Suppose that a finite-length sequence x[n] has the N-point DFT X [k], and suppose that the sequence satisfies the symmetry condition x[n] = − x[((n + N/2)) _N], 0 ≤ n ≤ N – 1, where N is even and x [n] is complex.

(a) Show that X [k] = 0 for k = 0, 2,…, N – 2.

(b) Show how to compute the odd-indexed DFT values X[k], K = 1, 3, …, N – 1 using only one N/2-point DFT plus a small amount of extra computation.