# OTFS-IM Modulation Based on Four-Dimensional Spherical Code in Air-to-Ground Communication
## 1 Introduction  
With the continuous development of information technology, the space range of information service has expanded from well-developed terrestrial communication to air-to-ground (A2G) communication and satellite communication. Unmanned aerial vehicles (UAVs) have become a very attractive research topic in A2G communication and satellite communication owing to their various advantages, such as mobile flexibility, small size, and low cost [1, 2, 3]. UAVs can have significant influence in agriculture, aerial surveys, and public rescue, and can even be used as a satellite communication relay to expand coverage and ensure communication quality. The drone network has also been developed comprehensively, such as through a novel offset feedback fractionally spaced equalization architecture, the design of drone-aided IoT networking, and a high-mobility support system for NTN connectivity [4, 5, 6]. In terms of profit, a UAV can be used as a cost-effective air platform for A2G communications. However, one major feature of A2G communication is its dynamic mobility. For example, in common UAV scenarios, speeds of 161 km/h may be reached. In certain scenarios, such as when using military UAVs and future aircraft, the flight speed can reach even more than 500 km/h, at which speeds they experience large path loss and serious Doppler spread, which may hinder the quality of the A2G link [7, 8]. Therefore, it is very important to study robust A2G wireless links for high-speed A2G communication. OFDM technology is widely used in terrestrial communication, such as 4G, because of the advantage of orthogonal subcarriers. However, with the increase in Doppler spread, orthogonality is no longer maintained [9]. Orthogonal time-frequency space (OTFS) modulation, a novel 2D modulation in the delay-Doppler (DD) domain, has been proposed to solve this problem; this allows the time and frequency diversity to be fully exploited by the 2D convolution between the modulation symbols and the effective channel in the DD domain [10]. The effective channel in the DD domain achieves near-constant channel gain, which is quite different from time-varying channels in the time-frequency (TF) domain. At the same time, the channel is sparse because there are limited reflectors in the propagation path, which is beneficial to the design of the receiver [11; 12]. According to the characteristics of the channel, many studies on signal detection have been carried out, such as on LMMSE, MP, MRC, and Bayes schemes [13; 14; 15; 16]. In practical signal detection, the correct channel state information is also indispensable. Some effective approaches have also been developed, such as the classical threshold-based method, orthogonal matching tracking (OMP), sparse Bayes, and our previous work on a DNN-based scheme [17; 18; 19; 20]. Additionally, some novel structure designs for OTFS have been proposed. Explicitly, in order to convey additional information in high-mobility scenarios, a new OTFS scheme with index modulation (OTFS-IM) is proposed in [21], which explores a new dimension. Compared with the conventional OTFS modulation, OTFS-IM modulation can convey additional index information by combining active and silent DD domain symbols to improve spectral efficiency. Moreover, the transmitted data in DD domain are sparser, which effectively reduces the interference and also leads to a significant gain in bit error rate (BER) performance. The authors in [22] explored the potential of OTFS-IM modulation and proposed an OTFS-IM with dual-mode modulation (OTFS-DM-IM), where the symbols in the DD domain are modulated by two constellation matrices to transmit more additional index bits, thus achieving a higher transmission rate. To increase the spectral efficiency of the existing OTFS-IM schemes, there is a new transmission scheme called OTFS-I/Q-IM [23]. By extending the grid index of OTFS systems to in-phase and quadrature dimensions, this new scheme has two index dimensions and \(L\)-ary PAM constellation symbols to transmit information.