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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) |
| − | + | 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 | |
<math> X(\omega) = 2cos(\omega) </math> | <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 <math> X(\omega) = 2cos(\omega) $
