# OTFS-IM Modulation Based on Four-Dimensional Spherical Code in Air-to-Ground Communication
## 4 Simulation Results  
In Sections 2 and 3, the general model of the proposed system and the mathematical expression of performance have been discussed in detail. Then, in this section, we simulate and examine the reliability and robustness of the proposed scheme and compare it with the conventional OTFS-IM scheme [18]. Additionally, the simulation results and mathematical analysis are also compared to verify the correctness of the derivation. As mentioned before, we focus on the four-dimensional spherical code \(K(4,16,67.19^{\circ})\), whose constellation matrix \(C_{4,16}^{\prime}\) is normalized, and two two-dimensional symbols convey one four-dimensional symbol, meaning that the total energy of a sub-block is \(k/2\). However, in the traditional OTFS-IM system, the total energy of \(k\) active symbols in a sub-block is \(k\). In order to maintain the same energy, the actual amplitude of each constellation point of \(\mathcal{A}\) should be \(\sqrt{2}\), which changes the constellation matrix as \(C_{4,16}=\sqrt{2}C_{4,16}^{\prime}\) and the minimum distance as \(d_{\min}=\sqrt{2}\times 1.1066\). An example of the index selection relationship is illustrated in Table 1, where a sub-block transmits a four-dimensional constellation point with the total bits \(p=6\). Therefore, the bit signal-to-noise ratio is \(E_{b}/N_{0}=k/(pN_{0})\). Furthermore, we consider an OTFS frame with \(M=64\) and \(N=8\). The operating carrier frequency is \(f_{c}=3\,\mathrm{GHz}\), and the subcarrier spacing is \(\Delta f=15\,\mathrm{kHz}\). As the number of paths in A2G communication is generally small, we model the channel as a two-path model, with reference to the model of the NTN-TDL series of the 3 GPP standard [30]. The maximum speed of the UAV is set to 500 km/h, and the maximum delay is about 4 us. In Figure 2, we compare the BER performance of our proposed OTFS-IM4 scheme with that of the traditional OTFS-IM2 scheme. According to reference [17], the number of transmitted bits in a sub-block of the OTFS-IM2 system is \(p=\log_{2}(C_{n}^{k})+k\log_{2}(M_{c})\). For fairness, sub-block parameters \(n=4,k=1\) and BPSK modulation are selected in the traditional two-dimensional modulation to transmit the same number of bits as the four-dimensional modulation. At this time, the lookup table in the OTFS-IM2 system has only four options to choose, which is the same as in the proposed scheme, and the spectral efficiencies are all \(\mathrm{SE}=3/4\). As discussed in Section 2, MMSE equalization and ML detection are performed successively for signal detection. It is obvious that the four-dimensional modulation has almost 2 dB gain compared with the two-dimensional modulation, which is consistent with the classic detection theory. Thus, the bit signal-to-noise ratio can be reduced under the acceptable BER, which plays an important role in saving the transmitting power and prolonging the flight time of the UAV. Since MMSE equalization needs to calculate the matrix inverse, the complexity is \(O(M^{3}N^{3})\), which exponentially increases as the OTFS frame size grows. In contrast, the complexity of the MRC algorithm is \(O(MN|L|n_{iter})\)[15], where \(|L|\) represents the number of distinct delay paths among \(P\) propagation paths, and \(n_{iter}<20\) denotes the number of iterations for the MRC algorithm. Therefore, the complexity of the MRC algorithm is lower than that of the MMSE algorithm. In addition, it can easily be used in the index modulation system for equalization by converting hard decisions into soft decisions. It can be seen that the performance of MRC is gradually better than MMSE when the \(E_{b}/N_{0}\) exceeds 10 dB. In particular, MRC equalization performs better than MMSE equalization, with more than 2 dB gain when the BER is \(10^{-4}\). However, there is zero padding in MRC equalization, which leads to a decrease in spectral efficiency [15]. In summary, there is a trade-off between application and performance. Therefore, an appropriate equalization method should be selected based on application scenarios in reality.