# Mode-Matching-Based Sound Field Recording and Synthesis with Circular Double-Layer Arrays
## 4 Computer Simulations
### Interior Synthesis Accuracy and Exterior Acoustic Contrast  
Figures 6 and 7, respectively, show the results of the primary sound field, synthesized sound field, interior sound field synthesis accuracy, and exterior acoustic contrast by the LS, GSVD, and proposed mode-matching methods with \(\Delta=0.25\) m in both the free-field and reverberant conditions for \(f=200\). The results in Figure 6 indicate that the LS and proposed methods realize a higher interior control accuracy than the GSVD method. This is because the conventional GSVD method reduces the exterior propagation above the spatial Nyquist frequency at the expense of the interior synthesis accuracy [39]. The results in Figure 6 also suggest that the proposed method can achieve the highest exterior acoustic contrast. As a result, the interior synthesis accuracies of the LS and GSVD methods are degraded by the reverberations caused by exterior propagations (Figure 7). On the other hand, only the proposed method can successfully control both the interior and exterior sound fields in the reverberant condition (Figure 7). Figures 8 and 9, respectively, plot the results of the spatially-averaged interior synthesis error and exterior acoustic contrast defined in Equations (35) and (36). As described above, the conventional GSVD approach reduces the exterior propagation beyond the spatial Nyquist frequency at the expense of the interior synthesis accuracy [39]. The results in Figure 9 show that the GSVD method successfully reduces the exterior propagation over the spatial Nyquist frequency compared with the LS and proposed methods. However, the interior field is no longer synthesized by the GSVD method at higher frequencies (Figure 8). The results in Figures 6-9 demonstrate the main impact of this paper; compared to conventional numerical approaches, the proposed method can more accurately synthesize the interior field while reducing the exterior propagation below the spatial Nyquist frequency in both the free-field and reverberant conditions. As described above, this is because the proposed method realizes the highest exterior acoustic contrast and successfully controls both the interior and exterior sound fields in the reverberant condition. These results also suggest the effectiveness of the spatial Fourier transform-based approach compared with the LS method [14; 34; 41; 42]. Figure 5: Computer simulations conditions: (**a**) double-layer microphone array; and (**b**) double-layer arrays of loudspeakers and control points. Each layer has 24 elements with equiangular sampling. Unlike the LS and GSVD methods, which depend on layer intervals \(\Delta\), the proposed method's interior field synthesis accuracy is independent of \(\Delta\) (Figure 8) because the proposed method can successfully estimate the interior sound field cylindrical harmonic spectrum from the circular double-layer microphone array using Equation (16) without the forbidden frequencies. The proposed method does not control the discretized sound pressures. Instead, it controls the estimated sound field in the cylindrical harmonic domain, as derived in Equations (30) and (31). However, the driving signals of the LS and GSVD methods obviously include the control point and loudspeaker locations, as described in Equations (6) and (11). Figure 6: Results of primary sound field, synthesized sound field, interior sound field synthesis accuracy and exterior acoustic contrast by LS, GSDV, and proposed mode-matching methods with \(\Delta=0.25\) m in free-field condition for \(f=200\) Hz. Blue \(\circ\), \(\triangle\) and \(+\) are microphones, loudspeakers and control points, respectively. Figure 7: Results of primary sound field, synthesized sound field, interior sound field synthesis accuracy and exterior acoustic contrast by LS, GSDV, and proposed mode-matching methods with \(\Delta=0.25\,\mathrm{m}\) in reverberant condition for \(f=200\) Hz. Blue \(\circ\), \(\triangle\) and \(+\) are microphones, loudspeakers and control points, respectively.