# Performance Analysis of RIS-Assisted SatComs Based on a ZFBF and Co-Phasing Scheme
## 2 System Model
### RIS-Assisted SatCom Channel Model  
To achieve high spectral efficiency and throughput in multibeam HTS systems, FFR is generally considered in SatCom systems [18]. However, the performance of the FFR-based SatCom system can be degraded by co-channel interference among inter-beam UAVs, particularly for UAVs at the beam edge. Consequently, the RISs installed on those UAVs reflect the incident signals, including the IBI, to the UEs. This results in severe performance degradation due to the phase-aligned and amplified IBI. Thus, in order to mitigate IBI, we assume that beamforming (i.e., active beamforming) is performed at the ES. Additionally, we consider that the feeder link is ideal, similar to most related works [5; 6; 7]. Then, the received signal at UE \(k\) is obtained as:  
\[y_{k}=\sqrt{\rho_{k}}f_{kk}^{\text{H}}\mathbf{\Phi}_{k}D_{k}B_{k}\mathbf{w}_{k}x_{k}+n_ {k}\]  
Figure 1: System model of the RIS-assisted SatCom system. \[+\sum_{i\
eq k}^{K}\underbrace{(\sqrt{\rho_{i}}f_{i,k}^{\text{H}} \mathbf{\Phi}_{k}D_{k}B_{k}\mathbf{w}_{i}x_{i}}_{\text{IBI}}+\underbrace{\sqrt{\rho_{i}} f_{i,k}^{\text{H}}\mathbf{\Phi}_{i}D_{i}B_{i}\mathbf{w}_{i}x_{i}}_{\text{IRI}}), \tag{1}\]  
where \(\rho_{k}\) and \(\mathbf{w}_{k}\in\mathbb{C}^{K\times 1}\) are, respectively, the transmit signal-to-noise ratio (SNR) and the transmit beamforming vector for UE \(k\), and \(\mathbf{f}_{i,k}\in\mathbb{C}^{N\times 1}\) and \(\mathbf{D}_{k}\in\mathbb{C}^{N\times N}\) are, respectively, the fading channels between RIS \(i\) and UE \(k\) and between the satellite and RIS \(k\). Also, \(x_{k}\) and \(n_{k}\) are the transmit signal and additive complex Gaussian noise with zero mean and unit variance, respectively, and \(\mathbf{\Phi}_{k}\in\mathbb{C}^{N\times N}\) is a reflection matrix that operates as a passive beamformer and can be controlled by the ES. The reflection matrix \(\mathbf{\Phi}_{k}\) can be obtained as follows:  
\[\mathbf{\Phi}_{k}=\text{diag}\Big{(}\sqrt{\overline{\alpha_{k,1}}}e^{\psi_{k,1}}, \sqrt{\overline{\alpha_{k,2}}}e^{\psi_{k,2}},\cdots,\sqrt{\overline{\alpha_{k,N}}}e^{\psi_{k,N}}\Big{)}, \tag{2}\]  
where \(\sqrt{\overline{\alpha_{k,n}}}\in[0,1]\) and \(\theta_{k,n}\in[0,2\pi)\) are, respectively, the amplitude and phase shifting response of the reflection element \(n\) in RIS \(k\). Assuming lossless reflection power at all the RISs, we consider that \(\alpha_{k,n}=1,\forall k,n\). In (1), the signals directly transmitted from the satellite to the UEs are neglected because of severe path loss and shadowing. For a practical air-to-ground (A2G) channel, \(\mathbf{f}_{i,k}\) is generated by a correlated Rician fading channel composed of a deterministic LoS path and a fading non-LoS (NLoS) path, as follows:  
\[\mathbf{f}_{i,k}=\bar{\beta}_{i,k}\mathbf{f}_{i,k}+\bar{\beta}_{i,k}\mathbf{\tilde{f}}_{i,k}, \tag{3}\]  
where  
\[\bar{\beta}_{i,k} =\sqrt{\frac{\overline{\alpha_{i,k}}\xi_{i,k}}{\overline{\alpha_ {i,k}}+1}}, \tag{4}\] \[\bar{\beta}_{i,k} =\sqrt{\frac{\overline{\zeta_{i,k}}}{\overline{\alpha_{i,k}}+1}}. \tag{5}\]  
Here, \(\kappa_{i,k}\) and \(\zeta_{i,k}\) are, respectively, the Rician factor and the path loss between RIS \(i\) and UE \(k\), and \(\mathbf{\tilde{f}}_{i,k}\in\mathbb{C}^{N\times 1}\) comprises the deterministic LoS components whose entries are determined depending on the locations of RIS \(i\) and UE \(k\). Defining \(\tilde{f}_{i,k,n}\) as the \(n\)th element of \(\mathbf{\tilde{f}}_{i,k}\), we have \(f_{i,k,n}=\exp(-j2\pi d_{i,k,n}/\lambda)\), where \(d_{i,k,n}\) denotes the distance between reflecting element \(n\) of RIS \(i\) and UE \(k\), and \(\lambda\) denotes the wavelength of a signal [19]. Also, \(\mathbf{\tilde{f}}_{i,k}=\mathbf{G}_{i}^{1/2}\mathbf{\tilde{f}}_{i,k}\in\mathbb{C}^{N \times 1}\) consists of the correlated NLoS components between RIS \(i\) and UE \(k\), where \(\mathbf{G}_{i}\in\mathbb{C}^{N\times N}\) is the spatial correlation matrix at RIS \(i\), and \(\mathbf{\hat{f}}_{i,k}\) is the independent NLoS component.