Question

What is the inverse z-transform of X(z)=\(\frac{1}{1-1.5z^{-1}+0.5z^{-2}}\) if ROC is |z|<0.5?

(a) [-2-0.5^n]u(n)

(b) [-2+0.5^n]u(n)

(c) [-2+0.5^n]u(-n-1)

(d) [-2-0.5^n]u(-n-1)

I got this question by my college director while I was bunking the class.

The doubt is from Inversion of Z Transform topic in portion Z Transform and its Application – Analysis of the LTI Systems of Digital Signal Processing

Answer 1

Correct option is (c) [-2+0.5^n]u(-n-1)

To explain: The partial fraction expansion for the given X(z) is

\(X(z)= \frac{2z}{z-1}-\frac{z}{z-0.5}\)

In case when ROC is |z|<0.5, the signal is anti causal. Thus both the terms in the above equation are anti causal terms. So, if we apply inverse z-transform to the above equation we get

x(n)= [-2+0.5^n]u(-n-1).