In probability theory and statistics , complex random variables are a generalization of real-valued random variables to complex numbers , i.e. the possible values a complex random variable may take are complex numbers.[ 1] distribution of one complex random variable may be interpreted as the joint distribution of two real random variables.
Some concepts of real random variables have a straightforward generalization to complex random variables—e.g., the definition of the mean of a complex random variable. Other concepts are unique to complex random variables.
Applications of complex random variables are found in digital signal processing ,[ 2] quadrature amplitude modulation and information theory .
A complex random variable
Z
{\displaystyle Z}
probability space
(
Ω
,
F
,
P
)
{\displaystyle (\Omega ,{\mathcal {F}},P)}
function
Z
:
Ω
→
C
{\displaystyle Z\colon \Omega \rightarrow \mathbb {C} }
ℜ
(
Z
)
{\displaystyle \Re {(Z)}}
ℑ
(
Z
)
{\displaystyle \Im {(Z)}}
random variables on
(
Ω
,
F
,
P
)
{\displaystyle (\Omega ,{\mathcal {F}},P)}
Consider a random variable that may take only the three complex values
1
+
i
,
1
−
i
,
2
{\displaystyle 1+i,1-i,2}
Probability
P
(
z
)
{\displaystyle P(z)}
Value
z
{\displaystyle z}
1
4
{\displaystyle {\frac {1}{4}}}
1
+
i
{\displaystyle 1+i}
1
4
{\displaystyle {\frac {1}{4}}}
1
−
i
{\displaystyle 1-i}
1
2
{\displaystyle {\frac {1}{2}}}
2
{\displaystyle 2}
The expectation of this random variable may be simply calculated:
E
[
Z
]
=
1
4
(
1
+
i
)
+
1
4
(
1
−
i
)
+
1
2
2
=
3
2
.
{\displaystyle \operatorname {E} [Z]={\frac {1}{4}}(1+i)+{\frac {1}{4}}(1-i)+{\frac {1}{2}}2={\frac {3}{2}}.}
Another example of a complex random variable is the uniform distribution over the filled unit circle, i.e. the set
{
z
∈
C
∣
|
z
|
≤
1
}
{\displaystyle \{z\in \mathbb {C} \mid |z|\leq 1\}}
probability density function is defined. The density function is shown as the yellow disk and dark blue base in the following figure.
Complex normal distribution [ edit ] Complex Gaussian random variables are often encountered in applications. They are a straightforward generalization of real Gaussian random variables. The following plot shows an example of the distribution of such a variable.
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
→
[
0
,
1
]
{\displaystyle F_{Z}:\mathbb {C} \to [0,1]}
joint distribution of their real and imaginary parts:
F
Z
(
z
)
=
F
ℜ
(
Z
)
,
ℑ
(
Z
)
(
ℜ
(
z
)
,
ℑ
(
z
)
)
=
P
(
ℜ
(
Z
)
≤
ℜ
(
z
)
,
ℑ
(
Z
)
≤
ℑ
(
z
)
)
{\displaystyle F_{Z}(z)=F_{\Re {(Z)},\Im {(Z)}}(\Re {(z)},\Im {(z)})=P(\Re {(Z)}\leq \Re {(z)},\Im {(Z)}\leq \Im {(z)})}
(Eq.1 )
Probability density function [ edit ] The probability density function of a complex random variable is defined as
f
Z
(
z
)
=
f
ℜ
(
Z
)
,
ℑ
(
Z
)
(
ℜ
(
z
)
,
ℑ
(
z
)
)
{\displaystyle f_{Z}(z)=f_{\Re {(Z)},\Im {(Z)}}(\Re {(z)},\Im {(z)})}
z
∈
C
{\displaystyle z\in \mathbb {C} }
(
ℜ
(
z
)
,
ℑ
(
z
)
)
{\displaystyle (\Re {(z)},\Im {(z)})}
An equivalent definition is given by
f
Z
(
z
)
=
∂
2
∂
x
∂
y
P
(
ℜ
(
Z
)
≤
x
,
ℑ
(
Z
)
≤
y
)
{\displaystyle f_{Z}(z)={\frac {\partial ^{2}}{\partial x\partial y}}P(\Re {(Z)}\leq x,\Im {(Z)}\leq y)}
x
=
ℜ
(
z
)
{\displaystyle x=\Re {(z)}}
y
=
ℑ
(
z
)
{\displaystyle y=\Im {(z)}}
As in the real case the density function may not exist.
