## 3 Applications
### Origin of the Idea and Proposed Method  
An analysis of Figure 5 shows that \(d_{m}\) is insufficient for accurate location estimation, and that the location estimates change along the same line, even with high estimation errors. This observation suggests that the DoA estimation based on the multilateration method can be performed with high accuracy, independent of \(d_{m}\). Figure 4: The TDoA model used in simulation studies. The multilateration equations for a microphone array consisting of three microphones, with the first microphone as the reference, are shown in Equation (13). Assuming that the distance from the sound source to the first microphone, \(d_{1}\), is known, Equation (13) can be rearranged as follows:  
\[ \begin{bmatrix}2(\mathbf{p}_{2}-\mathbf{p}_{1})^{T}\\ 2(\mathbf{p}_{3}-\mathbf{p}_{1})^{T}\end{bmatrix} \times \begin{bmatrix}\mathbf{p}_{k}\end{bmatrix} = \begin{bmatrix}\mathbf{p}_{2}^{T}\mathbf{p}_{2}-\mathbf{p}_{1}^{T}\mathbf{p}_{1}-v_{\rm s}\tau_{ 21}(v_{\rm s}\tau_{21}+2d_{1})\\ \mathbf{p}_{3}^{T}\mathbf{p}_{3}-\mathbf{p}_{1}^{T}\mathbf{p}_{1}-v_{\rm s}\tau_{31}(v_{\rm s} \tau_{31}+2d_{1})\end{bmatrix}  \tag{16}\]  
The simple representation of Equation (16) is as follows:  
\[\mathbf{A}_{1}\mathbf{x}_{1}=\mathbf{b}_{1} \tag{17}\]  
To perform localization on the plane, the distances of the microphones located on the perpendicular axes in the sensor array shown in Figure 3a to the reference microphone are taken as \(d_{mx}\) and \(d_{my}\). Since the reference microphone is located at the origin, \(d_{1}=d_{k}\). According to these, the \(\mathbf{A}_{1}\) matrix is as follows:  
\[\mathbf{A}_{1}=2\times \begin{bmatrix}d_{mx}&0\\ 0&d_{my}\end{bmatrix}  \tag{18}\]  
\(\mathbf{b}_{1}\) vector:  
\[\mathbf{b}_{1}= \begin{bmatrix}d_{mx}^{2}-v_{\rm g}^{2}\times\tau_{21}^{2}-2v_{\rm s }\tau_{21}d_{k}\\ d_{my}^{2}-v_{\rm s}^{2}\times\tau_{31}^{2}-2v_{\rm s}\tau_{31}d_{k}\end{bmatrix}  \tag{19}\]  
If we consider that the second microphone is on the \(x\)-axis and the third microphone is on the \(y\)-axis, we can write \(\tau_{21}=\tau_{x}\) and \(\tau_{31}=\tau_{y}\). It is clear that \(-d_{mx}\leq v_{\rm s}\tau_{x}\leq d_{mx}\) and \(-d_{my}\leq v_{\rm s}\tau_{y}\leq d_{my}\), depending on the direction of sound arrival. Therefore, the \(\mathbf{b}_{1}\) vector can be divided into two components and arranged as  
\[\mathbf{b}_{1}=\underbrace{ \begin{bmatrix}d_{mx}^{2}-v_{\rm g}^{2}\times\tau_{2} ^{2}\\ d_{my}^{2}-v_{\rm s}^{2}\times\tau_{y}^{2}\end{bmatrix} }_{\in \begin{bmatrix} 0,\,d_{mx}^{2}\\ 0,d_{my}^{2}\end{bmatrix} }- \begin{bmatrix}2v_{\rm s}\tau_{x}d_{k}\\ 2v_{\rm s}\tau_{y}d_{k}\end{bmatrix}  \tag{20}\]  
Figure 5: Localization of a sound source located at five different angles at a distance of 150 m by using different \(d_{m}\) distances. The first component in Equation (20) is neglected, because it has a very small value compared to the second component. Therefore, Equation (16) can be approximated as follows:  
\[ \begin{bmatrix}d_{mx}&0\\ 0&d_{my}\end{bmatrix} \times \begin{bmatrix}x_{k}\\ y_{k}\end{bmatrix} \approx-v_{s}d_{k} \begin{bmatrix}\tau_{x}\\ \tau_{y}\end{bmatrix}  \tag{21}\]  
The \(\mathbf{p}_{k}\) position vector can be isolated from Equation (21) as follows:  
\[ \begin{bmatrix}x_{k}\\ y_{k}\end{bmatrix} \approx-v_{s}d_{k} \begin{bmatrix}\frac{\tau_{x}}{d_{mx}}\\ \frac{\tau_{y}}{d_{my}}\end{bmatrix}  \tag{22}\]  
As shown in Equation (22), the position vector of the sound source varies according to the TDoA values between the microphones. When the ratio of the \(x_{k}\) and \(y_{k}\) positions specified in Equation (1) is calculated according to Equation (22), it is equal to the ratio of the TDoA times, and the \(\theta\) azimuth angle:  
\[\theta\approx atan2\bigg{(}-\frac{\tau_{y}}{d_{my}},-\frac{\tau_{x}}{d_{mx}} \bigg{)} \tag{23}\]  
Similarly, in the three-dimensional cartesian space, both the azimuth and elevation angles can be obtained using the four-microphone sensor array shown in Figure 3b.