Cross-correlation matrix

Concept in digital signal processing The cross-correlation matrix of two random vectors is a matrix containing as elements the cross-correlations of all pairs of elements of the random vectors. The cross-correlation matrix is used in various digital signal processing algorithms. Definition For two random vectors X = ( X 1 , … , X m ) T {\displaystyle \mathbf {X} =(X_{1},\ldots ,X_{m})^{ m {T}}} and Y = ( Y 1 , … , Y n ) T {\displaystyle \mathbf {Y} =(Y_{1},\ldots ,Y_{n})^{ m {T}}} , each containing random elements whose expected value and variance exist, the cross-correlation matrix of X {\displaystyle \mathbf {X} } and Y {\displaystyle \mathbf {Y} } is defined by R X Y ≜ E ⁡ [ X Y T ] {\displaystyle \operatorname {R} _{\mathbf {X} \mathbf {Y} }\triangleq \ \operatorname {E} [\mathbf {X} \mathbf {Y} ^{ m {T}}]} and has dimensions m × n {\displaystyle m\times n} . Written component-wise: R X Y = [ E ⁡ [ X 1 Y 1 ] E ⁡ [ X 1 Y 2 ] ⋯ E ⁡ [ X 1 Y n ] E ⁡ [ X 2 Y 1 ] E ⁡ [ X 2 Y 2 ] ⋯ E ⁡ [ X 2 Y n ] ⋮ ⋮ ⋱ ⋮ E ⁡ [ X m Y 1 ] E ⁡ [ X m Y 2 ] ⋯ E ⁡ [ X m Y n ] ] {\displaystyle \operatorname {R} _{\mathbf {X} \mathbf {Y} }={\begin{bmatrix}\operatornam