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Fourier Transform of delta functions | Fourier Transform of delta functions | ||
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| + | 1. | ||
<math> x(t) = \delta (t+1) + \delta (t-1) </math> | <math> x(t) = \delta (t+1) + \delta (t-1) </math> | ||
| − | <math> X(\omega) = \ | + | <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> |
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| + | <math> X(\omega) = e^{j \omega}+ e^{-j \omega} = \frac{1}{2} (e^ {j \omega} + e^ {-j \omega})^2 </math> | ||
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| + | <math> X(\omega) = 2cos(\omega) </math> | ||
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| + | 2. | ||
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| + | <math> x(t) = \frac{d}{dt} {u(-2-t) + u(t-2)} </math> | ||
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| + | <math> X(\omega) = \int_{-\infty}^{\infty} \frac{d}{dt} {u(-2-t) + u(t-2)} e^{-j \omega t} dt </math> | ||
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| + | <math> X(\omega) = \int_{-\infty}^{\infty} \delta (-2-t) e^{-j \omega t} dt + \int_{-\infty}^{\infty} \delta (t-2) e^{-j \omega t} dt</math> | ||
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| + | <math> X(\omega) = -e^{2j \omega} + e^{-2j \omega} </math> | ||
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| + | <math> X(\omega) = -2j (\frac{e^{2j \omega} - e^{-2j \omega}}{2j}) </math> | ||
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| + | <math> X(\omega) = -2j sin(2 \omega) </math> | ||
Latest revision as of 18:26, 24 October 2008
Fourier Transform of delta functions
1.
$ 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) $
2.
$ x(t) = \frac{d}{dt} {u(-2-t) + u(t-2)} $
$ X(\omega) = \int_{-\infty}^{\infty} \frac{d}{dt} {u(-2-t) + u(t-2)} e^{-j \omega t} dt $
$ X(\omega) = \int_{-\infty}^{\infty} \delta (-2-t) e^{-j \omega t} dt + \int_{-\infty}^{\infty} \delta (t-2) e^{-j \omega t} dt $
$ X(\omega) = -e^{2j \omega} + e^{-2j \omega} $
$ X(\omega) = -2j (\frac{e^{2j \omega} - e^{-2j \omega}}{2j}) $
$ X(\omega) = -2j sin(2 \omega) $
