## 3 Results
### Near Ground Path Loss Model  
The following step is to determine the path loss trend to determine the most suitable modelling. In all the path loss curves it was detected that the path loss modifies its trend at two points, namely critical points: one close to the transmitter (\(d_{C0}\)) and a second one (\(d_{Cf}\)) shortly after the previous. These points derive from the obstruction of the first Fresnel ellipsoid and the plane of the ground and split the propagation distance into three zones:  
* Zone #1: from the transmitter to the first critical distance \(d_{C0}\). In this first stage, the 2-ray propagation becomes dominant; however, the power combination reaching the receiver results in path loss rate equal to free-space for \(H_{2}\), or less restrictive for \(H_{1}\).
* Zone #2: this region is placed between both critical points and remains under obstruction larger than 60%; is the very zone of the near-ground scenario; the path loss increases its rate.
* Zone #3: beyond the second critical point, it is observed a more destructive path loss rate, due to the combination in the receiver antenna of diffracted and scattered rays coming out from the zone #2. Once this last region ends, the path loss remains constant.  
In order to determine the path loss trend, a log-distance fitting was applied to each of the three zones. The resulting path loss model represents the power attenuation, in dB, as a logarithmic function of the distance and the critical points occurrences and meets the expression given in (5)-(7):  
\[P_{L}(d)=10\cdot n_{1}\cdot\log_{10}(d),\quad d\leq\ d_{C0} \tag{5}\]  
\[P_{L}(d)=10\cdot n_{2}\cdot\log_{10}(d),\quad\quad\quad d_{C0}\leq\ d\leq\ d_{Cf} \tag{6}\]  
\[P_{L}(d)=10\cdot n_{3}\cdot\log_{10}(d),\quad d\geq\ d_{Cf} \tag{7}\]  
where the variables \(n_{1}\), \(n_{2}\) and \(n_{3}\) indicate the attenuation factor in dB/m for each propagation region. The values of the attenuation factors are summarized in Table 2. A value of 2 indicates that the attenuation matches the free-space model.  
The critical points \(d_{C0}\) and \(d_{Cf}\) are obtained as per (8):  
\[d_{C0}=\tfrac{H_{1}2^{2}}{p\cdot d}\quad,\quad d_{Cf}=2\cdot d_{C0} \tag{8}\]  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & \(\mathbf{F_{1}}\ =\ \mathbf{868}\ \mathbf{MHz}\) & \(\mathbf{F_{1}}\ =\ \mathbf{2.4\ GHz}\) & \(\mathbf{F_{1}}\ =\ \mathbf{5.8\ GHz}\) \\ \hline \(r_{1}=\sqrt{d\lambda}\ (\mathrm{m})\) & 2.94 & 1.77 & 1.14 \\ \% \(H_{1}/r_{1}\) & 93.2 & 88.7 & 82.4 \\ \% \(H_{2}/r_{2}\) & 86.4 & 77.4 & 64.8 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Obstruction of the First Fresnel Zone (FFZ) in percentage w.r.t antenna height.  
here \(H_{1,2}\) is the height of the transmitter and the receiver antennas, \(\lambda\) is the wavelength of the transmitted wave, and \(p\) is the ratio \(H_{1,2}/r_{1}\).  
The free space path loss \(L_{fs}\) is obtained in dB as per (4):  
\[L_{fs}=32.44\ +\ 20\ \log_{10}\Bigl{(}f\ \times\ 10^{-6}\Bigr{)}\ +\ 20\ \log_{10}\Bigl{(}d\ \times 10^{-3}\Bigr{)} \tag{9}\]  
where \(f\) is the transmitted signal frequency in Hz and \(d\) is the distance between the transmitter and the receiver in meters.  
In Figure 4, the measured results were compared to the proposed three-slope log-distance model resulting in a close approach between experimental and theoretical values. The proposed path loss model also indicates that the power attenuation due to the free-space model is not realistic for a correct prediction.