Question

Fourier transform of \( x(t) = e^{-at}u(t - 2) \) is

• A. \( \frac{e^{jw}}{a^2 + w^2} \)
• B. \( \frac{e^{jw}}{a^2 - w^2} \)
• C. \( \frac{e^{-jw}}{a^2 + w^2} \)
• D. \( \frac{e^{-jw}}{2a + w^2} \)

Hint:

For signals involving exponential terms with unit step functions, remember to account for the shift in time and use standard Fourier transform results.

Answer 1

The Correct Option is A

The Fourier transform of \( e^{-at}u(t-2) \), where \( u(t-2) \) is the unit step function shifted by 2, is given by: \[ X(jw) = \int_{-\infty}^{\infty} e^{-at}e^{-j\omega t} dt \] Since the signal is shifted by 2, the Fourier transform will include a shift factor. The standard result for this type of transform is: \[ X(jw) = \frac{e^{jw}}{a^2 + w^2} \] Therefore, the correct answer is option (1).