# Active Noise Control over Space: A Subspace Method for Performance Analysis
## 5 Simulation Results
### Simulation Setup  
In this simulation, we investigate the ANC performance in the reverberant environment. The reverberant environment is modelled as a cuboid room of 6 m \(\times\) 6 m \(\times\) 5 m. The reflection coefficients are set to [0.75, 0.8, 0.77, 0.85, 0.1, 0.1]. The reverberation is simulated by the image source method with the image order of 5. The origin of the room is on the left bottom corner. The region of interest is a spherical area with a radius of 0.5 m, and the center of the region is (3, 3, 1.5) with respect to the origin.  
We assume that the noise field only contains a single-frequency component. In the following investigations, the primary noise field is a spherical wave coming from a point source located at \((r,\phi,\psi)\) with respect to the center of the region, with a constant magnitude of 10. The locations of primary sources are varying in each case, as shown in Table 1.  
We assume the frequency of the noise field is 200 Hz. From (4), the region of interest in such a noise field can be represented by \(N=\lceil ekR_{1}/2\rceil=3\) modes. Thus, at least \((N+1)^{2}=16\) microphones must be placed on the boundary to capture the information of the residual noise field for each mode. In this simulation, we place 32 microphones on the spherical boundary, following the Gauss-Legendre sampling method. White Gaussian noise is added to each microphone recording to simulate the internal thermal noise of microphones.  
To control all the modes, 16 loudspeakers are required to be placed outside the control region. To emulate a practical scenario, however, in this simulation, only 12 loudspeakers are used. Among them, 8 loudspeakers are placed in the x-y plane with two different geometries, as shown in Table 1. Another four loudspeakers are placed on another plane, which is on the plane close to the ceiling. The loudspeaker positions for each case are shown in Table 2.  
We evaluate the ANC performance in terms of residual noise field, noise reduction on the sample points inside the region \(N_{r}^{\text{in}}\), and the energy of the driving signals \(E_{d}\).  
To evaluate the actual noise reduction performance within the control region, residual sound fields \(\epsilon_{in}\) at \(Z=1296\) points uniformly placed within the cross section between the region of interest and the x-y plane are examined. The noise reduction inside the region of interest over \(Z\) points \(N_{r}^{\text{in}}\) can be written by  
\[N_{r}^{\text{in}}\triangleq 10\log_{10}\frac{\sum_{z}E\{|e_{\text{in},z}|^{2} \}}{\sum_{z}E\{|e_{\text{in},z}(0)|^{2}\}}, \tag{34}\]  
where \(E\{|e_{\text{in},z}(0)|^{2}\}\) is the energy of the primary noise field at the \(z^{\text{th}}\) sample point, and \(E\{|e_{\text{in},z}|^{2}\}\) is the energy of the residual noise field at the \(z^{\text{th}}\) sample point.  
To evaluate the loudspeaker energy consumption, we compare the total energy of all the loudspeakers \(E_{d}\). The loudspeaker energy consumption can be represented by  
\[E_{d}=\mathbf{d}^{T}\mathbf{d}. \tag{35}\]  
For the subspace method, during the PCA process for the secondary path, we only reserve the principle components in \(\mathbf{u}\). Components in \(\mathbf{u}\) which correspond to \(\lambda_{i}\) occupying less than 5% of the largest eigenvalue \(\lambda_{1}\) are omitted in \(\mathbf{u}^{\circ}\).