Question

Let \( f(t) \) and \( g(t) \) represent continuous-time real-valued signals. If \( h(t) \) denotes the cross-correlation between \( f(t) \) and \( g(-t) \), its continuous-time Fourier transform \( H(j\omega) \) equals: Note: \( F(j\omega) \) and \( G(j\omega) \) denote the continuous-time Fourier transforms of \( f(t) \) and \( g(t) \), respectively.

• A. \( F(j\omega) \cdot G(j\omega) \)
• B. \( F(j\omega) \cdot G(j\omega) \)
• C. \( F(j\omega) \cdot \{G(-j\omega \)
• D. \( -\{F(j\omega) \cdot G(-j\omega) \)

Hint:

In continuous-time systems, cross-correlation in time domain corresponds to multiplication of the Fourier transform of one signal with the complex conjugate of the time-reversed version of the other.

Answer 1

The Correct Option is C

Step 1: Definition of cross-correlation.
The cross-correlation of two signals \( f(t) \) and \( g(t) \) is given by: \[ h(t) = \int_{-\infty}^{\infty} f^(\tau) g(\tau + t) \, d\tau \] This is equivalent to: \[ h(t) = f(-t) g(t) \] Taking the Fourier transform of this expression yields: \[ H(j\omega) = F(j\omega) \cdot \{G(-j\omega) \] Step 2: Justification for the conjugate and time-reversal.
Since we are given \( g(-t) \), its Fourier transform is \( G(-j\omega) \), and because cross-correlation includes complex conjugation, the resulting transform becomes: \[ F(j\omega) \cdot \{G(-j\omega) \]