## 4 Performance Evaluation  
In this section, the BER performance of the proposed SDF-OTFS scheme is evaluated and presented. Evaluations and simulations are performed in MATLAB R2018a.  
P = M = N = 2 and BPSK modulation are also used in all simulations. Obtained \(H\) and \(H_{eff}^{rect}\) matrices for the source-to-destination channel and the source-to-relay channel in Section 2 are used exactly the same. Maximum likelihood (ML) detection is used at the relay and the destination, since study [1] expressed that ML detection is convenient and easy to implement for the small numbers of MN in OTFS receivers.  
Figure 2 shows the BER performance of different SISO-OTFS input/output models according to the SNR values with the abovementioned parameters. Simulations have been completed with \(10^{6}\) iterations. In each iteration, bit errors are taken into account by  
regenerating the new random transmitted bit sequence, channel coefficient gains and noises added at the receiver. The first of the different models is the OTFS time domain model in which there is a direct transmission between the transmitter and receiver antennas given in Equation (3). The \(H\) channel matrix is used, and the simulated SNR-BER graph is obtained according to this model. The second of the different models is the model known as the OTFS delay-Doppler domain model. The \(H^{rect}_{eff}\) channel matrix given in Equation (8) is used, and the simulated SNR-BER graph is obtained according to this model. From Figure 2, it is understood that when these two channel models and the theoretical approximate average BER formula in Equation (45) are used, all of their SNR-BER performances overlap and are exactly the same. Furthermore, the same SNR-BER performances are also obtained for similar parameters of SISO-OTFS (Figure 4 of [27] and Figure 11 of [1] from the literature).  
In Figure 3, the BER performance of the proposed SDF-OTFS scheme is presented. Simulations were completed according to the end-to-end probability calculations given in the previous section. As shown in Figure 3, the proposed SDF-OTFS scheme outperforms the point-to-point SISO-OTFS models considered in non-cooperative cases. For the non-cooperative case, we can consider source-to-destination (S-D) transmissions. This is the case where there is no relay, and only point-to-point transmission occurs. The performance result obtained is the same as in Figure 2. For verification purposes, different point-to-point SISO transmissions, such as source-to-relay (S-R), relay-to-destination (R-D) transmissions and transmissions with theoretical approximate average BER obtained from the SISO channel are also included and their performance results are given as in Figure 2. It was shown that the proposed SDF-OTFS model offers better quality of communication service against the non-cooperative case. For example, while \(10^{-5}\) BER has been obtained at 22 dB SNR with the proposed SDF-OTFS scheme, nearly 4 \(\times\)\(10^{-3}\) BER has been obtained at 22 dB SNR with one of the SISO-OTFS models. To achieve \(10^{-4}\) BER performance, a 37 dB SNR is required in one of the SISO-OTFS models, while a 17 dB SNR is sufficient for the proposed SDF-OTFS scheme. For the \(10^{-4}\) BER performance, an approximately 100-times power efficiency is achieved when the proposed SDF-OTFS scheme is used instead of the point-to-point SISO-OTFS model.  
Figure 2: BER performances of OTFS time domain model, OTFS delay-Doppler domain model and theoretical approximate average BER in the SISO channel.