In probability theory and statistics , a complex random vector is typically a tuple of complex -valued random variables , and generally is a random variable taking values in a vector space over the field of complex numbers. If
Z
1
,
…
,
Z
n
{\displaystyle Z_{1},\ldots ,Z_{n}}
n -tuple
(
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1
,
…
,
Z
n
)
{\displaystyle \left(Z_{1},\ldots ,Z_{n}\right)}
Some concepts of real random vectors have a straightforward generalization to complex random vectors. For example, the definition of the mean of a complex random vector. Other concepts are unique to complex random vectors.
Applications of complex random vectors are found in digital signal processing .
A complex random vector
Z
=
(
Z
1
,
…
,
Z
n
)
T
{\displaystyle \mathbf {Z} =(Z_{1},\ldots ,Z_{n})^{T}}
probability space
(
Ω
,
F
,
P
)
{\displaystyle (\Omega ,{\mathcal {F}},P)}
function
Z
:
Ω
→
C
n
{\displaystyle \mathbf {Z} \colon \Omega \rightarrow \mathbb {C} ^{n}}
(
ℜ
(
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1
)
,
ℑ
(
Z
1
)
,
…
,
ℜ
(
Z
n
)
,
ℑ
(
Z
n
)
)
T
{\displaystyle (\Re {(Z_{1})},\Im {(Z_{1})},\ldots ,\Re {(Z_{n})},\Im {(Z_{n})})^{T}}
real random vector on
(
Ω
,
F
,
P
)
{\displaystyle (\Omega ,{\mathcal {F}},P)}
ℜ
(
z
)
{\displaystyle \Re {(z)}}
z
{\displaystyle z}
ℑ
(
z
)
{\displaystyle \Im {(z)}}
z
{\displaystyle z}
[ 1] : p. 292
Cumulative distribution function [ edit ] The generalization of the cumulative distribution function from real to complex random variables is not obvious because expressions of the form
P
(
Z
≤
1
+
3
i
)
{\displaystyle P(Z\leq 1+3i)}
P
(
ℜ
(
Z
)
≤
1
,
ℑ
(
Z
)
≤
3
)
{\displaystyle P(\Re {(Z)}\leq 1,\Im {(Z)}\leq 3)}
F
Z
:
C
n
↦
[
0
,
1
]
{\displaystyle F_{\mathbf {Z} }:\mathbb {C} ^{n}\mapsto [0,1]}
Z
=
(
Z
1
,
.
.
.
,
Z
n
)
T
{\displaystyle \mathbf {Z} =(Z_{1},...,Z_{n})^{T}}
F
Z
(
z
)
=
P
(
ℜ
(
Z
1
)
≤
ℜ
(
z
1
)
,
ℑ
(
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1
)
≤
ℑ
(
z
1
)
,
…
,
ℜ
(
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n
)
≤
ℜ
(
z
n
)
,
ℑ
(
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n
)
≤
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(
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n
)
)
{\displaystyle F_{\mathbf {Z} }(\mathbf {z} )=\operatorname {P} (\Re {(Z_{1})}\leq \Re {(z_{1})},\Im {(Z_{1})}\leq \Im {(z_{1})},\ldots ,\Re {(Z_{n})}\leq \Re {(z_{n})},\Im {(Z_{n})}\leq \Im {(z_{n})})}
(Eq.1 )
where
z
=
(
z
1
,
.
.
.
,
z
n
)
T
{\displaystyle \mathbf {z} =(z_{1},...,z_{n})^{T}}
As in the real case the expectation (also called expected value ) of a complex random vector is taken component-wise.[ 1] : p. 293
E
[
Z
]
=
(
E
[
Z
1
]
,
…
,
E
[
Z
n
]
)
T
{\displaystyle \operatorname {E} [\mathbf {Z} ]=(\operatorname {E} [Z_{1}],\ldots ,\operatorname {E} [Z_{n}])^{T}}
(Eq.2 )
Covariance matrix and pseudo-covariance matrix [ edit ] The covariance matrix second central moment )
K
Z
Z
{\displaystyle \operatorname {K} _{\mathbf {Z} \mathbf {Z} }}
n
×
1
{\displaystyle n\times 1}
n
×
n
{\displaystyle n\times n}
matrix whose
(
i
,
j
)
{\displaystyle (i,j)}
th element is the covariance between the i th and the j th random variables.[ 2] : p.372 complex conjugate of one of the two. Thus the covariance matrix is a Hermitian matrix .[ 1] : p. 293
K
Z
Z
=
cov
[
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,
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]
=
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[
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−
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[
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)
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[
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)
H
]
=
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[
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−
E
[
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]
E
[
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H
]
{\displaystyle {\begin{aligned}&\operatorname {K} _{\mathbf {Z} \mathbf {Z} }=\operatorname {cov} [\mathbf {Z} ,\mathbf {Z} ]=\operatorname {E} [(\mathbf {Z} -\operatorname {E} [\mathbf {Z} ]){(\mathbf {Z} -\operatorname {E} [\mathbf {Z} ])}^{H}]=\operatorname {E} [\mathbf {Z} \mathbf {Z} ^{H}]-\operatorname {E} [\mathbf {Z} ]\operatorname {E} [\mathbf {Z} ^{H}]\\[12pt]\end{aligned}}}
