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.
For two random vectors 
and 
, each containing random elements whose expected value and variance exist, the cross-correlation matrix of 
and 
is defined by[1]: p.337
![{\displaystyle \operatorname {R} _{\mathbf {X} \mathbf {Y} }\triangleq \ \operatorname {E} [\mathbf {X} \mathbf {Y} ^{\rm {T}}]}](//wikimedia.org/api/rest_v1/media/math/render/svg/1337f507d7cb6918d742ae2f029e0554013f7a58)
and has dimensions 
. Written component-wise:
![{\displaystyle \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}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/dfaf0f3923eafd144f762732bbaa951102ed00bb)
The random vectors and need not have the same dimension, and either might be a scalar value.
For example, if 
and 
are random vectors, then
is a 
matrix whose 
-th entry is ![{\displaystyle \operatorname {E} [X_{i}Y_{j}]}](//wikimedia.org/api/rest_v1/media/math/render/svg/e511d6ea021c2d4b0579f621faec897ec52a1999)
.
If 
and 
are complex random vectors, each containing random variables whose expected value and variance exist, the cross-correlation matrix of 
and 
is defined by
![{\displaystyle \operatorname {R} _{\mathbf {Z} \mathbf {W} }\triangleq \ \operatorname {E} [\mathbf {Z} \mathbf {W} ^{\rm {H}}]}](//wikimedia.org/api/rest_v1/media/math/render/svg/b96a25804b6f71adb2dd4366a18cc65e96553d88)
where 
denotes Hermitian transposition.
Relation to the cross-covariance matrix
The cross-correlation is related to the cross-covariance matrix as follows:
![{\displaystyle \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}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/1d58fc3b03892e013a545da3322c2d4942d7314e)
- Respectively for complex random vectors:
![{\displaystyle \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}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/e3b852cac5898146e76309d2f9d00ac163d1d65a)
Gubner, John A. (2006). Probability and Random Processes for Electrical and Computer Engineers. Cambridge University Press. ISBN 978-0-521-86470-1.
- Hayes, Monson H., Statistical Digital Signal Processing and Modeling, John Wiley & Sons, Inc., 1996. ISBN 0-471-59431-8.
- Solomon W. Golomb, and Guang Gong. Signal design for good correlation: for wireless communication, cryptography, and radar. Cambridge University Press, 2005.
- M. Soltanalian. Signal Design for Active Sensing and Communications. Uppsala Dissertations from the Faculty of Science and Technology (printed by Elanders Sverige AB), 2014.
![{\displaystyle \operatorname {R} _{\mathbf {X} \mathbf {Y} }\triangleq \ \operatorname {E} [\mathbf {X} \mathbf {Y} ^{\rm {T}}]}](http://wikimedia.org/api/rest_v1/media/math/render/svg/1337f507d7cb6918d742ae2f029e0554013f7a58)
![{\displaystyle \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}}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/dfaf0f3923eafd144f762732bbaa951102ed00bb)
![{\displaystyle \operatorname {E} [X_{i}Y_{j}]}](http://wikimedia.org/api/rest_v1/media/math/render/svg/e511d6ea021c2d4b0579f621faec897ec52a1999)
![{\displaystyle \operatorname {R} _{\mathbf {Z} \mathbf {W} }\triangleq \ \operatorname {E} [\mathbf {Z} \mathbf {W} ^{\rm {H}}]}](http://wikimedia.org/api/rest_v1/media/math/render/svg/b96a25804b6f71adb2dd4366a18cc65e96553d88)
![{\displaystyle \operatorname {E} [\mathbf {X} \mathbf {Y} ^{\rm {T}}]=\operatorname {E} [\mathbf {X} ]\operatorname {E} [\mathbf {Y} ]^{\rm {T}}.}](http://wikimedia.org/api/rest_v1/media/math/render/svg/1a8bb45997c59ac08d38c6073ad03c832d11abf6)
![{\displaystyle \operatorname {E} [\mathbf {Z} \mathbf {W} ^{\rm {H}}]=\operatorname {E} [\mathbf {Z} ]\operatorname {E} [\mathbf {W} ]^{\rm {H}}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/e653a5949cb26600c65a6d23c16310eae1e9867f)
![{\displaystyle \operatorname {E} [\mathbf {Z} \mathbf {W} ^{\rm {T}}]=\operatorname {E} [\mathbf {Z} ]\operatorname {E} [\mathbf {W} ]^{\rm {T}}.}](http://wikimedia.org/api/rest_v1/media/math/render/svg/500e14ec28154556c9b0aa741d1b036af7b7a67f)
![{\displaystyle \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}}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/1d58fc3b03892e013a545da3322c2d4942d7314e)
![{\displaystyle \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}}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/e3b852cac5898146e76309d2f9d00ac163d1d65a)