Inverse fourier transform

Jan 2010
133
7
Hello everyone!

I'm trying to IFT this fraction after I couldn't find it in a Fourier couples table.

\(\displaystyle F(j\omega)=\frac{250\pi \delta(\omega)}{30+125\omega}\).
When I integrated the latter, I got: \(\displaystyle \frac{1}{2\pi}\int_{0-}^{0+} \frac{250\pi \delta(\omega)}{30+125\omega}e^{-j\omega t}d\omega\) \(\displaystyle \ =\frac{1}{2\pi}\int_{0-}^{0+} \frac{250\pi \delta(\omega)}{30}d\omega\) \(\displaystyle =\frac{25}{6} u(t)\)??

Is there ever a \(\displaystyle u(t)\) in IFT ??
Thanks for any help!
 

CaptainBlack

MHF Hall of Fame
Nov 2005
14,972
5,271
someplace
Hello everyone!

I'm trying to IFT this fraction after I couldn't find it in a Fourier couples table.

\(\displaystyle F(j\omega)=\frac{250\pi \delta(\omega)}{30+125\omega}\).
When I integrated the latter, I got: \(\displaystyle \frac{1}{2\pi}\int_{0-}^{0+} \frac{250\pi \delta(\omega)}{30+125\omega}e^{-j\omega t}d\omega\) \(\displaystyle \ =\frac{1}{2\pi}\int_{0-}^{0+} \frac{250\pi \delta(\omega)}{30}d\omega\) \(\displaystyle =\frac{25}{6} u(t)\)??

Is there ever a \(\displaystyle u(t)\) in IFT ??
Thanks for any help!
For any function continuous in a neighbourhood of \(\displaystyle 0\) we have:

\(\displaystyle \int_A f(\omega) \delta(\omega)\;d\omega=f(0)\)

For any interval \(\displaystyle A\) containing \(\displaystyle 0\) as an interior point.

CB
 
Jan 2010
133
7
For any function continuous in a neighbourhood of we have:



For any interval containing as an interior point.
Isn't it that \(\displaystyle \int_A f(\omega)\delta(\omega) d\omega\) \(\displaystyle =\int_A f(0) \delta(\omega) d\omega = f(0)u(t)\)
At least that's what I know from circuit analysis...