## 1 Introduction  
Low-frequency noise, which is challenging to reduce using passive methods, can be reduced using a transducer-like loudspeaker or devices that generate vibration. Active structural acoustic control (ASAC), which reduces the sound power radiated into the air by controlling the vibration of the noise source [1, 2], and active noise control (ANC), which cancels the sound pressure by generating an \"anti\" signal that has an opposite phase but equal amplitude to those generated by the noise source, have been researched [3]. Given that ASAC reduces noise emission from a noise source, global noise reduction is possible; however, if the noise is radiated from several unspecified sources, an effective reduction is not possible unless the noise source is specified. However, because ANC eliminates sound pressure propagating in the air at the microphone position, it can be effectively used in situations where the noise source is unspecified [4, 5]. Local ANC is achieved by reducing the sound pressure at the microphone position; however, the wave field of the noise source must be suppressed to achieve global ANC. To achieve this, acoustic energy is selected as the cost function, and an additional microphone has been used to obtain the particle velocity. As a cost function, acoustic energy is more efficient than acoustic pressure for global noise reduction [6]. In an intuitive manner, if an \"anti\" wave noise-source field is generated, global ANC can be achieved. Wave field reproduction provides an immersive experience to all listeners within a large target area by creating a virtual wave field of the sound source using a densely placed loudspeaker configuration. Wave field reproduction can be divided into wave field synthesis, based on the Kirchhoff-Helmholtz integral, and higher order ambisonics, which uses harmonics coefficients [7]. However,to reproduce a three-dimensional (3D) wave field,a large number of microphones and loudspeakers must be placed on Gaussian, Lebedev, and Fliege sampling grids [8]. To address this limitation, 2.5-dimensional (2.5D) wave field reproduction, which uses only horizontal spherical harmonics and generates a wave field at the height of the human ear, has been primarily used [9; 10]. Given that 3D spatial information on the desired waves is prepared in advance, only a single circular loudspeaker array is required for 2.5D wave field reproduction. The circular array can be used in various fields, such as multi-sound zone reproduction and personal sound zones [11; 12]. Accordingly, research has been conducted on noise control over a horizontal plane combined with wave field reproduction and ANC, and a study was pursued on pulse noise removal via fixed control filtering based on wave field synthesis [13; 14]. To cope with the changing spatial information of the noise source, wave domain ANC, which minimizes spherical harmonics coefficients in 3D space using an adaptive filter, has been studied [15]. In the case of wave domain ANC, it has been proven that the global active noise control performance is better than that of the conventional ANC by using the harmonics coefficient as a cost function [16]. However, to obtain 3D spatial information and generate 3D wave fields, several challenges arise in practice owing to the large number of microphones and loudspeakers needed. A wave field, consisting of \(N^{2}\) spherical harmonics coefficients, is set to \(L=N\) to consider spatial aliasing. The relation between the highest order \(N\) of spherical harmonics coefficients, angular frequency \(w\), the speed of sound \(c\), and wave field generation radius \(r_{N-1}\) for expressing 3D wave field reproduction is expressed in Equations (1) and (2). \[(N-1) =\frac{w}{c}r_{N-1} \tag{1}\] \[N =L \tag{2}\]  
To obtain spherical harmonics coefficients of microphones and reproduce a sound field with loudspeakers, Gaussian sampling grids require \(2L^{2}\) microphones and loudspeakers, Lebedev grids require approximately \(1.3L^{2}\) samples, and Fliege grids require \(L^{2}\) samples [8].