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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:

\hbox{\sc FT}_{\omega}(x^\prime)
&\isdef & \int_{-\infty}^\infty x^\prime(t) e^{-j\omega t} dt\\
&=& \left. x(t)e^{-j\omega t}\right\vert _{-\infty}^{\infty} -
\int_{-\infty}^\infty x(t) (-j\omega)e^{-j\omega t} dt\\
&=& j\omega X(\omega)

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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``Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition'', by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8
Copyright © 2024-02-20 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University