The expectation of a complex random variable is defined based on the definition of the expectation of a real random variable:[ 3] : p. 112
E
[
Z
]
=
E
[
ℜ
(
Z
)
]
+
i
E
[
ℑ
(
Z
)
]
{\displaystyle \operatorname {E} [Z]=\operatorname {E} [\Re {(Z)}]+i\operatorname {E} [\Im {(Z)}]}
(Eq.2 )
Note that the expectation of a complex random variable does not exist if
E
[
ℜ
(
Z
)
]
{\displaystyle \operatorname {E} [\Re {(Z)}]}
E
[
ℑ
(
Z
)
]
{\displaystyle \operatorname {E} [\Im {(Z)}]}
If the complex random variable
Z
{\displaystyle Z}
f
Z
(
z
)
{\displaystyle f_{Z}(z)}
E
[
Z
]
=
∬
C
z
⋅
f
Z
(
z
)
d
x
d
y
{\displaystyle \operatorname {E} [Z]=\iint _{\mathbb {C} }z\cdot f_{Z}(z)\,dx\,dy}
If the complex random variable
Z
{\displaystyle Z}
probability mass function
p
Z
(
z
)
{\displaystyle p_{Z}(z)}
E
[
Z
]
=
∑
z
∈
Z
z
⋅
p
Z
(
z
)
{\displaystyle \operatorname {E} [Z]=\sum _{z\in \mathbb {Z} }z\cdot p_{Z}(z)}
Properties Whenever the expectation of a complex random variable exists, taking the expectation and complex conjugation commute:
E
[
Z
]
¯
=
E
[
Z
¯
]
.
{\displaystyle {\overline {\operatorname {E} [Z]}}=\operatorname {E} [{\overline {Z}}].}
The expected value operator
E
[
⋅
]
{\displaystyle \operatorname {E} [\cdot ]}
linear in the sense that
E
[
a
Z
+
b
W
]
=
a
E
[
Z
]
+
b
E
[
W
]
{\displaystyle \operatorname {E} [aZ+bW]=a\operatorname {E} [Z]+b\operatorname {E} [W]}
for any complex coefficients
a
,
b
{\displaystyle a,b}
Z
{\displaystyle Z}
W
{\displaystyle W}
independent .
Variance and pseudo-variance [ edit ] The variance is defined in terms of absolute squares as:[ 3] : 117
K
Z
Z
=
Var
[
Z
]
=
E
[
|
Z
−
E
[
Z
]
|
2
]
=
E
[
|
Z
|
2
]
−
|
E
[
Z
]
|
2
{\displaystyle \operatorname {K} _{ZZ}=\operatorname {Var} [Z]=\operatorname {E} \left[\left|Z-\operatorname {E} [Z]\right|^{2}\right]=\operatorname {E} [|Z|^{2}]-\left|\operatorname {E} [Z]\right|^{2}}
(Eq.3 )
Properties The variance is always a nonnegative real number. It is equal to the sum of the variances of the real and imaginary part of the complex random variable:
Var
[
Z
]
=
Var
[
ℜ
(
Z
)
]
+
Var
[
ℑ
(
Z
)
]
.
{\displaystyle \operatorname {Var} [Z]=\operatorname {Var} [\Re {(Z)}]+\operatorname {Var} [\Im {(Z)}].}
The variance of a linear combination of complex random variables may be calculated using the following formula:
Var
[
∑
k
=
1
N
a
k
Z
k
]
=
∑
i
=
1
N
∑
j
=
1
N
a
i
a
j
¯
Cov
[
Z
i
,
Z
j
]
.
