the cross-correlation between times and iswhere is the expected value operator. Note that this expression may be not defined. Cross-covariance function Subtracting the mean before multiplication yields the cross-covariance between times and :Note that this expression is not well-defined for all time series or processes, because the mean or variance may not exist. Definition for wide-sense stationary stochastic process Let represent a pair of stochastic processes that are jointly wide-sense stationary. Then the cross-covariance function and the cross-correlation function are given as follows. Cross-correlation function or equivalently Cross-covariance function or equivalently where and are the mean and standard deviation of the process , which are constant over time due to stationarity; and similarly for , respectively. indicates the expected value. That the cross-covariance and cross-correlation are independent of is precisely the additional information (beyond being individually wide-sense stationary) conveyed by the requirement that 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 sub-sampling of one of the signals). For a large number of samples, the average converges to the true cross-correlation. Normalization It is common practice in some disciplines (e.g. statistics and time series analysis) to normalize the cross-correlation function to get a time-dependent Pearson correlation coefficient. However, in other disciplines (e.g. engineering) the normalization is usually dropped and the terms "cross-correlation" and "cross-covariance" are used interchangeably. The definition of the normalized cross-correlation of a stochastic process isIf the function is well-defined, its value must lie in the range , with 1 indicating perfect correlation and −1 indicating perfect anti-correlation. For jointly wide-sense stationary stochastic processes, the definition isThe normalization is important both because the interpretation of the autocorrelation as a correlation provides a scale-free measure of the strength of statistical dependence, and because the normalization has an effect on the statistical properties of the estimated autocorrelations. Properties Symmetry property For jointly wide-sense stationary stochastic processes, the cross-correlation function has the following symmetry property:Respectively for jointly WSS processes: Time delay analysis Cross-correlations are useful for determining the time delay between two signals, e.g., for determining time delays for the propagation of acoustic signals