# Time Domain Spherical Harmonic Processing with Open Spherical Microphones Recording
## 5 Simulation Results and Analysis
### Comparison between the Time Domain and the Frequency Domain Spherical Harmonic Coefficients  
We use a narrow band signal at 1200 Hz to test if our proposed method can obtain time domain spherical harmonic coefficients \(a_{nm}(\
u)\) correctly with (26). In (11), we give the relationship between \(a_{nm}(\
u)\) and \(a_{nm}(k)\). We compare the Fourier transformation result of our obtained time domain spherical harmonic coefficients \(\mathcal{FT}\{a_{nm}(\
u)\}\) with the desired frequency domain spherical harmonic coefficients \(a_{nm}(k)\), obtained by Equation (4) in  
Figure 4: Time representation of the pre-designed order dependent FIR filter \(\rho_{n}(\
u,R_{Q})\) with length \(L=\) (a) 25, (b) 250 (c) 2500.  
frequency domain. Fourier transformations use \(J=1024\) points. We do not compare the phase of these coefficients since the group delay of the time domain method and the frequency domain method is different. Instead, we compare the phase difference, given by \(\alpha_{nm}(k)-\alpha_{n(m-1)}(k)\). The results of both amplitude and phase difference are shown in Figure 5.  
In Figure 5 we see that there is little to no difference on both amplitude and phase difference between the Fourier Transformed time domain coefficients and the frequency domain coefficients over all the order and modes. Thus, our proposed time domain method successfully obtained the time domain spherical harmonic coefficients, which can be related to the frequency domain coefficients by Fourier transformation.  
Next, we compare the coefficients over different frequencies with a wide band test signal within the frequency limited of \([20,1300]\) Hz. In Figure 6, we show the comparison of amplitude at \(FT\{a_{00}(\
u)\}\) and \(\alpha_{00}(k)\), \(FT\{a_{11}(\
u)\}\) and \(\alpha_{11}(k)\), and \(FT\{a_{31}(\
u)\}\) and \(\alpha_{31}(k)\) over frequencies respectively while Figure 7 shows the phase difference.  
A huge error is observed in Figure 5(a) at the 46 th frequency bin. This error is due to that there is a Bessel zero of the zeroth order at this frequency bin (around 1072 Hz). We see the frequency domain spherical harmonic coefficients \(\alpha_{00}(k)\) has a much higher amplitude, while our proposed method suppressed the amplitude at this certain frequency bin. Meanwhile, we can see in Figures 6 and 7 that the error at \(a_{31}(\
u)\) is higher compared to the other two modes. As order increases, the error increases. This error can be decreased by applying more microphones on the array. We also obtain a non-negligible error before the 30 th frequency bin of the coefficients amplitude for \((n,m)=(3,1)\) in Figures 5(c) and 5(c). This error is because our time domain proposed method and conventional frequency domain method have different processing on suppressing Bessel zeros. During the reconstruction process, the high pass property of spherical Bessel function removes the information at this frequency bin. Thus, this error will not influence the reconstruction of the sound field.  
Figure 5: (**a**) Amplitude and (**b**) phase difference comparison between the Fourier Transform of the time domain spherical harmonic coefficients \(a_{nm}(\
u)\) and frequency domain spherical harmonic coefficients \(\alpha_{nm}(k)\) at a single frequency \(f=1200\) Hz.  
Figure 6: Amplitude comparison between the Fourier Transform of the time domain spherical harmonic coefficients \(a_{nm}(\
u)\) and frequency domain spherical harmonic coefficients \(\alpha_{nm}(k)\) at mode (**a**) 00, (**b**) 11 and (**c**) 31 with a white Gaussian noise.