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.
01Definition
For two random vectors 
![\operatorname {R} _{\mathbf {X} \mathbf {Y} }\triangleq \ \operatorname {E} [\mathbf {X} \mathbf {Y} ^{\rm {T}}]](https://wikimedia.org/api/rest_v1/media/math/render/svg/1337f507d7cb6918d742ae2f029e0554013f7a58)
and has dimensions 
![\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dfaf0f3923eafd144f762732bbaa951102ed00bb)
The random vectors
02Example
For example, if 
![\operatorname {E} [X_{i}Y_{j}]](https://wikimedia.org/api/rest_v1/media/math/render/svg/e511d6ea021c2d4b0579f621faec897ec52a1999)
03Complex random vectors
If 
![\operatorname {R} _{\mathbf {Z} \mathbf {W} }\triangleq \ \operatorname {E} [\mathbf {Z} \mathbf {W} ^{\rm {H}}]](https://wikimedia.org/api/rest_v1/media/math/render/svg/b96a25804b6f71adb2dd4366a18cc65e96553d88)
where 
04Uncorrelatedness
Two random vectors
![\operatorname {E} [\mathbf {X} \mathbf {Y} ^{\rm {T}}]=\operatorname {E} [\mathbf {X} ]\operatorname {E} [\mathbf {Y} ]^{\rm {T}}.](https://wikimedia.org/api/rest_v1/media/math/render/svg/1a8bb45997c59ac08d38c6073ad03c832d11abf6)
They are uncorrelated if and only if their cross-covariance matrix 
In the case of two complex random vectors
![\operatorname {E} [\mathbf {Z} \mathbf {W} ^{\rm {H}}]=\operatorname {E} [\mathbf {Z} ]\operatorname {E} [\mathbf {W} ]^{\rm {H}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e653a5949cb26600c65a6d23c16310eae1e9867f)
and
![\operatorname {E} [\mathbf {Z} \mathbf {W} ^{\rm {T}}]=\operatorname {E} [\mathbf {Z} ]\operatorname {E} [\mathbf {W} ]^{\rm {T}}.](https://wikimedia.org/api/rest_v1/media/math/render/svg/500e14ec28154556c9b0aa741d1b036af7b7a67f)
05Properties
Relation to the cross-covariance matrix
The cross-correlation is related to the cross-covariance matrix as follows:
- Respectively for complex random vectors:
![\operatorname {K} _{\mathbf {X} \mathbf {Y} }=\operatorname {E} [(\mathbf {X} -\operatorname {E} [\mathbf {X} ])(\mathbf {Y} -\operatorname {E} [\mathbf {Y} ])^{\rm {T}}]=\operatorname {R} _{\mathbf {X} \mathbf {Y} }-\operatorname {E} [\mathbf {X} ]\operatorname {E} [\mathbf {Y} ]^{\rm {T}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1d58fc3b03892e013a545da3322c2d4942d7314e)
![\operatorname {K} _{\mathbf {Z} \mathbf {W} }=\operatorname {E} [(\mathbf {Z} -\operatorname {E} [\mathbf {Z} ])(\mathbf {W} -\operatorname {E} [\mathbf {W} ])^{\rm {H}}]=\operatorname {R} _{\mathbf {Z} \mathbf {W} }-\operatorname {E} [\mathbf {Z} ]\operatorname {E} [\mathbf {W} ]^{\rm {H}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e3b852cac5898146e76309d2f9d00ac163d1d65a)
Sources and credits
This article is adapted from the Wikipedia article “Cross-correlation matrix”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.
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