# Performance Analysis of UAV-IRS Relay Multi-Hop FSO/THz Link
## 2 System and Channel Models
### THz Link  
To improve the quality of the received signal in the THz link, the transmitting base station \(S\) is outfitted with \(W\) antennas. The signals emitted by these antennas are relayed to the subsequent IRS-UAV relay node, which is equipped with \(W\) reflective elements, thereby enhancing the signal quality in the THz link. On the receiving end, the user terminal \(D\) incorporates \(W\) receiving antennas and employs MRC as its reception strategy. The system comprises \(N+1\) links, with the transmitting base station represented as the 0-th node, and the receiving user terminal as the \((N+1)\)-th node. Consequently, the THz signal received at the \(i\)-th node of the \(i\)-th link can be expressed as  
\[y_{\text{THz\_}i}(t)=\left(P_{\text{THz\_}i}\sum\limits_{i=1}^{w}h_{\text{THz \_}i}\right)S_{i}+N_{\text{THz\_}i},\quad i=1,2,\ldots N+1. \tag{14}\]  
where \(P_{\text{THz\_}i}\) denotes the transmit power of the transmitting base station, and \(h_{\text{THz\_}i}\) represents the fading coefficient of the terahertz link at the \(i\)-th receiving antenna, with \(h_{\text{THz\_}i}=h_{\text{I},\text{THz\_}i}h_{\text{a},\text{THz\_}i}h_{\text {p},\text{THz\_}i}h_{\text{a}oa,\text{THz\_}i}\). This channel coefficient is influenced by the cumulative effects of path loss \(h_{\text{I},\text{THz\_}i}\), channel fading \(h_{a,\text{THz\_}i}\), antenna misalignment \(h_{p,\text{THz\_}i}\), and link disruption due to fluctuations in the angle of arrival \(h_{\text{a}oa,\text{THz\_}i}\). The signal-independent additive Gaussian white noise, denoted as \(N_{\text{THz\_}i}\), has a mean of 0 and a variance of \(\sigma_{0}^{2}\), with \(N_{\text{THz\_}i}\sim\mathcal{N}(0,\sigma_{0}^{2})\). Consequently, the SNR at the \(i\)-th node is given by \(\gamma_{\text{THz\_}i}=\left(P_{\text{THz\_}i}\sum\limits_{i=1}^{W}h_{\text{ THz\_}i}\right)^{2}/\sigma_{0}^{2}\). Subsequently, we analyze the PDF of the channel coefficients for the THz link, considering the impact of each aforementioned factor. The PDF of link disruption caused by AoA fluctuations was detailed in reference [15]. Based on this, the PDF of the overall channel gain can be derived using the approximate expression provided:  
\[ \begin{split}& f_{h_{\text{THz\_}i}}(h_{\text{THz\_}i})\simeq \exp\left(-\frac{\psi_{AoA,i}^{2}}{2m_{i}\sigma_{i-1j,angle}^{2}}\right)\delta (h_{\text{THz\_}i})+\left[1-\exp\left(-\frac{\psi_{AoA,i}^{2}}{2m_{i}\sigma_{i -1j,angle}^{2}}\right)\right]\\ &\times\frac{g_{ij}^{2}}{h_{\text{THz\_}i}\Gamma(\mu_{\text{THz \_}i})}\times\text{G}_{1,2}^{2,0}\left(\frac{\mu_{\text{THz\_}i}}{\bar{h}_{f}^ {\alpha_{\text{THz\_}i}}}\left(\frac{h_{\text{THz\_}i}}{A_{ij}\text{h}_{\text {THz\_}i}}\right)^{\alpha_{\text{THz\_}i}}\right|\begin{array}{c}1;1+\frac{g _{ij}^{2}}{4\pi\text{THz\_}i}\\ 0,\frac{\pi_{\text{THz\_}i}H_{\text{THz\_}i}}{A_{\text{THz\_}i}};0\end{array} \right).\end{split}  \tag{15}\]  
According to the CLT, after the signal traverses through \(W\) antennas and is reflected by \(W\) IRS elements at the IRS-UAV relay node, the PDF and CDF of the SNR can be derived through a methodology analogous to that previously employed for determining the SNR of the FSO link, as detailed below:  
\[ \begin{split} f_{\text{TSUMTHz\_}i}(\gamma_{\text{SUMTHz\_}i}) &\approx\frac{1}{\sqrt{2\pi\Omega_{\text{SUMTHz\_}i}^{2}}}e^{-\frac{ \left(\gamma_{\text{SUMTHz\_}i}-\mu_{\text{SUMTHz\_}i}\right)^{2}}{2\Omega_{ \text{SUMTHz\_}i}^{2}}},\end{split}  \tag{16}\]  
\[ \begin{split}& F_{\text{TSUMTHz\_}i}(\gamma_{\text{SUMTHz\_}i}) \approx 1-\frac{1}{2\sqrt{\pi}}\text{G}_{1,2}^{2,0}\left(\left(\frac{\gamma_{ \text{SUMTHz\_}i}-\mu_{\text{SUMTHz\_}i}}{\sqrt{2\Omega_{\text{SUMTHz\_}i}^{2} }}\right)^{2}\begin{array}{c}1\\ 0,\frac{1}{2}\end{array}\right)\\ &-\frac{1}{2}erf\left(-\frac{\mu_{\text{SUMTHz\_}i}}{\sqrt{2\Omega_{ \text{SUMTHz\_}i}}}\right),\end{split}