In probability and statistics, given two stochastic processes and, the cross-covariance is a function that gives the covariance of one process with the other at pairs of time points. With the usual notation ; for the expectation operator, if the processes have the mean functions and, then the cross-covariance is given by Cross-covariance is related to the more commonly used cross-correlation of the processes in question. In the case of two random vectors and, the cross-covariance would be a matrix with entries Thus the term cross-covariance is used in order to distinguish this concept from the covariance of a random vector, which is understood to be the matrix of covariances between the scalar components of itself. In signal processing, the cross-covariance is often called cross-correlation and is a measure of similarity of two signals, commonly used to find features in an unknown signal by comparing it to a known one. It is a function of the relative time between the signals, is sometimes called the sliding dot product, and has applications in pattern recognition and cryptanalysis.
The definition of cross-covariance of random vector may be generalized to stochastic processes as follows:
Definition
Let and denote stochastic processes. Then the cross-covariance function of the processes is defined by: where and. If the processes are complex stochastic processes, the second factor needs to be complex conjugated.
Definition for jointly WSS processes
If and are a jointly wide-sense stationary, then the following are true: and By setting , we may define The cross-covariance function of two jointly WSS processes is therefore given by: which is equivalent to
Uncorrelatedness
Two stochastic processes and are called uncorrelated if their covariance is zero for all times. Formally:
Cross-covariance of deterministic signals
The cross-covariance is also relevant in signal processing where the cross-covariance between two wide-sense stationaryrandom processes can be estimated by averaging the product of samples measured from one process and samples measured from the other. The samples included in the average can be an arbitrarysubset of all the samples in the signal. For a large number of samples, the average converges to the true covariance. Cross-covariance may also refer to a "deterministic" cross-covariance between two signals. This consists of summing over all time indices. For example, for discrete-time signals and the cross-covariance is defined as where the line indicates that the complex conjugate is taken when the signals are complex-valued. For continuous functions and the cross-covariance is defined as
Properties
The cross-covariance of two continuous signals is related to the convolution by and the cross-covariance of two discrete-time signals is related to the discrete convolution by
