## 2 Methodology
### Robust Adaptive Filtering Based on GMACC  
Robust adaptive filters build cost functions using statistics that can reduce the impact of impulsive noise, such as Huber statistics, fractional lower-order statistics, correntropy-induced metric statistics, etc. In the discussed situation of this paper, the system noise is not only impulsive but also asymmetric. In order to deal with this kind of system noise, this paper uses a generalized asymmetric correntropy as the cost function to build a robust linear adaptive filter. Based on the proposed generalized asymmetric Gaussian kernel, the cost function of the adaptive filter can be presented as:  
\[E[GAG_{\alpha,\sigma_{+},\sigma_{-}}(\varepsilon_{k})], \tag{6}\]  
where the estimation error can be expressed as  
\[\varepsilon_{k}=d_{k}-y_{k}=d_{k}-\Phi^{T}\Gamma_{k}, \tag{7}\]  
with \(d_{k}\in\mathbb{R}\) being the desired output at time \(k\), \(y_{k}=\Phi^{T}\Gamma_{k}\) being the output of the linear adaptive filter, \(\Phi=\left[\phi_{1},\phi_{2},\ldots,\phi_{n}\right]^{T}\in\mathbb{R}^{n}\) being the system parameter vector, and \(\Gamma_{k}\in\mathbb{R}^{n}\) being the input vector, which is composed by the sample value of the input \(x\) of the previous \(n\) times:  
\[\Gamma_{k}=[\gamma_{k},\gamma_{k-1},\ldots,\gamma_{k-n+1}]^{T}. \tag{8}\]  
The solution of the adaptive filter is the system parameter vector that maximizes the cost function:  
\[\Phi=\operatorname*{arg\,max}_{\Phi\in\mathbb{R}^{p}}\Bigg{\{}E[GAG_{\alpha, \sigma_{+},\sigma_{-}}(\varepsilon_{k})]=\frac{1}{N}\sum_{i=1}^{N}GAG_{\alpha, \sigma_{+},\sigma_{-}}(\varepsilon_{k})\Bigg{\}}, \tag{9}\]  
where the cost function is the expectation of the generalized asymmetric Gaussian kernel function, \(E[GAG_{\alpha,\sigma_{+},\sigma_{-}}(\varepsilon_{k})]\), which is equal to the average value of the kernel function at each time in the case of a discrete sample input, \(\frac{1}{N}\sum_{i=1}^{N}GAG_{\alpha,\sigma_{+},\sigma_{-}}(\varepsilon_{k})\). We use the gradient ascent method to solve the problem (9), which is similar to the gradient descent method. To simplify computation, the gradient of the kernel function corresponding to the instantaneous estimation error at each time is used as the instantaneous gradient. Then, the instantaneous-gradient-based adaptive algorithm, called in this work the GMACC algorithm, can be derived as  
\[\Phi_{k+1}=\Phi_{k}-\mu\
abla(k)=\Phi_{k}+\mu\phi_{GMACC}(\varepsilon_{k}) \varepsilon_{k}\Gamma_{k}, \tag{10}\]  
where the weighting function corresponding to the generalized maximum asymmetric correntropy criterion can be presented as  
\[\phi_{GMACC}(\varepsilon_{k})=\left\{ \begin{array}{ll}\exp\Bigl{(}-\left| \frac{\varepsilon_{k}}{\varepsilon_{k}}\right|^{\alpha}\Bigr{)}|\varepsilon_ {k}|^{\alpha-2},&\text{if }\varepsilon_{k}\geq 0\\ \exp\Bigl{(}-\left|\frac{\varepsilon_{k}}{\sigma_{-}}\right|^{\alpha}\Bigr{)}| \varepsilon_{k}|^{\alpha-2},&\text{if }\varepsilon_{k}<0\end{array} \right.. \tag{11}\]  
One can observe that:  
(1) When \(\alpha=2\), the GMACC algorithm becomes the MACC algorithm, which can be presented as:  
\[\Phi_{k+1}=\Phi_{k}-\mu\
abla(k)=\Phi_{k}+\mu\phi_{MACC}(\varepsilon_{k}) \varepsilon_{k}\Gamma_{k}, \tag{12}\]where the weighting function corresponding to the generalized maximum asymmetric correntropy criterion is  
\[\phi_{MACC}(\varepsilon_{k})=\left\{ \begin{array}{ll}\exp\biggl{(}-\left|\frac{ \varepsilon_{k}}{\sigma_{+}}\right|^{2}\biggr{)},&\text{if }\varepsilon_{k}\geq 0\\ \exp\biggl{(}-\left|\frac{\varepsilon_{k}}{\sigma_{-}}\right|^{2}\biggr{)},& \text{if }\varepsilon_{k}<0\end{array} \right.. \tag{13}\]  
(2) When \(\sigma_{+}=\sigma_{-}\), the GMACC algorithm becomes the GMCC algorithm [11]. When the two conditions \(\alpha=2\) and \(\sigma_{+}=\sigma_{-}\) are both satisfied, the GMACC algorithm becomes the MCC algorithm.