## Proposed Proportionate NMCC Algorithms
### Proportionate NMCC with a CIM
 \tag{21}\]  
The gradients on \(J(n+1)\) in Equation (20) by considering \(\mathbf{\tilde{w}}(n+1)\) and \(\lambda\) are given by  
\[\frac{\partial J(n+1)}{\partial\mathbf{\tilde{w}}(n+1)}=0\text{ and }\frac{ \partial J(n+1)}{\partial\lambda}=0 \tag{22}\]\[\widehat{\mathbf{w}}(n+1)=\widehat{\mathbf{w}}(n)+\lambda\mathbf{G}(n)\mathbf{x}(n) -\gamma_{\text{CIM}}\frac{1}{N\sigma_{1}^{3}\sqrt{2\pi}}\widehat{\mathbf{w}}(n) \exp\left(-\frac{\left\|\widehat{\mathbf{w}}(n)\right\|^{2}}{2\sigma_{1}^{2}} \right). \tag{23}\]  
Multiplying \(\mathbf{x}^{T}(n)\) on both sides of Equation (23), we get  
\[\mathbf{x}^{T}(n)\widehat{\mathbf{w}}(n+1)= \tag{24}\] \[\mathbf{x}^{T}(n)\widehat{\mathbf{w}}(n)+\lambda\mathbf{x}^{T}(n )\mathbf{G}(n)\mathbf{x}(n)-\gamma_{\text{CIM}}\mathbf{x}^{T}(n)\frac{1}{N \sigma_{1}^{3}\sqrt{2\pi}}\widehat{\mathbf{w}}(n)\exp\left(-\frac{\left\| \widehat{\mathbf{w}}(n)\right\|^{2}}{2\sigma_{1}^{2}}\right).\]  
Considering Equation (22), we obtain  
\[\hat{e}\left(n\right)=\left[1-\xi\exp\left(-\frac{e^{2}\left(n\right)}{2 \sigma^{2}}\right)\right]e\left(n\right). \tag{25}\]  
The Lagrange multiplier is obtained  
\[\lambda=\frac{\xi\exp\left(-\frac{e^{2}\left(n\right)}{2\sigma^{2}}\right)e \left(n\right)+\frac{\gamma_{\text{CIM}}\mathbf{x}^{T}\left(n\right)}{N\sigma _{1}^{3}\sqrt{2\pi}}\widehat{\mathbf{w}}\left(n\right)\exp\left(-\frac{ \left\|\widehat{\mathbf{w}}\left(n\right)\right\|^{2}}{2\sigma_{1}^{2}} \right)}{\mathbf{x}^{T}\left(n\right)\mathbf{G}\left(n\right)\mathbf{x}\left(n \right)}. \tag{26}\]  
Considering Equations (26) and (23), we get  
\[ \begin{split}\widehat{\mathbf{w}}(n+1)&=\widehat{ \mathbf{w}}(n)+\frac{\xi\exp\left(-\frac{e^{2}\left(n\right)}{2\sigma^{2}} \right)e\left(n\right)+\frac{\gamma_{\text{CIM}}\mathbf{x}^{T}\left(n\right) }{N\sigma_{1}^{3}\sqrt{2\pi}}\widehat{\mathbf{w}}\left(n\right)\exp\left(- \frac{\left\|\widehat{\mathbf{w}}\left(n\right)\right\|^{2}}{2\sigma_{1}^{2}} \right)}{\mathbf{x}^{T}\left(n\right)\mathbf{G}\left(n\right)\mathbf{x}\left( n\right)}\mathbf{G}(n)\mathbf{x}(n)\\ &-\gamma_{\text{CIM}}\frac{1}{N\sigma_{1}^{3}\sqrt{2\pi}} \widehat{\mathbf{w}}(n)\exp\left(-\frac{\left\|\widehat{\mathbf{w}}(n)\right\| ^{2}}{2\sigma_{1}^{2}}\right)\\ &=\widehat{\mathbf{w}}(n)+\frac{\xi e(n)\exp\left(-\frac{e^{2} \left(n\right)}{2\sigma^{2}}\right)\mathbf{G}(n)\mathbf{x}(n)}{\mathbf{x}^{T} (n)\mathbf{G}(n)\mathbf{x}(n)}\\ &-\gamma_{\text{CIM}}\left\{\mathbf{I}-\frac{\mathbf{G}(n) \mathbf{x}(n)\mathbf{x}^{T}(n)}{\mathbf{G}(n)\mathbf{x}(n)}\right\}\frac{1}{N \sigma_{1}^{3}\sqrt{2\pi}}\widehat{\mathbf{w}}(n)\exp\left(-\frac{\left\| \widehat{\mathbf{w}}(n)\right\|^{2}}{2\sigma_{1}^{2}}\right).\end{split}  \tag{27}\]  
From the updated Equation (27), we can see that the elements in \(\mathbf{G}(n)\mathbf{x}(n)\mathbf{x}^{T}(n)\{\mathbf{x}^{T}(n)\mathbf{G}(n) \mathbf{x}(n)\}^{-1}\) are very small in comparison with \(1\). Thus, we can ignore this term, and the proposed CIM-PNMCC's updating equation is  
\[\widehat{\mathbf{w}}(n+1)=\widehat{\mathbf{w}}(n)+\frac{\xi e(n)\exp\left(- \frac{e^{2}\left(n\right)}{2\sigma^{2}}\right)\mathbf{G}(n)\mathbf{x}(n)}{ \mathbf{x}^{T}(n)\mathbf{G}(n)\mathbf{x}(n)}-\gamma_{\text{CIM}}\frac{1}{N \sigma_{1}^{3}\sqrt{2\pi}}\widehat{\mathbf{w}}(n)\exp\left(-\frac{\left\| \widehat{\mathbf{w}}(n)\right\|^{2}}{2\sigma_{1}^{2}}\right). \tag{28}\]  
We introduce a step-size \(\beta\) and \(\epsilon_{\text{CIM}}=\delta_{x}^{2}/N\). Then, the proposed CIM-PNMCC's updating equation is changed to be  
\[ \begin{split}&\widehat{\mathbf{w}}(n+1)=\\ &\widehat{\mathbf{w}}(n)+\chi_{1}\frac{\exp\left(-\frac{e^{2} \left(n\right)}{2\sigma^{2}}\right)\mathbf{G}(n)e(n)\mathbf{x}(n)}{\mathbf{x}^ {T}(n)\mathbf{G}(n)\mathbf{x}(n)+\epsilon_{\text{CIM}}}-\rho_{\text{CIM}}\frac{1 }{N\sigma_{1}^{3}\sqrt{2\pi}}\widehat{\mathbf{w}}(n)\exp\left(-\frac{\left\| \widehat{\mathbf{w}}(n)\right\|^{2}}{2\sigma_{1}^{2}}\right),\end{split}