# Review of Advances in Active Impulsive Noise Control with Focus on Adaptive Algorithms
## 1 Introduction
The cost function in the FxLMS algorithm entails the square mean of the error signal, with \(E[*]\) denoting the expectation operator:  
\[J(n)=E\left[e^{2}(n)\right] \tag{3}\]  
The FxLMS algorithm employs gradient descent to derive the iterative update equation, which is as follows:  
\[\mathbf{w}(n+1)=\mathbf{w}(n)-\mu\frac{\partial J(n)}{\partial\mathbf{w}(n)}= \mathbf{w}(n)+\mu e(n)\mathbf{x}_{f}(n) \tag{4}\]  
The FxLMS algorithm possesses advantages such as simplicity, ease of implementation, and effective active control, making it particularly efficient in handling noise that follows a Gaussian distribution. However, its noise reduction performance significantly diminishes when dealing with impulse noise. The characteristics of impulse noise can be adequately illustrated using the noise model depicted in Figure 2, which follows a standard \(SaS\) distribution. As shown in Figure 2, in the presence of impulse noise, the \(l_{2}\) norm of the error signal exhibits unbounded characteristics. This implies that the \(e(n)\), rooted in the MSE, lacks a minimum value, hindering the algorithm from achieving stable convergence. Figure 1: Block diagram of an ANC system using the filtered-x structure. The S\(\alpha\)S distribution is characterized by the following expression for its characteristic function:  
\[\mathrm{V}(t)=\exp\{j\delta t-\gamma|t|^{\alpha}[1+j\beta sgn(t)S(t,\alpha)]\} \tag{5}\]  
where:  
\[S(t,\alpha)=\left\{ \begin{array}{cc}\tan\bigl{(}\frac{\alpha\pi}{2}\bigr{)}& \alpha=1\\ 2\log|y|/\pi&\alpha\
eq 1\end{array} \right. \tag{6}\]  
When \(\alpha\in(0,2)\), a decrease in \(\alpha\) results in a heavier-tailed probability distribution, increasing the likelihood of outliers in the signal. A smaller \(\alpha\) corresponds to a higher degree of noise impulses. In real-life situations, common values for the \(\alpha\) of pulse noise range from \((1,2)\). The graph distinctly illustrates that when \(\alpha=0.5\), the curve exhibits pronounced peaks and thicker tails, representing an uncommon occurrence in everyday noise. Conversely, when \(\alpha=2\), the distribution degenerates into a Gaussian distribution. \(\beta\) serves as the symmetry coefficient, where \(\beta=0\) indicates a symmetric distribution. The parameter \(\delta\) signifies the location parameter. \(\gamma\) is the scale parameter. Many everyday situations give rise to impulsive noise. Impulse noise is an unexpected, sharp sound that occurs instantaneously over the course of time. Figure 3 illustrates noise signals with varying intensities under different \(\alpha\) values of the \(\alpha\)-stable distribution. As observed in Figure 3, a decrease in \(\alpha\) leads to increased amplitude and discrete changes in the noise signal, thereby complicating the convergence of algorithms. Various robust adaptive algorithms, extensively discussed in prior literature, have been developed to improve the stability of ANC systems when confronted with abrupt noise interference. This study systematically organizes and categorizes adaptive algorithms designed explicitly for impulsive noise, classifying them based on their criteria. The aim is to provide a structured summary, offering valuable insights for research and applications in active impulsive noise control. Figure 2: Probability density distribution of S\(\alpha\)S signals with different intensities.