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})^{\rm {T}}}
and
Y
=
(
Y
1
,
…
,
Y
n
)
T
{\displaystyle \mathbf {Y} =(Y_{1},\ldots ,Y_{n})^{\rm {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[1] : p.337
R
X
Y
≜
E
[
X
Y
T
]
{\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}
. 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}\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} }
and
Y
{\displaystyle \mathbf {Y} }
need not have the same dimension, and either might be a scalar value.
Example
For example, if
X
=
(
X
1
,
X
2
,
X
3
)
T
{\displaystyle \mathbf {X} =\left(X_{1},X_{2},X_{3}\right)^{\rm {T}}}
and
Y
=
(
Y
1
,
Y
2
)
T
{\displaystyle \mathbf {Y} =\left(Y_{1},Y_{2}\right)^{\rm {T}}}
are random vectors, then
R
X
Y
{\displaystyle \operatorname {R} _{\mathbf {X} \mathbf {Y} }}
is a
3
×
2
{\displaystyle 3\times 2}
matrix whose
(
i
,
j
)
{\displaystyle (i,j)}
-th entry is
E
[
X
i
Y
j
]
{\displaystyle \operatorname {E} [X_{i}Y_{j}]}
.
Complex random vectors
If
Z
=
(
Z
1
,
…
,
Z
m
)
T
{\displaystyle \mathbf {Z} =(Z_{1},\ldots ,Z_{m})^{\rm {T}}}
and
W
=
(
W
1
,
…
,
W
n
)
T
{\displaystyle \mathbf {W} =(W_{1},\ldots ,W_{n})^{\rm {T}}}
are complex random vectors , each containing random variables whose expected value and variance exist, the cross-correlation matrix of
Z
{\displaystyle \mathbf {Z} }
and
W
{\displaystyle \mathbf {W} }
is defined by
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}}}
denotes Hermitian transposition .
Uncorrelatedness
Two random vectors
X
=
(
X
1
,
…
,
X
m
)
T
{\displaystyle \mathbf {X} =(X_{1},\ldots ,X_{m})^{\rm {T}}}
and
Y
=
(
Y
1
,
…
,
Y
n
)
T
{\displaystyle \mathbf {Y} =(Y_{1},\ldots ,Y_{n})^{\rm {T}}}
are called uncorrelated if
E
[
X
Y
T
]
=
E
[
X
]
E
[
Y
]
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} }}
matrix is zero.
In the case of two complex random vectors
Z
{\displaystyle \mathbf {Z} }
and
W
{\displaystyle \mathbf {W} }
they are called uncorrelated if
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
]
E
[
W
]
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
=
E
[
(
X
−
E
[
X
]
)
(
Y
−
E
[
Y
]
)
T
]
=
R
X
Y
−
E
[
X
]
E
[
Y
]
T
{\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
]
)
(
W
−
E
[
W
]
)
H
]
=
R
Z
W
−
E
[
Z
]
E
[
W
]
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.