## 4 Results
### O-FDTD for Spatially Resolved LoRa Signal Propagation Simulation  
The recently developed radiation-matter interaction O-FDTD algorithm [13], as implemented in the WaveBox toolbox for MATLAB, is used to simulate the propagation of LoRa signals in an urban environment, with building arrangements obtained from Google Maps. Here the O-FDTD method simulates the propagation of electric-field waves from an emitter (source) into a space characterized by different relative dielectric permittivity values, representing air \(\varepsilon_{\text{air}}=1\) or concrete buildings walls \(\varepsilon_{\text{wall}}\), respectively. Previous reports have shown many different (even disparate) dielectric permittivity values for concrete [56; 57]. The value considered in this work is derived as follows: We consider the real part of the optical refractive index \(n=2.5\) for concrete [52], and, since many historic buildings in Ourense are stone-walled, knowing that a signal crossing a 60 cm stone wall is attenuated by about \(-30\) dB [58], we calculated the absorption coefficient \(\alpha=0.115\) dB/cm, using [PERSON]'s law  
\[\mathrm{P}/\mathrm{P}_{0}=\mathrm{e}^{-\alpha l}, \tag{9}\]  
where \(\mathrm{P}/\mathrm{P}_{0}\) is the power decay across a length \(l\). The absorption coefficient as a function of the wavelength \(\lambda\) and the imaginary part of the refractive index \(\bar{n}=n+\mathrm{i}\kappa\) is expressed as  
\[\alpha=\frac{4\pi\kappa}{\lambda}. \tag{10}\]  
The dielectric permittivity is simply the square of the refractive index  
\[\epsilon=(n+\mathrm{i}\kappa)^{2}. \tag{11}\]  
Thus, considering a wavelength of \(\lambda=0.345\) m (f = 868 MHz), using Equations (9)-(11) we estimated the dielectric permittivity of buildings walls \(\epsilon_{\mathrm{wall}}=6.15+\mathrm{i}1.58\) (\(n_{\mathrm{wall}}=2.5+\mathrm{i}0.316\)), which agrees with the average reported values [53]. We performed the O-FDTD simulations for a portion of the city-map of Ourense, Spain, specifically, around the university campus, matching the realistic localization of the signal loss experimental characterization. Figure 4 displays the O-FDTD simulated RSSI maps. The emitter is located at the center of the simulation map, corresponding to the experimental position of the LoRa gateway. In the chosen false color map, the grayscale and colored areas identify the simulated RSSI values above and below the calculated \(P_{\mathrm{RX}}=-124\) dBm, respectively. In the close-up map Figure 3(a), interesting signal heterogeneities such as local hotspots and wave-ripples are observed, emerging due to reflections from the buildings. Conversely, Figure 3(b) shows larger-scale effects such as ray-like reflections between buildings. The LoRa signal often appears trapped, displaying wave-guiding between buildings and beam divergence effects at the end of narrow streets. Figure 3: Theoretical free-space Received Signal Strength Indicator (RSSI) distribution of an 868 MHz LoRa signal over distance with a transmitter gain of \(\mathrm{G_{Tx}}=+20\) dBm. (**a**) 2D representation of free-space RSSI in a radius of 300 m. (**b**) 2D histogram of combined RSSI and distance occurrences in the RSSI map shown in (**a**). The color scale indicates the relative occurrence rate, normalized to the number of map pixels. The dashed white lines represent the free-space RSSI (\(\mathrm{P_{FS}}\)) for \(\mathrm{G_{Tx}}\) of 0 and +20 dBm, and the receiver sensitivity \(\mathrm{P_{Rx}}\) of \(-124\) dBm, as indicated by the labels. The vertical dotted line indicates the 300 m spatial observation window used in the RSSI map representation in (**a**). Despite the significant RSSI decrease induced by building walls (about 15 dBm/wall on average), the signal often finds its way around them, either by reflection or diffraction effects, especially in sparsely-built regions. On the other hand, shadow regions of reduced signal strength are observed in densely-built areas.