## 4 Simulation Results and Discussion
### Simulation Results of the Noise Reduction for Each Method from 400 Hz to 1100 Hz  
In the third simulation, the noise reduction of the entire error points \(NR_{\text{e}}\), the noise reduction of the error points in target area 1 \(NR_{e\_aren1}\), the noise reduction of the error points in target area 2 \(NR_{e\_aren2}\), and the noise reduction of the far-field observing points \(NR_{ob\_far}\) for each method can be calculated using Equations (11)-(14). Figure 4a shows \(NR_{\text{e}}\) for each method, Figure 4b shows \(NR_{ob\_far}\) for each method, Figure 4c shows \(NR_{e\_aren1}\) for each method, and Figure 4d shows \(NR_{e\_aren2}\) for each method. The frequencies range from 400 Hz to 1100 Hz at a frequency interval of 100 Hz.  
Figure 4: Comparisons of the noise reduction for each method from 400 Hz to 1100 Hz: (**a**) the comparison of the noise reduction in the target area; (**b**) the comparison of the noise reduction in the far-field observing area; (**c**) the comparison of the noise reduction in the target area 1; and (**d**) the comparison of the noise reduction in the target area 2.  
As shown in Figure 4, the noise reduction of each algorithm in the entire target area 1, target area 2, and far-field observation area decreased with increasing frequency. Figure 4a illustrates that from 700 to 1100 Hz, the noise reduction of the three optimized algorithms in the entire target area is 2-13 dB greater than that of the Uni-RLS algorithm because the under-sampling issue of the uniformly placed secondary loudspeakers becomes gradually critical as the frequency increases. In contrast, the secondary loudspeaker spacing of the three optimized algorithms changes with increasing frequency. This provides better cancellation of the scattered sound field, resulting in improved noise reduction performance. Among the three optimized algorithms, the proposed CLASSSO-RLS algorithm outperforms the other algorithms. From 400 Hz-1100 Hz, the proposed strategy gains a 5 dB-16 dB reduction performance advantage in the target area compared to the Uni-RLS algorithm.  
As for the noise reduction performance in the far-field observation area, Figure 4b illustrates that the noise reduction of the proposed CLASSSO-RLS algorithm is better than that of the other methods from 400 Hz-1100 Hz because of the addition of the constraint points. The secondary loudspeaker selection process of the CLASSSO-RLS algorithm considers the noise reduction of the constraint points and the noise reduction of the entire target area. Thus, the noise reduction performance is better in the direction where the constraint points and the error points are located, and the noise reduction at the far-field observing points is also larger.  
As shown in Figure 4c, the CLASSSO-RLS algorithm achieves a noise reduction in target area1 that is 5-22 dB greater than that in the other two optimal methods from 400 Hz to 900 Hz. This is because the LASSO-RLS-\(\gamma_{C}\) and the LASSO-RLS-\(\gamma_{NL}\) methods have fewer secondary loudspeakers distributed in the backward region of the cylinder compared to the CLASSSO-RLS algorithm, and the distribution of the secondary loudspeakers is too concentrated to effectively cancel the scattered acoustic field.  
Figure 4d illustrates that the CLASSSO-RLS algorithm achieves better noise reduction in target area 2 compared to the other methods, because the change in the distribution of the selected secondary loudspeakers in the forward region more effectively matches the change in the wavelength of the sound as the frequency increases, leading to a better cancellation of the scattered sound field.