Digital Signal Processing Reference
In-Depth Information
10
1
10
0
(2)
(1)
(3)
10
−1
(0)
10
−2
0
50
100
150
200
250
300
350
400
450
500
Iteration number
2
Fro
) averaging 100
independent runs for the Oja's algorithm (1) and the smoothed Oja's algorithm with
a¼
Figure 4.2 Learning curves of the mean square error
E
(kP(k)P
k
1 (2) and a¼
0.3 (3) compared with mTr(C
P
) (0) in the same configuration [C
x
, W(0)]
as Figure 4.1.
defined in [25]. Note that when
x
(
k
)
¼
(
x
k
,
x
k
1
,
...
,
x
knþ
1
)
T
with
x
k
being an
ARMA stationary process, the covariance of the field (4.74) and thus
l
i,j
can be
expressed in closed form with the help of a finite sum [23].
The domain of learning rate
m
for which the previously described asymptotic
approach is valid and the performance criteria for which no analytical results could
be derived from our first-order analysis, such as the speed of convergence and the
deviation from orthonormality
d
2
(
m
)
¼
2
Fro
can be derived from
numerical experiments only. In order to compare Oja's algorithm and the smoothed
Oja's algorithm, the associated parameters
m
and (
a
,
m
) must be constrained to give
the same value of
m
Tr(
C
P
). In these conditions, it has been shown in [25] by numerical
simulations that the smoothed Oja's algorithm provides faster convergence and a
smaller deviation from orthonormality
d
2
(
m
) than Oja's algorithm. More precisely,
it has been shown that
d
2
(
m
)
/ m
2
[resp.
/m
4
] for Oja's [resp. the smoothed
Oja's] algorithm. This result agrees with the presentation of Oja's algorithm given
in Subsection 4.5.1 in which the term
O
(
m
k
) was omitted from the orthonormalization
of the columns of
W
(
k
).
Finally, using the theorem of continuity (e.g. [59, Th. 6.2a]), note that the behavior
of any differentiable function of
P
(
k
) can be obtained. For example, in DOA tracking
def
kW
T
(
k
)
W
(
k
)
I
r
k
Search WWH ::
Custom Search