{\displaystyle \operatorname {Var} \left[\sum _{k=1}^{N}a_{k}Z_{k}\right]=\sum _{i=1}^{N}\sum _{j=1}^{N}a_{i}{\overline {a_{j}}}\operatorname {Cov} [Z_{i},Z_{j}].}
The pseudo-variance is a special case of the pseudo-covariance and is defined in terms of ordinary complex squares , given by:
J
Z
Z
=
E
[
(
Z
−
E
[
Z
]
)
2
]
=
E
[
Z
2
]
−
(
E
[
Z
]
)
2
{\displaystyle \operatorname {J} _{ZZ}=\operatorname {E} [(Z-\operatorname {E} [Z])^{2}]=\operatorname {E} [Z^{2}]-(\operatorname {E} [Z])^{2}}
(Eq.4 )
Unlike the variance of
Z
{\displaystyle Z}
Z
{\displaystyle Z}
Covariance matrix of real and imaginary parts [ edit ] For a general complex random variable, the pair
(
ℜ
(
Z
)
,
ℑ
(
Z
)
)
{\displaystyle (\Re {(Z)},\Im {(Z)})}
covariance matrix of the form:
[
Var
[
ℜ
(
Z
)
]
Cov
[
ℑ
(
Z
)
,
ℜ
(
Z
)
]
Cov
[
ℜ
(
Z
)
,
ℑ
(
Z
)
]
Var
[
ℑ
(
Z
)
]
]
{\displaystyle {\begin{bmatrix}\operatorname {Var} [\Re {(Z)}]&\operatorname {Cov} [\Im {(Z)},\Re {(Z)}]\\\operatorname {Cov} [\Re {(Z)},\Im {(Z)}]&\operatorname {Var} [\Im {(Z)}]\end{bmatrix}}}
The matrix is symmetric, so
Cov
[
ℜ
(
Z
)
,
ℑ
(
Z
)
]
=
Cov
[
ℑ
(
Z
)
,
ℜ
(
Z
)
]
{\displaystyle \operatorname {Cov} [\Re {(Z)},\Im {(Z)}]=\operatorname {Cov} [\Im {(Z)},\Re {(Z)}]}
Its elements equal:
Var
[
ℜ
(
Z
)
]
=
1
2
Re
(
K
Z
Z
+
J
Z
Z
)
Var
[
ℑ
(
Z
)
]
=
1
2
Re
(
K
Z
Z
−
J
Z
Z
)
Cov
[
ℜ
(
Z
)
,
ℑ
(
Z
)
]
=
1
2
Im
(
J
Z
Z
)
{\displaystyle {\begin{aligned}&\operatorname {Var} [\Re {(Z)}]={\tfrac {1}{2}}\operatorname {Re} (\operatorname {K} _{ZZ}+\operatorname {J} _{ZZ})\\&\operatorname {Var} [\Im {(Z)}]={\tfrac {1}{2}}\operatorname {Re} (\operatorname {K} _{ZZ}-\operatorname {J} _{ZZ})\\&\operatorname {Cov} [\Re {(Z)},\Im {(Z)}]={\tfrac {1}{2}}\operatorname {Im} (\operatorname {J} _{ZZ})\\\end{aligned}}}
Conversely:
K
Z
Z
=
Var
[
ℜ
(
Z
)
]
+
Var
[
ℑ
(
Z
)
]
J
Z
Z
=
Var
[
ℜ
(
Z
)
]
−
Var
[
ℑ
(
Z
)
]
+
i
2
Cov
[
ℜ
(
Z
)
,
ℑ
(
Z
)
]
{\displaystyle {\begin{aligned}&\operatorname {K} _{ZZ}=\operatorname {Var} [\Re {(Z)}]+\operatorname {Var} [\Im {(Z)}]\\&\operatorname {J} _{ZZ}=\operatorname {Var} [\Re {(Z)}]-\operatorname {Var} [\Im {(Z)}]+i2\operatorname {Cov} [\Re {(Z)},\Im {(Z)}]\end{aligned}}}
Covariance and pseudo-covariance [ edit ] The covariance between two complex random variables
Z
,
W
{\displaystyle Z,W}
[ 3] : 119
K
Z
W
=
Cov
[
Z
,
W
]
=
E
[
(
Z
−
E
[
Z
]
)
(
W
−
E
[
W
]
)
¯
]
=
E
[
Z
W
¯
]
−
E
[
Z
]
E
[
W
¯
]
{\displaystyle \operatorname {K} _{ZW}=\operatorname {Cov} [Z,W]=\operatorname {E} [(Z-\operatorname {E} [Z]){\overline {(W-\operatorname {E} [W])}}]=\operatorname {E} [Z{\overline {W}}]-\operatorname {E} [Z]\operatorname {E} [{\overline {W}}]}
(Eq.5 )
Notice the complex conjugation of the second factor in the definition.
