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 x_i(n), i = 1, 2, 3, 4, that are all symmetric [i.e. x_i(N – n), 0 ≤ n ≤ N – 1]. Show that the sequences s_i(n) = x_i(n + 1) – x_i(n – 1) are antisymmetric [i.e. s_i(n) = -s_i(N – n) and s_i(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 X_i(k) of x_i(n), i = 1, 2, 3, 4 from the N-point DFT X(k) of x(n).

(d) Are there any frequency samples of X_i(k) that cannot be recovered from X(k)? Explain.