# OTFS-IM Modulation Based on Four-Dimensional Spherical Code in Air-to-Ground Communication
## 4 Simulation Results
Figure 3 presents the generalization of the OTFS-IM4 system at different speeds. To thoroughly investigate the effects of different Dopplers, we have taken into account both small UAV scenarios and higher-speed scenarios, such as those with military UAVs and future aircraft. Therefore, the selected flight speeds in simulation are 100 km/h and 500 km/h. As the speed increases, the BER performance behaves similarly, except there is a slight difference that proves the effectiveness of the OTFS-IM4 at different moving speeds. This means that under the same BER threshold and energy consumption, when transmitting signals are in the same communication range, a UAV in the OTFS-IM4 system can have a higher moving speed or have a larger communication range at the same speed. Figure 4 simulates the performance of an OTFS-IM system with a six-dimensional spherical code (denoted as OTFS-IM6) to analyze the robustness of the proposed system. According to the system model in Section 2, when the 64-point spherical code is used for six-dimensional modulation, the sub-block parameters are selected as \(n=6\) and \(k=3\). Since sub-blocks are organized along the Doppler axis, we choose the OTFS frame with \(M=64\) and \(N=12\) to ensure that the sub-block will not be separated. Other parameters remain the same as in Figure 2. As can be seen from Figure 4,  
Figure 3: BER performance for different speeds in two propagation paths. Figure 2: BER performance comparison between OTFS-IM4 and OTFS-IM2 in the two-path channel. equalization is about 4.5 dB lower than that of MRC when \(\text{BER}=10^{-4}\). Additionally, in the MRC equalization results, the six-dimensional modulation has a gain of more than 1.5 dB compared with the two-dimensional modulation. It is 0.5 dB higher than the gain between the OTFS-IM4 system and the OTFS-IM2 system in Figure 2. The conclusion drawn is that the higher the dimension, the better the performance. However, the detection is sub-block by sub-block in the proposed scheme using the ML method. The complexity of detecting a sub-block is \(O((M_{c})^{k/3}n/k)\) in the six-dimensional modulation and then the complexity of detecting an OTFS frame will be \(O(((M_{c})^{k/3}n/k)^{\delta})\). Therefore, the higher the dimension and the higher the number of constellation points, the higher the complexity. In this paper, four-dimensional modulation is chosen to take into account both performance and detection complexity. Additionally, we also simulate the BER performance of these two schemes in the EVA channel to verify the generality of the proposed scheme while the other parameters are the same as the two-path channel. It can be seen in Figure 5a that the four-dimensional modulation is still better than the two-dimensional modulation. For example, the OTFS-IM4 scheme achieves about 1.5 dB gain more than the OTFS-IM2 scheme when \(\text{BER}=10^{-4}\). Moreover, when QPSK is adopted in the two-dimensional modulation, the spectral efficiency is \(\text{SE}=1\), and the BER performance coincides with BPSK, which is the same conclusion as in the AWGN channel. Figure 5b demonstrates a comparison between the proposed scheme and BPSK and QPSK under the EVA channel, considering an increased number of symbols within an OTFS frame. The only difference from the conditions in Figure 5a is that the size of an OTFS frame is changed to \(M=64\), \(N=16\). At this time, the number of symbols in a frame is \(MN/n*k=64*16/8*2=256\). The same conclusion is reached as shown in Figure 5a. Due to the Doppler resolution \(\Delta\
u=1/NT\), the Doppler resolution improves when \(N\) increases. Therefore, the BER performance also improves accordingly, which explains the difference between the two figures. Results show that the proposed scheme is also applicable when the channel is more complex, which verifies the robustness and generality of the OTFS-IM scheme with four-dimensional spherical code. Figure 4: BER performance comparison between OTFS-IM6 and OTFS-IM2 in the two-path channel.