Question

The DFT is a sampled version of the DTFT of a finite-length sequence; i.e., N-1 (P9.25-1) Furthermore, an FFT algorithm is an efficient way to compute the values X Now consider a finite-length sequence xin] whose length is N samples.We want to evaluate X(z) the z-transform of the finite-length sequence, at the following points in the z-plane where ris a positive number. We have available an FFT algorithm (a) Plot the points z in the z-plane for the case N-8 and r0.9 (b) Write an equation [similar to Eq. (P9.25-1) above] for X(zk) that shows that X(zn) is the DFT of a modified sequence In.What is iln]? (c) Describe an algorithm for computing X(zk) using the given FFT function. (Direct evaluation is not an option.) You may describe your algorithm using any combination of English text and equations, but you must give a step-by-step procedure that starts with the sequence xIn] and ends with X(Zk).

Answer 1

We know that DT FT is DTFT: X[k] = sigma^N - 1_n = 0 x(n) e^- j (2 pi/N) k_n k = 0, 1, 2, Ellipsis N - 1 Z-transform: x(t) = sigma^infinity_n = -infinity x(n) z^- n z = re^j omega rightarrow Digital frequency if we sample the digital frequency then z_k = re^j (2 pi/N) k k = 0, 1, Ellipsis N - 1 Therefore x(z_k) = sigma^N - 1_n = 0 x(n) (z_k)^- n = sigma^n - 1_n = 0 x(n) (r e^j 2 pi/N h)^- n x(z_k) = sigma^N - 1_n = 0 x(n) r^- n e^- j (2 pi/N) kn X(z_k) = Dt FT of {x(n) r^- n} plotting of z_k = re^j 2 pi/N k, k = 0, Ellipsis N - 1 for N = 8, r = 0.9