Digital Signal Processing Reference
In-Depth Information
eigenvalues of the matrix
R
(
z
). In deference to [22], we shall call
l
i
(
i ¼
1,
...
,
k
) the
i
th circularity coefficients of
z
and we write
L ¼ L
(
z
)
¼
diag(
l
i
) for the
k k
matrix
of circularity coefficients. In [54], it has been shown that circularity coefficients are the
canonical correlations between
z
and its conjugate
z
. It is easy to show that circularity
coefficients are singular values of the symmetric matrix
K
(
z
)
W
B
(
z
)
P
(
z
)
B
(
z
)
T
(called
the coherence matrix in [54]), where
B
(
z
) is any square-root matrix of
C
(
z
)
1
(i.e.,
C
(
z
)
1
¼ B
(
z
)
H
B
(
z
)). This means that there exists a unitary matrix
U
such that
symmetric matrix
K
(
z
) has a special form of singular value decomposition (SVD),
called
Takagi factorization
, such that
K
(
z
)
¼ ULU
T
. Thus, if we now define matrix
W
[ C
kk
as
W¼ B
H
U
,where
B
and
U
are defined as above, then we observe
that the transformed data
s ¼W
H
z
satisfies
C
(
s
)
¼ U
H
BC
(
z
)
B
H
U ¼ I
and
P
(
s
)
¼ U
H
BP
(
z
)
B
T
U
¼ L
that is, transformed RV
s
has (strongly-)uncorrelated components. Hence the matrix
W
is called the strong-uncorrelating transform (SUT) [21, 22].
Note that
L ¼
0
,P¼
0
,R¼
0
:
As in the univariate case, circularity coefficients are invariant under the group of
invertible linear transformations {
z ! s ¼GzjG
[ C
kk
nonsingular }, that is,
l
i
(
z
)
¼ l
i
(
s
). Observe that the set of circularity coefficients {
l
i
(
z
),
i ¼
1,
...
,
k
}of
the RV
z
does not necessarily equal the set of circularity coefficient of the variables
{
l
(
z
i
),
i ¼
1,
...
,
k
} although in some cases (for example, when the components
z
1
,
...
,
z
k
of
z
are mutually statistically independent) they can coincide.
2.3 COMPLEX ELLIPTICALLY SYMMETRIC (CES) DISTRIBUTIONS
k
has
k
-variate circular CN distribution if its real and imaginary
part
x
and
y
have 2
k
-variate real normal distribution and a 2
k
2
k
real covariance
matrix with a special form (2.6), that is,
P
(
z
)
¼
0. Since the introduction of the circu-
lar CN distribution in [24, 61], the assumption (2.6) seems to be commonly
thought of as essential—although it was based on application specific reasoning—
in writing the joint pdf
f
(
x
,
y
)of
x
and
y
into a natural complex form
f
(
z
). In fact,
the prefix “circular” is often dropped when referring to circular CN distribution,
as it has due time become the commonly accepted complex normal distribution.
However, rather recently, in [51, 57], an intuitive expression for the joint density of
normal RVs
x
and
y
was derived without the unnecessary second-order circularity
assumption (2.6) on their covariances. The pdf of
z
with CN distribution is uniquely
parametrized by the covariance matrix
C
and pseudo-covariance matrix
P
, the case of
vanishing pseudo-covariance matrix,
P¼
0, thus indicating the (sub)class of circular
CN distributions.
Random vector
z
of
C
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