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F r o m W i k i p e d i a , t h e f r e e e n c y c l o p e d i a
( R e d i r e c t e d f r o m C o r r e l a t i o n f u n c t i o n s )
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 [ edit ]
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 [ edit ]
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 [ edit ]
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 [ edit ]
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 [ edit ]
Relation to the cross-covariance matrix [ edit ]
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 [ edit ]
References [ edit ]
^ Gubner, John A. (2006). Probability and Random Processes for Electrical and Computer Engineers . Cambridge University Press. ISBN 978-0-521-86470-1 .
Further reading [ edit ]
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.
R e t r i e v e d f r o m " https://en.wikipedia.org/w/index.php?title=Cross-correlation_matrix&oldid=1084900339 "
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● T h i s p a g e w a s l a s t e d i t e d o n 2 7 A p r i l 2 0 2 2 , a t 0 7 : 0 5 ( U T C ) .
● T e x t i s a v a i l a b l e u n d e r t h e C r e a t i v e C o m m o n s A t t r i b u t i o n - S h a r e A l i k e L i c e n s e 4 . 0 ;
a d d i t i o n a l t e r m s m a y a p p l y . B y u s i n g t h i s s i t e , y o u a g r e e t o t h e T e r m s o f U s e a n d P r i v a c y P o l i c y . W i k i p e d i a ® i s a r e g i s t e r e d t r a d e m a r k o f t h e W i k i m e d i a F o u n d a t i o n , I n c . , a n o n - p r o f i t o r g a n i z a t i o n .
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