# 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 produced by the double-layer circular monopole secondary source is then given:  
\[P(r,\phi,k) = \int_{0}^{2\pi}D(r_{1},\phi_{0},k)G(\mathbf{r},\mathbf{r}_{1}  
To synthesize the interior sound field recorded by the circular double-layer monopole receiver without exterior propagation, \(\hat{P}(r_{<},r_{1},k)\) and \(\hat{P}(r_{>},r_{2},k)\) are set to  
\[\hat{P}(r_{<},r_{1},k) = \hat{A}_{m}(k)J_{m}(kr), \tag{27}\] \[\hat{P}(r_{>},r_{2},k) = 0. \tag{28}\]  
Equations (27) and (28) are then represented in matrix form as in [31; 34]:  
\[2\pi\cdot\frac{j}{4}\left[ \begin{array}{cc}H_{m}(kr_{1})&H_{m}(kr_{2})\\ J_{m}(kr_{1})&J_{m}(kr_{2})\end{array} \right]\left[ \begin{array}{c}\hat{D}_ {m}(r_{1},k)\\ \hat{D}_{m}(r_{2},k)\end{array} \right]=\left[ \begin{array}{c}\hat{A}_{m}(k) \\ 0\end{array} \right], \tag{29}\]  
and the driving function of the circular double-layer monopole source is analytically derived:  
\[\hat{D}_{m}(r_{1},k) = \frac{-2 jJ_{m}(kr_{2})\hat{A}_{m}(k)}{\pi\left\{H_{m}(kr_{1})J_{m} (kr_{2})-H_{m}(kr_{2})J_{m}(kr_{1})\right\}}, \tag{30}\] \[\hat{D}_{m}(r_{2},k) = \frac{2 jJ_{m}(kr_{1})\hat{A}_{m}(k)}{\pi\left\{H_{m}(kr_{1})J_{m} (kr_{2})-H_{m}(kr_{2})J_{m}(kr_{1})\right\}}. \tag{31}\]  
Although the driving signals of the LS and GSVD methods derived in Equations (6) and (11) include the control point locations, those of the proposed method derived in Equations (30) and (31) are independent of the control point locations. The driving signals only depend on estimated cylindrical harmonic spectrum \(\hat{A}_{m}(k)\) and circular source radii \(r_{1}\) and \(r_{2}\) since \(J_{m}(kr)\) and \(H_{m}(kr)\) are completely cancelled in Equations (30) and (31). Unlike the LS and GSVD methods, the proposed method, therefore, does not require interior and exterior control points. Continuous circular double-layer receiver and source are finally discretized into circular double-layer arrays of loudspeakers and microphones. When the number of microphones of each layer is \(I_{\rm int}/2\) with equiangular sampling, the recorded sound fields for radii \(R_{1}\) and \(R_{2}\) in the cylindrical harmonic domain are calculated:  
\[\hat{P}_{m}(R_{1,2},k) \approx \frac{2\pi}{I_{\rm int}/2}\cdot\frac{1}{2\pi}\sum_{i=1}^{I_{\rm int }/2}P(R_{1,2},\phi_{i},k)e^{-jm\phi_{i}} \tag{32}\] \[= \frac{2}{I_{\rm int}}\sum_{i=1}^{I_{\rm int}/2}P(R_{1,2},\phi_{i},k)e^{-jm\phi_{i}}.\]  
When the number of loudspeakers of each layer is \(L/2\) with equiangular sampling, order \(m\) of the cylindrical harmonic spectrum in Equations (30) and (31) can be calculated up to \(M=\lfloor(L/2-1)/2\rfloor\), where \(\lfloor\cdot\rfloor\) is the floor function. However, the cylindrical harmonic spectrum at \(|n|>|kr_{1,2}|\) is the evanescent component and is quite large because of the inverse propagation when \(r>r_{1,2}\)[40]. To calculate a stable filter as in [42], \(m\) is up to \(M=\lfloor|kr_{1,2}|\rfloor\) if \(\lfloor|kr_{1,2}|\rfloor<\lfloor(L/2-1)/2\rfloor\) since using only the propagation component for \(|n|\leq|kr_{1,2}|\) is sufficient to control the target interior sound field. The driving signal of each loudspeaker at \(\phi_{l}\) for \(r_{1}\) and \(r_{2}\) in the frequency domain is obtained:  
\[D(r_{1,2},\phi_{l},k)=\sum_{m=-M}^{M}\hat{D}_{m}(r_{1,2},k)e^{jm\phi_{l}}, \qquad l=1,2,\cdots L/2. \tag{33}\]  
The purpose of introducing circular double-layer arrays of microphones and loudspeakers in the LS and GSVD methods differs from that in the proposed method. In the LS and GSVD methods, the circular double-layer arrays are introduced to satisfy the KH integral. Hence, an approximated dipole distribution is required. On the other hand, the approximated dipole distribution is not required because the proposed method is not based on the KH integral. Circular double-layer arrays are introduced in the proposed method to avoid the forbidden frequencies in the recording stage and to simultaneously control both the interior and exterior sound field cylindrical harmonic spectrums in the synthesis stage.