## 3 Air-to-Air Channel Models
### Theoretical Studies and Simulations for Channel Modeling
a direct influence when only single-bounce reflections were considered. In these cases, the wide-sense stationarity (WSS) of the channel cannot be easily preserved. Examining the time dispersion of the channel in terms of the frequency-correlation function, it was observed that no impact was induced by either movement direction (vertical or horizontal). On the other hand, the Doppler spectrum exhibited variations, which were affected by the UAV direction. When a UAV moved vertically, a sharper Doppler spectrum was created, whereas the Doppler shape widened if the horizontal movement was selected. In the second scenario, in which arbitrary movement was applied, different characteristics were observed in the time-correlation functions. These exhibited significant changes in both time and delay. The channel stationarity varied between 1 ms and 4 ms, indicating that when the random motion increased, the channel became non-stationary much more rapidly. Comparing the movement directions, both exhibited similar time correlations in the early period, but after a certain time period, the horizontal movement resulted in a correlation with faster time, and after a few more seconds, the vertical movement exhibited fast non-stationarity. This clearly indicated that the channel stationarity is different in variable time instants. Furthermore, the frequency-correlation function still preserved similar characteristics in the second scenario, demonstrating that the arbitrary UAV speed compromises WSS only in the time domain. Finally, assessing the Doppler spectrum in different time instants of the UAVs mobility, it was found that in the early period, the spectrum demonstrated similar characteristics in both the vertical and the horizontal direction of movement. However, at later time points, the form of the spectrum changed significantly (becoming wider), again suggesting that arbitrary mobility reduces the stationarity of the A2A channel. In general, movement directions do not have a considerable impact on the channel correlation or the Doppler spectrum when significant random mobility is introduced to the wireless link. The non-stationary characteristics of an A2A channel exploiting a GBSM model were also investigated in [57]. The specific model involved two UAVs that moved arbitrarily in a 3D space, incorporating two antenna elements in both the Tx and the Rx. The geometrical model comprised a LOS component, as well as scattering paths that arrived in clusters. These clusters underwent to an evolution process. Therefore, the non-stationarity of the channel was characterized by a birth-death process in those clusters, and a discrete Markov process was exploited for this purpose. The simulation was carried out at 2, 2.5, and 5.8 GHz, with an initial UAV separation of about 1 km. The velocity was set at 30 m/s for both the Tx and Rx, whereas 50 rays were assumed in each cluster. With an initial number of 10 clusters, it was observed that the random movement of both UAVs resulted in a fast cluster transition, the majority of which exhibited limited active duration. It was found that both the temporal and the spatial correlation functions decreased (from 20 ms down to 8 ms, and from 11 cm down to 4 cm, for 0.5 correlation levels) when the carrier frequency increased. The UAV antenna arrays accounted for this observation, since higher correlations were achieved when using lower carrier frequencies. Furthermore, the terminals' motion pattern was found to substantially affect the channel stationarity. When the UAVs were close to each other, the time correlation was smoother, resulting in a higher correlation and stationarity time (of more than 25 ms for a correlation level of 0.5), which could reduce the channel estimation costs. On the other hand, the channel exhibited quick non-stationary characteristics (lower than 2.5 ms) if the UAVs travelled in different directions. Finally, the stationarity interval was calculated for various Tx speeds in the range of 15-45 m/s.