## 3 Simulations
### Effect of the Secondary Source Location  
It is seen from Equation (13) that to achieve effective noise reduction in the target direction, the angular spectrum estimation errors of the sources need to be as small as possible. This section aims to investigate the effect of the position of the secondary source on the angular spectrum estimation error, as well as the performance of the active directivity control system. Figure 4 shows the schematic diagram of the simulation system. The secondary source is assumed to be placed at \(z\) = \(-\)0.6 m plane where the primary source is located. A 1.7 m \(\times\) 1.7 m error array with nine evenly spaced microphones distributed within the \(z\) = 0 plane is used to estimate the angular spectrum. The sound pressure level (SPL) reduction at a receiver which is 3 m away from the origin in the target direction is used to evaluate the noise reduction performance of the system. The evaluation plane is at the \(x\) = 0 plane. Figure 5a shows the angular spectrum estimation error when the sources are located at different positions in this plane and the target direction is \(\theta\) = 0. In this case, it can be seen from Equation (3) that the integrand is symmetric with respect to \(x\) and \(y\). Moreover, it can be observed from Equation (4) that within the \(x\)-\(y\) plane, the angular spectra of two point sources with the same strength are equal. Therefore, the distribution of estimation errors is bivariate symmetric within the \(x\)-\(y\) plane. Additionally, since the integrand can be decomposed into the form of sine and cosine integrals, the estimation errors will exhibit oscillatory characteristics in the \(x\)-\(y\) plane. The reason that grid-like patterns appear in areas with smaller estimation errors is that the microphone array is distributed in a square. The primary source is located at \(\mathbf{r_{p}}\) (0, 0.65, \(-\)0.6) m, and \(\mid\delta_{\mathrm{P}}\,\mid\) is \(-\)14.2 dB. When the secondary source is located at \(\mathbf{r_{s1}}\) = (\(-\)0.42, 0.42, \(-\)0.6) m and \(\mathbf{r_{s2}}\) = (0, \(-\)0.11, \(-\)0.6) m, \(\mid\delta_{\mathrm{S}}\,\mid\) is \(-\)10.8 \begin{table}
\begin{tabular}{c c c} \hline \(1\,\delta_{\mathrm{P}}\,\mathbf{1}\) **(dB)** & \(\theta\) **= 0** & \(\theta\) **= \(\pi\)/6** \\ \hline Microphone array, \(d\) = \(\lambda/2\) & \(-\)14.2 & \(-\)20.9 \\ \hline Microphone array, \(d\) = \(\lambda/3\) & \(-\)13.3 & \(-\)11.6 \\ \hline Integration & \(-\)10.4 & \(-\)13.7 \\ \hline \end{tabular}
\end{table} 
Table 1: Angular spectrum estimation error of the primary source located at (0, 0.65, \(-\)0.6) m when \(L\) = 1.7 m.  
Figure 4: Schematic diagram of the simulation system. The red rectangle represents the plane where the sound sources are located, the black circles represent the error microphones, and the green rectangle represents the evaluation plane. and 1.6 dB, and the corresponding noise reductions are 22.1 dB and 4.5 dB, respectively. The smaller the estimation error of the secondary source is, the larger the noise reduction will be, as shown in Figure 5b. Figure 6 shows the distribution of SPL and NR in the evaluation plane when the target direction is \(\theta=0\). When the secondary source is placed at \(\mathbf{r}_{\mathrm{s1}}\), a more significant directional quiet zone from the near field to the far field in the target direction is achieved, as shown in Figure 6b,e. For instance, the average SPL at the 11 orange points spaced 0.3 m apart in Figure 6c is reduced by 19.8 dB and 1.2 dB, respectively, when the secondary source is located at positions \(\mathbf{r}_{\mathrm{s1}}\) and \(\mathbf{r}_{\mathrm{s2}}\). Figure 7a shows the angular spectrum estimation errors when the target direction is chosen as \(\theta=\pi/6\) and \(\varphi=\pi/2\). The estimation errors of the secondary source are \(-3.9\) dB and \(-19.7\) dB when the secondary source is placed at \(\mathbf{r}_{\mathrm{s1}}\) and \(\mathbf{r}_{\mathrm{s2}}\), respectively.