# An Experimental Comparison of Basic Device Localization Systems in Wireless Sensor Networks
## 3 Sensor Localization
### Received Signal Strength Indicator--RSSI  
The term received signal strength indicator, or RSSI, refers to a numerical value that indicates the power of a received signal, thanks to which it is possible to estimate the distance from the station that transmitted it. This parameter is measured in dBm, or decibels-milliwatts, which is the unit of measure used to indicate a power ratio expressed in decibels (dB) with reference to a milliwatt (mW). In Table 1, a table associating typical RSSI values with received signal strengths is shown. The higher the RSSI value, the stronger the received signal.  
It is assumed that the received power follows the polynomial decay model, typical of the Friis' model [15]. This power is, in fact, assumed to depend on the transmitted power, on the path loss constants, and on the distance between the source and the receiver. The Friis equation evaluates the power \(\mathrm{P_{R}}\) received from a transmitter placed at distance \(d\):  
\[P_{R}=\frac{P_{T}G_{T}G_{R}\lambda^{2}}{(4\pi)2 dn} \tag{1}\]  
where the following apply:  
\begin{table}
\begin{tabular}{|c c c|} \hline
**Power (dBm)** & **Power (dBW)** & **Power (Wait)** \\ \hline \(-100\) & \(-130\) & \(0.1\) pW \\ \(-90\) & \(-120\) & \(1\) pW \\ \(-80\) & \(-110\) & \(10\) pW \\ \(-70\) & \(-100\) & \(100\) pW \\ \(-60\) & \(-90\) & \(1\) nW \\ \(-50\) & \(-80\) & \(10\) nW \\ \(-40\) & \(-70\) & \(100\) nW \\ \(-30\) & \(-60\) & \(1\) uW \\ \(-20\) & \(-50\) & \(10\) uW \\ \(-10\) & \(-40\) & \(100\) uW \\ \(1\) & \(-31\) & \(794\) uW \\ \(0\) & \(-30\) & \(1\) W \\ \hline \end{tabular}
\end{table}
Table 1: RSSI values relative to received signal power.  
Figure 3: Angle of Arrival—AoA.  
\(\mathrm{P_{R}}\): transmitted signal strength in watts;  
\(\mathrm{G_{R}}\): receiving antenna gain;  
\(\mathrm{G_{T}}\): gain of the transmitting antenna;  
\(\lambda\): signal wavelength; d: distance in meters;  
\(n\): propagation constant of the signal, also called propagation exponent, dependent on the environment.  
RSSI can be then calculated by converting the power from Watts to dBm:  
\[RSSI[\mathrm{dBm}]=10\mathrm{log}_{10}\,P[\mathrm{mW}] \tag{2}\]  
There is a different expression of RSSI, based on the beacon distance formula:  
\[RSSI=-10n\mathrm{log}\,d+A \tag{3}\]  
where the following apply:  
_A_: The RSSI value (in dBm) at a reference distance, typically 1 m from the beacon. This value is often provided by the beacon manufacturer and represents the signal strength at a known distance.  
RSSI: The current received signal strength indicator value (in dBm) measured by the receiver. This value indicates the strength of the beacon's signal at the receiver's location.  
\(n\): The path-loss exponent, which accounts for the environment's effect on signal propagation. The value of N typically ranges between 2 (representing free space) and 4 (representing indoor environments with obstacles such as walls and furniture).  
Figure 4 shows the ideal trend [20] of the RSSI in the open environment (\(A=-40\) dBm and \(n=2\)). The distance is then  
\[d=\frac{10^{(A-RSSI)}}{10\times n} \tag{4}\]  
With the use of this final formula, the distance between two nodes can be determined given the RSSI value and the parameters A and \(n\).  
Since the RSSI can be used without additional hardware in any devices, many research activities in recent years have focused on the use of this index for localization. Efforts were initially concentrated on the analysis of this indicator and on a better understanding of the causes of its very strong localization variability. Being widely used in the localization domain, this parameter is exploited for the purpose of this work.  
Figure 4: Ideal received power vs. distance.