## 4 Convergence Analysis of the Revised CIM-PNMCC
### Mean Convergence  
A misalignment vector is defined by  
\[\widehat{\mathbf{w}}\left(n\right)=\widehat{\mathbf{w}}\left(n\right)-\mathbf{ w}_{0}\left(n\right). \tag{31}\]Then, combining Equations (1), (2) and (30), we obtain  
\[ \begin{split}\hat{\mathbf{w}}\left(n+1\right)&=\hat{ \mathbf{w}}\left(n\right)+\frac{\chi_{1}\mathbf{G}\left(n\right)f\left(e\left( n\right)\right)e\left(n\right)\mathbf{x}\left(n\right)}{\mathbf{x}^{T}\left(n \right)\mathbf{G}\left(n\right)\mathbf{x}\left(n\right)+\varepsilon_{\text{ CIM}}}+\rho_{\text{CIM}}m\left(n\right)\\ &=\hat{\mathbf{w}}\left(n\right)+\frac{\chi_{1}\mathbf{G}\left(n \right)f\left(e\left(n\right)\right)}{\mathbf{x}^{T}\left(n\right)\mathbf{G} \left(n\right)\mathbf{x}\left(n\right)+\varepsilon_{\text{CIM}}}\left( \mathbf{x}^{T}\left(n\right)\mathbf{w}_{0}\left(n\right)+v\left(n\right)- \mathbf{x}^{T}\left(n\right)\hat{\mathbf{w}}\left(n\right)\right)\mathbf{x} \left(n\right)+\rho_{\text{CIM}}m\left(n\right)\\ &=\hat{\mathbf{w}}\left(n\right)+\frac{\chi_{1}\mathbf{G}\left(n \right)f\left(e\left(n\right)\right)v\left(n\right)\mathbf{x}\left(n\right)}{ \mathbf{x}^{T}\left(n\right)\mathbf{G}\left(n\right)\mathbf{x}\left(n\right)+ \varepsilon_{\text{CIM}}}-\frac{\chi_{1}\mathbf{G}\left(n\right)f\left(e \left(n\right)\right)\mathbf{x}\left(n\right)\mathbf{x}\left(n\right)\mathbf{ x}\left(n\right)+\varepsilon_{\text{CIM}}}{\mathbf{x}^{T}\left(n\right) \mathbf{G}\left(n\right)\mathbf{x}\left(n\right)+\varepsilon_{\text{CIM}}}+ \rho_{\text{CIM}}m\left(n\right)\\ &=\left[\mathbf{I}-\frac{\chi_{1}\mathbf{G}\left(n\right)f\left(e \left(n\right)\right)\mathbf{x}\left(n\right)\mathbf{x}^{T}\left(n\right)}{ \mathbf{x}^{T}\left(n\right)\mathbf{G}\left(n\right)\mathbf{x}\left(n\right) \mathbf{x}\left(n\right)+\varepsilon_{\text{CIM}}}\right]\hat{\mathbf{w}} \left(n\right)+\frac{\chi_{1}\mathbf{G}\left(n\right)f\left(e\left(n\right) \right)v\left(n\right)\mathbf{x}\left(n\right)}{\mathbf{x}^{T}\left(n\right) \mathbf{G}\left(n\right)\mathbf{x}\left(n\right)+\varepsilon_{\text{CIM}}}+ \rho_{\text{CIM}}m\left(n\right).\end{split}  \tag{32}\]  
Taking the expectation on Equation (32) and using the A1, A2, A3 and A5, we get  
\[ \begin{split}\text{E}\left[\hat{\mathbf{w}}\left(n+1\right) \right]&=\left[I-\frac{\chi_{1}\mathbf{G}\left(n\right)\text{E} \left[f\left(e\left(n\right)\right)\mathbf{x}\left(n\right)\mathbf{x}^{T} \left(n\right)\right]}{\text{E}\left[\mathbf{x}^{T}\left(n\right)\mathbf{G} \left(n\right)\mathbf{x}\left(n\right)+\varepsilon_{\text{CIM}}\right]}\right] \text{E}\left[\hat{\mathbf{w}}\left(n\right)\right]+\rho_{\text{CIM}}\text{E} \left[m\left(n\right)\right]\\ &=\left[I-\frac{\chi_{1}\mathbf{G}\left(n\right)\mathbf{R}}{N \sigma_{x}^{2}+\varepsilon_{\text{CIM}}}\right]\text{E}\left[\hat{\mathbf{w}} \left(n\right)\right]+\rho_{\text{CIM}}\text{E}\left[m\left(n\right)\right], \end{split}  \tag{33}\]  
where \(\mathbf{R}=\text{E}\left[f\left(e\left(n\right)\right)\mathbf{x}\left(n\right) \mathbf{x}^{T}\left(n\right)\right]\) represents the auto-correlation matrix. Based on the symmetric and positive semidefinite property of the \(\mathbf{R}\), we have  
\[\mathbf{R}=\mathbf{Q}\mathbf{\Lambda}\mathbf{Q}^{T}, \tag{34}\]  
where \(\mathbf{\Lambda}=\text{diag}\left(\lambda_{1},\lambda_{2},\cdots,\lambda_{k}\right)\) is the eigenvalue matrix. Next, premultiplying \(\mathbf{Q}^{T}\) on both sides of Equation (33), and considering \(\mathbf{Q}\) unitarians, we get  
\[\hat{\mathbf{w}}^{\prime}\left(n+1\right)=\left[\mathbf{I}-\frac{\chi_{1} \mathbf{G}\left(n\right)\mathbf{\Lambda}}{N\sigma_{x}^{2}+\varepsilon_{\text{ CIM}}}\right]\hat{\mathbf{w}}^{\prime}\left(n\right)+\rho_{\text{CIM}}\mathbf{Q}^{T} \text{E}\left[m\left(n\right)\right], \tag{35}\]  
where \(\hat{\mathbf{w}}^{\prime}\left(n\right)=\mathbf{Q}^{T}\text{E}\left[\hat{ \mathbf{w}}\left(n\right)\right]\), and \(\left[\mathbf{I}-\frac{\chi_{1}\mathbf{G}\left(n\right)\mathbf{\Lambda}}{N \sigma_{x}^{2}+\varepsilon_{\text{CIM}}}\right]\) represents a diagonal.