(Eq.3 )
K
Z
Z
=
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−
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¯
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⋯
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]
{\displaystyle \operatorname {K} _{\mathbf {Z} \mathbf {Z} }={\begin{bmatrix}\mathrm {E} [(Z_{1}-\operatorname {E} [Z_{1}]){\overline {(Z_{1}-\operatorname {E} [Z_{1}])}}]&\mathrm {E} [(Z_{1}-\operatorname {E} [Z_{1}]){\overline {(Z_{2}-\operatorname {E} [Z_{2}])}}]&\cdots &\mathrm {E} [(Z_{1}-\operatorname {E} [Z_{1}]){\overline {(Z_{n}-\operatorname {E} [Z_{n}])}}]\\\\\mathrm {E} [(Z_{2}-\operatorname {E} [Z_{2}]){\overline {(Z_{1}-\operatorname {E} [Z_{1}])}}]&\mathrm {E} [(Z_{2}-\operatorname {E} [Z_{2}]){\overline {(Z_{2}-\operatorname {E} [Z_{2}])}}]&\cdots &\mathrm {E} [(Z_{2}-\operatorname {E} [Z_{2}]){\overline {(Z_{n}-\operatorname {E} [Z_{n}])}}]\\\\\vdots &\vdots &\ddots &\vdots \\\\\mathrm {E} [(Z_{n}-\operatorname {E} [Z_{n}]){\overline {(Z_{1}-\operatorname {E} [Z_{1}])}}]&\mathrm {E} [(Z_{n}-\operatorname {E} [Z_{n}]){\overline {(Z_{2}-\operatorname {E} [Z_{2}])}}]&\cdots &\mathrm {E} [(Z_{n}-\operatorname {E} [Z_{n}]){\overline {(Z_{n}-\operatorname {E} [Z_{n}])}}]\end{bmatrix}}}
The pseudo-covariance matrix relation matrix ) is defined replacing Hermitian transposition by transposition in the definition above.
J
Z
Z
=
cov
[
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,
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¯
]
=
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[
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[
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−
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)
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]
=
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[
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−
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[
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]
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[
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T
]
{\displaystyle \operatorname {J} _{\mathbf {Z} \mathbf {Z} }=\operatorname {cov} [\mathbf {Z} ,{\overline {\mathbf {Z} }}]=\operatorname {E} [(\mathbf {Z} -\operatorname {E} [\mathbf {Z} ]){(\mathbf {Z} -\operatorname {E} [\mathbf {Z} ])}^{T}]=\operatorname {E} [\mathbf {Z} \mathbf {Z} ^{T}]-\operatorname {E} [\mathbf {Z} ]\operatorname {E} [\mathbf {Z} ^{T}]}
(Eq.4 )
J
Z
Z
=
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−
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]
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[
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[
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)
]
⋯
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]
{\displaystyle \operatorname {J} _{\mathbf {Z} \mathbf {Z} }={\begin{bmatrix}\mathrm {E} [(Z_{1}-\operatorname {E} [Z_{1}])(Z_{1}-\operatorname {E} [Z_{1}])]&\mathrm {E} [(Z_{1}-\operatorname {E} [Z_{1}])(Z_{2}-\operatorname {E} [Z_{2}])]&\cdots &\mathrm {E} [(Z_{1}-\operatorname {E} [Z_{1}])(Z_{n}-\operatorname {E} [Z_{n}])]\\\\\mathrm {E} [(Z_{2}-\operatorname {E} [Z_{2}])(Z_{1}-\operatorname {E} [Z_{1}])]&\mathrm {E} [(Z_{2}-\operatorname {E} [Z_{2}])(Z_{2}-\operatorname {E} [Z_{2}])]&\cdots &\mathrm {E} [(Z_{2}-\operatorname {E} [Z_{2}])(Z_{n}-\operatorname {E} [Z_{n}])]\\\\\vdots &\vdots &\ddots &\vdots \\\\\mathrm {E} [(Z_{n}-\operatorname {E} [Z_{n}])(Z_{1}-\operatorname {E} [Z_{1}])]&\mathrm {E} [(Z_{n}-\operatorname {E} [Z_{n}])(Z_{2}-\operatorname {E} [Z_{2}])]&\cdots &\mathrm {E} [(Z_{n}-\operatorname {E} [Z_{n}])(Z_{n}-\operatorname {E} [Z_{n}])]\end{bmatrix}}}
Properties The covariance matrix is a hermitian matrix , i.e.[ 1] : p. 293
K
Z
Z
H
=
K
Z
Z
{\displaystyle \operatorname {K} _{\mathbf {Z} \mathbf {Z} }^{H}=\operatorname {K} _{\mathbf {Z} \mathbf {Z} }}
The pseudo-covariance matrix is a symmetric matrix , i.e.
