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The cross-correlation of a signal with itself gives the autocorrelation function of that signal:

$\displaystyle r_{x}(l) \isdef r_{xx}(l) = E\{\overline{x(n)}x(n+l)\}$

Note that the autocorrelation function is Hermitian:

$\displaystyle r_x(-l) = \overline{r_x(l)}$

When is real, its autocorrelation is symmetric. More specifically, it is real and even.

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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