Question

Problem 3. Discovering the System from the Output. 25 points. x[n] yln] Figure 2: A cascade of two LTI systems. yIn] 2 2 -6-5-4-3 4 5 6 7 Figure 3: The system output y[n] (a) 20 points. Consider the system in Figure 2 which is a cascade of two LTI systems, with hn n]26[n 1]. For input signal [n]-6[n] 1+n -1], the output y[n] appears in Figure 3. Determine the impulse response h2[n].

Answer 1

Now redraw the signal Now, from y(n) we can observe two things (1) y(n) = x_1 (n - 5) + x_1 (n + 5) (2) H_2 (n) must contain two shifted impulse Now, use another property of discrete convolution (1) x(n) times delta (n) = x(n) (2) x(n) times delta (n - n_0) = x(n - n_0) where delta (n) rightarrow impulse now using above properties H_2 (n) = delta (n + 5) + delta (n - 5) \r\n convolution between H_2 (n) lambda x_1 (n) result in y(n). \r\n Now first perform convolution between x(n) and h_1 (n) x_1 (n) and h_1 (n) x_1 (n) = x(n) times h_1 (n) method adopt here for convolution is linear convolution x(n) = [1, 1, 1] h(n) = [1, 2] x_1 (n) = 1 1 1 times 1 2 2 2 2 1 1 1 times 1 3 3 2 x_1 (n) = [1 3 3 2] now x_1 (n) times h_2 (n) = y(n)