## 4 Simulations
### Performance Comparison  
#### 4.3.1 Comparison with MM-JAD (the Method in [29])  
Compared to the MM-JAD method, the proposed method improves the robustness of the solution by adding a regularization parameter. Figure 5. shows that the regularization parameter mainly affects the performance of the beamformers below 1 kHz. To better evaluate the differences between the proposed method and the MM-JAD method at low frequencies, Figure 6 presents comparisons between the synthesized beampattern by the  
Figure 5: Performance comparison of the sixth-order synthesized beamformer with different regularization parameters: (**a**) WNG and (**b**) DI. MM-JAD method, the proposed method, and the ideal beampattern at frequencies of 250 Hz, 500 Hz, 750 Hz, and 1 kHz. The regularization parameter is set to 0.01. As can be seen at 250 Hz and 500 Hz, compared with the proposed method, the beampattern synthesized by MM-JAD is closer to the ideal beampattern. The beamformer designed by the proposed method cannot generate the target sixth-order broadside beampattern but forms a second-order and a fourth-order broadside beampattern, respectively. At 750 Hz, the proposed method can form a sixth-order beampattern, and the match with the sidelobe of the ideal beampattern is not as good as the MM-JAD. At 1 kHz, the beampatterns produced by both MM-JAD and the proposed method are similar, with slight difference from the ideal beampattern. The performance comparison between the beamformers synthesized by MM-JAD and those synthesized by the proposed method is listed in Table 1. As seen, at 250 Hz, the proposed method obtained a higher WNG than that of the MM-JAD method by sacrificing the performance of DI. At 500 Hz, the proposed method improves the WNG by adding a regularization parameter. Meanwhile, the DI of the proposed method is comparable to that of the MM-JAD method. At 750 Hz, the WNG of the proposed method has been further improved and the DI is also higher than that of the MM-JAD method. At 1 kHz, the WNG and the DI values of the two methods are similar, indicating the MM-JAD method is robust enough at high frequencies, and the improvement due to adding the regularization parameter is not significant. Hence, it can be found that, compared to the MM-JAD method, the proposed method balances system robustness and directivity by adding a regularization parameter to trade-off between the WNG and DI at low frequencies. 3.2 Performance Comparison between MM-JAD and the Proposed Method with the Perturbations Added to the Spatial Responses of the Loudspeakers  
To demonstrate the proposed method is more robust than the MM-JAD method at low frequencies, the performance of the two methods are evaluated under uncertainties in the frequency response of the loudspeakers, for which the error has a multiplicative form with uniform distribution between \(-3\) and \(+3\) dB in magnitude and uniform distribution  
Figure 6: The simulated and ideal beampatterns of the sixth-order broadside differential beamformers synthesized by MM-JAD and the proposed method at different frequencies. (**a**) 250 Hz, (**b**) 500 Hz, (**c**) 750 Hz, (**d**) 1 kHz. between \(-10^{\circ}\) and \(+10^{\circ}\) in phase. The performance averages over 5000 Monte Carlo trails describe the performance of the system with robustness. Figure 7 shows comparisons of the desired beampattern and the averaged simulated beampatterns with the two methods. It can be seen from Figure 7a that, at 250 Hz, the beampattern synthesized by the MM-JAD method deviates significantly from the ideal beampattern and we cannot discern any directionality of the beampattern from the radiated beampattern. Although the beampattern radiated by the proposed method does not form the target sixth-order beampattern, the fact that the main lobe is 10 dB higher than the sidelobes indicates that the beamformer already exhibits directionality. At 500 Hz, the MM-JAD method still lacks directionality, but the main lobe of the beampattern radiated by the proposed method already resembles that of the ideal beampattern.