# A Study on the Maximum Reliability of Multi-UAV Cooperation Relay Systems
## 2 System Model and Problem Formulation
### Modeling UAV Communication System  
In this paper, we study a multi-hop communication system relating to a tethered multi-rotor UAV low-altitude emergency communication platform, where the UAV acts as the air MBSs and MRs and \(M\geq 1\) UAVs are deployed in a given geographical area within a given limited duration \(T\) to collaboratively serve a group of \(K>1\) GUs at altitude \(h\). Each UAV is equipped with a micro base station to provide communication coverage for GUs within the circle of the maximum coverage radius \(r_{max}\) in Figure 1, where the radius of GUs in polar coordinates is \(r_{\text{G}}\) and the angle is \(\alpha_{\text{G}}\). If the UAV is static, it hovers over the center of the circular coverage area; if the UAV is mobile, it circles over the circular coverage area with radius \(r_{\text{U}}\) and angle \(\alpha_{\text{U}}\), and flies in one direction at its maximum speed \(V\). This multi-UAV collaborative communication system equipped with dual antennas adopts a 2.4 GHz microwave communication frequency band and space-time block coding (STBC). Assuming the use of \(M\)-base modulation, each \(m\) bits first in the binary information bit stream transmitted by the information source are divided into a group, and two consecutive sets of bits are modulated and mapped; that is, each set of bits is mapped to a modulating symbol on the constellation diagram. Therefore, we obtain two modulation signals, \(x_{1}\) and \(x_{2}\), and the method used for encoding is as follows:  
\[X= \begin{bmatrix}x_{1}&x_{2}\\ -x_{2}^{*}&x_{1}^{*}\end{bmatrix} , \tag{7}\]  
where \(X\) is the encoding matrix and \(x_{1}^{*}\) represents the complex conjugate of \(x_{1}\). UAVs and GUs constitute a multi-source multi-destination multi-hop relay, as shown in Figure 1, where it is assumed that \(\text{GU}_{i}\) is the source node or the destination node, \(\text{GU}_{j}\) acts as the destination node or source node according to the communication direction, while \(\text{UAV}_{m}\) and \(\text{UAV}_{m+1}\) act as relays in both directions, and the center of the circular coverage area of \(\text{UAV}_{m}\) is the origin of the coordinate system. The coordinates for the \(\text{GU}_{i}\) and \(\text{GU}_{j}\) are \((r_{i}\cos\alpha_{i},r_{i}\sin\alpha_{i},\ 0)\) and \((D+r_{j}\cos\alpha_{j},\ r_{j}\sin\alpha_{j},\ 0)\), respectively, where \(D\) denoting the distance between the centers of the coverage areas of \(\text{UAV}_{m}\) and \(\text{UAV}_{m+1}\). For a static UAV, the coordinates for the \(\text{UAV}_{m}\) and \(\text{UAV}_{m+1}\) are \((0,\,0,h_{m})\) and \((D,\,0,\ h_{m+1})\), respectively. For a mobile UAV, the coordinates for the \(\text{UAV}_{m}\) and \(\text{UAV}_{m+1}\) are \((r_{m}\cos\alpha_{m},r_{m}\sin\alpha_{m},\ h_{m})\) and \((D+r_{m+1}\cos\alpha_{m+1},r_{m+1}\sin\alpha_{m+1},\ h_{m+1})\), respectively, where \(\alpha_{m}\) and \(\alpha_{m+1}\) is the angle between the UAV and the X axis. Using CC, \(\text{UAV}_{m}\) demodulates and decodes the information sent by \(\text{GU}_{i}\), and then encodes the channel and forwards it to \(\text{UAV}_{m+1}\). \(\text{UAV}_{m+1}\) demodulates and decodes the received information, then encodes the channel and then forwards it to \(\text{GU}_{j}\). The communication link's signal-to-noise ratio (SNR) is as follows:  
\[\gamma_{n}=\frac{P_{\text{TX}}\|g_{n}\|^{2}}{\sigma^{2}Q_{n}}, \tag{8}\]  
where \(g_{n}\) is the channel fading coefficient, \(\sigma^{2}\) is the noise variance, and the power loss \(\mathcal{Q}_{n}\) satisfies \(PL_{n}=10\text{log}_{10}\,\mathcal{Q}_{n}\), while \(n\in\{1,2,3\}\), \(n=1\), \(n=2\), and \(n=3\) correspond to the uplink, AtA, and downlink conditions, respectively.