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Fourier Transform of delta functions | Fourier Transform of delta functions | ||
| − | <math> x(t) = \delta (t+1) + \delta (t-1) < | + | <math> x(t) = \delta (t+1) + \delta (t-1) </math> |
| − | <math> X(\omega) = \int_{-\infty}^{\infty} \delta (t+1)e^{-j \omega t} + \int_{-\infty}^{\infty} \delta (t-1)e^{-j \omega t} dt < | + | <math> X(\omega) = \int_{-\infty}^{\infty} \delta (t+1)e^{-j \omega t} + \int_{-\infty}^{\infty} \delta (t-1)e^{-j \omega t} dt </math> |
| − | <math> X(\omega) = e^{j \omega}+ e^{-j \omega} = \frac{1}{2} (e^ {j \omega} + e^ {-j \omega})^2 < | + | <math> X(\omega) = e^{j \omega}+ e^{-j \omega} = \frac{1}{2} (e^ {j \omega} + e^ {-j \omega})^2 </math> |
| − | <math> X(omega) = 2cos(\omega) < | + | <math> X(omega) = 2cos(\omega) </math> |
Revision as of 17:11, 24 October 2008
Fourier Transform of delta functions
$ x(t) = \delta (t+1) + \delta (t-1) $
$ X(\omega) = \int_{-\infty}^{\infty} \delta (t+1)e^{-j \omega t} + \int_{-\infty}^{\infty} \delta (t-1)e^{-j \omega t} dt $
$ X(\omega) = e^{j \omega}+ e^{-j \omega} = \frac{1}{2} (e^ {j \omega} + e^ {-j \omega})^2 $
$ X(omega) = 2cos(\omega) $
