## 4 Numerical Simulations  
In this section, three examples are given to verify the performance of SLMS-RL1 algorithm by using proposed reweighted factor \(\varepsilon=0.005\) in the scenarios of \(SNR=10\) dB, impulsive-noise strength \(T\in\{200,400,600,800\}\), mixture parameters \(\phi=0.1\) as well as channel sparsity \(K\in\{2,4,8,12,16\}\). For achieving average performance, \(M=1000\) independent Monte-Carlo runs are adopted as well. Detailed parameters for computer simulation are listed in Table 2.  
Figure 1: Monte Carlo simulations averaging for evaluating sign least mean square with reweighted L1-norm constraint (SLMS-RL1) algorithm over 1000 runs with respect to reweighted factors \(\varepsilon\in\{0.001,0.005,0.01,0.05,0.1,0.5\}\) in the scenarios of mixture parameter \(\phi=0.1\), impulsive-noise strength \(T=400\), channel length \(N=80\), channl sparsity \(K=4\) and \(SNR=5\) dB. Figure 2: Monte Carlo simulations averaging for evaluating SLMS-RL1 algorithm over 1000 runs with respect to reweighted factors \(\varepsilon\in\{0.001,0.005,0.01,0.05,0.1,0.5\}\) in the scenarios of mixture parameter \(\phi=0.1\), impulsive-noise strength \(T=400\), channel length \(N=80\), channl sparsity \(K=4\) and \(SNR=10\) dB. In the first example, average MSE curves of different algorithms are depicted in Figure 3. Under the certain circumstance, channel sparsity \(K=4\), \(SNR=10\)dB, GMM noise with impulsive-noise parameter \(T=400\) as well as mixture parameter \(\phi=0.1\), one can find that proposed SLMS-RL1 algorithm can achieve at least 5 dB and 10 dB performance gain in contrast to SLMS algorithm and LMS-type algorithms, respectively. Because SLMS algorithm does not exploit the channel sparsity while LMS-type algorithms do not stable under GMM noise environments. Hence, the proposed SLMS-RL1 can exploit channel sparsity and can keep stability in the presence of GMM noises. In the second example, average MSE curves of SLMS-RL1 algorithm with respect to channel sparsity \(K\) are depicted in Figure 4. Under the certain circumstance, e.g., \(SNR=10\)dB, GMM noise with impulsive-noise parameter \(T=400\) as well as mixture parameter \(\phi=0.1\), one can find that that convergence speed of SLMS-RL1 depends on channel sparsity (\(K\)) while steady-state MSE curves of corresponding algorithms are very close. For different channel sparsity, in other words, the adaptive sparse algorithms may differ from conventional compressive sensing based sparse channel estimation algorithms [15], [16], [25], [26] which depend highly on channel sparsity. Hence, SLMS-RL1 using MC-based reweighted factor is expected to deal with different sparse channels stably even non-sparse cases. \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 & \(K\in\{2,4,8,16\}\) \\ Distribution of nonzero coefficient & Random Gaussian distribution \(\mathcal{CN}(0,1)\) \\ Received SNR & \(SNR\in\{5\) dB, \(10\) dB\} \\ GMM noise distribution & \(\alpha_{1}=\alpha_{2}=0\), \(\sigma_{1}^{2}=10^{(-SNR/10)}\) \\ (\(T\) controls impulsive noise strength) & \(\sigma_{2}^{2}=T\sigma_{1}^{2}\), \(T=\{200,400,600,800\}\) \\ Step-size & \(\mu=0.01\) \\ Regularization parameters for sparse penalties & \(\lambda=0.008\) \\ Threshold of the SLMS-RL1 & \(\varepsilon=0.005\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Simulation parameters. Figure 3: Monte Carlo simulations averaging for different algorithms over 1000 runs in the scenarios of mixture parameter \(\phi=0.1\), impulsive-noise strength \(T=400\), channel length \(N=80\), channl sparsity \(K=4\) and \(SNR=10\) dB.