## 4 RLS Adaptive Algorithm
### Performance of the Iterative Wiener Filter  
In this first subsection, the performance of the proposed iterative Wiener filter (developed in Section 3) is investigated, as compared to the conventional Wiener filter. The performance measure is the normalized misalignment (in dB), which is evaluated as \(20\text{log}_{10}\|\mathbf{h}-\mathbf{g}_{\text{w}}\|/\|\mathbf{h}\|\), where \(\|\cdot\|\) denotes the Euclidean norm. In case of the iterative Wiener filter, the estimate of the impulse response is obtained as explained in the end of Section 3, while the conventional Wiener solution results in \(\mathbf{g}_{\text{W}}=\mathbf{R}_{\mathbf{x}}^{-1}\mathbf{r}_{\text{x}d}\). The statistics \(\mathbf{R}_{\mathbf{x}}\) and \(\mathbf{r}_{\text{x}d}\) are required within both the conventional and iterative Wiener filters. These can be estimated by averaging over \(N\) available data samples of \(x(t)\) and \(d(t)\), so that the covariance matrix can be evaluated as \((1/N)\sum_{t=1}^{N}\mathbf{x}(t)\mathbf{x}^{T}(t)\), whereas the cross-correlation vector results in \((1/N)\sum_{t=1}^{N}\mathbf{x}(t)d(t)\). In the experiments reported in this subsection, the input signal is an AR(1) process with a pole at 0.9. The accuracy of the statistics' estimates reflects the accuracy of the Wiener solution, so that a higher value of \(N\) is desirable, i.e., \(N\gg L\). However, in practice, only a limited set of data could be available, which raises significant challenges. Because the proposed iterative Wiener filter combines the solution of two shorter filters (of lengths \(PL_{1}\) and \(PL_{2}\), with \(P\ll L_{2}\)), an improved accuracy of the solution is expected in such challenging scenarios, as outlined in the end of Section 3. In the first experiment, we consider that a large amount of data is available (i.e., \(N=10L=5000\) samples) for the identification of the network echo paths from Figure 1a,b. In addition, we set \(\text{SNR}=20\) dB, so that the influence of the external noise on the accuracy of the statistics' estimates is mild. The results are provided in Figure 3, comparing the conventional Wiener filter with the solution obtained by the proposed iterative algorithm using different values of \(P\). As we can notice, the iterative Wiener filter with \(P=L_{2}/10=2\) is able to match the accuracy of the conventional Wiener solution, whereas for \(P=3\) it even outperforms its conventional counterpart. This supports the discussion related to Figure 2a,b, where the singular values of \(\mathbf{H}\) decrease very quickly to zero, so that the rank of this matrix is much smaller than \(L_{2}=20\). Figure 2: Singular values of the matrix \(\mathbf{H}\) of size \(L_{1}\times L_{2}\) (normalized with respect to the maximum one), with \(L_{1}=25\) and \(L_{2}=20\), corresponding to: (**a**) the impulse response from Figure 1a; (**b**) the impulse response from Figure 1b; (**c**) the impulse response from Figure 1c; and (**d**) the impulse response from Figure 1d. A similar experiment is performed in Figure 4, for the identification of the acoustic echo paths from Figure 1c,d. The other conditions remain the same, i.e., \(N=5000\) and \(\text{SNR}=20\) dB. As previously indicated in Figure 2c,d, the rank of matrix \(\mathbf{H}\) is larger for the acoustic impulse responses. Nevertheless, a reasonably small value of \(P\) (as compared to \(L_{2}\)) can still be used even in such scenarios. The results provided in Figure 4 show that a value of \(P<L_{2}/2\) is sufficient for the iterative Wiener filter to achieve the accuracy of the conventional Wiener solution. In addition, a reasonable attenuation of the misalignment is obtained by the iterative algorithm even for a smaller value of its specific parameter, e.g., \(P<L_{2}/3\). Figure 4: Performance of the conventional Wiener filter (WF) and the iterative Wiener filter (IWF) for the identification of (**a**) the acoustic echo path from Figure 1c and (**b**) the acoustic echo path from Figure 1d.