A finite-length signal of length L = 500 (x[n] = 0 for n < 0 and n > L – 1) is obtained by sampling a continuous-time signal with sampling rate 10,000 samples per second. We wish to compute samples of the z-transform of x[n] at the N equally spaced points z_k = (0.8) e ^j2πk/N, for 0 ≤ k ≤ N – 1, with an effective frequency spacing of 50 Hz or less.

(a) Determine the minimum value for N if N = 2^v.

(b) Determine a sequence y[n] of length N, where is as determined in part (a), such that its DFT Y[k] is equal to the desired samples of the z-transform of x[n].