Difference between revisions of "Fourier transform"
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Which is just to say that 
Which is just to say that 

<math> 
<math> 

−  \beta(t)=\int_{\infty}^\infty 
+  \beta(t)=\int_{\infty}^\infty \Beta(f) e^{j2\pi fkt}\,df 
</math> 
</math> 

<br> 
<br> 

<br> 
<br> 

−  So we have that 
+  So we have that 
<math> 
<math> 

−  \beta(t) 

+  \mathcal{F}[\beta(t)]=\Beta(f)=\int_{\infty}^{\infty} \beta(t) e^{j2\pi ft}\, dt 

</math> 
</math> 

−  is 

+  <br> 

+  Further 

<math> 
<math> 

−  +  \mathcal{F}^{1}[\Beta(f)]=\beta(t)=\int_{\infty}^\infty \Beta(f) e^{j2\pi fkt}\,df 

</math> 
</math> 
Revision as of 11:32, 9 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
Now remember that
and
Which means that
Which is just to say that
So we have that
Further