DFT if real sequences with special symmetries (a) Using the symmetry properties of section 5.2 (especially the
Question:
DFT if real sequences with special symmetries
(a) Using the symmetry properties of section 5.2 (especially the decomposition properties), explain how we can compute the DFT of two real symmetric (even) and two real antisymmetric (odd) sequences simultaneously using an N-point DFT only.
(b) Suppose now that we are given four real sequences xi(n), i = 1, 2, 3, 4, that are all symmetric [i.e. xi(N – n), 0 ≤ n ≤ N – 1]. Show that the sequences si(n) = xi(n + 1) – xi(n – 1) are antisymmetric [i.e. si(n) = -si(N – n) and si(0) = 0].
(c) Form a sequence x(n) using x1(n), x2(n), s3(n), and s4(n) and show how to compute the DFT Xi(k) of xi(n), i = 1, 2, 3, 4 from the N-point DFT X(k) of x(n).
(d) Are there any frequency samples of Xi(k) that cannot be recovered from X(k)? Explain.
Step by Step Answer:
Digital Signal Processing
ISBN: ?978-0133737622
3rd Edition
Authors: Jonh G. Proakis, Dimitris G.Manolakis