J
Z
Z
T
=
J
Z
Z
{\displaystyle \operatorname {J} _{\mathbf {Z} \mathbf {Z} }^{T}=\operatorname {J} _{\mathbf {Z} \mathbf {Z} }}
The covariance matrix is a positive semidefinite matrix , i.e.
a
H
K
Z
Z
a
≥
0
for all
a
∈
C
n
{\displaystyle \mathbf {a} ^{H}\operatorname {K} _{\mathbf {Z} \mathbf {Z} }\mathbf {a} \geq 0\quad {\text{for all }}\mathbf {a} \in \mathbb {C} ^{n}}
Covariance matrices of real and imaginary parts [ edit ] By decomposing the random vector
Z
{\displaystyle \mathbf {Z} }
X
=
ℜ
(
Z
)
{\displaystyle \mathbf {X} =\Re {(\mathbf {Z} )}}
Y
=
ℑ
(
Z
)
{\displaystyle \mathbf {Y} =\Im {(\mathbf {Z} )}}
Z
=
X
+
i
Y
{\displaystyle \mathbf {Z} =\mathbf {X} +i\mathbf {Y} }
(
X
,
Y
)
{\displaystyle (\mathbf {X} ,\mathbf {Y} )}
covariance matrix of the form:
[
K
X
X
K
X
Y
K
Y
X
K
Y
Y
]
{\displaystyle {\begin{bmatrix}\operatorname {K} _{\mathbf {X} \mathbf {X} }&\operatorname {K} _{\mathbf {X} \mathbf {Y} }\\\operatorname {K} _{\mathbf {Y} \mathbf {X} }&\operatorname {K} _{\mathbf {Y} \mathbf {Y} }\end{bmatrix}}}
The matrices
K
Z
Z
{\displaystyle \operatorname {K} _{\mathbf {Z} \mathbf {Z} }}
J
Z
Z
{\displaystyle \operatorname {J} _{\mathbf {Z} \mathbf {Z} }}
X
{\displaystyle \mathbf {X} }
Y
{\displaystyle \mathbf {Y} }
K
X
X
=
E
[
(
X
−
E
[
X
]
)
(
X
−
E
[
X
]
)
T
]
=
1
2
Re
(
K
Z
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+
J
Z
Z
)
K
Y
Y
=
E
[
(
Y
−
E
[
Y
]
)
(
Y
−
E
[
Y
]
)
T
]
=
1
2
Re
(
K
Z
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−
J
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Z
)
K
Y
X
=
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[
(
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−
E
[
Y
]
)
(
X
−
E
[
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]
)
T
]
=
1
2
Im
(
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+
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)
K
X
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=
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[
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−
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[
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]
)
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−
E
[
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]
)
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]
=
1
2
Im
(
J
Z
Z
−
K
Z
Z
)
{\displaystyle {\begin{aligned}&\operatorname {K} _{\mathbf {X} \mathbf {X} }=\operatorname {E} [(\mathbf {X} -\operatorname {E} [\mathbf {X} ])(\mathbf {X} -\operatorname {E} [\mathbf {X} ])^{\mathrm {T} }]={\tfrac {1}{2}}\operatorname {Re} (\operatorname {K} _{\mathbf {Z} \mathbf {Z} }+\operatorname {J} _{\mathbf {Z} \mathbf {Z} })\\&\operatorname {K} _{\mathbf {Y} \mathbf {Y} }=\operatorname {E} [(\mathbf {Y} -\operatorname {E} [\mathbf {Y} ])(\mathbf {Y} -\operatorname {E} [\mathbf {Y} ])^{\mathrm {T} }]={\tfrac {1}{2}}\operatorname {Re} (\operatorname {K} _{\mathbf {Z} \mathbf {Z} }-\operatorname {J} _{\mathbf {Z} \mathbf {Z} })\\&\operatorname {K} _{\mathbf {Y} \mathbf {X} }=\operatorname {E} [(\mathbf {Y} -\operatorname {E} [\mathbf {Y} ])(\mathbf {X} -\operatorname {E} [\mathbf {X} ])^{\mathrm {T} }]={\tfrac {1}{2}}\operatorname {Im} (\operatorname {J} _{\mathbf {Z} \mathbf {Z} }+\operatorname {K} _{\mathbf {Z} \mathbf {Z} })\\&\operatorname {K} _{\mathbf {X} \mathbf {Y} }=\operatorname {E} [(\mathbf {X} -\operatorname {E} [\mathbf {X} ])(\mathbf {Y} -\operatorname {E} [\mathbf {Y} ])^{\mathrm {T} }]={\tfrac {1}{2}}\operatorname {Im} (\operatorname {J} _{\mathbf {Z} \mathbf {Z} }-\operatorname {K} _{\mathbf {Z} \mathbf {Z} })\\\end{aligned}}}
Conversely:
K
Z
Z
=
K
X
X
+
K
Y
Y
+
i
(
K
Y
X
−
K
X
Y
)
J
Z
Z
=
K
X
X
−
K
Y
Y
+
i
(
K
Y
X
+
K
X
Y
)
{\displaystyle {\begin{aligned}&\operatorname {K} _{\mathbf {Z} \mathbf {Z} }=\operatorname {K} _{\mathbf {X} \mathbf {X} }+\operatorname {K} _{\mathbf {Y} \mathbf {Y} }+i(\operatorname {K} _{\mathbf {Y} \mathbf {X} }-\operatorname {K} _{\mathbf {X} \mathbf {Y} })\\&\operatorname {J} _{\mathbf {Z} \mathbf {Z} }=\operatorname {K} _{\mathbf {X} \mathbf {X} }-\operatorname {K} _{\mathbf {Y} \mathbf {Y} }+i(\operatorname {K} _{\mathbf {Y} \mathbf {X} }+\operatorname {K} _{\mathbf {X} \mathbf {Y} })\end{aligned}}}
Cross-covariance matrix and pseudo-cross-covariance matrix [ edit ] The cross-covariance matrix between two complex random vectors
Z
,
W
{\displaystyle \mathbf {Z} ,\mathbf {W} }
K
Z
W
=
cov
[
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,
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]
=
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[
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[
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]
)
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−
