# Deep Learning-Based Signal Detection for Underwater Acoustic OTFS Communication
## 4 Numerical Results
### Computational Complexity Analysis  
As shown in Table 5, the computational complexity of the proposed SC-CNN-BiLSTM signal detection and other methods are compared.  
The complexity of SC-CNN-BiLSTM can be calculated as the summation of the CNN and LSTM. According to [36], the computational complexity of the CNN can be defined as \(O(\sum_{l=1}^{N_{l}}N_{inc_{l}}\cdot w^{2}\cdot N_{f_{l}}(NM))\), where \(N_{l}\) is the number of CNN layers used to construct the model, \(l\) is the index of a convolutional layer, \(N_{f_{l}}\) is the number of filters (also known as the width) in the \(l\)th layer, \(N_{inc_{l}}\) is the number of input channels of the \(l\)th layer, \(w^{2}\) is the kernel size and \(NM\) is the OTFS frame size for the network input. As all the other parameters are constant during the training and testing phase, the overall complexity is a linear function of \(NM\) expressed as \(O(NM)\). Meanwhile, the computational complexity of the BiLSTM can be defined as \(O(\sum_{l=1}^{d}n^{2}d_{c})\), where \(d_{c}\) is the dimension of each cell, \(n_{in}\) is the size of the input and \(n_{in}=NM\) in our system. The complexity of BiLSTM can be expressed as \(O((NM)^{2})\). Thus, the SC-CNN-BiLSTM complexity is a linear function expressed as \(O((NM)^{2})\).  
The complexity of the 2D-CNN is \(O(NM)\)[32]. The complexity of the MP-based method is \(O(n_{iter}MNn_{cl}\Theta)\), where \(n_{iter}\) is the number of iterations, \(n_{cl}\) is the number of non-zero channel and \(\Theta\) is the modulation bit size. Thus, the complexity of MP depends on the sparsity level of the channels. The complexity of LMMSE and ZF is \(O((MN)^{3})\).  
From Table 5, we can see that the complexity of SC-CNN-BiLSTM was lower than those of the MP, LMMSE and ZF methods but higher than that of the 2D-CNN. Let us take the consideration of complexity and BER performance together. Both the complexity and BER performance of SC-CNN-BiLSTM outperformed FC-DNN, MP, LMMSE and ZF signal detection. Although the proposed method has higher complexity than the 2D-CNN, it can achieve better BER performance. The complexity of SC-CNN-BiLSTM is a linear function expressed as \(O((NM)^{2})\), and the complexity of the 2D-CNN is \(O(NM)\). Therefore, the proposed SC-CNN-BiLSTM signal detection method has higher complexity than the 2D-CNN. For BER performance, in Figure 6, at a BER of \(5\times 10^{-3}\), the proposed  
Figure 13: BER performance comparison under non-ideal channel estimation for KAU1 experiment.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**ZF** & **LMMSE** & **MP** & **2D-CNN** & **SC-CNN-LSTM** \\ \hline \(O((MN)^{3})\) & \(O((MN)^{3})\) & \(O(n_{iter}MNn_{cl}\Theta)\) & \(O(MN)\) & \(O((MN)^{2})\) \\ \hline \hline \end{tabular}
\end{table}
Table 5: Computational complexity comparison.  
SC-CNN-BiLSTM could achieve about 1.5 dB improvement compared with the 2D-CNN under a statistic simulation channel model. In Figure 9, at a BER of \(1\times 10^{-3}\), the proposed method could achieve about 2 dB of improvement compared with the 2D-CNN under the NOF channel. In Figure 12, at a BER of \(1.5\times 10^{-3}\), the proposed method could achieve about 1.5 dB of improvement under the KAU1 channel. The proposed method of SC-CNN-BiLSTM had higher complexity than the 2D-CNN, but it could achieve better BER performance in various channel conditions.