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account of complex second-order moments and the Gaussian probability density function
for the complex scalar
x
=
u
+
j
v
. A more general account for vector-valued
x
will be
given in Chapter
2
.
1.6.1
Bivariate Gaussian distribution
The real components
u
and
v
of the complex scalar random variable
x
=
u
+
j
v
, which
]
T
, are said to be bivariate Gaussian distributed,
with mean zero and covariance matrix
R
zz
, if their joint probability density function
(pdf) is
may be arranged in a vector
z
=
[
u
,v
exp
u
v
2
u
v
R
−
1
zz
1
1
p
u
v
(
u
,v
)
=
−
det
1
/
2
R
zz
π
2
exp
−
)
.
1
1
=
2
q
u
v
(
u
,v
(1.44)
det
1
/
2
R
zz
2
π
Here the quadratic form
q
u
v
(
u
,v
) and the covariance matrix
R
zz
of the composite vector
z
are defined as follows:
u
v
=
u
R
−
1
zz
q
u
v
(
u
,v
)
v
,
(1.45)
E
(
u
2
)
√
R
uu
√
R
vv
ρ
u
v
E
(
u
v
)
R
uu
E
(
zz
T
)
√
R
uu
√
R
vv
ρ
u
v
R
zz
=
=
=
.
(1.46)
2
)
E
(
v
u
)
E
(
v
R
vv
In the right-most parameterization of
R
zz
,thetermsare
E
(
u
2
)
R
uu
=
,
variance of the random variable
u
2
)
R
vv
=
v
v,
E
(
variance of the random variable
R
uu
√
R
vv
ρ
u
v
=
R
u
v
=
E
(
u
v
)
correlation of the random variables
u
,v,
R
u
v
√
R
uu
√
R
vv
ρ
u
v
=
correlation coefficient of the random variables
u
,v.
As in (
A1.38
), the inverse of the covariance matrix
R
−
1
zz
may be factored as
1
1
(
√
R
uu
/
√
R
vv
)
u
1
0
/
[
R
uu
(1
−
ρ
)]
0
−
ρ
u
v
R
−
1
(
√
R
uu
/
√
R
vv
)
v
zz
=
.
−
ρ
u
v
1
0
1
/
R
vv
0
1
(1.47)
Using (
A1.3
) we find det
R
zz
=
R
uu
(1
−
ρ
u
v
)
R
vv
, and from here the bivariate pdf
p
u
v
(
u
,v
) may be written as
exp
2
R
uu
u
1
1
2
R
uu
(1
√
R
vv
ρ
u
v
v
p
u
v
(
u
,v
)
=
−
−
(2
π
R
uu
(1
−
ρ
u
))
1
/
2
−
ρ
u
)
v
v
exp
2
1
1
2
R
vv
v
×
−
.
(1.48)
(2
π
R
vv
)
1
/
2
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