Laplace Transforms

Step 1: Consider a real-valued signal.
Let be a real-valued time-domain signal with Fourier transform .
Step 2: Apply Fourier symmetry property.
For real signals, the Fourier transform satisfies the conjugate symmetry condition:

Step 3: Interpretation of symmetry.
This property is known as conjugate symmetry and is characteristic of real-valued signals.
Step 4: Eliminate incorrect options.
Odd or even symmetry alone is insufficient to describe .
Step 5: Final conclusion.
Therefore, the Fourier transform of a real-valued signal has conjugate symmetry.

Step 1: Properties of Fourier transform.
Fourier transform symmetry properties relate the nature of to .
Step 2: Given condition.
The signal 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, is an imaginary and odd function of .