## 1 Introduction
Figure 1 shows a Gaussian sampling grid with \(L=8\)[10]. To overcome this problem, research was conducted to obtain 3D spatial information by placing a multi-circular microphone array on a horizontal plane or by placing a multi-circular array in a 3D space [17; 18]. In addition, a 3D sound field was reproduced with a simplified system using a multi-circular loudspeaker array. Unlike wave field reproduction, which can acquire 3D spatial information in advance, wave domain ANC must obtain 3D spatial information based on the real-time wave field analysis process. To execute this process, [PERSON] et al. [19] used the aforementioned multi-circular array for wave domain ANC. The performance of the wave domain ANC was verified experimentally. Twelve microphones and loudspeakers were used for a noise source operated at 400 Hz to control a spherical region with a 0.25 m radius. This is a simpler system compared with the existing system, which places microphones and loudspeakers on Gaussian or Lebedev sampling grids. However, using 12 speakers and microphones to control a spherical region with a 0.25 m radius is still not suitable for global ANC commercialization. If there are sufficient sound field components parallel to the plane of interest, a horizontal wave field with a 0.25 m radius can be reproduced using only five speakers and microphones from two-dimensional (2D) wave field reproduction [20]. With this approach, if the noise control area is reduced to a horizontal plane at the height of the human ear, instead of the entire 3D space, noise reduction is possible through the ANC system with a 2D geometry, which is a simpler system compared with the multi-circular array system. Figure 1 also shows how the system can be compared with Gaussian sampling grids using only 2D spatial information when \(N=8\). For ANC, omnidirectional microphones must be used, which leads to the Bessel zero problem, which in turn degrades the spatial acoustic resolution when extracting wave field information [21]. However, it is necessary to investigate to what extent the Bessel zero problem degrades the performance in the ANC region where only sound pressure reduction is of interest through wave field reproduction error. Therefore, in this study, the target area was assumed to be a horizontal plane at the height of the human ear, as shown in Figure 2, and we approached the noise reduction performance in terms of wave field reproduction using a circular array that had a remarkably reduced number of microphones and speakers while using only 2D spatial information. This approach, in terms of wave field reproduction, provides information regarding the wave field that the system can express, and it is easy to confirm the reduction in ANC performance owing to the Bessel zero problem. The ANC system was efficiently constructed according to the variables affecting the wave field reproduction error. The experimental results and simulation results were compared, and the validity of this approach was verified. The findings of this study suggests the possibility of using a single circular array for ANC systems, which adds a significant contribution to global ANC commercialization because of the simplified system.