## 5 Numerical and Simulation Results
### Simulation Results  
Figure 2 demonstrates the product of the achievable rate and the expected number of users versus the number of iterations for the proposed scheme when the UAV's height \(h_{uav}\) is set to 10 m initially. The figure compares the performance under three different signal-to-noise ratio (SNR) conditions. It shows that in higher SNR, the system converges within more iterations, whereas in lower SNR, convergence requires fewer iterations. This fast convergence can be explained as follows: In low-SNR scenarios, the communication link between the STAR-RIS and users suffers from severe fading, leading to limited channel gains. As a result, the optimization space becomes effectively constrained, and the algorithm quickly reaches a point where further iterations yield negligible improvement. In Figure 3, we investigate the achievable rate of all three schemes versus the number of reflection elements when the position of the UAV is fixed. From this figure, it is first observed that the proposed scheme achieves the best performance for all cases of \(M\). For instance, when \(M=24\), the achievable rate of the proposed scheme can achieve 2.7 bps/Hz, which is almost triple that of the baseline 2 scheme and slightly better than that in the baseline 1 scheme. The baseline 2 scheme lacks the optimization of several STAR-RIS parameters, including the beamforming vector and phase shifts. In contrast, the proposed scheme optimizes these parameters, resulting in superior performance. This further validates that the proposed me can effectively adjust the STAR-RIS phase shifters to find an optimal solution. In addition, the scheme with RIS performs worse than the proposed scheme because the STAR-RIS has both reflective and transmissive elements, allowing the proposed scheme to adjust the phases of its elements more precisely. \begin{table}
\begin{tabular}{l c c} \hline
**Description of Parameter** & **Parameter** & **Value** \\ \hline U1 position & \((L,0)\) & \((10,0)\) \\ \hline U2 position & \((-L,0)\) & \((-10,0)\) \\ \hline Number of antennas in BS & \(N\) & 2 \\ \hline Power coefficient of the U1 & \(a_{r}\) & 0.3 \\ \hline Power coefficient of the U2 & \(a_{t}\) & 0.7 \\ \hline Transmission coefficients & \(\beta_{m}^{t},\forall m\in\mathcal{M}\) & 0.5 \\ \hline Reflection coefficients & \(\beta_{m}^{r},\forall m\in\mathcal{M}\) & 0.5 \\ \hline Learning rate & \(\Delta\) & 0.1 \\ \hline Convergence condition & \(\eta\) & \(1\times 10^{-5}\) \\ \hline Maximum number of iterations & \(\eta_{\text{max.iter}}\) & 1000 \\ \hline \end{tabular}
\end{table} 
Table 2: _Cont._In Figure 4, we analyze the BER of the three schemes as the number of reflection elements varies. The baseline 2 scheme, using a random phase approach, shows that the BER remains relatively high for both users. As \(M\) increases, the BER of user 1 and user 2 slightly decreases. This decreased performance is partly due to the complexity of adjusting multiple STAR-RIS parameters, including the beamforming vector and phase shifts, in the baseline 2 scheme. In contrast, the proposed scheme demonstrates a significant reduction in BER for both users as \(M\) increases. Meanwhile, user 1 and user 2 also experience a rapid decline in BER, indicating the overall superior performance of the proposed scheme. For the baseline 1 scheme, which involves phase shift adjustment, the BER performance is better than that of baseline 2 but worse than that of the proposed scheme. This performance gap is mainly due to the physical characteristics of RIS and STAR-RIS, in which the STAR-RIS allows for more parameters to be adjusted, providing greater flexibility and finer control, which leads to enhanced performance. Overall, the proposed scheme outperforms the baseline 1 and 2 schemes, highlighting the benefits of using beamforming and phase shift optimization to reduce BER. Figure 3: Achievable rate of the proposed scheme, baseline 2 scheme and baseline 1 scheme versus number of STAR-RIS elements when \(h_{uw}=10\) and \(r=5\).