# Deep Learning-Based Signal Detection for Underwater Acoustic OTFS Communication
## 2 UWA-OTFS System Model  
Compared with wireless radio communication, the Doppler shift in UWA communication is more severe, which is mainly determined by the transmission characteristics of the UWA channel [33]. Table 1 shows the characteristics comparison between wireless radio and the UWA channel. The propagation speed of UWA waves is five orders of magnitude slower than that of radio waves. Due to the severe distance- and frequency-dependent attenuation, the available frequency for long-range communication is only in roughly the KHz range. Due to these factors, even a slight movement can cause obvious Doppler shifts in the UWA channel. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Wireless Radio Communication** & **UWA Communication** \\ \hline Propagation speed \(c\) & \(3\times 10^{8}\) m/s & 1500 m/s \\ Typical carrier frequency \(f_{c}\) & Several GHz & Several kHz \\ Subcarrier spacing \(\Delta f\) & Several KHz & Several Hz \\ CFO \(f_{D}\) with \(v_{t}=1\) m/s & Several Hz & Several Hz \\ Normalized CFO \(\theta\) with \(v_{t}=1\) m/s & \(\approx\)1\(\times 10^{-3}\) & \(\approx\)1 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparision of radio and UWA communications. In Table 1, the carrier frequency offset can be calculated as \(f_{D}=(v_{t}f_{c})/c\), where \(v_{t}\) is the speed of movement between the transceivers. The normalized CFO can be calculated as \(\theta=f_{D}/\Delta f\), which can represent the impact of the CFO referenced to subcarrier spacing. For example, as shown in Table 1, for a relative moving speed of 1 m/s, the normalized CFO is about \(10^{-3}\) for radio communication and 1 for UWA communication. Doppler shifts on subcarriers have more severe impact on UWA communication performance than radio systems. In the UWA OTFS system, the Doppler effect will cause severe ISI and ICI. This paper will enhance the performance of the OTFS system with the power of deep learning. Figure 1 shows the block diagram of the UWA OTFS system. At the transmitter, the modulation module can map the one-dimensional constellation symbols \(x=[x_{1},...,x_{NM}]\) into 2D transmission symbols \(x[k,l]\), \(k=0,...,N-1,l=0,...,M-1\) with the specified modulation mode (e.g., BPSK or QPSK). The 2D symbols are distributed over \(N\times M\) OTFS delay-Doppler data grids. Then, the symbols in the DD domain are converted to the TF domain by an inverse symplectic finite Fourier transform (ISFFT) as  
\[X[n,m]=\frac{1}{\sqrt{MN}}\sum_{k=0}^{N-1}\sum_{l=0}^{M-1}x[k,l]e^{j2\pi\pi( \frac{\mu}{N}-\frac{\mu}{M})}. \tag{1}\]  
The TF domain signal is further transformed into time domain signal \(x(t)\) by Heisenberg transform as  
\[x(t)=\sum_{n=0}^{N-1}\sum_{m=0}^{M-1}X[n,m]g_{tx}(t-nT)e^{j2\pi m\Delta f(t- \Delta f)}, \tag{2}\]  
where \(\Delta f\) is the subcarrier spacing, \(T=1/\Delta f\) is the symbol duration and \(g_{tx}\) is the transmit pulse-shaping filter. The channel impulse response (CIR) \(h(\tau,v)\) in the DD domain can be expressed as  
\[h(\tau,v)=\sum_{i=1}^{p}h_{i}\delta(\tau-\tau_{i})\delta(v-v_{i}), \tag{3}\]  
where \(h_{i}\) is the channel coefficient of path \(i\) and \(v_{i}\) and \(\tau_{i}\) are the frequency bias and time delay of path \(i\), respectively. We assume that the CIRs are perfectly known at the receiver. The transmitted signal will go through the UWA channel, which is represented by the CIR and additive noise. The received signal can be expressed as  
\[r(t)=\sum_{i=1}^{p}h_{i}e^{j2\pi v_{i}(t-\tau_{i})}x(t-\tau_{i})+w(t), \tag{4}\]  
where \(w(t)\) is the additive Gaussian white noise. At the receiver, the received time domain signal \(r(t)\) is converted into a TF domain signal through a Wigner transform as  
\[Y(n,m)=\left[\int g_{rx}^{*}(t-\tau)y(t)e^{-j2\pi\tau/(t-\tau)}dt\right], \tag{5}\]  
Figure 1: Block diagram of the UWA OTFS system. where \(\tau=nT\), \(v=m\Delta f\).