Digital Signal Processing Reference
In-Depth Information
Circularity Properties of a Complex Random Variable and
Random Vector
An important property of complex-valued random variables
is related to their circular nature.
A zero-mean complex random variable is called
second-order circular
[91] (or proper
[81, 101]) when its pseudo-covariance is zero, that is,
E
{
X
2
}
¼
0
which implies that
s
X
r
¼ s
X
i
and
E
{
X
r
X
i
}
¼
0 where
s
X
r
and
s
X
i
are the standard
deviations of the real and imaginary parts of the variable.
For a random vector
X
, the condition for second-order circularity is written in terms
of the pseudo-covariance matrix as
P ¼
0, which implies that
E
{
X
r
X
r
}
¼ E
{
X
i
X
i
}
and
E
{
X
r
X
i
}
¼E
{
X
i
X
r
}.
A stronger condition for circularity is based on the pdf of the random variable.
A random variable
X
is called
circular in the strict-sense
, or simply
circular
,if
X
and
Xe
ju
have the same pdf, that is, the pdf is rotation invariant [91].
In this case, the phase is non-informative and the pdf is a function of only themagnitude,
f
X
(
x
)
¼ g
(
jxj
) where
g
:
R
7!
R
, hence the pdf can be written as a function of
zz
rather
than
z
and
z
separately. A direct consequence of this property is that
EfX
p
(
X
)
q
g¼
0
for all
p
=
q
if
X
is circular. Circularity is a strong property, preserved under linear
transformations, and since it implies noninformative phase, a real-valued approach
and a complex-valued approach for this case are usually equivalent [109].
As one would expect, circularity implies second-order circularity, and only for a
Gaussian-distributed random variable, second-order circularity implies (strict sense)
circularity. Otherwise, the reverse is not true.
For random vectors, in [91], three different types of circularity are identified. A
random vector
X
[ C
N
is called
† marginally circular
if each component of the random vector
X
n
is a circular
random variable;
† weakly circular
if
X
and
Xe
ju
have the same distribution for any given
u
; and
† strongly circular
if
X
and
X
0
have the same distribution where
X
0
is formed by
rotating the corresponding entries (random variables) in
X
by
u
n
, such that
X
0
n
¼ X
n
e
ju
n
. This condition is satisfied when
u
k
are independent and identically
distributed random variables with uniform distribution in [
2
p
,
p
] and are
independent of the amplitude of the random variables,
X
n
.
As the definitions suggest, strong circularity implies weak circularity, and weak
circularity implies marginal circularity.
Differential Entropy of Complex Random Vectors
The differential
entropy of a zero mean random vector
X
[ C
N
is given by the joint entropy
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