pseudo-covariance (also called complementary variance ):
J
Z
W
=
Cov
[
Z
,
W
¯
]
=
E
[
(
Z
−
E
[
Z
]
)
(
W
−
E
[
W
]
)
]
=
E
[
Z
W
]
−
E
[
Z
]
E
[
W
]
{\displaystyle \operatorname {J} _{ZW}=\operatorname {Cov} [Z,{\overline {W}}]=\operatorname {E} [(Z-\operatorname {E} [Z])(W-\operatorname {E} [W])]=\operatorname {E} [ZW]-\operatorname {E} [Z]\operatorname {E} [W]}
(Eq.6 )
The second order statistics are fully characterized by the covariance and the pseudo-covariance.
Properties The covariance has the following properties:
Cov
[
Z
,
W
]
=
Cov
[
W
,
Z
]
¯
{\displaystyle \operatorname {Cov} [Z,W]={\overline {\operatorname {Cov} [W,Z]}}}
Cov
[
α
Z
,
W
]
=
α
Cov
[
Z
,
W
]
{\displaystyle \operatorname {Cov} [\alpha Z,W]=\alpha \operatorname {Cov} [Z,W]}
Cov
[
Z
,
α
W
]
=
α
¯
Cov
[
Z
,
W
]
{\displaystyle \operatorname {Cov} [Z,\alpha W]={\overline {\alpha }}\operatorname {Cov} [Z,W]}
Cov
[
Z
1
+
Z
2
,
W
]
=
Cov
[
Z
1
,
W
]
+
Cov
[
Z
2
,
W
]
{\displaystyle \operatorname {Cov} [Z_{1}+Z_{2},W]=\operatorname {Cov} [Z_{1},W]+\operatorname {Cov} [Z_{2},W]}
Cov
[
Z
,
W
1
+
W
2
]
=
Cov
[
Z
,
W
1
]
+
Cov
[
Z
,
W
2
]
{\displaystyle \operatorname {Cov} [Z,W_{1}+W_{2}]=\operatorname {Cov} [Z,W_{1}]+\operatorname {Cov} [Z,W_{2}]}
Cov
[
Z
,
Z
]
=
Var
[
Z
]
{\displaystyle \operatorname {Cov} [Z,Z]={\operatorname {Var} [Z]}}
Uncorrelatedness: two complex random variables
Z
{\displaystyle Z}
W
{\displaystyle W}
uncorrelated if
K
Z
W
=
J
Z
W
=
0
{\displaystyle \operatorname {K} _{ZW}=\operatorname {J} _{ZW}=0}
uncorrelatedness (probability theory) ).
Orthogonality: two complex random variables
Z
{\displaystyle Z}
W
{\displaystyle W}
orthogonal if
E
[
Z
W
¯
]
=
0
{\displaystyle \operatorname {E} [Z{\overline {W}}]=0}
Circular symmetry of complex random variables is a common assumption used in the field of wireless communication. A typical example of a circular symmetric complex random variable is the complex Gaussian random variable with zero mean and zero pseudo-covariance matrix.
