Digital Signal Processing Reference
In-Depth Information
kk
is a unitary matrix (i.e., the Takagi factor of
P
(
v
)) and
L
is the circularity matrix (i.e., the singular values of
P
(
v
) are the circularity
coefficients
l
i
¼
[
P
(
s
)]
ii
appearing in (2.42)).
(c) Set
W
(
z
)
¼ U
(
v
)
H
B
(
z
).
where
U ¼ U
(
v
)
[ C
In step (a), the data is
whitened
in the sense that
C
(
v
)
¼ I
. Naturally, if the selected
scatter matrix is the covariance matrix, then the data is whitened in the conventional
sense. Since the whitening transform
B
is unique only up to left-multiplication by a
unitary matrix, GUT matrix
W ¼ U
H
B
is also a whitening transform in the conven-
tional sense but with an additional property that it diagonalizes the selected spatial
pseudo-scatter matrix.
As revealed in step (b) of the algorithm, the circularity matrix
L ¼
diag(
l
i
) has
singular values
l
1
,
...
,
l
k
of
P
(
v
) as its diagonal elements. It has been shown in
Theorem 3 of [47] that the circularity coefficient is
l
i
¼j
[
P
(
s
)]
ii
j
where
s ¼ C
(
s
)
1
=
2
s
denotes the whitened source whose
i
th component
is
p
[
C
(
s
)]
ii
s
i
¼ s
i
=
. For example, consider the SUT functional (i.e.,
C ¼C
,
P ¼P
).
Then
l
i
¼j
[
P
(
s
)]
ii
j¼jt
(
s
i
)
j¼jt
(
s
i
)
j=s
2
(
s
i
)
¼ l
(
s
i
)
(2
:
44)
that is, the
i
th circularity coefficient is equal to the circularity coefficient of the
i
th source
s
i
. Next consider the case that the GUT functional employs
C ¼ C
and
P ¼ P
kur
. Then
l
i
¼j
[
P
kur
(
s
)]
ii
j¼j
[
E
[(
s
H
s
)
ss
T
]]
ii
j¼jE
[
js
i
j
2
s
i
]
þ
(
k
1)
t
(
s
i
)
j:
Hence the
i
th circularity coefficient
l
i
is the modulus of a weighted sum of a 4th-order
and 2nd-order moment of the
i
th whitened source ˜
i
.
The following result has been proved in [47].
Theorem 7 Under the assumptions
G1 : C
(
s
)
and P
(
s
)
exists, and
G2 : circularity coefficients l
1
, ..., l
k
(the singular values of P
(
v
)
are distinct, that
is, l
i
=
l
j
for all
i
=
j
[
f
1
, ...,kg, the GUT functional W
(
z
)
¼ U
(
v
)
H
B
(
z
)
is
a separating matrix for the complex-valued ICA model.
As an example, consider the SUT functional. Then assumption G1 is equivalent with
the assumption that sources has finite variances. Assumption G2 implies (due to
(2.44)) that sources have distinct circularity coefficients.
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