# 2D Triangulation of Signals Source by Pole-Polar Geometric Models
## 1 Introduction
### Standard Methods and Considerations
#### 1.2.4 How to Solve the System
\[\mathbf{X}_{k+1}=\mathbf{X}_{k}-F(\mathbf{X}_{k})\Big{(}[J(\mathbf{X}_{k})]^{t}[J(\mathbf{X}_{k})] \Big{)}^{-1}[J(\mathbf{X}_{k})]^{t} \tag{12}\]  
Algebraic Solution--An algebraic 3D solution to locate a receiver ([PERSON] and [PERSON], 2000) aimed to locate a receiver using TDOA measurements and four satellites emitters) was found out by [PERSON] (2000), [PERSON]'s method, which reduces the problem to solve a quadratic equation and yields the 3D Cartesian coordinate of the receiver as well as the common time of signal transmission. It also performs several algebraic manipulations to reduce the equations to a least-squares problem. The presented solution (13) considers that  
\[\widetilde{r}_{i}^{2}=(r_{i}-\xi_{i})^{2}=\|p_{i}-\rho\|^{2}=(x-x_{i})^{2}+(y -y_{i})^{2}+(z-z_{i})^{2}, \tag{13}\]  
where, \(r_{i}\) and \(\widetilde{r}_{i}\) are the observed and actual radial distances (pseudorange and actual range, respectively); \(\xi_{i}\) is the measurement noise at the receiver corresponding to the measurement between the \(i^{th}\) satellite and the receiver, respectively and \(\rho\) is the receiver position \((x,y,z)\) to determine. A generally acceptable modeling of the ranging error \(\xi_{i}\) was described in [PERSON] and [PERSON] (2000). Thus, using the receiver clock bias \(-b=\xi_{i}\) as the only noise measurement and applying some algebra the following equation is obtained  
\[\left(x_{i}^{2}+y_{i}^{2}+z_{i}^{2}-r_{i}^{2}\right)-2(x_{i}x+y_{i}y+z_{i}z-r_ {i}b)+\left(x^{2}+y^{2}+z^{2}-r^{2}\right)=0.\]  
Let \(\rho=[xyz]^{t}\) denotes the receiver position vector and \(p_{i}=\left[x_{i}y_{i}z_{i}r_{i}\right]^{t}\) denotes the satellite position and range vectors. According to Lorentz inner product for 4-space (given by \(\langle\overrightarrow{u},\overrightarrow{v}\rangle=u_{1}v_{1}+u_{2}v_{2}+u_ {3}v_{3}-u_{4}v_{4}\)), can be rewritten as  
\[\frac{1}{2}\langle p_{i},p_{i}\rangle-\langle p_{i},\rho\rangle+\frac{1}{2} \langle\rho,\rho\rangle=0.\]  
Now, the solution can be found utilizing least-squares estimation to the organized satellites equations in the matrix \(\mathbf{B}_{m\times 4}=\left[x_{k}y_{k}z_{k}-r_{k}\right],k=1\ldots m\) and the vector \(\mathbf{a}_{m\times 1}=\frac{1}{2}[\langle p_{k},p_{k}\rangle]\). The final solution demands a consistency analysis of the partial results found. Analytical Solution--When two circles with centers at \((x_{1},y_{1})\) and \((x_{2},y_{2})\) and radius \(r_{1}\) and \(r_{2}\) intersect at one or two points \(P\big{(}x_{p},y_{p}\big{)}\) and \(Q\big{(}x_{q},y_{q}\big{)}\). This intersection do not always occurs due to noise and underestimation of ranges [21]. The analytical solution \((x,y)\) is found at the median of the straight line segment \(s\) that connects \(P\) and \(Q\) points (Figure 1a), given by (14)  
\[(x,y)=\left(\frac{x_{p}+x_{q}}{2},\frac{y_{p}+y_{q}}{2}\right), \tag{14}\]  
where  
\(x_{p}=\frac{-E+G}{H}\); \(x_{q}=\frac{-E-G}{H}\); \(y_{p}=\frac{A-Bx_{p}}{\text{C}}\); \(y_{q}=\frac{A-Bx_{q}}{\text{C}}\);  
\(A={r_{1}}^{2}-{r_{2}}^{2}-{y_{1}}^{2}+{x_{2}}^{2}+{y_{2}}^{2}\); \(B=-2(x_{1}-x_{2})\); \(C=-2(y_{1}-y_{2})\); \(L=\frac{p}{\text{C}}\); \(D=1+L^{2}\);  
\(E=2(L[1-A]-x_{1})\); \(F=L(L-2y_{1})+{x_{1}}^{2}+{y_{1}}^{2}-{r_{1}}^{2}\); \(G=\sqrt{E^{2}-4 DF}\); \(H=2L\). For a normal equation system another analytical/algebraic solution for (8) can be obtained performing the linearization by row (equations) subtraction and then applying the Gauss-Jordan elimination method [18]. Despite the method chosen to solve this problem (analytical, algebraic, least-squares, numerical or NLLS), there are invariable errors associated with the acquired data. These errors are propagated by computational arithmetical operators, mainly multiplication and division. It is necessary to minimize the number of arithmetic operations involved in the solution in order to improve the accuracy of the results and, therefore, reduce the computational cost to provide facilities for designing of real time systems.