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Correlation

The cross-correlation of $ x$ and $ y$ in $ \mathbb{C}^N$ is defined as:

$\displaystyle (x \star y)(n) \;\mathrel{\stackrel{\mathrm{\Delta}}{=}}\;\sum_{m=0}^{N-1}\overline{x}(m)y(n+m), \quad
x,y \in \mathbb{C}^N
$

Using this definition we have the correlation theorem:

$\displaystyle \zbox{(x \star y) \leftrightarrow \overline{X(\omega_k)}Y(\omega_k)}
$

The correlation theorem is often used in the context of spectral analysis of filtered noise signals.


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``Review of the Discrete Fourier Transform (DFT)'', by Julius O. Smith III, (From Lecture Overheads, Music 421).
Copyright © 2018-04-10 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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