Editing Cross-correlation (section)

==== Cross-covariance function====
<math display="block">\operatorname{K}_{XY}(\tau) \triangleq\ \operatorname{E}\left[\left(X_t - \mu_X\right)\overline{\left(Y_{t+\tau} - \mu_Y\right)}\right]</math> or equivalently <math display="block">\operatorname{K}_{XY}(\tau) = \operatorname{E}\left[\left(X_{t-\tau} - \mu_X\right)\overline{\left(Y_{t} - \mu_Y\right)}\right]</math>where <math>\mu_X</math> and <math>\sigma_X</math> are the mean and standard deviation of the process <math>(X_t)</math>, which are constant over time due to stationarity; and similarly for <math>(Y_t)</math>, respectively. <math>\operatorname{E}[\ ]</math> indicates the [[expected value]]. That the cross-covariance and cross-correlation are independent of <math>t</math> is precisely the additional information (beyond being individually wide-sense stationary) conveyed by the requirement that <math>(X_t, Y_t)</math> are ''jointly'' wide-sense stationary.

The cross-correlation of a pair of jointly [[wide sense stationary]] [[stochastic processes]] can be estimated by averaging the product of samples measured from one process and samples measured from the other (and its time shifts). The samples included in the average can be an arbitrary subset of all the samples in the signal (e.g., samples within a finite time window or a [[sampling (statistics)|sub-sampling]]{{which|date=May 2015}} of one of the signals). For a large number of samples, the average converges to the true cross-correlation.