# Modeling and Spatial Diversity-Based Receiving Improvement of In-Flight UAV FSO Communication Links
## 2 Theoretical Model of the In-Flight UAV FSO Communication Link  
When the UAV at the receiving end is flying horizontally, the arrival angle between the centerline of the beam and the receiving plane is not only related to the pointing deviation caused by the jitter of the receiving end, but also needs to consider the tilt angle of the receiving plane. In a wireless optical communication link, the received signal at the receiving end can be expressed as [11]:  
\[y=hx+n \tag{1}\]  
where \(x\) is the signal emitted at the transmitting end and \(n\) is the additive white Gaussian noise (AWGN). \(h\) is the channel coefficient that can be given by the following equation:  
\[h=h_{t}h_{a}h_{g} \tag{2}\]  
In Equation (2), \(h_{t}\) is the attenuation and turbulence coefficient of the atmospheric channel; \(h_{a}\) represents the link interrupt coefficient, which indicates whether the spot is detected on the detection plane; and \(h_{g}\) is the geometric and pointing error coefficient, which represents the position deviation between the spot center and the lens center at the receiving end. As for the attenuation of the optical signal by the influence of atmospheric turbulence, the distribution of \(h_{t}\) can be described by the Gamma-Gamma model [16]:  
\[f(h_{t})=\frac{2(\alpha\beta)^{\frac{(\alpha+\beta)}{2}}}{\Gamma(\alpha)\Gamma (\beta)}h_{t}!^{\frac{(\alpha+\beta)}{2}|-1}K_{\alpha-\beta}\Big{(}2\sqrt{ \alpha\beta h_{t}}\Big{)} \tag{3}\]  
where \(K_{n}(\cdot)\) is the \(n\)-th order modified Bessel function of the second kind, \(\Gamma(\cdot)\) is the Gamma function, \(\alpha\) and \(\beta\) represent the effective number of the large-scale and small-scale eddies, respectively:\[\alpha=\left\{exp\left[\frac{0.49\sigma_{R}^{2}}{\left(1+1.11\sigma_{R}^{\frac{12}{ 6}}\right)^{\frac{2}{6}}}\right]-1\right\}^{-1} \tag{4}\]  
\[\beta=\left\{exp\left[\frac{0.51\sigma_{R}^{2}}{\left(1+0.69\sigma_{R}^{\frac{ 12}{6}}\right)^{\frac{2}{6}}}\right]-1\right\}^{-1} \tag{5}\]  
where the Rytov variance is expressed as:  
\[\sigma_{R}^{2}=1.23k^{\frac{2}{6}}C_{n}^{2}L^{\frac{11}{6}} \tag{6}\]  
In Equation (3), \(k\) is the wave number, \(C_{n}^{2}\) is the atmospheric refractive index structural parameter, and \(L\) is the propagation distance. In this work, the phase screen method is used to simulate the atmospheric turbulence in the Gamma-Gamma model [17; 18; 19]. When a multi-rotor UAV is flying in a straight and horizontal flight, a small tilt will exist between the fuselage and the \(y\) axis, as shown in Figure 1. The size of the tilt angle is related to the speed of the UAV, which affects the link performance through the coefficient \(h_{a}\). As shown in Figure 2a, assuming the UAV at the receiving end is flying horizontally, the included angle between the central axis and the \(y\) axis is denoted as \(\varphi\). Since the plane of the lens at the receiving end is parallel to the central axis of the UAV, the angle between the plane of the lens and the \(y\) axis is also \(\varphi\). The angle between the centerline of the beam and the perpendicular direction of the lens plane is defined as the arrival angle \(\theta_{a}\), and the pointing deviation caused by the jitter inherent in the receiving and transmitting end is defined as \(\theta\). Here, the UAV is assumed to fly at the same altitude, and it is reasonable to choose a case without loss of generality to describe the relationship between \(\theta_{a}\) and \(\varphi\). As shown in Figure 2b, the direction of the light beam is parallel to the \(y\)-\(z\) plane, and the angle between the beam axis and the \(z\) axis is \(\theta\). According to the diffraction theory, the communication link will be interrupted when the main lobe of the Airy disk is outside of the detection region of the detector. In the state of horizontal flight, \(\theta_{a}\) can be expressed as \(\theta_{a}=\theta-\varphi\). Therefore, the link will be interrupted when \(\theta_{a}\) is larger than the receiver's field angle \(\theta_{FOV}\).