Question

(5) Consider the following discrete time signal zln-S[n] + δ[n-1] + a[n-2] + δ n-3] a) Compute the DTFT of n b) Compute the DFT4 (DFT with N 4) coefficients for zn], i.e., Xk for k 0,1,2,3. c) Compute the DFTs (DFT with N-8) coefficients for r[n], i.e., Xk for k 0,1,...,7.

Answer 1

G T x [n] = delta [n] + delta [n - 1] + delta [n - 2] + delta [n - 3] DTFT = x (e^jw) = sigma_n = -infinity^infinity x (n) e^-jwn x (e^jw) = sigma_n = -infinity^infinity [delta (n) + delta (n - 1) + delta (n - 2) + delta (n - 3)] x (e^jw) = sigma_n = -infinity^infinity delta (n) e^-jwn + sigma_n = -infinity^infinity delta (n - 1) e^-jw e^-jwn + sigma_n = -infinity^infinity delta (n - 2) e^-jwn + sigma_n = -infinity^infinity (n - 3) e^-jwn x (e^jw) = e^-jwn Vertical-bar _n = 0 + e^-jwn Vertical-bar _n = 1 + e^-jwn Vertical-bar _n = 2 + e^-jwn Vertical-bar _n = 3 x (e^jw) = e^-jw (0) + e^-jw (1) + e^-2jw + e^-3jw x (e^-jw) = 1 + e^-jw + e^-2jw + e^-3jw which is know as DTFT Compute the D_F T_4 (DFT with N = 4) x (k) for k = 0, 1, 2, 3 \r\n x (n) = {1, 1, 1, 1} uparrow n = 0