## 3 Multichannel Feedforward ANC System with Optimal Reference Microphones Based on TDOA
### Proposed Multichannel Feedforward ANC System  
To solve the above problem, a multichannel feedforward ANC system with an optimal reference microphone selector based on the TDOA is proposed. The proposed ANC system has \(J\) reference microphones and only a few reference microphones that satisfy the causality constraint are selected. A block diagram of the proposed system is shown in Figure 4. In Figure 4, \(\mathbf{v}(n)=[v_{1,1}(n)\cdots v_{m,j}(n)\cdots v_{M,J}(n)]\) represents the switch vector for each reference microphone. In the proposed system, the TDOA is obtained by calculating the cross-correlation between the reference signal obtained at the reference microphone \(x_{j}(n)\) and the unwanted noise obtained at the error microphone \(d_{m}(n)\) as follows:  
Figure 3: Block diagram of multichannel ANC system with \(J\) reference microphones, \(K\) secondary sources, and \(M\) error microphones. Figure 2: Structure of multichannel feedforward ANC system with \(J\) reference microphones, \(K\) secondary sources, and \(M\) error microphones. Then, the causality constraint (5) is rewritten as:  
\[\text{TDOA}\left(x_{j}(n),d_{m}(n)\right)>D_{\text{C}}+D_{\text{S}}. \tag{18}\]  
Here, the time delay of the secondary path \(D_{\text{S}}\) cannot be obtained directly. Fortunately, the secondary path model \(\hat{S}\) is identified before operating the ANC system and \(D_{\text{S}}\) can be estimated from the group delay of \(\hat{S}\) as:  
\[D_{\text{S}}\approx\min\left\{-\frac{d\theta_{\hat{S}}(\omega)}{d\omega} \right\}, \tag{19}\]  
where \(\omega\) is the angular frequency. \(D_{\text{C}}\) is the delay due to the calculation of the filter coefficients of the noise control filter. In other words, \(D_{\text{C}}\) depends on the performance of the digital signal processor (DSP). The switch vector \(\mathbf{v}(n)\) is obtained as follows:  
Figure 4: Block diagram of proposed multichannel feedforward ANC system (case \((J,K,M)\)) with reference microphone selector based on TDOA (time difference of arrival). \[v_{m,j}(n)= \begin{cases}1,\left(\text{TDOA}\left(x_{j}(n),d_{m}(n)\right)>D_{\text{ C}}+D_{\text{S}}\right),\\ 0,\left(\text{otherwise}\right).\end{cases}  \tag{20}\]  
When \(v_{m,j}(n)=1\), the \(j\)th reference microphone is selected for the \(m\)th error microphone, and when \(v_{m,j}(n)=0\), the \(j\)th reference microphone is not selected for the \(m\)th error microphone. In the proposed method, the multiple reference microphones are selected using Equation (20). In other words, all the reference microphones that satisfy the causality constraint are selected. Here, the noise control filter is updated as follows:  
\[\mathbf{w}_{k,j}(n+1)=\mathbf{w}_{k,j}(n)+\frac{\alpha\sum_{m=1}^{M}\mathbf{u }_{m,k,j}(n)e_{m}(n)v_{m,j}(n)}{\beta+\sum_{k=1}^{K}\sum_{j=1}^{J}\left\| \mathbf{u}_{m,k,j}(n)\right\|^{2}}. \tag{21}\]  
If the noise environment is changed and the noise reduction performance deteriorates, the reference microphones are reselected. To reselect the reference microphones, the proposed ANC system must detect the change in the noise environment. This is achieved by using the reduction defined by:  
\[\text{Reduction}=10\log_{10}\frac{\sum_{m=1}^{M}P_{d,m}(n)}{\sum_{m=1}^{M}P_ {e,m}(n)}, \tag{22}\]  
where  
\[P_{d,m}(n) =\sum_{l=0}^{L_{\text{P}}-1}d_{  
the optimal noise control filter is changed after the noise environment changes. In general, the optimal noise control filter of the single-channel feedforward ANC system is expressed as  
\[W_{\text{opt}}(z)=\frac{P(z)}{R(z)S(z)}, \tag{28}\]  
where \(W_{\text{opt}}(z)\), \(P(z)\), \(R(z)\), and \(S(z)\) are the optimal noise control filter, primary path, reference path, and secondary path in the \(z\)-transform representation, respectively. If the noise environment changes, then the primary path and reference path also change. Therefore, the optimal filters before and after changing the noise environment are different. When the update of the noise control filter is stopped, the unwanted noise can be directly obtained by the error microphone.