(→Fourier Transform) |
(→Fourier Transform) |
||
| Line 13: | Line 13: | ||
| − | = <math> \frac{e^{-j \omega}}{2 + j \omega} + \frac{e^{-j \omega}}{2 - j \omega} | + | = <math> \frac{e^{-j \omega}}{2 + j \omega} + \frac{e^{-j \omega}}{2 - j \omega} </math> |
| + | |||
| + | |||
| + | = <math> \frac{4e^{-j \omega}}{4 + \omega ^2} | ||
Revision as of 15:43, 7 October 2008
Fourier Transform
Signal: x(t) = $ e^{3|t-1|} $
$ X(j \omega) = \int_{-\infty}^{\infty} x(t) e^{-j\omega t} dt \! $
$ = \int_{-\infty}^{\infty} e^{2|t-1|} e^{-j\omega t} dt \! $
$ = \int_{1}^{\infty} e^{2|t-1|} e^{-j\omega t} dt \! $ + $ \int_{-\infty}^{1} e^{2|t-1|} e^{-j\omega t} dt \! $
... LOTS OF MATH...
= $ \frac{e^{-j \omega}}{2 + j \omega} + \frac{e^{-j \omega}}{2 - j \omega} $
= $ \frac{4e^{-j \omega}}{4 + \omega ^2} $
