## 5 Results and Discussions of the PNMCC and CIM-PNMCC Algorithms
The CIM-PNMCC algorithm only needs 200 iterations to reach a steady-state MSD, while the previously reported RZA-MCC algorithm requires more than 400 iterations to get its steady-state MSD. Thereby, the CIM-PNMCC algorithm converges much quicker than the RZA-MCC algorithm for achieving the same MSD. Figure 4: Behaviors the CIM-PNMCC algorithm with different \(\rho_{\text{CIM}}\). Figure 5: Convergence of the PNMCC and CIM-PNMCC algorithms. Next, the effects of the sparsity level are analyzed by using the non-zero coefficients \(K\) in the FIR system. Here, we still use a 16 length sparse system that has \(K\) dominant coefficients. The simulated parameters for the mentioned adaptive filtering algorithms are \(\chi_{\text{MCC}}=0.03\), \(\chi_{\text{NMCC}}=0.4\), \(\mu_{\text{ZA}}=\mu_{\text{RZA}}=0.03\), \(\rho_{\text{ZA}}=8\times 10^{-5}\), \(\rho_{\text{RZA}}=2\times 10^{-4}\), \(\mu_{\text{PNLMS}}=0.27\), \(\chi=0.24\), \(\chi_{1}=0.3\), \(\rho_{\text{CIM}}=5\times 10^{-5}\). The estimation behavior of our developed PNMCC and CIM-PNMCC algorithms is discussed for various \(K\), and their estimation behaviors are given in Figures 6-9, respectively. It is noted that our developed PNMCC achieves a little gain compared with the PNLMS because they have comparable complexity and similar updating equation except the exponential weighting. However, our PNMCC algorithm achieves lower steady-state bias for \(K=1\). As for the CIM-PNMCC, it provides the lowest steady-state misalignment because it has a zero attraction term that can quickly force inactive coefficients to zero. With an increase of \(K\), the MSD floor is increased. However, our developed CIM-PNMCC still provides the lowest steady-state misalignment, indicating that the CIM-PNMCC is more useful. Even for \(K=8\), our CIM-PNMCC algorithm can still get an MSD level of \(10^{-4}\), which is a very low estimation bias. As we know, the ZA-MCC is a regular zero attraction algorithm. However, with the increasing of \(K\), we can see that the performance of the ZA-MCC has degraded because of its uniform zero attraction on all the channel coefficients. Then, we construct an example to discuss the tracking behavior of our devised PNMCC and CIM-PNMCC algorithms. Herein, we investigate their tracking performance over network echo channels with different sparsities, and the channel length is set to be 256. We use \(\zeta_{12}(\mathbf{w_{o}})=\frac{N}{N-\sqrt{N}}\left(1-\frac{\left\|\mathbf{w_{o} }\right\|_{1}}{\sqrt{N}\left\|\mathbf{w_{o}}\right\|_{2}}\right)\) to measure the sparsity of the designated network echo channel [37]. A typical network echo channel is given in Figure 10. The simulation parameters used in this experiment are \(\chi_{\text{MCC}}=0.0055\), \(\chi_{\text{NMCC}}=1.3\), \(\mu_{\text{ZA}}=\mu_{\text{RZA}}=0.0055\), \(\rho_{\text{ZA}}=4\times 10^{-6}\), \(\rho_{\text{RZA}}=1\times 10^{-5}\), \(\mu_{\text{PNLMS}}=1\), \(\chi=0.9\), \(\chi_{1}=0.8\), \(\rho_{\text{CIM}}=6\times 10^{-8}\). The tracking behavior for \(\zeta_{12}(\mathbf{w_{o}})=0.8222\) and \(\zeta_{12}(\mathbf{w_{o}})=0.7362\) is illustrated in Figure 2.0 Our PNMCC algorithm outperforms the PNLMS, MCC and NMCC algorithms in terms of the estimation bias for \(\zeta_{12}(\mathbf{w_{o}})=0.8222\). However, our PNMCC algorithm achieves the similar tracking behavior to the PNLMS algorithm because of the weighted step-size assignment scheme. For our developed CIM-PNMCC, it possesses the fastest convergence speed and lowest MSD for both \(\zeta_{12}(\mathbf{w_{o}})=0.8222\) and \(\zeta_{12}(\mathbf{w_{o}})=0.7362\). Therefore, a conclusion is given that our developed PNMCC and CIM-PNMCC algorithms can be used to effectively handle the sparse system identification. Furthermore, the CIM-PNMCC algorithm has little effect on the sparsity of \(\mathbf{w_{o}}\) compared with the mentioned algorithms above. Figure 2.1 Behaviors of the PNMCC and CIM-PNMCC algorithms for \(K=4\). Figure 2.2 Behaviors of the PNMCC and CIM-PNMCC algorithms for \(K=8\).