## 2 Review of the MCC and Its Related Sparse Algorithms  
We consider a typical channel estimation system based on the MCC algorithm, which is given in Figure 1. \(\mathbf{x}\left(n\right)\) is the input training signal with a length of \(M\), which is surveyed to an unknown sparse channel whose vector form is \(\mathbf{g}=\left[g_{1},g_{2},\cdots,g_{M}\right]^{T}\). As a sparse channel, most of the channel coefficients in the unknown sparse channel \(\mathbf{g}\) are zeros or near-zeros. Herein, we use \(K\) to represent the number of nonzero coefficients in the unknown sparse channel \(\mathbf{g}\). Then, the desired signal can be written as  
\[d\left(n\right)=\mathbf{g}^{T}\mathbf{x}\left(n\right)+r\left(n\right), \tag{1}\]  
where \(r\left(n\right)\) is a mixed Gaussian noise which is independent of the input training signal \(\mathbf{x}\left(n\right)\). Moreover, the instantaneous estimation error at \(n\)-th iteration is defined as  
\[e\left(n\right)=d\left(n\right)-\mathbf{\hat{g}}^{T}\left(n\right)\mathbf{x} \left(n\right), \tag{2}\]  
where \(\mathbf{\hat{g}}\left(n\right)\) denotes the estimated channel vector. The MCC-based channel estimation is to obtain the minimization of the iterative instantaneous estimation error \(e\left(n\right)\), and hence, we can estimate the unknown sparse channel \(\mathbf{g}\). The MCC algorithm is using the localized similarity to solve the following problem  
\[ \begin{array}{l}\min\frac{1}{2}\left\|\mathbf{\hat{g}}\left(n+1\right)- \mathbf{\hat{g}}\left(n\right)\right\|^{2}\\ \text{subject to }\hat{e}\left(n\right)=\left[1-\alpha\exp\left(-\frac{e^{2} \left(n\right)}{2\sigma^{2}}\right)\right]e\left(n\right)\end{array} , \tag{3}\]  
where \(\hat{e}\left(n\right)=d\left(n\right)-\mathbf{\hat{g}}^{T}\left(n+1\right) \mathbf{x}\left(n\right)\), and \(\sigma>0\) is a trade off parameter. Additionally, \(\left\|\bullet\right\|^{2}\) represents the Euclidean norm [36], and \(a=\beta_{\text{MCC}}\left\|\mathbf{x}\left(n\right)\right\|^{2}\), where \(\beta_{\text{MCC}}\) is the step-size of the MCC algorithm. Based on Equation (3), we can write MCC's cost function as  
\[J_{0}\left(n\right)=\frac{1}{2}\left\|\mathbf{\hat{g}}\left(n+1\right)- \mathbf{\hat{g}}\left(n\right)\right\|^{2}+\lambda_{\text{MCC}}\left(\hat{e} \left(n\right)-\left[1-\alpha\exp\left(-\frac{e^{2}\left(n\right)}{2\sigma^{2 }}\right)\right]e\left(n\right)\right), \tag{4}\]  
where \(\lambda_{\text{MCC}}\) is the multiplier parameter. By utilizing the Lagrange multiplier method (LMM), we can obtain the partial derivatives of \(\mathbf{\hat{g}}\left(n+1\right)\) and \(\lambda_{\text{MCC}}\) in Equations (5) and (6)  
\[\mathbf{\hat{g}}\left(n+1\right)-\mathbf{\hat{g}}\left(n\right)-\lambda_{\text {MCC}}\mathbf{x}\left(n\right)=0, \tag{5}\]  
\[\hat{e}\left(n\right)-\left[1-\alpha\exp\left(-\frac{e^{2}\left(n\right)}{2 \sigma^{2}}\right)\right]e\left(n\right)=0. \tag{6}\]  
Then, we get \(\lambda_{\text{MCC}}\)  
\[\lambda_{\text{MCC}}=\alpha\frac{\exp\left(-\frac{e^{2}\left(n\right)}{2 \sigma^{2}}\right)e\left(n\right)}{\left\|\mathbf{x}\left(n\right)\right\|^{2}}. \tag{7}\]  
Substituting (7) into (5), the updating equation with the vector formed of the MCC algorithm is  
\[\mathbf{\hat{g}}\left(n+1\right)=\mathbf{\hat{g}}\left(n\right)+\beta_{\text {MCC}}\exp\left(-\frac{e^{2}\left(n\right)}{2\sigma^{2}}\right)e\left(n\right) \mathbf{x}\left(n\right). \tag{8}\]It can be seen that the recursion of the MCC algorithm is similar to the LMS algorithm, except the exponential term. Similarly, the MCC algorithm does not utilize the intrinsic sparsity property of the sparse channels. Motivated by the ZA-LMS and RZA-LMS algorithms, sparse MCC algorithms have been presented and they are named as zero attracting MCC (ZA-MCC) and reweighting ZA-MCC (RZA-MCC) [33]. For the ZA-MCC algorithm, it is implemented by integrating a \(l_{1}\)-norm into MCC's cost function, and solving [33]  
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