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
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{\displaystyle \mathbf {X} =(X_{1},\ldots ,X_{m})^{\rm {T}}}
Y
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T
{\displaystyle \mathbf {Y} =(Y_{1},\ldots ,Y_{n})^{\rm {T}}}
random elements whose expected value and variance exist, the cross-correlation matrix of
X
{\displaystyle \mathbf {X} }
Y
{\displaystyle \mathbf {Y} }
[ 1] : p.337
R
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Y
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[
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Y
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{\displaystyle \operatorname {R} _{\mathbf {X} \mathbf {Y} }\triangleq \ \operatorname {E} [\mathbf {X} \mathbf {Y} ^{\rm {T}}]}
and has dimensions
m
×
n
{\displaystyle m\times n}
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{\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}}}
The random vectors
X
{\displaystyle \mathbf {X} }
Y
{\displaystyle \mathbf {Y} }
Example
For example, if
X
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,
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2
,
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3
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{\displaystyle \mathbf {X} =\left(X_{1},X_{2},X_{3}\right)^{\rm {T}}}
Y
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{\displaystyle \mathbf {Y} =\left(Y_{1},Y_{2}\right)^{\rm {T}}}
R
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{\displaystyle \operatorname {R} _{\mathbf {X} \mathbf {Y} }}
3
×
2
{\displaystyle 3\times 2}
(
i
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j
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{\displaystyle (i,j)}
E
[
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{\displaystyle \operatorname {E} [X_{i}Y_{j}]}
Complex random vectors
If
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m
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T
{\displaystyle \mathbf {Z} =(Z_{1},\ldots ,Z_{m})^{\rm {T}}}
W
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W
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,
…
,
W
n
)
T
{\displaystyle \mathbf {W} =(W_{1},\ldots ,W_{n})^{\rm {T}}}
complex random vectors , each containing random variables whose expected value and variance exist, the cross-correlation matrix of
Z
{\displaystyle \mathbf {Z} }
W
{\displaystyle \mathbf {W} }
R
Z
W
≜
E
[
Z
W
H
]
{\displaystyle \operatorname {R} _{\mathbf {Z} \mathbf {W} }\triangleq \ \operatorname {E} [\mathbf {Z} \mathbf {W} ^{\rm {H}}]}
where
H
{\displaystyle {}^{\rm {H}}}
Hermitian transposition .
Two random vectors
X
=
(
X
1
,
…
,
X
m
)
T
{\displaystyle \mathbf {X} =(X_{1},\ldots ,X_{m})^{\rm {T}}}
Y
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Y
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,
…
,
Y
n
)
T
{\displaystyle \mathbf {Y} =(Y_{1},\ldots ,Y_{n})^{\rm {T}}}
uncorrelated if
E
[
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Y
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=
E
[
X
]
E
[
Y
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T
.
{\displaystyle \operatorname {E} [\mathbf {X} \mathbf {Y} ^{\rm {T}}]=\operatorname {E} [\mathbf {X} ]\operatorname {E} [\mathbf {Y} ]^{\rm {T}}.}
They are uncorrelated if and only if their cross-covariance matrix
K
X
Y
{\displaystyle \operatorname {K} _{\mathbf {X} \mathbf {Y} }}
In the case of two complex random vectors
Z
{\displaystyle \mathbf {Z} }
W
{\displaystyle \mathbf {W} }
E
[
Z
W
H
]
=
E
[
Z
]
E
[
W
]
H
{\displaystyle \operatorname {E} [\mathbf {Z} \mathbf {W} ^{\rm {H}}]=\operatorname {E} [\mathbf {Z} ]\operatorname {E} [\mathbf {W} ]^{\rm {H}}}
and
E
[
Z
W
T
]
=
E
[
Z
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E
[
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T
.
{\displaystyle \operatorname {E} [\mathbf {Z} \mathbf {W} ^{\rm {T}}]=\operatorname {E} [\mathbf {Z} ]\operatorname {E} [\mathbf {W} ]^{\rm {T}}.}
Properties
Relation to the cross-covariance matrix
The cross-correlation is related to the cross-covariance matrix as follows:
K
X
Y
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E
[
(
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−
E
[
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(
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−
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[
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=
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{\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}}}
Respectively for complex random vectors:
K
Z
W
=
E
[
(
Z
−
E
[
Z
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)
(
W
−
E
[
W
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)
H
]
=
R
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−
E
[
Z
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E
[
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H
{\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}}}
See also
References
^ Gubner, John A. (2006). Probability and Random Processes for Electrical and Computer Engineers . Cambridge University Press. ISBN 978-0-521-86470-1
Further reading
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}}]}](/media/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}}}](/media/api/rest_v1/media/math/render/svg/dfaf0f3923eafd144f762732bbaa951102ed00bb)
![{\displaystyle \operatorname {E} [X_{i}Y_{j}]}](/media/api/rest_v1/media/math/render/svg/e511d6ea021c2d4b0579f621faec897ec52a1999)
![{\displaystyle \operatorname {R} _{\mathbf {Z} \mathbf {W} }\triangleq \ \operatorname {E} [\mathbf {Z} \mathbf {W} ^{\rm {H}}]}](/media/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}}.}](/media/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}}}](/media/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}}.}](/media/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}}}](/media/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}}}](/media/api/rest_v1/media/math/render/svg/e3b852cac5898146e76309d2f9d00ac163d1d65a)