## 4 Simulation Results  
This section presents simulation results to validate the performance of our proposed algorithms. We consider 4 BSs serving a square region on the ground with a size of 1 km \(\times\) 1 km. The coordinates of the four vertices of the considered square region are (0 km, 0 km), (1 km, 0 km), (1 km, 1 km), and (0 km, 1 km), respectively. The four BSs are uniformly placed in the square region, with coordinates of (0.25 km, 0.25 km), (0.25 km, 0.75 km), (0.75 km, 0.75 km), and (0.75 km, 0.25 km), respectively. A low Earth orbit (LEO) satellite atan altitude of 300 km is deployed to serve the considered region. All the UEs are evenly and randomly distributed within the considered square region. The channels between UEs and BSs are modeled as Rayleigh channels [23], and the channels between UEs and the satellite are modeled as far-field line-of-sight channels [24]. The path loss exponents for the two types of links are set to 3 and 2, respectively. The maximum transmit powers \(P_{k}^{max}\) for all \(k\in\mathcal{K}\) are set to be the same. Table 1 summarizes the simulation parameters. The simulations were conducted using Matlab 2021 B software, yielding numerical results from Monte Carlo simulations with 100 independent channel realizations. Figure 2 gives the convergence of the proposed algorithm under different settings of \(K\) and \(N\). The main iteration steps occur in the inner and outer iterations of stage 2 in Algorithm 1. During the inner iteration, parameters \(\gamma_{m,k}\), \(\gamma_{m,k}\) are updated. The outer iteration involves the alternative optimization of \(\mathbf{v}_{m,k}\) and \(p_{m,k}\). Consequently, in the revised manuscript, we incorporated the convergence behavior for both the inner and outer iterations of stage 2 in Algorithm 1. It is seen that for both inner and outer iterations, the convergence is monotonic under all settings of \(K\), \(N\). For the inner iteration, it takes approximately 15 iterations to reach convergence. For the outer iteration, about four iterations are required to converge. Figure 3 presents the achievable sum rate versus the maximum transmit power. The total number of UEs is set to \(K=10\), \(K=20\), \(K=30\). The number of receive antennas is set to \(N=20\). It is seen that the achievable sum rate increases with the maximum transmit power. The NOMA scheme is obviously better than the OMA scheme, which indicates the benefits of adopting NOMA in uplink ISTNs. The performance degrades when the number of UEs is set to 30. This is because, when the number of receive antennas exceeds the number of UEs, the system has a sufficient degree of freedom to distinguish signals \begin{table}
\begin{tabular}{l l} \hline \hline
**Parameter** & **Setting** \\ \hline Path loss model & \(\beta(\frac{d}{d_{0}})^{-\alpha}\)[15] \\ Parameters in path loss & \(\beta=\)–30 dB, \(d_{0}=1\) m, \(\alpha=3\), 2 [15] \\ Noise power & \(-174\) dBm/Hz [15] \\ Altitude of satellite & 300 km [1] \\ Number of BSs & \(M=4\) \\ Coordinates of BSs & (0.75 km, 0.75 km), (0.75 km, 0.25 km) \\  & (0.75 km, 0.75 km), (0.75 km, 0.25 km) \\ Backhaul link capacity for BS & \(C_{1}=C_{2}=\cdots=C_{M}=C=50\) bps/Hz \\ Number of receive antennas at BS & \(N_{1}=N_{2}=\cdots=N_{M+1}=N=10\), 20, 40 \\ Number of users & \(K=10\), \(20\), \(30\), \(40\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Simulation parameters. Figure 2: Convergence of the proposed algorithm under different settings of \(K\) and \(N\). from different UEs and minimize interference among them. However, once the number of UEs exceeds the number of antennas, the system lacks the necessary degree of freedom for leveraging multiuser diversity, resulting in the degradation of system performance. Figure 4 presents the achievable sum rate versus the number of UEs. The maximum transmit power is set as 26 dBm. The numbers of receive antennas for different APs are set to the same constant \(N\).