## 2 Methodology
### Robust Adaptive Filtering Based on GMACC
Furthermore, according to the properties of the GMCC, when \(\sigma_{+}=\sigma_{-}\rightarrow\infty\), the GMACC algorithm becomes the least mean p-norm algorithm (LMP), while when \(\alpha=2\) and \(\sigma_{+}=\sigma_{-}\rightarrow\infty\), the GMACC algorithm becomes the LMS algorithm. It can be found that GMACC is a broad concept, which includes many algorithms as its special cases. The relationship between the GMACC and other algorithms as its special cases are presented in Figure 1.  
(3) The kernel width controls the balance between convergence rate and robustness. When the kernel size is small, the estimation error can affect the value of the weighting function significantly, thus reducing the effect of the input data that cause larger estimation error. In contrast, when the kernel size is large, the weighting function stays close to \(|\varepsilon_{k}|^{\alpha-2}\), and the convergence occurs at a high rate. Moreover, the kernel widths corresponding to the positive and negative parts of the error variable, \(\sigma_{+}\) and \(\sigma_{-}\), determine the robustness and convergence rate of the algorithm faced with a positive and negative error, respectively. Compared with the symmetric Gaussian kernel, the asymmetric Gaussian kernel can flexibly choose the widths of the kernel, \(\sigma_{+}\) and \(\sigma_{-}\), so as to deal with asymmetric noise better. For example, the system noise in a network delay prediction follows a right-skewed distribution due to the sharp increase of queuing delay caused by a user's burst request, in which case a small \(\sigma_{+}\) is needed because data that bring about large positive errors are usually corrupted by impulsive noise. (4) The GMACC algorithm can be considered as one kind of least mean square algorithm with variable step size, \(\mu_{GMACC,k}\), which can be presented as  
\[\mu_{GMACC,k}=\left\{ \begin{array}{ll}\mu\exp\biggl{(}-\left|\frac{ \varepsilon_{k}}{\sigma_{+}}\right|^{\alpha}\biggr{)}|\varepsilon_{k}|^{ \alpha-2},&\text{if }\varepsilon_{k}\geq 0\\ \mu\exp\biggl{(}-\left|\frac{\varepsilon_{k}}{\sigma_{-}}\right|^{\alpha} \biggr{)}|\varepsilon_{k}|^{\alpha-2},&\text{if }\varepsilon_{k}<0\end{array} \right.. \tag{14}\]  
(5) The difference between the GMACC and other algorithms can be also attributed to the difference of the weighting functions of the algorithms. The weighting functions of robust adaptive filters usually assign less weight to data outliers or large estimation error caused by impulsive noise, which keeps the algorithm robust in a non-Gaussian noise environment. The weighting functions of different algorithms are presented in Figure 2, which shows that ITL-inspired criteria assign very little weight on sample data that cause a large estimation error compared with the MMSE criterion and least p-norm criterion, and  
Figure 1: Relationship between the GMACC and other algorithms. the asymmetric criteria assign different weights on the positive and negative estimation errors even though they have the same absolute value. (6) The computational complexity of the proposed GMACC algorithm is almost the same as that of the LMS algorithm, and the only extra computational effort needed is to calculate the weighting function (11), which is obviously not expensive. For a detailed description of the computational complexity of the algorithm, we analyzed the number of operations per iteration, which is presented in Table 1. One can conclude that the computational complexity of the GMACC is \(O(n)\).