## 4 WL-DR-RLS-DCD  
In the recent literature, for the process of determining \(\widetilde{\mathbf{h}}(n)\) from \(\widetilde{\mathbf{h}}(n-1)\) when employing RLS adaptive algorithms, the DR approach was introduced in order to improve their corresponding tracking speeds [23; 24]. Correspondingly, in order to update the filter's coefficients multiple times at the same time index \(n\) (i.e., for the same input data), Steps 3-7 from Table 1 will be run in a loop for a number of \(N_{it}\) iterations (including the DCD method). We call the proposed algorithm the WL-DR-RLS-DCD, which is equivalent to the WL-RLS-DCD for \(N_{it}=1\). For each of the \(N_{it}\) DR iterations, the algorithm will determine a new solution vector \(\Delta\widetilde{\mathbf{h}}_{q}(n)\), \(q=0,\ldots,N_{it}-1\) based on the same input data [i.e., \(\widetilde{\mathbf{x}}(n)\) and \(\mathbf{R}(n)\)] and add its contribution to the filter estimate \(\widetilde{\mathbf{h}}(n-1)\) accumulated with all previous DR iterations from time index \(n\). The complex reference \(d(n)\) will remain the same for all the \(N_{it}\) steps, as no updated value will be made available from outside the adaptive system. However, the error signal and the residual contribution will be updated accordingly and will go through \(N_{it}\) stages. We will distinguish them using the notations \(e_{q}(n)\) and \(\mathbf{p}_{0,q}(n)\), with \(q=0,\ldots,N_{it}-1\). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Step** & **Actions** & \(\times\) & \(+\) \\ \hline \multirow{6}{*}{0} & Initialization : & & \\  & Set: \(\Delta\widetilde{\mathbf{h}}=\mathbf{0}_{2L\times 1}\); \(\mathbf{r}_{aux}=\mathbf{p}_{0}(n)\) & & \\  & \(\alpha=H\), \(m=0\) & & \\  & For \(k=1,2,\ldots,N_{it}:\) & & \\
1 & \((val_{1};pos_{1})=\max\{|\mathfrak{Re}\{\mathbf{r}_{aux}\}|\}\) & \(2L-1\) & \\  & \((val_{2};pos_{2})=\max\{|\mathfrak{Im}\{\mathbf{r}_{aux}\}|\}\) & \(2L-1\) & \\  & \((p;\eta)=(val_{1}>val_{2})\)? \((pos_{1};1)\) : \((pos_{2};j)\) & \(1\) \\  & Jump to Step 4 & & \\
2 & \(\alpha=\alpha/2;m=m+1\) & & \\
3 & If \(m>M_{b}\)\(\xrightarrow{\text{return}}\) & \(\left\{\Delta\widetilde{\mathbf{h}};\mathbf{r}_{aux}\right\}\) as \(\left\{\Delta\widetilde{\mathbf{h}}(n);\mathbf{r}(n)\right\}\) & \\
4 & \(r_{temp}=(\eta==1)\)? \(\mathfrak{Re}\{\mathbf{r}_{aux,p}\}\) : \(\mathfrak{Im}\{\mathbf{r}_{aux,p}\}\) & \\  & If \(\left|r_{temp}\right|\leq\frac{\epsilon}{2}R_{p,p}\), \(\xrightarrow{\text{jump}}\) & Step 2 & 1 \\
5 & \(\Delta\widetilde{\mathbf{h}}_{p}=\Delta\widetilde{\mathbf{h}}_{p}+\text{ sign}\{r_{temp}\}\eta\alpha\) & \(1\) \\
6 & \(\mathbf{r}_{aux}=\mathbf{r}_{aux}-\text{sign}\{r_{temp}\}\eta\alpha\mathbf{R}_ {p}\) & \(4L\) \\  & Return \(\left\{\Delta\widetilde{\mathbf{h}};\mathbf{r}_{aux}\right\}\) as \(\left\{\Delta\widetilde{\mathbf{h}}(n);\mathbf{r}(n)\right\}\) & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Complex valued DCD iterations with a leading element at time index \(n\). At the beginning of the DR process (i.e., for \(q=0\)), the form of the complex output signal remains the same as (18), corresponding to the estimate \(\widetilde{\mathbf{h}}(n-1)\). For the next iterations (\(q=1,\ldots,N_{it}-1\)), the output is adjusted according to the new updates performed by the DCD, which leads to the analytical expression:  
\[\widetilde{y}_{q}(n)= \begin{cases}\widetilde{\mathbf{h}}^{H}(n-1)\widetilde{ \mathbf{x}}(n),&q=0,\\ \left[\widetilde{\mathbf{h}}(n-1)+\sum_{k=0}^{q-1}\Delta\widetilde{\mathbf{h}} _{k}(n)\right]^{H}\mathbf{x}(n),&q=1,\ldots,N_{it}-1.\end{cases}