Question

The inverse Fourier transform of $X\left(\omega \right)=e^{-\omega }u\left(\omega \right)$is

• A. $\frac{2\pi }{1-jt}$
• B. $\frac{\pi }{2(1-jt)}$
• C. $\frac{1}{2\pi (1-jt)}$
• D. $\frac{1}{1-jt}$

Answer 1

The correct option is C $\frac{1}{2\pi (1-jt)}$

We know that
$FT\left[e^{-t}u\right(t\left)\right]=\frac{1}{1+j\omega }$

Using duality property
$x\left(t\right)\stackrel{FT}{⟷}X\left(\omega \right)$

$x\left(t\right)\stackrel{FT}{⟷}2\pi X(-\omega )$

We have $FT\left[\frac{1}{1+jt}\right]\stackrel{FT}{⟷}2\pi e^{-(-\omega )}u(-\omega )$

$\stackrel{FT}{⟷}2\pi e^{\omega }u(-\omega )$

Using the time reversal property,
$i.e.,x(-t)=X(-\omega ),$ we have

$FT\left[\frac{1}{1-jt}\right]\stackrel{FT}{⟷}2\pi e^{-\omega }u\left(\omega \right)$

$\therefore e^{-\omega }\stackrel{IFT}{⟷}\frac{1}{2\pi }\left[\frac{1}{1-jt}\right]$

$\therefore IFT\left[e^{-\omega }u\right(\omega \left)\right]=\frac{1}{2\pi (1-jt)}$