## 4 Experimental Results  
Several experiments were constructed to give an analysis on the performance of the PAPMCC algorithm for implementing the sparse CE. Since the \(\alpha\)-stable distribution can well construct the non-Gaussian phenomenon, which is ubiquitous in practice, it was chosen to model the impulsive noise in the simulations. The \(\alpha\)-stable distribution function is defined as  
\[f(t)=\exp\left\{j\chi t-\gamma|t|^{\alpha}\left[1+j\beta\mathrm{sgn}(t)S(t, \alpha)\right]\right\}, \tag{21}\]  
where  
\[S(t,\alpha)=\left\{ \begin{array}{ll}\tan\frac{\alpha\gamma}{2}&\text{if} \quad\alpha\
eq 1\\ \frac{2}{\pi}\log|t|&\text{if}\quad\alpha=1,\end{array} \right. \tag{22}\]  
in which \(\alpha\in(0,2]\) represents the characteristic index, which controls the behavior of the impulsive distribution. When parameter \(\alpha\) is smaller, the impulsive intensity becomes larger. \(\beta\in[-1,1]\) is the symmetric parameter, \(\chi\) denotes positional parameter, and \(\gamma>0\) represents the dispersion parameter. Furthermore, the \(\alpha\)-stable distribution is given by \(V_{\alpha-\mathrm{stable}}(\alpha,\beta,\gamma,\chi)\). Herein, \(V_{\alpha-\mathrm{stable}}(1.5,0,0.2,0)\) is chosen to implement the impulsive noise. In all simulation experiments, \(K=1024\), and \(\sigma=1\) were selected, and the input signal power was \(1\). The network echo channel used for the experiments, which is classical sparse channel presented in Figure 2, whose active coefficients distributed in \([257,272]\), was considered to evaluate the proposed PAPMCC algorithm. The related parameters were set to be \(\delta_{\mathrm{AP}}=\delta_{\mathrm{ZA-AP}}=\delta_{\mathrm{RZA-AP}}=0.01\) and \(\delta_{\mathrm{PAP}}=\delta_{\mathrm{PAPMCC}}=\frac{1}{\xi}\delta_{\mathrm{AP}}\)[48]. The performance for all used algorithms was evaluated by normalized misalignment (NM), which is written as \(10\mathrm{log}_{10}(\left\|\mathbf{w}-\mathbf{\hat{w}}\right\|_{2}^{2}/\left\| \mathbf{w}\right\|_{2}^{2})\). \begin{table}
\begin{tabular}{l c c c} \hline \hline
**Algorithm** & **Addition** & **Multiplication** & **Division** \\ \hline AP & \((2M^{2}+M)K\) & \((2M^{2}+3M)K+M^{2}\) & 0 \\ ZA-AP & \((2M^{2}+M+1)K\) & \((2M^{2}+3M+1)K+M^{2}\) & 0 \\ RZA-AP & \((2M^{2}+M+2)K\) & \((2M^{2}+3M+2)K+M^{2}\) & \(K\) \\ PAP & \(2 MK^{2}+(2M^{2}-M+1)K-1\) & \(2 MK^{2}+(2M^{2}+3M+1)K+M^{2}\) & \(K\) \\ PAPMCC & \(2 MK^{2}+(2M^{2}-M+1)K-1\) & \(2 MK^{2}+(2M^{2}+3M+1)K+M^{2}+2M\) & \(K+M\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Computational complexity in each iteration. Performance of the PAPMCC Algorithm with Various Projection Orders M, Step-Sizes \(\mu\) and Kernel Width \(\sigma\)  
Firstly, the effects of the projection order \(M\) on the convergence for the PAPMCC algorithm was investigated. The colored noise, which was obtained from white Gaussian noise (WGN) filtering through an autoregressive with a pole at 0.8, was used as the input signal. Herein, \(\mu=0.05\). The results given in Figure 3 point out that increasing the projection order \(M\) could speed up the convergence, while the steady-state misalignment was increased. Therefore, a trade-off between the convergence speed and steady-state misalignment should be taken into consideration. Secondly, the effects of kernel width \(\sigma\) on the convergence for the PAPMCC algorithm was analyzed and discussed. Herein, \(M=4\) was selected. From Equation (20), the parameter \(\sigma\) affects the estimation behaviors of the PAPMCC algorithm, while \(\sigma\) is an important parameter for Gaussian kernel to suppress noise interference. Given the diversity and complexity of the target signal and noise, it is not easy to get the optimal solution of the kernel width \(\sigma\) from the theoretical derivation. Therefore, the simulation experiments were used to determine the appropriate value \(\sigma\). The results given in Figure 4 point out that the steady-state error of the PAPMCC algorithm increased with the  
Figure 3: The effects of the projection orders on PAPMCC algorithm.