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[
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]
)
H
]
=
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[
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]
−
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[
Z
]
E
[
W
H
]
{\displaystyle \operatorname {K} _{\mathbf {Z} \mathbf {W} }=\operatorname {cov} [\mathbf {Z} ,\mathbf {W} ]=\operatorname {E} [(\mathbf {Z} -\operatorname {E} [\mathbf {Z} ]){(\mathbf {W} -\operatorname {E} [\mathbf {W} ])}^{H}]=\operatorname {E} [\mathbf {Z} \mathbf {W} ^{H}]-\operatorname {E} [\mathbf {Z} ]\operatorname {E} [\mathbf {W} ^{H}]}
(Eq.5 )
K
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W
=
[
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[
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−
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[
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1
]
)
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]
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]
⋯
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]
{\displaystyle \operatorname {K} _{\mathbf {Z} \mathbf {W} }={\begin{bmatrix}\mathrm {E} [(Z_{1}-\operatorname {E} [Z_{1}]){\overline {(W_{1}-\operatorname {E} [W_{1}])}}]&\mathrm {E} [(Z_{1}-\operatorname {E} [Z_{1}]){\overline {(W_{2}-\operatorname {E} [W_{2}])}}]&\cdots &\mathrm {E} [(Z_{1}-\operatorname {E} [Z_{1}]){\overline {(W_{n}-\operatorname {E} [W_{n}])}}]\\\\\mathrm {E} [(Z_{2}-\operatorname {E} [Z_{2}]){\overline {(W_{1}-\operatorname {E} [W_{1}])}}]&\mathrm {E} [(Z_{2}-\operatorname {E} [Z_{2}]){\overline {(W_{2}-\operatorname {E} [W_{2}])}}]&\cdots &\mathrm {E} [(Z_{2}-\operatorname {E} [Z_{2}]){\overline {(W_{n}-\operatorname {E} [W_{n}])}}]\\\\\vdots &\vdots &\ddots &\vdots \\\\\mathrm {E} [(Z_{n}-\operatorname {E} [Z_{n}]){\overline {(W_{1}-\operatorname {E} [W_{1}])}}]&\mathrm {E} [(Z_{n}-\operatorname {E} [Z_{n}]){\overline {(W_{2}-\operatorname {E} [W_{2}])}}]&\cdots &\mathrm {E} [(Z_{n}-\operatorname {E} [Z_{n}]){\overline {(W_{n}-\operatorname {E} [W_{n}])}}]\end{bmatrix}}}
And the pseudo-cross-covariance matrix is defined as:
J
Z
W
=
cov
[
Z
,
W
¯
]
=
E
[
(
Z
−
E
[
Z
]
)
(
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−
E
[
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]
)
T
]
=
E
[
Z
W
T
]
−
E
[
Z
]
E
[
W
T
]
{\displaystyle \operatorname {J} _{\mathbf {Z} \mathbf {W} }=\operatorname {cov} [\mathbf {Z} ,{\overline {\mathbf {W} }}]=\operatorname {E} [(\mathbf {Z} -\operatorname {E} [\mathbf {Z} ]){(\mathbf {W} -\operatorname {E} [\mathbf {W} ])}^{T}]=\operatorname {E} [\mathbf {Z} \mathbf {W} ^{T}]-\operatorname {E} [\mathbf {Z} ]\operatorname {E} [\mathbf {W} ^{T}]}
(Eq.6 )
J
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W
=
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[
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−
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[
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1
]
)
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)
]
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)
]
⋮
⋮
⋱
⋮
E
[
(
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n
−
E
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)
(
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1
−
E
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E
[
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⋯
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[
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{\displaystyle \operatorname {J} _{\mathbf {Z} \mathbf {W} }={\begin{bmatrix}\mathrm {E} [(Z_{1}-\operatorname {E} [Z_{1}])(W_{1}-\operatorname {E} [W_{1}])]&\mathrm {E} [(Z_{1}-\operatorname {E} [Z_{1}])(W_{2}-\operatorname {E} [W_{2}])]&\cdots &\mathrm {E} [(Z_{1}-\operatorname {E} [Z_{1}])(W_{n}-\operatorname {E} [W_{n}])]\\\\\mathrm {E} [(Z_{2}-\operatorname {E} [Z_{2}])(W_{1}-\operatorname {E} [W_{1}])]&\mathrm {E} [(Z_{2}-\operatorname {E} [Z_{2}])(W_{2}-\operatorname {E} [W_{2}])]&\cdots &\mathrm {E} [(Z_{2}-\operatorname {E} [Z_{2}])(W_{n}-\operatorname {E} [W_{n}])]\\\\\vdots &\vdots &\ddots &\vdots \\\\\mathrm {E} [(Z_{n}-\operatorname {E} [Z_{n}])(W_{1}-\operatorname {E} [W_{1}])]&\mathrm {E} [(Z_{n}-\operatorname {E} [Z_{n}])(W_{2}-\operatorname {E} [W_{2}])]&\cdots &\mathrm {E} [(Z_{n}-\operatorname {E} [Z_{n}])(W_{n}-\operatorname {E} [W_{n}])]\end{bmatrix}}}
Two complex random vectors
Z
{\displaystyle \mathbf {Z} }
W
{\displaystyle \mathbf {W} }
uncorrelated if
K
Z
W
=
J
Z
W
=
0
{\displaystyle \operatorname {K} _{\mathbf {Z} \mathbf {W} }=\operatorname {J} _{\mathbf {Z} \mathbf {W} }=0}
Two complex random vectors
Z
=
(
Z
1
,
.
