## 3 The Proposed Group-Constrained Sparse MCC Algorithms
From Equations (20) and (21), we can obtain \(\lambda_{\text{GC}}\)  
\[\lambda_{\text{GC}}=\frac{\alpha\exp\left(-\frac{e^{2}\left(n\right)}{2\sigma^ {2}}\right)e\left(n\right)+\theta_{\text{GC}}\frac{p_{i}\cdot\text{sgn}\left[ \hat{g}_{i}\left(n+1\right)\right]}{\left\|\hat{g}_{i}\left(n+1\right)\right\|^ {1-p_{i}}}x_{i}^{\text{T}}\left(n\right)}{\left\|x_{i}\left(n\right)\right\|^{ 2}}. \tag{22}\]  
Substituting \(\lambda_{\text{GC}}\) into Equation (20), the updated equation of the proposed GC-MCC algorithm is  
\[\hat{g}_{i}\left(n+1\right)=\hat{g}_{i}\left(n\right)+\beta_{\text{GC}}\exp \left(-\frac{e^{2}\left(n\right)}{2\sigma^{2}}\right)e\left(n\right)x_{i} \left(n\right)-\theta_{\text{GC}}\frac{p_{i}\cdot\text{sgn}\left[\hat{g}_{i} \left(n+1\right)\right]}{\left\|\hat{g}\left(n+1\right)\right\|^{1-p_{i}}} \left(1-\frac{x_{i}\left(n\right)x_{i}^{\text{T}}\left(n\right)}{\left\|x_{i} \left(n\right)\right\|^{2}}\right) \tag{23}\]  
where \(\beta_{\text{GC}}\) is a step-size for GC-MCC algorithm. It can be seen from Equation (23), the matrix \(\mathbf{p}=\left[p_{1},p_{2},\cdots,p_{M}\right]\) can assign different \(p_{i}\) to each channel coefficients. To better exert the \(p_{i}\) to the channel coefficients, the channel coefficients are classified according to their magnitudes. From the measurement and the previous investigations of the sparse channels [2; 6; 7; 10; 11; 12; 13; 14; 15; 16; 21; 22; 23; 24; 26; 27], we found that few channel coefficients are active non-zero ones, while most of the channel coefficients are inactive zero or near-zero ones. Thus, we propose a threshold to categorize the channel coefficients into two groups. Herein, a classify criterion which is used as the threshold is proposed based on the absolute value expectation of \(\hat{\mathbf{g}}\left(n\right)\) and it is defined as  
\[y(n)=\mathrm{E}\left[\left|\hat{g}_{i}\left(n\right)\right|\right],\forall 1 \leq i<M. \tag{24}\]  
Then, we categorize the channel coefficients into two groups in terms of the criterion in (24). When the channel coefficients \(\hat{g}_{i}(n)>y(n)\), the channel coefficients belong to the \"large\" group, while \(\hat{g}_{i}(n)<y(n)\), the channel coefficients are located in the \"small\" group. In fact, a threshold is proposed to split the variable norm into the \"large\" group and the \"small\" group, where the threshold is implemented using the mean of the estimated channel coefficients. If the channel coefficients are greater than this threshold, they are the \"large\" group. Otherwise, the channel coefficients belong to the \"small\" group when the channel coefficients are smaller than this threshold. For the \"large\" group, \(l_{0}\)-norm penalty is used to count the number of active channel coefficients, and \(l_{1}\)-norm penalty is adopted to uniformly attract inactive coefficients to zero for the \"small\" group. To effectively integrate these two groups into (23), we define [37]  
\[f_{i}=\frac{\text{sgn}\left[y(n)-\left|\hat{g}_{i}(n)\right|\right]+1}{2}, \forall 1\leq i<M. \tag{25}\]  
Therefore, \(f_{i}\) is set to be 0 when \(\hat{g}_{i}(n)>y(n)\), while \(f_{i}\) will be 1 for \(\hat{g}_{i}(n)<y(n)\). Finally, the updated equation of the GC-MCC is  
\[\hat{g}_{i}\left(n+1\right)=\hat{g}_{i}\left(n\right)+\beta_{\text{GC}}\exp \left(-\frac{e^{2}\left(n\right)}{2\sigma^{2}}\right)e\left(n\right)x_{i} \left(n\right)-\theta_{\text{GC}}f_{i}\text{sgn}\left[\hat{g}_{i}\left(n+1 \right)\right]\left(1-\frac{x_{i}\left(n\right)x_{i}^{\text{T}}\left(n\right)} {\left\|x_{i}\left(n\right)\right\|^{2}}\right). \tag{26}\]  
In Equation (26), \(\frac{x_{i}\left(n\right)x_{i}^{\text{T}}\left(n\right)}{\left\|x_{i}\left(n \right)\right\|^{2}}\) is far less than 1, so it can be ignored.