## 4 RLS Adaptive Algorithm
### Performance of the RLS-SF
Even when using the smallest value \(P=2\), the proposed algorithm is able to match the misalignment of the conventional RLS filter. In addition, the tracking capabilities of the RLS-SF are generally still better as compared to its conventional counterpart. The next experiment considers a more challenging scenario, wherein the input signal is a speech signal and the SNR is reduced to 10 dB. The other settings remain the same as in the previous simulation. As we can see in Figure 11, the RLS-SF outperforms the conventional RLS algorithm in terms of the main performance criteria, for all the values of \(P\ll L_{2}\). This behavior also supports the results reported in the previous subsection (for the iterative Wiener filter), which indicate that the proposed decomposition-based approach with the symmetric filters performs better (as compared to the conventional benchmarks) especially in challenging conditions, e.g., in noisy environments. The normalized misalignment represents the accuracy of the solution provided by the adaptive filter, i.e., the distance between the true impulse response of the system and its estimate. Consequently, the misalignment level is influenced by the SNR. This can be seen when comparing the results from Figures 10 and 11, where most of the conditions are similar, but the SNRs are different. Although in Figure 10 the misalignment range is between \(-30\) to \(-20\) dB, in Figure 11 we can see a higher misalignment level, i.e., around \(-15\) to \(-10\) dB. Nevertheless, in both  
Figure 9: 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 network echo paths from Figure 1a,b. The input signal is an AR(1) process, \(\text{SNR}=20\) dB, and the echo path changes after 2.5 s.  
scenarios, the proposed RLS-SF reaches lower misalignment levels as compared to the conventional RLS filter. Figure 11: 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 network echo paths from Figure 1a,b. The input signal is a speech sequence, \(\text{SNR}=10\) dB, and the echo path changes after 2.5 s.  
Figure 10: 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 network echo paths from Figure 1a,b. The input signal is an AR(1) process, \(\text{SNR}=20\) dB, and the echo path changes after 2.5 s.  
The performance gain is also reflected in the accuracy of the estimated acoustic impulse responses (obtained by the conventional RLS filter and the RLS-SF), as indicated in Figure 12. In this figure, the RLS-SF uses \(P=3\), whereas the other conditions are the same as in the experiment reported in Figure 11. As we can see, the accuracy of the estimate provided by the RLS-SF is superior to the estimate of the conventional RLS filter. Even if the estimated impulse response of the echo path is not really required in echo cancellation (where the error signal is transmitted back to the far-end), the results from Figure 12 validate the performance features of the proposed RLS-SF. The last set of simulations focuses on the identification of the acoustic echo paths from Figure 1c,d. Because the background noise is usually considered stronger in the context of acoustic echo cancellation, while it is also amplified by the microphone of the hands-free device [2], we set \(\text{SNR}=10\) dB in all the following experiments. In the simulation provided in Figure 13, the input signal is an AR(1) process and the acoustic impulse response from Figure 1c is considered for the first 2.5 s, then the echo path abruptly changes into the impulse response from Figure 1d.