Sketch the analysis window  w[n] and specify the temporal-sampling factor L and the frequency-sampling factor M for  Xw [n, k] . Justify your answers.

(b) If  x[n] = 0 for  n > 0, does it follow that  Xw [n, k]= 0 for  n > 0 ? Justify your answer.

Problem 6.3

Let  Yv [n, k]= Yv [n, 2πk /10) represent the discrete TDFT of a signal  y[n] with respect to an

analysis window   1 −  ejn +δ[n] . Is it true that αYv [n, k]= y[n] for some

real number α whose value is independent of n? Justify your answer