# A Service-Constrained Positioning Strategy for an Autonomous Fleet of Airborne Base Stations
## 2 System Overview
### Channel Model  
Several UAV-user channel models have been proposed and analyzed in the literature. The simplest channel model consists in a direct path between the transmitter and the receiver. It is described by the well known free space path loss [1] through Friis' equation [19] that depends only on the distance, the frequency and the gains of the antennas. In the ABBS context, it is commonly expressed by the altitude of the UAV and by the elevation angle between the user and the UAV [2; 3; 4]. Moreover, in [5], a relation between the coverage and the altitude of the UAV is extracted. In [2; 3; 4; 6; 7], more realistic channels models are proposed. Specifically, they include the presence of buildings. This leads to the identification of two different propagation conditions: line-of-sight (LOS) and non-light-of-sight (NLOS). In the first case, a direct communication link between source and destination is possible, whereas, in the latter case, it is not, which leads to additional losses. Each of these cases happens with a given probability of occurrence. In some of these works, the authors also obtained realistic numerical values for the parameters involved in the model. In some of those works, four scenarios are analyzed depending on the distribution of the buildings (suburban, urban, dense urban and high-rise urban) and fading is also taken into account. Finally, another aspect considered in the literature is the movement of the UAVs, which affects the channel coherence time due to the Doppler effect [20]. In this paper, the selected model is the one proposed in [4]. It consists in Friis' equation plus a fading term due to the presence of buildings in a urban environment. According to this, the received power \(P_{r}\) can be expressed as  
\[P_{r}=G_{T}G_{R}\left(\frac{\lambda_{c}}{4\pi d}\right)^{2}\frac{1}{\xi}P_{t}, \tag{2}\]  
where \(P_{t}\) is the transmission power; \(G_{T}\) and \(G_{R}\) are the gains of the transmit and receive antennas, respectively; \(\lambda_{c}\) is the carrier wavelength; \(d\) is the transmitter-receiver distance; and \(\xi\) is the loss term due to fading in linear scale. Among all of these variables, two require deeper analysis: the antenna gains and the fading. Regarding the antenna gains, and for the sake of simplicity of the analysis, the user terminal antenna has been selected as isotropic, although the proposed strategy could be extended easily to any other radiation pattern. According to this assumption, the user antenna gain is constant regardless  
Figure 1: Example of a complete scenario: three ABBSs linked through a wireless connection to a backhaul ground station. the relative position of the mobile terminal and the ABBS. However, for the UAV antenna, and to obtain more realistic results, the following antennas have been evaluated: omnidirectional in azimuth, half-wave dipole [21; 22], Yagi-Uda [21], patch [21] and, finally, a specific antenna used by 3 GPP to simulate macro-cell BSs (annex A.2.1.1.1 of [23]). All these antennas lead to a very similar average rate per user when applied in the ABBS scenario. Therefore, the omnidirectional antenna in azimuth has been selected for the UAV, as it presents the simplest gain expression. Its radiation pattern is:  
\[G_{\rm T}(\theta)=\sin^{2}(\theta)=\frac{h_{i}^{2}}{(x-x_{i})^{2}+(y-y_{i})^{2 }+h_{i}^{2}}, \tag{3}\]  
where \((x,y,0)\) is the position of the user, \((x_{i},y_{i},h_{i})\) is the position of the UAV serving that user (which is denoted by index \(i\)) and \(\theta\) is the elevation angle that can be expressed as  
\[\theta=\tan^{-1}\left(\frac{h_{i}}{\sqrt{(x-x_{i})^{2}+(y-y_{i})^{2}}}\right). \tag{4}\]  
In reference to the fading \(\xi\), as it has been said, two fading terms, \(\xi_{LOS}\) and \(\xi_{NLOS}\), are possible depending on whether we are in LOS or NLOS conditions, respectively.