(New page: == Inverse F.T'ing == Given <math>X(\omega) = 3\pi\delta(\omega-\pi)+\delta(\omega-2\pi)-2\pi\delta(\omega-3\pi)\!</math> ____ <math>x(t)= \frac{1}{2\pi}\int_{-\infty}^{\infty}X(\omega...) |
(No difference)
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Revision as of 18:36, 8 October 2008
Inverse F.T'ing
Given
$ X(\omega) = 3\pi\delta(\omega-\pi)+\delta(\omega-2\pi)-2\pi\delta(\omega-3\pi)\! $
____
$ x(t)= \frac{1}{2\pi}\int_{-\infty}^{\infty}X(\omega)e^{jwt}dw $
$ x(t)= \frac{1}{2\pi}\int_{-\infty}^{\infty}[3\pi\delta(\omega-\pi)+\delta(\omega-2\pi)-2\pi\delta(\omega-3\pi)\!]e^{jwt}dw $
$ x(t)= \frac{1}{2\pi}\int_{-\infty}^{\infty}3\pi\delta(\omega-\pi)e^{jwt}dw+\frac{1}{2\pi}\int_{-\infty}^{\infty}\delta(\omega-2\pi)e^{jwt}dw-\frac{1}{2\pi}\int_{-\infty}^{\infty}2\pi\delta(\omega-3\pi)\!e^{jwt}dw $
$ x(t)= \frac{3}{2}\int_{-\infty}^{\infty}\delta(\omega-\pi)e^{jwt}dw+\frac{1}{2\pi}\int_{-\infty}^{\infty}\delta(\omega-2\pi)e^{jwt}dw-\int_{-\infty}^{\infty}\delta(\omega-3\pi)\!e^{jwt}dw $
$ x(t)= \frac{3*e^{j\pi*t}}{2}e^{jwt}+\frac{e^{j2\pi*t}}{2\pi}e^{jwt}-e^{j3*\pi*t}e^{jwt} $