A complex random variable
Z
{\displaystyle Z}
ϕ
∈
[
−
π
,
π
]
{\displaystyle \phi \in [-\pi ,\pi ]}
e
i
ϕ
Z
{\displaystyle e^{\mathrm {i} \phi }Z}
Z
{\displaystyle Z}
Properties By definition, a circularly symmetric complex random variable has
E
[
Z
]
=
E
[
e
i
ϕ
Z
]
=
e
i
ϕ
E
[
Z
]
{\displaystyle \operatorname {E} [Z]=\operatorname {E} [e^{\mathrm {i} \phi }Z]=e^{\mathrm {i} \phi }\operatorname {E} [Z]}
ϕ
{\displaystyle \phi }
Thus the expectation of a circularly symmetric complex random variable can only be either zero or undefined.
Additionally,
E
[
Z
Z
]
=
E
[
e
i
ϕ
Z
e
i
ϕ
Z
]
=
e
2
i
ϕ
E
[
Z
Z
]
{\displaystyle \operatorname {E} [ZZ]=\operatorname {E} [e^{\mathrm {i} \phi }Ze^{\mathrm {i} \phi }Z]=e^{\mathrm {2} i\phi }\operatorname {E} [ZZ]}
ϕ
{\displaystyle \phi }
Thus the pseudo-variance of a circularly symmetric complex random variable can only be zero.
If
Z
{\displaystyle Z}
e
i
ϕ
Z
{\displaystyle e^{\mathrm {i} \phi }Z}
Z
{\displaystyle Z}
[
−
π
,
π
]
{\displaystyle [-\pi ,\pi ]}
Z
{\displaystyle Z}
[ 4]
Proper complex random variables [ edit ] The concept of proper random variables is unique to complex random variables, and has no correspondent concept with real random variables.
A complex random variable
Z
{\displaystyle Z}
E
[
Z
]
=
0
{\displaystyle \operatorname {E} [Z]=0}
Var
[
Z
]
<
∞
{\displaystyle \operatorname {Var} [Z]<\infty }
E
[
Z
2
]
=
0
{\displaystyle \operatorname {E} [Z^{2}]=0}
This definition is equivalent to the following conditions. This means that a complex random variable is proper if, and only if:
E
[
Z
]
=
0
{\displaystyle \operatorname {E} [Z]=0}
E
[
ℜ
(
Z
)
2
]
=
E
[
ℑ
(
Z
)
2
]
≠
∞
{\displaystyle \operatorname {E} [\Re {(Z)}^{2}]=\operatorname {E} [\Im {(Z)}^{2}]\neq \infty }
E
[
ℜ
(
Z
)
ℑ
(
Z
)
]
=
0
{\displaystyle \operatorname {E} [\Re {(Z)}\Im {(Z)}]=0}
Theorem — Every circularly symmetric complex random variable with finite variance is proper.
For a proper complex random variable, the covariance matrix of the pair
(
ℜ
(
Z
)
,
ℑ
(
Z
)
)
{\displaystyle (\Re {(Z)},\Im {(Z)})}
[
1
2
Var
[
Z
]
0
0
1
2
Var
[
Z
]
]
{\displaystyle {\begin{bmatrix}{\frac {1}{2}}\operatorname {Var} [Z]&0\\0&{\frac {1}{2}}\operatorname {Var} [Z]\end{bmatrix}}}
I.e.:
Var
[
ℜ
(
Z
)
]
=
Var
[
ℑ
(
Z
)
]
=
1
2
Var
[
Z
]
Cov
[
ℜ
(
Z
)
,
ℑ
(
Z
)
]
=
0
{\displaystyle {\begin{aligned}&\operatorname {Var} [\Re {(Z)}]=\operatorname {Var} [\Im {(Z)}]={\tfrac {1}{2}}\operatorname {Var} [Z]\\&\operatorname {Cov} [\Re {(Z)},\Im {(Z)}]=0\\\end{aligned}}}
Cauchy-Schwarz inequality [ edit ] The Cauchy-Schwarz inequality for complex random variables, which can be derived using the Triangle inequality and Hölder's inequality , is
|
E
[
Z
W
¯
]
|
2
≤
|
E
[
|
Z
W
¯
|
]
|
2
≤
E
[
|
Z
|
2
]
E
[
|
W
|
2
]
{\displaystyle \left|\operatorname {E} \left[Z{\overline {W}}\right]\right|^{2}\leq \left|\operatorname {E} \left[\left|Z{\overline {W}}\right|\right]\right|^{2}\leq \operatorname {E} \left[|Z|^{2}\right]\operatorname {E} \left[|W|^{2}\right]}
Characteristic function [ edit ] The characteristic function of a complex random variable is a function
C
→
C
{\displaystyle \mathbb {C} \to \mathbb {C} }
φ
Z
(
ω
)
=
E
[
e
i
ℜ
(
ω
¯
Z
)
]
=
E
[
e
i
(
ℜ
(
ω
)
ℜ
(
Z
)
+
ℑ
(
ω
)
ℑ
(
Z
)
)
]
.
