# OTFS-IM Modulation Based on Four-Dimensional Spherical Code in Air-to-Ground Communication
## 2 System Model
The index set can then be represented as \(\mathbf{I}=\{k(j-1)+1,k(j-1)+2,\cdots,k(j-1)+k\}\), where \(j\) comes from the set \(\{1,2,\cdots,n/k\}\) and is an integer in general. For example, the index selection lookup table when \(n=8,k=2\) is depicted in Table 1. For the \(\beta\)-th sub-block, the index selection is a subset of the index set \(\mathbf{I}\), denoted as \(\mathbf{I}_{\beta}\). The number of index selection bits is \(p_{1}=\lfloor\log_{2}(n/k)\rfloor\). So far, the total bits in a sub-block are as follows:  
\[p=p_{1}+p_{2}=\left\lfloor\log_{2}(\frac{n}{k})\right\rfloor+\frac{k}{2}\log_{ 2}(M_{c}) \tag{2}\]  
so the spectral efficiency is \(\mathrm{SE}=p/n\). Finally, the symbols of the \(\beta\)-th sub-block can be written as:  
\[\mathbf{X}_{\beta}=\{x(1),x(2),\cdots,x(n)\},x(i)=\left\{ \begin{array}{l} \mathbf{s}_{\beta},i\in\mathbf{I}_{\beta}\\ 0,i\
otin\mathbf{I}_{\beta}\end{array} \right. \tag{3}\]  
For the subsequent OTFS modulation, the modulated sub-blocks are organized into the OTFS frame \(\mathbf{X}_{\rm DD}\) along the Doppler axis through the frame cre  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Bits** & **Indices** & **Sub-Blocks** \\ \hline
00 & [1 2] & [s\({}_{1}\),\({}_{5}\),\({}_{2}\),0,0,0,0,0] \\
01 & [3 4] & [0,0,\({}_{5}\),\({}_{1}\),\({}_{5}\),0,0,0,0] \\
10 & [5 6] & [0,0,0,0,\({}_{5}\),\({}_{1}\),\({}_{5}\),2,0] \\
11 & [7 8] & [0,0,0,0,0 TF domain is obtained from \(\mathbf{X}_{\text{DD}}\) via Inverse Symplectic Fast Fourier Transform (ISSFT). Then, it is converted to the time domain expressed as \(\mathbf{S}\) by Heisenberg transform and shaping pulse:  
\[\mathbf{S}=\mathbf{G}_{tx}\mathbf{F}_{M}^{H}(\mathbf{F}_{M}\mathbf{X}_{\text{DD }}\mathbf{F}_{N}^{H})=\mathbf{G}_{tx}\mathbf{X}_{\text{DD}}\mathbf{F}_{N}^{H} \tag{4}\]  
where \(\mathbf{G}_{tx},\mathbf{F}_{num},(\cdot)^{H}\) are the transmitted shaping pulse matrix, \(num\)-point normalized Fourier transform matrix, and Hermitian transformation, respectively. Thus, the transmitted signal \(\mathbf{s}\) in the time domain can be expressed as:  
\[\mathbf{s}=vec(\mathbf{S})=(\mathbf{F}_{N}^{H}\otimes\mathbf{G}_{tx})\mathbf{x} \tag{5}\]  
where \(vec(\cdot)\) is the column-wise operation, \(\otimes\) is the Kronecker product, and \(\mathbf{x}\) is the column-wise vector of \(\mathbf{X}_{\text{DD}}\). The signal \(\mathbf{s}\) is transmitted over the time-varying channel, and then the received signal \(\mathbf{r}\) can be expressed as:  
\[\mathbf{r}=\mathbf{H}\mathbf{s}+\mathbf{n} \tag{6}\]  
where \(\mathbf{n}\in\mathbb{C}^{MN\times 1}\) is the additive Gaussian white noise with mean 0 and variance \(N_{0}\), and \(\mathbf{H}\in\mathbb{C}^{MN\times MN}\) is the channel matrix in the time domain. The expression of \(\mathbf{H}\) is as follows:  
\[\mathbf{H}=\sum_{i=1}^{p}h_{i}\mathbf{\Pi}^{i}\mathbf{\Delta}^{(k_{i})} \tag{7}\]  
where \(P\) is the number of propagation paths, \(h_{i}\) is the channel gain of the \(i\)-th path, \(\mathbf{\Pi}=circ([0,1,\cdots,0]_{MN\times 1}^{T})\in\mathbb{C}^{MN\times MN}\) is the permutation matrix (forward cyclic shift) reflecting the delay, and \(\Delta=diag\Big{\{}e^{2\pi(0)/MN},e^{2\pi(1)/MN},\cdots,e^{2\pi(MN-1)/MN} \Big{\}}\in\mathbb{C}^{MN\times MN}\) is the diagonal matrix reflecting the Doppler. The received signal \(\mathbf{r}\) can also be rewritten as an \(M\times N\) matrix \(\mathbf{R}\) by the inverse column-wise operation.