# Hamming Distance Optimized Underwater Acoustic OTFS-IM Systems
## 4 Simulation Analysis  
In this section, we first verify the trade-off between BER performance benefits and the spectral efficiency of OTFS-IM modulation techniques in underwater acoustic communication systems by simulations. Then, the performance gain of the Hamming distance optimization model in the OTFS-IM underwater acoustic communication system under two demodulation algorithms is analyzed, and the performance gain of the Hamming distance optimization model is verified. The channel parameters used in the simulations are shown in Table 3, and the time-varying underwater acoustic channel impulse response is shown in Figure 4.  
Based on the characteristics of the time-varying underwater acoustic channel, we set the parameters of the OTFS-IM system as \(M=256\) and \(N=128\); the signal bandwidth is set to 4 kHz; the subcarrier spacing is \(\Delta f=15.63\) Hz; the symbol spacing is set to \(T_{s}=0.25\) ms; and the CP prefix length is set to \(T_{cp}=16\) ms. The duration of an OTFS-IM frame can therefore be expressed as \(T_{OTFS-IM}=10.24\) s. The QPSK modulation mode is used in the simulation, while the channel is assumed to be completely known at the receiver end. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Index Bit** & **Active Sub-Grid** & **Sub-Grid Group** \\ \hline \(\left[0\quad 0\right]\) & \(\left[1\quad 2\right]\) & \(\left[s_{\beta,1}\quad s_{\beta,2}\quad 0\quad 0\right]^{T}\) \\ \(\left[0\quad 1\right]\) & \(\left[1\quad 3\right]\) & \(\left[s_{\beta,1}\quad 0\quad s_{\beta,2}\quad 0\right]^{T}\) \\ \(\left[1\quad 0\right]\) & \(\left[2\quad 4\right]\) & \(\left[0\quad s_{\beta,1}\quad 0\quad s_{\beta,2}\right]^{T}\) \\ \(\left[1\quad 1\right]\) & \(\left[3\quad 4\right]\) & \(\left[0\quad 0\quad s_{\beta,1}\quad s_{\beta,2}\right]^{T}\) \\ redundancy & \(\left[2\quad 3\right]\) & \(\left[0\quad s_{\beta,1}\quad s_{\beta,2}\quad 0\right]^{T}\) \\ redundancy & \(\left[1\quad 4\right]\) & \(\left[s_{\beta,1}\quad 0\quad 0\quad s_{\beta,2}\right]^{T}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Index mapping table for Hamming distance optimization. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Value** \\ \hline Surface height (m) & 100 \\ Transmitter height (m) & 20 \\ Receiver height (m) & 50 \\ Channel distance (m) & 1000 \\ Spread factor (kHz) & 1.7 \\ Doppler factor (v/c) & \(10^{-4}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Time-varying underwater acoustic channel parameters. The data transmitted by the OTFS-IM system consists of index bits and constellation modulation bits; therefore, the spectrum efficiency of OTFS-IM can be expressed as:  
\[\eta=\frac{\left|\log_{2}C_{n}^{k}\right|+k\log_{2}W}{n} \tag{19}\]  
It can be seen that the spectral efficiency of the OTFS-IM system is related to the packet size \(n\), the number of active sub-grids \(k\), the constellation mapping dimension \(W\), and other parameters. We first simulated the BER performance of the proposed OTFS-IM underwater acoustic communication system under different spectrum efficiencies. Dual validation of the relationship between BER performance and spectrum efficiency uses two different receiver side detection algorithms--the ML detection algorithm and the GD detection algorithm--and the receiver side uses MMSE equalization to counteract the effects of the channel on the transmitted signal. Figure 5 shows the spectral efficiency plots for different values of \((n,k)\). Figure 6 shows a comparison of the simulations of the two detection methods with different numbers of active sub-grids. The sub-grid group size is set to \(n=4\). Three conclusions can be drawn from the combination of Figures 4 and 5. Firstly, the spectrum efficiency of conventional OTFS systems can be achieved or even exceeded by means of index modulation for certain combinations of activation sub-grid numbers. At the same time, its system BER performance is better than that of conventional OTFS systems.