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<math>X(\omega) = \int_{-\infty}^{\infty}\frac{1}{2}(e^{jt}+e^{-jt})e^{-j\omega t}dt</math> | <math>X(\omega) = \int_{-\infty}^{\infty}\frac{1}{2}(e^{jt}+e^{-jt})e^{-j\omega t}dt</math> | ||
| − | <math>X(\omega) = \frac{1}{2}(\int_{-\infty}^{\infty}e^{jt(1-\omega)}+\int_{-\infty}^{\infty}e^{-jt(1+\omega)})</math> | + | <math>X(\omega) = \frac{1}{2}(\int_{-\infty}^{\infty}e^{jt(1-\omega)}dt+\int_{-\infty}^{\infty}e^{-jt(1+\omega)}dt)</math> |
| + | |||
| + | <math>X(\omega) = \frac{1}{2}(\int_{-\infty}^{\infty}e^{jt(1-\omega)}dt+\int_{-\infty}^{\infty}e^{-jt(1+\omega)}dt)</math> | ||
| + | |||
| + | <math>X(\omega)={\left. \frac{e^{jt(1-\omega)}}{j(1-\omega))}\right]_{-\infty}^{\infty}} + {\left. \frac{e^{-jt(1-\omega)}}{-j(1+\omega))}\right]_{-\infty}^{\infty}} | ||
Revision as of 06:57, 8 October 2008
Let x(t)= $ cos(t) $
Then
$ X(\omega) = \int_{-\infty}^{\infty}x(t)e^{-j\omega t}dt $
$ X(\omega) = \int_{-\infty}^{\infty}cos(t)e^{-j\omega t}dt $
$ X(\omega) = \int_{-\infty}^{\infty}\frac{1}{2}(e^{jt}+e^{-jt})e^{-j\omega t}dt $
$ X(\omega) = \frac{1}{2}(\int_{-\infty}^{\infty}e^{jt(1-\omega)}dt+\int_{-\infty}^{\infty}e^{-jt(1+\omega)}dt) $
$ X(\omega) = \frac{1}{2}(\int_{-\infty}^{\infty}e^{jt(1-\omega)}dt+\int_{-\infty}^{\infty}e^{-jt(1+\omega)}dt) $
$ X(\omega)={\left. \frac{e^{jt(1-\omega)}}{j(1-\omega))}\right]_{-\infty}^{\infty}} + {\left. \frac{e^{-jt(1-\omega)}}{-j(1+\omega))}\right]_{-\infty}^{\infty}} $
