The GATE Grind

GATE 2020 EE – Question 47

Signals and Systems · Linear time invariant and causal systems · 2 marks · Multiple choice

Suppose for input $x(t)$ a linear time-invariant system with impulse response $h(t)$ produces output $y(t)$, so that $x(t)*h(t)=y(t)$. Further, if $|x(t)|*|h(t)|=z(t)$, which of the following statements is true?

  1. For all $t\in(-\infty,\infty)$, $z(t)\le y(t)$
  2. For some but not all $t\in(-\infty,\infty)$, $z(t)\le y(t)$
  3. For all $t\in(-\infty,\infty)$, $z(t)\ge y(t)$
  4. For some but not all $t\in(-\infty,\infty)$, $z(t)\ge y(t)$

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Correct answer: (C) For all $t\in(-\infty,\infty)$, $z(t)\ge y(t)$

Explanation

$|y(t)|=\left|\int x(\tau)h(t-\tau)d\tau\right|\le\int|x(\tau)||h(t-\tau)|d\tau=z(t)$, and since $z(t)\ge0$ we also have $z(t)\ge y(t)$ for all $t$.