## 3 Proposed Method  
### Position Estimation Principle  
Figure 2 depicts the assignment of two microphone arrays for position estimation. The direction perpendicular to the array was set to \(0^{\circ}\) of the beam. Let \((x,y)\) be the coordinate of an estimation target position based on origin \(O\). The two microphone arrays, designated as M1 and M2, are installed at distance \(p_{x}\) horizontally and at distance \(p_{y}\) vertically.  
Assuming \(\theta_{1}\) and \(\theta_{2}\) as the angles between the lines perpendicular to \(p_{x}\) and \(p_{y}\) and the straight line to \((x,y)\), and letting \(L_{1}\) and \(L_{2}\) represent the straight distances between the respective microphone array systems and \((x,y)\), then using trigonometric functions, \(p_{x}\) and \(p_{y}\) are given as presented below [98].  
\[p_{x} = L_{1}\cos\theta_{1}+L_{2}\sin\theta_{2}, \tag{1}\] \[L_{1} = \frac{p_{x}-L_{2}\sin\theta_{2}}{\cos\theta_{1}},\] (2) \[p_{y} = L_{1}\sin\theta_{1}+L_{2}\cos\theta_{2}\] (3) \[= \frac{p_{x}-L_{2}\sin\theta_{2}}{\cos\theta_{1}}\sin\theta_{1}+L_ {2}\frac{\cos\theta_{1}\cos\theta_{2}}{\cos\theta_{1}}\] (4) \[= L_{2}\frac{\cos\theta_{1}\cos\theta_{2}}{\cos\theta_{1}}-L_{2} \frac{\sin\theta_{1}\sin\theta_{2}}{\cos\theta_{1}}+p_{x}\frac{\sin\theta_{1}} {\cos\theta_{1}}. \tag{5}\]  
The following expanded equation is obtained by solving for \(L_{2}\).  
\[L_{2}\frac{\cos\theta_{1}\cos\theta_{2}-\sin\theta_{1}\sin\theta _{2}}{\cos\theta_{1}} = p_{y}-p_{x}\frac{\sin\theta_{1}}{\cos\theta_{1}} \tag{6}\] \[L_{2}\frac{-\cos(\theta_{1}+\theta_{2})}{\cos\theta_{1}} = \frac{p_{y}\cos\theta_{1}-p_{x}\sin\theta_{1}}{\cos\theta_{1}}\] (7) \[L_{2} = \frac{p_{x}\sin\theta_{1}-p_{y}\cos\theta_{1}}{\cos(\theta_{1}+ \theta_{2})}. \tag{8}\]  
One can obtain \((x,y)\) as follows:  
\[\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}p_{x}-L_{2}\cos\theta_{2}\\ p_{y}-L_{2}\sin\theta_{2}\end{array}\right]. \tag{9}\]  
Figure 2: Arrangement of two microphone arrays M1 and M2 for position estimation.  
Alternatively, using \(L_{2}\), one can obtain \(p_{y}\) as follows:  
\[L_{2} = \frac{p_{x}-L_{1}\cos\theta_{1}}{\sin\theta_{2}}, \tag{10}\] \[p_{y} = L_{1}\sin\theta_{1}+\frac{p_{x}-L_{1}\cos\theta_{1}}{\sin\theta_{ 2}}\cos\theta_{2}\] (11) \[= L_{1}\frac{\sin\theta_{1}\sin\theta_{2}}{\sin\theta_{2}}+p_{x} \frac{\cos\theta_{2}}{\sin\theta_{2}}-L_{1}\frac{\cos\theta_{1}\cos\theta_{2} }{\sin\theta_{2}}. \tag{12}\]  
The following expanded equation is obtained by solving for \(L_{1}\).  
\[L_{1}\frac{\sin\theta_{1}\sin\theta_{2}-\cos\theta_{1}\cos \theta_{2}}{\sin\theta_{2}} = p_{y}-p_{x}\frac{\cos\theta_{2}}{\sin\theta_{2}} \tag{13}\] \[L_{1}\frac{-\cos(\theta_{1}+\theta_{2})}{\sin\theta_{2}} = \frac{p_{y}\sin\theta_{2}-p_{x}\cos\theta_{2}}{\sin\theta_{2}}\] (14) \[L_{1} = \frac{p_{x}\sin\theta_{2}-p_{y}\sin\theta_{2}}{\cos(\theta_{1}+ \theta_{2})}. \tag{15}\]  
Using \(L_{1}\) and \(\theta_{1}\), one can obtain \((x,y)\) as follows:  
\[\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}p_{x}-L_{1}\cos\theta_{1}\\ p_{y}-L_{1}\sin\theta_{1}\end{array}\right]. \tag{16}\]  
Angles \(\theta_{1}\) and \(\theta_{2}\) are calculated using the DAS beamforming method, as shown in Figure 3.