# OTFS-IM Modulation Based on Four-Dimensional Spherical Code in Air-to-Ground Communication
## 4 Simulation Results
The comparison of theoretical values and simulation results for BER and PAPR performance in the OTFS-IM4 system are well illustrated in the following. First, the BER comparison is shown in Figure 6. Since the derived BER theoretical value needs to traverse the entire signal space, the calculation complexity of the theoretical value is relatively high. In order to facilitate the solution, the simulation parameters are specified as follows. The OTFS frames are simplified to \(M=4\) and \(N=2\). Since the symbol duration \(T\) and the sub-carrier spacing \(\Delta f\) determine the maximum delay \(\tau_{\text{max}}\) and maximum Doppler \(\
u_{\text{max}}\) that can be supported for the channel [31], we adapt the channel state to \([l_{i},k_{i}]=\{[0,0],[1,1]\}\) and \(h_{i}\sim CN(0,1/P)\). Other parameters are the same as in the four-dimensional modulation. There is exactly one sub-block in a frame in this case, i.e., \(g=1\), \(p=6\), and \(n_{\text{x}}=64\). It can be seen that the simulation results and the theoretical values become closer as \(E_{b}/N_{0}\) increases, which verifies the correctness of the derived BER performance. Figure 5: BER performance comparison between OTFS-IM4 and OTFS-IM2 in the EVA channel. Figure 6: BER comparison of theoretical values and simulation results in the OTFS-IM4 system. Next, Figure 7 simulates the CCDF of the PAPR for different OTFS schemes and different sub-block situations. The theoretical analysis results show that PAPR in the OTFS system is related to the parameter \(N\), while in the OTFS-IM system, PAPR is also related to the proportion of the active symbols in the sub-block. Therefore, different \(N\) values are simulated to verify the analysis results. We take \(N=16\) and \(N=32\) for an example. Meanwhile, BPSK modulation is still used in the OTFS-IM2 system. When \(N=16\), the simulation results are as shown in Figure 7, and there is the following specific analysis. The spectral efficiency of the two solid curves is \(\mathrm{SE}=3/4\). It can be seen that the PAPR performance of the OTFS-IM2 is similar to that of the OTFS-IM4. When the spectral efficiency is \(\mathrm{SE}\approx 1\), as shown by the two dashed lines, the PAPR performance of the OTFS-IM4 is about 0.1 dB better than that of the OTFS-IM2. As for the theoretical analysis in Section 3, the PAPR performance of these two schemes is only slightly different. In addition, it is proved that the PAPR increases linearly with the proportion \(k/n\) of active symbols in a sub-block. Theoretically, the PAPR with \(k/n=1/4\) is 3 dB smaller than with \(k/n=1/2\), which corresponds to the simulation results. For \(N=32\), the simulation results are depicted in Figure 7. Although there are slight differences in performance compared with that in Figure 7, the trend is the same. When \(\mathrm{CCDF}=10^{-3}\), the PAPR with \(\mathrm{SE}=3/4\) is approximately equal in OTFS-IM2 and OTFS-IM4, and the four-dimensional modulation has 0.1 dB gain over two-dimensional modulation at \(\mathrm{SE}\approx 1\). Comparing Figure 7,b horizontally, PAPR increases by 3 dB with the doubling of \(N\) when spectral efficiency is the same, which is also consistent with the theoretical analysis. Additionally, the PAPR of the OTFS system is proportional to \(N\) but has a threshold.