Pole-Zero Plot and ROC for LTI System

[hxluo]

Consider a discrete-time LTI system with transfer function
H(z) =(1-3z^-1) / ( 2-z^-1)
(a) Sketch the pole-zero plot of H(z).
(b) Suppose the system is stable. Determine all possible regions of convergence (ROC) for H(z) under this condition, or state that none exists.
(c) Repeat part (b) assuming the system is causal instead of stable.
(d) Can this system have a causal and stable inverse? If so, determine
H^(-1)(z) including its ROC. If not, explain why not.

 

[LeBrad]

Consider a discrete-time LTI system with transfer function
H(z) =(1-3z^-1) / ( 2-z^-1)
(a) Sketch the pole-zero plot of H(z).
(b) Suppose the system is stable. Determine all possible regions of convergence (ROC) for H(z) under this condition, or state that none exists.
(c) Repeat part (b) assuming the system is causal instead of stable.
(d) Can this system have a causal and stable inverse? If so, determine
H^(-1)(z) including its ROC. If not, explain why not.

1.) Find the poles and zeros of H. They will be easy to see if you multiply the top and bottom by z.

2.) Find how stability and causality relate to ROC of your plot. Stability is related to the unit circle and causality is related to inside or outside of the extreme poles.

3.) Draw a pair of axes. Draw a circle centered at the origin. Use knowledge gained in steps 1 and 2 to draw the pole-zero diagram.