# Robust Beamforming Based on Graph Attention Networks for IRS-Assisted Satellite IoT Communications
## 2 System Model
### Channel Model  
To realistically model the propagation characteristics of the satellite channel, the impact of path loss, atmospheric attenuation, and satellite beam gain should be accounted for. The downlink channel between the satellite and the ground device \(k\) can be expressed as [26]:  
\[h_{k}=\sqrt{C_{k}}b_{k}^{\frac{1}{2}}\circ r_{k}^{\frac{1}{2}}\ \bar{o}\ \widetilde{h}_{k},\ k=0,1,\cdots,K \tag{3}\]  
where \(C_{k}\) is the large-scale fading efficient, which can be calculated by:  
\[C_{k}=\left(\frac{\lambda}{4\pi d_{k}}\right)^{2}\frac{G_{k}}{\kappa TB} \tag{4}\]  
where \(d_{k}\), \(\lambda\) and \(G_{k}\) represent the propagation distance, carrier wavelength and receive antenna gain, respectively. \(\kappa=1.38\times 10^{-23}\)J/m is Boltzman's constant, \(B\) is the carrier bandwidth, and \(T\) represents the receive noise temperature. \(b_{k}\)= \(\left[b_{k,1},\cdots,b_{k,M}\right]^{T}\) in Equation (3) is an \(M\)-dimensional beam radiation pattern vector, where the \(m\)-th element \(b_{k}\) can be approximated by [26; 27]:  
\[b_{k,m}=b_{\max}\Bigg{(}\frac{I_{1}(u_{k,m})}{2u_{k,m}}+36\frac{I_{3}(u_{k,m}) }{2u_{k,m}^{2}}\Bigg{)}^{2} \tag{5}\]  
where \(u_{k,m}=2.07123\sin(\varphi_{k,m})/\sin(\varphi_{k,3 dB})\), \(\varphi_{k,m}\) is the angle between the \(m\)-th satellite beam centre and user \(k\), \(\varphi_{k,3 dB}\) is the 3-dB angle for the \(k\)-th user. \(b_{\max}\) is the maximal satellite antenna gain. \(J_{1}(\cdot)\) and \(J_{3}(\cdot)\) are the first and third orders of the first-kind Bessel function, respectively. Moreover, \(r_{k}\) in Equation (3) is also an \(M\)-dimensional vector in which represents the rain attenuation coefficient and its form of dB follows lognormal random distribution \(\ln(20\text{lg}(r_{k,m}))\sim(\mu_{m},\sigma_{m}^{2})\). Moreover, we adopt shadowed Rician fading as  
Figure 1: IRS-assisted downlink LEO satellite communication system. the satellite channel fading model, which is proposed in [28] and has been widely used in prior studies. In this model, the probability density function of \(\left|\widetilde{h}_{k}\right|^{2}\) can be expressed as:  
\[f_{|\widetilde{h}_{k}|^{2}}(x)=\left(\frac{2\lambda_{k}m_{k}}{2\lambda_{k}m_{k}+ \Omega_{k}}\right)^{m_{k}}\frac{1}{2\lambda_{k}}exp(-\frac{x}{2\lambda_{k}}) \cdot{}_{1}F_{1}\left(m_{k},1,\frac{\Omega_{k}x}{2\lambda_{k}(2\lambda_{k}m_{k }+\Omega_{k})}\right),x\geq 0 \tag{6}\]  
where \({}_{1}F_{1}(a;b;c)\) is the confluent hypergeometric function and \(2\lambda_{k}\) and \(\Omega_{k}\) are the average power of the scatter component and LoS component, respectively. \(m_{k}\geq 0\) denotes the Nakagami-\(m\) parameter. Therefore, the channel fading coefficient \(\widetilde{h}_{k}\) can be represented as \(\widetilde{h}_{k}=(\lambda_{k},m_{k},\Omega_{k})\). When \(k=0\), we denote \(h_{0}\) represents the channel between the satellite and IRS. In this paper, we assume that the channels of satellite-IRS links undergo infrequent light shadowing (ILS), while satellite-user links experience frequent heavy shadowing (FHS) [29], i.e., \(h_{0}>h_{k},\forall k\in\{1,2,\cdots,K\}\). For the channel model between IRS and user \(k\), both LoS and no line-of-sight (NLoS) components are considered, so we model channel \(h_{k}^{r}\) as Rician fading channels:  
\[h_{k}^{r}=\beta_{k}\left(\sqrt{\frac{\xi}{1+\xi}}\widetilde{h}_{k}^{los}+\sqrt {\frac{1}{1+\xi}}\widetilde{h}_{k}^{los}\right) \tag{7}\]  
where \(\beta_{k}\) is the path loss from the IRS to user \(k\) and can be modelled as \(30+22\log(d_{k})\), \(d_{k}\) is the distance between the RIS and the \(k\)-th user. \(\xi\) is the Rician factor, and \(\widetilde{h}_{k}^{nlos}\) is the NLoS component vector, which is a complex Gaussian distributed with zero mean and unit variance.