## 2 System Model
### Phase 1: Source broadcasts Symbol to Destination and Relay (for Destination)  
The source broadcasts to the receiver (destination) and relay in this phase. It is assumed that there are two paths in each transmission in this system model. One of them is the line-of-sight (LOS) path and the other is the reflected path. In this phase, the direct path from the source to the destination (s,\(d\)) is considered as the LOS path, and the path from the source to the relay and then to the destination is considered as the reflected path (s,\(r\),\(d\)). For the destination, the received time domain signal is expressed as the point-to-point single-input single-output (SISO) model, as follows:  
\[ \begin{array}{l}r_{d}(t)=h_{s,d}s(t-\tau_{s,d})+\\ h_{s,r,d}e^{j2\pi\tau_{s,r,d}(t-\tau_{s,r,d})}s(t-\tau_{s,r,d})\end{array}  \tag{1}\]  
where \(h_{s,d}\) is the channel gain of the LoS path, \(\tau_{s,d}\) is the delay shift of the LoS path, \(s(t)\) is the transmitted signal from the source, \(h_{s,r,d}\) is the channel gain of the NLoS (reflected) path, \(\
u_{s,r,d}=f_{c}\frac{v_{\text{CoS}}}{c}=\
u_{r,d}\) is the Doppler frequency (shift) of the reflected path and \(\tau_{s,r,d}\) is the delay shift of the reflected path. For two propagation paths, a specific 2-tap doubly selective time-variant channel impulse response (CIR) for this transmission is given below. When \(s(t)\) passes through this channel, Equation (1) is obtained:  
\[ \begin{array}{l}\mathrm{g}(t,\tau)=h_{s,d}\delta(t-\tau_{s,d})+\\ h_{s,r,d}e^{j2\pi\tau_{s,r,d}(t-\tau_{s,r,d})}\delta(t-\tau_{s,r,d})\end{array}  \tag{2}\]  
Figure 1: System model of the SDF-OTFS system. To make the mathematical expressions for the time-varying channel above easier to understand, we present below the received linearly modeled signal from the source to the destination, in time domain, in vector and matrix forms in the context of OTFS modulation:  
\[\overline{r}_{s,d}=H_{s,d}\overline{s}+\overline{n} \tag{3}\]  
where \(\overline{s}\) is the transmitted signal vector in time domain and \(\overline{n}\) is the additive Gaussian noise vector. The general mathematical expression of the channel matrix \(H_{s,d}\) is expressed as  
\[H_{s,d}=\sum_{i=1}^{p}h_{i}\Pi_{MN}^{l_{i}}\Delta_{MN}^{k_{i}} \tag{4}\]  
where P is number of channel paths, \(l_{i}=(M\Delta f)\tau_{i}\) represents the delay taps (bins as integer numbers), \(k_{i}=(N/\Delta f)\gamma_{i}\) represents the Doppler taps (bins as integer numbers), \(\Delta f\) is subcarrier spacing, M is the number of subcarriers and N is the size of the time slots. \(\Pi_{MN}^{l_{i}}\) denotes the MN \(\times\) MN cyclic-shift matrix while \(\Delta_{MN}^{k_{i}}\) denotes the MNxMN diagonal matrix, and they are given below, respectively. \[\Pi_{MN}^{l_{i}} \triangleq \begin{bmatrix}0&\dots&0&1\\ 1&\cdots&0&0\\ \vdots&\ddots&\vdots&\vdots\\ 0&\dots&1&0\end{bmatrix}  \tag{5}\] \[\Delta_{MN}^{k_{i}} \triangleq diag\Big{[}z^{0}\,\ z^{1}\,\ \dots,\ z^{(n-1)}\Big{]}\] (6) \[\text{with}\quad z \triangleq e^{(\frac{B\pi}{MN})}.\]  
In vectorized form, the received signal from the source to destination in the delay-Doppler domain can be written as  
\[\overline{y}_{s,d}=H_{eff_{s,d}}\overline{x}+\overline{\widetilde{n}} \tag{7}\]  
where \(H_{eff_{s,d}}=(F_{N}\otimes G_{rx})H_{s,d}\left(F_{N}^{H}\otimes G_{tx}\right)\) indicates an effective channel matrix, \(\overline{x}\) is the transmitted signal in the delay-Doppler domain, and \(\overline{\widetilde{n}}=(F_{N}\otimes G_{rx})\overline{n}\) is the noise vector. \(F_{N}\) and \(F_{N}^{H}\) denote the N-point FFT matrix and the N-point IFFT matrix, respectively. For the rectangular waveforms, \(H_{eff_{s,d}}^{rect}=(F_{N}\otimes I_{M})H_{s,d}(F_{N}^{H}\otimes I_{M})\) is obtained using \(G_{rx}=G_{tx}=I_{M}\). Then, Equation (7) can be transformed below. \[\overline{y}_{s,d}=H_{eff_{s,d}}^{rect}\overline{x}+\overline{\widetilde{n}} \tag{8}\]  
First Path (LoS Path): Since the vehicle moves in the \(V\) direction, the receiver moves perpendicular to the source.