Biomedical Engineering Reference
In-Depth Information
H(
)
Note that we use a simplified notation
x
in this topic to indicate the entropy of
3
(
)
H
[
(
)
]
the probability distribution
p
x
, which should formally be written as
p
x
C.2
Complex Gaussian Distribution
We derive the probability distribution when the random variable is a complex-valued
Gaussian. The arguments in this section follows those of Neeser and Massey [1].
Let us define a column vector of
N
complex random variables as
z
, and define
z
and
x
and
y
are real-valued N-dimensional
Gaussian random vectors. Here, we assume that
E
=
x
+
i
y
where
x
=
(
z
)
,
y
=
(
z
)
(
z
)
=
0, and as a result,
E
(
x
)
=
0
E
zz
H
.
and
E
(
y
)
=
0. The covariance matrix of
z
is defined as
ʣ
zz
=
x
T
y
T
T
. The covariance
We next define a
(
2
N
×
1
)
vector
ˆ
such that
ˆ
=[
,
]
matrix of
ˆ
,
ʣ
ˆˆ
, is given by
E
x
y
x
T
y
T
ʣ
xx
ʣ
T
yx
ʣ
yx
ʣ
yy
ʣ
ˆˆ
=
,
=
,
(C.10)
xx
T
yx
T
yy
T
where
ʣ
xx
=
E
(
)
,
ʣ
yx
=
E
(
)
, and
ʣ
yy
=
E
(
)
. We assume that the joint
distribution of
x
and
y
is given by
2
exp
1
1
2
ˆ
T
ʣ
−
1
p
(
x
,
y
)
=
p
(
ˆ
)
=
−
ˆˆ
ˆ
.
(C.11)
/
(
2
ˀ)
N
|
ʣ
ˆˆ
|
1
In this section, we show that this joint distribution is equal to
exp
z
1
z
H
ʣ
−
1
zz
p
(
z
)
=
−
,
(C.12)
ˀ
N
|
ʣ
zz
|
which is called the complex Gaussian distribution.
To show this equality, we assume a property, called “proper”, on the complex
random variable
z
. The complex random variable
z
is proper if its pseudo covariance
ʣ
zz
is equal to zero. The pseudo covariance is defined such that
E
zz
T
,
ʣ
zz
=
which is written as
E
T
i
yx
ʣ
zz
=
T
(
x
+
i
y
)(
x
+
i
y
)
=
ʣ
xx
−
ʣ
yy
+
ʣ
yx
+
ʣ
.
ʣ
zz
=
Therefore,
0
is equivalent to the relationships
T
ʣ
xx
=
ʣ
yy
and
−
ʣ
yx
=
ʣ
yx
.
(C.13)
3
The notation
H
(
x
)
may look as if the entropy
H
(
x
)
is a function of
x
, but the entropy is a functional
of the probability distribution
p
(
x
)
, not a function of
x
.
