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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
*estimator*^{8.8} 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 theorem
(§

7.4.7).

**Previous:** Correlation Analysis**Next:** Unbiased Cross-Correlation

**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 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.