# Difference between revisions of "Fourier transform"

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Suppose that we have some function, say <math> \beta (t) </math>, that is nonperiodic and finite in duration.<br> |
Suppose that we have some function, say <math> \beta (t) </math>, that is nonperiodic and finite in duration.<br> |
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This means that <math> \beta(t)=0 </math> for some <math> T_\alpha < \left | t \right | </math> |
This means that <math> \beta(t)=0 </math> for some <math> T_\alpha < \left | t \right | </math> |
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+ | <br><br> |
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+ | Now let's make a periodic function <math> \gamma(t) </math> by repeating <math> \beta(t) </math> with a fundamental period <math> T_\zeta </math>. |
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+ | <br> |
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+ | The Fourier Series representation of <math> \gamma(t) </math> is |
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+ | <br> |
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+ | <math> \gamma(t)=\sum_{k=-\infty}^\infty \alpha_k e^{j2\pi fkt} </math> where <math> f={1\over T_\alpha} |
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+ | </math> <br>and <math> \alpha_k={1\over T_\alpha}\int_{-{T_\alpha\over 2}}^{{T_\alpha\over 2}} \gamma(t) e^{-j2\pi kt}\,dt</math> |

## Revision as of 09:44, 8 December 2004

An initially identity that is useful:

Suppose that we have some function, say , that is nonperiodic and finite in duration.

This means that for some

Now let's make a periodic function by repeating with a fundamental period .

The Fourier Series representation of is

where

and