## 3 End-to-End BER Calculation for the Proposed SDF Scheme  
The system model expressions for the three channels are given below, respectively, for the delay-Doppler domain:  
\[\overline{y}_{s,d}=H^{rect}_{eff_{s,d}}\overline{x}+\overline{\widetilde{n}} _{s,d} \tag{38}\]  
\[\overline{y}_{s,r}=H^{rect}_{eff_{s,r}}\overline{x}+\overline{\widetilde{n}} _{s,r} \tag{39}\]  
\[\overline{y}_{r,d}=H^{rect}_{eff_{r,d}}\overline{x}+\overline{\widetilde{n}} _{r,d} \tag{40}\]  
In our proposed method, we calculated the end-to-end bit error rate (BER) at the destination with OTFS waveform transmissions by implementing the SDF scheme, as presented below. Using the total probability formula, the end-to-end BER at the destination can be expressed as follows:  
\[Pr(e)=Pr(e\cap\,\emptyset)+Pr(e\,\cap\overline{\emptyset}\,) \tag{41}\]  
where the event \(\emptyset\) indicates the error at relay, the event \(\overline{\phi}\) indicates no error at relay, and the event \(e\) indicates the end-to-end error at the destination. Then, the total probability formula considering conditional probabilities is  
\[Pr(e)=Pr(e/\emptyset)Pr(\emptyset)+Pr(e\,/\overline{\emptyset})Pr(\overline{ \emptyset}) \tag{42}\]  
where \(Pr(e/\emptyset)\), related with Equation (38), is the probability of error at the destination in event error at relay; \(\Pr(\emptyset)\), related with Equation (39), is the probability of error at relay; \(\Pr(e\,/\overline{\emptyset})\) is the probability of error at the destination when the relay decodes accurately and \(\Pr(\overline{\emptyset})\) is the probability of no error at relay. The probability of error at relay at high SNR is assumed to be zero as in below. Hence, the probability of no error at relay at high SNR is assumed to be one, as in below:  
\[\Pr(\emptyset)\approx 0\]  
\[Pr(\overline{\emptyset})\approx 1-Pr(\emptyset)\approx 1\]Then, Equation(42) is transformed to the equation below:  
\[\Pr(\e)\approx\Pr(\e/\emptyset)\Pr(\emptyset)+\Pr(\e/\emptyset) \tag{43}\]  
Each model in Equations (38)-(40) can be accepted as the same as the classical multiple-input multiple-output (MIMO) system model structure with vector-matrix notation because the channel matrices are square matrices with \(MN\times MN\) dimensions in all structures. The approximate average BER expression for a Zero Forcing (ZF) MIMO system with BPSK modulation in a Rayleigh fading channel is presented as  
\[BER_{MIMO}=\binom{2L-1}{L}\left(\frac{1}{2 SNR}\right)^{L} \tag{44}\]  
where SNR is the signal-to-noise ratio at the receiver, \(L=r-t+1\) in which \(r>t\) is the diversity order of ZF, \(r\) is the number of receiver antennas and \(t\) is the transmit antennas in MIMO systems [26]. The approximate average BER \(txt\) MIMO (r = t and L = 1) channel in Rayleigh fading with ZF equalization is the same as the BER derived for the \(1\times 1\) (SISO) channel in Rayleigh fading, which is given as  
\[BER_{SISO}\approx\frac{1}{2 SNR} \tag{45}\]  
where \(L\) equals to 1. Because the channel for the symbol transmitted from each spatial dimension is similar to a \(1\times 1\) (SISO) Rayleigh fading channel, Equation (44) is transformed to Equation (45) when L = 1 is taken into account. Theoretical approximate average BER expressions for \(Pr(e/\emptyset)\) and \(\Pr(\emptyset)\) can be obtained from the source-to-destination transmission given in Equation (38) and the source-to-relay transmission given in Equation (43), respectively. Furthermore, the theoretical approximate average BER expression for the \(\Pr(e/\emptyset)\) can be obtained using the concatenated single-input multiple-output (SIMO) model given below because the transmitted signal is the same but the received signal is different in each transmission. \[ \begin{bmatrix}\yvcenter{s,d}\\ \yvcenter{r,d}\end{bmatrix} = \begin{bmatrix}H_{eff,d}^{rect}\\ H_{eff,d}^{rect}\end{bmatrix} \mp+ \begin{bmatrix}\widetilde{n}_{s,d}\\ \widetilde{n}_{r,d}\end{bmatrix}