(New page: ==Inverse Fourier Transforms== If we have a Fourier series <math>X(\omega)</math>, then <math>x(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}X(\omega)e^{j\omega t}d\omega</math> ==Example== <...) |
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Revision as of 15:44, 7 October 2008
Inverse Fourier Transforms
If we have a Fourier series $ X(\omega) $, then
$ x(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}X(\omega)e^{j\omega t}d\omega $
Example
$ X(\omega)=4\pi\delta(\omega-\frac{3\pi}{2})+4\pi\delta(\omega+\frac{3\pi}{2}) $
$ x(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}4\pi\delta(\omega-\frac{3\pi}{2})e^{j\omega t}+4\pi\delta(\omega+\frac{3\pi}{2})d\omega $
$ x(t)=2\int_{-\infty}^{\infty}\delta(\omega-\frac{3\pi}{2})e^{j\omega t}+delta(\omega+\frac{3\pi}{2})d\omega $
$ x(t)=2e^{j\frac{3\pi}{2}t}+2e^{j\frac{-3\pi}{2}} $
$ x(t)=4[\frac{e^{j\frac{3\pi}{2}t}+e^{-j\frac{3\pi}{2}}}{2}] $
$ x(t)=4cos(\frac{3\pi}{2}t) $
