## 2 System and Channel Models
### Average Bit Error Rate  
For UAV-assisted hybrid systems, the binary modulation scheme is used in FSO or RF transmission. Therefore, according to [16; 39], the average BER can be mathematically expressed as:  
\[P_{b}=\frac{q^{p}}{2\Gamma(p)}\int_{0}^{\infty}\left(\gamma\right)^{p-1}\exp( -q\gamma)F_{\gamma}(\gamma)d\gamma, \tag{16}\]  
where \(p\) and \(q\) are parameters used to describe different binary modulation schemes, as shown in Table 2.  
By substituting Equation (15) into Equation (16), the average BER of our proposed hybrid scheme without PE can be obtained:  
\[P_{b}^{RA} =\frac{q^{p}}{2\Gamma(p)}\int_{0}^{\infty}\left(\gamma\right)^{p- 1}\exp(-q\gamma)\frac{1}{\Gamma(m_{s,d})}G_{1,2}^{1,1}\left(\frac{m_{s,d} \gamma}{\overline{r}_{s,d}^{\text{SC}}}\left| \begin{array}{c}1\\ m_{s,d},0\end{array} \right.\right)\left\{1-\exp\left[-\left(\frac{1}{\overline{n }_{s,d}}\sqrt{\frac{\gamma}{\overline{r}_{s,d}^{\text{SC}}_{d}}}\right)^{ \beta_{s,d}}\right]\right\}^{\alpha_{s,d}} \tag{17}\] \[\times\left\{1-\left\{1-\frac{1}{\Gamma(m_{ij})}G_{1,2}^{1,1} \left(\frac{m_{s,j}\gamma}{\overline{r}_{s,d}^{\text{SC}}}\left| \begin{array} []{c}1\\ m_{i,j},0\end{array} \right.\right)\left\{1-\exp\left[-\left(\frac{1}{\overline{n }_{ij}}\sqrt{\frac{\gamma}{\overline{r}_{s,d}^{\text{SC}}_{d}}}\right)^{\beta_ {i,j}}\right]\right\}\right\}^{\alpha_{s,j}}\right\}^{M}\right\}^{N}d\gamma.\]  
According to Equation (3.6.1) in [40], we have the approximate solution for the Generalized Gaussian-Laguerre quadrature function: \(\int_{a}^{\infty}\left(x-a\right)^{c}\exp(-b(x-a))f(x)dx\approx\sum\limits_{ \tau=1}^{n}w_{\tau}f(x_{\tau})\), where \(w_{\tau}\) and \(x_{\tau}\) are the weight and a special point called the abscissa, respectively. Note that \(w_{\tau}\) and \(x_{\tau}\) are both determined by parameters \(a,b,c\), and \(n\). When \(a=b=1\), it can be seen from [41] that \(x_{\tau}\) is the \(\tau\)-th root of the Generalized Laguerre polynomial \(L_{n}^{(-1/2)}(x)\), the corresponding weight coeffificient \(w_{\tau}=\Gamma[n+(1/2)]x_{t}/\left\{n! (n+1)^{2}\Big{[}L_{n+1}^{(-1/2)}(x_{t} )\Big{]}^{2}\right\}\). Therefore, Equation (17) can be simplified as: \begin{table}
\begin{tabular}{c c c} \hline \hline
**Binary Modulation Scheme** & \(p\) & \(q\) \\ \hline Coherent binary phase shift keying (CBPSK) & 0.5 & 1 \\ \hline Differential binary phase shift keying (DBPSK) & 1 & 1 \\ \hline Coherent binary frequency shift keying(CBPSK) & 0.5 & 0.5 \\ \hline Non-coherent binary frequency shift keying (NBPSK)(NBPSK) & 1 & 0.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters \(p\) and \(q\) for the various binary modulation scheme [16; 39]. \[\begin{array}{ll}P_{b}^{RA}&=\frac{q^{p}}{2\Gamma(p)}\sum\limits_{\tau=1}^{n}w_{ \tau}\frac{1}{\Gamma\left(m_{s,d}\right)}G_{1,2}^{1,1}\bigg{(}\frac{m_{s,d}x_{ \tau}}{\overline{\gamma}_{s,d}^{RF}}\bigg{|} \begin{array}{c}1\\ m_{s,d},0\end{array} \bigg{)}\Bigg{\{}1-\exp\left[-\left(\frac{1}{\overline{ \eta}_{s,d}}\sqrt{\frac{3x_{\tau}}{\overline{\gamma}_{s,d}^{TS0}}}\right)^{ \beta_{s,d}}\right]\Bigg{\}}^{\alpha_{s,d}}\\ &\times\left\{1-\left\{1-\frac{1}{\Gamma\left(m_{ij}\right)}G_{1,2}^{1,1} \bigg{(}\frac{m_{ij,\gamma}}{\overline{\gamma}_{ij}^{RF}}\bigg{|} \begin{array} []{c}1\\ m_{i,j},0\end{array} \bigg{)}\right\}\Bigg{\{}1-\exp\left[-\left(\frac{1}{ \overline{\eta}_{ij}}\sqrt{\frac{\gamma}{\overline{\gamma}_{ij}^{TS0}}} \right)^{\beta_{ij}}\right]\Bigg{\}}^{\alpha_{s,i}}\Bigg{\}}^{M}\Bigg{\}}^{M} \Bigg{\}}^{M}\Bigg{\}}^{M}\Bigg{\}}^{M}\Bigg{\}}^{M}\Bigg{\}}^{M}\Bigg{\}^{M} \Bigg{\}}^{M}\Bigg{\}}^{M}\Bigg{\}}^{M}\Bigg{\}}^{M}\Bigg{\}^{M}}\Bigg{\}^{M} \Bigg{\}}^{M}}\Bigg{\}^{M}}\Bigg{\}^{M}}\Bigg{\}^{M}}\Bigg{\}^{M}}\Bigg{}^{M} \Bigg{\}}^{M}}\Bigg{\}^{M}}}\Bigg{}^{M}}} \tag{18}\]  
Similarly, the average BER of our proposed hybrid scheme with PE can be derived as:  
\[