## 4 Results
### Free Space Path Loss (FSPL) Model and Receiver Sensitivity  
For this section, the LoRa network coverage was estimated using the RSSI predictions of the FSPL model. For this purpose, we determined the distance from the emitter at which the RSSI drops below the receiver's decoding sensitivity. One can obtain more reliable signal path loss-based estimations for real-world applications by considering the detectability threshold approximately 10 dBm above the decoding sensitivity to account for the high packet losses for RSSI values close to the sensitivity limit. In the FSPL model, the attenuation of an electromagnetic signal can be expressed as:  
\[\text{FSPL}= (4\pi fd/c)^{2}, \tag{5}\]  
where \(f\), \(c\), and \(d\) are the signal frequency, the speed of light, and the distance from the emitter, respectively. The receiver sensitivity \(\text{P}_{\text{Rx}}\) quantifies the minimum RSSI that the receiver can decode and can be expressed as  
\[\text{P}_{\text{Rx}}=(\text{S}/\text{N})\text{k}_{0}\text{T}(\text{BW})\text {F}, \tag{6}\]  
where S/N is the signal-to-noise ratio, \(\text{k}_{\text{b}}\) is Boltzmann's constant, T is the temperature in Kelvin, BW is the signal bandwidth, F is the noise factor, and the product \(\text{k}_{0}\times\text{T}\times(\text{BW})\times\text{F}\) corresponds to the noise power. Table 3 summarizes the signal parameters used in this work, whereby \(NF=10\text{log}_{10}(F)\) (dB) is the noise figure of the receiver architecture. Using Equation (6), we estimated the receiver sensitivity in dBm:  
\[\text{P}_{\text{Rx}}=-7.5_{[\text{dB}]}+10\log_{10}\left(1.38\times 10^{-23} \times 290\times 1000\right)_{[\text{dB}]}+10\log_{10}(125,000)_{[\text{dB}]}+6_{[ \text{dB}]}=-124\text{ dBm}, \tag{7}\]  
which is in good agreement with reference values [54] and the product specifications. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{ \text  
Using the FSPL model, we can express the free-space RSSI, P\({}_{\rm FS}\) as  
\[\rm P_{FS}\ [dBm]=G_{Tx}+20\log_{10}\biggl{(}\frac{c}{4\pi fd}\biggr{)}. \tag{8}\]  
Figure 2 shows free-space RSSI expressed in Equation (8) for two different transmitter gains \(G_{Tx}\) of 0 and +20 dBm. By finding the distance at which the RSSI drops below the sensor sensitivity, the maximum theoretical ranges of about 45 and 435 km can be estimated for \(G_{Tx}\) of 0 (1 mW) and +20 dBm (100 mW), respectively. The relatively long coverage range achieved by LoRa networks is one of its main attractive features. However, no matter how impressing the free-space coverage ranges may be, they are hardly realistic in urban environments, where absorption, scattering, and interference effects induced by dense and intricate buildings can severely decrease the coverage range. To simplify the comparison between free-space and cityscape scenarios, we plotted in Figure 3a a 2D representation of the RSSI (P\({}_{\rm FS}\)) decaying from the central position. Figure 3b shows a 2D histogram of combined RSSI and distance occurrences in the theoretical P\({}_{\rm FS}\) map from Figure 3a. As expected, the 2D histogram shows a single line, which is in perfect overlap with \(P_{\rm FS}(G_{Tx}=+20\) dBm). For additional reference, and similarly to Figure 2, we plotted the \(P_{\rm FS}\)(0 dBm) and receiver sensitivity, which shows that the free-space RSSI is far above the sensitivity threshold in these relatively short distances. It is worth noticing that the increase in occurrence rate (color gradient) for distances from 0 to 300 m is due to the larger circumference perimeter associated with longer radii. Furthermore, even though we show distances up to 350 m, the FSPL map is side-edge limited at a distance of 300 m from the center, which explains the decrease of the occurrence rate (dimmer color) for distances larger than 300 m. Nevertheless, the line continues to follow the P\({}_{\rm FS}\) dashed line.