{\displaystyle \varphi _{Z}(\omega )=\operatorname {E} \left[e^{i\Re {({\overline {\omega }}Z)}}\right]=\operatorname {E} \left[e^{i(\Re {(\omega )}\Re {(Z)}+\Im {(\omega )}\Im {(Z)})}\right].}
^ Eriksson, Jan; Ollila, Esa; Koivunen, Visa (2009). Statistics for complex random variables revisited . 2009 IEEE International Conference on Acoustics, Speech and Signal Processing. Taipei, Taiwan: Institute of Electrical and Electronics Engineers . pp. 3565–3568. doi :10.1109/ICASSP.2009.4960396 . ^ Lapidoth, A. (2009). A Foundation in Digital Communication . Cambridge University Press. ISBN 9780521193955 ^ a b c Park,Kun Il (2018). Fundamentals of Probability and Stochastic Processes with Applications to Communications . Springer. ISBN 978-3-319-68074-3 ^ Peter J. Schreier, Louis L. Scharf (2011). Statistical Signal Processing of Complex-Valued Data . Cambridge University Press. ISBN 9780511815911
![{\displaystyle \operatorname {E} [Z]={\frac {1}{4}}(1+i)+{\frac {1}{4}}(1-i)+{\frac {1}{2}}2={\frac {3}{2}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c524ba54cdf76d00057ff202c9ef13d276797808)
![{\displaystyle F_{Z}:\mathbb {C} \to [0,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a49f4f55d06b18465dbc269884ebf7747b9156f3)
![{\displaystyle \operatorname {E} [Z]=\operatorname {E} [\Re {(Z)}]+i\operatorname {E} [\Im {(Z)}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f6e1ff26c4d56190d3bcbbc0f5aaede2132ea4de)
![{\displaystyle \operatorname {E} [\Re {(Z)}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e844e16f165052318326c84418f6f397eef22c61)
![{\displaystyle \operatorname {E} [\Im {(Z)}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/db17b20ce3199ef25f805f27c8473a08d3753378)
![{\displaystyle \operatorname {E} [Z]=\iint _{\mathbb {C} }z\cdot f_{Z}(z)\,dx\,dy}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ce36d7e5aca36e4baed05d1ce7eb8be351a3da68)
![{\displaystyle \operatorname {E} [Z]=\sum _{z\in \mathbb {Z} }z\cdot p_{Z}(z)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8bd6feced28ac4cb07956d41e9b0476422f8b168)
![{\displaystyle {\overline {\operatorname {E} [Z]}}=\operatorname {E} [{\overline {Z}}].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d0f389d24b8d094d5cbcfe0587f73462936d18ff)
![{\displaystyle \operatorname {E} [\cdot ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a71518eb57ffaf54c0c31bf94de5ac9d7ab11a1)
![{\displaystyle \operatorname {E} [aZ+bW]=a\operatorname {E} [Z]+b\operatorname {E} [W]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e642bbe3e97314e03a0804a0bcfeaaf4a49c7bc)
![{\displaystyle \operatorname {K} _{ZZ}=\operatorname {Var} [Z]=\operatorname {E} \left[\left|Z-\operatorname {E} [Z]\right|^{2}\right]=\operatorname {E} [|Z|^{2}]-\left|\operatorname {E} [Z]\right|^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/20d080a97d05f3891fad48566952956c8bc33616)
![{\displaystyle \operatorname {Var} [Z]=\operatorname {Var} [\Re {(Z)}]+\operatorname {Var} [\Im {(Z)}].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/add16b3176754caaa4cf67a75a16b4ac92b34c76)
