## 5 Simulation Results  
Simulations were performed in the context of SAEC using the WL framework. The input signal was filtered through two distinct real acoustic impulse responses in order to simulate the effect of the _left_ and _right_ channels and generate the samples \(x_{\mathbf{l}_{\mathrm{L}}}(n)\) and \(x_{\mathbf{R}_{\mathrm{c}}}(n)\). The input signal source was chosen as a Gaussian noise filtered through an auto-regressive (AR) system with a single pole and a speech sequence [25]. In order to overcome the problem of the correlation between the sequences \(x_{\mathbf{l}_{\mathrm{L}}}(n)\) and \(x_{\mathbf{R}_{\mathrm{c}}}(n)\), we employed the nonlinear pre-distortion approach presented in [4; 18]. The method implies the addition of a specific level of distortion to the input signals, \(x_{\mathbf{l}_{\mathrm{L}}}(n)\) and \(x_{\mathbf{R}_{\mathrm{c}}}(n)\), respectively, in order to obtain a unique solution. The namely _half-wave rectifier_ generates two new signals using the expressions [4; 9; 18]  
\[\mathbf{x}^{\prime}_{\mathbf{l}_{\mathrm{L}}}(n)=\mathbf{x}_{\mathbf{l}_{ \mathrm{c}}}(n)+\alpha_{\mathrm{r}}\frac{\mathbf{x}_{\mathbf{l}_{\mathrm{L}}} (n)+|\mathbf{x}_{\mathbf{l}_{\mathrm{L}}}(n)|}{2} \tag{35}\] \begin{table}
\begin{tabular}{l l l l} \hline \hline
**Step** & **Actions** & \(\times\) & \(+\) \\ \hline \multirow{4}{*}{0} & \multicolumn{2}{l}{Initialization :} & \multirow{4}{*}{\(\times\)} & \multirow{4}{*}{\(+\)} \\  & Set: \(\widetilde{\mathbf{h}}(0)=\mathbf{0}_{2L\times 1}\); & & \\  & \(\mathbf{R}(0)=\mathbf{\epsilon}_{2L}\), & \(\epsilon>0\), & \(q=0\) & \\  & & For \(n=1,2,\ldots\),number of iterations : & \\
1 & \multicolumn{2}{l}{Update \(\widetilde{\mathbf{x}}(n)\) using (22)} & \multirow{4}{*}{\(12L\)} & \multirow{4}{*}{\(8L\)} \\  & & Update \(\widetilde{\mathbf{x}}(n)\) using time-shift & \\  & & \(\mathbf{R}^{(1)}(n)=\lambda\mathbf{R}^{(1)}(n-1)+x^{*}(n)\widetilde{\mathbf{x} }(n)\) & & \\
3 & \(q=q+1\) & & \\
4 & \multicolumn{2}{l}{Determine \(e_{q}(n)\) using (32)} & \multirow{2}{*}{\(\leq 8L+4(N_{it}-1)N_{u}\)} & \multirow{2}{*}{\(\leq 6L-2+2N_{it}\)} \\  & & & \(+4N_{u}(N_{it}-1)\) \\ \hline
5 & \multicolumn{2}{l}{Determine \(\mathbf{p}_{0q}(n)\) using (33)} & \(4L(2N_{it}+1)\) & & \(4 LN_{it}\) \\ \hline
6 & \(\mathbf{R}(n)\Delta\widetilde{\mathbf{h}}_{q}(n)=\mathbf{p}_{0q}(n)\xrightarrow{ \mathrm{DCD}}\Delta\widetilde{\mathbf{h}}_{q}(n),\mathbf{r}_{q}(n)\) & & \(\leq M_{b}+\) \\  & & & \(+N_{it}[(8L+1)N_{u}]\) \\ \hline
7 & \multicolumn{2}{l}{Determine \(\widetilde{\mathbf{h}}_{q}(n)\) using (34)} & \multirow{2}{*}{\(\leq N_{it}N_{it}\)} \\  & & If \(q<N_{it}\xrightarrow{\mathrm{jump}}\) & Step 3 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: WL-DR-RLS-DCD algorithm. \[\mathbf{x}_{\mathrm{R_{c}}}^{\prime}(n)=\mathbf{x}_{\mathrm{R_{c}}}(n)+a_{\mathrm{ r}}\frac{\mathbf{x}_{\mathrm{R_{c}}}(n)-\left|\mathbf{x}_{\mathrm{R_{c}}}(n) \right|}{2}, \tag{36}\]  
where the parameter \(\alpha_{\mathrm{r}}<1\) controls the amount of nonlinearity. In practice, the stereo effect is considered not to be drastically affected for the range \(0<\alpha_{\mathrm{r}}\leq 0.5\). The performance of the proposed WL-DR-RLS-DCD algorithm has been validated in the context of the identification of the interleaved impulse response \(\widetilde{\mathbf{h}}(n)\), with respect to the results provided by the version with \(N_{it}=1\) DR iterations (i.e., the WL-RLS-DCD). The performance has been measured using the normalized misalignment, having the following expression in the WL context:  
\[\mathrm{Mis}(n)=20\log_{10}\frac{\left\|\widetilde{\mathbf{h}}_{\mathrm{t}}- \widetilde{\mathbf{h}}(n)\right\|_{2}}{\left\|\widetilde{\mathbf{h}}_{\mathrm{ t}}\right\|_{2}}\quad\mathrm{[dB]}, \tag{37}\]  
where \(\left\|\cdot\right\|_{2}\) denotes the \(\ell_{2}\) norm [4]. The forgetting factor was chosen with the form \(\lambda=1-1/(KL)\), and the parameters specific to the DCD iterations were set to \(H=1\), \(N_{u}=4\), and \(M_{b}=16\).