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<math>=[\frac{t^2 j}{\omega}e^{-j\omega t} + \frac{2}{j \omega}(\frac{tj}{\omega}e^{-j\omega t}+\frac{1}{\omega ^2}e^{-j\omega t})]_{1}^{\infty}</math> | <math>=[\frac{t^2 j}{\omega}e^{-j\omega t} + \frac{2}{j \omega}(\frac{tj}{\omega}e^{-j\omega t}+\frac{1}{\omega ^2}e^{-j\omega t})]_{1}^{\infty}</math> | ||
| − | <math>=(0) - (jt^ | + | <math>=(0) - (jt^2 e^{-jt} + 2te^{-j)+\frac{2}{j}e^{-jt})</math> |
<math>=\frac{2j}{\omega ^3}</math> | <math>=\frac{2j}{\omega ^3}</math> | ||
Revision as of 09:07, 3 October 2008
Fourier Transform
$ X(\omega)=\int_{-\infty}^{\infty}x(t)e^{-j\omega t}dt $
$ x(t)=t^2 u(t-1) $
$ X(\omega)=\int_{-\infty}^{\infty}t^2 u(t-1) e^{-j\omega t}dt \; = \int_{1}^{\infty}t^2 e^{-j\omega t}dt $
Integration by Parts
$ \int u \; dv = uv - \int v \; du $
$ u=t^2 \; \; \; \; \; \; \; \; \; \; \; \; \; dv = e^{-j \omega t} $
$ du=2t \; dt \; \; \; \; \; \; \; \; v = \frac{1}{-j\omega}e^{-j \omega t} $
$ X(\omega)=\frac{t^2 j}{\omega}e^{-j\omega t}|_{1}^{\infty} + \frac{2}{j \omega}\int_{1}^{\infty}t^2 e^{-j\omega t}dt $
Integration by Parts
$ u=t \; \; \; \; \; \; \; \; \; \; \; \; \; dv = e^{-j \omega t} $
$ du=1 \; dt \; \; \; \; \; \; \; \; v = \frac{1}{-j\omega}e^{-j \omega t} $
$ X(\omega)=\frac{t^2 j}{\omega}e^{-j\omega t}|_{1}^{\infty} + \frac{2}{j \omega}[\frac{tj}{\omega}e^{-j\omega t}|_{1}^{\infty}+\frac{1}{j \omega}\int_{1}^{\infty}e^{-j\omega t}dt] $
$ =[\frac{t^2 j}{\omega}e^{-j\omega t} + \frac{2}{j \omega}(\frac{tj}{\omega}e^{-j\omega t}+\frac{1}{\omega ^2}e^{-j\omega t})]_{1}^{\infty} $
$ =(0) - (jt^2 e^{-jt} + 2te^{-j)+\frac{2}{j}e^{-jt}) $
$ =\frac{2j}{\omega ^3} $
