Fourier transform: Difference between revisions

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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
</math>
</math>
==Some Useful Fourier Transform Identities==
==Some Useful Fourier Transform Pairs==
<math>
\mathcal{F}[\alpha(t)]=\Alpha(f)
</math>
<br>
{|
|-
|<math>\mathcal{F}[c_1\alpha(t)+c_2\beta(t)]</math>
|<math>=\int_{-\infty}^{\infty} (c_1\alpha(t)+c_2\beta(t)) e^{-j2\pi ft}\, dt</math>
|-
|
|<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>
|-
|
|<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>
|-
|}
<br>
<math>
\mathcal{F}[\alpha(t-\gamma)]=e^{-j2\pi f\gamma}\Alpha(f)
</math>
<br>
<math>
<math>
\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)
</math>
</math>
<br>


==A Second Approach to Fourier Transforms==
==A Second Approach to Fourier Transforms==

Revision as of 11:41, 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






A Second Approach to Fourier Transforms