## 4 RLS Adaptive Algorithm
### Performance of the RLS-SF  
This subsection is dedicated to the proposed RLS-SF (from Section 4), which is compared in the following to its conventional counterpart, i.e., the RLS adaptive filter [34; 35; 36]. The performance measure is the normalized misalignment (in dB), which is now evaluated as \(20\log_{10}\lVert\mathbf{h}-\widehat{\mathbf{g}}(t)\rVert/\lVert\mathbf{h}\rVert\), where \(\widehat{\mathbf{g}}(t)\) represents the adaptive filter estimate at time index \(t\). In order to assess the tracking capabilities of the algorithms, an abrupt change of the echo path is introduced in most of the following experiments, by changing the impulse response in the middle of simulation. In Figure 9, the identification of the network echo paths from Figure 1a,b is evaluated. Initially, the impulse response from Figure 1a is used for the first \(2.5\,\mathrm{s}\). Then, there is an abrupt change of the system and the impulse response from Figure 1b is involved. The forgetting factors of the algorithms are set in order to achieve the same misalignment level, so that we can focus only on the comparisons related to the convergence rate and tracking behavior. Consequently, the conventional RLS adaptive filter uses \(\lambda=1-1/(KL)\), whereas the parameters of the RLS-SF are similarly set (according to the filters' lengths) to \(\lambda_{1}=1-1/(KPL_{1})\) and \(\lambda_{2}=1-1/(KPL_{2})\), with \(K=10\), \(L=500\), \(L_{1}=25\), and \(L_{2}=20\). As we can see in Figure 9, the initial convergence rate is almost the same for all the algorithms, but the tracking capability of the RLS-SF is much better as compared to the conventional benchmark, for all the values of \(P\) that are used in this simulation. Most importantly, we should note that all these values satisfy the condition \(P\ll L_{2}\), which leads to a significant gain in terms of the computational complexity. As expected, for a value of \(P\) smaller than the rank of the matrix \(\mathbf{H}\) (e.g., \(P=2\)), the RLS-SF pays with a higher misalignment level as compared to the conventional RLS algorithm. However, even in this case, it fully compensates in terms of the computational complexity and tracking capability. Figure 8: Performance of the conventional Wiener filter (WF) and the iterative Wiener filter (IWF) using \(P=4\), for the identification of the acoustic echo paths from Figure 1c,d. (**a**) The true acoustic impulse response from Figure 1c; (**b**) the true acoustic impulse response from Figure 1d; (**c**) estimated impulse response with the conventional WF, corresponding to Figure 1c; (**d**) estimated impulse response with the conventional WF, corresponding to Figure 1d; (**e**) estimated impulse response with the IWF, corresponding to Figure 1c; and (**f**) estimated impulse response with the IWF, corresponding to Figure 1d. Other conditions are the same as in Figure 7.  
These results suggest another strategy for compromising between the main performance criteria of the RLS-SF. Because its tracking behavior is better as compared to the conventional RLS algorithm, we could increase the forgetting factors of the RLS-SF, targeting a lower misalignment, while paying with a slightly reduced tracking capability. On the other hand, the forgetting factor of the conventional RLS algorithm could be reduced in order to improve its tracking capability, but this leads to a higher misalignment level. Such a scenario is considered in Figure 9, where the conditions are similar to the previous experiment from Figure 10, but the forgetting factors are set as discussed before. Consequently, the conventional RLS algorithm uses a smaller forgetting factor, i.e., \(\lambda=1-1/[(K/2)L]\), whereas the forgetting factors of the RLS-SF are increased to \(\lambda_{1}=1-1/(2 KPL_{1})\) and \(\lambda_{2}=1-1/(2 KPL_{2})\), with \(K=10\). As expected, the results obtained in Figure 10 show that the RLS-SF outperforms the conventional RLS algorithm in terms of accuracy, reaching a lower misalignment level for most of the values of \(P\) used in this experiment.