## 2 Proposed Algorithms
### Statistical Analysis of the Proposed Recursive Detection Method  
To give further explanation of the recursive state detector in Section 2.3, the probability problem analysis is shown in this section. To explain the detection problem briefly, we use the binary detection mode to describe it. Assuming that the impulsive signal \(\tan\left(\theta\right)s\left(n\right)\) follows the Gaussian distribution with mean \(a_{1}=E\left[\tan\left(\theta\right)s\left(n\right)+\varpi\left(n\right)\right]\) and variance \(\sigma_{\eta}^{2}\). For the smaller amplitude Gaussian noise \(\varpi\left(n\right)\), its mean value is \(a_{0}=E\left[\varpi\left(n\right)\right]\) and variance is also \(\sigma_{\eta}^{2}\). The hypotheses H\({}_{0}\) means that is no impulsive noise exists, and H\({}_{1}\) for impulsive noise to occur. The data length is \(N\). The binary detection problem is expressed as  
\[\begin{array}{l}\text{H}_{0}:x\left(n\right)=\varpi\left(n\right)\\ \text{H}_{1}:x\left(n\right)=\tan\left(\theta\right)s\left(n\right)+\varpi \left(n\right)\end{array}\Bigg{\}}\ \forall n\in\left[1,2,\cdots,N\right] \tag{38}\]Define \(P_{FA}\) is false alarm probability as  
\[P_{FA}=P\left(T\left(x\right)>\gamma_{s};\mathrm{H}_{0}\right) \tag{39}\]  
and define \(P_{D}\) is the detection probability as  
\[P_{D}=P\left(T\left(x\right)>\gamma_{s};\mathrm{H}_{1}\right). \tag{40}\]  
To minimize \(P_{FA}\) and maximize \(P_{D}\), using the method in [16], the statistic threshold for the impulsive noise detection is set as  
\[T\left(x\right)=\frac{1}{N_{w}}\sqrt{\sum_{n=1}^{N_{w}}\left[x\left(n\right) \right]^{2}-\sum_{n=1}^{N_{w}}\left[x\left(n\right)-\tan\left(\theta\right)s \left(n\right)\right]^{2}}. \tag{41}\]  
But to minimize the computational load, the threshold in Equation (41) is calculated recursively as mentioned in [15]. It means the threshold \(\sigma_{B,s}^{2}\) is the same one as \(T\left(x\right)\) in theory.  
For the second threshold \(\gamma_{s}\), it is calculated using the maximum the posteriori estimation of the Bayes method. Based on the assumption that \(\tan\left(\theta\right)s\left(n\right)\) and \(\varpi\left(n\right)\) follows the Gaussian distribution, the probability distribution function (PDF) is shown as follows  
\[p\left(x|\mathrm{H}_{k}\right)=\left(2\pi\sigma_{\eta}^{2}\right)^{-N_{w}/2} \exp\left(-\sum_{i=1}^{N_{w}}\frac{\left(x_{i}-a_{k}\right)^{2}}{2\sigma_{ \eta}^{2}}\right)\left(k=0,1\right). \tag{42}\]  
Therefore, the likelihood ratio function is written as  
\[\Lambda\left(x\right)=\frac{p\left(x\right|\mathrm{H}_{1}\right)}{p\left(x \right|\mathrm{H}_{0})}=\exp\left[\frac{\left(a_{1}-a_{0}\right)}{\sigma_{ \eta}^{2}}\sum_{i=1}^{N_{w}}x_{i}-\frac{N\left(a_{1}^{2}-a_{0}^{2}\right)}{2 \sigma_{\eta}^{2}}\right] \tag{43}\]  
and the posterior probability threshold is given by  
\[\gamma_{s}=\frac{\left(a_{1}+a_{0}\right)}{2}+\frac{\sigma^{2}\ln\Lambda_{0}} {N_{w}\left(a_{1}-a_{0}\right)}, \tag{44}\]  
where the \(\Lambda_{0}\) is the optimal likelihood ratio threshold.  
It is clear that if the data length \(N\) is long enough, then \(\gamma_{s}\rightarrow\left(a_{1}+a_{0}\right)/2\), which means that the \(P_{FA}\) is decreasing as the increasing of length \(N_{w}\). To calculate the \(\gamma_{s}\) recursively, Equations (33)-(36) are used in the proposed DMPExNLMS algorithm.