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<math> x(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} (\delta(\omega - 1)e^{j\omega t} + \delta(\omega - 3)e^{j\omega t} d\omega \ </math> | <math> x(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} (\delta(\omega - 1)e^{j\omega t} + \delta(\omega - 3)e^{j\omega t} d\omega \ </math> | ||
| + | |||
| + | <math> x(t) = \frac{1}{2\pi}[e^{jt}+ e^{3jt}]</math> | ||
Revision as of 17:09, 8 October 2008
INVERSE FOURIER TRANSFORM
$ X(\omega) = \delta(\omega - 1) + \delta(\omega - 3) $
Knowing the formula for the Inverse Fourier transform
$ x(t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}X(\omega - 1)e^{j\omega t}d\omega \, $
We can proceed to compute its inverse
$ x(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} (\delta(\omega - 1)e^{j\omega t} + \delta(\omega - 3)e^{j\omega t} d\omega \ $
$ x(t) = \frac{1}{2\pi}[e^{jt}+ e^{3jt}] $
