Question

A linear time invariant system is characterized by the system function H(z)=\(\frac{1}{1-0.5z^{-1}}+\frac{2}{1-3z^{-1}}\). What is the h(n) if the system is stable?

(a) (0.5)^nu(n)-2(3)^nu(n)

(b) (0.5)^nu(-n-1)-2(3)^nu(-n-1)

(c) (0.5)^nu(-n-1)-2(3)^nu(n)

(d) (0.5)^nu(n)-2(3)^nu(-n-1)

The question was posed to me in homework.

This interesting question is from Analysis of LTI System in Z Domain topic in section Z Transform and its Application – Analysis of the LTI Systems of Digital Signal Processing

Answer 1

The correct choice is (d) (0.5)^nu(n)-2(3)^nu(-n-1)

Easy explanation: The system has poles at z=0.5 and at z=3.

Since the system is stable, its ROC must include unit circle and hence it is 0.5<|z|<3 . Consequently, h(n) is non causal and is given as

h(n)= (0.5)^nu(n)-2(3)^nu(-n-1).