Digital Signal Processing Reference
In-Depth Information
H
(
X
r
,
X
i
), and satisfies [81]:
H
(
X
)
log[(
pe
)
N
det(
C
)]
(1
:
21)
with equality if, and only if,
X
is second-order circular and Gaussian with zero mean.
Thus, it is a
circular
Gaussian random variable that maximizes the entropy for the
complex case. It is also worthwhile to note that orthogonality and Gaussianity,
together do not imply independence for complex Gaussian random variables, unless
the variable is circular.
For a noncircular Gaussian random vector, we have [33, 100]
2
log
Y
N
1
H
noncirc
¼
log[(
pe
)
N
det(
C
)]
(1
k
n
)
þ
|
{z
}
H
circ
n¼
1
where
k
n
are the singular values of
P
as defined and
k
n
¼
0 when the random vector
is circular. Hence, the circularity coefficients provide an attractive measure for quan-
tifying circularity and a number of those measures are studied in [100]. Since
k
n
1
for all
n
, the second term is negative for noncircular random variables decreasing the
overall differential entropy as a function of the circularity coefficients.
Complex Random Processes
In [8, 27, 81, 90, 91], the statistical character-
ization and properties of complex random processes are discussed in detail. In particu-
lar, [91] explores the strong relationship between stationarity and circularity of a
random process through definitions of circularity and stationarity with varying degrees
of assumptions on the properties of the process.
In our introduction to complex random processes, we focus on discrete-time pro-
cesses and primarily use the notations and terminology adopted by [81] and [91].
The covariance function for a complex discrete-time random process
X
(
n
) is written as
c
(
n
,
m
)
¼ E
{
X
(
n
)
X
(
m
)}
E
{
X
(
n
)}
E
{
X
(
m
)}
and the correlation function as
EfX
(
n
)
X
(
m
)
g
.
To completely define the second-order statistics, as in the case of random variables,
we also define the pseudo-covariance function [81]—also called the complementary
covariance [101] and the relation function [91]—as
p
(
n
,
m
)
¼ E
{
X
(
n
)
X
(
m
)}
E
{
X
(
n
)}
E
{
X
(
m
)}
:
In the sequel, to simplify the expressions, we assume zero mean random processes,
and hence, the covariance and correlation functions coincide.
Stationarity and Circularity Properties of Random Processes
A
random signal
X
(
n
) is stationary if all of its statistical properties are invariant to any
given time shift (translations by the origin), or alternatively,
if the family of
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