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

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 the ``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).

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University