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 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).