Digital Signal Processing Reference
In-Depth Information
1. When
f
(
x
) is Gaussian, it can be verified that each of the marginals
f
X
k
(
x
k
) is Gaussian.
2. Defining the
linearly transformed
vector
y
=
Ax
,
where
A
is a pos-
sibly rectangular real matrix, it can be shown [Feller, 1965], [Therrien
and Tummala, 2004] that
y
is Gaussian. It has mean
m
y
=
Am
x
and
covariance
C
yy
=
T
.
Since the correlation matrix has the form
AC
xx
A
y
,wealsohave
R
yy
=
AR
xx
A
T
.
R
yy
=
C
yy
+
m
y
m
3. A
complex
random vector
x
=
x
re
+
j
x
im
is said to be Gaussian if the
real vector
u
=
x
re
x
im
is Gaussian. The linearly transformed version
y
=
Ax
is also Gaussian for
any (possibly complex and rectangular)
A
. The means and covariances are
related as
m
y
=
Am
x
and
C
yy
=
AC
xx
A
†
.
A complex Gaussian vector
is said to be
circularly symmetric
if it satisfies some additional properties
as described in Sec. 6.6.
4. Note in particular that, when
x
is Gaussian, the sum of its individual
components
y
=
x
0
+
x
1
+
...
+
x
N−
1
(E
.
58)
is Gaussian. More generally, any linear combination of
x
k
's is Gaussian,
as this has the form
y
=
Ax
,
where
A
is a row vector.
E.4.2 Jointly Gaussian random variables
Let
x
0
and
x
1
be two real random variables.
We say that they are jointly
Gaussian if the vector
x
=
x
x
1
(E
.
59)
has Gaussian pdf
f
(
x
) (also denoted as
f
(
x
0
,x
1
)). In this case the marginal
pdfs
f
X
0
(
x
0
)and
f
X
1
(
x
1
) are also Gaussian, as mentioned above. This raises
the following question: is it possible to have two random variables
x
0
and
x
1
that are individually Gaussian, but not jointly Gaussain? In other words, is it
possible that
f
X
k
(
x
k
) are Gaussian but that
f
(
x
) does not take the form (E.56)?
This is indeed possible! A number of examples can be found in Feller [1965]. An
amusing example due to E. Nelson is given below.
Example E.1: Gaussian variables that are not jointly Gaussian
Let
g
(
y
) be the Gaussian pdf
g
(
y
)=
e
−y
2
/
2
√
2
π
,
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