The GATE Grind

GATE 2025 IN – Question 48

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

Let $f(t)$ and $g(t)$ be continuous-time real signals. If $h(t)$ is their cross-correlation using $f(t)$ and $g(-t)$, its Fourier transform $H(j\omega)$ equals ______. Here $F(j\omega)$ and $G(j\omega)$ are the Fourier transforms of $f(t)$ and $g(t)$.

  1. $F(j\omega)G(j\omega)$
  2. $F(-j\omega)G(j\omega)$
  3. $F(j\omega)G(-j\omega)$
  4. $-F(j\omega)G(-j\omega)$

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Show answer and explanation

Correct answer: (A) $F(j\omega)G(j\omega)$

Explanation

Using the correlation convention in which rfg(τ)=∫f(t)g(t−τ)dt, correlation with g(−t) becomes convolution f*g.

Its Fourier transform is therefore **F(jω)G(jω), A**.

The real-signal assumption permits the usual reversal/conjugation identities.