## 4 RLS Adaptive Algorithm
### Performance of the Iterative Wiener Filter
The input signal is an AR(1) process, \(N=5000\) (available data samples), and \(\text{SNR}=20\) dB. Figure 3: Performance of the conventional Wiener filter (WF) and the iterative Wiener filter (IWF) for the identification of (**a**) the network echo path from Figure 1a and (**b**) the network echo path from Figure 1b. The input signal is an AR(1) process, \(N=5000\) (available data samples), and \(\text{SNR}=20\) dB. The performance gain of the iterative Wiener filter becomes more apparent in challenging scenarios, e.g., for a small value of \(N\) and/or in a low SNR environment [43]. It is known that the accuracy of the conventional Wiener filter is highly influenced in such adverse conditions [34; 35; 36]. In the following experiments from this subsection, we consider both these challenges, by using only \(N=1000\) data samples to estimate the statistics and setting \(\text{SNR}=10\) dB. In this framework, the identification of the network echo paths from Figure 1a,b is analyzed in Figure 5. As we can see, the conventional Wiener filter is not able to provide an accurate solution in this case. On the other hand, the iterative Wiener filter using \(P\ll L_{2}\) still achieves a reasonable attenuation of the misalignment. Moreover, it can be remarked that the iterative algorithm even with \(P=1\) clearly outperforms its conventional counterpart. These performances are also supported in Figure 6, where the estimated impulse responses provided by the conventional Wiener filter and the proposed iterative version (using \(P=2\)) are depicted, as compared to the true network impulse responses, corresponding to the previous experiment from Figure 5. As we can see, the iterative Wiener filter is able to obtain more accurate estimates as compared to its conventional counterpart. A similar behavior can be observed in Figures 7 and 8, where the identification of the acoustic echo paths from Figure 1c,d is assessed. As we can see in Figure 7, even if a larger value of \(P\) should be used here (as compared to the previous experiment from Figure 5), this value is much smaller as compared to \(L_{2}=20\) (e.g., \(P=L_{2}/4\) or \(L_{2}/5\)). In fact, the conventional Wiener filter is outperformed by the proposed iterative algorithm even for very small values of \(P\) (e.g., \(P=3\)). Finally, it is important to outline that smaller values of \(P\) can be used in these difficult scenarios (presented in Figures 5 and 7), as compared to the experiments reported in Figures 3 and 4, which were performed in more favorable conditions (i.e., \(N\gg L\) and good SNR). Figure 8 also supports the performance gain, showing the estimated acoustic impulse responses obtained with the conventional and iterative versions of the Wiener filter. In this figure, the iterative Wiener filter uses \(P=4\), whereas the other conditions are the same as in the experiment reported in Figure 7. It can be noticed that more accurate estimates can be obtained by using the proposed iterative Wiener filter, as compared to the solutions provided by the conventional Wiener benchmark. Figure 5: Performance of the conventional Wiener filter (WF) and the iterative Wiener filter (IWF) for the identification of (**a**) the network echo path from Figure 1a and (**b**) the network echo path from Figure 1b. The input signal is an AR(1) process, \(N=1000\) (available data samples), and \(\text{SNR}=10\) dB. Figure 6: Performance of the conventional Wiener filter (WF) and the iterative Wiener filter (IWF) using \(P=2\), for the identification of the network echo paths from Figure 1a,b. (**a**) The true network impulse response from Figure 1a; (**b**) the true network impulse response from Figure 1b; (**c**) estimated impulse response with the conventional WF, corresponding to Figure 1a; (**d**) estimated impulse response with the conventional WF, corresponding to Figure 1b; (**e**) estimated impulse response with the IWF, corresponding to Figure 1a; and (**f**) estimated impulse response with the IWF, corresponding to Figure 1b.