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Spectral Audio Signal Processing
    Fourier Transforms for Continuous/Discrete Time/Frequency
       Fourier Theorems for the DTFT
          Differentiation Theorem Dual

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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}) \isdef \frac{d}{d\omega} X(e^{j\omega}) $

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

\begin{eqnarray*} \hbox{\sc IDTFT}_{n}(X^\prime) &\isdef & \frac{1}{2\pi}\int_{... ...i}^\pi X(e^{j\omega}) (jn)e^{j\omega n} d\omega\\ &=& -jn x(n). \end{eqnarray*}

An alternate method of proof is given in §2.4.3.

Corollary: Perhaps a cleaner statement is as follows:

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

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Next: Selected Continuous Fourier Theorems
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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