The GATE Grind

GATE 2024 EE – Question 15

Signals and Systems · Linear time invariant and causal systems · 1 mark · Multiple choice

Suppose signal $y(t)$ is obtained by the time-reversal of signal $x(t)$, i.e., $y(t)=x(-t)$, $-\infty<t<\infty$. Which one of the following options is always true for the convolution of $x(t)$ and $y(t)$?

  1. It is an even signal.
  2. It is an odd signal.
  3. It is a causal signal.
  4. It is an anti-causal signal.

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: (A) It is an even signal.

Explanation

$z(t)=\int x(\tau)\,x(\tau-t)\,d\tau$ is the autocorrelation of $x$. Substituting $\tau\to\tau+t$ shows $z(-t)=z(t)$, so it is always even.