Consider a finite-length sequence x[n] of length N: i.e., 

x[n] = 0           outside 0 ≤ n ≤ N – 1.

X(e^jω) denotes the Fourier transform of x[n]. X[k] denotes the sequence of 64 equally spaced samples of X(e^jω), i.e., 

X[k] = X(e^jω)|_ω=2πk/64.

It is known that in the range 0 ≤ k ≤ 63, X[32] = 1 and all the other values of X[k] are zero.

(a) If the sequence length is N = 64, determine one sequence x[n] consistent with the given information. Indicate whether the answer is unique. If it is, clearly explain why. If it is not, give a second distinct choice.

(b) If the sequence length is N = 192 = 3 × 64, determine one sequence x[n] consistent with the constraint that in the range 0 ≤ k ≤ 63, X[32] = 1 and all other values in that range are zero. Indicate whether the answer is unique. If it is, clearly explain why. If it is not, give a second distinct choice.