The circular cross-correlation
of two signals
may be defined by
(Note that the ``lag''
is an integer variable, not the constant
.) The DFT correlation operator
' was first defined in
The term ``cross-correlation'' comes from
, and what we have defined here is more properly
called a ``sample cross-correlation.''
of the true
which is an assumed statistical property
of the signal itself. This definition of a sample cross-correlation is only valid for
stochastic processes, e.g.
, ``steady noises
sound unchanged over time. The statistics of a stationary stochastic
process are by definition time invariant
, thereby allowing
to be used for estimating statistics such
as cross-correlations. For brevity below, we will typically
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
, or ``cross-power spectrum
,'' or even simply ``cross-spectrum'':
The last equality above follows from the correlation theorem
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