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GATE 2014 EC (EC1) – Question 53

Networks, Signals and Systems · Discrete-time Signals · 2 marks · Multiple choice

Find the z-transform ROC of $x[n]=(-1/9)^nu[n]-(-1/3)^nu[-n-1]$.

  1. $|z|>1/9$
  2. $|z|<1/3$
  3. $1/9<|z|<1/3$
  4. The ROC does not exist

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Correct answer: (C) $1/9<|z|<1/3$

Explanation

The right-sided term(−1/9)ⁿu[n] converges for |z|>1/9. The left-sided term−(−1/3)ⁿu[−n−1] converges for |z|<1/3. Both conditions must hold, so the combined ROC is **1/9<|z|<1/3,C**. The pole magnitudes determine the annular boundaries.