## 3 Diversity Performance  
Compared to MISO based systems, the MIMO systems have multiple \(N\) transceivers at the UAV. The misalignment is detected at the UAV with \(M\) transmitted beams from the ground and \(N\) positioning sensors. Then, the \(N\) transmitted beams with the pointing error propagate to \(M\) receivers at the ground terminal. Figure 1 shows the footprint of the transmitted beam and the arrangement of the \(M\) receivers at the ground. The pointed direction of the transmitted beam is assumed to aim at the center of the receiver distributions. The channel gain for \(M\times N\) MIMO based FSO transmissions can be expressed as  
\[H=\sum_{m=1}^{M}\sum_{n=1}^{N}H_{mn}=\sum_{m=1}^{M}\sum_{n=1}^{N}A_{0}e^{Z_{mn }-2R_{mn}^{2}/\omega^{2}}, \tag{8}\]  
where \(\omega\) and \(R_{mn}\) are the beam width and the radial displacement of the \(m\)-th transmitted beam at the \(n\)-th receiver due to pointing error [11]. Because the pointing directions of the transmitted beams are aimed at the center of the receiver plane, the center of the beam footprints \(Q_{n}\) are equal to \(Q\), similar to the coordinates shown in Figure 1:  
\[Q_{n}=Q=\begin{bmatrix}X^{\prime}\\ Y^{\prime}\end{bmatrix}\text{ for all }n\in\{1,\ \cdots,N\}. \tag{9}\]  
\(X^{\prime}\) and \(Y^{\prime}\) are the displacements due to the pointing error. Thus, the combined channel gain for multiple transceivers in Equation (8) can be approximated with the random variables \(G\), \(T\), as referred to by [10] for MIMO transmissions:  
\[H=A_{0}e^{-2\hat{d}^{2}/\{4\omega^{2}\sin^{2}{(\pi/M)}\}}e^{G-T}, \tag{10}\]  
where \(G\) is the Gaussian random variable with the mean \(\mu_{G}\) and the variance \(\sigma_{G}^{2}\) to represent the sum of the multiple atmospheric channels with the coordinates of transceivers as described in Appendix A. The beam parameters \(A_{0}\approx 2a_{r}^{2}/\omega^{2}\). \(\omega\) and \(a_{r}\) are the beam width and the aperture size of the receiver. \(T\) is the random variable for the term of the pointing error, \(T=2R_{0}^{2}/\omega^{2}\).  
Figure 1: Beam footprint of the transmitter and receiver arrangement with pointing error.  
For the diversity performance analysis of MIMO based FSO systems, we assumed an intensity modulation and direct detection (IM/DD) system with non-return-to-zero on-off keying (NRZ-OOK) modulation. The outage of the transmitted signal is defined as the received power \(P_{r}\) becoming less than the receiver sensitivity \(P_{rs}\). Then, the outage probability from Equation (10) can be derived from the joint probability density function of \(G\) and \(T\) as referred to by [18]:  
\[p_{out}(P_{r}) =\mathrm{Prob}(P_{r}<P_{rs})=\mathrm{Prob}(H\cdot P_{t}<P_{rs}) \tag{11}\] \[=e^{\gamma\epsilon^{2}\varphi_{C}^{2}/2-\gamma^{2}P_{G}}\times \left(\frac{e^{2\beta^{2}/\left\{4\omega^{2}\sin^{2}\left(\pi/M\right)\right\}} }{A_{0}}\frac{P_{rs}}{P_{r}}\right)\gamma^{2}\]  
where \(P_{t}\) is the transmitted power and \(\gamma\) is the beam parameter with the statistical misalignment model in Equation (7) given by  
\[\gamma=\frac{\omega}{2\sqrt{\sigma_{c}^{2}+\sigma_{s}^{2}}}. \tag{12}\]  
The required power to achieve the target outage probability \(p_{out}^{tg}\) for the system requirements can be derived from Equation (11) as  
\[P_{t}=\frac{P_{rs}}{A_{0}\left(p_{out}^{tg}\right)^{-\omega^{2}/\left\{4( \sigma_{c}^{2}+\sigma_{c}^{2})\right\}}}e^{2\beta^{2}/\left\{4\omega^{2}\sin^ {2}\left(\pi/M\right)\right\}+\omega^{2}\sigma_{G}^{2}/\left\{8(\sigma_{s}^{2 }+\sigma_{c}^{2})\right\}-\mu_{G}}. \tag{13}\]