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

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Correlation Theorem for the DTFT

We define the correlation of discrete-time signals $ x$ and $ y$ by

$\displaystyle \zbox {(x\star y)_n \isdef \sum_m \overline{x(m)} y(m+n)} $

The correlation theorem for DTFTs is then

$\displaystyle \zbox {x\star y \;\longleftrightarrow\;\overline{X}\cdot Y} $



Proof:

\begin{eqnarray*} (x\star y)_n &\isdef & \sum_m \overline{x(m)}y(n+m) \\ &=&... ...t y\right)_n \\ &\;\longleftrightarrow\;& \overline{X} \cdot Y \end{eqnarray*}

where the last step follows from the convolution theorem of §2.3.5 and the symmetry result $ \hbox{\sc Flip}(\overline{x}) \;\longleftrightarrow\; \overline{X}$ of §2.3.2.

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Previous: Convolution Theorem for the DTFT
Next: Autocorrelation
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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