# Model-Driven Deep-Learning-Based Underwater Acoustic OTFS Channel Estimation
## 2 UWA OTFS System Model  
### UWA OTFS System Model  
In comparison with OFDM employing TF domain modulation, OTFS modulates information in the DD domain, which has the advantages of Doppler and delay resilience, reduced signaling latency, and reduced complexity of implementation [25]. Figure 1 illustrates the block diagram of a UWA OTFS system.  
Figure 1: UWA OTFS communication system.  
The transmitter constructs a DD grid as \(\Lambda=\{(I\Delta\tau,kv),l=0,...,M-1,k=0,...,N-1\}\), where \(\Delta\tau\) is the grid interval in delay dimension, and \(\Delta v\) is the grid interval in Doppler dimension. The number of grids in Doppler dimension is \(N\), and the number of grids in delay dimension is \(M\). The modulation module maps the one-dimensional constellation symbols \(x=[x_{1},...,x_{NM}]\) into two-dimensional transmission symbols using the specified modulation mode. The two-dimensional symbols are distributed across OTFS DD data grids. A pilot symbol is inserted in the two-dimensional symbols. And in the DD domain, the pilot is placed in the middle of the DD grids, while the guard intervals exist between the pilot symbol and the data frame for interference mitigation. Using inverse symplectic finite Fourier transform (ISFFT), the DD domain symbols can be converted into the TF domain as  
\[X[n,m]=\frac{1}{\sqrt{MN}}\sum_{k=0}^{N-1}\sum_{l=0}^{M-1}x[k,l]e^{j2\pi(\frac{ ik}{N}-\frac{ml}{M})}, \tag{1}\]  
where \(m=0,1,2,...,M-1\), and \(n=0,1,2,...,N-1\). Next, a time-frequency modulator converts the samples \(X[n,m]\) to a continuous time waveform \(s(t)\) with a shaping waveform \(g_{tx}(t)\) 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 symbol duration, and \(g_{tx}\) represents the transmit pulse shaping filter.  
The channel can be represented by CIR in the DD domain, which is 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 \(i\), \(v_{i}\) is the frequency bias of path \(i\), and \(\tau_{i}\) is time delay of path \(i\).  
The received signal is represented by the transmitted signal propagating through the UWA channel. The channel is modeled by CIR and additive noise. Therefore, the signal at the receiver can be expressed as  
\[r(t)=\sum_{i=1}^{p}h_{i}e^{j2\pi v_{i}(t-\tau_{i})}x(t-\tau_{i})+w(t). \tag{4}\]  
At the receiver, the received time-domain signal undergoes a series of steps to be transformed into the delay-Doppler (DD) domain; then, channel estimation and signal detection processes are subsequently executed to recover the transmitted data. At the receiver, the received time-domain signal \(r(t)\) first undergoes a transformation into a TF domain signal, using the Wigner transform as  
\[Y(n,m)=\left[\int g_{rx}^{*}(t-\tau)r(t)e^{-j2\pi f(t-\tau)}dt\right], \tag{5}\]  
where \(\tau=nT\), \(v=m\Delta f\). Then, the TF domain signal is transformed into DD domain using the symplectic finite Fourier transform (SFFT) as  
\[y[k,l]=\frac{1}