# 2D Triangulation of Signals Source by Pole-Polar Geometric Models
## 1 Introduction
### Considerations about Signal Data
As the wireless signal radiates out of the emitter antenna, it may interact with reflective surfaces and be reflected in different directions. If a fraction of this reflected signal reaches the receiver antenna, it will acquire signals from the same emitter that traveled by different paths. The reflected signal may be out of phase with the Line-Of-Sight (LOS) signal that can result in destructive interference and fade the receiver signal. The received signal is affected by different environment components (e.g., the own air) that promote the signal attenuation. If the signal does not suffer external interference, it is expected an inverse exponential relationship between the radial distance \(d\) that the emitter device is there from the position of an AP and the value RSSI \(\rho\) of this signal measured in that AP [8]. In the dry air, the signal attenuation \(A\) is given by (2). \[A=20 log_{10}\bigg{(}\frac{4\pi d}{\lambda}\bigg{)} \tag{2}\]  
Therefore, for a signal propagation in an environment consisting of dry air, we can argue that the signal strength \(\rho_{r}\) received by the receiver \(r\) is dependent of the signal strength \(\rho_{e}\) that is produced by the emitter \(e\), of the gains \(g_{r}\) and \(g_{e}\) promoted by sensors antennas and of the attenuation \(A\), (3), promoted by the dry air. \[\rho_{r}=\rho_{e}+\rho_{g}-A \tag{3}\]  
RSSI is used as a metric in most of distance measurement algorithms that run in indoor environments. Nevertheless, at a more basic level the reliability and precision of using RSSI to determine a spatial position in an outdoor environment has not been extensively experimented and there can be interference caused by structures. This monotonous behavior between the radial distance \(d\) and the signal strength \(\rho\) is not always observed. Time of Flight (TOF) are time-based methods for measuring distance using signals data that propagate in a given environment with a known speed \(v\). They consist in measuring the time interval that a wave (e.g., light, radio, or sound) needs to travel a spatial distance. Measuring radio or light wave propagations across small distance requires a clock with subnanosecond time measure; for sound it must be measured by millisecond. TOF-based methods are Time of Arrival (TOA), Time Differential of Arrival (TDOA), and Time of Transfer (TOT). The emitter and the receiver clocks must be perfectly synchronized for using TOA approach. When an emitter sends a signal at the instant \(t_{E}\), the receiver acquire this signal at instant \(t_{R}>t_{E}\), so, the time interval spent for the signal arrival from emitter to receiver, \(\Delta t=t_{R}-t_{E}\), allows computing the radial distance \(d\) that separates the emitter from the receiver by \(d=v\Delta t\). In the 2D case, knowing the emitter distance from two receivers is sufficient to perform the triangulation of emitter location estimation. The TOT method is an unsynchronized-time approach that computes the radial distance between emitter and receivers based on the time delay of a standardized signal \(\xi\) (message) exchanged between the emitter and those receivers. Accuracy of time acquisition is very important for any type of TOF measurements. An EMR wave travels 1 m in 3 ns in air, thus if a locating system with 1m of accuracy is desired, a 300 MHz clock (least) must be used for time acquisition. Clock accuracy depends on clock granularity, which is associated with the size of the smallest time interval measured by a clock. Larger granularity implies a small time interval, then greater precision [9]. A granularity of 20 ppm has an inaccuracy of 1.2 ms per minute, a rubidium clock has an inaccuracy of 0.06 ns per minute, and a caesium clock has an inaccuracy of 6 ps per minute. To sum up, clock precision is a source of error in TOF methods. The sources of errors can be intrinsic (i.e., related to the hardware) or extrinsic (i.e., related to the environment).