## 2 Proposed Algorithms
### PFxNLMS with M-Estimate Function  
For automation selection of an appropriate value of \(\rho\) to enhance the tracking of conventional method in [13], PFxNLMS with M-estimate function (MPFxNLMS) algorithm is proposed in this section. A Gaussian kernel of information entropy criterion is used in PFxNLMS to act as the new M-function. Referring to [15], the adaptive Gaussian kernel of information entropy criterion is defined as  
\[\kappa\left(n\right)=\exp\left(-\frac{e^{2}\left(n\right)}{2\sigma^{2}\left(n \right)}\right). \tag{1}\]  
Apparently, the Gaussian kernel of information entropy criterion is an M-estimate function with adaptive adjustment threshold. To calculate \(\sigma\left(n\right)\) recursively, the  
to ensure that the adaptive algorithm works in robust modes under severe noise conditions [16]. To solve this problem, an on-line detection method is combined with the MPFxNLMS algorithm to enhance the robust performance, called the detector based M-estimate function post normalized filtered-x least mean square (DMPFxNLMS) algorithm, where \"D\" represents the new state detector. The block diagram of the DMPFxNLMS algorithm is shown in Figure 2.  
In the DMPFxNLMS, the forgetting coefficient in Equation (3) is redefined as  
\[\varepsilon\left(n\right)=\left\{ \begin{array}{l}\kappa\left(n\right)-1, \ \ \ \ \ if\ \ \textbf{fast tracking mode}\\ 1-\kappa\left(n\right),\ \ \ \ \ if\ \ \ \textbf{robust mode}\end{array} \right., \tag{14}\]  
where \(\varepsilon\left(n\right)<0\) is defined as the **fast tracking mode** and \(0\leq\varepsilon\left(n\right)<1\) is defined as the **robust mode**. Inspired by [17] combining the statistical theories and the two-class division method to select an interested target for two-dimensional digital graphics, we developed a novel recursive impulsive noise detection method for the ANC system. The detection factor is calculated as follows  
\[\eta\left(n\right)=\frac{k\left[1-\kappa\left(n\right)\right]}{L},\ \ if\ \frac{k}{L}\leq\eta\left(n\right)<\frac{k+1}{L}\left(k=0,1,\cdots,L-1\right), \tag{15}\]  
where \(\eta\left(n\right)\) is divided into \(L\) levels \(\left[1,2,\cdots,L\right]\). A sliding window of length \(N_{w}\) is used to restore the quantization value of \(\eta\left(n\right)\). To illustrate the adaptive detection process more briefly, define  
\[\Phi_{1}=\eta\left(n\right) \tag{16}\]  
\[\Phi_{2}=\eta\left(n-N_{w}+1\right) \tag{17}\]  
where \(\Phi_{1}\) represents the input value of detection factor, and \(\Phi_{2}\) represents the output value of detection factor. Based on [17], the optimal threshold is calculated as follows. Assuming that the values of \(L\) levels of \(\eta\left(n\right)\) can be divided into two groups as non-impulsive and impulsive noise, the number of amplitudes at level \(i\) is denoted by \(n_{i}\), and the total number of amplitudes is denoted by \(N_{w}=n_{1}+n_{2}+\cdots+n_{L}\). The probability at the \(i\)-th level is normalized and expressed by  
\[p_{i}=\frac{n_{i}}{N_{w}} \tag{18}\]  
Figure 2: Block diagram of the proposed DMPFxNLMS algorithm. \[p_{i}\geq 0,\sum_{i=1}^{L}p_{i}=1. \tag{19}\]  
On the basis of the different impulsiveness of amplitudes, the data is split into two classes \(C_{0}\) and \(C_{1}\) with a threshold at level \(k\). \(C_{0}\) represents amplitudes with levels \([1,\cdots,k]\), which denotes the non-impulsiveness class. The parameter \(C_{1}\) represents amplitudes with levels \([k+1,\cdots,L]\), which denotes the impulsiveness class. Then the probabilities of occurrence of a class and its mean level are given by Equations (20)-(23):  
\[P_{0,k}=\Pr\left(C_{0}\right)=\sum_{i=1}^{k}p_{i}=P_{k} \tag{20}\]  
\[P_{1,k}=\Pr\left(C_{1}\right)=\sum_{i=k+1}^{L}p_{i}=1-P_{k} \tag{21}\]  
and  
\[m_{0,k}=\sum_{i=1}^{k}i\Pr\left( \begin{array}{cc}i&|C_{0}\end{array} \right)= \sum_{i=1}^{k}\frac{ip_{i}}{P_{0,k}}=\frac{m_{k}}{P_{k}} \tag{22}\]  
\[m_{1,k}=\sum_{i=k+1}^{L}i\Pr\left( \begin{array}{cc}i&|C_{1}\end{array}