# Performance Analysis of UAV-IRS Relay Multi-Hop FSO/THz Link
## 3 Performance Analysis
### Average Bit Error Rate  
The average BER serves as a direct indicator of the modulation method's influence on the performance of the communication system. The equation for determining the BER is provided in reference [33] as follows:  
\[P_{e}=\frac{q^{p}}{2\Gamma(p)}\int_{0}^{\infty}\gamma^{p-1}e^{-q\gamma}F_{ \gamma}(\gamma)d\gamma. \tag{31}\]  
In this study, according to reference [34], BPSK modulation is employed, setting the parameters as \(p=0.5\) and \(q=1\). To calculate the BER for the FSO link, we first process the CDF of the FSO link; according to Equation (13), it can be obtained that the expression \(\frac{1}{2}\text{erf}\bigg{(}-\frac{\mu_{\text{SUMMSO},i}}{\sqrt{2}\Omega_{ \text{SUMMSO},j}}\bigg{)}\) is considered a constant. To simplify the computation process, this entire expression is denoted by \(T\). Consequently, the CDF of the FSO link is reformulated into a novel expression:  
\[F_{\gamma_{\text{SUMMSO},j}}(\gamma_{\text{SUMMSO},j})=1-\frac{1}{2\sqrt{ \pi}}\quad\times\text{G}_{1,2}^{2,0}\left(\left(\frac{\gamma_{\text{SUMMSO },i}-\mu_{\text{SUMMSO},i}}{\sqrt{2}\Omega_{\text{SUMMSO},i}}\right)^{2} \bigg{|} \begin{array}{c}1\\ 0,\frac{1}{2}\end{array} \right)+T. \tag{32}\]  
The end-to-end PDF for the FSO link is as follows:  
\[ \begin{split}& F_{\gamma_{\text{SUMMSO}}}(\gamma_{\text{SUMMSO }})=1-\big{(}1-F_{\gamma_{\text{SUMMSO},i}}(\gamma_{\text{SUMMSO},i})\big{)} ^{N}\\ &=1-\sum_{k=0}^{N}\left(\begin{array}{c}N\\ k\end{array}\right)(-T)^{N-k}\times\frac{1}{2\sqrt{\pi}}\left(\text{G}_{1,2}^ {2,0}\left(\left(\frac{\gamma_{\text{SUMMSO},i}-\mu_{\text{SUMMSO},i}}{ \sqrt{2}\Omega_{\text{SUMMSO},i}}\right)^{2}\bigg{|}\begin{array}{c}1\\ 0,\frac{1}{2}\end{array}\right)\right)^{k}.\end{split}  \tag{33}\]  
So the average symbol BER of the FSO link is  
\[P_{efso}=\frac{q^{p}}{2\Gamma(p)}\int_{0}^{\infty}\gamma_{\text{SUMFSO},i}p^{ -1}e^{-q\gamma_{\text{SUMFSO},j}}\quad\times F_{\gamma_{\text{SUMFSO}}}( \gamma_{\text{SUMFSO}})d\gamma_{\text{SUMFSO},i}. \tag{34}\]  
It can further be written as \(P_{efso}=I_{1}-I_{2}\), where \(I_{1}\) and \(I_{2}\) are denoted as follows:  
\[I_{1}=\frac{q^{p}}{2\Gamma(p)}\int_{0}^{\infty}\gamma^{p-1}e^{-q\gamma}d\gamma, \tag{35}\]\[I_{2}=\frac{q^{p}}{2\Gamma(p)}\int_{0}^{\infty}\gamma_{SUMFSO\_i}p^{-1}e^{-q \gamma_{SUMFSO\_i}}\sum_{k=0}^{N}\binom{N}{k}\times(-T)^{N-k} \tag{36}\]  
Drawing on the methodology outlined in reference [35], the calculation of the \(I_{1}\) integral can be performed as follows:  
\[I_{1}=\frac{1}{q^{p}}\Gamma(p). \tag{37}\]  
Computing the integral of \(I_{2}\) is inherently complex, making the derivation of its closed-form expression particularly challenging when relying on conventional integration methods. To address this issue, the Gauss-Laguerre integration criterion is employed herein to approximate the expression for the average BER in an end-to-end closed scenario:  
\[P_{efso}\approx\frac{1}{q^{p}}\Gamma(p)-\frac{q^{p}}{2\Gamma(p)} \sum_{k=0}^{N}\binom{N}{k}(-T)^{N-k}\bigg{(}\frac{1}{2}\bigg{)}^{k}\times \bigg{(}\frac{1}{q}\bigg{)}^{p} \tag{38}\] \[\times\sum_{i=1}^{n}\varphi_{FSO\_i}\times\Bigg{(}\mathrm{G}_{1,2}^{2,0}\Bigg{(}\Bigg{(}\frac{\gamma_{\mathrm{SUMFSO\_i}}-\mu_{\mathrm{SUMFSO \_i}}}{\sqrt{2}\Omega_{SUMFSO\_i}}\Bigg{)}^{2}\bigg{|} \begin{array}{c}1\\ 0,\frac{1}{2}\end{array} \Bigg{)}\Bigg{)}^{k},\]  
\[\varphi_{FSO\_i}=\frac{\Gamma(n+p)\gamma_{SUMFSO\_i}}{n!(n+1)^{2}[L_{n+1}^{(p- 1)}(\gamma_{SUMFSO\_i})]^{2}}. \tag{39}\]  
The same reasoning is available:  
\[P_{eTHz}\simeq\frac{1}{q^{p}}\Gamma(p)-\frac{q^{p}}{2\Gamma(p)} \sum_{k=0}^{N}\bigg{(} \begin{array}{c}N\\ k\end{array} \bigg{)}(-T)^{N-k}\bigg{(}\frac{1}{2}\bigg{)}^{k}\times\bigg{(} \frac{1}{q}\bigg{)}^{p} \tag{40}\] \[\times\sum_{i=1}^{n}\varphi_{THz\_i}\times\Bigg{(}\mathrm{G}_{1,2}^{2,0}\Bigg{(}\Bigg{(}\frac{\gamma_{SUM\_i}{THz\_i}-\mu_{SUM\_i}{THz\_i}} {\sqrt{2}\Omega_{SUM\_i}{THz\_i}}\Bigg{)}^{2}\bigg{|} \begin{array}{c}1\\ 0,\frac{1}{2}\end{array}