## 3 Multichannel Feedforward ANC System with Optimal Reference Microphones Based on TDOA  
### Multichannel Feedforward ANC System  
If the multiple noise sources are located in a wide area, it is difficult to reduce the unwanted noises using the single-channel feedforward ANC system. A multichannel feedforward ANC system is used to reduce the unwanted noises in this situation. In general, a multichannel feedforward ANC system has multiple reference microphones, secondary sources, and error microphones. Figures 2 and 3 respectively show the structure and a block diagram of a multichannel feedforward ANC system. In Figures 2 and 3, \(J,K,\) and \(M\) represent the numbers of reference microphones, secondary sources, and error microphones, respectively. Here, this composition is called a case \((J,K,M)\) ANC system. Although there are feedback paths from the secondary sources to the reference microphones in a real situation, the feedback paths are neglected to simplify the discussion. In Figure 3, \(\mathbf{P},\mathbf{R},\mathbf{W},\) and \(\mathbf{S}\) depict the primary path matrix, reference path matrix, coefficient matrix of the noise control filters, and secondary path matrix, respectively. Here, the multichannel feedforward ANC system uses the multiple reference FXNLMS (MRFXNLMS) algorithm to update the filter coefficient vector \(\mathbf{w}_{k,j}(n)\), and the update is given by:  
\[\mathbf{w}_{k,j}(n+1)=\mathbf{w}_{k,j}(n)+\frac{\alpha\sum_{m=1}^{M}\mathbf{u}_ {m,k,j}(n)e_{m}(n)}{\beta+\sum_{k=1}^{K}\sum_{j=1}^{I}\left\|\mathbf{u}_{m,k,j} (n)\right\|^{2}}, \tag{6}\]  
where  
\[\mathbf{w}_{k,j}(n) =[w_{k,j,0}(n)\ w_{k,j,1}(n)\cdots w_{k,j,N-1}(n)]^{\text{T}}, \tag{7}\] \[\mathbf{u}_{m,k,j}(n) =\left[u_{m,k,j}(n)\ u_{m,k,j}(n-1)\cdots u_{m,k,j}(n-N+1)\right] ^{\text{T}},\] (8) \[u_{m,k,j}(n) =\mathbf{\hat{s}}_{m,k}^{\text{T}}\mathbf{x}_{j}(n),\] (9) \[\mathbf{x}_{j}(n) =\left[x_{j}(n)\ x_{j}(n-1)\ \cdots\ x_{j}(n-N-1)\right]^{\text{T}},\] (10) \[\mathbf{\hat{s}}_{m,k} =\left[s_{m,k,0}\ \mathbf{\hat{s}}_{m,k,1}\cdots\mathbf{\hat{s}}_{m,k,L-1} \right]^{\text{T}},\] (11) \[e_{m}(n) =d_{m}(n)-y_{m}^{\prime}(n),\] (12) \[y_{m}^{\prime}(n) =\sum_{k=1}^{K}\mathbf{s}_{m,k}^{\text{T}}\mathbf{y}_{k}(n),\] (13) \[\mathbf{s}_{m,k}(n) =\left[s_{m,k,0}(n)\ s_{m,k,1}(n)\cdots s_{m,k,L-1}(n)\right]^{ \text{T}},\] (14) \[\mathbf{y}_{k}(n) =\left[y_{k}(n)\ y_{k}(n-1)\ \cdots\ y_{k}(n-L-1)\right]^{\text{T}},\] (15) \[y_{k}(n) =\sum_{j=1}^{J}\mathbf{w}_{k,j}^{\text{T}}(n)\mathbf{x}_{j}(n). \tag{16}\]  
Hereafter, we consider the case \((J,1,1)\) system with \(J\) noise controllers. In general, the direction of the unwanted noise is unknown. One of the solutions to this problem is for the reference microphones to be located around the noise control point. An example of this situation is noise environment that occurred in the medical room. In the medical room, a lot of machines are located and they generate the noise. In this case, the timing for generating the noise from these machines are different from each other, i.e., the direction of the unwanted noise changes from time to time. Then, the reference microphone may not satisfy the causality constraint. Even if the multichannel ANC system is used, some reference microphones may not satisfy the causality constraint. Then, the noise reduction performance of the ANC system degrades.