# Difference between revisions of "Fourier transform"

From Class Wiki

(→Some Useful Fourier Transform Identities) |
(→Some Useful Fourier Transform Identities) |
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\mathcal{F}^{-1}[\Beta(f)]=\beta(t)=\int_{-\infty}^\infty \Beta(f) e^{j2\pi fkt}\,df |
\mathcal{F}^{-1}[\Beta(f)]=\beta(t)=\int_{-\infty}^\infty \Beta(f) e^{j2\pi fkt}\,df |
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</math> |
</math> |
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− | ==Some Useful Fourier Transform |
+ | ==Some Useful Fourier Transform Pairs== |

+ | <math> |
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+ | \mathcal{F}[\alpha(t)]=\Alpha(f) |
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+ | </math> |
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+ | <br> |
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+ | {| |
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+ | |- |
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+ | |<math>\mathcal{F}[c_1\alpha(t)+c_2\beta(t)]</math> |
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+ | |<math>=\int_{-\infty}^{\infty} (c_1\alpha(t)+c_2\beta(t)) e^{-j2\pi ft}\, dt</math> |
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+ | |- |
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+ | | |
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+ | |<math>=\int_{-\infty}^{\infty}c_1\alpha(t)e^{-j2\pi ft}\, dt+\int_{-\infty}^{\infty}c_2\beta(t)e^{-j2\pi ft}\, dt</math> |
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+ | |- |
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+ | | |
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+ | |<math>=c_1\int_{-\infty}^{\infty}\alpha(t)e^{-j2\pi ft}\, dt+c_2\int_{-\infty}^{\infty}\beta(t)e^{-j2\pi ft}\, dt=c_1\Alpha(f)+c_2\Beta(f)</math> |
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+ | |- |
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+ | |} |
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+ | <br> |
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+ | <math> |
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+ | \mathcal{F}[\alpha(t-\gamma)]=e^{-j2\pi f\gamma}\Alpha(f) |
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+ | </math> |
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+ | <br> |
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<math> |
<math> |
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\mathcal{F}[\alpha(t)*\beta(t)]=\Alpha(f)\Beta(f) |
\mathcal{F}[\alpha(t)*\beta(t)]=\Alpha(f)\Beta(f) |
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\mathcal{F}[\alpha(t)\beta(t)]=\Alpha(f)*\Beta(f) |
\mathcal{F}[\alpha(t)\beta(t)]=\Alpha(f)*\Beta(f) |
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</math> |
</math> |
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+ | <br> |
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==A Second Approach to Fourier Transforms== |
==A Second Approach to Fourier Transforms== |

## Revision as of 10:41, 10 December 2004

## From the Fourier Transform to the Inverse Fourier Transform

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

Now remember that
and

Which means that

Which is just to say that

So we have that

Further

## Some Useful Fourier Transform Pairs