## 6 Simulation Analysis  
In this section, we analyze the effectiveness of the proposed modulation scheme and verify the switch probability of the modulation scheme when the user moves. For convenience, we assume there are five available frequency sub-windows for a certain distance, i.e., \(M=5\). Each sub-window is divided into 10 non-overlapped sub-bands, i.e., \(N1=N2=N3=N4=N5=10\). The total available THz bandwidth is \(B=2.087\) THz [12]. We use MATLAB software to verify the theoretic analysis. The simulation parameter settings are shown in Table 3[12]. \begin{table}
\begin{tabular}{c l c c} \hline \hline
**Notation** & **Parameter Definition** & **Value** & **Unit** \\ \hline \(P_{\mathrm{Tx}}\) & Maximum transmit power & 10 & dBm \\ \hline \(G_{t}\) & Transmit antenna gain & 30 & dBi \\ \hline \(G_{r}\) & Receive antenna gain & 30 & dBi \\ \hline \(d_{1}\) & Distance between UAV\({}_{1}\) (UE\({}_{1}\)) and BS & 1 & m \\ \hline \(d_{2}\) & Distance between UAV\({}_{2}\) (UE\({}_{2}\)) and BS & 5 & m \\ \hline \(N_{0}\) & Noise power & \(-80\) & dBm/THz \\ \hline \(B\) & Total bandwidth & 2.087 & THz \\ \hline \(M\) & Number of sub-windows of frequencies & 5 & - \\ \hline \(N_{1}\)-\(N_{5}\) & Sub-band in each sub-window & 10 & - \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Simulation Parameters. On the multiple sub-bands, we adopt different power allocation strategies, i.e., equal power allocation, water-filling power allocation, and the proposed power optimization strategy. The simulation results are shown in Figures 2 and 3. The data rate of our proposed joint optimization strategy is compared with that of the equal power allocation strategy and water-filling power optimization scheme, as shown in Figure 2. The results show that the data rate of the proposed joint optimization adaptive modulation strategy is close to the upper bound of water-filling strategy. The data rate of equal-power allocation strategy is lowest due to under-utilization of channel varying and distance-dependent frequency characteristics. Figure 3 illustrates the convergence behavior of the proposed BCD method for different \(P_{t}\). To speed up the convergence of the BCD method, we initialize the frequency \(f^{(0)}\) by equally dividing the bandwidth in each frequency sub-window. The time complexities of the two convex sub-problems in each iteration of the BCD method are polynomial. Therefore, from the figure, we can observe that the BCD method needs about 10 iterations to converge, which can also be deduced by the complexity analysis in Algorithm 2, i.e., \(O(\sqrt{MN_{m}}log(1/\delta_{s}))=O(\sqrt{5\times 10}log(1/0.1))\approx 12.2\). Therefore, the proposed method overall has a relatively fast convergence speed. From Figure 4, when the user moves farther from the base station, the modulation switch probability is higher than in the other two situations. When the user moves closer to the base station, the modulation switch probability is slightly higher than that of not moving as expected. Moreover, when the switch threshold \(\gamma_{th}\) or \(B(d)_{min}\) increases, the modulation  
Figure 3: Complexity vs Pt when \(B_{available}=2.087\times 10^{12}\) Hz. Figure 2: The data rate comparison of three different power optimization schemes. switch probability becomes smaller because the range of the satisfying modulation rule threshold enlarges. From Figure 5, we can see the data rate of the proposed HM-QAM scheme is higher than the other two modulation schemes. The data rate of HBM is higher than HM, as expected. However, we can see from Figure 6 that the bit error rate (BER) of HM-MQAM is higher than that of HBM and HM modulation strategies in low SNR regions. This is because the system trades off the bit error rate for an increase in the data rate. It represents a compromise between reliability and efficiency. Figure 4: The modulation adjustment probability with user’s mobility. Figure 5: The data rate of different modulation schemes. Figure 6: BER comparison under different modulation schemes.