# 2D Triangulation of Signals Source by Pole-Polar Geometric Models
## 1 Introduction
### Standard Methods and Considerations  
#### 1.2.1 RSSI Approach  
The first system used for tracking and locating a mobile device by radio frequency in an indoor environment was RADAR [10]. Previously, [11; 12] used the infrared to locate devices. Nevertheless, there are many limitations in the latter when using this range of the electromagnetic spectrum energy to establish communication among sensors and limited results were obtained. In contrast, the RADAR system operates by processing signal strength data at multiple base stations (AP, Access Point) geometrically positioned to provide overlapping coverage in the area of interest. The system enables estimating the location of an emitter by applying empirical measurements with signal propagation modeling.  
The empirical models proposed by Equations (4) and (5) present significant disparities when confronted by actual data due to signal attenuation promoted by the floor and walls of an indoor environment. Reference [10] observed that sampled data could be filtered to achieve better results. Considering data median, model (5) provides a precision of 4.3 m, compared to 2.94 m for model (4) and 8.16 m for the strongest signal data values. For the 25 th percentile, model (5) provides a precision of 1.86 m, whereas model (4) provides 1.92 m and 4.94 m for the strongest values. To summarize, this evaluation [10] shows that signals can suffer from low and high frequencies interferences (noise). Besides, [10] performed experiments to determine the impact of the data sample size on the result precision. He shows that using two data values by sample, the accuracy worsens by 11%, while using three data values it worsens by 4% (the larger the sample is, the more accurate the result is).  
\[\rho(d)=\rho_{0}-10\mathcal{L}log_{10}\left(\frac{d}{d_{0}}\right)-w(nW,C), \tag{4}\]  
Or  
\[\rho(d)=\rho_{0}-20 log_{10}\left(\frac{4\pi d}{\lambda}\right)+x_{\sigma}, \tag{5}\]  
where,  
\(\rho(d)\) is the signal strength value (dB) expected by an AP located at a radial distance \(d\) from the signal origin.  
\(\rho_{0}\) is the signal strength value (dB) at some reference distance \(d_{0}\).  
\(\mathcal{L}\) indicates the rate at which the path loss increases with distance (empirical value).  
\(w(nW,C)\) is the signal attenuation factor promoted by walls.  
\(\lambda\) is the signal wavelength.  
\(x_{\sigma}\) is a Gaussian noise with zero-mean and variance \(\sigma^{2}\).  
Considering \(w(nW,C)=0\), in free obstructions outdoor environment, and measure \(\rho_{0}\) at \(d_{0}=1\) m from the AP, by (4), the radial distance \(d\) can be computed by (6).  
\[d=10^{\frac{\rho_{0}-\rho(d)}{10C}}. \tag{6}\]The parameter path-loss exponent \(\mathcal{L}\) (6) depends on the frequency of signal and environment interferences (e.g., walls, buildings, cars, rain, air moisture, or people); for open free space, \(\mathcal{L}\cong 2\), and for environments with obstructions, \(\mathcal{L}>2\). A more appropriate \(\mathcal{L}\) value can be determined in the system calibration phase [13]. Experiments carried out by [14] showed that in an urban area, the path loss variation ranges \(2.7\leq\mathcal{L}\leq 3.5\), using a cellular radio.