# 2D Triangulation of Signals Source by Pole-Polar Geometric Models
## 1 Introduction
### Considerations about Signal Data  
In this section we briefly describe about the signal data acquisition and limitations, including the restrictive conditions involved in this acquisition. We try to associate the acquisition of data with the usual methods applied to the spatial location of a point of interest. What locates what? There are applications which needed estimating the location of an emitter device (e.g., an airplane black box). There are applications which a receiver device needs to be located (e.g., a GPS unit). All methods presented in this paper are able to operate in any of these applications. The experimental cases (Section 3) adopt the convention for locating a receiver device. The standard techniques range-based applied to location estimation uses a server to that aim. They perform the localization in two steps: (1) ranging, where the radial distance between two sensors (one with known positions and another with unknown position) is obtained, by some methods such as TOA (Time of Arrival), TDOA (Time Difference of Arrival), RSSI (Received Signal Strength Indicator) or AOA (Angle of Arrival) and (2) location computation, where the unknown location of a device is calculated by estimation through methods such as triangulation using the radial distance or angle data [1]. A Received Signal Strength Indicator (RSSI) is a measure of the power contained in a propagated signal. In an IEEE 802.11 system, RSSI refers to received radio signal strength (power level) in a wireless environment, measured in arbitrary units, that a particular signal is received by a receiver. Therefore, the higher the RSSI value is, the stronger the signal is received, and closer are the emitter and the receiver. RSSI-based techniques require the value of the signal magnitude attenuation to find the radial distance among a set of receiver devices. It is a practical and self-organized method that utilizes more power to send a large influx of data to the central server. Received Signal Strength (RSS)-based fingerprinting approaches have been widely exploited for mobile device localization. The received signal strength is a function of radial distance between emitter and receiver devices. The strength magnitude of a received signal can vary because there are fonts of interferences (noise) in the signal propagation path, especially in outdoor environments [2]. Reference [3] explored signal strength data to feed a Perceptron and Decision Systems based on Learning Vector Quantization to obtain a fingerprint map of that radio strength signal. Reference [4] investigated the effects of hardware orientation, location, measurement time and duration, as well as the interference of the user presence on the RSS acquired data. In the simplest case of free space propagation, electromagnetic waves radiate out of an isotropic antenna in all directions without being perturbed. The received power, any point in space, for such a case is provided by Friis equation [5; 6] given by (1). \[\rho_{r}=\rho_{e}g_{e}g_{r}\bigg{(}\frac{\lambda}{4\pi d}\bigg{)}^{2}, \tag{1}\]  
\(\rho_{r}\) and \(\rho_{e}\) are the receiver and the emitter power  
\(g_{r}\) and \(g_{e}\) are the receiver and emitter antennas gains  
\(d\) is the radial distance between an emitter and a receiver in meters. \(\lambda\) is the wavelength. For IEEE 802.11b and g to 2.4 GHz, \(\lambda\approx\) 0.125 m and, to 5.7 GHz, \(\lambda\approx\) 0.06 m.  
The wave propagation of Electromagnetic Radiation (EMR) in the Earth's free space conditions is compromised primarily by three phenomena [7]: (1) reflection. It happens when EMR interacts with a smooth surface with larger dimension than the wavelength; (2) diffraction, or shadowing. It takes place when the wave is obstructed by objects larger than the wavelength (a secondary wave is created behind the object); and (3) scattering. It occurs when the waves encounter a rough surface whose dimensions are of the same order as the wavelength [6].