# 2D Triangulation of Signals Source by Pole-Polar Geometric Models
## 3 Experimental Cases  
### Methodology Applied to Real Data Acquisition  
In order to carry out our experiments, a structure composed of five APs devices to compose the emitters system (AP4, AP5, AP6, AP7, and AP8) is set up in an outdoor environment that covers an area of \(150\) m\({}^{2}\) on the ground (Figure 14). Each AP is positioned at the Cartesian coordinate \((x,y)\) and the respective value (in meters) is registered alongside the respective AP (e.g., the AP5 is at \((9.60,-2.84)\)--Figure 14). These Cartesian coordinate (2D) design the set of C\({}_{k}\) points (centers of range signals circles computed for the _kth_-AP) and defines the geometric arrangement of the covered area on the ground.  
Figure 13: Geometric incoherence. The receiver is placed inside the range of another receiver. The geometric models TLI and MAI are able to overcome this data inconsistency and produce a satisfactory result.  
An AP is the interface between wireless and wired section of the network. Its task is supported by the IEEE 802.11 Network Infrastructures and mainly consists in a translation of the frames from wireless framing format into a wired network framing format like Ethernet. This experiment uses the AP Tp-Link Wireless N 300 Mbps TL-WR849N to compose the system of emitters that emits signals of radio frequency of 2.4 GHz.  
The receiver device is a commercial cell phone operating with the operational system Android. A previously developed application capable of receiving, identifying and measuring the strength of the signal sent by each emitter (APs of interest) was installed in this cell phone (our mobile device).  
To calibrate the mobile device it is placed at a distance of one meter away from each AP to acquire the respective signal strength of reference \(\left(\rho_{0}^{k}\right)\) sent by the \(kth\)-AP.  
Next, a set of coordinates \(E_{p}\), marked and identified as known ground points, is located on the covered region. The mobile device is positioned on each of these marked ground points to receive the signals sent by each AP of interest. In this position, the strength of the signal \(\rho_{k}(d)\) sent by the \(kth\)-AP is measured by the mobile device application.  
By means of Equation (6), the radial signal-range \(d=r_{kp}\) is computed for each \(k\)-AP located at \(C_{k}\) using the signal strength \(\rho_{kp}\) sent by the respective \(k\)-AP and received by the mobile device located at \(E_{p}\) at each ground coordinate. The value \(\mathcal{L}=2.2\) is adopted for the path-loss factor in order to use the Equation (6),  
Thus, we know the position of each emitter \(k\) given by \(C_{k}(x_{k},y_{k})\); the radial signal-range at each AP given by \(r_{kp}\) and; the real receiver position given by \(E_{p}(x_{p},y_{p})\) (Figure 15).  
Now, through the proposed geometrical models, using \(C_{k}(x_{k},y_{k})\) and \(r_{k}\) as input data, the location estimation \(E_{xy}\big{(}E_{x},E_{y}\big{)}\) of the receiver can be computed and the distance between this emitter location approximation \(E_{xy}\) to its known (real) position \(E_{p}\) can be calculated.  
The acquired data in this analysis cases are not preprocessed. Raw data are used to verify the behavior of the proposed geometric models in cases of works with data incoherency.