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System identification was first performed by an MLP with
neurons in its hidden layer and then using the sliding-window
kernel RLS algorithm, for two different window sizes . For both
methods we applied time-embedding techniques in which the length
of the linear channel was known. More specifically, the used
MLP was a time-delay MLP with inputs, and the input vectors
for the kernel RLS algorithm were time-delayed vectors of length
,
. In
each iteration, system identification was performed by estimating
the output sample corresponding to the next input sample, and
comparing it to the actual output. The mean square error (MSE) for
both approaches is shown in Fig. 2. Most
noticeable is the fast convergence of the kernel RLS algorithm:
convergence time is of the order of the window length.
Further note that the structure of the nonlinear system has not
been exploited while performing identification. Obviously, the
presented kernel RLS method can be extended and used as the basis
of a more complex algorithm that models better the known system
structure. For instance, the solution to the nonlinear Wiener
identification problem could be found as the solution to two
coupled LS problems, where the first one applies a linear kernel
on the input data and the second one applies a nonlinear kernel on
the output data. Also, the implications of not knowing the correct
linear channel length remain to be studied. We will consider this
and other extensions as future research lines.
Figure 2:
MSE of the identification of the nonlinear Wiener system
of Fig. 1, for the standard method
using an MLP and for the window-based kernel RLS algorithm with
window length (thick curve) and (thin curve). A
change in filter coefficients of the nonlinear Wiener system was
introduced after iterations. The results were averaged out
over Monte-Carlo simulations.
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Next: Conclusions
Up: Example Problem: Identification of
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Pdf version (187 KB)
Steven Van Vaerenbergh
Last modified: 2006-03-08