Question

Problem 3) Two discrete-time LTI systems are connected in cascade. The first system is defined by its frequency response: H(e-1+and the second system is (a) Determine the frequency response for the overall cascade system. Simplify your (c) Write down the difference equation that relates the output y[n] to the input x[n]. defined by its impulse response: hln]-n-n-+n-2]-n-3] answer as far as possible. (b) Determine and plot the impulse response h[n] for the overall cascade system.

Answer 1

H_1 (e^jw) = 1 + 2 e^-jw + e^-jw2 H_1 (Z) = 1 + 2 Z^-1 + Z^-2 h_2(n) = delta(n) - delta(n - 1) + delta(n - 2) - delta(n - 3) H_2(Z) = 1 - Z^-1 + Z^-2 - Z^-3 a) For Cascade system, h(n) = h_1 (n) * h_2 (n) H(Z) = H_1 (Z) middot H_2 (Z) H(Z) = (1 + 2 Z^-1 + Z^-2) (1 - Z^-1 + Z^-2 - Z^-3) = 1 - Z^-1 + Z^-2 - Z^-3 + 2 Z^-1 + 2Z^-3 - 2Z^-2 -2Z^-4 + Z^-2 - Z^-3 + Z^-4 - Z^-5 H(Z) = 1 + Z^-1 - Z^-4 - Z^-5 H (e^jw) = 1 + e^-jw - e^-jw4 - e^-jw5 b) Inverse Z_transform, h(n) = delta(n) + delta(n - 1) - delta(n - 4) - delta(n - 5) Impulse Response. \r\n c) H(Z) = Y(Z)/X(Z) = 1 + Z^-1 - Z^-4 - Z^-5 Y(Z) = X(Z) (1 + Z^-1 - Z^-4 - Z^-5) y(n) = x(n) + x(n - 1) - x(n - 4) - x(n - 5) Difference equation.