Cross covariance

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 fi and gi 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


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

where the integral is over the appropriate values of t.

The cross-covariance is similar in nature to the convolution of two functions.

Properties
The cross-covariance is related to the convolution by:


 * $$f(t)\star g(t) = f^*(-t)*g(t)$$

so that


 * $$(f\star g) = f*g$$

if either f or g is an even function. Also:


 * $$(f\star g)\star(f\star g)=(f\star f)\star (g\star g)$$