Editing Cross-correlation (section)

==Cross-correlation of random vectors==
{{main|Cross-correlation matrix}}

===Definition===
For [[random vector]]s <math>\mathbf{X} = (X_1,\ldots,X_m)</math> and <math>\mathbf{Y} = (Y_1,\ldots,Y_n)</math>, each containing [[random element]]s whose [[expected value]] and [[variance]] exist, the '''cross-correlation matrix''' of <math>\mathbf{X}</math> and <math>\mathbf{Y}</math> is defined by<ref name=Gubner>{{cite book |first=John A. |last=Gubner |year=2006 |title=Probability and Random Processes for Electrical and Computer Engineers |publisher=Cambridge University Press |isbn=978-0-521-86470-1}}</ref>{{rp|p.337}}<math display="block">\operatorname{R}_{\mathbf{X}\mathbf{Y}} \triangleq\ \operatorname{E}\left[\mathbf{X} \mathbf{Y}\right]</math>and has dimensions <math>m \times n</math>. Written component-wise:<math display="block">\operatorname{R}_{\mathbf{X}\mathbf{Y}} =
\begin{bmatrix}
\operatorname{E}[X_1 Y_1] & \operatorname{E}[X_1 Y_2] & \cdots & \operatorname{E}[X_1 Y_n] \\ \\
\operatorname{E}[X_2 Y_1] & \operatorname{E}[X_2 Y_2] & \cdots & \operatorname{E}[X_2 Y_n] \\ \\
 \vdots & \vdots & \ddots & \vdots \\ \\
\operatorname{E}[X_m Y_1] & \operatorname{E}[X_m Y_2] & \cdots & \operatorname{E}[X_m Y_n]
\end{bmatrix}
</math>The random vectors <math>\mathbf{X}</math> and <math>\mathbf{Y}</math> need not have the same dimension, and either might be a scalar value.
Where <math>\operatorname{E}</math> is the [[expectation value]].

===Example===
For example, if <math>\mathbf{X} = \left( X_1,X_2,X_3 \right)</math> and <math>\mathbf{Y} = \left( Y_1,Y_2 \right)</math> are random vectors, then <math>\operatorname{R}_{\mathbf{X}\mathbf{Y}}</math> is a <math>3 \times 2</math> matrix whose <math>(i,j)</math>-th entry is <math>\operatorname{E}[X_i Y_j]</math>.

===Definition for complex random vectors===
If <math>\mathbf{Z} = (Z_1,\ldots,Z_m)</math> and <math>\mathbf{W} = (W_1,\ldots,W_n)</math> are [[complex random vector]]s, each containing random variables whose expected value and variance exist, the cross-correlation matrix of <math>\mathbf{Z}</math> and <math>\mathbf{W}</math> is defined by<math display="block">\operatorname{R}_{\mathbf{Z}\mathbf{W}} \triangleq\ \operatorname{E}[\mathbf{Z} \mathbf{W}^{\rm H}]</math>where <math>{}^{\rm H}</math> denotes [[Hermitian transpose|Hermitian transposition]].