# Performance Analysis of UAV-IRS Relay Multi-Hop FSO/THz Link
## 2 System and Channel Models
### FSO Link
This study adopts an indirect method due to the mathematical complexitiesinvolved in directly deriving a closed-form expression for the overall SNR. Specifically, the PDF of \(h_{FSO\_i}^{2}\) is calculated first, followed by leveraging this PDF to ascertain the PDF of \(f_{\gamma_{SUMFSO\_i}}(\gamma_{SUMFSO\_i})\). The PDF of \(h_{FSO\_i}^{2}\) is derived by setting \(K=h_{FSO\_i}^{2}\):  
\[ \begin{split}& f_{K}(x)=\frac{1}{2\sqrt{x}}exp\left(-\frac{\psi_{A \alpha A,i}^{2}}{2m_{i}\sigma_{i-1j,angle}^{2}}\right)\delta\left(x^{\frac{1}{2 }}\right)+\left[1-exp\left(-\frac{\psi_{A\alpha A,i}^{2}}{2m_{i}\sigma_{i-1j,angle }^{2}}\right)\right]\times\frac{g_{ij}^{2}}{2x\Gamma(\alpha_{i})\Gamma(\beta_{ i})}\\ &\times\mathrm{G}_{1,3}^{3,0}\left(\frac{\alpha_{i}\beta_{i}}{A_{ ij}\eta_{i,FSO\_i}}x^{\frac{1}{2}}\left|\begin{array}{c}\delta_{ij}^{2}+1\\ \delta_{ij}^{2},\alpha_{i},\beta_{i}\end{array}\right.\right).\end{split}  \tag{7}\]  
To simplify the computation of the complex SNR expression, this study employs the central limit theorem (CLT) [28], positing that the SNR distribution can be approximated by a Gaussian distribution. Based on this approximation, the expectation (mean), \(\mu_{FSO}\), and variance, \(\Omega_{FSO}^{2}\), of \(K=h_{FSO\_i}^{2}\) are initially calculated. The specific outcomes of these calculations are presented as follows:  
\[ \begin{split}&\mu_{FSO}=E[K]=\frac{(h_{l\_i}A_{i})^{4}g_{ij}^{2} \Gamma(2+\alpha_{i})\Gamma(2+\beta_{i})}{(2+g_{ij}^{2})\Gamma(\alpha_{i}) \Gamma(\beta_{i})\left(\alpha_{i}\beta_{i}\right)^{2}}\times\left[1-\exp \left(-\frac{\psi_{A\alpha A,i}^{2}}{2m_{i}\sigma_{i-1j,angle}^{2}}\right) \right]\\ &+\exp\left(-\frac{\psi_{A\alpha A,i}^{2}}{2m_{i}\sigma_{i-1j,angle }^{2}}\right),\end{split}  \tag{8}\]  
\[ \begin{split}&\Omega_{FSO}^{2}=E[K^{2}]-(E[K])^{2}=\frac{(h_{l\_i} \sigma_{i})^{4}g_{ij}^{2}\Gamma(4+\alpha_{i})\Gamma(4+\beta_{i})}{(4+g_{ij}^{ 2})\Gamma(\alpha_{i})\Gamma(\beta_{i})\left(\alpha_{i}\beta_{i}\right)^{4}} \times\left[1-\exp\left(-\frac{\psi_{A\alpha A,i}^{2}}{2m_{i}\sigma_{i-1j,angle }^{2}}\right)\right]\\ &+\exp\left(-\frac{\psi_{A\alpha A,i}^{2}}{2m_{i}\sigma_{i-1j,angle }^{2}}\right)-\mu_{FSO}^{2}.\end{split}  \tag{9}\]  
The CLT can be effectively applied to accurately estimate the mean and variance of signals reflected by a reflective element in an IRS-UAV system. \[\mu_{SUMFSO\_i}=\gamma_{FSO\_i}\times\mu_{FSO}\times W, \tag{10}\]  
\[\Omega_{SUMFSO\_i}^{2}=\gamma_{FSO\_i}^{2}\times\Omega_{FSO}^{2}\times W. \tag{11}\]  
Consequently, the PDF of \(f_{\gamma_{SUMFSO\_i}}(\gamma_{SUMFSO\_i})\) can be derived as follows:  
\[ \begin{split}& f_{\gamma_{SUMFSO\_i}}(\gamma_{SUMFSO\_i}) \approx\frac{1}{\sqrt{2\pi\Omega_{SUMFSO\_i}^{2}}}e^{-\mathrm{u}},\\ & u=\frac{(\gamma_{SUMFSO\_i}-\mu_{SUMFSO\_i})^{2}}{2\Omega_{ SUIMFSO\_i}^{2}}.\end{split}  \tag{12}\]  
Following the methodology outlined in reference [29], Equation (12) undergoes further mathematical manipulation and integration, resulting in the formulation of Equation (13) as follows:  
\[ \begin{split}& F\gamma_{SUMFSO\_i}(\gamma_{SUMFSO\_i})\approx 1 -P1-P2,\\ & P1=\frac{1}{2\sqrt{\pi}}\mathrm{G}_{1,2}^{2,0}\left(\left(\frac{ \gamma_{SUMFSO\_i}-\mu_{SUMFSO\_i}}{\sqrt{2}\Omega_{SUMFSO\_i}}\right)^{2} \right|\begin{array}{c}1\\ 0,\frac{1}{2}\end{array}\right),\\ & P2=\frac{1}{2}\;erf\left(-\frac{\mu_{SUMFSO\_i}}{\sqrt{2}\Omega_{SUMFSO\_ i}}\right).\end{split}  \tag{13}\]