## 2 2D Wave Field Reproduction in 3D Space Using a Single Circular Array  
Figure 2 shows the 2D array system and target area. An arbitrary point \(x\) consists of information related to the distance between the origin and the arbitrary point \(r\) and angle \(\phi\). In the free field, we assumed that it is an internal problem in which a noise source exists outside the microphone and loudspeaker array, and the noise source, microphone, and loudspeaker array exist on the \(x-y\) plane. The desired wave field \(S_{D}\) can be represented by truncated summation [10] according to Equation (3). \[S_{D}(x,k)\approx\sum_{m=-M}^{M}\bar{S}(r,k,m)exp(im\phi) \tag{3}\]  
Figure 1: (**a**) Gaussian sampling grids for three-dimensional (3D) wave field information. (**b**) proposed circular array for two-dimensional (2D) wave field information when \(N=8\). Figure 2: Proposed active noise control system based on the use of a single circular array with a quiet zone (blue region). where \(k=2\pi f/c\) is the wavenumber, \(c\) is the wave propagation speed, \(f\) is the frequency, \(m\) is the order, \(M\) is the highest order, \(exp\) is the exponential function, and \(\tilde{S}\) is the circular wave spectrum. The circular wave spectrum can be expressed by the circular harmonics coefficient and the Bessel function [22]. To avoid spatial aliasing owing to the discretization of the continuous array in Figure 2, the angular bandwidth \((M=N)\) and the number of microphones and loudspeakers can be determined via Equations (1) and (4). \[\text{Number of Microphone and loudspeaker}\geq 2M+1 \tag{4}\]  
From Equation (4), any 2D wave field can be expressed if the \(2M+1\) circular spectrum is known. \(\tilde{S}(r,k,m)\) can be obtained using the discrete spatial Fourier transform (Equation (5)) by assuming the orthogonal property of the basis function \(exp\). \[\tilde{S}_{mic}(R_{mic},k,m)=\frac{1}{Q}\sum_{q=1}^{Q}S(\mathbf{x}_{q},k)exp(-i2\pi m (q-1)/Q), \tag{5}\]  
where \(\tilde{S}_{mic}\) is the circular wave spectrum obtained using the microphone, \(R_{mic}\) is the radius from the origin to the microphone array, \(\mathbf{x}_{q}\) is the position of each microphone, and \(Q\) is the number of microphones. If the radial part \(J(kR_{mic})\) is divided into both sides of Equation (5), it is called a circular harmonics coefficient and can be used as the cost function of the wave domain ANC [23]. When the Bessel function becomes zero, an error occurs in the process of extracting the circular harmonics coefficient, and this error affects the wave field reproduction error. The sound pressure of the desired wave field can be matched only within a circle with a radius of \(r_{ref}\) (reference circle) owing to the dimensional mismatch that occurs because the 3D wave field is expressed as a 2D array. The relationship between the circular harmonics coefficients of \(r_{ref}\) and those of \(R_{mic}\) can be expressed as [22],  
\[\tilde{S}_{D}(r_{ref},k,m)/J_{m}(kr_{ref})=\tilde{S}_{mic}(R_{mic},k,m)/J_{m}( kR_{mic}), \tag{6}\]  
where \(\tilde{S}_{D}(r_{ref},k,m)\) is the circular wave spectrum of the desired wave field at \(r_{ref}\). Because only the 2D sound field information was used in 3D space, wave field reproduction has a lower performance than that of 2.5D wave field reproduction based on spherical harmonics coefficients. However, as aforementioned, a spherical microphone array or a multi-circular microphone array are required to obtain information on the 3D wave field [17].