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.

Statistics

For random vectors, X and Y, each containing random elements whose expected value and variance exist, the cross-covariance matrix of X and Y is defined by

$\operatorname{cov}(X,Y)=\operatorname{E}[(X-\mu_X)(Y-\mu_Y)'],$

where μ_X and μ_Y are vectors containing the expected values of X and Y. The vectors X and Y need not have the same dimension, and either might be a scalar value. Any element of the cross-covariance matrix is itself a "cross-covariance".

Signal processing

Main article: Cross-correlation

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 (x) 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 of two signals 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:

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

See also

• Convolution
• Correlation
• Autocovariance

External links

• Cross Correlation from Mathworld
• http://scribblethink.org/Work/nvisionInterface/nip.html
• http://www.phys.ufl.edu/LIGO/stochastic/sign05.pdf
• http://www.staff.ncl.ac.uk/oliver.hinton/eee305/Chapter6.pdf