## 4 RLS Adaptive Algorithm
### Performance of the RLS-SF
The forgetting factors of the conventional RLS filter and the proposed RLS-SF are set to \(\lambda=1-1/(KL)\) and \(\lambda_{j}=1-1/(KPL_{j})\), \(j=1,2\), respectively. Because the corresponding matrix \(\mathbf{H}\) of the acoustic impulse response is closer to full-rank, higher values of \(P\) are required in this framework. Nevertheless, as we can see in Figure 13, the RLS-SF using a value of \(P\) reasonably smaller as compared to \(L_{2}\) can reach a misalignment level closer to the conventional RLS algorithm. On the other hand, the tracking capability of the RLS-SF (for all the values of \(P\)) is better as compared to its conventional counterpart. In addition, we can further tune the forgetting factors of the RLS-SF in order to improve its accuracy (i.e., to reach a lower misalignment level), while reducing the forgetting factor of the conventional RLS algorithm for a faster tracking. Such a scenario is considered in Figure 14, where these parameters are set to \(\lambda_{j}=1-1/(2 KPL_{j})\), \(j=1,2\) and \(\lambda=1-1/[(K/2)L]\), respectively, by using \(K=10\). The other conditions are the same as in the previous experiment reported in Figure 13. As we can see in Figure 14, the initial  
Figure 12: Performance of the conventional RLS filter [with \(\lambda=1-1/(KL/2)\)] and the RLS-SF [with \(\lambda_{1}=1-1/(2 KPL_{1})\), \(\lambda_{2}=1-1/(2 KPL_{2})\), and \(P=3\)], using \(K=10\), for the identification of the network echo path from Figure 1b. (**a**) The true network impulse response, (**b**) estimated impulse response (after 5 s) with the conventional RLS filter, and (**c**) estimated impulse response (after 5 s) with the RLS-SF. Other conditions are the same as in Figure 11.  
convergence rate and the tracking reaction is basically the same for all the algorithms, but the proposed RLS-SF (for all the values of \(P\)) outperforms the conventional RLS filter in terms of the accuracy of its solution, achieving a lower misalignment level. Figure 7.0 Performance of the conventional RLS filter [with \(\lambda=1-1/(KL/2)\)] and the RLS-SF [with \(\lambda_{1}=1-1/(2 KPL_{1})\) and \(\lambda_{2}=1-1/(2 KPL_{2})\)], using \(K=10\), for the identification of the acoustic echo paths from Figure 1c,d. The input signal is an AR(1) process, SNR \(=10\) dB, and the echo path changes after 2.5 s.  
Figure 7.1 Performance of the conventional RLS filter [with \(\lambda=1-1/(KL)\)] and the RLS-SF [with \(\lambda_{1}=1-1/(KPL_{1})\) and \(\lambda_{2}=1-1/(KPL_{2})\)], using \(K=10\), for the identification of the acoustic echo paths from Figure 1c,d. The input signal is an AR(1) process, SNR \(=10\) dB, and the echo path changes after 2.5 s.  
Finally, in Figure 15, the robustness of the algorithms is assessed. To this purpose, we consider a variation of the background noise at time 2.5 s, for a period of 0.5 s, by reducing the SNR from 10 dB to 0 dB. Such variations can often happen in acoustic environments, e.g., due to different movements of objects and bodies, the presence of ventilation systems or other electrical devices, etc. In this experiment, the input signal is a speech sequence and the acoustic echo path from Figure 1c is considered. The forgetting factors of the algorithms are set as in the previous simulation (from Figure 14). It is interesting to note that the RLS-SF has the same robustness features despite the value of \(P\). Most importantly, the proposed algorithm reaches a lower misalignment level as compared to the conventional RLS benchmark. The estimated acoustic impulse responses corresponding to the previous experiment are depicted in Figure 16, where the RLS-SF uses \(P=8\). The results are compared to the true impulse response of the acoustic echo path from Figure 1c. It can be seen that a better accuracy of the estimate is obtained by the RLS-SF, as compared to the conventional RLS counterpart.