## 4 Computational Simulations and Discussion of Results
Then, the steady-state error of the GC-MCC algorithm becomes worse when the \(\theta_{\text{GC}}\) continues to reduce. When \(\theta_{\text{GC}}\!=\!3\times 10^{-4}\), the GC-MCC algorithm achieves the smallest steady-state MSD. As for the RGC-MCC algorithm, the MSD is reduced when \(\theta_{\text{RGC}}\) ranges from \(\theta_{\text{RGC}}\!=\!7\times 10^{-3}\) to \(\theta_{\text{RGC}}\!=\!7\times 10^{-4}\). Then, the MSD of the RGC-MCC algorithm is increased with a continuous decrement of \(\theta_{\text{RGC}}\). The RGC-MCC algorithm achieves the smallest steady-state MSD for \(\theta_{\text{RGC}}\!=\!7\times 10^{-4}\). Therefore, we take \(\theta_{\text{GC}}\!=\!3\times 10^{-4}\) and \(\theta_{\text{RGC}}\!=\!7\times 10^{-4}\) into consideration to ensure the best channel estimation performance of the proposed GC-MCC and RGC-MCC algorithms. Next, the effect of the \(\beta\) that includes \(\beta_{\text{GC}}\) and \(\beta_{\text{RGC}}\) on the proposed GC-MCC and RGC-MCC algorithms is given in Figure 5. It can be seen that the MSD of the RGC-MCC is worse than the GC-MCC when the step size is less than 0.013, while the RGC-MCC is better than the GC-MCC when \(\beta>0.013\). It is noted that the steady-state errors of the GC-MCC and RGC-MCC algorithms become worse as the parameter \(\beta\) increases. Therefore, we should carefully choose the  
Figure 4: Effects of \(\theta_{\text{RGC}}\) on the MSD of the proposed RGC-MCC algorithm. Figure 3: Effects of \(\theta_{\text{GC}}\) on the mean square deviation (MSD) of the proposed GC-MCC algorithm. step size and the zero attraction controlling parameters to achieve a good estimation performance for handling sparse channel estimations. The effects of the reweighting controlling parameter are given in Figure 6. It can be seen that the reweighting controlling parameter mainly affects the small channel coefficients when \(\varepsilon_{1}\) increases from 4 to 25. It means that the reweighted zero attractor mainly exerts effects on the magnitudes which are comparable to \(1/\varepsilon_{1}\), while little shrinkage is penalized on the channel coefficients that are far greater than \(1/\varepsilon_{1}\). In the RGC-MCC algorithm, the reweighting controlling parameter exerts strong zero attracting on the small group to provide a fast convergence. As we know, adaptive filters have been extensively investigated and applied for channel estimation, and have been used in real time systems. Moreover, adaptive filters algorithms have been further developed for sparse channel estimations. Similar to the previous investigations [2; 6; 7; 10; 11; 12; 13; 14; 15; 16; 21; 22; 23; 24; 26; 27], the proposed GC-MCC and RGC-MCC algorithms are investigated and their performance is compared  
Figure 5: MSD performance of the proposed GC-MCC and RGC-MCC algorithms with different \(\beta\). \[\left(1-\chi\right)N\left(\mu_{1},\
u_{1}^{2}\right)+\chi N\left(\mu_{2},\
u_{2 }^{2}\right)\,, \tag{29}\]0 Effects of \(\varepsilon_{1}\) on the MSD of the proposed GC-MCC and RGC-MCC algorithm. with the SPF-MCC algorithm. In the following experiment, the proposed GC-MCC and RGC-MCC algorithms are investigated in different SNR environments. In this experiment, the step size of the GC-MCC and RGC-MCC algorithms is 0.026, and \(\theta_{\rm GC}=3\times 10^{-4}\) and \(\theta_{\rm RGC}=7\times 10^{-4}\). The MSD performance at different SNRs is shown in \[\left(1-\chi\right)N\left(\mu_{1},\
u_{1}^{2}\right)+\chi N\left(\mu_{2},\
u_{2 }^{2}\right)\,, \tag{29}\]1 It can be seen that the performance of the proposed GC-MCC and RGC-MCC algorithms is improved with the increment of SNR. It is worth noting that the performance of the RGC-MCC is always better than that of the GC-MCC at the same SNR. This is because the reweighting factor provides a selective zero attracting in the RGC-MCC algorithm. Next, the convergence of the GC-MCC and RGC-MCC algorithms is studied and it is compared with the MCC, NMCC, ZA-MCC, RZA-MCC and SPF-MCC algorithms at SNR = 30 dB.