| Line 4: | Line 4: | ||
<font "size"=4> | <font "size"=4> | ||
| − | <math>x(t)=t^2 u(t)</math> | + | <math>x(t)=t^2 u(t-1)</math> |
</font> | </font> | ||
| − | <math>X(\omega)=\int_{-\infty}^{\infty}t^2 u(t) e^{-j\omega t}dt \; = \int_{ | + | <math>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</math> |
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<math>du=2t \; dt \; \; \; \; \; \; \; \; v = \frac{1}{-j\omega}e^{-j \omega t}</math> | <math>du=2t \; dt \; \; \; \; \; \; \; \; v = \frac{1}{-j\omega}e^{-j \omega t}</math> | ||
| − | <math>X(\omega)=\frac{t^2 j}{\omega}e^{-j\omega t}|_{ | + | <math>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</math> |
''Integration by Parts'' | ''Integration by Parts'' | ||
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<math>du=1 \; dt \; \; \; \; \; \; \; \; v = \frac{1}{-j\omega}e^{-j \omega t}</math> | <math>du=1 \; dt \; \; \; \; \; \; \; \; v = \frac{1}{-j\omega}e^{-j \omega t}</math> | ||
| − | <math>X(\omega)=\frac{t^2 j}{\omega}e^{-j\omega t}|_{ | + | <math>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]</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})]_{ | + | <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) - [(\frac{2}{j \omega})(\frac{1}{\omega ^2})]</math> | <math>=(0) - [(\frac{2}{j \omega})(\frac{1}{\omega ^2})]</math> | ||
<math>=\frac{2j}{\omega ^3}</math> | <math>=\frac{2j}{\omega ^3}</math> | ||
Revision as of 09:03, 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) - [(\frac{2}{j \omega})(\frac{1}{\omega ^2})] $
$ =\frac{2j}{\omega ^3} $
