(a) X(e j ω ) is the DTFT of the discrete-time signal x[n] = (1/2)nu[n].
Find a length-5 sequence g[n] whose five-point DFT G[k] is identical to samples of the DTFT of x[n] at ωk= 2πk/5, i.e.,
g[n] = 0 for n < 0, n>4
and
G[k] = X(ej2πk/5) for k = 0, 1, . . . , 4.
(b) Let w[n] be a sequence that is strictly nonzero for 0 ≤ n ≤ 9 and zero elsewhere, i.e.,
w[n] ≠ 0, 0 ≤ n ≤ 9
w[n] = 0 otherwise
Determine a choice for w[n] such that its DTFT W(ejω) is equal to X(ejω) at the frequencies ω = 2πk/5, k = 0, 1, . . . , 4, i.e.,
W(e j 2 π k/ 5 ) = X(ej2πk/5) for k = 0, 1, . . . , 4.
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