# 2D Triangulation of Signals Source by Pole-Polar Geometric Models
## 1 Introduction
### Standard Methods and Considerations
#### 1.2.4 How to Solve the System  
The following are some possible solutions for the system of nonlinear equations given in (8). Solution by Least-Squares Method--For this solution [16] proposes a linearization of the non-linear equation system (8). This linearization is achieved when an equation of the system (8) is used to subtract all the others. By performing this subtraction, the system (8) with non-linear equations results in a system of \(m-1\) linear equations. In this case, to solve a 2D location problem, \(m\geq 3\) receivers are necessary. For example, if the equation 1 of (8) is chosen to perform this linearization, then the resulting linear system, written in the form \(\mathbf{M}[x,y]^{t}=\mathbf{b}\), is given by (9). The solution of \(\mathbf{M}[x,y]^{t}=\mathbf{b}\) is given by \(\left[x,y\right]^{t}=M^{-1}b\), if \(m=3\), or by \(\left[x,y\right]^{t}=\left(M^{t}M\right)^{-1}M^{t}b\), if \(m>3\). \[\mathbf{M}=\left[ \begin{array}{ccc}x_{2}-x_{1}&y_{2}-y_{1}\\ x_{3}-x_{1}&y_{3}-y_{1}\\ x_{4}-x_{1}&y_{4}-y_{1}\\ \vdots&\vdots\\ x_{m}-x_{1}&y_{m}-y_{1}\end{array} \right]_{m-1\times 2}\text{ and }\mathbf{b}=\frac{1}{2}\left[ \begin{array}{c}b_{2}-b_{1}\\ b_{3}-b_{1}\\ b_{4}-b_{1}\\ \vdots\\ b_{m}-b_{1}\end{array} \right]_{m-1\times 1} \tag{9}\]  
where \(b_{k}=r_{k}^{2}-\left(x_{k}^{2}+y_{k}^{2}\right)\), for \(k=1,2,3,\ldots,m\). Note that these solutions have any mechanism to handle errors involved with the acquired data. The position solution obtained by (9) was based upon a fundamental assumption that the ranges have no error in them. However, the same approach is valid when the range errors for all the reference points are statistically independent and identical. However, this is not true in real cases. Often, this error is neither independent nor identical. Under such conditions, the least squares position estimate does not hold good as an optimal estimate. If the range errors are Gaussian and the covariance of ranging errors is given by the matrix \(\Sigma\), then the optimal estimation for location is given by the Weighted Least Squares (WLSm) provide by \(\left[x,y\right]^{t}=\left(M^{t}\Sigma^{-1}M\right)^{-1}\left(M^{t}\Sigma^{-1 }b\right)\)[7; 16; 17]. Solution by Numerical Method--For this specific case, the Jacobian matrix \(\mathbf{J}\left(\ref{eq:J}\right)\) is the matrix of first order partial derivatives of \(\mathbf{F}(x,y)\), such that, \(\mathbf{J}(\mathbf{F}(x,y))\) is given by \[J(F(x,y))=\left[ \begin{array}{cc}\frac{\partial f_{0}}{\partial x}&\frac{ \partial f_{0}}{\partial y}\\ \frac{\partial f_{1}}{\partial x}&\frac{\partial f_{1}}{\partial y}\\ \vdots&\vdots\\ \frac{\partial f_{m-1}}{\partial x}&\frac{\partial f_{m-1}}{\partial y}\\ \end{array} \right],\]  
applying in (8)  
\[J(F(x,y))=2\left[ \begin{array}{cc}x-x_{0}&y-y_{0}\\ x-x_{1}&y-y_{1}\\ \vdots&\vdots\\ x-x_{m-1}&y-y_{m-1}\\ \end{array} \right] \tag{10}\]  
We look for \(F(x,y)=0\). For \(m=2\), an interactive solution for \(F(x,y)=0\) is found by [PERSON]'s method [PERSON] (1958), by way (11)  
\[\left[ \begin{array}{c}x_{k+1}\\ y_{k+1}\\ \end{array} \right]_{2\times 1}=\left[ \begin{array}{c}x_{k}\\ y_{k}\\ \end{array} \right]_{2\times 1}-\left[J(x_{k},y_{k})\right]_{2\times 1}^{-1} \left[F(x_{k},y_{k})\right]_{2\times 1} \tag{11}\]  
where \(k=0,1,2,\ldots\) is \(k^{th}\) iteration, \(\left[F(x_{k},y_{k})\right]_{m\times 1}\) is vector function, \(\left[J(x_{k},y_{k})\right]_{2\times m}^{-1}\) is the inverse of the Jacobian matrix, and \((x_{0},y_{0})\) is the first estimative for the solution. Such solution is found when \(\left[x_{k+1},y_{k+1}\right]^{t}\leq\left[\varepsilon,\varepsilon\right]^{t}\), and \(\varepsilon\) stands for the precision value specified a priori. If \(m>2\) and \(\mathbf{X}_{k}=\left[x_{k},y_{k}\right]^{t}\), the system (8) can be solved by (12).