## 2 Summarize \(l_{1}\)-RLS  
In the sparse channel estimation problem of interest, the system observes a signal represented by an \(M\times 1\) vector \(\mathbf{x}(k)=\left[x_{k},\cdots,x_{k-M+1}\right]^{T}\) at time instant \(\mathbf{n}\), performs filtering, and obtains the output \(y(i)=\mathbf{x}^{T}(k)\mathbf{w}_{o}(k)\), where \(\mathbf{w}_{o}(k)=\left[w_{k},\cdots,w_{k-M+1}\right]^{T}\) is the M dimensional actual system with finite impulse response (FIR) type. For system estimation, an adaptive filter system applies with M dimensional vector \(\mathbf{w}(k)\) to the same signal vector \(\mathbf{x}(k)\) and produces an estimated output \(\hat{y}(k)=\mathbf{x}^{T}(k)\mathbf{w}(k)\), and calculates the error signal \(e(k)=y(k)+n(k)-\hat{y}(k)=\widetilde{y}(k)-\hat{y}(k)\), where \(n(k)\) is the measurement noise, \(y(k)\) is the output of the actual system, and \(\hat{y}(k)\) is the estimated output. In order to estimate the channel impulse response, an adaptive algorithm minimizes the cost function defined by  
\[\mathbf{w}=\underset{\mathbf{w}}{\text{argmin}}\ \frac{1}{2}\sum\limits_{m=0}^{k} \ \lambda^{k-m}(e(m))^{2}. \tag{1}\]  
From the gradient based minimization, Equation (1) becomes  
\[\mathbf{R}(k)\mathbf{w}(k)=\mathbf{r}(k), \tag{2}\]  
where \(\mathbf{R}(k)=\sum\limits_{m=0}^{k}\ \lambda^{k-m}\mathbf{x}(m)\mathbf{x}^{T}(m)\) and \(\mathbf{r}(k)=\sum\limits_{m=0}^{k}\ \lambda^{k-m}\widetilde{y}(m)\mathbf{x}(m)\). This equation is the normal equation for the least squares solution. Especially, \(\mathbf{w}_{o}(k)\) is considered as a sparse system when the number of nonzero coefficients \(K\) is less than the system order of \(M\). In order to estimate the sparse system, most estimation algorithms exploit non-zero coefficients in the system [11; 12; 13; 14; 15; 16; 17]. In [17], [PERSON] proposed a full recursive \(l_{1}\)-regularized algorithm by the minimization of the object function as shown in Equation (3).  
\[J_{k}=\frac{1}{2}\varepsilon_{k}+\gamma_{k}\|\mathbf{w}\|_{1}, \tag{3}\]  
where \(\varepsilon_{k}=\sum\limits_{m=0}^{k}\ \lambda^{k-m}(e(m))^{2}\). From the minimization of Equation (3), a modified normal equation was derived as shown in Equation (4).  
\[\mathbf{R}(k)\mathbf{w}(k)=\mathbf{r}(k)-\gamma_{k}\
abla^{s}\|\mathbf{w}(k) \|_{1}=\hat{\mathbf{p}}(k). \tag{4}\]  
When we solve Equation (4), we should select the regularization factor as shown in Equation (5).  
\[\gamma_{k}=\frac{2^{\mathit{tr}(\mathbf{R}^{-1}(k))}}{\frac{M}{\|\mathbf{R}^{ -1}(k)\
abla^{s}f(\mathbf{w}(k))\|_{2}^{2}}}\times[(f(\mathbf{w}(k))-\rho)+ \
abla^{s}f(\mathbf{w}(k))\mathbf{R}^{-1}(k)\varepsilon(k)], \tag{5}\]where \(f(\mathbf{w}(k))=\left\|\mathbf{w}(k)\right\|_{1}\) and the subgradient of \(f(\mathbf{w})\) is \(\
abla^{s}\left\|\mathbf{w}\right\|_{1}=\text{sgn}(\mathbf{w})\). In Equation (5), the regularization factor has the parameter, \(\rho\), which should be set beforehand. In [17], the parameter was set as \(\rho=f(\mathbf{w}_{true})=\left\|\mathbf{w}_{true}\right\|_{1}\) with \(\mathbf{w}_{true}\) indicating the impulse response of the true channel. There was no further discussion about how to set \(\rho\). However, it is not practical to know the true channel in advance.