## 5 Results and Discussions of the PNMCC and CIM-PNMCC Algorithms  
We will discuss the behavior of our proposed PNMCC and CIM-PNMCC for identifying sparse systems. Furthermore, their system identification performance is analyzed by comparing with NMCC, MCC, RZA-MCC and ZA-MCC algorithms. Additionally, the PNLMS algorithm is also employed for estimation performance comparison since the MCC is an LMS-like algorithm and the PNLMS is a proportionate-typed NLMS, which is also a PNMCC-like algorithm. The MSD is employed for evaluating the behaviors of the devised PNMCC and CIM-PNMCC algorithms, defining by  
\[\mathrm{MSD}(\tilde{\mathbf{w}}(n))=\mathrm{E}[\|\mathbf{w}_{o}(n)-\tilde{ \mathbf{w}}(n)\|^{2}]. \tag{48}\]  
Based on the derivation of the PNMCC and CIM-PNMCC algorithms, \(\chi\) gives an important effect on the PNMCC algorithm, while \(\chi_{1}\) and \(\rho_{\mathrm{CIM}}\) play important effects on the CIM-PNMCC algorithm. Thereby, we first investigate their effects on the estimation behaviors over sparse systems with impulsive noise. In this paper, \((1-\theta)N(\iota_{1},\
u_{1}^{2})+\theta N(\iota_{2},\
u_{2}^{2})=(0,0.01,0,20,0.05)\) is used for generating the desired noise, where \(N(\iota_{i},\
u_{i}^{2})(i=1,2)\) are the Gaussian distributions with means of \(\iota_{i}\) and variances of \(\upsilon_{i}^{2}\), where \(\theta\) is a mixture parameter. Herein, a sparse system with a length of \(N=16\) is adopted. One non-zero tap randomly distributes in the system. To investigate the effects of \(\chi\) on the PNMCC algorithm, the simulation parameters are \(\vartheta=0.01\) and \(\sigma=1000\) with corresponding results presented in Figure 2. The developed PNMCC converges faster when \(\chi\) increases from \(\chi=0.3\) to \(\chi=1\), which has a similar function in the PNLMS algorithm to implement the step-size. Similarly, the parameter \(\chi_{1}\) is also selected to understand the performance of the derived CIM-PNMCC, whose performance is described in Figure 3. The CIM-PNMCC's convergence speed rate becomes faster when \(\chi_{1}\) increases from \(0.2\) to \(0.7\). However, the estimation misalignment is getting worse with an increment of \(\chi_{1}\). Additionally, \(\rho_{\mathrm{CIM}}\) is a zero attraction controlling parameter, which is designed to trade off the sparsity development and estimation bias. The effects of \(\rho_{\mathrm{CIM}}\) on the CIM-PNMCC estimation behavior is illustrated in Figure 4. It is observed that the estimation bias of the CIM-PNMCC is reduced as \(\rho_{\mathrm{CIM}}\) ranges from \(9\times 10^{-4}\) to \(5\times 10^{-5}\). If \(\rho_{\mathrm{CIM}}\) is reduced from \(5\times 10^{-5}\) to \(5\times 10^{-7}\), the estimation bias rebounds towards the opposite direction. Therefore, we should choose proper parameter value to obtain good performance of the PNMCCFigure 3: Effects on the CIM-PNMCC algorithm with different \(\chi_{1}\). Figure 2: Effects on the PNMCC algorithm with different \(\chi\). Then, these parameters are used to obtain the convergence speed investigation of the developed PNMCC and CIM-PNMCC algorithms. Here, \(\chi_{\text{MCC}}=0.0052\), \(\chi_{\text{NMCC}}=0.085\), \(\mu_{\text{ZA}}=0.01\), \(\mu_{\text{RZA}}=0.015\), \(\rho_{\text{ZA}}=3\times 10^{-5}\), \(\rho_{\text{RZA}}=7\times 10^{-5}\), \(\mu_{\text{PNLMS}}=0.072\), \(\chi=0.088\), \(\chi_{1}=0.4\), \(\rho_{\text{CIM}}=2\times 10^{-5}\), where \(\chi_{\text{NMCC}}\), \(\chi_{\text{MCC}}\), \(\mu_{\text{RZA}}\), \(\mu_{\text{ZA}}\), \(\mu_{\text{PNLMS}}\) are step-sizes for NMCC, MCC, RZA-MCC, ZA-MCC and PNLMS algorithms. \(\rho_{\text{RZA}}\) and \(\rho_{\text{ZA}}\) are the regularization parameters for RZA-MCC and ZA-MCC, respectively. The convergence comparisons for PNMCC and CIM-PNMCC algorithms are given in Figure 5. From Figure 5, we observe that the PNMCC converges faster than the PNLMS and NMCC algorithms. It is worth noting that our developed CIM-PNMCC gets the fastest convergence speed for adaptive system identifications.