Cross-Correlation

Definition: The cross-correlation of two signals and may be defined by

I.e., it is the expected value of the lagged products in random signals and . In principle, the expected value must be computed by averaging over many realizations of the stochastic process and . That is, for each ``roll of the dice'' we obtain and for all time, and we can average across all realizations to estimate the expected value of . This is called an ``ensemble average'' across realizations of a stochastic process.

If the signals are stationary (which means their statistics are time-invariant), then we may replace ensemble averaging by averaging across time. In other words, for stationary stochastic processes, time averages equal ensemble averages.

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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.