## 1 Introduction
To evaluate the effectiveness of the proposed scheme, we simulate single-channel and multichannel ANC systems. The results demonstrate that the proposed algorithm exhibits a good convergence rate, a small MSE and a reduced computational burden when compared with existing schemes based on linear active noise control. This paper is organized as follows. In Section 2, the conventional filtered-x structure is described. The proposed combination with switching selection is described in Section 3. In Section 4, the analysis of the computational complexity of the derived algorithm is presented. In Section 5, some experiments to evaluate the proposal are described. Finally, the conclusions are presented in Section 7. A Brief Introduction to the Multichannel Filtered-x Affine Projection-Like (FXAPL-I) and Filtered-x Least Mean Square (FXLMS) Algorithms  
In this section, the general background on adaptive filters based on filtered-x affine projection-like (FXAPL-I) and filtered-x least mean square (FXLMS) algorithms is briefly presented because they are used in the development of the multichannel ANC systems in this work. The generic structure of a multichannel ANC system is shown in Figure 1. The blocks \(p\) and \(s\) are the primary and secondary paths, respectively; \(K\) is the number of error sensors; \(J\) is the number of actuators; \(x_{i}(n)\) is the \(i\)th reference signal obtained from \(I\) reference sensors; and \(y_{j}(n)\) is the adaptive filter output as follows:  
\[y_{j}(n)=\mathbf{w}_{ji}^{t}(n)\mathbf{x}_{i}(n) \tag{1}\]The reference signal is defined as follows:  
\[\mathbf{x}_{i}(n)=\left[x_{i}(n)\quad x_{i}(n-1)\quad\ldots\quad x_{i}(n-N+1) \right]^{t} \tag{2}\]  
and the adaptive filter coefficients are as follows:  
\[\mathbf{w}_{ji}(n)=\left[w_{j,i,1}(n)\quad w_{j,i,2}(n)\quad\ldots\quad w_{j,i,N }\right]^{t} \tag{3}\]  
where \(N\) is the length of the adaptive filter. To compensate for the effects caused by the secondary paths \(\mathbf{s}_{kj}(n)\), the reference signal \(x_{i}(n)\) is filtered using the following expression:  
\[x^{\prime}_{ijk}(n)=\mathbf{\hat{s}}^{t}_{kj}(n)\mathbf{v}_{i}(n) \tag{4}\]  
where \(\mathbf{\hat{s}}_{kj}\) is a fixed FIR filter of length \(M\) that models \(\mathbf{s}_{kj}(n)\) and \(v_{i}(n)=\left[x_{i}(n)\quad x_{i}(n-1)\quad\ldots\quad x_{i}(n-M+1)\right]^{t}\). In practical ANC systems, the desired signal \(\mathbf{d}(n)\), which is used for the error computation, is unavailable. However, the error signal \(\mathbf{e}(n)\) can be obtained. As a consequence, the error signal vector \(\mathbf{e}(n)\) is estimated by means of an approximation. The estimation can thus be performed by employing past samples of the error signals \(\mathbf{e}(n)\) as follows:  
\[\mathbf{e}_{k}(n)\approx\left[e_{k}(n)\quad e_{k}(n-1)\quad\ldots\quad e_{k}( n-L+1)\right]^{t} \tag{5}\]  
As shown in Figure 1, the generic structure of the ANC system uses a block that is known as an adaptive algorithm. This block can use either the FXAPL-I or the FXLMS algorithm. * FXLMS The FXLMS algorithm updates the coefficients as follows: \[\mathbf{w}_{ji}(n+1)=\mathbf{w}_{ji}(n)+\sum_{k=1}^{K}\mu_{ijk}x^{\prime}_{ijk }(n)e_{k}(n),\qquad j=1,2,\ldots,J,\quad i=1,2,\ldots,I\] (6) where \(\mu_{ijk}\) is a fixed value between [0,1] and is usually chosen by trial and error. The FXLMS algorithm is widely used because of its simplicity, robustness and steady-state MSE. However, its convergence speed is low, which can restrict its use in some practical applications. The algorithms based on AP overcome the aforementioned problem because its convergence speed is very fast by paying a penalty in terms of computational complexity and a large MSE at the steady state. Figure 1: Block diagram of the conventional filtered-x multichannel active noise control.