## 3 Air-to-Air Channel Models
### Theoretical Studies and Simulations for Channel Modeling
Finally, \(X_{\xi}\), stands for the shadow fading, which is lognormally distributed with zero mean and standard deviation \(\sigma_{\xi}\). Similarly, the standard deviation depended on the elevation angle, according to (12). The path-loss exponents and the shadow fading were 1.98 and 2.54, respectively, and 0.9 and 9.43 dB for the LOS and NLOS conditions, respectively. Applying (11) and (12) in (13) for a dense urban environment, the path-loss variation for Tx-Rx distances up to \(5\) km is illustrated in Figure 8 for the LOS and NLOS scenarios, respectively. The Rx height was set between 10 m and 100 m. Higher losses were encountered in the NLOS condition and the shadow fading presented greater dynamic range, as can be observed in Figure 9b. At a distance of 5 km, the link in the NLOS conditions demonstrated a path loss between 130 dB and 150 dB (depending on the Rx elevation), which was 10-25 dB higher than in the LOS case. The flight altitude influenced the path loss less in the LOS condition, in which the Rx was close to the Tx and became more intense at greater Tx-Rx separations (the path loss was almost stable with the Rx flight at short distances, but varied linearly in the range of 5 dB at distances greater than 2 km). Similar tendencies were observed in the NLOS condition, although at greater distances, the path loss was affected more by the Rx height, demonstrating a linear variation in the order of 20 dB at distances greater than 2 km. Another interesting approach is the probability of the LOS component between the two UAVs, which is directly associated with the altitude of the A2A link. Considering a geometrical deployment of buildings, the LOS probability is approximated by  
\[p_{\text{LOS}}(h_{\text{Rx}},\theta)\approx\exp(-\kappa Q(\frac{h_{\text{Rx}}} {\gamma})\cot(\theta)), \tag{14}\]  
where \(\theta\) indicates the elevation angle in degrees between the two UAVs, \(h_{\text{Rx}}\) stands for the height of the Rx, and \(Q(\cdot)\) designates the \(Q\)-function. The parameter \(\kappa=4\gamma\sqrt{2\alpha\beta/\pi}\) denotes the decay factor, which is correlated with the empirical parameters in the ITU-R model [49], which suggests that the probability of a LOS component depends on the building distribution. In the \(k\) parameter, \(\alpha\) indicates the ratio of the land area covered with buildings to the total land area (dimensionless), \(\beta\) denotes the mean number of buildings per unit area (buildings/km\({}^{2}\)), and \(\gamma\) is a variable that regulates the building-height distribution. Taking into consideration Athens' city center (dense urban environment), selecting \(a=0.5\), \(\beta=300\), and \(\gamma=20\)[49], and applying (14), the LOS probability for the specific terrain is indicatively presented in Figure 10. The elevation angle is set in the range of 0\({}^{\circ}\)-90\({}^{\circ}\) and the Rx flight altitude is set between 20 and 200 m.  
Figure 9: Path loss versus distance for various t Rx drone heights: (**a**) LOS condition; (**b**) NLOS condition. The produced path-loss samples included the shadowing factor. Figure 10 indicates that the probability of achieving LOS between the two UAVs increases exponentially with the elevation angle, although it depends on the Rx height relative to the buildings' height distribution. The highest probability is achieved for Rx heights above 80 m, which is about four times the height distribution of the buildings in the area under study (20-21 m, in Athens), and it is also independent of the elevation angle. To achieve, for example, an 80% probability for LOS links, an elevation angle close to 90\({}^{\circ}\) is required for flight altitudes below 60 m. However, to sustain a similar probability for lower elevation angles, the UAV has to fly higher, approaching altitudes of 80 m and above. According to Figure 10, it can be inferred that a safe approach to achieving a robust link between two UAVs in an urban environment is to maintain a flight altitude much higher than the average height of the buildings in the area.