# Model-Driven Deep-Learning-Based Underwater Acoustic OTFS Channel Estimation
## 4 Numerical Results
### System Performance under Simulated UWA Channel  
The system performance is analyzed using the simulated UWA channel with the statistical channel model. The parameters of the simulated UWA channel are specified as presented in Table 3. For the analysis of the Doppler shift's impact on OTFS channel estimation in UWA channel, the following parameters are employed: The number of multipaths is set at 4 and 8, with a maximum multipath delay of 100 ms. These eight paths are randomly distributed within the maximum delay range, and their channel gains conform to independent Rayleigh distributions. The speed of sound, denoted as \(c\), is fixed at 1500 m/s. To investigate the influence of Doppler shift, node movement speeds in water are set to 1, 2, and 3 knots, corresponding to maximum Doppler expansions of 2, 4, and 6 Hz. The Doppler coefficients for each path are generated uniformly from the range of \([-f_{D},f_{D}]\) with equal probability. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Input Size** & **Output Size** \\ \hline Conv + ReLU & (32, \(3\times 3\)) & (8, \(M\), \(N\)) \\ Conv + BN + ReLU & (32, \(3\times 3\)) & (8, \(M\), \(N\)) \\ Conv + BN + ReLU & (32, \(3\times 3\)) & (8, \(M\), \(N\)) \\ Conv + BN + ReLU & (32, \(3\times 3\)) & (8, \(M\), \(N\)) \\ Conv & (2, \(3\times 3\)) & (2, \(M\), \(N\)) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters of DnCNN for UWA OTFS channel estimation. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Value** \\ \hline Number of multipaths & 4, 8 \\ Maximum multipath delay/\(\tau_{\text{max}}\) & 100 ms \\ Channel gain of multipaths/\(h_{i}\) & Rayleigh distribution \\ Speed of sound/\(c\) & 1500 m/s \\ Moving speed/\(v_{\text{max}}\) & 1, 2, 3 knots \\ Maximum Doppler spread/\(f_{D}\) & 2, 4, 6 Hz \\ Doppler distribution & equal probability \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameters for simulation of UWA channel. Figure 3 shows the MSE performance under simulated UWA channel with four multipaths. The proposed DnCNN-based channel estimation outperforms the OMP channel estimation at various Doppler shifts. Compared with OMP channel estimation, the proposed DnCNN-based channel estimation has about 0.9 dB gain at 4 Hz Doppler shift and 1.7 dB gain at 6 Hz Doppler shift, at an MSE of \(0.8\times 10^{-3}\). The heuristic nature of OMP makes it difficult to find the optimal solution. The DnCNN-based method can overcome this limitation by data training fitting to obtain the final optimal solution. With the well-trained neural network, the proposed method can address non-Gaussian noise and improve the estimation accuracy by denoising. Comparing the performance of different scales of Doppler shifts for OMP channel estimation, at an MSE of \(0.8\times 10^{-3}\), the SNR required for a 2 Hz Doppler shift is about 1.4 dB, which is 2.8 dB lower than that for 4 Hz and 6 Hz Doppler shift, respectively. At the same MSE, for the proposed DnCNN-based OTFS channel estimation, the required SNR for a 2 Hz Doppler shift is about 1.3 dB, which is 1.9 dB lower than that for 4 Hz and 6 Hz Doppler shift, respectively. It can be seen that the DnCNN-based channel estimation with different Doppler shifts shows a narrower performance gap compared with the OMP channel estimation. In UWA channels, a larger Doppler shift can lead to channel expansion and introduce fractional delay, diminishing the sparsity of channel information in the Doppler domain. This poses a challenge for accurate channel estimation using the OMP algorithm, especially when the Doppler shift is large. The proposed DL-based method has better accuracy and robustness in various Doppler scales under complex UWA channels. Figure 4 shows the MSE performance under simulated channel with eight multipaths. Similarly, the proposed DnCNN-based channel estimation outperforms the OMP channel estimation at various Doppler shifts.