The GATE Grind

GATE 2024 EE – Question 37

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

The input $x(t)$ and the output $y(t)$ of a system are related as

$$y(t)=e^{-t}\int_{-\infty}^te^\tau x(\tau)\,d\tau,\qquad-\infty<t<\infty.$$

The system is

  1. nonlinear.
  2. linear and time-invariant.
  3. linear but not time-invariant.
  4. noncausal.

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: (B) linear and time-invariant.

Explanation

$y(t)=\int_{-\infty}^te^{-(t-\tau)}x(\tau)\,d\tau=\big(x*h\big)(t)$ with $h(t)=e^{-t}u(t)$. It is a convolution, so the system is linear and time-invariant, and since $h(t)=0$ for $t<0$ it is also causal.