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
(
Z
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 .
Definition
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
(
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1
)
,
…
,
ℜ
(
Z
n
)
,
ℑ
(
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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
Expectation
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
[
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1
]
,
…
,
E
[
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n
]
)
T
{\displaystyle \operatorname {E} [\mathbf {Z} ]=(\operatorname {E} [Z_{1}],\ldots ,\operatorname {E} [Z_{n}])^{T}}
Eq.1
Covariance matrix and pseudo-covariance matrix
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. Unlike in the case of real random variables, the covariance between two random variables involves the 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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−
E
[
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[
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)
H
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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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{\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.2
K
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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[
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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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]
E
[
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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 (also called relation matrix) is defined as follows. In contrast to the covariance matrix defined above transposition gets replaced by Hermitian transposition in the definition.
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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]
E
[
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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.3
J
Z
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=
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−
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]
E
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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}}}
Cross-covariance matrix and pseudo-cross-covariance matrix
The cross-covariance matrix between two complex random vectors
Z
,
W
{\displaystyle \mathbf {Z} ,\mathbf {W} }
K
Z
W
=
cov
[
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,
W
]
=
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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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[
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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.4
K
Z
W
=
[
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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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⋯
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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
[
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,
W
¯
]
=
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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]
=
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[
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W
T
]
−
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[
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.5
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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}}}
Circular symmetry
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} }
[ 2] : pp. 500–501
The expectation of a circularly symmetric complex random vectors is either zero or it is not defined.[ 2] : p. 500
Proper complex random vectors
Definition
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
(
Z
1
,
Z
2
,
…
,
Z
m
,
W
1
,
W
2
,
…
,
W
n
)
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
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 covariace matrix is zero, i.e. if
K
Z
W
=
0
{\displaystyle \operatorname {K} _{\mathbf {Z} \mathbf {W} }=0}
Characteristic function
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
Z
(
ω
)
=
E
[
e
i
ℜ
(
ω
H
Z
)
]
=
E
[
e
i
(
ℜ
(
ω
1
)
ℜ
(
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1
)
+
ℑ
(
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1
)
ℑ
(
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1
)
+
⋯
+
ℜ
(
ω
n
)
ℜ
(
Z
n
)
+
ℑ
(
ω
n
)
ℑ
(
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n
)
)
]
{\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]}
See also
References
^ a b c d e f g h Lapidoth, Amos, A Foundation in Digital Communication , Cambridge University Press, 2009.
^ a b Tse, David, Fundamentals of Wireless Communication , Cambridge University Press, 2005.
![{\displaystyle \operatorname {E} [\mathbf {Z} ]=(\operatorname {E} [Z_{1}],\ldots ,\operatorname {E} [Z_{n}])^{T}}](/media/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}}}](/media/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}}}](/media/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}]}](/media/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}}}](/media/api/rest_v1/media/math/render/svg/50248ec68758163e693b5fa474537f0d953110ee)
![{\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}]}](/media/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}}}](/media/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}]}](/media/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}}}](/media/api/rest_v1/media/math/render/svg/240fc2c9717f84911d95a271d7ca87595b185cc5)
![{\displaystyle \operatorname {E} [\mathbf {Z} ]=0}](/media/api/rest_v1/media/math/render/svg/d136e6dc4d72ba213940846421b1f8c00160bc8d)
![{\displaystyle \operatorname {var} [Z_{1}]<\infty ,\ldots ,\operatorname {var} [Z_{n}]<\infty }](/media/api/rest_v1/media/math/render/svg/ca262a1fa11c0e897e89da432119076df0575dc4)
![{\displaystyle \operatorname {E} [\mathbf {Z} \mathbf {Z} ^{T}]=0}](/media/api/rest_v1/media/math/render/svg/e567ff4fa3785973cc8d53106ec6a0e075d969e9)
![{\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]}](/media/api/rest_v1/media/math/render/svg/55ef4b68890cb6b51eea0081d9a757a4c9f88849)