## 5 New Modeling for \(l_{1}\)-RLS with Uncertainty in the Regularization Factor  
If we set \(\rho=\) constant, the regularization factor becomes  
\[ \begin{array}{ll}\widetilde{\gamma}_{k}&=\frac{2^{\frac{\
u\left(\mathbf{R} ^{-1}(k)\right)}{M}}}{\left\|\mathbf{R}^{-1}(k)\
abla^{s}f(\mathbf{w}(k)) \right\|_{2}^{2}}\times\left[f(\mathbf{w}(k))-\text{constant}\right)+\
abla^ {s}f(\mathbf{w}(k))\times\mathbf{R}^{-1}(k)\epsilon(k)\right]\\ &=\frac{2^{\frac{\
u\left(\mathbf{R}^{-1}(k)\right)}{M}}}{\left\|\mathbf{R}^ {-1}(k)\
abla^{s}f(\mathbf{w}(k))\right\|_{2}^{2}}\times\left(f(\mathbf{w}(k) )-\left\|\mathbf{h}\right\|_{1}+\left\|\mathbf{h}\right\|_{1}-\text{constant} \right)+\frac{2^{\frac{\
u\left(\mathbf{R}^{-1}(k)\right)}{M}\
abla^{s}f( \mathbf{w}(k))\mathbf{R}^{-1}(k)\epsilon(k)}}{\left\|\mathbf{R}^{-1}(k) \
abla^{s}f(\mathbf{w}(k))\right\|_{2}^{2}}.\end{array}  \tag{7}\]  
Then,  
\[\widetilde{\gamma}_{k}=\gamma_{k}+\frac{2^{\frac{\
u\left(\mathbf{R}^{-1}(k) \right)}{M}}(\left\|\mathbf{h}\right\|_{1}-\text{constant})}{\left\|\mathbf{R} ^{-1}(k)\
abla^{s}f(\mathbf{w}(k))\right\|_{2}^{2}}=\gamma_{k}+\Delta\gamma. \tag{8}\]  
Using Equation (8), Equation (4) becomes  
\[\mathbf{R}(k)\mathbf{w}(k)=\mathbf{r}(k)-(\gamma_{k}+\Delta\gamma)\
abla^{s} \left\|\mathbf{w}(k)\right\|_{1}\quad. \tag{9}\]  
\(\
abla^{s}\left\|\mathbf{w}\right\|_{1}=\text{sgn}(\mathbf{w})\) is represented as  
\[\
abla^{s}\left\|\mathbf{w}(k)\right\|_{1}=\left[ \begin{array}{ccc}\ddots&& &\\ &\frac{1}{\left|\mathbf{w}_{i}\right|}&&\\ &&\ddots&\end{array} \right]\mathbf{w}(k). \tag{10}\]  
By applying Equation (10) to Equation (9),  
\[\left(\mathbf{R}(k)+\Delta\gamma\left[ \begin{array}{ccc}\ddots&&\\ &\frac{1}{\left|\mathbf{w}_{i}\right|}&&\\ &&\ddots&\end{array} \right]\right)\mathbf{w}(k)=\mathbf{r}(k)-\gamma_{k} \left\|\mathbf{w}(k)\right\|_{1}, \tag{11}\]  
where \(\mathbf{w}_{i}\) is \(i\)-th element of \(\mathbf{w}(k)\). Then it is simplified as  
\[\left(\mathbf{R}(k)+\Delta\gamma\left[ \begin{array}{ccc}\ddots&&\\ &\frac{1}{\left|\mathbf{w}_{i}\right|}&&\\ &&\ddots&\end{array} \right]\right)\mathbf{w}(k)=\hat{\mathbf{p}}(k). \tag{12}\] \begin{table}
\begin{tabular}{c c} \hline Step 1 & Sparsity: \(\chi=\frac{L}{L-\sqrt{L}}\left(1-\frac{\left\|\hat{\mathbf{w}}\right\|_{1}}{ \sqrt{L}\left\|\hat{\mathbf{w}}\right\|_{2}}\right)\)[20] \\  & where \(L\) is the length of the impulse response. \\ \hline Step 2 & \(\rho(k)=0.99\rho(k-1)+0.01\left(e^{c-1}\times\left\|\hat{\mathbf{w}}\right\|_{1}\right)\) \\ \hline Step 3 & \(\rho(k)=\left\{\begin{array}{ll}min(\rho(k),0.98\|\hat{\mathbf{w}}\|_{1}), \text{if }x>0.75\\ min(\rho(k),0.999\|\hat{\mathbf{w}}\|_{1}),\text{otherwise}\end{array}\right.\) \\ \hline \end{tabular}
\end{table} 
Table 1: \(\rho\) selection method in the sparsity regularization constant \(\gamma_{k}\). Equation (12) is very similar to the system model in Figure 2 that is contaminated by noise both in input and in output. Suppose that an example of the system in Figure 2 is represented as  
\[\left[ \begin{array}{cccc}x_{k}+n_{i,k}&\cdots&x_{k-N+1}+n_{i,k-N+1}\\ x_{k-1}+n_{i,k-1}&\cdots&x_{k-N}+n_{i,k-N}\\ \vdots&\ddots&\vdots\\ x_{k-N+1}+n_{i,k-N+1}&\cdots&x_{k-2N+2}+n_{i,k-2N+2}\end{array} \right]\times \mathbf{w}(k)=\left[ \begin{array}{c}y_{k}+n_{o,k}\\ y_{k-1}+n_{o,k-1}\\ \vdots\\ y_{k-N+1}+n_{o,k-N+1}\end{array} \right], \tag{13}\]  
where \(x_{k}\) is \(x(k)\), \(n_{i,k}\) is \(n_{i}(k)\), and \(n_{o,k}\) is \(n_{o}(k)\). Equation (13) is simplified as  
\[\mathbf{Aw}(k)=\mathbf{b}. \tag{14}\]  
If we multiply Equation (14) by \(\mathbf{A}^{H}\) and average it, we get  
\[E\Big{(}\mathbf{A}^{H}\mathbf{A}\Big{)}\mathbf{w}(k)=E\Big{(}\mathbf{A}^{H} \mathbf{b}\Big{)}. \tag{15}\]  
We can rewrite Equation (15) as follows  
\[\left[ \begin{array}{cccc}r_{xx}(0)+\sigma_{n}^{2}&r_{xx}(1)&\cdots&r_{xx}(N- 1)\\ r_{xx}(1)&r_{xx}(0)+\sigma_{n}^{2}&\cdots&r_{xx}(N-2)\\ \vdots&\vdots&\ddots&\vdots\\ r_{xx}(N-1)&r_{xx}(N-2)&\cdots&r_{xx}(0)+\sigma_{n}^{2}\end{array} \right] \mathbf{w}(k)=\left[