# Difference between revisions of "Fourier transform"

From Class Wiki

Line 11: | Line 11: | ||

<br><br> |
<br><br> |
||

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>. |
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>. |
||

+ | Note that <math> \lim_{T_\zeta \to \infty}\gamma(t)=\beta(t) </math> |
||

<br> |
<br> |
||

The Fourier Series representation of <math> \gamma(t) </math> is |
The Fourier Series representation of <math> \gamma(t) </math> is |
||

<br> |
<br> |
||

− | <math> \gamma(t)=\sum_{k=-\infty}^\infty \alpha_k e^{j2\pi fkt} </math> where <math> f={1\over T_\ |
+ | <math> \gamma(t)=\sum_{k=-\infty}^\infty \alpha_k e^{j2\pi fkt} </math> where <math> f={1\over T_\zeta} |

− | </math> <br>and <math> \alpha_k={1\over T_\ |
+ | </math> <br>and <math> \alpha_k={1\over T_\zeta}\int_{-{T_\zeta\over 2}}^{{T_\zeta\over 2}} \gamma(t) e^{-j2\pi kt}\,dt</math> |

+ | <br> |
||

+ | <math> \alpha_k </math> can now be rewritten as <math> \alpha_k={1\over T_\zeta}\int_{-\infty}^{\infty} \beta(t) e^{-j2\pi kt}\,dt </math> |
||

+ | <br>From our initial identity then, we can write <math> \alpha_k </math> as |
||

+ | <math> |
||

+ | \alpha_k={1\over T_\zeta}\Beta(kf) |
||

+ | </math> |
||

+ | <br> and |
||

+ | <math> |
||

+ | \gamma(t) |
||

+ | </math> |
||

+ | becomes |
||

+ | <math> |
||

+ | \gamma(t)=\sum_{k=-\infty}^\infty {1\over T_\zeta}\Beta(kf) e^{j2\pi fkt} |
||

+ | </math> |

## Revision as of 10:02, 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 .
Note that

The Fourier Series representation of is

where

and

can now be rewritten as

From our initial identity then, we can write as

and
becomes