Problem 5.1

(A) If x[n] has TDFT  Xw [n,ω) with respect to analysis window  w[n], derive the expression for the TDFT of each of the following signals:

a)  x[n − n0 ]

b)  ejω0nx[n]

(B) Throughout this problem, assume that  w[n] =  and  Bw [n,ω) is the TDFT of

b[n] with respect to analysis window  w[n] .

(a) Sketch the Fourier transform of  (−1)n w[ −n] . Justify your answer.

(b) Sketch  Bw [n, π) as a function of n when  b[n] =  . Justify your

answer.

Problem 5.2

(A) In this problem, we consider a 128-point signal  q[n]with TDFT denoted by  Qw [n,ω) with respect to the analysis window  w[n] = (0 .5)n u[n].

(a)  Determine all values of n such that it is guaranteed that  Qw [n,ω)= 0 for all ω, regardless of the values of  q[n] in the range  0 三 n < 128 . Justify your answer.

(b)  Determine all values of ω such that it is guaranteed that  Qw [n,ω)= 0 for all n, regardless of the values of  q[m] in the range  0 三 m < 128 . Justify your answer.

(B) Throughout this problem,  Xw [n,ω) denotes the TDFT of  x[n]with respect to the

analysis window  w[n] .

(a) If  x[n] and  w[n]are both even signals, does it follow that each column (for fixed n) of Xw [n,ω)is even, i.e.,  Xw [n,ω)= Xw [n, −ω)? Justify your answer.

(b) If  x[n] is a periodic signal, does it follow that each row (for fixed ω) of  Xw [n,ω)is periodic? Justify your answer.

(c) Determine and sketch w[n]such that Xw [n,ω)= x[n + 3]e− j3ω . Justify your answer.