Question

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

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

Answer 1

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

We know that

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

Using duality property

$x\left(t\right)\stackrel{}{\leftarrow }\stackrel{FT}{\to }X\left(\omega \right)$

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

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

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

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

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