.
.
,
Z
m
)
T
{\displaystyle \mathbf {Z} =(Z_{1},...,Z_{m})^{T}}
W
=
(
W
1
,
.
.
.
,
W
n
)
T
{\displaystyle \mathbf {W} =(W_{1},...,W_{n})^{T}}
independent if
F
Z
,
W
(
z
,
w
)
=
F
Z
(
z
)
⋅
F
W
(
w
)
for all
z
,
w
{\displaystyle F_{\mathbf {Z,W} }(\mathbf {z,w} )=F_{\mathbf {Z} }(\mathbf {z} )\cdot F_{\mathbf {W} }(\mathbf {w} )\quad {\text{for all }}\mathbf {z} ,\mathbf {w} }
(Eq.7 )
where
F
Z
(
z
)
{\displaystyle F_{\mathbf {Z} }(\mathbf {z} )}
F
W
(
w
)
{\displaystyle F_{\mathbf {W} }(\mathbf {w} )}
Z
{\displaystyle \mathbf {Z} }
W
{\displaystyle \mathbf {W} }
Eq.1
F
Z
,
W
(
z
,
w
)
{\displaystyle F_{\mathbf {Z,W} }(\mathbf {z,w} )}
Z
{\displaystyle \mathbf {Z} }
W
{\displaystyle \mathbf {W} }
Z
⊥
⊥
W
{\displaystyle \mathbf {Z} \perp \!\!\!\perp \mathbf {W} }
Z
{\displaystyle \mathbf {Z} }
W
{\displaystyle \mathbf {W} }
F
Z
1
,
…
,
Z
m
,
W
1
,
…
,
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n
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{\displaystyle F_{Z_{1},\ldots ,Z_{m},W_{1},\ldots ,W_{n}}(z_{1},\ldots ,z_{m},w_{1},\ldots ,w_{n})=F_{Z_{1},\ldots ,Z_{m}}(z_{1},\ldots ,z_{m})\cdot F_{W_{1},\ldots ,W_{n}}(w_{1},\ldots ,w_{n})\quad {\text{for all }}z_{1},\ldots ,z_{m},w_{1},\ldots ,w_{n}}
A complex random vector
Z
{\displaystyle \mathbf {Z} }
φ
∈
[
−
π
,
π
)
{\displaystyle \varphi \in [-\pi ,\pi )}
e
i
φ
Z
{\displaystyle e^{\mathrm {i} \varphi }\mathbf {Z} }
Z
{\displaystyle \mathbf {Z} }
[ 3] : pp. 500–501
Properties The expectation of a circularly symmetric complex random vector is either zero or it is not defined.[ 3] : p. 500
The pseudo-covariance matrix of a circularly symmetric complex random vector is zero.[ 3] : p. 584 Proper complex random vectors [ edit ] A complex random vector
Z
{\displaystyle \mathbf {Z} }
proper if the following three conditions are all satisfied:[ 1] : p. 293
E
[
Z
]
=
0
{\displaystyle \operatorname {E} [\mathbf {Z} ]=0}
var
[
Z
1
]
<
∞
,
…
,
var
[
Z
n
]
<
∞
{\displaystyle \operatorname {var} [Z_{1}]<\infty ,\ldots ,\operatorname {var} [Z_{n}]<\infty }
E
[
Z
Z
T
]
=
0
{\displaystyle \operatorname {E} [\mathbf {Z} \mathbf {Z} ^{T}]=0}
Two complex random vectors
Z
,
W
{\displaystyle \mathbf {Z} ,\mathbf {W} }
jointly proper is the composite random vector
(
Z
1
,
Z
2
,
…
,
Z
m
,
W
1
,
W
2
,
…
,
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n
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T
{\displaystyle (Z_{1},Z_{2},\ldots ,Z_{m},W_{1},W_{2},\ldots ,W_{n})^{T}}
Properties A complex random vector
Z
{\displaystyle \mathbf {Z} }
c
∈
C
n
{\displaystyle \mathbf {c} \in \mathbb {C} ^{n}}
c
T
Z
{\displaystyle \mathbf {c} ^{T}\mathbf {Z} }
[ 1] : p. 293
Linear transformations of proper complex random vectors are proper, i.e. if
Z
{\displaystyle \mathbf {Z} }
n
{\displaystyle n}
A
{\displaystyle A}
m
×
n
{\displaystyle m\times n}
A
Z
{\displaystyle A\mathbf {Z} }
[ 1] : p. 295
Every circularly symmetric complex random vector with finite variance of all its components is proper.[ 1] : p. 295
There are proper complex random vectors that are not circularly symmetric.[ 1] : p. 504
A real random vector is proper if and only if it is constant.
