The GATE Grind

GATE 2016 EC – Question 18

Networks, Signals and Systems · Discrete-time Signals · 1 mark · Multiple choice

Consider the sequence $x[n] = a^n u[n] + b^n u[n]$, where $u[n]$ denotes the unit-step sequence and $0 < |a| < |b| < 1$. The region of convergence (ROC) of the z-transform of $x[n]$ is

  1. $|z| > |a|$
  2. $|z| > |b|$
  3. $|z| < |a|$
  4. $|a| < |z| < |b|$

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Correct answer: (B) $|z| > |b|$

Explanation

Each term is a right-sided sequence, so it converges outside a circle: $a^n u[n]$ for $|z| > |a|$ and $b^n u[n]$ for $|z| > |b|$. The sum converges where both do, which is $|z| > |b|$ because $|b|$ is the larger radius.