## 3 Reweighted Factor Selection for SLMS-RL1 Algorithm  
MC-based reweighted factor selection method is developed for SLMS-RL1 algorithm in different SNR regimes, impulsive-noise strength = 400, mixture parameters \(\phi=0.1\) as well as channel sparsity \(K=4\). For achieving average performance, \(M=1000\) independent Monte-Carlo runs are adopted.  
The simulation setup is configured according to the typical broadband wireless communication system [10]. The signal bandwidth is 50 MHz located at the central radio frequency of 2.1 GHz. The maximum delay spread of \(0.8\,\mu\mathrm{s}\). Hence, the maximum length of channel vector \(\mathbf{w}\) is \(N=80\). In addition, each dominant channel tap follows random Gaussian distribution as \(\mathcal{CN}\left(0,\sigma_{w}^{2}\right)\) which is subject to \(E\{\|\mathbf{w}\|_{2}^{2}\}=1\) and their positions are randomly decided within \(\mathbf{w}\). To evaluate SLMS-RL1 algorithm using different factors, we adopt the average mean square error (MSE) metric which is defined as  
\[\text{Average MSE}\{\mathbf{w}(n)\}\triangleq 10\log_{10}E\left\{\frac{\| \mathbf{w}(n)-\mathbf{w}\|_{2}^{2}}{\|\mathbf{w}\|_{2}^{2}}\right\} \tag{14}\]  
where \(\mathbf{w}\) and \(\mathbf{w}(\mathrm{n})\) are the actual signal vector and reconstruction vector, respectively; \(E\{\cdot\}\) denotes mathematical expectation operator. The received SNR is defined as \(P_{0}/\sigma_{z}^{2}\), where \(P_{0}\) is the received power of the pseudo-random (PN) binary sequence for training signal. Detailed parameters for computer simulation are listed in Table 1.  
First of all, MC based reweighted factor selection method is performed in Figures 1 and 2. Average MSE curves of the SLMS-RL1 algorithm are depicted under two SNR regimes, _i.e._, \(\in\{5\;\mathrm{dB},10\;\mathrm{dB}\}\). To confirm the effectiveness of the proposed method, standard SLMS [1] is considered as a performance benchmark. As we discussed in Section 2, one can see that the MSE performance of SLMS-RL1 algorithms depends highly on the reweighted factor \(\varepsilon\). In these two figures, the lowest MSE performance of SLMS-RL1 is achieved when reweighted factor \(\varepsilon\) is set as 0.005 in two SNR regimes. On the one hand, too big reweighted factor \(\varepsilon\) may suppress noise excessively and hence it result in lossy exploitation of channel sparsity. On the other hand, a too small reweighted factor \(\varepsilon\) may mitigate noise insufficiently and it causes inefficient exploitation of channel sparsity. Therefore, suitable reweighted factor could balance the noise suppression and channel sparsity exploitation.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Values** \\ \hline Training signal structure & Pseudo-random Binary sequences \\ Channel length & \(N=80\) \\ No. of nonzero coefficients & \(K=4\) \\ Distribution of nonzero coefficient & Random Gaussian distribution \(\mathcal{CN}\left(0,1\right)\) \\ Received SNR & \(SNR\in\{5\;\mathrm{dB},10\;\mathrm{dB}\}\) \\ GMM noise distribution & \(\alpha_{1}=\alpha_{2}=0\), \(\sigma_{1}^{2}=10^{\left(-SNR/10\right)}\), \(T=400\) \\ Step-size & \(\mu=0.01\) \\ Regularization parameters for sparse penalties & \(\lambda=0.008\) \\ Thresholds of the SLMS-RL1 algorithms & \(\varepsilon\in\{0.5,0.1,0.05,0.01,0.005,0.001\}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Simulation parameters.