Biomedical Engineering Reference
In-Depth Information
FIgURE 3.5:
Wiener filter testing. The actual hand trajectory (dotted red line) and the estimated hand
trajectory (solid black line) in the
x
,
y
, and
z
coordinates for a 3D hand-reaching task during a sample
part of the test data set.
where τ is the settling time (four times the time constant), λ
min
is the smallest of the eigenvalues of
the input correlation matrix, and
M
is the misadjustment. Therefore, to speed up the convergence
(i.e., less samples to reach the neighborhood of the optimal weight vector), the misadjustment
increases proportionally (i.e., the algorithm rattles around the optimal weight value with a larger
radius).
Practically, the LMS algorithm has other difficulties, in particular, when the input signal has
large dynamical range and the stepsize is chosen for fast convergence. Indeed, in any adaptive algo-
rithm, the weight vector contains all the information extracted from previous data. If the algorithm
diverges for some reason, all the past data information is effectively lost. Therefore, avoiding diver-
gence is critical. Since Because the theory for setting the stepsize is based on temporal or statistical
averages, an algorithm with a stepsize that converges on average, can transiently diverge. This will
happen in times of large errors (difficult to identify) or when the input signal has abrupt large values.
To avoid this transient behavior, the normalize stepsize algorithm was developed [
2
].
The normalized stepsize algorithm is the solution of an optimization problem where the
norm of the weight update is minimized subject to the constraint of zero error, that is,
2
( )
(3.20)
J
=
w n
(
+
1
)
−
w n
( )
+
λ
( ( )
d n w n
−
(
+
1
)
x n
)
