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GATE 2020 EE – Question 14

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

Consider a signal $x[n]=\left(\frac12\right)^n\mathbb 1[n]$, where $\mathbb 1[n]=0$ if $n<0$, and $\mathbb 1[n]=1$ if $n\ge0$. The z-transform of $x[n-k]$, $k>0$ is $\dfrac{z^{-k}}{1-\frac12z^{-1}}$ with region of convergence being

  1. $|z|<2$
  2. $|z|>2$
  3. $|z|<1/2$
  4. $|z|>1/2$

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Correct answer: (D) $|z|>1/2$

Explanation

The signal $x[n]$ is causal, so its ROC is the exterior of a circle through the pole at $z=\frac12$, i.e. $|z|>\frac12$. Shifting by $k$ only adds factors of $z^{-k}$, which do not change the ROC (apart from possibly $z=\infty$).