## 2 System Model
### OTFS Modulation and Demodulation  
The transmitter transforms the information symbol \(x[k,l]\in\mathbb{C}^{M\times N}\) from the DD domain to the time-frequency domain using the inverse symplectic finite Fourier transform (ISFFT).  
\[X[n,m]=\frac{1}{\sqrt{MN}}\sum_{k=0}^{N}\sum_{l=0}^{M}x[k,l]e^{2\pi(\frac{\pi}{ N}-\frac{\pi}{M})} \tag{1}\]  
At this time, \(X[n,m]\) generates a time waveform \(s(t)\) using a transmission pulse \(g_{tx}(t)\). This transform is called the Heisenberg transform.  
\[s(t)=\sum_{n=0}^{N}\sum_{m=0}^{M}X[n,m]g_{tx}(t)e^{j2\pi mAf(t-nT)} \tag{2}\]  
where \(\Delta f\) and \(T\) are the subcarrier spacing and symbol period, respectively.  
After that, the signal \(s(t)\) is transmitted through a fast time-varying channel with a complex baseband channel impulse response \(h(\tau,\
u)\), which specifies the channel response to an impulse with delay \(\tau\) and Doppler \(\
u\).  
\[r(t)=\int\int h(\tau,\
u)s(t-\tau)e^{j2\pi\
u(t-\tau)} \tag{3}\]  
The channel only has a few parameters in the delay-Doppler domain. The sparse representation of channel \(h(\tau,\
u)\) is given as  
\[h(\tau,\
u)=\sum_{i=1}^{P}h_{i}\delta(\tau-\tau_{i})\delta(\
u-v_{i}) \tag{4}\]  
Figure 1: OTFS modulation frame.  
where \(P\) denotes the number of paths for propagation. \(\delta()\) denotes the Dirac delta function, whereas \(h_{i}\), \(\tau_{i}\) and \(\
u_{i}\) stand for the path gain, delay, and Doppler shift associated with the \(i\)-th path, respectively. We choose the delay and Doppler taps for the \(i\)-th path as follows:  
\[\tau_{i}=\frac{l_{\tau_{i}}}{M\Delta f},\
u_{i}=\frac{k_{\
u_{i}}+\kappa_{\
u_{ i}}}{NT} \tag{5}\]  
where the integers \(l_{\tau_{i}}\), \(k_{\
u_{i}}\) represent the delay tap and Doppler tap indices corresponding to the delay \(\tau_{i}\) and Doppler frequency \(\
u_{i}\), respectively, and \(\kappa_{\
u_{i}}\in[-0.5,0.5]\). We do not need to examine the impact of the fractional delay because the resolution of the time axis is sufficient to approximate the path delay to the nearest integer grid point.  
The received signal \(r(t)\) is given by the Wigner transform and is realized by the receiving matching filter of impulse response \(g_{rx}\), obtaining the time-frequency domain symbol \(Y[n,m]\).  
\[Y[n,m]=\int r(t)g_{rx}(t-nT)e^{i2\pi m\Delta f(t-nT)}dt \tag{6}\]  
Finally, the symplectic finite Fourier transform (SFFT) is used to convert the information symbols \(Y[n,m]\) in the TF domain into the symbols \(y[k,l]\) in the DD domain.  
When the transmitter pulse \(g_{tx}\) and the receiver pulse \(g_{rx}\) are ideal pulse functions, the relationship between the transmitted signal and the received signal in the delay-Doppler domain can be written as  
\[y[k,l]=\sum_{i=1}^{P}\sum_{k^{\prime}=\left[k-k_{\
u_{i}}-Q\right] _{N}}^{\left[k-k_{\
u_{i}}+Q\right]_{N}} \tag{7}\] \[\left(\frac{e^{-j2\pi\left(k-k_{\
u_{i}}-k^{\prime}-\kappa_{\
u_{ i}}\right)}-1}{Ne^{-j\frac{2\pi}{N}\left(k-k_{\
u_{i}}-k^{\prime}-\kappa_{ \
u_{i}}\right)}-N}\right)+v[k,l]\] \[=\sum_{i=1}^{P}\sum_{q=-Q}^{Q}\left(\frac{e^{-j2\pi\left(-q- \kappa_{\
u_{i}}\right)}-1}{Ne^{-j\frac{2\pi}{N}\left(-q-\kappa_{\
u_{i}} \right)}-N}\right)h_{i}e^{-j2\pi\
u_{i}\tau_{i}}\] \[\times x\left[\left[k-k_{\
u_{i}}+q\right]_{N^{\prime}}\left[l- \left.t_{\tau_{i}}\right]_{M}\right]+v[k,l]\]  
where \(v[k,l]\) denotes the system noise in the DD domain. We set the proper \(Q\) to counteract the impact of fractional Doppler. The delay-Doppler plane is discretized to an \(M\times N\) grid. According to Equation (7), due to the influence of the doubly dispersive channel, the signal received at the \(k\)-th Doppler point is subjected to the superposition of signals from the \(2Q+1\) (\(k-Q\) to \(k+Q\)) Doppler points. In other words, the received signal \(y[k,l]\) is a linear combination of the sent signal \(\Omega=\sum_{i=1}^{P}2Q+1\).