Question

A signal $x(t)$ has a Fourier transform $X(\omega)$. If $x(t)$ is a real and odd function of $t$, then $X(\omega)$ is

• A. a real and even function of $\omega$
• B. an imaginary and odd function of $\omega$
• C. an imaginary and even function of $\omega$
• D. a real and odd function of $\omega$

Hint:

Real–odd signals always produce imaginary–odd Fourier transforms.

Answer 1

The Correct Option is B

Step 1: Properties of Fourier transform.
Fourier transform symmetry properties relate the nature of $x(t)$ to $X(\omega)$.
Step 2: Given condition.
The signal $x(t)$ is real and odd.
Step 3: Apply known Fourier properties.
For a real and odd time-domain signal, the Fourier transform is purely imaginary and odd.
Step 4: Verification of options.
Only option (B) satisfies both conditions: imaginary and odd.
Step 5: Final conclusion.
Thus, $X(\omega)$ is an imaginary and odd function of $\omega$.