A discrete-time signal x[n] is sketched below. Sketch and label carefully each of the following signals:...

 

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A discrete-time signal x[n] is sketched below. Sketch and label carefully each of the following signals: (a) x[n - 2]; (b) x [2 -n]; (d) x[-1-2n]; (c) x [2n]; (e) xod[n] (the odd part of x[n]) 1 -3-2-1 (b) Sketch and label carefully the DTFT X (eu) of x[n] obtained by using the smallest sampling rate. (c) Determine the frequency response of the ideal recon- struction filter needed to fully recover xa (t). 0 1 -0.5 -16 2. (10 points) For each of the following signals, determine if it is periodic. If yes, specify the fundamental period; if not, explain why: (a) xi[n] = cos(n); (b) x2 [n] = sin(); (d) x4 [n] = 1 + sin² (2n). (c) x3[n] = 3 cos(n) + 2 sin(0.27mn + π); 3. (16 points) Let x[n] = {1, -2, 1, 2}, 0 ≤ n ≤ 3 and h[n] = {2, 1,0,0, -1},0 ≤n ≤ 4. (a) (6 points) Determine y[n] = x[n] h[n]. Show details for full credit. (b) (6 points) Determine the 5-point circular convolution z[n] = x[n] h[n]. Show details for full credit. (c) (4 points) What is the smallest value of N, such that x[n] h[n] = x[n] h[n]? Verify your choice of N by carrying out the N-point circular convolution. 4. (18 points) Consider a causal LTI system described by the following linear difference equation: 1 1 1 y[n] – y[n − 1] = x[n] — — x[n − 1] +x[n − 2] - -22₂-22₁ 0.5 2 3 4 5 6 7 (a) (5 points) Determine the frequency response of the system. (b) (5 points) Determine the impulse response of the system. (c) (8 points) An input x[n] = 1 + cos(n +n/3) is applied to the system. Determine the corresponding steady-state response y[n]. 5. (15 points) An analog signal xa(t) has a spectrum X₁ (jn) sketched below, where ₁ = 50T, N₂ = 90. (a) Find the smallest sampling frequency so that xa (t) can be fully recovered from its sampled version x[n]? Xa(jn) 1 n 1 821 222 Ω A discrete-time signal x[n] is sketched below. Sketch and label carefully each of the following signals: (a) x[n - 2]; (b) x [2 -n]; (d) x[-1-2n]; (c) x [2n]; (e) xod[n] (the odd part of x[n]) 1 -3-2-1 (b) Sketch and label carefully the DTFT X (eu) of x[n] obtained by using the smallest sampling rate. (c) Determine the frequency response of the ideal recon- struction filter needed to fully recover xa (t). 0 1 -0.5 -16 2. (10 points) For each of the following signals, determine if it is periodic. If yes, specify the fundamental period; if not, explain why: (a) xi[n] = cos(n); (b) x2 [n] = sin(); (d) x4 [n] = 1 + sin² (2n). (c) x3[n] = 3 cos(n) + 2 sin(0.27mn + π); 3. (16 points) Let x[n] = {1, -2, 1, 2}, 0 ≤ n ≤ 3 and h[n] = {2, 1,0,0, -1},0 ≤n ≤ 4. (a) (6 points) Determine y[n] = x[n] h[n]. Show details for full credit. (b) (6 points) Determine the 5-point circular convolution z[n] = x[n] h[n]. Show details for full credit. (c) (4 points) What is the smallest value of N, such that x[n] h[n] = x[n] h[n]? Verify your choice of N by carrying out the N-point circular convolution. 4. (18 points) Consider a causal LTI system described by the following linear difference equation: 1 1 1 y[n] – y[n − 1] = x[n] — — x[n − 1] +x[n − 2] - -22₂-22₁ 0.5 2 3 4 5 6 7 (a) (5 points) Determine the frequency response of the system. (b) (5 points) Determine the impulse response of the system. (c) (8 points) An input x[n] = 1 + cos(n +n/3) is applied to the system. Determine the corresponding steady-state response y[n]. 5. (15 points) An analog signal xa(t) has a spectrum X₁ (jn) sketched below, where ₁ = 50T, N₂ = 90. (a) Find the smallest sampling frequency so that xa (t) can be fully recovered from its sampled version x[n]? Xa(jn) 1 n 1 821 222 Ω