Question

Problem 4. (20 points): Consider a causal LTI system that is described by the difference equation Find the impulse response sequence h[n] by computing the system function H(S2)

Answer 1

Y(n) - y(n - 1) + 3/16 y(n - 2) = 2 x(n) - x(n - 1) taking z-transform y(z) - z^-1 y(z) + 3/16 z^-2 y(z) = 2 times (z) - z^-1 times (z) y(z)[1 - z^-1 + 3/16 z^-2] = (2 - z^-1) times (z) y(z)/x(z) = 2 - z^-1/1 - z^-1 + 3/16 z^-2 downarrow H(z) = 2 - z^-1/1 - z^-1 + 3/16 z^-2 H(z) = (2 z - 1) times 16 z^2/z(16 z^2 - 16 z + 3) H(z) = 32 z^2 - 16 z/16 z^2 - 16^z + 3 H(z)/z = 32 z - 16/16 z^2 + 3 - 16 z H(z)/z = 32 z - 16/(z - 1/4) (z - 3/4) \r\n H(z)/z = A/(Z - 1/4) + B/(z - 3/4) A = H(z)/z times (z - 1/4) Vertical-bar _z = 1/4 A = 32 z - 16/z - 3/4 Vertical-bar _z = 1/4 A = -B/-1/2 = 16 Similarly, B = H(z)/z times (z - 3/4) Vertical-bar _z = 3/4 B = 32 z - 16/z - 1/4 Vertical-bar _z = 3/4 B = B/1/2 = 16 H(z)/z = 16/z - 1/4 + 16/z - 3/4 H(z) = 16 (z/z - 1/4 +