Cross-correlation: Encyclopedia - Cross-correlation

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

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-correlation is defined as

where the integral is over the appropriate values of t.

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

Cross-correlation - Properties

The cross-correlation is related to the convolution by:

so that

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

See also

• Convolution
• Correlation
• Autocovariance

Adapted from the Wikipedia article "Cross-correlation", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki