## 2 System Model  
### OTFS Modulation  
Two-dimensional base functions with properties including nonlocal TF domain and local DD domain are used in OTFS modulation. These functions map the symbol information onto a grid in the DD domain, where the \(M\times N\) transmitted signals occupy time duration, \(NT\), and bandwidth, \(B=\Delta fM\), in which \(T\) and \(\Delta f=\frac{1}{T}\) describe the OTFS block duration and the subcarrier spacing, respectively. The information symbols \(x_{l,k},l=0,\cdots,M-1\), \(k=0,\cdots,N-1\) are discretized into a grid in the DD domain, \(\Gamma=\{(\frac{l}{\Delta M},\frac{k}{NT}),\) for \(l=0,\cdots,M-1,k=0,\cdots,N-1\}\), where \(\frac{1}{\Delta M}\) and \(\frac{1}{NT}\) represent the resolutions of delay and the resolutions of Doppler axes, respectively. Figure 1 shows the UAC system diagram, mainly including two modules, namely, the transmitter for modulation and the receiver for demodulation. Transmitter maps \(x_{l,k}\) into samples \(X_{m,n}\) in the TF domain through inverse symplectic finite Fourier transform (ISFFT), as shown in Figure 1.  
By employing the Heisenberg transform, one can convert the TF domain signals \(X_{m,n}\) to time domain signal \(\mathbf{S^{\prime}}\). \[\mathbf{S^{\prime}}=\mathbf{G}_{tx}\mathbf{F}_{M}^{H}\mathbf{X} \tag{1}\]  
where \(\mathbf{F}\) is the fast Fourier transform (FFT) matrix, \(\mathbf{G}_{tx}=\text{diag}[g_{tx}(0),g_{tx}(T/M),\)\(\cdots,\)\(g_{tx}((M-1)T/M)]\), \(g_{tx}(t)\) is the transmit pulse-shaping waveform. In this paper, a rectangular waveform is employed to shape the Heisenberg transform, leading to inverse finite Fourier transform (IFFT), which will greatly reduce the computational complexity. In order to mitigate ISI, we obtain the time-domain signal after adding a cyclic prefix (CP), then  
\[\mathbf{S}=\mathbf{A}^{CP}\mathbf{S^{\prime}} \tag{2}\]  
where \(\mathbf{A}^{CP}\in\mathbb{C}^{(M+N_{\text{CP}})\times N}\) represents a CP addition matrix, and \(N_{\text{CP}}\) represents the length of added CP. To be transmitted by the antenna, vectorization operation changes the paralleled data matrices \(\mathbf{S}\) to data vector \(\mathbf{s}\), which is listed as follows:  
\[\mathbf{s}=\text{vec}\{\mathbf{S}\} \tag{3}\]  
According to [10], the time-varying channels in the DD domain can be sparsely represented as  
Figure 1: Underwater acoustic communication system based on OTFS. \[h(\tau,\
u)=\sum_{i=1}^{p}h_{i}\delta(\tau-\tau_{i})\delta(\
u-v_{i}) \tag{4}\]  
where \(P\) is the total number of delay taps; \(h_{i}\), \(\tau_{i}\) and \(v_{i}\), respectively, represent the propagation gain, delay, and Doppler shift associated with the \(i\)-th path, for \(i=1,\cdots,P\). In the same DD grid as the OTFS signals, \(\tau_{i}\) and \(v_{i}\) are defined by the following:  
\[\tau_{i}=\frac{l_{\tau_{i}}}{M\Delta f},\;v_{i}=\frac{k_{\tau_{i}}}{NT}, \tag{5}\]  
As can be seen in Equation (5), given large enough \(M\) and \(N\), the delay \(l_{\tau_{i}}\) and Doppler shifts \(k_{\tau_{i}}\) may nearly be integers. At the receiver, we denote the time domain signal of the OTFS system through underwater acoustic channels as \(r\). The received vector form of the time-domain signal is first parallelized into a matrix,  
\[\mathbf{R^{\prime}}=\text{invec}\{\mathbf{r}\} \tag{6}\]  
Then, by removing the CP, the obtained matrix can be written as  
\[\mathbf{R}=\mathbf{R^{CP}}\mathbf{R^{\prime}} \tag{7}\]  
where \(\mathbf{R^{CP}}\in\mathbb{C}^{M\times(M+N_{CP})}\) represents the CP-removing matrix. Similarly with transmitting, the received signal \(\mathbf{Y}\) in the DD domain can be converted in the TF domain by using a Wigner transformer. Since the received pulse is a rectangular pulse waveform, the Wigner transform degenerates into the discrete Fourier transform. Then, applying SFFT, \(\mathbf{Y}\) tends to be a 2D periodic convolution of \(\mathbf{X}^{\text{DD}}\).