DFI of real sequences with odd harmonics only Let x(n) be an N-point real sequence with N-point DFT X(k) (N even). In addition, x(n) satisfied the following symmetry property: 

x(n + N/2) = -x(n)       n = 0, 1 … N/2 – 1

that is, the upper half of the sequence is the negative of the lower half.

(a) Show that

X(k) = 0          k even

that is, the sequence has a spectrum with odd harmonic.

(b) Show that the values of this odd-harmonic spectrum can be computed by evaluating the N/2-point DFT of a complex modulated version of the original sequence x(n).