## 3 The Proposed Group-Constrained Sparse MCC Algorithms  
From the above discussion, we found that the zero attractor \(-\theta_{\mathrm{ZASgn}}\left[\hat{\mathbf{g}}\left(n\right)\right]\) is realized by incorporating the \(l_{1}\)-norm penalty in the MCC's cost function, and it can speed up the convergence and reduce the channel estimation error. Moreover, the \(l_{0}\)-norm-constrained MCC algorithm can further improve the performance since the \(l_{0}\)-norm can count the number of the non-zero channel coefficients. However, the complexity is significantly increased due to the calculation of the \(l_{0}\)-norm approximation and its constraint. In order to fully exploit the sparsity property of the multi-path channel, we propose a group-constrained MCC algorithm by exerting the \(l_{0}\)-norm penalty on the group of large channel coefficients and forcing the \(l_{1}\)-norm penalty on the group of small channel coefficients. Herein, a non-uniform norm is used to split the non-uniform penalized algorithms into a large group and a small group, and the non-uniform norm is defined as [37; 38]  
\[\left\|\hat{\mathbf{g}}(n)\right\|_{p}^{p}=\sum_{i=1}^{M}|\hat{\mathbf{g}}_{i} (n)|^{p},0\leq p\leq 1, \tag{13}\]  
which is a \(l_{0}\)-norm when \(p\to 0\)  
\[\lim_{p\to 0}\left\|\hat{\mathbf{g}}(n)\right\|_{p}^{p}=\left\|\hat{\mathbf{g}} (n)\right\|_{0}, \tag{14}\]  
and it is a \(l_{1}\)-norm when \(p\) is infinitely close to \(1\)  
\[\lim_{p\to 1}\left\|\hat{\mathbf{g}}(n)\right\|_{p}^{p}=\left\|\hat{\mathbf{g}} (n)\right\|_{1}. \tag{15}\]  
Herein, the uniform-norm given in Equation (13) is a variable norm which is controlled by the parameter \(p\). When \(p\) is very close to zero, the proposed norm in Equation (13) can be regarded as a \(l_{0}\)-norm. As for \(p=1\), the norm in Equation (13) is the \(l_{1}\)-norm. Then, the constructed non-uniform norm in Equation (13) is introduced into the MCC's cost function to devise our proposed GC-MCC algorithm, and GC-MCC is to solve  
\[ \begin{array}{l}\min\frac{1}{2}\left\|\hat{\mathbf{g}}\left(n+1\right)-\hat {\mathbf{g}}\left(n\right)\right\|^{2}+\theta_{\mathrm{GC}}\left\|\hat{ \mathbf{g}}(n+1)\right\|_{p,M}^{p}\\ \mathrm{subject\,\,\,to\,\,}\delta\left(n\right)=\left[1-\alpha\exp\left(- \frac{e^{2}\left(n\right)}{2\sigma^{2}}\right)\right]e\left(n\right)\end{array} , \tag{16}\]  
where \(\left\|\hat{\mathbf{g}}(n+1)\right\|_{p,M}^{p}\) is a kind of \(\left\|\hat{\mathbf{g}}(n+1)\right\|_{p}^{p}\), which uses a different value of \(p\) for each channel coefficient at \(M\)-th position in the sparse channels, here with the introduction of \(\mathbf{p}=\left[p_{1},p_{2},...,p_{M}\right]\), and \(\theta_{\mathrm{GC}}\) is a regularization parameter. Then, the cost function of the GC-MCC algorithm is  
\[J_{\mathrm{GC}}\left(n\right)=\frac{1}{2}\left\|\hat{\mathbf{g}}\left(n+1 \right)-\hat{\mathbf{g}}\left(n\right)\right\|^{2}+\theta_{\mathrm{GC}}\left\| \hat{\mathbf{g}}(n+1)\right\|_{p,M}^{p}+\lambda_{\mathrm{GC}}\left(\delta \left(n\right)-\left[1-\alpha\exp\left(-\frac{e^{2}\left(n\right)}{2\sigma^{ 2}}\right)\right]e\left(n\right)\right). \tag{17}\]  
In Equation (17), \(\lambda_{\mathrm{GC}}\) is a multiplier. Based on the LMM, the gradients of \(J_{\mathrm{GC}}\left(n\right)\) with respect to \(\hat{\mathbf{g}}\left(n+1\right)\) and \(\lambda_{\mathrm{GC}}\) are  
\[\frac{\partial J_{\mathrm{GC}}\left(n\right)}{\partial\hat{\mathbf{g}}\left(n +1\right)}=0, \tag{18}\]  
and  
\[\frac{\partial J_{\mathrm{GC}}\left(n\right)}{\partial\lambda_{\mathrm{GC}}}=0. \tag{19}\]  
Then, we can get  
\[\hat{\mathbf{g}}_{i}\left(n+1\right)=\hat{\mathbf{g}}_{i}\left(n\right)-\theta _{\mathrm{GC}}\frac{p_{i}\cdot\mathrm{sgn}\left[\hat{\mathbf{g}}_{i}\left(n+1 \right)\right]}{\left|\hat{\mathbf{g}}_{i}\left(n+1\right)\right|^{1-p_{i}}}+ \lambda_{\mathrm{GC}}x_{i}\left(n\right), \tag{20}\]\[\hat{e}\left(n\right)=\left[1-\alpha\exp\left(-\frac{e^{2}\left(n\right)}{2\sigma^{ 2}}\right)\right]e\left(n\right), \tag{21}\]  
where \(p_{i}\) is the \(i\)-th element of matrix \(\mathbf{p}\).