## 4 Results and Discussion
### Simulation Conditions  
The experimental data provided with [3] are used to obtain FIR models for the acoustic paths, and their characteristics are shown in Figure 4. It is worth mentioning that these acoustic paths were measured for a practical ANC setup and have been used in many previous works, including the seminal paper on impulsive ANC systems [24]. All adaptive and fixed filters in various methods are also considered to be of the FIR type. \begin{table}
\begin{tabular}{c c} \hline \hline
**while** & \(\{x(n)\}\) available & \\  & 1. \(\boldsymbol{x}(n)=[x(n),x(n-1),\cdots,x(n-L+1)]^{T}\); & \% Update signal vector \\  & 2. \(y(n)=\boldsymbol{x}^{T}(n)\boldsymbol{w}(n)\); & \% ANC filter output \\  & 3. \(x_{f}(n)=s(n+x_{i})\); & \% Fitered-x signal \\  & 4. \(d(n)=e(n)+s(n)*y(n)\); & \% Estimate of disturbance signal \\  & 5. \(\boldsymbol{x}_{f}(n)=[x_{f}(n),x_{f}(n-1),\cdots,x_{f}(n-L+1)]^{T}\); & \% Update filtered-x signal vector \\  & 6. \(g(n)=\bar{d}(n)-\boldsymbol{x}_{f}^{T}(n)\boldsymbol{w}(n-1)\); & \% Error signal for adaptation \\  & 7. \(m(n)=\left\{\frac{1}{(1+\beta^{2}(n))^{2}}\right\}\); & \% Parameter in objective function \\  & 8. \(\boldsymbol{u}_{m}(n)=\boldsymbol{\Phi}_{m}(n-1)\boldsymbol{x}_{f}(n)\); & \% vector for Kalman gain computation \\  & 9. \(\boldsymbol{k}_{m}(n)=\frac{\boldsymbol{u}_{m}(n)}{\frac{\lambda}{m(n)}+ \boldsymbol{x}_{f}^{T}(n)\boldsymbol{u}_{m}(n)}\); & \% Kalman gain vector \\  & 10. \(\boldsymbol{\Phi}_{m}(n)=\boldsymbol{\zeta}\boldsymbol{\Phi}_{m}(n-1)+(1- \zeta)\left[\lambda^{-1}\boldsymbol{\Phi}_{m}(n-1)-\lambda^{-1}\boldsymbol{ k}_{m}(n)\boldsymbol{u}_{m}^{T}(n)\right]\); & \% Inverse correlation matrix \\  & 11. \(\boldsymbol{w}(n)=\boldsymbol{w}(n-1)+pg(n)\boldsymbol{k}_{m}(n)\); & \% FS-MFxRRLS update equation \\  & 12. \(\mu(n)=\boldsymbol{w}(n-1)+(1-\gamma(n))\boldsymbol{k}_{m}\); & \% Time-varying step-size \\  & 13. \(\boldsymbol{w}(n)=\boldsymbol{w}(n-1)+\mu(n)(n)(n)\boldsymbol{k}_{m}(n)\); & \% CCS-MFxRRLS update equation \\  & 14. \(b(n)=b(n)+1+\mu_{b}\boldsymbol{s}\boldsymbol{s}\boldsymbol{p}(g(n-1))g(n-1)[ \gamma(n-1)(1-\gamma(n-1))]\boldsymbol{x}_{f}^{T}(n)\boldsymbol{k}_{m}(n-1)\); & \% Adaptive parameter for \(\gamma(n)\) \\  & 15. \(b(n)=\left\{\begin{array}{ll}-2.49,&b(n)<-2.49\\ +2.49,&b(n)>2.49\end{array}\right.\) & \% Threshold adaptive parameter \(b(n)\) \\  & 16. \(\gamma(n)=\frac{1}{1+\exp\{-b(n)\}}\); & \% Sigmoidal activation function \\ \hline \end{tabular}
\end{table} 
Table 1: The implementation summary of the proposed algorithms presented in this paper. Figure 3: Variation of sigmoidal activation function \(\gamma(n)\) vs. adaptive parameter \(b(n)\) (and selected threshold values) in the proposed algorithm. The impulsive noise source is assumed to be modeled as a standard symmetric \(\alpha\)-stable (SaS) distribution. The standard SaS distribution has no unique probability density distribution (PDF) and can be better described by using the characteristic function of the form [31]  
\[\varphi(t)=e^{-|t|^{\alpha}}, \tag{47}\]  
where \(\alpha\) is called the characteristics exponent which signifies the 'impulsiveness' of the standard SaS distribution and dictates the corresponding PDF. In fact, \(\alpha\) may take any value in the range \(0<\alpha\leq 2\), where \(\alpha=2\) corresponds to Gaussian PDF with zero impulsiveness, and increased impulsiveness (PDF with outliers) is expected if the value of \(\alpha\) is small. Figure 5 presents one realization of the input signal \(x(n)\) obtained from a standard SaS process with \(\alpha=1.1\), \(\alpha=1.4\), and \(\alpha=1.6\), which are termed as severely impulsive, strongly impulsive, and mildly impulsive for the results presented and explained in the following sections. The variances of typical signals shown in Figure 5 are found to be \(2.6313\times 10^{3}\), \(0.2991\times 10^{3}\), and \(0.0168\times 10^{3}\), for \(\alpha=1.1\), \(\alpha=1.4\), and \(\alpha=1.6\), respectively.