Cross-correlation

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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

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

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

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

where the integral is over the appropriate values of t.

The cross-correlation is similar in nature to the convolution of two functions. Whereas convolution involves reversing a signal, then shifting it and multiplying by another signal, correlation only involves shifting it and multiplying (no reversing).

If <math>X</math> and <math>Y</math> are two independent random variables with probability distributions f and g, respectively, then the probability distribution of the difference <math> -X + Y</math> is given by the cross-correlation f <math> \star </math> g. In contrast, the convolution f <math> * </math> g gives the probability distribution of the sum <math>X + Y</math>

[edit] Properties

The cross-correlation is related to the convolution by:

so that if either f or g is an even function

Also: <math>(f\star g)\star(f\star g)=(f\star f)\star (g\star g)</math>

In analogy with the convolution theorem, the cross-correlation satisfies

<math>\mathcal{F}[f*g]=(\mathcal{F}[f^*]) \cdot (\mathcal{F}[g])</math>

where <math>\mathcal{F}</math> denotes the Fourier transform, and an asterisk again indicates the complex conjugate. Coupled with fast Fourier transform algorithms, this property is often exploited for the efficient numerical computation of cross-correlations.

The cross-correlation is related to the spectral density. See Wiener–Khinchin theorem