# Multiple-Mode Orthogonal Time Frequency Space with Index Modulation
## 2 System Model  
### The Transmitter  
We considered an MM-OTFS-IM system with \(M\) subcarriers and \(N\) symbols, where the subcarrier spacing and symbol duration are \(\Delta f\) and \(T\), respectively. The block diagram of the MM-OTFS-IM transmitter is depicted in Figure 1. At each time instant, a number of \(p\) bits arrive to the transmitter and are equally split into \(g\) groups with \(p_{a}=p/g\) bits in each group. Each group of \(p_{a}\) bits is then divided into \(p_{1}\) bits and \(p_{a}\) bits, separately referred to as index bits and information bits, to generate an OTFS subblock containing \(s\) points on the DD domain, where \(s=MN/\)g. \(N\) and \(M\) are the number of resource units along the Doppler dimension and delay dimension, respectively. As shown in Figure 1, the first \(p_{1}\) bits flow into the index selector to select the constellation mode carrying the modulation information of each grid point resource in the subblock, and the remaining \(p_{2}\) bits are the multi-constellation modulation information carried by all grid point resources in the subblock. The size of each subblock is \(s=mn\), where the number of grids along the delay dimension of the subblock is \(m\), and the number of grids along the Doppler dimension is \(n\). The frame architecture of the MM-OTFS-IM in the DD domain is shown in Figure 2; the left side is the signal frame of the MM-OTFS-IM system in the DD domain, which is composed of \(g\) subblocks, and the structure of each subblock is shown on the right side of Figure 2, each grid in the subblock carries the modulation signal of a different constellation mode. Since each subblock has the same processing procedure, for brevity, we take the \(\beta\)-th subblock as an example, where \(\beta=1,\cdots,g\). The index selector determines the order of elements in set \(\Omega=\{1,\cdots,mn\}\) according to the incoming \(p_{1}\) bits and obtains  
Figure 1: Block diagram of the MM-OTFS-IM transmitter. Figure 2: Frame architecture of the MM-OTFS-IM in the DD domain. the combination information of different constellation patterns adopted by \(mn\) grids in the subblock  
\[\mathbf{S}_{w^{\beta}}=\left\{S_{w_{1}^{\beta}},\cdots,S_{w_{mn}^{\beta}}\right\} \tag{1}\]  
where \(S_{w_{i}^{\beta}}\) represents the constellation mode adopted by the \(i\)-th lattice point in the \(\beta\)-th subblock, and the average power of \(\{S_{w_{1}^{\beta}},\cdots,S_{w_{mn}^{\beta}}\}\) has been normalized to unit power. The remaining \(p_{2}\) bits use the corresponding constellation mode to generate \(mn\) modulation signals. To achieve index modulation, the different constellation modes need to satisfy \(S_{i}\cap S_{j}=\varnothing\), where \(\forall i,j\in\{1,\cdots,mn\}\), \(i\
eq j\). By rotating the constellation diagram of \(M_{mod}\)-ary phase shift keying (PSK) \(mn\) times, where the rotation angle is \(2\pi(r-1)/mmM_{mod}\), \(r\in\{1,\cdots,mn\}\) the signal constellations of \(M_{mod}\)-ary PSK with \(mn\) different modes can be obtained [14]. Figure 3 shows the BPSK signal constellation diagram of eight different modes, which is obtained by rotating the original BPSK signal constellation eight times. Therefore, \(p_{1}\) and \(p_{2}\) can be calculated as \(p_{1}=\lfloor\log_{2}((mn)! )\rfloor\) and \(p_{2}=mn(\log_{2}(M_{\text{mod}}\ ))\), where \(\lfloor\cdot\rfloor\) is the floor function. Afterward, the OTFS block creator generates the DD domain signal of all the subblocks  
\[\mathbf{X}^{DD}= \begin{bmatrix}x_{0,0}&\cdots&x_{0,N-1}\\ \vdots&\ddots&\vdots\\ x_{M-1,0}&\cdots&x_{M-1,N-1}\end{bmatrix}  \tag{2}\]  
where \(x_{x,\iota}\in\{S_{1},\cdots,S_{mn}\}\), \(\kappa=0,...,M-1\) and \(\iota=1,...,N-1\).