## 2 System and Channel Models
### Signal Models  
The signal is firstly modulated at the OGS and then transmitted to the LEO satellite. At the LEO satellite, the received signal over the FSO feeder link during the uplink transmission is expressed by  
\[y_{5f}=\left(\frac{P_{6}G_{f}^{Gt}G_{f}^{Sr}}{FL_{G}}I_{GS}\right)^{\frac{b}{2 }}x_{f}+n_{s} \tag{1}\]where \(P_{G}\) is the transmit power at OGS, \(G_{f}^{Gt}\) is the transmit telescope gain at OGS, \(G_{f}^{St}\) is the receive telescope gain at LEO satellite. \(FL_{G}\) is the free-space loss defined as \(FL_{G}=\frac{4\pi LT_{GS}}{\lambda_{f}}\)[9], \(L_{GS}\) is the link length between OGS and LEO satellite, \(\lambda_{f}\) is the optical wavelength. \(I_{GS}\) denotes the irradiance of FSO link calculated by \(I_{GS}=I_{GS}^{l}I_{GS}^{a}I_{GS}^{p}\), \(I_{GS}^{l}\) denotes the atmospheric attenuation due to complex weather conditions, \(I_{GS}^{a}\) denotes atmospheric turbulence, \(I_{GS}^{p}\) denotes pointing errors. \(b=1\) is associated with heterodyne detection (HD) and \(b=2\) is associated with intensity modulation with direct-detection (IM/DD). \(x_{f}\) denotes the transmit signal at OGS, \(n_{s}\) is the additive white Gaussian noise (AWGN) with variance \(\sigma_{s}^{2}\). Then, the instantaneous received SNR at the LEO satellite is given by  
\[\gamma_{GS}^{f}=\frac{\left(P_{G}G_{f}^{Gt}G_{f}^{Gr}\right)^{b}}{FL_{G}^{b} \sigma_{s}^{2}}I_{GS}^{b}=\overline{\gamma}_{GS}^{f}I_{GS}^{b} \tag{2}\]  
where \(\overline{\gamma}_{GS}^{f}=\frac{\left(P_{G}G_{f}^{Gt}G_{f}^{Gr}\right)^{b}} {FL_{G}^{b}\sigma_{s}^{2}}\) is the average received SNR at LEO satellite. The received signal at the LEO satellite is amplified and then forwarded to the HAP. The received signal at the HAP over the FSO relay link is written as  
\[y_{Hf}=\left(\frac{\eta G_{f}^{St}G_{f}^{Hr}}{FL_{H}}I_{SH}\right)^{\frac{b}{2 }}y_{Sf}+n_{H} \tag{3}\]  
where \(\eta\) is the photoelectric conversion ratio, \(G_{f}^{St}\) is the transmit telescope gain at satellite and \(G_{f}^{Hr}\) is the receive telescope gain at HAP. \(FL_{H}\) is the free-space loss, \(I_{SH}\) denotes the irradiance of satellite-to-HAP FSO relay link, \(n_{H}\) is the AWGN with variance \(\sigma_{H}^{2}\). The instantaneous received SNR at HAP is defined as  
\[\gamma_{SH}^{f}=\frac{\left(P_{G}G_{f}^{Gt}G_{f}^{St}\eta G_{f}^{St}G_{f}^{Hr} \right)^{b}}{\eta G_{f}^{St}G_{f}^{Hr}I_{SH}^{b}\tau E_{H}^{b}\sigma_{s}^{2}+FL _{H}^{b}\sigma_{H}^{2}}I_{GS}^{b}I_{SH}^{b} \tag{4}\]  
The received signal at the HAP is firstly decoded and then forwarded to ground users through the parallel FSO/RF links. The received signal at the AP through the FSO link is expressed as  
\[y_{A}=\left(\frac{\rho\eta P_{H}G_{f}^{Ht}G_{f}^{Ar}}{FL_{A}}I_{HA}\right)^{ \frac{b}{2}}\hat{y}_{Hf}+n_{f} \tag{5}\]  
where \(\rho\) is the power allocation coefficient, \(\eta\) is the photoelectric conversion ratio, \(P_{H}\) is the total transmit power of HAP, \(G_{f}^{Ht}\) is the transmit telescope gain at HAP, \(G_{f}^{Ar}\) is the receive telescope gain at AP. \(FL_{A}\) is the free-space loss, \(I_{HA}\) is the irradiance of the HAP-to-AP FSO link, \(\hat{y}_{Hf}\) is the decoded version \(y_{Hf}\), \(n_{f}\) is the AWGN with variance \(\sigma_{f}^{2}\). The instantaneous received SNR at the AP is given as  
\[\gamma_{f}=\frac{\left(\rho\eta P_{H}G_{f}^{Ht}G_{f}^{Ar}\right)^{b}}{FL_{A}^{ b}\sigma_{f}^{2}}I_{HA}^{b}=\overline{\gamma}_{f}I_{HA}^{b} \tag{6}\]  
where \(\overline{\gamma}_{f}=\frac{\left(\rho\eta P_{H}G_{f}^{Ht}G_{f}^{Ar}\right)^{ b}}{FL_{A}^{b}\sigma_{f}^{2}}\) is the average received SNR at AP.