Digital Signal Processing Reference
In-Depth Information
(RLS) based on the matrix inversion lemma have been proposed.
7
We note that because
of the approximation of
J
(
W
)by
J
0
(
W
), the columns of
W
(
k
) are not exactly orthonor-
mal. But this lack of orthonormality does not mean that we need to perform a reortho-
normalization of
W
(
k
) after each update. For this algorithm, the necessity
of orthonormalization depends solely on the post processing method which uses this
signal subspace estimate to extract the desired signal information (see e.g. Section
4.8). It is shown in the numerical experiments presented in [71] that the deviation of
W
(
k
) from orthonormality is very small and for a growing sliding window (
b¼
1),
W
(
k
) converges to a matrix with exactly orthonormal columns under stationary
signal. Finally, note that a theoretical study of convergence and a derivation of
the asymptotic distribution of the recursive subspace estimators have been presented
in [73, 74] respectively. Using the ODE associated with this algorithm (see
Section 4.7.1) which is here a pair of coupled matrix differential equations, it is
proved that under signal stationarity and other weak conditions, the PAST algorithm
converges to the desired signal subspace with probability one.
To speed up the convergence of the PAST algorithm and to guarantee the ortho-
normality of
W
(
k
) at each iteration, an orthonormal version of the PAST algorithm
dubbed OPAST has been proposed in [1]. This algorithm consists of the PAST algor-
ithm where
W
(
k þ
1) is related to
W
(
k
)by
W
(
k þ
1)
¼ W
(
k
)
þp
(
k
)
q
(
k
), plus an
orthonormalization step of
W
(
k
) based on the same approach as those used in the
FRANS algorithm (see Subsection 4.5.1) which leads to the update
W
(
k þ
1)
¼
W
(
k
)
þp
0
(
k
)
q
(
k
).
Note that the PAST algorithm cannot be used to estimate the noise subspace by
simply changing the sign of the stepsize because the associated ODE is unstable.
Efforts to eliminate this instability were attempted in [4] by forcing the orthonormality
of
W
(
k
) at each time step. Although there was a definite improvement in the stability
characteristics, the resulting algorithm remains numerically unstable.
4.5.3 Additional Methodologies
Various generalizations of criteria (4.7) and (4.14) have been proposed (e.g. in [41]),
which generally yield robust estimates of principal subspaces or eigenvectors that
are totally different from the standard ones. Among them, the following novel
information criterion (NIC) [48] results in a fast algorithm to estimate the principal
subspace with a number of attractive properties
def
Tr[ln(
W
T
CW
)]
Tr(
W
T
W
)
ma
W
{
J
(
W
)} with
J
(
W
)
¼
(4
:
46)
given that
W
lies in the domain
fW
such that
W
T
CW
.
0
g
, where the matrix
logarithm is defined, for example, in [35, Chap. 11]. It is proved in [48] (see also
7
For possible sudden signal parameter changes (see Subsection 4.3.1), the use of a sliding exponential
window (4.21) version of the cost function may offer faster convergence. In this case,
W
(
k
) can be calculated
recursively as well [71] by applying the general form of the matrix inversion lemma (
AþBDC
T
)
1
¼
A
1
A
1
B
(
D
1
þC
T
A
1
B
)
1
C
T
A
1
which requires inversion of a 2
2 matrix.
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