# Mode-Matching-Based Sound Field Recording and Synthesis with Circular Double-Layer Arrays
## 3 Proposed Analytical Formulation
### Interior and Exterior Sound Field Control with Circular Double-Layer Monopole Source and Receiver  
A sound field recorded by a continuous circular monopole receiver distribution that is centered at the origin with radius \(R_{0}\) is converted into the cylindrical harmonic domain [40]:  
Figure 3: Definition of spherical coordinates relative to Cartesian coordinates. \[\hat{P}_{m}(R_{0},k)=\frac{1}{2\pi}\int_{0}^{2\pi}P(R_{0},\phi,k)e^{-im\phi}d\phi. \tag{14}\]  
The cylindrical harmonic spectrum of the interior sound field is then obtained:  
\[\hat{A}_{m}(k)=\frac{\hat{P}_{m}(R_{0})}{J_{m}(kR_{0})}. \tag{15}\]  
Equation (15), however, has a forbidden frequency (Bessel-zero) problem where \(J_{m}(kR_{0})=0\) and is dependent on receiver radius \(R_{0}\) and wavenumber \(k\)[40]. To avoid the forbidden frequency problem for open circular monopole receivers, a circular double-layer monopole receiver centered on the origin with radii \(R_{1}\) and \(R_{2}\) is introduced (Figure 4a). Sound fields recorded by both layers are converted into the cylindrical harmonic domain by Equation (14) as \(\hat{P}_{m}(R_{1},k)\) and \(\hat{P}_{m}(R_{2},k)\). By applying a method with spherical double-layer monopole receivers [43] into circular double-layer monopole receivers in the proposed method, the cylindrical harmonic spectrum is calculated:  
\[\hat{A}_{m}(k)=\frac{(1-\beta_{m}(R_{1},R_{2},k))\hat{P}_{m}(R_{1},k)+\beta_{m }(R_{1},R_{2},k)\hat{P}_{m}(R_{2},k)}{(1-\beta_{m}(R_{1},R_{2},k))J_{m}(kR_{1} )+\beta_{m}(R_{1},R_{2},k)J_{m}(kR_{2})}, \tag{16}\]  
where  
\[\beta_{m}(R_{1},R_{2},k)=\left\{ \begin{array}{ll}0,&|J_{m}(kR_{1})|\geq|J_{m }(kR_{2})|\\ 1,&|J_{m}(kR_{1})|<|J_{m}(kR_{2})|\end{array} \right.. \tag{17}\]  
Equation (16) indicates that \(\hat{A}_{m}(k)\) is obtained from a single layer of \(J_{m}(kR_{1})\) or \(J_{m}(kR_{2})\) with a large absolute value based on the assumption that \(J_{m}(kR_{1})\) and \(J_{m}(kR_{2})\) do not simultaneously become zero. In the proposal, therefore, the absence of mutual zeros needs to hold for all cylindrical harmonic orders. Figure 4: Arrangements: (**a**) circular double-layer monopole receiver for recording a primary sound field; and (**b**) circular double-layer monopole source for synthesizing recorded primary sound field at a secondary field. Alternatively, two methods with circular single-layer receivers have been proposed. The first one introduces an open circular single-layer directional receiver with radius \(R_{0}\) and the cylindrical harmonic spectrum is obtained [44]:  
\[\hat{A}_{m}(k)=\frac{\hat{Q}_{m}(R_{0})}{\delta J_{m}(kR_{0})-j(1-\delta)J^{ \prime}_{m}(kR_{0})}, \tag{18}\]  
where \(J^{\prime}_{m}\) is the radial derivative of \(J_{m}\), \(\delta\) is the dipole weighting factor and \(\hat{Q}_{m}(R_{0})\) is the cylindrical harmonic spectrum of the sound pressure and its radial derivative received by the circular directional receiver with radius \(R_{0}\). The other method employs a circular single-layer receiver mounted on a rigid baffle with radius \(R_{0}\), and the cylindrical harmonic spectrum is obtained [45]:  
\[\hat{A}_{m}(k)=\frac{\hat{P}_{m}(R_{0})}{J_{m}(kR_{0})-\frac{J^{\prime}_{m}(kR _{0})}{H^{\prime}_{m}(kR_{0})}H_{m}(kR_{0})}, \tag{19}\]  
where \(H^{\prime}_{m}\) is the radial derivative of \(H_{m}\). Compared with the method involving double-layer receivers, these methods can be realized using half the number of microphones in actual implementations. However, they cannot be directly compared with the pressure matching-based LS and GSVD methods. To directly compare the proposed mode-matching method with conventional pressure matching-based approaches, a circular double-layer array of microphones is introduced in the computer simulations conducted in the next section. For simultaneously controlling both the interior and exterior sound fields, a circular double-layer monopole source centered the origin with radii \(r_{1}\), and \(r_{2}\) is introduced (Figure 4b).