## 4 Results
### Analysis of Simulation Versus Experimental Results  
Figure 9a shows the overlay of the experimental coverage area onto the O-FDTD simulation map. In Figure 9b, a color classification represents the match between simulated and experimental coverage regions. Matching regions are yellow, exclusively simulated are red, and exclusively experimental are green. The interpolation algorithm described above ignores the presence of buildings, which leads to inaccurate building edge descriptions. For a more accurate comparison, an adjusted/improved experimental coverage area was obtained by subtracting the building prints from the original experimental signal coverage map--see the difference between Figure 9a,b. To compare the coverage prediction with the experimental result, we calculated the total area overlap \(\eta\) as the ratio between the  
Figure 7: Mapping of time gap between packet arrivals around the LoRa gateway receiver. (**a**) Time gap between packet arrivals measured along the trajectory used to determine the edge of the signal coverage area. (**b**) Interpolated map of time gap between packet arrivals, obtained by expanding the results in (**a**) in all directions. The displayed city maps were retrieved from Google Maps [51]. overlapping area \(C_{overlap}\) and the sum of the simulation \(C_{sim}\) and experimental \(C_{exp}\) spatial coverage areas as follows:  
\[\eta_{area}=\frac{2\sum C_{overlap}}{\sum C_{sim}+\sum C_{exp}}. \tag{12}\]  
A total area overlap \(\eta_{area}=84\%\) can be identified. For the statistical analysis of the signal coverage as a function of the gateway distance, we defined the histogram \(H(r)\) of distance values within the coverage map \(C\) as  
\[H(r)=\text{Hist}(C,r)=\sum_{x,y}R(r)\ \odot\ C, \tag{13}\]  
where Hist(_A_,_r_) is the histogram of r values within a matrix \(A\), \(R(r)\) is the binary map of the circumference with radius across the \(xy\) space and \(\odot\) is the element-wise (Hadamard) product. Then, we defined the radial coverage \(H_{\text{r}}(r)\) as the coverage fraction over each circumference of radius \(r\) by normalizing \(H(r)\) by the perimeter of the rim, \(\sum_{x,y}R(r)\)  
\[H_{R}(r)=\frac{\sum_{x,y}R(r)\odot C}{\sum_{x,y}R(r)}=\frac{H(r)}{\sum_{x,y}R( r)} \tag{14}\]  
Figure 8: Expansion of experimental time gaps between packet arrivals measured along a line trajectory around the LoRa gateway into a map of time gap between packet arrivals, using an iterative flood-like interpolation algorithm. The four subplots show the results after 0 (starting point), 20, 40, and 80 iterations. The dashed line indicates the edge of the coverage area, defined as the interface where the time gap between packet arrivals equals 20 s.  
The distance histogram \(H(r)\) and the circular coverage \(H_{\text{R}}(r)\) are plotted in Figure 9c,d, for the experimental (green) and the simulated (red) coverage maps. The red and green shades identify the differences between the profiles and are directly related to the mismatching areas identified in Figure 9b. An average of 95 m of radial coverage range is obtained, which agrees with the transition point in Figure 5b, where the mean RSSI drops below the receiver sensitivity. A maximum LoRa signal coverage distance of about 340 m can be obtained from both approaches. We analyzed the agreement between simulations and experiments at the edge of the determined signal coverage areas in more detail, see Figure 10. Figure 10a represents the differences between the experimental (green) and O-FDTD simulations (red). We aimed to compare with a simplified path-loss model, for which we allowed the coverage radius to fit the experimental values optimally (see Figure 10b). Projecting radius-dependent deviations over the detection angle \(\Delta r(\theta)\), the Root Mean Squared Deviation (RMSD) can be defined as  
\[\text{RMSD}=\sqrt{\frac{\Sigma(\Delta r(\theta))^{2}}{N}} \tag{15}\]  
Figure 9: Correlation between simulation and experimental results.