GATE 2026 EE – Question 11

Signals and Systems · Applications of Fourier Transform for continuous and discrete time signals, Laplace Transform and Z transform · 1 mark · Multiple choice

Consider the infinite-length, discrete-time sequence $x[n]=0.9^{|n|}$, where $n$ is an integer. The region of convergence of its Z-transform $X(z)$ is given by:

(Note: $z$ is a complex variable)

$|z|>0.9$
$|z|<0.9$
$0.9<|z|<1/0.9$
$\{z\text{ such that }|z|<0.9\}\cup\{z\text{ such that }|z|>1/0.9\}$

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Correct answer: (C) $0.9<|z|<1/0.9$

Explanation

$x[n]=0.9^n$ for $n\ge0$ converges for $|z|>0.9$. The left-sided part $0.9^{-n}$ for $n<0$ converges for $|z|<1/0.9$. The ROC of the two-sided sequence is the intersection: $0.9<|z|<1/0.9$.