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Simulation Results

System identification was first performed by an MLP with $ 8$ neurons in its hidden layer and then using the sliding-window kernel RLS algorithm, for two different window sizes $ N$. For both methods we applied time-embedding techniques in which the length $ L$ of the linear channel was known. More specifically, the used MLP was a time-delay MLP with $ L$ inputs, and the input vectors for the kernel RLS algorithm were time-delayed vectors of length $ L$, $ \textbf{s}(n) = [\textbf{s}(n-L+1),\dots,\textbf{s}(n)]$. 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 $ N=150$ (thick curve) and $ N=75$ (thin curve). A change in filter coefficients of the nonlinear Wiener system was introduced after $ 500$ iterations. The results were averaged out over $ 250$ Monte-Carlo simulations.
MSE of the identification of the nonlinear Wiener system
of Fig. 1


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Next: Conclusions Up: Example Problem: Identification of Previous: Experimental Setup
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Steven Van Vaerenbergh
Last modified: 2006-03-08