Question

A discrete-time LTI system has the system function H(z) given below: 2 H(z (a) Sketch the pole-zero plot for this system. How many possible (ROCs) are there for H(z). List the possible ROCs and indicate what type of sequence (left-sided, right-sided, two-sided, finite-length) they correspond to (b) Which ROC (or ROCs) correspond to a stable system? Why? (c) Which ROC (or ROCs) correspond to a causal system? Why? (d) Write a difference equation that relates the input to the output of the system (e) Determine the impulse response h[n] associated with the stable system defined by this system function (f) Make a careful sketch of the frequency response magnitude, i.e., |H(e), of this system for lwl T. Label your sketch!

Answer 1

"Given H(z) = z^2/z^2 - 1/4 = z^2/(z + 1/2)(z - 1/2) (a) pole - zero plot: ROC: Vertical-bar z Vertical-bar > 1/2 rightarrow Right - sided sequence Vertical-bar z Vertical-bar < 1/2 rightarrow Left sided sequence (b) For a stable system ROC must include ""unit circle "" the ROC Vertical-bar z Vertical-bar > 1/2 includes unit circle, Thus ROC correspond to stable system. (c) ROC: Vertical-bar z Vertical-bar > 1/2 corresponds to causal system, since the sequence of this system is Right - sided. (d) H (z) = z^2/z^2 - 1/4 rightarrow Y(z)/X(z) = z^2/z^2 - 1/4 Y(z) (Z^2 - 1/4) = x(z) z^2 rightarrow Y(z) z^2 - 1/4 Y(z) = z^2 x(z) rightarrow y(n + 2) - 1/4 y(n) = x(n + 1) [Because x(n + n_0) = x(z) z^n 0] e) H(z) = z^2/z^2 - 1/4 H (z)/z^2 = 1/z^2 - 1/4 = 1/(z + 1/2) (z - 1/2) H(z)/z^2 = -1/z + 1/2 + 1/z - 1/2 H (z) = -z^2/z + 1/2 + z^2/z - 1/2 H(z) = -z/1 + 1/2 z^- 1 + z/1 - 1/2 z^- 1 downarrow Inverse z transform"