Identification of a Time-varying Nonlinear System

For the second experiment, we consider a nonlinear communications channel composed of a linear filter followed by a static nonlinearity, also known as a Wiener system. As the system input we choose a binary signal, $x_i \in \{-1,+1\}$, and $30$dB of white Gaussian noise is added at its output. The binary input signal is time-embedded with $4$ taps, resulting in the input space showing $16$ clusters. The static nonlinearity is $f(x) = \tanh(x)$. During the first $500$ iterations the linear filter is fixed as $H_1(z) = 1 -0.2663z^{-1} -0.5541z^{-2} + 0.1420z^{-3}$. On iteration $501$, the channel is abruptly switched to $H_2 = 1 + 0.1050z^{-1} - 0.3760z^{-2} -0.4284z^{-3}$, which is then changed linearly until becoming $H_3(z) = 1 -0.4326z^{-1} - 0.1656z^{-2} -0.3153z^{-3}$ on the $1500$-th iteration.

The algorithms use the following parameters: A Gaussian kernel with $\sigma=0.1$ and $\lambda = 0.01$ are chosen for ALD-KRLS and FB-KRLS. Both methods only require $16$ points in their dictionary, which is obtained for ALD-KRLS by setting $\nu=0.1$, and for FB-KRLS by fixing $M=16$. ALD-KRLS and RLS use a forgetting factor $\beta=0.99$, and for FB-KRLS $\mu$ is set to $0.8$. SW-KRLS uses a Gaussian kernel with $\sigma = 2$ and $M=50$. The results, averaged out over $100$ Monte Carlo simulations, are shown in Fig. 2. FB-KRLS obtains better performance than SW-KRLS, since the latter does not actively select significant patterns. Moreover, due to the fact that ALD-KRLS is not designed to be a tracking algorithm, it performs worst in this experiment. Table 1 displays the MSE averaged out over the last $500$ iterations.

Figure 2: Performance on a time-varying Wiener system. In the first zone (``$H_1$'') all algorithms reach optimal steady-state performance, and ALD-KRLS and FB-KRLS coincide. After the abrupt change at iteration $500$, only the tracking algorithms are able to recover.

Table 1: Performance comparison of final MSE values.

Algorithm	MSE	memory size
ALD-KRLS	$0.0951 \pm 0.1363$	$16$
RLS	$0.0084 \pm 0.0099$	n/a
SW-KRLS	$0.0020 \pm 0.0087$	$50$
FB-KRLS	$0.0012 \pm 0.0018$	$16$