## 2 General Properties of Method
### Multilateration Method  
In order to perform localization in a two-dimensional Cartesian plane, a sensor array consisting of at least three microphones placed at different positions is required. For the three-microphone sensor array shown in Figure 3a, let the planar position of the microphones be denoted by \(\mathbf{p}_{i}=[x_{i},y_{i}]^{T}\), \(i=1,\ 2,\ 3\) and the position of the sound source be denoted by \(\mathbf{p}_{k}=[x_{k},y_{k}]^{T}\). Let us denote the arrival times of sound signals at the microphones as \(\tau_{i}\) and the time difference between the arrivals at two microphones as \(\tau_{ij}=\tau_{i}-\tau_{j}\). In this case, the  
Figure 3: (**a**) Microphone array utilized in the plane and (**b**) microphone array utilized in the Cartesian space. distance between the sound source and the microphones, depending on the speed of sound propagation in the air, is given as follows:  
\[d_{i}=\left\|\mathbf{p_{k}}-\mathbf{p_{i}}\right\|=\tau_{i}\times v_{s} \tag{6}\]  
Therefore, the distance between the \(i\)-th microphone and the source is as follows:  
\[d_{i}=\sqrt{\left(x_{k}-x_{i}\right)^{2}+\left(y_{k}-y_{i}\right)^{2}} \tag{7}\]  
By taking the first microphone as a reference, we can write Equation (6) for the \(j\)-th microphone and the reference microphone, and then take the difference and rearrange as follows:  
\[\tau_{j1}\times v_{s}=\left\|\mathbf{p_{k}}-\mathbf{p_{j}}\right\|-\left\|\mathbf{p_{k}}- \mathbf{p_{1}}\right\| \tag{8}\]  
By squaring both sides of Equation (8), we obtain the following:  
\[\tau_{j1}^{2}\times{v_{s}}^{2}+\left\|\mathbf{p_{k}}-\mathbf{p_{1}}\right\|^{2}+2 \times v_{s}\times\tau_{j1}\times\left\|\mathbf{p_{k}}-\mathbf{p_{1}}\right\|=\left\| \mathbf{p_{k}}-\mathbf{p_{j}}\right\|^{2}\ \ \ j=2,\ 3,\ldots,N \tag{9}\]  
With the expansion of the quadratic expressions formed in Equation (9):  
\[\tau_{j1}^{2}\times{v_{s}}^{2}+\mathbf{p_{k}^{T}}\mathbf{p_{k}}-2\mathbf{p_{1}^{T}}\mathbf{p_{ k}}+\mathbf{p_{1}^{T}}\mathbf{p_{1}}+2\times v_{s}\times\tau_{j1}\times\left\|\mathbf{p_{k}}- \mathbf{p_{1}}\right\|=\mathbf{p_{k}^{T}}\mathbf{p_{k}}-2\mathbf{p_{j}^{T}}\mathbf{p_{k}}+\mathbf{p_{ j}^{T}}\mathbf{p_{j}} \tag{10}\]  
By performing the necessary simplifications in Equation (10), and by moving the unknowns to the left-hand side of the equation and the knowns to the right-hand side, we obtain the following:  
\[2\mathbf{p_{j}^{T}}\mathbf{p_{k}}-2\mathbf{p_{1}^{T}}\mathbf{p_{k}}+2\times v_{s}\times\tau_{ j1}\times\left\|\mathbf{p_{k}}-\mathbf{p_{1}}\right\|=\mathbf{p_{j}^{T}}\mathbf{p_{j}}-\mathbf{p_{1}^{T}} \mathbf{p_{1}}-\tau_{j1}^{2}\times{v_{s}}^{2} \tag{11}\]  
By substituting the distance between the reference microphone and the sound source, \(d_{1}=\left\|\mathbf{p_{k}}-\mathbf{p_{1}}\right\|\), into Equation (11):  
\[2\mathbf{p_{j}^{T}}\mathbf{p_{k}}-2\mathbf{p_{1}^{T}}\mathbf{p_{k}}+2\times v_{s}\times\tau_{ j1}\times d_{1}=\mathbf{p_{j}^{T}}\mathbf{p_{j}}-\mathbf{p_{1}^{T}}\mathbf{p_{1}}-\tau_{j1}^{2} \times{v_{s}}^{2}\ \ j=2,3,\ldots,N \tag{12}\]  
The linear equation between the first microphone, which is taken as the reference, and the \(j\)-th microphone, with \(\mathbf{p_{k}}\) and \(d_{1}\) as unknowns, is defined in Equation (12). For \(N\) microphones, \(N-1\) equations are written using the same reference microphone. For a sensor array shown in Figure 3a, the equations are rearranged as follows:  
\[ \begin{bmatrix}2(\mathbf{p_{2}}-\mathbf{p_{1}})^{T}&2v_{s}\tau_{21}\\ 2(\mathbf{p_{3}}-\mathbf{p_{1}})^{T}&2v_{s}\tau_{31}\end{bmatrix} \times \begin{bmatrix} \mathbf{p_{k}}\\ d_{1}\end{bmatrix} = \begin{bmatrix}\mathbf{p_{j}^{T}}\mathbf{p_{2}}-\mathbf{p_{1}^{T}}\mathbf{ p_{1}}-v_{s}^{2}\tau_{31}^{2}\\ \mathbf{p_{3}^{T}}\mathbf{p_{3}}-\mathbf{p_{1}^{T}}\mathbf{p_{1}}-v_{s}^{2}\tau_{31}^{2}\end{bmatrix}