# OTFS-IM Modulation Based on Four-Dimensional Spherical Code in Air-to-Ground Communication
## 2 System Model  
We consider an OTFS-IM frame with \(M\) subcarriers and \(N\) symbols. The subcarrier spacing and symbol duration are denoted as \(\Delta f\) and \(T\), respectively. Therefore, an OTFS-IM frame with duration \(NT\) and bandwidth \(M\Delta f\) could be treated as an \(M\times N\) grid in the DD domain. The block diagram of the OTFS-IM system with four-dimensional spherical code modulation is shown in Figure 1. It is assumed that there are \(n\) symbols and \(k\) active symbols in a sub-block, where \"symbols\" here represents bins in the DD grid. Then, an OTFS-IM4 frame can be divided into \(g=MN/n\) groups. The transmitted bit set \(\mathbf{u}\) is evenly divided into \(g\) groups, where each sub-block transmits \(p=p_{1}+p_{2}\) bits. Specifically, the \(p_{1}\) bits and \(p_{2}\) bits are used for index selection and constellation mapping, respectively. Since all OTFS-IM4 sub-blocks are independent of each other, the \(\beta\)-th sub-block is taken as an example for description. As mentioned before, increasing the dimensions enables a larger Euclidean distance between constellation points when the design is elegant. We mainly consider four-dimensional spherical code modulation as constellation mapping in this paper. To begin with, \(\Omega_{L}\) represents the surface of the unit sphere in \(L\)-dimensional space:  
\[\Omega_{L}=\left\{\mathbf{c}=\{c_{1},c_{2},\cdots,c_{L}\}\in\mathbb{R}^{L} \middle|\sum_{l=1}^{L}c_{l}^{2}=1\right\} \tag{1}\]  
where \(\mathbf{c}\) is a point on \(\Omega_{L}\), and \(c_{l}\) is the amplitude of the \(l\)-th dimension. Taking \(M_{c}\) points of the surface as the modulation constellation alphabet \(\mathcal{A}\), the spherical code can be expressed as \(K(L,M_{c},\theta)\), where \(\theta\in(0,\pi]\) represents the angle between two point vectors. According to the Law of Cosines, the minimum Euclidean distance between two points of \(K\) is \(d_{\min}=\sqrt{2-2\cos\theta}\) when \(\theta\) reaches the maximum attainable value. [PERSON], [PERSON], and [PERSON] have given the arrangements in three, four, and five dimensions with \(M_{c}\)= 4, 5,..., 130 [27]. For the four-dimensional spherical code, we choose \(L=4\), \(M_{c}=16\), and  
Figure 1: Block diagram of OTFS-IM4 system. \(\theta=67.19^{\circ}\), which maximizes the minimum distance \(d_{\rm min}=1.1066\) between constellation points [27]. Among the 16 points generated by spherical code \(K(4,16,67.19^{\circ})\), each constellation is a four-dimensional vector representing four information bits and is listed as a column of the constellation matrix \(C_{4,16}^{\prime}\):  
\[C_{4,16}^{\prime}=\left[ \begin{array}{cccc}0.2719&-0.9228&-0.2660&-0.0610\\ 0.1438&-0.3165&0.0030&-0.9376\\ 0.1566&-0.5843&0.7734&-0.1897\\ -0.2404&-0.2245&-0.9414&0.0747\\ 0.7397&-0.3101&0.2502&0.5423\\ 0.7252&-0.0080&-0.6189&-0.3015\\ 0.2421&0.6419&0.5150&0.5139\\ -0.6990&0.4390&-0.3000&0.4781\\ 0.7625&0.2618&0.4506&-0.3834\\ -0.2276&-0.4922&-0.1710&0.8226\\ -0.3001&0.3434&0.6808&-0.5732\\ -0.6143&0.3162&-0.3791&-0.6156\\ 0.2293&0.9096&-0.2554&-0.2340\\ 0.3613&0.3580&-0.5359&0.6739\\ -0.5966&-0.0533&0.6839&0.4165\\ -0.7727&-0.6169&-0.0775&-0.1278\end{array} \right]^{T}\)  
Because the four-dimensional constellation point \(\mathbf{c}_{i}=(c_{i,1},c_{i,2},c_{i,3},c_{i,4})\) of \(\mathcal{A}\) is a four-dimensional vector, two two-dimensional symbols will be used jointly to carry a four-dimensional symbol in the OTFS-IM4 system. \(\mathbf{c}_{i}\) is transmitted by two-dimensional symbols \(s_{1}=c_{i,1}+jc_{i,2}\) and \(s_{2}=c_{i,3}+jc_{i,4}\). These two symbols are denoted as \(\mathbf{s}_{\beta}\). Therefore, the number of active symbols \(k\) should be a multiple of 2, which means there are \(k/2\) four-dimensional symbols in a sub-block. Consequently, the number of constellation mapping bits is \(p_{2}=k/2\log_{2}(M_{c})\). To facilitate signal detection, we consider \(k\) as continuous, and non-overlap symbols are selected together as the active symbols among \(n\) symbols in a sub-block.