## 3 The Proposed Group-Constrained Sparse MCC Algorithms
Thus, the updating recursion of the GC-MCC algorithm is rewritten as  
\[\frac{\partial J_{\mathrm{GC}}\left(n\right)}{\partial\hat{\mathbf{g}}\left(n +1\right)}=0, \tag{18}\]0  
The last term \(-\theta_{\text{GC}}\text{Fsgn}\left[\hat{\mathbf{g}}\left(n\right)\right]\) is the proposed zero attraction term which exerts different zero attracting on the two grouped channel coefficients. We can see that both the \(l_{0}\)-norm and \(l_{1}\)-norm constraints are implemented on the channel coefficients in the GC-MCC algorithm, which is different from the ZA-MCC and \(l_{0}\)-MCC algorithms. For the GC-MCC channel estimation, our proposed zero attractor can distinguish the value of coefficients and categorize the channel coefficients into two groups. The proposed GC-MCC algorithm can achieve small steady-state error and fast convergence due to the term \(-\theta_{\text{GC}}\text{FSgn}\left[\hat{\mathbf{\xi}}\left(n\right)\right]\). To further enhance the performance of our proposed GC-MCC algorithm, a reweighting factor is introduced into the GC-MCC's update equation to implement the RGC-MCC algorithm, which is similar to the RZA-LMS algorithm. Thus, the RGC-MCC's updating equation is  
\[\frac{\partial J_{\mathrm{GC}}\left(n\right)}{\partial\hat{\mathbf{g}}\left(n +1\right)}=0, \tag{18}\]1  
where \(\theta_{\text{RGC}}\) is the zero attraction controlling parameter, \(\varepsilon_{1}\) is the reweighting controlling parameter and \(\beta_{\text{RGC}}\) is a step-size of the RGC-MCC algorithm. With the help of \(-\theta_{\text{RGC}}\mathbf{F}\frac{\text{sgn}\left[\hat{\mathbf{\xi}}\left(n \right)\right]}{1+\varepsilon_{1}\left|\hat{\mathbf{\xi}}\left(n\right)\right|}\), our proposed RGC-MCC algorithm can assign well different values \(p_{i}\) to channel coefficients according to the magnitudes of the channel coefficients. Moreover, we can properly select the value of \(\varepsilon_{1}\) to obtain a better channel estimation performance. In fact, the proposed GC-MCC and RZA-GC algorithms are the extension of the ZA-MCC and RZA-MCC algorithms. However, the proposed GC-MCC and RGC-MCC are different with the proposed ZA-MCC, RZA-MCC and \(l_{0}\)-norm penalized MCC algorithms. Our contributions are summarized herein. The GC-MCC algorithm is realized by incorporating a variable norm in Equation (13) into the cost function of the traditional MCC algorithm, where the variable norm is controlled by the parameter \(p\). To distinguish the large channel coefficients and the small channel coefficients, a mean of the estimated channel coefficients shown in Equation (24) is proposed to provide a threshold. Then, the variable norm is split into two groups by comparing the channel coefficients with the threshold. As a result, a large group is given when \(\hat{g}_{i}(n)>y(n)\), while a small group is created when \(\hat{g}_{i}(n)<y(n)\). For the channel coefficients in the large group, we use a \(l_{0}\)-norm to count the number of the non-zero channel coefficients. As for the channel coefficients in the small group, \(l_{1}\)-norm penalty is used for attracting the zero or near-zero channel coefficients to zero quickly. It is found that a norm penalty matrix with values in its diagonal is proposed in Equation (25) to implement the \(l_{0}\) and \(l_{1}\) in the GC-MCC algorithm, which is also different with the conventional \(l_{0}\)- and \(l_{1}\)- norm constraints. Then, we use a reweighting factor to enhance the zero attracting ability in the GC-MCC algorithm to generate the RGC-MCC algorithm. Both the GC-MCC and RGC-MCC algorithms are developed to exploit well the inherent sparseness properties of the multi-path channels due to the expected zero-attraction terms in their iterations. The channel estimation behaviors will be discussed over sparse channels in mixed Gaussian noise environments in the next section.