## 4 Computational Simulations and Discussion of Results  
In this section, we will construct several experiments to verify the GC- and RGC-MCC's channel estimation performance. The steady-state channel estimation error and the convergence are considered to give an evaluation of the proposed GC- and RGC-MCC algorithms. The results are also compared with the MCC, NMCC, ZA-, RZA- and SPF-MCC algorithms. From the discussion, we know that the CIM and SPF are \(l_{0}\)-norm approximations. Thus, we first set up an experiment to discuss the zero attraction ability of the CIM, SPF and \(l_{0}\)-norm approximation. The zero attraction ability of these \(l_{0}\)-norm approximations is shown in Figure 2. The related parameters in these \(l_{0}\)-norm approximations are \(\tau_{1}\), \(\sigma_{1}\) and \(\tau\). It can be seen that the zero attraction ability of the SPF is close to the \(l_{0}\)-norm for larger \(\tau_{1}\), while the zero attraction ability of the CIM approximates to \(l_{0}\)-norm when \(\sigma_{1}\) is small. From these results, we found that the SPF can give an accurate approximation and can provide a better zero attraction when \(\tau_{1}=\tau=5\) and \(\sigma_{1}=0.05\). For \(\tau_{1}=\tau=1\) and \(\sigma_{1}=0.08\), the zero attraction ability of SPF is weak. However, it can give an adaption for a wider range of sparse channel estimation applications. It can be concluded that the zero attraction ability of the zero attractor produced by SPF is superior. Thus, we will choose SPF-MCC to give a comparison with our proposed GC-MCC and RGC-MCC algorithms in terms of the steady-state error and convergence. All of the experiments are constructed under a mix-Gaussian noise environment, and the mixed noise model has been used in the previous study and is a better description of the real wireless communication environment. The mixed noise model is given by [34; 36]  
\[\left(1-\chi\right)N\left(\mu_{1},\
u_{1}^{2}\right)+\chi N\left(\mu_{2},\
u_{2 }^{2}\right)\,, \tag{29}\]  
where \(N\left(\mu_{i},\
u_{i}^{2}\right)\left(i=1,2\right)\) represent the Gaussian distribution, and parameters \(\mu_{i}\), \(\
u_{i}^{2}\) and \(\chi\) are the means, variances and mixing parameter, respectively. In all the experiments, we set \(\left(\mu_{1},\mu_{2},\
u_{1}^{2},\
u_{2}^{2},\chi\right)=\left(0,0,0.05,20,0.05\right)\). The estimated performance of our proposed GC-MCC and RGC-MCC algorithms is given by mean square deviation (MSD), which is defined as  
\[\text{MSD}\left(\mathbf{\hat{g}}\left(n\right)\right)=\text{E}\left[\left\| \mathbf{g}-\mathbf{\hat{g}}\left(n\right)\right\|^{2}\right]. \tag{30}\]  
For all the simulations, 300 Monte Carlo runs are performed for obtaining each point in all mentioned algorithms. Herein, the total length of the unknown channel is \(M=16\) and the number of the non-zero channel coefficients is \(K\). The signal-to-noise ratio (SNR) is 30 dB. Those non-zero channel coefficients are randomly distributed within the length of the channel. Since the regularization parameters have an important effect on the performance of the proposed GC-MCC and RGC-MCC algorithms, the effects of regularization parameters \(\theta_{\text{GC}}\), \(\theta_{\text{RGC}}\) and \(\beta_{\text{GC/RGC}}\) are investigated and given in Figures 3-5. Figure 2: Zero attraction ability of the zero attracting term produced by soft parameter function (SPF), \(l_{0}\)-norm and correntropy-induced metric (CIM) penalties. In this experiment, the step size of the GC-MCC and RGC-MCC is 0.026, \(\varepsilon_{1}\!=\!5\) and \(K\!=\!1\). The influence of \(\theta_{\text{GC}}\) on the MSDs of the GC-MCC algorithm is shown in Figure 3, while the influence of \(\theta_{\text{RGC}}\) on the MSDs of the RGC-MCC algorithm is shown in Figure 4. From Figures 3 and 4, the steady-state error with respect to the MSDs of the GC-MCC is reduced with the decrease of the \(\theta_{\text{GC}}\) ranging from \(\theta_{\text{GC}}\!=\!3\times 10^{-3}\) to \(\theta_{\text{GC}}\!=\!3\times 10^{-4}\).