Differentiation Theorem

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Differentiation Theorem

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

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

where $X(\omega)$ denotes the Fourier transform of

. In operator notation:

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

Proof: This follows immediately from integration by parts:

$\begin{eqnarray*} \hbox{\sc FT}_{\omega}({\dot x}) &\isdef & \int_{-\infty}^\in... ...infty x(t) (-j\omega)e^{-j\omega t} dt\\ &=& j\omega X(\omega), \end{eqnarray*}$

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

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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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Spectral Audio Signal Processing
    Fourier Transforms for Continuous/Discrete Time/Frequency
       Selected Continuous Fourier Theorems
          Differentiation Theorem