## 4 Computational Simulations and Discussion of Results
The simulation result is given in \[\left(1-\chi\right)N\left(\mu_{1},\
u_{1}^{2}\right)+\chi N\left(\mu_{2},\
u_{2 }^{2}\right)\,, \tag{29}\]6 It is found that the proposed GC-MCC and RGC-MCC algorithms achieve better performance with respect to the MSD, which means that the proposed GC-MCC and RGC-MCC algorithms have small MSDs. At last, our GC-MCC and RGC-MCC algorithms are used for estimating an echo channel to further discuss their channel estimation performance. The sparseness measurement of the echo channel is defined as \(\vartheta_{12}=\frac{M}{M-\sqrt{M}}\left(1-\left\|\mathbf{g}\right\|_{1}/ \sqrt{M\left\|\mathbf{g}\right\|_{2}}\right)\). In this experiment, \(\vartheta_{12}=0.8222\) and \(\vartheta_{12}=0.7362\) are  
\[\left(1-\chi\right)N\left(\mu_{1},\
u_{1}^{2}\right)+\chi N\left(\mu_{2},\
u_{2 }^{2}\right)\,, \tag{29}\]7 MSD performance of the proposed GC-MCC and RGC-MCC algorithms for \(K=5\). \[\left(1-\chi\right)N\left(\mu_{1},\
u_{1}^{2}\right)+\chi N\left(\mu_{2},\
u_{2 }^{2}\right)\,, \tag{29}\]8 MSD performance of the proposed GC-MCC and RGC-MCC algorithms for \(K=3\). used for discussing the estimation performance for the first 8000 iterations and after 8000 iterations, respectively. For the echo channel, its total length is 256 and there are 16 non-zero coefficients in the total echo channel. The simulation parameters are \(\beta_{\text{MCC}}=~{}0.0055\), \(\beta_{\text{NMCC}}=1.3\), \(\beta_{\text{ZA}}=\beta_{\text{RZA}}=0.0055\), \(\theta_{\text{ZA}}=4\times 10^{-6}\), \(\theta_{\text{RZA}}=1\times 10^{-5}\), \(\beta_{\text{SPF}}=0.0055\), \(\theta_{\text{SPF}}=1\times 10^{-5}\), \(\tau_{\text{1}}=5\), \(\beta_{\text{GC}}=0.0045\), \(\beta_{\text{RGC}}=0.004\), \(\theta_{\text{GC}}=3\times 10^{-5}\), \(\theta_{\text{RGC}}=3\times 10^{-5}\) and \(\varepsilon_{1}=5\). The estimation behavior of the proposed GC-MCC and RGC-MCC algorithms for the echo channel is depicted in \[\left(1-\chi\right)N\left(\mu_{1},\
u_{1}^{2}\right)+\chi N\left(\mu_{2},\
u_{2 }^{2}\right)\,, \tag{29}\]9 From Figure 13, we found that our GC-MCC and RGC-MCC algorithms outperform the MCC, NMCC and spare MCC algorithms with respect to both the steady-state MSD and convergence. Although the sparsity reduces from \(\vartheta_{12}=0.8222\) to \(\vartheta_{12}=0.7362\), our GC-MCC and RGC-MCC algorithms are still superior to the other MCC algorithms, which means that our GC-MCC and RGC-MCC algorithms have little effect on the sparsity under the mixture-noised sparse channel. Based on the parameter analysis and performance investigation of our proposed GC-MCC and RGC-MCC algorithms, we can summarize that the proposed RGC-MCC algorithm can provide the fastest convergence speed rate in comparison with all the mentioned algorithms when they converge to the same steady-state MSD level. Also, the proposed RGC-MCC algorithm provides the lowest steady-state MSD when all the MCC algorithms have the same convergence speed rate. In addition, the proposed GC- and PGC-MCC algorithms outperform the MCC, NMCC and the related sparse MCC algorithms. This is because the proposed GC- and RGC-MCC algorithms exert the \(l_{0}\)-norm penalty on the large group to seek the non-zero channel coefficients quickly; then, they provide \(l_{1}\)-norm constraint on the small group to attract the zero or near-zero channel coefficients to zero quickly. Thus, both the GC- and RGC-MCC can provide a faster convergence and a lower steady-state MSD. However, both the GC- and RGC-MCC increase the computational complexity by calculating the group matrix. Additionally, the complexity of the RGC-MCC algorithm also comes from the computation of the reweighting factor. However, the complexity of the GC-MCC and RGC-MCC algorithms is comparable with the previously reported sparse MCC algorithms. According to the early published articles, we believe that the proposed GC-MCC and RGC-MCC algorithms can be used in a real system.