## 3 Proposed Scheme with Switching Filter Selection  
In this section, the proposed scheme based on the alternating selection of parallel FXAPL-I and FXLMS filters to achieve high convergence speed with low MSE is described. The proposed strategy is based on alternating affine projection algorithm (A-AP) [18]. In this study, we propose a new strategy to be used in the development of ANC systems. Specifically, the selection criteria are based on the comparison between the error power of the combined filters and the steady-state MSE. We use the computational capabilities of the FXAPL-I algorithm to achieve high convergence speed, while the FXLMS is used to guarantee a low steady-state MSE. The proposed block diagram is shown in Figure 2, in which the outputs of the filters \(y_{1 ji}(n)\) and \(y_{2 ji}(n)\) are combined using the following equation:  
\[y_{ji}(n)=\lambda(n)\textbf{y}_{1 ji}(n)+[1-\lambda(n)]y_{2 ji}(n) \tag{14}\]  
where \(\lambda(n)\) is a mixing parameter, which is in the range [0,1]. This parameter determines the combination of the two filters (FXAPL-I and FXLMS) during the entire filtering process at each iteration. According to Equation (14), the parameter \(\lambda(n)\) determines the overall output \(y_{ji}(n)\). If \(\lambda(n)\) is near one, the overall output \(y_{ji}(n)\) is equal to the output \(y_{1 ji}(n)\) (FXAPL-I's output). However, if \(\lambda(n)\) is near zero, the overall output \(y_{ji}(n)\) is equal to the output \(y_{2 ji}(n)\) (FXLMS's output). A sigmoid function is used to compute \(\lambda(n)\) as follows:  
\[\lambda(n)=\frac{1}{1+\mathrm{e}^{-\textbf{a}(n)}} \tag{15}\]  
where \(\textbf{a}(n)\) is an auxiliary parameter used to minimize the instantaneous square error of the filters and is computed as follows:  
\[\textbf{a}(n+1)=\textbf{a}(n)+\mu_{a}\textbf{e}_{k}(n)[\textbf{y}_{1 ji}(n)-y_{ 2 ji}(n)]\cdot\lambda(n)[1-\lambda(n)]+\rho[\textbf{a}(n)+\textbf{a}(n-1)] \tag{16}\]  
where \(\rho\) is a positive constant, \(\mu_{a}\) is the step size for \(\textbf{a}(n)\) and \(\textbf{e}_{k}(n)\) represents the error signal of the overall filter and is presented as follows:  
\[\textbf{e}_{k}(n)=\lambda(n)\textbf{e}_{1k}(n)+[1-\lambda(n)]e_{2k}(n) \tag{17}\]  
To speed up the convergence of the FXLMS filter, the weights of the FXAPL-I filter are partially transferred to the FXLMS filter when abrupt change occurs. The adaptation rule to transfer these weights is defined as follows:  
\[\textbf{w}_{2 ji}(n)=\alpha\textbf{w}_{2 ji}(n)+(1-\alpha)\textbf{w}_{2 ji}(n-1) \text{ for }\lambda(n)>t\]  
where the parameters \(t\) and \(\alpha\) are chosen as 0.98 and 0.9, respectively [17]. Analysis of the prior research shows that the convex combinations significantly reduce the steady-state MSE but with a penalty in terms of computational complexity because two filters operate simultaneously. We propose implementing switching selection criteria to reduce the computational complexity of the convex combination because the filters (FXAPL-I and FXLMS) need not operate simultaneously during the entire filtering process. We propose selection criteria that indicate when the coefficients must be updated by analyzing the state of the overall error signal \(\textbf{e}_{k}(n)\), i.e., the selection criteria are determined by comparing the instantaneous error power with the steady-state MSE of the FXAPL-I algorithm, which is the faster filter. The theoretical MSE is determined as follows [19]:  
\[\delta=\frac{2L}{2L-1}\sigma_{\mathbf{v}}^{2} \tag{18}\]  
where \(\sigma_{\mathbf{v}}^{2}\) is the variance of a noise signal. The updates of the coefficients for each filter are determined by the following rule:  
\[\mathbf{w}_{1 ji}(n+1)= \begin{cases}\mathbf{w}_{1 ji}(n)+\sum_{k=1}^{K}\mu_{1 ij }\mathbf{X}_{ijk}(n)\mathbf{e}_{1k}(n),&\text{if }e_{k}^{2}(n)\geq\delta\\ \mathbf{w}_{1 ji}(n),&\text{otherwise}\end{cases}  \tag{19}\]  
\[\mathbf{w}_{2 ji}(n+1)=