Question

What is the z-transform of the signal x(n)=(0.5)^nu(n)?

(a) \(\frac{1}{1-0.5z^{-1}};ROC |z|>0.5\)

(b) \(\frac{1}{1-0.5z^{-1}};ROC |z|<0.5\)

(c) \(\frac{1}{1+0.5z^{-1}};ROC |z|>0.5\)

(d) \(\frac{1}{1+0.5z^{-1}};ROC |z|<0.5\)

The question was asked during an interview.

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

Answer 1

Correct choice is (a) \(\frac{1}{1-0.5z^{-1}};ROC |z|>0.5\)

For explanation: For a given signal x(n), its z-transform X(z)=\(\sum_{n=-\infty}^{\infty} x(n)z^{-n}\)

Given x(n)=(0.5)^nu(n)=(0.5)^n  for n≥0

So, X(z)=\(\sum_{n=0}^{\infty} 0.5^n z^{-n}=\sum_{n=0}^{\infty} (0.5z^{-1})^n\)

This is an infinite GP whose sum is given as

X(z)=\(\frac{1}{1-0.5z^{-1}}\) under the condition that |0.5z^-1|<1

=> X(z)=\(\frac{1}{1-0.5z^{-1}}\) and ROC is |z|>0.5.