## 3 Simulations
### Angular Spectrum of a Point Source  
Figure 2 shows the estimated angular spectrum by integrating the sound pressure over the error plane with the finite size \(L\times L\) and the analytical solution provided by Equation (4) for a primary source located at (0, 0.65, \(-\)0.6) m with a volume velocity of 1 m\({}^{3}\)/s. The center of the integration domain is at the origin of the coordinate system. When the side length of the integration plane in Equation (3) is much smaller than the wavelength, the phases of the sound pressures in the plane are similar. Therefore, as the size of the integration plane increases, the magnitude of the integration increases at first, while the phase remains almost unchanged. As the size of the plane further increases, the phase distribution of the sound pressures in the plane becomes complicated, resulting in the oscillatory characteristics of the result. Nevertheless, both the magnitude and phase of the integration converge to the analytical solution, which demonstrates that the derived analytical solution of the angular spectrum in Equation (4) is correct.  
Figure 3 illustrates the variation in estimation errors with the error microphone spacing \(d\), the location of source \(z_{\text{p}}\), and the target direction \(\theta\), respectively, when Equation (8) is used. In Figure 3a, the estimation error of the angular spectrum is generally smaller with a smaller \(d\), although sometimes the estimation error of \(d=\lambda/2\) is smaller than that of \(d=\lambda/3\) due to the finite size of the microphone array. In Figure 3b, the estimation error increases with the distance between the source and the microphone array when the microphone spacing is \(\lambda/3\). For example, when \(L=1.7\) m, the estimation errors are \(-13.3\), \(-0.6\), and \(-0.2\) dB when the distance \(z_{\text{p}}\) is \(-0.6\), \(-5.6\), and \(-10.6\) m, respectively. As the source moves away from the array, the array appears to shrink because the proportion of the array that lies on the spherical surface centered at the source decreases. In Figure 3c, the overall angular spectrum estimation error increases with the target angle \(\theta\). The reason is that as \(\theta\) increases, the projection of the array in the plane perpendicular to \(\mathbf{k}\) decreases.  
Figure 3: Variation in the angular spectrum estimation error (**a**) with the microphone spacing when \(\mathbf{r_{\text{p}}}=(0,\,0.65,\,-0.6)\) m and \(\theta=0\), (**b**) with the distance between the array and source when \(x_{\text{p}}=0\), \(y_{\text{p}}=0.65\) m, \(d=\lambda/3\) and \(\theta=0\), and (**c**) with the target direction \(\theta\) when \(\mathbf{r_{\text{p}}}=(0,\,0.65,\,-0.6)\) m, \(d=\lambda/3\) and \(\varphi=\pi/2\).  
Figure 2: The angular spectrum estimated with Equation (3) and the analytical solution in Equation (4).  
Table 1 lists the angular spectrum estimation error of the primary source located at (0, 0.65, \(-\)0.6) m with different error microphone spacings when the size of error plane \(L\) = 1.7 m. When \(d\) = \(\lambda/2\), smaller estimation errors can be obtained for both \(\theta\) compared with the other two configurations; therefore, the microphone spacing is \(\lambda/2\) and the size of the microphone array is chosen as \(L\) = 1.7 m in the following simulations.