# Hamming Distance Optimized Underwater Acoustic OTFS-IM Systems
## 2 OTFS-IM for Underwater Acoustic Communication System  
### System Model  
The OTFS-IM system improves the spectral efficiency of the system by making the signal transmitted in the DD domain carry a portion of the index bits through index modulation. When defining an OTFS data frame size as \(M\times N\), a data frame occupies a duration of \(NT\) and a bandwidth of \(B=M\Delta f\) while \(\Delta f\) denotes the subcarrier interval and \(T=1/\Delta f\) denotes the duration of each group of symbols. As shown in Figure 1, the \(A\) bit information is divided into \(g\) groups with \(p=A/g\) bit information in each group by a bit splitter at the system transmitter, and the OTFS data frames in the corresponding DD domain are also divided into \(g\) grid groups. The size of each DD domain grid group is \(n=MN/g\). The \(p\) bit information of each data group is divided into two parts: \(p_{1}\) bit and \(p_{2}\) bit. \(p_{1}\) bit is the index bit used to index the position of the corresponding active sub-grid in the DD domain sub-grid group. When using \(k\) to denote the number of grids in a sub-grid group that are activated by a sub-grid, a total of \(C_{n}^{k}\) (\(C\) is a binomial coefficient) seed grid activation combinations are available, and the number of bits transmitted by the index selector can be expressed as:  
\[p_{1}=\left\lfloor\log_{2}C_{n}^{k}\right\rfloor \tag{1}\]  
\(\left\lfloor\cdot\right\rfloor\) is rounded down. Taking the \(\beta\)th group of the sub-grids as an example, and selecting \(k\) sub-grids from the \(n\) sub-grids to activate the bearer information, the index selector output can be expressed as:  
\[I_{\beta}=\left\{i_{\beta,1},i_{\beta,2},\cdots,i_{\beta,k}\right\} \tag{2}\]  
where \(i_{\beta,l}\in[1,2,\cdots,n]\), \(\beta=1,2,\cdots,g,l=1,2,\cdots,k\). The other part of the \(p_{2}\) bit is used for the traditional constellation symbol mapping and can be represented as:  
\[p_{2}=k\log_{2}W \tag{3}\]  
\(W\) represents the dimension of constellation mapping, and the output of constellation mapper in group \(\beta\) can be expressed as:  
\[S_{\beta}=\left\{s_{\beta,1},s_{\beta,2},\cdots,s_{\beta,k}\right\} \tag{4}\]  
where \(s_{\beta,\alpha}\in W,\beta=1,2,\cdots g\alpha=1,2,\cdots,k\).The constellation symbols of \(S_{\beta}\) are mapped correspondingly according to the position of the activated sub-grid selected by \(I_{\beta}\). The data output from the \(\beta\)th sub-grid groups after the index mapping can be expressed as:  
\[X_{\beta}=\left[x_{\beta}(1),x_{\beta}(2),\cdots,x_{\beta}(n)\right]^{T} \tag{5}\]  
\[x_{\beta}(\gamma)=\left\{\begin{matrix}s_{\beta,\gamma},&\gamma\in I_{\beta} \\ 0,&\gamma\
ot\in I_{\beta}\end{matrix}\right. \tag{6}\]  
Figure 1: Block diagram of the underwater acoustic OTFS-IM system.  
After completing index modulation for each sub-grid group in the DD domain, \(X_{\beta},\beta=1,2,\cdots,g\) is obtained. After the OTFS block generator, the DD-domain signal is the output, and after the inverse symplectic Fourier transform (ISFFT), the time frequency domain signal is obtained:  
\[X^{FT}=F_{M}X^{DD}F_{N}^{H} \tag{7}\]  
\(F_{M}\) and \(F_{N}\) are the Fourier transform matrices, and the superscript \(H\) is the conjugate matrix. The time frequency domain signal is obtained in the time domain by the Heisenberg transform:  
\[S=F_{M}^{H}X^{FT}=X^{DD}F_{N}^{H} \tag{8}\]  
Finally, to avoid inter-symbol interference (ISI) between the OTFS blocks, a cyclic prefix (CP) is added to each data frame to obtain the complete transmit signal.