
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 theorem
(§
7.4.7).
Next Section: Unbiased Cross-CorrelationPrevious Section: Phase Response