The GATE Grind

GATE 2023 IN – Question 44

Signals and Systems · Fourier analysis and sampling · 2 marks · Multiple select

A continuous real-valued signal $x(t)$ has finite positive energy and is zero for all $t<0$. Select all signals whose continuous-time Fourier transform is purely imaginary.

  1. $x(t)+x(-t)$
  2. $x(t)-x(-t)$
  3. $j[x(t)+x(-t)]$
  4. $j[x(t)-x(-t)]$

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Show answer and explanation

Correct answer: (B) $x(t)-x(-t)$; (C) $j[x(t)+x(-t)]$

Explanation

For real x, $x(t)-x(-t)$ is real odd, with purely imaginary Fourier transform. The sum is real even, with real transform; multiplication by j makes that transform imaginary. Multiplication of the odd difference by j instead makes its transform real. Correct choices: **B, C**.