# Model-Driven Deep-Learning-Based Underwater Acoustic OTFS Channel Estimation
## 3 DL-Based OTFS Channel Estimation  
The OMP channel estimation algorithm relies on a sparse channel matrix. However, in UWA channels, the limited bandwidth and frame duration constraints lead to restricted resolution in the DD grid. Consequently, fractional Doppler shifts or delay shifts may span multiple lattices instead of aligning precisely with a single lattice. This phenomenon introduces channel expansion in the DD domain, making the UWA channel matrix nonsparse. As a result, the performance of the OMP channel estimation algorithm is degraded. This paper incorporates a residual network [26], referred to as the DnCNN network, into OTFS channel estimation to alleviate the performance degradation of the OMP channel estimation algorithm with nonstrict sparse channel information. DnCNN and ResNet share a common feature of residual learning. However, DnCNN does not incorporate connections between various layers of the neural network. Instead, it directly modifies the output of the network learning process to represent residuals. This unique characteristic enables faster convergence of the entire network and reduces training time compared with ResNet. Figure 2 illustrates the system structure of the proposed DnCNN-based OTFS channel estimation method. The pilot symbols in the received signal are input into the OMP channel estimation module, which generates raw channel estimation results. Then, the results are input into the DnCNN network for further processing. The DnCNN network for UWA OTFS channel estimation includes the following layers:  
* Convolutional (Conv.) layer with Rectified Linear Unit (ReLU) activation: This initial layer is responsible for extracting the channel information features. * Conv.+batch normalization(BN)+ReLU layer: This combination of layers forms a convolution block that focuses on signal reconstruction. The BN layer enhances the flow of gradients through the network, improving training speed and enhancing the network's learning capabilities. * Conv. layer: The final layer solely consists of a convolutional layer, which is utilized for information feature extraction. * Residual block: After the last convolutional layer, the residual noise \(n\) is output. This residual noise is then processed by a residual block, and the channel estimation results \(\hat{H}_{DL}\) is obtained by the raw estimation and the residual noise. The DnCNN network uses residual learning with skip connections to mitigate gradient disappearance and gradient explosion while denoising, thus achieving better performance than traditional CNNs. In DnCNN-based OTFS channel estimation, the DnCNN learns the correlation between the input raw estimated channel information and the residual noise using the residual block. As shown in Figure 2, the output channel estimation information \(\hat{H}_{DL}\) obtained after the residual block can be expressed as  
\[\hat{H}_{DL}=\hat{H}_{QMP}-n,\]where \(n\) is the trained residual noise, \(n\) and \(\hat{H}_{OMP}\) are the inputs of the residual block, and \(\hat{H}_{DL}\) is the output of the residual block. The residual block learns to minimize the difference between the input \(n\) and the output \(\hat{H}_{DL}\). So, the DnCNN learns more accurate channel information. The DnCNN-based channel estimation algorithm is initially trained offline and then deployed online. In the offline training phase, the input of the DnCNN training is the raw channel estimation result, the training labels are the actual channel responses, and the objective of the training is to minimize the difference between the network output and the labels. During the DnCNN training process, the mean-square error (MSE) serves as the loss function, which can be denoted as  
\[L=MSE(H,\hat{H}_{DL})=\frac{1}{n_{B}}\sum_{1}^{n_{B}}{(H-\hat{H}_{DL})}^{2}, \tag{8}\]  
where \(n_{B}\) is the batch size. In the online deployment phase, a test set is generated, and the previously trained network model is loaded. The DnCNN network module is deployed on the receiver to output the estimated channel information \(\hat{H}_{DL}\).