## 2 System Model  
This paper evaluates and analyzes the multi-beam QEFC LEO satellite system with inter-beam interference. We use the NTN large-scale channel model defined in [16]. We assume all users are transmitting with the same power \(P_{l}\). The receiving power \(P_{s,l}(t,m)\) from user \(s\) to beam \(l\) in satellite with equivalent antenna aperture \(m\) at time \(t\) is given by  
\[P_{s,l}(t,m)=P_{l}-L_{s}(t)-\kappa_{e_{s,l}(t)}-\zeta_{e_{s,l}(t)}+G\left(\sigma _{l}^{s}(t),m\right). \tag{1}\]  
Path loss \(L_{s}\) of user \(s\) consists of free-space path loss (FSPL), ionospheric loss of 2.2 dB [17], atmospheric loss, building entry loss, and polarization loss of 3 dB [17]. This paper assumes \begin{table}
\begin{tabular}{c c} \hline \hline
**Notation** & **Distriptions** \\ \hline \(\mathcal{F}\), \(\mathcal{T}\) & The set of the simulated frequency band and time. \(|\mathcal{F}|=3\) for FRF3 and \(|\mathcal{F}|=1\) for non-frequency reuse. \\ \hline \(\mathcal{L}_{f}\), \(S_{l}\) & The set of beams in frequency band \(f\) (\(f\in\mathcal{F}\)), UE serving by beam \(l\) (\(l\in\mathcal{L}_{f}\)). \\ \hline \(\epsilon_{l}\), \(e_{s,l}\) & Elevation angle of beam \(l\), elevation angle of user \(s\) served by beam \(l\). \\ \hline \(\zeta_{c}\) & Clutter loss. Clutter loss is 0 for the LOS state. The values for the (NLOS state at elevation angle \(\epsilon\) are defined in [16]. \\ \hline \(\kappa_{c}\) & Shadow fading at elevation angle \(\epsilon\). Shadow fading is zero mean Gaussian variable. \\ \hline \(\sigma_{\text{LOS},\rho}\), \(\sigma_{\text{NLOS},\rho}\) & The variation in shadow fading for lien-of-sight (LOS) and non line-of-sight (NLOS) states at elevation angle \(\epsilon\)[16]. \\ \hline \(p_{\text{LOS},\rho}\), \(p_{\text{NLOS},\rho}\) & LOS probability and NLOS probability at elevation angle \(\epsilon\)[16]. \\ \hline \(\varphi_{k,l}\) & Angle difference of boresight direction between beam \(l\) and beam \(k\) (\(l,k\in\mathcal{L}\) and \(l\
eq k\)). \\ \hline \(\varphi_{l}^{s}\) & Angle difference of UE \(s\) and boresight direction of beam \(l\). \\ \hline \(\mathbb{P}(x),\mathbb{P}_{y}(x)\) & The probability of the condition \(x\) to be satisfied, \(\mathbb{P}_{y}(x)=\mathbb{P}(x|y)\). \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The description of notations. all users are outdoors, and atmospheric loss is negligible in the S-band. Atmospheric loss and building entry loss are equal to zero. The path loss is obtained as follows [16]:  
\[L_{s}(t)=FSPL_{s}(t)+2.2+3 \tag{2}\]  
\[FSPL_{s}(t)=32.45+20\log_{10}f_{c}+20\log_{10}d_{s}(t) \tag{3}\]  
\[d_{s}(t)=||\mathbf{x}_{sat}(t)-\mathbf{x}_{s}(t)|| \tag{4}\]  
where \(\mathbf{x}_{sat}\) is the location vector of satellite, \(\mathbf{x}_{s}\) is the location vector of UE \(s\), \(d_{s}\) is the distance between user \(s\) and the satellite, \(h\) is the satellite height, \(R_{E}\) is the earth radius, and \(f_{c}\) is carrier frequency. In the 3 GPP satellite communication system, the equivalent antenna aperture is used to model the satellite antenna. The larger equivalent antenna aperture represents more active antenna elements, which form a narrower and higher power gain beam. The antenna gain is calculated as follows [16; 18]:  
\[G(\phi,m)=10\log_{10}\eta\pi^{2}m^{2}\lambda^{-2}g(\phi,m) \tag{5}\]  
\[g(\phi,m)= \begin{cases}1,&\text{if }\phi=0\\ 4|\frac{\hbar(\pi m\lambda^{-1}\sin\phi)}{\pi m\lambda^{-1}\sin\phi}|^{2},& \text{otherwise}\end{cases}  \tag{6}\]  
where \(m\) is equivalent antenna aperture, \(\phi\) is the angle between the boresight of the beam and UE, \(J_{1}(. )\) is the first-kind and first-order Bessel function, \(\lambda\) is the wave length, and \(\eta\) is the antenna aperture efficiency 0.57. The maximum antenna gain is \(10\log_{10}\eta\pi^{2}m^{2}\lambda^{-2}\) dBi.