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<math> = \int_{1}^{\infty} e^{2|t-1|} e^{-j\omega t} dt \!</math> + <math> \int_{-\infty}^{1} e^{2|t-1|} e^{-j\omega t} dt \!</math> | <math> = \int_{1}^{\infty} e^{2|t-1|} e^{-j\omega t} dt \!</math> + <math> \int_{-\infty}^{1} e^{2|t-1|} e^{-j\omega t} dt \!</math> | ||
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Revision as of 16:29, 8 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 \! $
= $ \frac{e^{-j \omega}}{2 + j \omega} + \frac{e^{-j \omega}}{2 - j \omega} $
= $ \frac{4e^{-j \omega}}{4 + \omega ^2} $
