Question

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

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

(b) (2+0.5^n)u(n)

(c) (2^n-0.5^n)u(n)

(d) None of the mentioned

I got this question in quiz.

The question is from Inversion of Z Transform in division Z Transform and its Application – Analysis of the LTI Systems of Digital Signal Processing

Answer 1

The correct answer is (a) (2-0.5^n)u(n)

Explanation: 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|>1, the signal x(n) is causal and both the terms in the above equation are causal terms. Thus, when we apply inverse z-transform to the above equation, we get

x(n)=2(1)^nu(n)-(0.5)^nu(n)=(2-0.5^n)u(n).