## 3 Optimal Multichannel ANC for Multiple Noise Scenarios
### Optimal Control for Multiple Incidences  
Let \(\theta_{i}\) be the \(i\)th incidence angle of the impinging noise. First, we obtain the optimal secondary outputs \(\mathbf{Y}_{\mathrm{opt}}(\theta_{i})\) using (3). Clearly, our aim here is to obtain a single control filter \(\mathbf{W}\) that can always yield optimal secondary outputs, which is given by  
\[\mathbf{Y}_{\mathrm{opt}}(\theta_{i})=\mathbf{W}_{\mathrm{opt}}\mathbf{X}( \theta_{i}). \tag{15}\]  
Revisiting the subproblem 2 in Figure 2, we know that for each incident angle, (15) is underdetermined. Therefore, we may cast more constraints, which in this case, means different incident angles. Consider \(J\) reference microphones and \(K\) secondary sources, we have \(J^{*}K\) unknown elements in \(\mathbf{W}\). Therefore, we could combine \(J\) equations of (15) with different incident angles, and arrive at  
Figure 3: (**a**) Setup of a 2-2-4 ANC system and (**b**) noise reduction performance using the fully-coupled multichannel FxLMS, the colocated FxLMS, and the theoretical optimal performance.  
\[\dot{\mathbf{Y}}_{J}=\mathbf{W}_{\text{full-rank}}\dot{\mathbf{X}}_{I}, \tag{16}\]  
where \(\dot{\mathbf{Y}}_{I}=\left[\begin{array}{cccc}\mathbf{Y}_{\text{opt}}(\theta_ {1})&\ldots&\mathbf{Y}_{\text{opt}}(\theta_{j})&\ldots&\mathbf{Y}_{\text{opt}}( \theta_{j})\end{array}\right]\), and \(\dot{\mathbf{X}}_{J}=\left[\begin{array}{cccc}\mathbf{X}(\theta_{1})& \ldots&\mathbf{X}(\theta_{j})&\ldots&\mathbf{X}(\theta_{I})\end{array}\right]\). Equation (16) yields a fully-determined system, where one unique solution can be obtained by  
\[\mathbf{W}_{\text{full-rank}}=\dot{\mathbf{Y}}_{J}\dot{\mathbf{X}}_{J}^{ \text{\tiny{$\leftrightarrow$}}\mathbf{H}}\left(\dot{\mathbf{X}}_{J}^{\text{ \tiny{$\leftrightarrow$}}\mathbf{H}}\right)^{-1}. \tag{17}\]  
We will henceforth refer to this method as the full-rank optimal fixed filter.  
Furthermore, considering that in a practical scenario, the direction of noise could vary from \(-90^{\circ}\) to \(90^{\circ}\), a full-rank optimal filter could only cover \(J\) incidences, resulting in a degraded performance in those directions that are not selected. With this consideration, we can extend the full-rank factorization into an overdetermined factorization, where \(J^{\prime}\geq J\) directions are considered. Similar to (16), the overdetermined optimal filter can be derived as  
\[\mathbf{W}_{\text{overdetermined}}=\dot{\mathbf{Y}}_{I^{\prime}}\dot{\mathbf{ X}}_{I^{\prime}}^{\text{\tiny{$\leftrightarrow$}}\mathbf{H}}\left(\dot{ \mathbf{X}}_{I^{\prime}}\dot{\mathbf{X}}_{I^{\prime}}^{\text{\tiny{$\leftrightarrow$ }}\mathbf{H}}\right)^{-1}, \tag{18}\]  
where \(\dot{\mathbf{Y}}_{I^{\prime}}=\left[\begin{array}{cccc}\mathbf{Y}_{\text{ opt}}(\theta_{1})&\ldots&\mathbf{Y}_{\text{opt}}(\theta_{j})&\ldots&\mathbf{Y}_{ \text{opt}}(\theta_{I^{\prime}})\end{array}\right]\), and \(\dot{\mathbf{X}}_{I^{\prime}}=\left[\begin{array}{cccc}\mathbf{X}(\theta_{1} )&\ldots&\mathbf{X}(\theta_{j})&\ldots&\mathbf{X}(\theta_{I^{\prime}})\end{array}\right]\). It is not difficult to understand that the overdetermined optimal filter could yield a more balanced noise reduction performance than the full-rank optimal filter in broader incidence angles, at the cost of sacrificing performance in the angles selected in the full-rank method.