Question

Find the inverse Fourier transform of \( F(f) = \frac{3\pi f}{1 + j\pi f} \)

• A. \( 3\delta(t) - 6e^{-2t} u(t) \)
• B. \( 3e^{-t} u(t) \)
• C. \( 2\delta(t) + 3e^{-t} u(t) \)
• D. \( 6e^{-3t} u(t) \)

Hint:

Use linearity of Fourier inverse and known transforms of rational functions.

Answer 1

The Correct Option is A

Break the function: \[ F(f) = \frac{3\pi f}{1 + j\pi f} = 3 - \frac{3}{1 + j\pi f} \] Now take inverse transform: \[ \mathcal{F}^{-1}\left[3\right] = 3\delta(t),\quad \mathcal{F}^{-1}\left[\frac{1}{1 + j\pi f}\right] = e^{-2t} u(t) \] So, \[ x(t) = 3\delta(t) - 6e^{-2t} u(t) \]