In Section 9.2, we used the fact that W^-kN_N = 1 to derive a recurrence algorithm for computing a specific DFT value X [k] for a finite-length sequence x[n], = 0, 1, ,?, N ? 1.

(a) Using the fact that W^kN_N = W^Nn_N = 1, show that X[N ? k] can be obtained as the output after N iterations of the difference equation depicted in Figure. That is, show that X[N ? k] = YK[n].

(b) Show that X[N ? k] is also equal to the output after N iterations of the difference equation depicted in Figure. Note that the system of Figure has the same poles as the system in Figure, bur the coefficient required to implement the complex zero in figure is the complex conjugate of the corresponding coefficient in figure, W^-k_N = (W^k_N)*.

Transcribed Image Text:

Part A x[r] Part B x[r] yalr) cos() (2nk) -w