Mathematics of the DFTExample Applications of the DFTCorrelation AnalysisCross-Correlation

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Cross-Correlation

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

(Note that the lag'' is an integer variable, not the constant .) The DFT correlation operator ' was first defined in §7.2.5.

The term cross-correlation'' comes from statistics, and what we have defined here is more properly called a sample cross-correlation.'' That is, is an estimator8.8 of the true cross-correlation which is an assumed statistical property of the signal itself. This definition of a sample cross-correlation is only valid for stationary stochastic processes, e.g., steady noises'' that sound unchanged over time. The statistics of a stationary stochastic process are by definition time invariant, thereby allowing time-averages to be used for estimating statistics such as cross-correlations. For brevity below, we will typically not include sample'' qualifier, because all computational methods discussed will be sample-based methods intended for use on stationary data segments.

The DFT of the cross-correlation may be called the cross-spectral density, or cross-power spectrum,'' or even simply `cross-spectrum'':

The last equality above follows from the correlation theorem7.4.7).

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About the Author: 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 (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.