Differentiation Theorem
Let denote a function differentiable for all
such that
and the Fourier Transforms (FT) of both
and
exist, where
denotes the time derivative
of
. Then we have
![$\displaystyle \zbox {x^\prime(t) \;\longleftrightarrow\;j\omega X(\omega)}
$](http://www.dsprelated.com/josimages_new/mdft/img1726.png)
![$ X(\omega)$](http://www.dsprelated.com/josimages_new/mdft/img466.png)
![$ x(t)$](http://www.dsprelated.com/josimages_new/mdft/img4.png)
![$\displaystyle \zbox {\hbox{\sc FT}_{\omega}(x^\prime) = j\omega X(\omega)}
$](http://www.dsprelated.com/josimages_new/mdft/img1727.png)
Proof:
This follows immediately from integration by parts:
![\begin{eqnarray*}
\hbox{\sc FT}_{\omega}(x^\prime)
&\isdef & \int_{-\infty}^\in...
...\infty x(t) (-j\omega)e^{-j\omega t} dt\\
&=& j\omega X(\omega)
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/mdft/img1728.png)
since
.
The differentiation theorem is implicitly used in §E.6 to show that audio signals are perceptually equivalent to bandlimited signals which are infinitely differentiable for all time.
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Fourier Series (FS) and Relation to DFT