Let x[n] = u[n + 2] u[n 3] (a) Find the DTFT X(e jÏ ) of x[n]
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(a) Find the DTFT X(ejÏ) of x[n] and sketch |X(ejÏ)| vs Ï giving its value at Ï = ± Ï, ± Ï/2, 0.
(b) If x1[n] = x[2n], i.e., x[n] is down-sampled with M = 2, find its DTFT X1 (ejÏ). Carefully sketch x1 [n] and |X1 (ejÏ)| indicating its values at Ï = ±Ï, ± Ï/2, 0. Is X1 (ejÏ) = 0.5X(ejÏ/2)? If not, how would you process x[n] so that when x1 [n] = x[2n] you would satisfy this condition? Explain.
(c) Consider now the up-sampled signal
Find the DTFT X2(ejÏ) of x2[n], and carefully sketch both (in particular, when plotting X2(ejÏ) indicate the values at frequencies Ï = ± Ï, ± Ï/2, 0). Explain the differences between this case the down-sampling cases.
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