## 2 Separation of the Multichannel Active Noise Control Problem into Two Subproblems
### The Underdetermined Subproblem 2 and Possible Solutions  
After we obtain the optimal secondary outputs \(\mathbf{Y}_{\mathrm{opt}}\), we can solve the subproblem 2 to get optimal solutions for control filters \(\mathbf{W}\). In this case, we have the following equation,  
\[\mathbf{Y}_{\mathrm{opt}}=\mathbf{W}\mathbf{X}. \tag{5}\]  
Figure 1: Block diagram of the \(J\times K\times M\) multichannel active noise control system. Figure 2: A separation of the multichannel ANC problem into 2 subproblems. Given \(K\) equations with \(K^{\prime}\!\!f\) unknowns in \(\mathbf{W}\), this is obviously an underdetermined problem (for \(J>1\)), which means that there are no unique solutions. The conventional solution employing pseudoinverse of \(\mathbf{X}\) yields [22]  
\[\mathbf{W}=\mathbf{Y}_{\mathrm{opt}}\mathbf{X}^{+}, \tag{6}\]  
where \(\mathbf{X}^{+}=\left(\mathbf{X}^{\mathrm{H}}\mathbf{X}\right)^{-1}\!\! \mathbf{X}^{\mathrm{H}}.\) This solution, though valid, is not the only solution and it assumes an implicit constraint. With reference to the pseudoinverse in underdetermined systems, the additional constraint in solution (6) is such that the \(l2\)-norm power of \(\mathbf{W}\) is minimized, which can be proven using the Lagrange multiplier. To find alternative solutions to the underdetermined situation in the subproblem 2, other constraints can be introduced. One of the methods is to reduce the number of unknown elements in \(\mathbf{W}\). This can be achieved by, for example, letting \(J=1\), which is the case for one single reference microphone. Another alternative method to reduce the number of unknowns in \(\mathbf{W}\) is to consider a so-called colocated method, where we assume the reference microphones are located extremely close to the positions of each secondary source (and also assume their quantities are equal, i.e., \(J=K\)). This method was adopted in [PERSON]'s active acoustic shielding (AAS) system to tackle noise from open windows [11]. In this case, we have  
\[diag(\mathbf{W})=\mathbf{Y}_{\mathrm{opt}}\cdot/\mathbf{X}, \tag{7}\]  
where \(\cdot/\) represents element-wise division and the non-diagonal elements in \(\mathbf{W}\) are zero-valued. It can be found in the literature [1; 22] that the pseudoinverse solution of the optimal control filter in (6) can be realized using the fully-coupled multichannel FxLMS adaptive filtering algorithm. Similar to [24], we can express the control filter update in the frequency domain as  
\[\mathbf{W}_{n+1}=\mathbf{W}_{n}-\mu\mathbf{S}^{\mathrm{H}}\mathbf{E}_{n} \mathbf{X}^{\mathrm{H}}, \tag{8}\]  
where \(n\) is the iteration index, \(\mu\) is the step size, and \(\mathbf{E}_{n}=\mathbf{D}+\mathbf{S}\mathbf{W}_{n}\mathbf{X}\) is the updated complex error spectra. Further derivation of (8) leads us to the steady-state control filter  
\[\mathbf{W}_{\infty}=-\Big{(}\mathbf{S}^{\mathrm{H}}\mathbf{S}\Big{)}^{-1} \mathbf{S}^{\mathrm{H}}\mathbf{D}\Big{(}\mathbf{X}^{\mathrm{H}}\mathbf{X} \Big{)}^{-1}\mathbf{X}^{\mathrm{H}} \tag{9}\]  
with the convergence condition as \(\mu<\frac{1}{X^{\mathrm{H}}\mathbf{X}}\frac{1}{\lambda_{\mathrm{max}}}\), where \(\lambda_{\mathrm{max}}\) is the maximum eigenvalue of \(\mathbf{S}^{\mathrm{H}}\mathbf{S}\). Comparing \(\mathbf{W}_{\infty}\) with the optimal pseudoinverse solution given in (6), we verify that the fully-coupled multichannel FxLMS converges to the pseudoinverse solution. Similarly, we can derive the adaptive filter using the colocation method, where we only consider the diagonal elements in the square matrix (i.e., \(J=K\)) \(\mathbf{W}\), and assume the rest to be 0. We represent such a diagonalized matrix as \(\widetilde{\mathbf{W}}_{n}\) and introduce a reference signal matrix as  
\[\widetilde{\mathbf{X}}(k_{1},k_{2})=\left\{ \begin{array}{ll}X(k_{1}),&\forall 1 \leq k_{1}=k_{2}\leq K\\ 0,&\text{otherwise}\end{array}