# Performance Analysis of UAV-IRS Relay Multi-Hop FSO/THz Link
## 2 System and Channel Models
### FSO Link  
To enhance the received signal quality and system robustness, the transmitting base station \(S\) is outfitted with \(W\) laser transmitters, establishing \(W\) optical links for data transmission. Each relay node adjusts the direction and phase of incoming signals via its reflective elements, thereby ensuring efficient propagation to the subsequent node. The receiver employs \(W\) antennas, utilizing the maximum ratio combination (MRC) technique for optimal signal reception. The system encompasses \(N+1\) links, with the base station as node 0 and the user terminal (\(D\)) as node \(N+1\). The optical signal at the \(i\)-th node on the \(i\)-th link is mathematically represented as follows:  
\[y_{FSO\_i}(t)=(P_{FSO\_i}\eta\sum_{i=1}^{W}h_{FSO\_i})S_{i}+N_{FSO\_i},i=1\dots N +1, \tag{1}\]  
In this framework, \(P_{FSO\_i}\) signifies the optical power transmitted by the base station; \(\eta\) represents the photoelectric conversion factor; \(h_{FSO\_i}\) corresponds to the total channel gain of the \(i\)-th link, defined by \(h_{FSO\_i}=h_{l,FSO\_i}h_{a,FSO\_i}\eta_{l,P_{FSO\_i}}h_{aa,FSO\_i}\eta_{i,IFSO\_i}\) quantifies atmospheric loss; \(h_{a,FSO\_i}\) measures atmospheric turbulence; \(h_{p,FSO\_i}\) assesses pointing error; and \(h_{aa,FSO\_i}\) reflects link disruption caused by AOA fluctuations. Let \(S_{i}\) denote the signal received from the preceding node, \(N_{FSO\_i}\) represent the signal-independent additive Gaussian white noise characterized by a mean of 0 and a variance of \(\sigma_{0}^{2}\), and \(N_{FSO\_i}\sim\mathcal{N}(0,\sigma_{0}^{2})\) and \(\tau_{FSO\_i}=\left(P_{FSO\_i}\eta\Sigma_{i=1}^{W}h_{FSO\_i}\right)^{2}/\sigma _{0}^{2}\) signify the instantaneous SNR at the \(i\)-th node. Next, examine the four parameters mentioned above in terms of their statistical properties. Derived from reference [27], the PDF for channel coefficients, influenced by pointing errors, atmospheric turbulence, and atmospheric attenuation, is obtained as follows:  
\[f_{h_{FSO\_i}}{}^{1}\left(h_{FSO\_i}{}^{1}\right)=\frac{\alpha_{i}\beta_{i} \delta_{ij}^{2}}{\Lambda_{ij}h_{l,FSO\_i}\Gamma(\alpha_{i})\Gamma(\beta_{i})} \times\mathrm{G}_{1,3}^{3,0}\left(\frac{\alpha_{i}\beta_{i}h_{FSO\_i}}{A_{ij}h _{l,FSO\_i}}\left| \begin{array}{c}-;g_{ij}^{2}\\ g_{ij}^{2}-1,\alpha_{i}-1,\beta_{i}-1;-\end{array} \right.\right), \tag{2}\] \begin{table}
\begin{tabular}{l l} \hline
**Parameter** & **Definition** \\ \hline \(P_{FSO\_i}\) & Optical power transmitted at S \\ \(\eta_{THz\_i}\) & THz power transmitted at S \\ \(\eta\) & Photoelectric conversion factor \\ \(S_{i}\) & Transmitted signal \\ \(W\) & Number of antennas at S and D, number of IRS elements \\ \(N\) & Number of UAV \\ \(\alpha_{i},\beta_{i}\) & Shape parameters of the G-G fading \\ \(A_{ij}\) & Fraction of power collected at the focal point \(r=0\) \\ \(g_{ij}^{2}\) & Ratio of the radii of equivalent beamwidths \\ \(o_{s\_ij}\) & Variance of the pointing error \\ \(L_{FSO\_i-1\_LTH\_i-j-1}\) & (\(i-1\))th link distance \\ \(\Psi_{AOA\_i}\) & Field of view (FoV) of the \(i\)-th link \\ \(m_{i}\) & Variable dependent on link type \\ \(\sigma_{ij,u}^{2}\) & Displacement variance due to the positional offset of the \(i\)-th node \\ \(\sigma_{i-1j\_a}^{2}\) & Displacement variance due to the (\(i-1\))-th node direction offset \\ \(\sigma_{i-1j\_angle}^{2}\) & Displacement variance due to the (\(i-1\))-th node position offset \\ \(\mu_{THz\_i},\mu_{THz\_i}\) & Fading parameters of THz \\ \(h_{f}^{4}\) & \(\alpha\)-root mean value of fading channel envelope \\ \(\tau_{T_{h}}^{FSO}\) & SNR threshold of FSO link \\ \(\tau_{T_{h}}^{HHz}\) & SNR threshold of THz link \\ \(\tau_{T_{h}SO}^{FSO}\) & Upper SNR threshold of FSO link \\ \(\tau_{H_{h},l}^{FSO}\) & Lower SNR threshold of FSO link \\ \(p,q\) & Represent different binary modulations schemes \\ \hline \end{tabular}
\end{table} 
Table 1: List of parameters.