Let x[n] = 0, n<0, n> 7, be a real eight-point sequence, and let X [k] be its eigh...

Let x[n] = 0, n<0, n> 7, be a real eight-point sequence, and let X [k] be its eight-point DFT.

(a) Evaluate

in terms of x[n].

(b) Let v[n] = 0, n<0, n> 7, be an eight-point sequence, and let V[k] be its eight-point DFT.

If V [k] = X(z) at z = 2 exp(j (2πk + π)/8) for k = 0, . . . , 7, where X(z) is the z transform of x[n], express v[n] in terms of x[n].

(c) Let w[n] = 0, n<0, n > 3, be a four-point sequence, and let W[k] be its four-point DFT. If W[k] = X [k] + X[k + 4], express w[n] in terms of x[n].

(d) Let y[n] = 0, n<0, n > 7, be an eight-point sequence, and let Y [k] be its eight-point DFT.

If

express y[n] in terms of x[n].