# Mode-Matching-Based Sound Field Recording and Synthesis with Circular Double-Layer Arrays
## 4 Computer Simulations
### Simulation Conditions  
In all the simulations, two-dimensional sound fields were assumed and both free-field and reverberant conditions were evaluated. The speed of sound \(c\) was 343.36 m/s. In the reverberant condition, the two-dimensional room size was \(15\) m \(\times\) 12 m and the room center was centered at the origin. The transfer functions in the reverberant condition were calculated by the image method [47] in the frequency domain with reflection coefficient 0.3 and reflection order 10. The primary sound field was calculated as a cylindrical wave propagating in a free-field from a point at \(\mathbf{x}=[5,\ 3]^{\mathrm{T}}\).  
Both the amounts of microphones, interior and exterior control points for the LS and GSVD methods, and loudspeakers for each layer were \(I_{\mathrm{int}}/2=L/2=24\), and their total numbers were \(I_{\mathrm{int}}=L=48\). Then, each layer had 24 elements with equiangular sampling. The central radii of the circular double-layer arrays of microphones and loudspeakers were \((R_{1}+R_{2})/2=1.0\) m and \((r_{1}+r_{2})/2=2.0\) m. Two circular double-layer arrays of control points with radii \(R_{1}\), \(R_{2}\), \(R_{3}\) and \(R_{4}\) were introduced for the LS and GSVD methods and their central radii were \((R_{1}+R_{2})/2=1.0\) m and \((R_{3}+R_{4})/2=3.0\) m. Three types of layer intervals \(\Delta=R_{2}-R_{1}=R_{4}-R_{3}=r_{2}-r_{1}=0.05\), 0.25, and 0.5 m were evaluated. The arrangements of microphones, loudspeakers and control points are illustrated in Figure 5. The case with \(\Delta=0.05\) m is the same condition with previous simulations for the LS and GSVD [38]. The spatial Nyquist frequency of the circular secondary source was approximately derived as [31; 39]:  
\[f_{\mathrm{Nyq}}\approx\frac{c(L/2)}{4\pi r_{2}}. \tag{34}\]  
In the simulations, the \(f_{\mathrm{Nyq}}\) values were about 324, 309, and 291 Hz for \(\Delta=0.05\), 0.25, and 0.50 m, respectively.  
To evaluate the synthesized sound field, the synthesis error and exterior acoustic contrast at position \(\mathbf{r}\) were defined as  
\[E(\mathbf{r}) =10\log_{10}\frac{|P_{\mathrm{org}}(\mathbf{r})-P_{\mathrm{syn}}(\bm {r})|^{2}}{|P_{\mathrm{org}}(\mathbf{r})|^{2}}, \tag{35}\] \[C(\mathbf{r}) =10\log_{10}\frac{|P_{\mathrm{syn}}(\mathbf{r})|^{2}}{|P_{\mathrm{org }}(\mathbf{r})|^{2}}, \tag{36}\]  
where \(P_{\mathrm{org}}(\mathbf{r})\) and \(P_{\mathrm{syn}}(\mathbf{r})\) are, respectively, the primary and synthesized sound pressures at position \(\mathbf{r}\). To evaluate the interior sound field synthesis accuracy and exterior acoustic contrast, the spatially-averaged synthesis error within \(r\leq 1.5\) m and exterior acoustic contrast with \(2.5\) m \(\leq r\leq 4.5\) m were, respectively, calculated for \(10\) Hz \(\leq f\leq 500\) Hz.  
In the GSVD method, singular value truncation parameter \(K\) in Equation (11) was also automatically chosen to satisfy that all the diagonal terms of \(C_{K}(k)\) are smaller than 0.01 [39]. The maximum cylindrical harmonic order was \(M=\lfloor(L/2-1)/2\rfloor=11\) in the proposed approach.