# Deep Learning-Based Signal Detection for Underwater Acoustic OTFS Communication
## 3 SC-CNN-BiLSTM-Based Signal Detection for UWA-OTFS
### Architecture of the Proposed SC-CNN-BiLSTM Detector
Finally, at each moment, the final output can be achieved by combining the corresponding output results of the forward layer and the backward layer, which can be expressed as  
\[h_{t}^{\text{F}}=f(w_{1}\tilde{x}_{t}+w_{2}h_{t-1}) \tag{9}\]  
\[h_{t}^{\text{B}}=f(w_{3}\tilde{x}_{t}+w_{4}h_{t+1}) \tag{10}\]  
\[x_{t}^{Bi}=o_{t}\left(w_{5}h_{t}^{\text{F}}+w_{6}h_{t}^{\text{B}}\right)*\text {tanh}(C_{t}) \tag{11}\]  
where \(h_{t}^{\text{F}}\) and \(h_{t}^{\text{B}}\) represent the output of the forward calculation and backward calculation at time \(t\), respectively, \(x_{t}^{Bi}\) represents the final output of the BiLSTM, \(\tilde{x}_{t}\) represents the input of the current LSTM, \(h_{t-1}\) represents the output of the last LSTM, \(h_{t+1}\) indicates the output of LSTM in opposite directions, \(w_{1},w_{2},w_{3},w_{4},w_{5},w_{6}\) are the corresponding weights of the variables, \(C_{t}\) is the cell state in LSTM and \(o_{t}\) is the forgetting factor. BiLSTM can learn more comprehensive intrinsic correlation of the input series signal by learning from the past to the future and from the future to the past. Therefore, it can improve the performance of signal detection in UWA OTFS. BiLSTM consists of mutiple LSTM cells. Figure 4 shows the basic structure of the LSTM cell. With collaboration of the input gates, forget gates and output gates, LSTM can memorize important information and solve the problem of long-term dependence on data in the learning process. Figure 3: BiLSTM structure. First, the forget gate of LSTM decides which information to forget to have \(f_{t}\) for the cell state update. Then, the input gate updates the important information \(i_{t}\) in the learning process and determines the useful information retained in cell state \(\bar{C}_{t}\). The cell state is calculated as  
\[C_{t}=f_{t}*C_{t-1}+i_{t}*\bar{C}_{t} \tag{12}\]  
Finally, the output gate calculates the forgetting factor \(o_{t}\) according to \(h_{t-1}\) and \(\bar{x}_{t}\), and it obtains the final output \(h_{t}\) according to \(o_{t}\) and the cell state \(C_{t}\):  
\[o_{t}=\sigma(w_{o}\cdot(h_{t-1},\bar{x}_{t})+b_{o}) \tag{13}\]  
\[\tilde{x}=f_{6}(f_{5}(...f_{1}(y))), \tag{7}\]0  
where \(w_{o}\) is the weight of \(o_{t}\), \(b_{o}\) is the bias of \(o_{t}\) and \(\tanh(\cdot)\) is the hyperbolic tangent activation function. The BiLSTM layer in SC-CNN-BiLSTM has a strong ability to capture the correlations of times series data. It can not only remember correlated and ignore uncorrelated information by the gates structure, but it also can extract features from both the forward and backward directions. Therefore, for the signal detection of sequential data with interference, BiLSTM can enhance the neural network to better memorize and extract the effective information with sequentially transmitted symbols from the past to the future, which can mitigate the ICI and ISI in UWA communication. **FC layer:**  
For the output of the SC-CNN-BiLSTM network, there is one FC-DNN layer with 32 neurons and a logistic sigmoid activation function. The 32 neurons correspond to 32 bits to be estimated from 32 consecutive subcarriers. The logistic sigmoid function mapped the output values between [0,1] as soft bits, and then the soft bits will be processed to obtain the target sent bits, which can be expressed as  
\[\tilde{x}=f_{6}(f_{5}(...f_{1}(y))), \tag{7}\]1  
where \(x^{Bi}\) represents the output of BiLSTM. Finally, the transmitted bits are obtained according to the decision formula as  
\[\hat{x}=\left\{ \begin{array}{ll}1,&\hat{x}^{\prime}-0.5\geq 0\\ 0,&\hat{x}^{\prime}-0.5<0\end{array} \right. \tag{16}\]