Cross covariance

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In statistics, the term cross-covariance is sometimes used to refer to the covariance cov(X, Y) between two random vectors X and Y, in order to distinguish that concept from the "covariance" of a random vector X, which is understood to be the matrix of covariances between the scalar components of X.

In signal processing, the cross-covariance (or sometimes "cross-correlation") 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.

For discrete functions f_i and g_i the cross-covariance is defined as

$(f\star g)_i \ \stackrel{\mathrm{def}}{=}\ \sum_j f^*_j\,g_{i+j}$

where the sum is over the appropriate values of the integer j and an asterisk indicates the complex conjugate. For continuous functions f (x) and g _i the cross-covariance is defined as

$(f\star g)(x) \ \stackrel{\mathrm{def}}{=}\ \int f^*(t) g(x+t)\,dt$