![{\displaystyle \operatorname {Var} \left[\sum _{k=1}^{N}a_{k}Z_{k}\right]=\sum _{i=1}^{N}\sum _{j=1}^{N}a_{i}{\overline {a_{j}}}\operatorname {Cov} [Z_{i},Z_{j}].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f99e419701670b4886082740417b566f5c3f3123)
![{\displaystyle \operatorname {J} _{ZZ}=\operatorname {E} [(Z-\operatorname {E} [Z])^{2}]=\operatorname {E} [Z^{2}]-(\operatorname {E} [Z])^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4ff6bdd87afbafd84fa355854e5d4d7884b7cb5f)
![{\displaystyle {\begin{bmatrix}\operatorname {Var} [\Re {(Z)}]&\operatorname {Cov} [\Im {(Z)},\Re {(Z)}]\\\operatorname {Cov} [\Re {(Z)},\Im {(Z)}]&\operatorname {Var} [\Im {(Z)}]\end{bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ac330d152bea21f53db8f3ff194d894157b6fc3c)
![{\displaystyle \operatorname {Cov} [\Re {(Z)},\Im {(Z)}]=\operatorname {Cov} [\Im {(Z)},\Re {(Z)}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4c2fd64c83c920bd9f44dfcb5281682245f7d103)
![{\displaystyle {\begin{aligned}&\operatorname {Var} [\Re {(Z)}]={\tfrac {1}{2}}\operatorname {Re} (\operatorname {K} _{ZZ}+\operatorname {J} _{ZZ})\\&\operatorname {Var} [\Im {(Z)}]={\tfrac {1}{2}}\operatorname {Re} (\operatorname {K} _{ZZ}-\operatorname {J} _{ZZ})\\&\operatorname {Cov} [\Re {(Z)},\Im {(Z)}]={\tfrac {1}{2}}\operatorname {Im} (\operatorname {J} _{ZZ})\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/660a39d44691cd1f64d313944c0046eeaa5fe2cc)
![{\displaystyle {\begin{aligned}&\operatorname {K} _{ZZ}=\operatorname {Var} [\Re {(Z)}]+\operatorname {Var} [\Im {(Z)}]\\&\operatorname {J} _{ZZ}=\operatorname {Var} [\Re {(Z)}]-\operatorname {Var} [\Im {(Z)}]+i2\operatorname {Cov} [\Re {(Z)},\Im {(Z)}]\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/82090510b637bbd7629cffdef53b712e564ec908)
![{\displaystyle \operatorname {K} _{ZW}=\operatorname {Cov} [Z,W]=\operatorname {E} [(Z-\operatorname {E} [Z]){\overline {(W-\operatorname {E} [W])}}]=\operatorname {E} [Z{\overline {W}}]-\operatorname {E} [Z]\operatorname {E} [{\overline {W}}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f7a961696e5eb2780adc679459b7cf2c5000dd1e)
![{\displaystyle \operatorname {J} _{ZW}=\operatorname {Cov} [Z,{\overline {W}}]=\operatorname {E} [(Z-\operatorname {E} [Z])(W-\operatorname {E} [W])]=\operatorname {E} [ZW]-\operatorname {E} [Z]\operatorname {E} [W]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f0850e20d08a280f61bd0e8db2c108656c09f163)
![{\displaystyle \operatorname {Cov} [Z,W]={\overline {\operatorname {Cov} [W,Z]}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e6c45f52a23a45d8c0489113054f2899fc7abe05)
![{\displaystyle \operatorname {Cov} [\alpha Z,W]=\alpha \operatorname {Cov} [Z,W]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/36d563b3f4a69e63da7c74774262faa56e9d8dbb)
![{\displaystyle \operatorname {Cov} [Z,\alpha W]={\overline {\alpha }}\operatorname {Cov} [Z,W]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cdae2df03771a6783a07fce94daafe717183d542)
![{\displaystyle \operatorname {Cov} [Z_{1}+Z_{2},W]=\operatorname {Cov} [Z_{1},W]+\operatorname {Cov} [Z_{2},W]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/58046998dba3171529973d2cea6df46bd55cd24b)
