Question

A discrete-time LTI system has the system function \(H(z)\) given below:

$$ H(z)=\frac{z^{2}}{z^{2}-\frac{1}{4}} $$

(a) Sketch the pole-zero plot for this system. How many possible regions of convergence (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., \(\left|H\left(e^{j \omega}\right)\right|\), of this system for \(|\omega| \leq \pi\). Label your sketch!

Answer 1

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