## 4 Numerical Experiment Results and Discussions
### Performance of the Full-Rank Optimal Filter  
Based on the optimal secondary source output computed from (3), we can obtain the theoretical optimal noise reduction performance for the setup specified in Figure 4. The noise reduction at five angles of incidence with frequencies from 0 to 2000 Hz is illustrated in Figure 5. Noise reduction performance reduces as frequency increases, as dictated by the control source separation distances and its relative position in the aperture [27; 29]. The result presented in Figure 5 is consistent with the upper frequency limit of control defined in [27] as \(f<c/d(1+\left|\sin\theta\right|)\). The attenuation performance is identical for all incidence angles at low frequencies but deteriorates rapidly as they approach their upper frequency limits of control. Next, we construct the full-rank optimal filter in the same setup. Since the number of secondary sources is \(K=8\), we can include eight incidence angles in (16) to obtain the full-rank optimal filter \(\mathbf{W}\). Six sets of the eight incidence angles are listed in Table 1 and illustrated in Figure 6. We calculate  
In Figure 7, we compare the full-rank optimal filter #1 with the exact optimal filter obtained for each incidence individually. The full-rank filter could only achieve identical performance for those angles selected in the full-rank factorization, except for lower frequencies (e.g., \(f\) = 100, 200 Hz), where the noise reduction is significantly large, indicating that the residue noise pressure is close to zero and might be affected by the numerical precision of the simulation. For incidence angles other than the eight incidences used in formulating the filter, the noise reduction performance is degraded. The degradation increases in severity when the incidence angles are farther away from the selected angles, especially for higher frequencies. To examine the effect of the different sets of the eight incidence angles, we illustrate the performance of the four cases (#1, #2, #3, #4 from Table 1) in Figure 8. Since these cases employ different angles, their performance varies significantly. By visual inspection, it is clear that when more incidences from the negative region are used in the formulation (from case 1 to case 4), better attenuation is observed in the negative incidences than the positive directions. One extreme case is case 4, where all eight selected directions are in the zero and negative region, which yields good attenuation for all negative incidences in general but significantly degrades for positive incidences. In Figure 9, we further compare two such selections that only cover the negative directions, i.e., case 4 and case 5 in Table 1. The key difference between these two cases is that case 5 ignores the extreme directions from \(-90^{\circ}\) to \(-70^{\circ}\) and hence occupies more directions in \(-70^{\circ}\)to \(0^{\circ}\) range. This selection is motivated by the observation that at extreme incidence angles, the noise that enters the window are at lower levels than higher incidences. Overall, results in Figure 9 indicate that with the selection in case 5, we can achieve optimal performance for almost all negative directions. Figure 10 illustrates the noise reduction performance for the full-rank optimal filters with formulation cases 5 and 6, where only negative and positive directions are covered, respectively. The interval of \(10^{\circ}\) is used in both cases. A distinct symmetrical performance is observed where each optimal filter performs optimally in its respective half of the entire incidence range. This result implies  
Figure 6: Illustration of the eight directions selected in the six cases of the full-rank optimal filter. that if the ANC system permits, two full-rank optimal filters can be applied to cover all angles of noise incidence. Note that a priori information of the approximate noise incidence (positive or negative) is required for switching the filters.