Correlation

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The correlation operator for two signals and in ${\bf C}^N$ is defined as

$\displaystyle \zbox {(x\star y)_n \isdef \sum_{m=0}^{N-1}\overline{x(m)} y(m+n)}$

We may interpret the correlation operator as

$\displaystyle (x\star y)_n = \left<\hbox{\sc Shift}_{-n}(y), x\right>$

which is $\vert\vert\,x\,\vert\vert ^2=N$ times the coefficient of projection onto

advanced by

samples (shifted circularly to the left by

samples). The time shift

is called the correlation lag, and $\overline{x(m)} y(m+n)$ is called a lagged product. Applications of correlation are discussed in §8.4.

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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