The GATE Grind

GATE 2026 EC – Question 46

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

Consider a real baseband signal $x(t)=e^{-2t}$, for $t$ (in seconds) $\ge0$. If 99% of energy of $x(t)$ lies within $B$ Hz, then which of the following options is TRUE for the value of $B$?

  1. $B > 1\ \text{kHz}$
  2. $63/\pi\ \text{Hz} < B < 64/\pi\ \text{Hz}$
  3. $126/\pi\ \text{Hz} < B < 128/\pi\ \text{Hz}$
  4. $B < 1\ \text{Hz}$

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Correct answer: (B) $63/\pi\ \text{Hz} < B < 64/\pi\ \text{Hz}$

Explanation

$|X(f)|^2=1/(4+4\pi^2f^2)$, so the energy in $[-B,B]$ is $\frac1{2\pi}\arctan(\pi B)$ out of a total of $1/4$. Setting $\arctan(\pi B)=0.99\cdot\pi/2$ gives $\pi B=\cot(0.005\pi)\approx63.66$, so $B\approx63.66/\pi$ Hz, which lies between $63/\pi$ and $64/\pi$ Hz.