Two jointly proper complex random vectors are uncorrelated if and only if their covariance matrix is zero, i.e. if
K
Z
W
=
0
{\displaystyle \operatorname {K} _{\mathbf {Z} \mathbf {W} }=0}
Cauchy-Schwarz inequality [ edit ] The Cauchy-Schwarz inequality for complex random vectors is
|
E
[
Z
H
W
]
|
2
≤
E
[
Z
H
Z
]
E
[
|
W
H
W
|
]
{\displaystyle \left|\operatorname {E} [\mathbf {Z} ^{H}\mathbf {W} ]\right|^{2}\leq \operatorname {E} [\mathbf {Z} ^{H}\mathbf {Z} ]\operatorname {E} [|\mathbf {W} ^{H}\mathbf {W} |]}
Characteristic function [ edit ] The characteristic function of a complex random vector
Z
{\displaystyle \mathbf {Z} }
n
{\displaystyle n}
C
n
→
C
{\displaystyle \mathbb {C} ^{n}\to \mathbb {C} }
[ 1] : p. 295
φ
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[
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i
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+
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⋯
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]
{\displaystyle \varphi _{\mathbf {Z} }(\mathbf {\omega } )=\operatorname {E} \left[e^{i\Re {(\mathbf {\omega } ^{H}\mathbf {Z} )}}\right]=\operatorname {E} \left[e^{i(\Re {(\omega _{1})}\Re {(Z_{1})}+\Im {(\omega _{1})}\Im {(Z_{1})}+\cdots +\Re {(\omega _{n})}\Re {(Z_{n})}+\Im {(\omega _{n})}\Im {(Z_{n})})}\right]}
^ a b c d e f g h i j Lapidoth, Amos (2009). A Foundation in Digital Communication . Cambridge University Press. ISBN 978-0-521-19395-5 ^ Gubner, John A. (2006). Probability and Random Processes for Electrical and Computer Engineers . Cambridge University Press. ISBN 978-0-521-86470-1 ^ a b c Tse, David (2005). Fundamentals of Wireless Communication . Cambridge University Press.
![{\displaystyle F_{\mathbf {Z} }:\mathbb {C} ^{n}\mapsto [0,1]}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/8661e6d29a4d779324e5c3978506838f8c244b65)
![{\displaystyle \operatorname {E} [\mathbf {Z} ]=(\operatorname {E} [Z_{1}],\ldots ,\operatorname {E} [Z_{n}])^{T}}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/9bad8687a134b34189596765316018f79fdca495)
![{\displaystyle {\begin{aligned}&\operatorname {K} _{\mathbf {Z} \mathbf {Z} }=\operatorname {cov} [\mathbf {Z} ,\mathbf {Z} ]=\operatorname {E} [(\mathbf {Z} -\operatorname {E} [\mathbf {Z} ]){(\mathbf {Z} -\operatorname {E} [\mathbf {Z} ])}^{H}]=\operatorname {E} [\mathbf {Z} \mathbf {Z} ^{H}]-\operatorname {E} [\mathbf {Z} ]\operatorname {E} [\mathbf {Z} ^{H}]\\[12pt]\end{aligned}}}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/ebd5472da7a2129480754c26bf1527c36e0e1aef)
![{\displaystyle \operatorname {K} _{\mathbf {Z} \mathbf {Z} }={\begin{bmatrix}\mathrm {E} [(Z_{1}-\operatorname {E} [Z_{1}]){\overline {(Z_{1}-\operatorname {E} [Z_{1}])}}]&\mathrm {E} [(Z_{1}-\operatorname {E} [Z_{1}]){\overline {(Z_{2}-\operatorname {E} [Z_{2}])}}]&\cdots &\mathrm {E} [(Z_{1}-\operatorname {E} [Z_{1}]){\overline {(Z_{n}-\operatorname {E} [Z_{n}])}}]\\\\\mathrm {E} [(Z_{2}-\operatorname {E} [Z_{2}]){\overline {(Z_{1}-\operatorname {E} [Z_{1}])}}]&\mathrm {E} [(Z_{2}-\operatorname {E} [Z_{2}]){\overline {(Z_{2}-\operatorname {E} [Z_{2}])}}]&\cdots &\mathrm {E} [(Z_{2}-\operatorname {E} [Z_{2}]){\overline {(Z_{n}-\operatorname {E} [Z_{n}])}}]\\\\\vdots &\vdots &\ddots &\vdots \\\\\mathrm {E} [(Z_{n}-\operatorname {E} [Z_{n}]){\overline {(Z_{1}-\operatorname {E} [Z_{1}])}}]&\mathrm {E} [(Z_{n}-\operatorname {E} [Z_{n}]){\overline {(Z_{2}-\operatorname {E} [Z_{2}])}}]&\cdots &\mathrm {E} [(Z_{n}-\operatorname {E} [Z_{n}]){\overline {(Z_{n}-\operatorname {E} [Z_{n}])}}]\end{bmatrix}}}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/445eafa0ab8d1d761c1c34d3bb993b4598ec4b15)
