# Performance of UAV-to-Ground FSO Communications with APD and Pointing Errors
## 2 System and Channel Modeling  
### System Modeling  
We consider a typical UAV-based FSO communication system with applications in the surveillance and monitoring of specific working environments, such as early natural disaster and environmental pollution detection: that of the water environment survey in estuarine and coastal environments, as depicted in Figure 1. The hovering UAV/drone navigates thanks to GNSS satellites for its positioning. The UAV communicates with the ground IoT sensor nodes via multiple-access RF when flying over cluster 1 and cluster 2 of the wireless sensor networks. The head node communicates with ground IoT nodes using any wireless connectivity, including short-range (e.g., Bluetooth, \(Z\)-wave), medium-range (e.g., Wi-Fi, ZigBee), and long-range (e.g., LoRa, NB-IoT) means. The UAV collects various data from ground surfaces such as environmental air and water data, captured images or videos recorded from monitoring cameras. In such scenarios, the UAV acts as a mobile gateway in the IoT network. We also consider an FSO transmitter (Tx) mounted on the UAV/drone. The FSO Tx uses the SI-QAM to compensate for the limitations of the OOK and pulse-position modulation (PPM) schemes. Firstly, data collected from ground nodes will be used for electrical modulation to generate the electrical signal. The SIM modulates each block of data bits including \(M_{I}\)-PAM symbols and \(M_{Q}\)-PAM symbols using the subcarrier \(f_{c}\) in both in-phase (\(M_{I}\)) amplitude and quadrature amplitude (\(M_{Q}\)) in the signal space. Then, the modulated signal is employed to modulate the intensity of the laser before transmitting toward to the Rx. Consequently, the transmitted optical signal can be formulated as [13]  
\[s(t)=P_{t}\big{\{}1+m\big{(}s_{I}(t)\cos(2\pi f_{c}t)-s_{Q}(t)\sin(2\pi f_{c}t) \big{)}\big{\}}, \tag{1}\]  
where \(P_{t}\) and \(m\) are the average transmitted power per symbol and the modulation index, respectively. At the FSO Rx side, the received optical signal at the APD receiver can be obtained as [13]  
\[r(t)=X(t)P_{t}\big{\{}1+m\big{(}s_{I}(t)\cos(2\pi f_{c}t)-s_{Q}(t)\sin(2\pi f_{ c}t)\big{)}\big{\}}, \tag{2}\]  
where \(X(t)\) denotes the instantaneous turbulence channel of the FSO link that is the stationary random process of the optical signal scintillation, link loss, position and orientation fluctuation due to the hovering UAV. In Equation (2), the filter can be used for filtering out the component of \(\big{\{}X(t)P_{t}\big{\}}\). Consequently, the electrical signal received at the APD receiver output can be written as  
\[r(t) =G\Re X(t)P_{t}\{1+m(s_{I}(t)\cos(2\pi f_{c}t)-\] \[-s_{Q}(t)\sin(2\pi f_{c}t))\}+\underbrace{n_{I}(t)+n_{Q}(t)}_{n( t)}, \tag{3}\]  
where \(G\) and \(\Re\) denote the average gain of the APD and responsivity (also called the quantum efficiency), respectively. \(n(t)\) is the total noise at the output of the APD including the in-phase \(n_{I}(t)\) and quadrature \(n_{Q}(t)\). They can be modeled as zero-mean Gaussian distribution with the assumption that the variance is the same. They can be expressed as  
\[\sigma_{n}^{2}=\sigma_{s}^{2}+\sigma_{th}^{2}+\sigma_{bg}^{2}, \tag{4}\]  
Figure 1: A proposed UAV-based free-space optical communication system. where \(\sigma_{s}^{2}\), \(\sigma_{th}^{2}\), and \(\sigma_{sh}^{2}\) are the noise variance of the shot noise due to the desired received signal, the noise variance of thermal noise, and the noise variance due to undesired background radiation, respectively. Particularly, the \(\sigma_{s}^{2}\) depends on the average APD gain \(G\), the excess noise factor \(F\), the electrical bandwidth of the APD \(B_{e}\), and the average transmitted power \(P_{t}\). \(\sigma_{s}^{2}\) can be given by [22]  
\[\sigma_{s}^{2}=2 eGFB_{e}\eta P_{t}. \tag{5}\]  
In Equation (5), \(\eta=\frac{eG\Re}{h_{Planck}\
u}\), where \(h_{Planck}\), \(\
u=c/\lambda\), \(c\), and \(e\) denote the Planck constant, the optical frequency, the speed of light and the electron charge, respectively.