## 4 Simulation and Lake Experiment Results  
### Results of Simulations  
In this section, the performance of the proposed channel estimation methods in the OTFS system is verified by simulations. It should be noted that all simulations and experiments are run on the MATLAB 2022 platform. Complex-value UWA channels are simulated according to [43]. The system parameters of the UWAc system are set as Table 1.  
According to the characteristics of the channel, we consider an OTFS system with the parameters shown in Table 2. The parameters explained in the previous section of the proposed algorithm are set in Table 3.  
Figure 3 describes the channel structure in the time domain (Figure 3a) and the scattering function of the UWA channel (Figure 3b). It is clear that at least nine clusters of multipath with obvious time variance are shown in Figure 3a. The Doppler offset of each path in the channel in the DD domain is plotted in Figure 3b. We can find that it tends to be different with taps. Moreover, the maximum range of the Doppler spread is near \([-0.7,0.7]\) in the first cluster of this channel. In Figure 4, we give the estimation accuracy comparison of three different single-direction adaptive algorithms. IPNLMS represents that no denoising method is used in the estimate. IPNLMS-TA represents that the estimate of each filter is denoised by averaging the \(N_{h}\) estimates. From Figure 4, we find that the effect of the denoising processing is positive, especially in low-noise ratio (SNR) situations. The performance gain is about 0.02 dB for the denoising method 1. Therefore, we adopt denoising method 1 in all of the following simulations. Figure 5 plots an estimation accuracy versus SNR of various adaptive algorithms. To clarify the impact of denoising method 2 on single-directional IPNLMS, we first compare the normalized mean square error (NMSE) among several methods, as shown in Figure 5a. IPNLMS-single represents that the estimate of each filter is denoised by denoising method 1, only averaging the \(N_{h}\) estimates. IPNLMS-MA-single and IPNLMS-EMA-single represent the adaptive channel estimation technique that includes an MA or EMA filter with a step size of 5 in the IPNLMS-single basis, respectively. The performance improvement of adding an MA or EMA filter to IPNLMS-single is evident from the graph, both ranging from 0 to 0.1 dB. By comparing IPNLMS-MA-single and IPNLMS-EMA-single, there is little difference in performance between the two MA filters. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Water Depth & 500 (m) \\ Height of Transmitter & 200 (m) \\ Height of Receiver & 200 (m) \\ Distance between Transmitter to Receiver & 1000 (m) \\ Underwater Sound Velocity & 1500 (m/s) \\ Center Frequency \(f_{c}\) & 14.5 (kHz) \\ Bandwidth \(B\) & 5 (kHz) \\ Vessel Speed & 0 (m/s) \\ Spreading Factor & 1.7 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of the UWAc system. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline \(M\) & 64 \\ \(N\) & 32 \\ Bandwidth \(B\) & 5 (kHz) \\ Subcarrier Interval \(\Delta f\) & 78.1 (Hz) \\ Symbol Interval & 0.2 (ms) \\ Modulation Type & QPSK \\ \(M_{\tau}\) & 32 \\ \(N_{\
u}\) & 32 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters of OTFS. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline \(\mu\) & 0.5 \\ \(\alpha\) & 0.9 \\ \(\beta\) & 0.1 \\ \(\gamma\) & \(6\times 10^{-7}\) \\ \(\delta\) & 0.01 \\ \(\epsilon\) & 0.01 \\ \(N_{h}\) & 500 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameters of Proposed Adaptive Channel Estimation Method. After establishing that the MA filter can be employed to enhance the accuracy of the adaptive channel estimation algorithm, we compare the multidirectional moving average IPNLMS-TA method with the adaptive channel estimation technique algorithm, in which the MA filter is unemployed in Figure 5b. IPNLMS-MuD represents the performance of the multidirectional IPNLMS-TA algorithm.