The GATE Grind

GATE 2024 IN – Question 38

Signals and Systems · Laplace and z-transforms · 2 marks · Multiple choice

For $x[n]=\begin{cases}(0.2)^n,&0\le n\le7\\0,&\text{otherwise}\end{cases}$, the region of convergence of its z-transform consists of

  1. all values of $z$ except $z=0.2$
  2. all values of $z$
  3. all values of $z$ except $z=0$
  4. all values of $z$ except $z=\infty$

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Correct answer: (C) all values of $z$ except $z=0$

Explanation

The transform is the finite polynomial $X(z)=1+0.2z^{-1}+\cdots+0.2^7z^{-7}$. It converges at every finite nonzero $z$ and at infinity, but is singular at $z=0$.

Thus the ROC is **all z except zero, C**. The infinite-geometric-series ROC does not apply to this finite sequence.