Let x[n] = u[n + 2] u[n 3] (a) Find the DTFT X(e jÏ ) of x[n] and sketch |X(e jÏ )| vs Ï giving

Let x[n] = u[n + 2] ˆ’ u[n ˆ’ 3]

(a) Find the DTFT X(e^jω) of x[n] and sketch |X(e^jω)| vs ω giving its value at ω = ± Ï€, ± Ï€/2, 0.

(b) If x₁[n] = x[2n], i.e., x[n] is down-sampled with M = 2, find its  DTFT X₁ (e^jω). Carefully sketch x₁ [n] and |X₁ (e^jω)| indicating its values at ω = ±Ï€, ± Ï€/2, 0. Is X₁ (e^jω) = 0.5X(e^jω/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(e^jω) of x₂[n], and carefully sketch both (in  particular, when plotting X₂(e^jω) indicate the values at frequencies  Ï‰ = ± Ï€, ± Ï€/2, 0). Explain the differences between this case the  down-sampling cases.

x[n/2] n even x2[n] = otherwise