Digital Signal Processing Reference
In-Depth Information
presented in [18], that it is not necessary to have
W
(0) orthogonal to ensure the
convergence, since the MALASE algorithm steers
W
(
k
) towards the manifold of
orthogonal matrices. The MALASE algorithm seems to involve high computational
cost, due to the matrix exponential that applies in (4.67). However, since
exp{
m
k
[
L
1
(
k
)
y
(
k
)
y
T
(
k
)
y
(
k
)
y
T
(
k
)
L
1
(
k
)]} is the exponential of a sum of two
rank-one matrices, the calculation of this matrix requires only
O
(
n
2
) operations
[18]. Originally, this algorithm which updates the EVD of the covariance matrix
C
x
(
k
) can be modified by a simple preprocessing to estimate the principal or minor
r
signal eigenvectors only, when the remaining
n
2
r
eigenvectors are associated
with a common eigenvalue
s
2
(
k
) (see Subsection 4.3.1). This algorithm, denoted
MALASE(
r
) requires
O
(
nr
) operations by iteration. Finally, note that a theoretical
analysis of convergence has been presented in [18]. It is proved that in stationary
environments, the stationary stable points of the algorithm (4.66), (4.67) correspond
to the EVD of
C
x
. Furthermore, the covariance of the asymptotic distribution of the
estimated parameters is given for Gaussian independently distributed data
x
(
k
)
using general results of Gaussian approximation (see Subsection 4.7.2).
4.6.5 Particular Case of Second-order Stationary Data
Finally, note that for
x
(
k
)
¼
[
x
(
k
),
x
(
k
1),
...
,
x
(
k nþ
1)]
T
comprising of time
delayed versions of scalar valued second-order stationary data
x
(
k
), the covariance
matrix
C
x
(
k
)
¼
E[
x
(
k
)
x
T
(
k
)] is Toeplitz and consequently centro-symmetric. This
property occurs in important applications—temporal covariance matrices obtained
from a uniform sampling of a second-order stationary signals, and spatial covariance
matrices issued from uncorrelated and band-limited sources observed on a centro-
symmetric sensor array (e.g. on uniform linear arrays). This centro-symmetric
structure of
C
x
allows us to use for real-valued data, the property
9
[14] that its EVD
can be obtained from two orthonormal eigenbases of half-size real symmetric
matrices. For example if
n
is even,
C
x
can be partitioned as follows
"
#
C
2
C
1
C
x
¼
C
2
JC
1
J
where
J
is a
n
/
2
n
/
2 matrix with ones on its antidiagonal and zeroes elsewhere.
Then, the
n
unit 2-norm eigenvectors
v
i
of
C
x
are given by
n
/
2 symmetric and
n
/
2
where
1
i
¼+
1, respectively issued from
1
2
u
i
1
i
Ju
i
p
skew symmetric vectors
v
i
¼
the unit 2-norm eigenvectors
u
i
of
C
1
þ1
i
JC
2
¼
2
E{[
x
0
(
k
)
þ1
i
Jx
00
(
k
)][
x
0
(
k
)
þ
1
i
Jx
00
(
k
)]
T
} with
x
(
k
)
¼
[
x
0T
(
k
),
x
00T
(
k
)]
T
. This property has been exploited [23, 26]
1
9
Note that for Hermitian centro-symmetric covariance matrices, such property does not extend. But any
eigenvector
v
i
satisfies the relation [
v
i
]
k
¼ e
if
i
[
v
i
]
nk
, that can be used to reduce the computational cost
by a factor of two.
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