# Performance Analysis of UAV-IRS Relay Multi-Hop FSO/THz Link
## 3 Performance Analysis
### Average Channel Capacity  
According to the findings in references [36; 37], the formula for average channel capacity is represented by the variable \(\bar{C}=E[\log_{2}(1+\gamma)]\), where \(E(\cdot)\) serves as the mathematical expectation operator. Consequently, this allows for the definition of the average channel capacity:  
\[C=\int_{0}^{\infty}\log_{2}(1+\gamma)f_{\gamma}(\gamma)d\gamma=\frac{1}{\ln 2 }\int_{0}^{\infty}\frac{F_{\gamma}^{C}(\gamma)}{1+\gamma}d\gamma, \tag{48}\]  
where \(F_{\gamma}^{C}(\gamma)=1-F_{\gamma}(\gamma)\). The channel capacity of the FSO link is derived from Equation (48) as presented above:  
\[ \begin{split}& C_{FSO}(\gamma_{th}^{FSO})=\frac{1}{\ln 2}\int_{ \gamma_{th}^{FSO}}^{\infty}\log_{2}(1+\gamma)f_{\gamma}(\gamma)d\gamma=\frac{ 1}{\ln 2}\int_{\gamma_{th}^{FSO}}^{\infty}\frac{1}{1+\gamma_{SUMFSO\_j}}\\ &\times\Big{(}\big{(}1-F_{\gamma_{SUMFSO\_j}}(\gamma_{SUMFSO \_j})\big{)}^{N}\Big{)}d\gamma_{SUMFSO\_i}.\end{split}  \tag{49}\]  
Transforming the aforementioned equation in its entirety to satisfy the condition \(\gamma_{SUMFSO\_j}-\gamma_{th}^{FSO}=t\) yields the equation  
\[ \begin{split}& C_{FSO}(\gamma_{th}^{FSO})=\frac{1}{\ln 2} \times\sum_{k=0}^{N}\left(\begin{array}{c}N\\ k\end{array}\right)(-T)^{N-k}\times\int_{0}^{\infty}\frac{1}{1+t+\gamma_{th}^ {FSO}}\\ &\left(\frac{1}{2\sqrt{\pi}}\mathrm{G}_{1,2}^{2,0}\Bigg{(}\left( \frac{t+\gamma_{th}^{Thz}-\mu_{SUMTHz\_i}}{\sqrt{2}\Omega_{SUMTHz\_i}}\right) ^{2}\bigg{|}\begin{array}{c}1\\ 0,\frac{1}{2}\end{array}\right)\Bigg{)}^{k}d\gamma_{SUMFSO\_i}.\end{split}  \tag{50}\]  
Modifying the preceding equation results in  
\[ \begin{split}& C_{FSO}(\gamma_{th}^{FSO})=\frac{1}{\ln 2} \sum_{k=0}^{N}\left(\begin{array}{c}N\\ k\end{array}\right)\times(-T)^{N-k}\times\int_{0}^{\infty}e^{-t}\times e^{t} \times\frac{1}{t+\gamma_{th}^{FSO}+1}\\ &\times\Bigg{(}\frac{1}{2\sqrt{\pi}}\mathrm{G}_{1,2}^{2,0}\Bigg{(} \left(\frac{t+\gamma_{th}^{THz}-\mu_{SUMTHz\_i}}{\sqrt{2}\Omega_{SUMTHz\_i}} \right)^{2}\bigg{|}\begin{array}{c}1\\ 0,\frac{1}{2}\end{array}\right)\Bigg{)}^{k}dt.\end{split}  \tag{51}\]  
The following steps are taken to process the above equation, as follows:  
\[ \begin{split}& g_{\text{FSO}}(t)=e^{t}\times\frac{1}{t+\gamma_{ th}^{FSO}+1}\times\Bigg{(}\frac{1}{2\sqrt{\pi}}\mathrm{G}_{1,2}^{2,0}\Bigg{(} \big{(}\frac{t+\gamma_{th}^{FSO}-\mu_{SUMFSO\_i}}{\sqrt{2}\Omega_{SUMFSO\_ i}}\big{)}^{2}\bigg{|}\begin{array}{c}1\\ 0,\frac{1}{2}\end{array}\Bigg{)}\Bigg{)}^{k}.\end{split}  \tag{52}\]  
Its approximate expression is derived employing the Gaussian-Laguerre product formula:  
\[ \begin{split}& C_{FSO}=\frac{1}{\ln 2}\sum_{k=0}^{N}\left( \begin{array}{c}N\\ k\end{array}\right)(-T)^{N-k}\int_{0}^{\infty}e^{-t}g_{FSO}(t)dt\approx\sum_{k=0 }^{N}\left(\begin{array}{c}N\\ k\end{array}\right)(-T)^{N-k}\sum_{i=1}^{n}w_{CFSO\_i}\mathcal{S}_{FSO}(t_{i}), \end{split}  \tag{53}\]Similarly, the expression for the channel capacity closure of THz link multi-hop configuration can be derived as follows:  
\[C_{\rm THz}=\frac{1}{\ln 2}\sum_{k=0}^{N}\left( \begin{array}{c}N\\ k\end{array} \right)\left(-T\right)^{N-k}\int_{0}^{\infty}e^{-t}g_{THz}(t)dt=\sum_ {k=0}^{N}\left( \begin{array}{c}N\\ k\end{array} \right)\left(-T\right)^{N-k}\sum_{i=1}^{n}w_{CTHz,j}g_{THz}(t_{i}), \tag{55}\]  
\[w_{CTHz,j}=\frac{\Gamma(n)t_{i}}{n!(n+1)^{2}[L_{n+1}^{(-1)}(t_{i})]^{2}}, \tag{56}\]  
\[g_{THE}(t_{i})=\frac{1}{t_{i}+\gamma_{th}^{THz}+1}e^{t_{i}}\times\ \left(\frac{1}{2}\mathrm{G}_{1,2}^{2,0}\left(\left(\frac{t+\gamma_{th}^{ THz}-\mu_{SUMTHz-i}}{\sqrt{2}\Omega_{SUMTHz-i}}\right)^{2}\right| \begin{array} []{c}1\\ 0,\frac{1}{2}\end{array} \right)\right)^{k}.