Question

uestion A causal, linear time-invariant system is excited with an input x (n) described as x(n) 3u(n) with the output y(n) of the system as follows: 7l n) -2"u(n) y(n)- a) Determine z-transform X(z) and Y (z) (4 marks) b) Determine the transfer function H(z). (3 marks) Based on (b), determine the impulse response h(n). Based on (b), sketch the z-plane for the transfer function of the system Based on (d), determine the stability of the system and discuss the result c) (11 marks) d) (4 marks) e) (3 marks) [Total: 25 marks]

Answer 1

x(n) = 3u(n) y(n) = (1/2)^n u(n) - 2^n u(n) we know z-transform oF a^n u(n) doubleheadarrow z/z - a a^n-1 u(n) doubleheadarrow 1/z - a so, X(z) = 3z/z - 1 Y(z) = z/z - 1/2 - z/z - 2 H(z) = Y(z)/X(z) = z/z - 1/2 - z/z - 2/3z/z - 1 = (z^2 - 2z - z^2 + z/2)/3z[z - 1/2][z-2] (z-1) H(z) = z^3 - 2z^2 - z^3 + z^2/3 - z^2 + 2z + z^2 - z/2/3z(z-1/2)(z-2) = -1.5z^2 + 1.5z/3z(z-1/2)(z-2) H(z) = -1/2 [z-1]/(z-1/2)(z-2) From partial fractions A/z-1/2 + B/z-2 put z = 1/2 -5A = 1/4 A = -1/6 put z = 2, 1.5B = -1/2 B = -1/3 So, H(z) = -1/6 1/z - 1/2 - 1/3 1/z - 2 h(n) = -1/6 (1/2)^n-1 u(n-1) - 1/3 (2)^n-1 u(n-1) H(z) = -1/4 (z-1)/(z-1/2)(z-2) In order to get stability of the system, Roc must include imaginary axis so, Vertical-bar z Vertical-bar > 1/2 Vertical-bar z Vertical-bar < 2 1/2 < Vertical-bar z Vertical-bar < 2 rightarrow Roc of the system so system stable, so, impulse response H(z) =