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Embedded Systems

Mathematics of the DFT
    Selected Continuous-Time Fourier Theorems
       Differentiation Theorem

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Differentiation Theorem

Let $ x(t)$ denote a function differentiable for all $ t$ such that $ x(\pm\infty)=0$ and the Fourier Transforms (FT) of both $ x(t)$ and $ x^\prime(t)$ exist, where $ x^\prime(t)$ denotes the time derivative of $ x(t)$. Then we have

$\displaystyle \zbox {x^\prime(t) \;\longleftrightarrow\;j\omega X(\omega)} $

where $ X(\omega)$ denotes the Fourier transform of $ x(t)$. In operator notation:

$\displaystyle \zbox {\hbox{\sc FT}_{\omega}(x^\prime) = j\omega X(\omega)} $



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*}

since $ x(\pm\infty)=0$.

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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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


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