## 3 Air-to-Air Channel Models
### Theoretical Studies and Simulations for Channel Modeling  
Path-loss predictions for A2A links, exploiting the performance of machine-learning algorithms, were investigated in [47]. Simulations were carried out at 2.4 GHz, with Tx and Rx UAVs flying on predetermined routes over an urban environment. The Tx heights were set as 60, 70, and 80 m, whereas the Rx flew between 10 m and 40 m in steps of 10 m. Random forest (RF) and k-nearest neighbor (kNN) algorithms were employed to predict the path loss between the flying UAVs. When applying RF (using a tree-depth of three and one hundred and forty ensemble trees), the root-mean-square error (RMSE) between the simulated and the predicted path loss was 3.1 dB. On the other hand, the kNN (selecting k = 5 nearest neighbors) demonstrated a deteriorated RMSE, reaching up to 8.9 dB. However, both machine-learning methods outperformed the COST231 Walfish-Ikegami (28.5 dB) and Stanford University Interim (SUI), (13.4 dB) empirical models. Another significant finding was that the most important input feature that affected the predictability of the path loss was the path visibility (i.e., LOS or NLOS condition), with the minimum contribution observed for the Rx altitude. The propagation characteristics of the A2A channel for urban and dense environments were studied in [48]. Full 3D ray-tracing simulations were conducted at 800 MHz and 2.4 GHz, in an area of 1.5 km \(\times\) 1.5 km, transmitting a 100-MHz signal with a power of 0 dBm. The Tx UAV was set at 300 m above the ground and different Rx altitudes were studied in the range of 2-40 m. During the simulations, the direct, the reflected (up to 10 reflections), and the diffracted components were considered. The large-scale fading of the simulated channel was expressed in excess loss, given by  
\[PL_{\xi}(d,h_{\mathrm{Tx}},h_{\mathrm{Rx}})=20\log_{10}(\frac{4\pi d}{\lambda} )+n_{\xi}, \tag{10}\]  
where \(d\) is the Tx-Rx separation, \(h_{\mathrm{Tx}}\) and \(h_{\mathrm{Rx}}\) denote the flight altitudes of the Tx and Rx UAVs, respectively, and \(\lambda\) stands for the transmitted wavelength. In the following, the term \(\xi\) indicates whether there is LOS or NLOS in the propagation channel. The excess loss is normally distributed \(n_{\xi}\)-\(N(\mu_{\xi}\sigma_{\xi})\), with a mean value \(\mu_{\xi}\) and a standard deviation \(\sigma_{\xi}\), which are both were expressed as a function of the elevation angle and the UAVs' altitude for either the LOS or the NLOS condition. The elevation angle between the Tx and the Rx is \(\theta=\sin^{-1}((h_{\mathrm{Tx}}-h_{\mathrm{Rx}})/d)\). It was found that \(\mu_{\xi}\) increases exponentially with the Rx altitude, according to  
\[\mu_{\xi}=a_{\xi}\exp(b_{\xi}h_{\mathrm{Rx}}), \tag{11}\]  
which stands for both the LOS and the NLOS condition, whereas the constant parameters were found to fit the data. On the other hand, the standard deviation was provided by  
\[\sigma_{\mathrm{LOS}}=a_{\mathrm{LOS}}\theta+b_{\mathrm{LOS}} \tag{12}\] \[\sigma_{\mathrm{NLOS}}=a_{\mathrm{NLOS}}(\theta-b_{\mathrm{LOS} })^{2}+c_{\mathrm{NLOS}}\,\]  
where \(a\), \(b\), and \(c\) were yielded from fitting the simulated data and can be found in [48]. Equation (12) indicates that the standard deviation varied in a manner that was inversely proportional to the elevation angle. At low elevation angles, the wireless link between the UAVs was more likely to encounter shadow fading from the surrounding buildings, thus increasing the \(\sigma_{\xi}\). Furthermore, the CI model was found to approximate the simulated path loss well. \[PL_{\xi}(d,h_{\mathrm{Tx}},h_{\mathrm{Rx}})=20\log_{10}(\frac{4\pi d_{0}}{ \lambda})+10n_{\xi}\log_{10}(\frac{d}{d_{0}})+X_{\xi}, \tag{13}\]  
where \(d_{0}\) denotes the reference distance (usually set as 1 m) and \(n_{\xi}\) indicates the path-loss exponent. The latter was found to be related to the Rx height, with the same relationship as (11).