# 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
The probability density function (PDF) of \(\theta_{a}\) is given by [10]:  
\[f(\theta_{a})=\frac{|\theta-\varphi|}{\sigma_{r_{0}}^{2}+\sigma_{l_{0}}^{2}} exp\left(-\frac{(\theta-\varphi)^{2}}{2\left(\sigma_{r_{0}}^{2}+\sigma_{l_{0}}^{2} \right)}\right) \tag{7}\]  
Figure 1: Diagram of the in-flight unmanned aerial vehicle (UAV) free-space optical (FSO) communication. where \(\sigma_{to}^{2}\) and \(\sigma_{ro}^{2}\) are the variances of the angle deviation of the transmitter and the receiver respectively. \(\theta\) can be expressed as:  
\[\theta=\sqrt{\left(\theta_{tx}+\theta_{rx}\right)^{2}+\left(\theta_{ty}+\theta_{ yy}\right)^{2}} \tag{8}\]  
where \(\theta_{tx}\) and \(\theta_{rx}\) represent the angle deviations of the transmitting end and the receiving end in the \(x\) direction respectively, \(\theta_{ty}\) and \(\theta_{ry}\) represent the angle deviations of the transmitting end and the receiving end in the \(y\) direction, respectively. According to the above discussion, \(h_{a}\) has two possible values. The communication will be interrupted when \(\theta_{a}\) is greater than \(\theta_{FOV}\), i.e., \(h_{a}\)= 0; the communication interruption will not be induced when \(\theta_{a}\) is less than or equal to \(\theta_{FOV}\), i.e., \(h_{a}\) = 1. The value condition of \(h_{a}\) is given by the following equation:  
\[h_{a}=\left\{ \begin{array}{l}0\ \theta>\theta_{FOV}-\varphi\ \text{or}\ \theta_{a}<-(\theta_{FOV}+\varphi)\\ 1\ \theta\leq\theta_{FOV}-\varphi\ \text{and}\ \theta_{a}\geq-(\theta_{FOV}+ \varphi)\end{array} \right. \tag{9}\]  
Therefore, the probability density function (PDF) of \(h_{a}\) is given by:  
\[ \begin{array}{l}f(h_{a})=\left[\frac{1}{2}exp(-\frac{(\theta_{ FOV}-\varphi)^{2}}{2\left(\sigma_{0}^{2}+\sigma_{0}^{2}\right)})+\frac{1}{2}exp(- \frac{(\theta_{FOV}+\varphi)^{2}}{2\left(\sigma_{0}^{2}+\sigma_{0}^{2}\right) })\right]\delta(h_{a})\\ +\Bigg{\{}1-\left[\frac{1}{2}exp(-\frac{(\theta_{FOV}-\varphi)^{2}}{2\left( \sigma_{0}^{2}+\sigma_{0}^{2}\right)})+\frac{1}{2}exp(-\frac{(\theta_{FOV}+ \varphi)^{2}}{2\left(\sigma_{0}^{2}+\sigma_{0}^{2}\right)})\right]\Bigg{\}} \delta(h_{a}-1)\end{array}  \tag{10}\]  
where \(\delta(\cdot)\) is the Dirac function. In the hovering UAV optical communication, the position deviation of the transceiver UAV, and the angle deviation of the transmitter UAV are the main factors that cause the deviation of the center of the spot and the center of the lens. The geometric and pointing error coefficient \(h_{g}\) is defined as the proportion of the light intensity of the coincidental part  
Figure 2: Schematic diagram of the receiving plane at the receiving end UAV. **(a)** 3D schematic diagram; **(b)** \(y\)= plane diagram. of the spot and the lens to the total light intensity. Here we assume that the transmitting UAV is hovering while the receiving UAV is yawing at the same altitude. In this case, the UAV at the receiving end will be affected by the random lateral wind resistance in the horizontal direction, which will cause the fuselage to wobble around the \(y\) axis, thus causing the lens plane to be no longer orthogonal to the centerline of the beam. As shown in Figure 3, the projection of the lens on the x-y plane becomes an ellipse instead of a circle. At this point, the intensity of the part where the lens and spot coincide is different from that in the hovering state. At a certain moment, suppose that the jitter induced by lateral wind resistance causes the lens plane to rotate by an angle \(\alpha\) around the \(y\) axis. The ellipse formed by the projection of the lens on the \(x\)-\(y\) plane can be expressed as:  
\[\xi^{2}x^{2}+y^{2}=a^{2} \tag{11}\]  
where \(\xi=1/\cos\alpha\). The long axis of the ellipse is \(a\), and the light intensity of the Gaussian beam section can be written as:  
\[I(\mathbf{r},Z)=\frac{2}{\pi w_{z}^{2}}exp\left(-\frac{2|\mathbf{r}|^{2}}{w_{z}^{2}}\right) \tag{12}\]  
where \(\mathbf{r}\) is the radial vector with r distance from the center of the light spot and \(w_{z}\) is the beam waist at distance \(Z\).