Cross-correlation

In statistics, the term cross-correlation 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-correlation (or sometimes "cross-covariance") 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-correlation is defined as

(f\star g)_{i}\equiv \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 (x) the cross-correlation is defined as

(f\star g)(x)\equiv \int f^{*}(t)g(x+t)\,dt

where the integral is over the appropriate values of t.

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

Properties

The cross-correlation is related to the convolution by:

f(t)\star g(t)=f^{*}(-t)*g(t)

so that if either f or g is an even function

(f\star g)=f*g

Also: $(f\star g)\star (f\star g)=(f\star f)\star (g\star g)$

External links

Properties

See also

External links