Question

Find the inverse Fourier transform of \( e^{j2\pi t} \):

• A. \( 2\pi \delta(\omega - 2) \)
• B. \( \pi \delta(\omega - 2) \)
• C. \( \pi \delta(\omega + 2) \)
• D. \( 2\pi \delta(\omega + 2) \)

Hint:

For a function of the form \( e^{j\omega_0 t} \), its inverse Fourier transform is \( 2\pi \delta(\omega - \omega_0) \).

Answer 1

The Correct Option is A

The inverse Fourier transform of \( e^{j2\pi t} \) can be found using the standard Fourier transform pair: \[ \mathcal{F}^{-1}\{ e^{j\omega_0 t} \} = 2\pi \delta(\omega - \omega_0) \] In this case, \( \omega_0 = 2 \), so the inverse Fourier transform is: \[ \mathcal{F}^{-1}\{ e^{j2\pi t} \} = 2\pi \delta(\omega - 2) \] Thus, the correct answer is \( 2\pi \delta(\omega - 2) \), which corresponds to option (1).