![{\displaystyle \operatorname {J} _{\mathbf {Z} \mathbf {Z} }=\operatorname {cov} [\mathbf {Z} ,{\overline {\mathbf {Z} }}]=\operatorname {E} [(\mathbf {Z} -\operatorname {E} [\mathbf {Z} ]){(\mathbf {Z} -\operatorname {E} [\mathbf {Z} ])}^{T}]=\operatorname {E} [\mathbf {Z} \mathbf {Z} ^{T}]-\operatorname {E} [\mathbf {Z} ]\operatorname {E} [\mathbf {Z} ^{T}]}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/073e290548d019170600567c2a29b7feaaf98a51)
![{\displaystyle \operatorname {J} _{\mathbf {Z} \mathbf {Z} }={\begin{bmatrix}\mathrm {E} [(Z_{1}-\operatorname {E} [Z_{1}])(Z_{1}-\operatorname {E} [Z_{1}])]&\mathrm {E} [(Z_{1}-\operatorname {E} [Z_{1}])(Z_{2}-\operatorname {E} [Z_{2}])]&\cdots &\mathrm {E} [(Z_{1}-\operatorname {E} [Z_{1}])(Z_{n}-\operatorname {E} [Z_{n}])]\\\\\mathrm {E} [(Z_{2}-\operatorname {E} [Z_{2}])(Z_{1}-\operatorname {E} [Z_{1}])]&\mathrm {E} [(Z_{2}-\operatorname {E} [Z_{2}])(Z_{2}-\operatorname {E} [Z_{2}])]&\cdots &\mathrm {E} [(Z_{2}-\operatorname {E} [Z_{2}])(Z_{n}-\operatorname {E} [Z_{n}])]\\\\\vdots &\vdots &\ddots &\vdots \\\\\mathrm {E} [(Z_{n}-\operatorname {E} [Z_{n}])(Z_{1}-\operatorname {E} [Z_{1}])]&\mathrm {E} [(Z_{n}-\operatorname {E} [Z_{n}])(Z_{2}-\operatorname {E} [Z_{2}])]&\cdots &\mathrm {E} [(Z_{n}-\operatorname {E} [Z_{n}])(Z_{n}-\operatorname {E} [Z_{n}])]\end{bmatrix}}}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/50248ec68758163e693b5fa474537f0d953110ee)
![{\displaystyle {\begin{aligned}&\operatorname {K} _{\mathbf {X} \mathbf {X} }=\operatorname {E} [(\mathbf {X} -\operatorname {E} [\mathbf {X} ])(\mathbf {X} -\operatorname {E} [\mathbf {X} ])^{\mathrm {T} }]={\tfrac {1}{2}}\operatorname {Re} (\operatorname {K} _{\mathbf {Z} \mathbf {Z} }+\operatorname {J} _{\mathbf {Z} \mathbf {Z} })\\&\operatorname {K} _{\mathbf {Y} \mathbf {Y} }=\operatorname {E} [(\mathbf {Y} -\operatorname {E} [\mathbf {Y} ])(\mathbf {Y} -\operatorname {E} [\mathbf {Y} ])^{\mathrm {T} }]={\tfrac {1}{2}}\operatorname {Re} (\operatorname {K} _{\mathbf {Z} \mathbf {Z} }-\operatorname {J} _{\mathbf {Z} \mathbf {Z} })\\&\operatorname {K} _{\mathbf {Y} \mathbf {X} }=\operatorname {E} [(\mathbf {Y} -\operatorname {E} [\mathbf {Y} ])(\mathbf {X} -\operatorname {E} [\mathbf {X} ])^{\mathrm {T} }]={\tfrac {1}{2}}\operatorname {Im} (\operatorname {J} _{\mathbf {Z} \mathbf {Z} }+\operatorname {K} _{\mathbf {Z} \mathbf {Z} })\\&\operatorname {K} _{\mathbf {X} \mathbf {Y} }=\operatorname {E} [(\mathbf {X} -\operatorname {E} [\mathbf {X} ])(\mathbf {Y} -\operatorname {E} [\mathbf {Y} ])^{\mathrm {T} }]={\tfrac {1}{2}}\operatorname {Im} (\operatorname {J} _{\mathbf {Z} \mathbf {Z} }-\operatorname {K} _{\mathbf {Z} \mathbf {Z} })\\\end{aligned}}}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/02b2d81c9eb40be718176122b197e7384c6ca81c)
![{\displaystyle \operatorname {K} _{\mathbf {Z} \mathbf {W} }=\operatorname {cov} [\mathbf {Z} ,\mathbf {W} ]=\operatorname {E} [(\mathbf {Z} -\operatorname {E} [\mathbf {Z} ]){(\mathbf {W} -\operatorname {E} [\mathbf {W} ])}^{H}]=\operatorname {E} [\mathbf {Z} \mathbf {W} ^{H}]-\operatorname {E} [\mathbf {Z} ]\operatorname {E} [\mathbf {W} ^{H}]}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/9844313bcca0b6e7e67e2c61bae3559a8714b4e4)
