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Differentiation Theorem Dual

Theorem: Let $ x(n)$ denote a signal with DTFT $ X(e^{j\omega})$ , and let

$\displaystyle X^\prime(e^{j\omega}) \isdefs \frac{d}{d\omega} X(e^{j\omega})$ (3.40)

denote the derivative of $ X$ with respect to $ \omega$ . Then we have

$\displaystyle \zbox {-jn x(n) \;\longleftrightarrow\;\frac{d}{d\omega}X(e^{j\omega})}

where $ X(e^{j\omega})$ denotes the DTFT of $ x(n)$ .

Proof: Using integration by parts, we obtain

\hbox{\sc IDTFT}_{n}(X^\prime)
&\isdef & \frac{1}{2\pi}\int_{-\pi}^\pi X^\prime(e^{j\omega}) e^{j\omega n} d\omega\\
&=& \left. \frac{1}{2\pi}X(e^{j\omega})e^{j\omega t}\right\vert _{-\pi}^{\pi} -
\frac{1}{2\pi}\int_{-\pi}^\pi X(e^{j\omega}) (jn)e^{j\omega n} d\omega\\
&=& -jn x(n).

An alternate method of proof is given in §B.3.

Corollary: Perhaps a cleaner statement is as follows:

$\displaystyle \zbox {- n x(n) \;\longleftrightarrow\;\frac{d}{d(j\omega)}X(e^{j\omega})}

This completes our coverage of selected DTFT theorems. The next section adds some especially useful FT theorems having no precise counterpart in the DTFT (discrete-time) case.

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``Spectral Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2011, ISBN 978-0-9745607-3-1.
Copyright © 2019-07-26 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University