## 4 Computational Simulations and Discussion of Results
The corresponding simulation parameters for the mentioned algorithms are \(\beta_{\rm MCC}=0.0052\), \(\beta_{\rm NMCC}=0.085\), \(\beta_{\rm ZA}=0.01\), \(\theta_{\rm ZA}=3\times 10^{-5}\), \(\beta_{\rm RZA}=0.015\), \(\theta_{\rm RZA}=7\times 10^{-5}\), \(\beta_{\rm SPF}=0.016\), \(\theta_{\rm SPF}=2.7\times 10^{-5}\), \(\tau_{1}=100\), \(\beta_{\rm GC}=0.026\), \(\theta_{\rm GC}=3\times 10^{-4}\), \(\beta_{\rm RGC}=0.032\), \(\theta_{\rm RGC}=7\times 10^{-4}\), \(\epsilon_{1}=5\) and \(\sigma=1000\). Herein, \(\beta_{\rm NMCC}\) is the step size of the NMCC algorithm. The convergence of the proposed GC-MCC and RGC-MCC algorithms is given in \[\left(1-\chi\right)N\left(\mu_{1},\
u_{1}^{2}\right)+\chi N\left(\mu_{2},\
u_{2 }^{2}\right)\,, \tag{29}\]2 It can be seen that the convergence of the proposed GC-MCC and RGC-MCC algorithms is better than that of the MCC, NMCC, ZA-MCC, RZA-MCC and SPF-MCC algorithms at the same MSD level. Moreover, our RGC-MCC has the fastest convergence speed rate. Next, the channel estimation performance of our presented GC-MCC and RGC-MCC algorithms is analyzed under a different sparsity level \(K\) that is also the number of the non-zero channel coefficients of the sparse channel. Firstly, only one non-zero channel coefficient is randomly distributed within the unknown sparse channel. This means that the sparsity level is \(K=1\). In this experiment, the related parameters are the following \(\beta_{\rm MCC}=0.03\), \(\beta_{\rm NMCC}=0.4\), \(\beta_{\rm ZA}=\beta_{\rm RZA}=0.03\), \(\theta_{\rm ZA}=8\times 10^{-5}\), \(\theta_{\rm RZA}=2\times 10^{-4}\), \(\beta_{\rm SPF}=0.03\), \(\theta_{\rm SPF}=3\times 10^{-5}\), \(\tau_{1}=100\), \(\beta_{\rm GC}=\beta_{\rm RGC}=0.026\), \(\theta_{\rm GC}=3\times 10^{-4}\), \(\theta_{\rm RGC}=7\times 10^{-4}\) and \(\epsilon_{1}=5\). The MSD performance of the GC-MCC and RGC-MCC algorithms is demonstrated in Figure 9 for \(K=1\). It can be seen in Figure 9 that the steady-state MSD of our presented GC-MCC and RGC-MCC algorithms is lower than that of the mentioned MCC algorithms with the same convergence speed. The channel estimations with respect to the MSD for \(K=3\) and \(K=5\) are given in Figures 10 and 11, respectively. When the sparsity level increases from \(K=3\) to \(K=5\), the MSD floor is increased in comparison with \(K=1\) because the sparseness is reduced. However, it is worth noting that our proposed GC-MCC and RGC-MCC algorithms still outperform  
\[\left(1-\chi\right)N\left(\mu_{1},\
u_{1}^{2}\right)+\chi N\left(\mu_{2},\
u_{2 }^{2}\right)\,, \tag{29}\]3 Estimation behavior of the proposed GC-MCC and RGC-MCC algorithms with different SNR. the existing MCC and its sparse forms. In addition, the RGC-MCC always achieves the lowest MSD for different \(K\). \[\left(1-\chi\right)N\left(\mu_{1},\
u_{1}^{2}\right)+\chi N\left(\mu_{2},\
u_{2 }^{2}\right)\,, \tag{29}\]4 Convergence of the proposed GC-MCC and RGC-MCC algorithms. \[\left(1-\chi\right)N\left(\mu_{1},\
u_{1}^{2}\right)+\chi N\left(\mu_{2},\
u_{2 }^{2}\right)\,, \tag{29}\]5 MSD performance of the proposed GC-MCC and RGC-MCC algorithms for \(K=1\). Then, a realistic IEEE 802.15.4a channel model, which can be downloaded from [39], that works in CM1 mode is employed to discuss the effectiveness of the proposed GC-MCC and RGC-MCC algorithms. The simulation parameters are \(\beta_{\text{MCC}}=~{}0.002\), \(\beta_{\text{NMCC}}=0.8\), \(\beta_{\text{ZA}}=0.0011\), \(\beta_{\text{RZA}}=0.001\), \(\theta_{\text{ZA}}=~{}6\times 10^{-6}\), \(\theta_{\text{RZA}}=~{}9\times 10^{-5}\), \(\theta_{\text{SPF}}=8\times 10^{-6}\), \(\tau_{1}=100\), \(\beta_{\text{SPF}}=\beta_{\text{GC}}=\beta_{\text{RGC}}=0.001\), \(\theta_{\text{GC}}=5\times 10^{-4}\), \(\theta_{\text{RGC}}=3\times 10^{-4}\) and \(\varepsilon_{1}=5\).