## 5 New Modeling for \(l_{1}\)-RLS with Uncertainty in the Regularization Factor
\begin{array}{c}r_{xy}(0)\\ r_{xy}(1)\\ \vdots\\ r_{xy}(N-1)\end{array} \right] \tag{16}\]  
Then, it can be represented as  
\[\Big{(}\mathbf{R}+\sigma_{n}^{2}\mathbf{I}\Big{)}\mathbf{w}(k)=\mathbf{ \widetilde{p}}(k). \tag{17}\]  
When we compare Equation (12) with Equation (17), the two system models have almost the same form. Therefore, it is feasible that the TLS method can be applied to Equation (12) [24; 25; 26; 27; 28; 29; 30]. Therefore, we expect to obtain almost the same performance as \(l_{1}\)-RLS with the true channel response if we apply the TLS method by the regularization factor with the new \(\rho\) in Table 1. In the next section, we summarize \(l_{1}\)-RTLS (recursive total least squares) algorithm in [29]. Figure 2: The model of a noisy input and noisy output system. Summarize \(l_{1}\)-RTLS for the Solution of \(l_{1}\)-RLS with Uncertainty in the Regularization Factor  
[PERSON], one of the authors of this paper, has proposed the TLS solution for \(l_{1}\)-RLS known as \(l_{1}\)-RTLS [30]. In this section, we summarize \(l_{1}\)-RTLS in [30] for the solution of Equation (11). The TLS system model assumes that both input and output are contaminated by additive noise as Figure 2. The output is given by  
\[\widetilde{y}(k)=\widetilde{\mathbf{x}}^{T}(k)\mathbf{w}_{o}+n_{o}(k), \tag{18}\]  
where the output noise \(n_{o}(k)\) is the Gaussian white noise with variance \(\sigma_{o}^{2}\). The noisy input vector in the system is modeled by  
\[\widetilde{\mathbf{x}}(k)=\mathbf{x}(k)+\mathbf{n}_{i}(k)\in C^{M\times 1}, \tag{19}\]  
where \(\mathbf{n}_{i}(k)=\left[n_{i}(k),n_{i}(k-1),\cdots n_{i}(k-M+1)\right]^{T}\) and the input noise \(n_{i}(k)\) is the Gaussian white noise with variance \(\sigma_{i}^{2}\). For the TLS solution, we set the augmented data vector as  
\[\widetilde{\mathbf{x}}(k)=\left[\widetilde{\mathbf{x}}^{T}(k),\widetilde{y}( k)\right]^{T}\in\mathbb{R}^{(M+1)\times 1}. \tag{20}\]  
The correlation matrix is represented as  
\[\overline{\mathbf{R}}=\left[ \begin{array}{cc}\widetilde{\mathbf{R}}& \mathbf{p}\\ \mathbf{p}^{T}&c\end{array} \right], \tag{21}\]  
where \(\mathbf{p}=E\{\widetilde{\mathbf{x}}(k)y(k)\}\), \(c=E\{y(k)y(k)\}\), \(\mathbf{R}=E\{\mathbf{x}(k)\mathbf{x}^{T}(k)\}\) and \(\widetilde{\mathbf{R}}=E\Big{\{}\widetilde{\mathbf{x}}(k)\widetilde{\mathbf{x} }^{T}(k)\Big{\}}=\mathbf{R}+\sigma_{i}^{2}\mathbf{I}\). In [27; 28], the TLS problem becomes to find the eigenvector associated with the smallest eigenvalue of \(\overline{\mathbf{R}}\). Equation (22) is the typical cost function to find the eigenvector associated with the smallest eigenvalue of \(\overline{\mathbf{R}}\). \[J(k)=\frac{1}{2}\widetilde{\mathbf{w}}^{T}(k)\overline{\mathbf{R}}(k) \widetilde{\mathbf{w}}(k), \tag{22}\]  
where \(\overline{\mathbf{R}}(k)\) is a sample correlation matrix at \(k\)-th instant, and \(\widetilde{\mathbf{w}}(k)=\left[\hat{\mathbf{w}}^{T}(k),-1\right]^{T}\) in which \(\hat{\mathbf{w}}(\mathrm{k})\) is the estimation result for the unknown system at \(k\)-th instant. We modify the cost function by adding a penalty function in order to reflect prior knowledge about the true sparsity system. \[J(k)=\frac{1}{2}\widetilde{\mathbf{w}}^{T}(k)\overline{\mathbf{R}}(k) \widetilde{\mathbf{w}}(k)+\lambda\Big{(}\widetilde{\mathbf{w}}^{T}(k) \widetilde{\mathbf{w}}(k-1)-1\Big{)}+\gamma_{k}f(\widetilde{\mathbf{w}}(k)), \tag{23}\]  
where \(\lambda\) is the Lagrange multiplier and \(\gamma_{k}\) is the regularized parameter in [13]. We solve the equations by \(\
abla_{\hat{\mathbf{w}}}J(k)=0\) and \(\
abla_{\lambda}J(k)=0\) simultaneously.