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) {….62,30,14,6,2}

(b) {…..62,30,14,6,2,0,0}

(c) {0,0,2,6,14,30,62…..}

(d) {2,6,14,30,62…..}

I have been asked this question in homework.

I need to ask this question from Inversion of Z Transform in chapter Z Transform and its Application – Analysis of the LTI Systems of Digital Signal Processing

Answer 1

The correct answer is (b) {…..62,30,14,6,2,0,0}

To explain: In this case the ROC is the interior of a circle. Consequently, the signal x(n) is anti causal. To obtain a power series expansion in positive powers of z, we perform the long division in the following way:

Thus X(z)=\(\frac{1}{1-1.5z^{-1}+0.5z^{-2}}=2z^2+6z^3+14z^4+30z^5+62z^6+…\)

In this case x(n)=0 for n≥0.Thus we obtain x(n)= {…..62,30,14,6,2,0,0}