## 3 Simulations
### Random Impulsive Interference  
The stable noise environment was simulated by the mixture of sinusoidal signal of 150 Hz, 300 Hz, 450 Hz and additional Gaussian noise. In this simulation, using [18; 19], the impulsive interference process was produced by a random noise function of a symmetric \(\alpha\)-stable (S\(\alpha\)S) distribution. The probability function of the random noise producer is given by  
\[\varphi\left(t\right)=\exp\left(-|t|^{\alpha}\right). \tag{47}\]  
The \(\alpha\) changed with different noise environments. To make comments of the proposed DMPFxNLMS completely, the Gaussian noise (\(\alpha=2\)) field and the variable-\(\alpha\) environment should also be considered. According to this idea, as shown in Figure 5, in the noise interference environment where \(\alpha=1.2\) and \(\alpha=1.7\), the M-estimation method based FxlogLMS, the PFxNLMS(Opt) and FxRMC were used to compare with DMPFxNLMS. The impulsive interferences began at the 25,000-th iteration and end with the 31,000-iteration. The changing process of \(\varepsilon\left(n\right)\) is also shown to give further information to understand DMPFxNLMS.  
In Figure 5, PFxNLMS(Opt) diverges at about 30,000 iterations. The FxRMC has the best initiate tracking ability, when impulsive interference is strong enough (\(\alpha=1.2\)), it diverges at about 25,000 iterations. The FxlogLMS performs worse on noise reduction level, and it cannot keep steady when interference occurs. The proposed DMPFxNLMS had a lower computational load than FxRMC, and it had similar initial tracking performance and better robust performance when strong interference  
Figure 4: Selection of optimal \(\rho\) for PFxNLMS. (**a**) Reference signal in time domain; (**b**) ANR performance.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Algorithms** & **Parameters** \\ \hline FxLMS & \(\mu=2.5\times 10^{-4}\) \\ FxlogLMS & \(\mu=2.5\times 10^{-4}\), \(\Delta=1\times 10^{-10}\) \\ PFxNLMS & \(\mu=0.01,\rho=-0.5\) \\ FxRMC & \(\lambda=0.999,N_{w}=2,\delta=0.02\) \\ DMPFxNLMS & \(\mu=0.01,N_{q}=2,N_{w}=100,L=10\), initial \(\theta=0.4\pi\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: Parameters for different methods in this paper.  
happened. In Figure 5b, it is noted that in the iterations between 30,000 and 31,000, DMPFxNLMS works in robust state, which shows the reason of that DMPFxNLMS dose not diverge at the 30,000 iterations like the PFxNLMS(Opt).  
In a lower degree impulsiveness noise environment as shown in Figure 6a, the DMPFxNLMS had better tracking performance than that of the PFxNLMS(Opt). Although the FxRMC has superior performance than other methods, it has bigger computational load. The Figure 6b reveals that DMPFxNLMS works most of the time in fast tracking mode, which proves the conclusion again that DMPFxNLMS could detect the impulsive degrees and find optimal parameters for adaptive filters.  
Based on theory analysis, it is noted that the PFxNLMS used the certain \(\rho\) to fit a certain environment. Considering Figures 5a and 6a, the impulsive process always has a variable \(\alpha\), which may decrease the performance of PFxNLMS. The FxlogLMS was able to deal with burst interference, but it had lower noise reduction level in Gaussian environment, which was also demonstrated in [9]. For FxRMC, the RLS based algorithm had better tracking capability than DMPFxNLMS, but it cannot  
Figure 5: Performance comparison when impulsive interference \(\alpha=1.2\) (**a**) ANR curves; (**b**) changing process of \(\varepsilon\left(n\right)\).  
Figure 6: Performance comparison when impulsive interference \(\alpha=1.7\) (**a**) ANR curves; (**b**) changing process of \(\varepsilon\left(n\right)\).  
be used when severe impulsive interference occurs. Simulations in this section show that the proposed DMPFxNLMS works well in both burst and less-burst noise field.