Digital Signal Processing Reference
In-Depth Information
from Ex. 6.2. So
f
u
(
u
) can be rewritten as
1
2
πσ
x
/
2
1
2
πσ
x
/
2
e
−x
re
/
2(
σ
x
/
2)
×
e
−x
im
/
2(
σ
x
/
2)
.
f
u
(
u
)=
(6
.
62)
Since
σ
x
re
=
σ
x
im
=
σ
x
/
2, this is nothing but the product of the individual
pdfs of
x
re
and
x
im
. This happens because, in the Gaussian case, the uncor-
relatedness of
x
re
and
x
im
(induced by the circular symmetry of
x
) implies
that they are statistically independent as well.
Indeed, given any pdf of the form (6.60), it can always be rewritten in the
form (6.62) which shows that such an
x
has uncorrelated real and imaginary
parts with identical variances, showing that the expression (6.60) always
represents a circularly symmetric complex Gaussian
x.
The pdf of a more
general complex zero-mean Gaussian
x
would have to be expressed in the
form
x
re
x
im
,
det(2
π
C
uu
)
e
−
2
[
x
re
x
im
]
C
−
1
uu
1
f
(
x
re
,x
im
)=
(6
.
63)
where
C
uu
=
σ
x
re
ρ
,
ρ
x
im
with
ρ
=
E
[
x
re
x
im
].
6.6.4 Entropy of Gaussian random vectors
The differential entropy [Cover and Thomas, 1991] of a real random vector
u
with pdf
f
u
(
u
) is defined as
f
u
(
u
)ln
f
u
(
u
)
d
u
,
H
(
f
u
)=
−
(6
.
64)
where the integration is over all the components of
u
.
When there is no confusion
of notations we indicate
(
u
).
If
x
is a complex Gaussian vector, then its pdf can be expressed as in Eq.
(6.54) where
u
is as in Eq. (6.53) and
C
uu
is its covariance. Then the entropy
evaluated using Eq. (6.64) is given by
H
(
f
u
) by the simpler notation
H
(
f
u
)=ln
det (2
πe
C
uu
)
.
H
(6
.
65)
Note that the mean value of the random vector plays no role in this expression.
For the special case where
x
−
m
x
is circularly symmetric, Eq. (6.46) holds. So,
when a complex Gaussian vector
x
is such that
x
−
m
x
is circularly symmetric,
the differential entropy is
H
(
f
x
)=lndet(
πe
C
xx
)
.
(6
.
66)
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