![{\displaystyle \operatorname {Cov} [Z,W_{1}+W_{2}]=\operatorname {Cov} [Z,W_{1}]+\operatorname {Cov} [Z,W_{2}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/21ba4eef59808fe6d651f74824fa7354b63de613)
![{\displaystyle \operatorname {Cov} [Z,Z]={\operatorname {Var} [Z]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1d0dd5a038210af5d291b2baf8d6f5f4017130ba)
![{\displaystyle \operatorname {E} [Z{\overline {W}}]=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a9158c9d3364ee6a00a491d275b5aaa59fb06b08)
![{\displaystyle \phi \in [-\pi ,\pi ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bd67c7af835707a32f70e8ceb8388953705d2ab3)
![{\displaystyle \operatorname {E} [Z]=\operatorname {E} [e^{\mathrm {i} \phi }Z]=e^{\mathrm {i} \phi }\operatorname {E} [Z]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7a0c0eaaf719d6a7f539ee634a8ada78a25fb814)
![{\displaystyle \operatorname {E} [ZZ]=\operatorname {E} [e^{\mathrm {i} \phi }Ze^{\mathrm {i} \phi }Z]=e^{\mathrm {2} i\phi }\operatorname {E} [ZZ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d399819a5ae5dbfd8d504a1771a23da9424712fe)
![{\displaystyle [-\pi ,\pi ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cb064fd6c55820cfa660eabeeda0f6e3c4935ae6)
![{\displaystyle \operatorname {E} [Z]=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e6c80bd7c75894b696a91f03059d601eed06827f)
![{\displaystyle \operatorname {Var} [Z]<\infty }](https://wikimedia.org/api/rest_v1/media/math/render/svg/a9598b3723f10a6af6b01d1a9e70dac49eaa054e)
![{\displaystyle \operatorname {E} [Z^{2}]=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e0142e4405a348fd70966a8f62d4c260f5b30498)
![{\displaystyle \operatorname {E} [\Re {(Z)}^{2}]=\operatorname {E} [\Im {(Z)}^{2}]\neq \infty }](https://wikimedia.org/api/rest_v1/media/math/render/svg/a24c7509a56aab23c94a541282cc12c0f2793e70)
![{\displaystyle \operatorname {E} [\Re {(Z)}\Im {(Z)}]=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/38f594a6d61e9568109957798cff69ca1239516b)
![{\displaystyle {\begin{bmatrix}{\frac {1}{2}}\operatorname {Var} [Z]&0\\0&{\frac {1}{2}}\operatorname {Var} [Z]\end{bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/409bedd03d9fcbd63fb8d7338d6dc4ad4fe69e86)
![{\displaystyle {\begin{aligned}&\operatorname {Var} [\Re {(Z)}]=\operatorname {Var} [\Im {(Z)}]={\tfrac {1}{2}}\operatorname {Var} [Z]\\&\operatorname {Cov} [\Re {(Z)},\Im {(Z)}]=0\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9cc188b8b1273b1a3c41d381002ba1095ce961ec)
![{\displaystyle \left|\operatorname {E} \left[Z{\overline {W}}\right]\right|^{2}\leq \left|\operatorname {E} \left[\left|Z{\overline {W}}\right|\right]\right|^{2}\leq \operatorname {E} \left[|Z|^{2}\right]\operatorname {E} \left[|W|^{2}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d22338d330f3e5c32c1fb25b14de2d80569cac86)
![{\displaystyle \varphi _{Z}(\omega )=\operatorname {E} \left[e^{i\Re {({\overline {\omega }}Z)}}\right]=\operatorname {E} \left[e^{i(\Re {(\omega )}\Re {(Z)}+\Im {(\omega )}\Im {(Z)})}\right].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/88cf0381ac2f47d34cdf5a23923e0699a8b6515d)