# Active Noise Control over Space: A Subspace Method for Performance Analysis
## 5 Simulation Results
### Cancellation Performance Using Different Methods  
We first compare cancellation performance using the subspace method and the WDLS method for two different noise source positions. In this simulation, white Gaussian noise with signal to noise ratio (SNR) (Here, the SNR level is with respect to the primary noise field level on virtual microphone in the center of the region.) of 60 dB is added to each microphone recording. For simplicity of plotting,  
\begin{table}
\begin{tabular}{c c c} \hline
**Noise Source Position** & **Loudspeaker Array** & **Non-Symmetry** & **Symmetry** \\ \hline \((2,315^{\circ},45^{\circ})\) & case 1 & \\ \((2,315^{\circ},90^{\circ})\) & case 2 & \\
24 position candidates & case 3 & case 4 \\ \hline \end{tabular}
\end{table}
Table 1: Loudspeaker array setup and noise source location in 4 cases, which are given in Figures 2 and 5, respectively.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multicolumn{3}{c}{**Loudspeakers in the x-y Plane**} & \multicolumn{3}{c}{**Loudspeakers Outside the x-y Plane**} \\ \hline
**No.** & **Non-Symmetry** & **Symmetry** & **No.** & **Non-Symmetry or Symmetry** \\ \hline
1 & \((4,3,2.5)\) & \((4.5,3,2.5)\) & 9 & \((0.5,0.5,4.5)\) \\
2 & \((1.8,3,2.5)\) & \((1.5,3,2.5)\) & 10 & \((5.5,5.5,4.5)\) \\
3 & \((3,2,2.5)\) & \((3,15,2.5)\) & 11 & \((5.5,0.5,4.5)\) \\
4 & \((3,4,2.5)\) & \((3,4,5.2,5)\) & 12 & \((0.5,5.5,4.5)\) \\
5 & \((4.3,3.2,2.5)\) & \((4.2,1.8,2.5)\) & & \\
6 & \((1.7,2.8,2.5)\) & \((1.8,1.8,2.5)\) & & \\
7 & \((3.2,1.7,2.5)\) & \((4.2,4.2,2.5)\) & & \\
8 & \((2.8,4.2,2.5)\) & \((1.8,4.2,2.5)\) & & \\ \hline \hline \end{tabular}
\end{table}
Table 2: Loudspeaker positions for non-symmetric placement and symmetric placement.  
the cancellation performance over the region is confined to horizontal planes at elevation of 90\({}^{\circ}\) (x-y plane) of the 3-D region.  
As shown in Figure 2, we assume noise field is generated by a point source located at (2, 315\({}^{\circ}\), 45\({}^{\circ}\)) (case 1) or (2, 315\({}^{\circ}\), 90\({}^{\circ}\)) (case 2) in the spherical coordinates. The geometry of loudspeakers in the x-y plane is not symmetrical. The energy of the primary noise field is shown in Figure 3.  
Figure 2: ANC system setup, where the pink point is the noise source position, blue points are loudspeaker positions, and red points are microphone positions: (**a**) case 1; (**b**) case 2.  
Figure 4 demonstrates the energy of the residual noise field in the x-y plane. As we expected, since the number of loudspeakers (12) cannot cover all the modes (16) in the region, in all four figures, the primary noise field in the region of interest cannot be fully cancelled. In case 2, compared with the primary noise field (Figure 3b), both the WDLS method and the subspace method can achieve significant noise reduction in the region of interest, which are dark areas in the middle of Figure 4b,d. In case 1, since the noise source is in a different hemisphere from the loudspeaker array, compared with Figure 3a, cancellation performance inside the region is fairly limited for both WDLS and the subspace method, as shown in Figure 4a,c.  
Meanwhile, compared with Figure 4a,b, in Figure 4c,d, the subspace method results in lower energy of the residual noise field outside the region of interest. The WDLS method on the other hand produces increased sound energy outside the region, especially when the noise reduction level is fairly limited inside the region. Using the subspace method, we analyze physically achievable performance and avoid sound amplification outside the control area.  
Figure 4: _Cont._  
Figure 3: Energy of the primary noise field, where pink point is the projection of the primary source on the x-y plane, blue points are the loudspeaker points located on the x-y plane, and the red dashed circle is the boundary of the region of interest: (**a**) case 1; (**b**) case 2.