The GATE Grind

GATE 2024 EC – Question 21

Networks, Signals and Systems · LTI Systems · 1 mark · Multiple choice

A causal and stable LTI system with impulse response $h(t)$ produces an output $y(t)$ for an input signal $x(t)$. A signal $x(0.5t)$ is applied to another causal and stable LTI system with impulse response $h(0.5t)$. The resulting output is ______.

  1. $2y(0.5t)$
  2. $4y(0.5t)$
  3. $0.25y(2t)$
  4. $0.25y(0.25t)$

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Correct answer: (A) $2y(0.5t)$

Explanation

The output is $\int h(0.5\tau)\,x\big(0.5(t-\tau)\big)d\tau$. Substituting $u=0.5\tau$ ($d\tau=2\,du$) gives $2\int h(u)\,x(0.5t-u)\,du=2\,y(0.5t)$.