![{\displaystyle \operatorname {K} _{\mathbf {Z} \mathbf {W} }={\begin{bmatrix}\mathrm {E} [(Z_{1}-\operatorname {E} [Z_{1}]){\overline {(W_{1}-\operatorname {E} [W_{1}])}}]&\mathrm {E} [(Z_{1}-\operatorname {E} [Z_{1}]){\overline {(W_{2}-\operatorname {E} [W_{2}])}}]&\cdots &\mathrm {E} [(Z_{1}-\operatorname {E} [Z_{1}]){\overline {(W_{n}-\operatorname {E} [W_{n}])}}]\\\\\mathrm {E} [(Z_{2}-\operatorname {E} [Z_{2}]){\overline {(W_{1}-\operatorname {E} [W_{1}])}}]&\mathrm {E} [(Z_{2}-\operatorname {E} [Z_{2}]){\overline {(W_{2}-\operatorname {E} [W_{2}])}}]&\cdots &\mathrm {E} [(Z_{2}-\operatorname {E} [Z_{2}]){\overline {(W_{n}-\operatorname {E} [W_{n}])}}]\\\\\vdots &\vdots &\ddots &\vdots \\\\\mathrm {E} [(Z_{n}-\operatorname {E} [Z_{n}]){\overline {(W_{1}-\operatorname {E} [W_{1}])}}]&\mathrm {E} [(Z_{n}-\operatorname {E} [Z_{n}]){\overline {(W_{2}-\operatorname {E} [W_{2}])}}]&\cdots &\mathrm {E} [(Z_{n}-\operatorname {E} [Z_{n}]){\overline {(W_{n}-\operatorname {E} [W_{n}])}}]\end{bmatrix}}}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/9941690a38b08ff9f3b9be4bd1de0bb2fb9a67b5)
![{\displaystyle \operatorname {J} _{\mathbf {Z} \mathbf {W} }=\operatorname {cov} [\mathbf {Z} ,{\overline {\mathbf {W} }}]=\operatorname {E} [(\mathbf {Z} -\operatorname {E} [\mathbf {Z} ]){(\mathbf {W} -\operatorname {E} [\mathbf {W} ])}^{T}]=\operatorname {E} [\mathbf {Z} \mathbf {W} ^{T}]-\operatorname {E} [\mathbf {Z} ]\operatorname {E} [\mathbf {W} ^{T}]}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/3e6a151a8db17da379cb32babce0e737bca9a8b9)
![{\displaystyle \operatorname {J} _{\mathbf {Z} \mathbf {W} }={\begin{bmatrix}\mathrm {E} [(Z_{1}-\operatorname {E} [Z_{1}])(W_{1}-\operatorname {E} [W_{1}])]&\mathrm {E} [(Z_{1}-\operatorname {E} [Z_{1}])(W_{2}-\operatorname {E} [W_{2}])]&\cdots &\mathrm {E} [(Z_{1}-\operatorname {E} [Z_{1}])(W_{n}-\operatorname {E} [W_{n}])]\\\\\mathrm {E} [(Z_{2}-\operatorname {E} [Z_{2}])(W_{1}-\operatorname {E} [W_{1}])]&\mathrm {E} [(Z_{2}-\operatorname {E} [Z_{2}])(W_{2}-\operatorname {E} [W_{2}])]&\cdots &\mathrm {E} [(Z_{2}-\operatorname {E} [Z_{2}])(W_{n}-\operatorname {E} [W_{n}])]\\\\\vdots &\vdots &\ddots &\vdots \\\\\mathrm {E} [(Z_{n}-\operatorname {E} [Z_{n}])(W_{1}-\operatorname {E} [W_{1}])]&\mathrm {E} [(Z_{n}-\operatorname {E} [Z_{n}])(W_{2}-\operatorname {E} [W_{2}])]&\cdots &\mathrm {E} [(Z_{n}-\operatorname {E} [Z_{n}])(W_{n}-\operatorname {E} [W_{n}])]\end{bmatrix}}}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/240fc2c9717f84911d95a271d7ca87595b185cc5)
![{\displaystyle \operatorname {E} [\mathbf {Z} ]=0}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/d136e6dc4d72ba213940846421b1f8c00160bc8d)
![{\displaystyle \operatorname {var} [Z_{1}]<\infty ,\ldots ,\operatorname {var} [Z_{n}]<\infty }](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/ca262a1fa11c0e897e89da432119076df0575dc4)
![{\displaystyle \operatorname {E} [\mathbf {Z} \mathbf {Z} ^{T}]=0}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/e567ff4fa3785973cc8d53106ec6a0e075d969e9)
![{\displaystyle \left|\operatorname {E} [\mathbf {Z} ^{H}\mathbf {W} ]\right|^{2}\leq \operatorname {E} [\mathbf {Z} ^{H}\mathbf {Z} ]\operatorname {E} [|\mathbf {W} ^{H}\mathbf {W} |]}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/e6675b4efe08db65ef119042110564db71520472)
![{\displaystyle \varphi _{\mathbf {Z} }(\mathbf {\omega } )=\operatorname {E} \left[e^{i\Re {(\mathbf {\omega } ^{H}\mathbf {Z} )}}\right]=\operatorname {E} \left[e^{i(\Re {(\omega _{1})}\Re {(Z_{1})}+\Im {(\omega _{1})}\Im {(Z_{1})}+\cdots +\Re {(\omega _{n})}\Re {(Z_{n})}+\Im {(\omega _{n})}\Im {(Z_{n})})}\right]}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/55ef4b68890cb6b51eea0081d9a757a4c9f88849)