## 5 Numerical Simulations and Analysis of Results
### Simulations and Analyses of the Proposed PCSDF Cooperation
We let the distance of link S-R\({}_{opt}\) be \(d\), and the distance of link R\({}_{opt}\)-D be \((1-d)\); then, the BER performance of the PCSDF scheme with different distances \(d\) is as shown in Figure 10.  
The gain increases with the distance \(d\). For the PCSDF scheme, there is a gain of about 0.3 dB at a \(d\) of 0.8, compared to a gain of of 0.5 dB at a BER of \(10^{-3}\). The proposed scheme has about 0.5 dB gain compared to the DF one at a BER of \(10^{-2}\) and \(d\) of 0.8. Thus, the BER performance is increased as the distance of link R\({}_{opt}\)-D decreases. This phenomenon can be explained as follows. This scheme selects the best equivalent relay to forward the decoded messages in the SDF cooperation. With a short distance between the best relay and the destination, the cooperation ability of the former is decreased due to the ease of its absence in cooperation, which is caused by the equalization of the diversity gains and spectrum efficiency. Then, the priority of spectrum efficiency is obtained at the cost of acceptable BER loss. On the contrary, the cooperative participation of the relay is important for large diversity gains and low BER when there is a long distance between the best relay and the destination. In addition, the error propagation, which usually appears in the DF scheme, can be effectively avoided. Only the best relay forwarding procedure can ensure that the received information at the relay nodes is sent to the destination node effectively. Thus, a shorter distance between the best relay and the destination leads to lower error probability or better BER performance. Therefore, in the multi-relay system, a relay located near the destination node can obtain much better performance than other counterparts. To compare the obtained results with recent research, we compare the frame error rate (FER) of our polar coded cooperation scheme with that of the LDPC coded one under different relay numbers. The simulation parameters are configured as follows: code length \(N=4028\), code rate \(R=0.5\), \(\alpha=0.2\), and \(M=32\) for the optimized polar encoding and SC decoding. The FER performances of polar codes using different average SNRs per link are shown in Figure 11. In Figure 11, the FER curves represent the constructed polar codes with cooperation and the counterpart of the LDPC coded cooperation [33]. At high average SNRs per link, the FER performance of our scheme is almost the same as that of the LDPC coded cooperation with the increased SNR. For instance, our scheme has a performance gap of only 0.3-2 dB compared to those of counterparts at SNRs from 0 dB to 12 dB. Figure 10: The BER performance of the proposed schemes with different S-R\({}_{opt}\) distances. In Figure 11, while our scheme exhibits slightly poor performance, it obtains both low complexity and low latency. The performance difference can be explained from two viewpoints, as follows. First, the channel codes adopted in the coded cooperation mean that in the comparison experiments, the LDPC cooperation counterpart uses a rather long code length of 4096 with code rate of 1/2 in cooperation. The constructed of the codewords is improved by using optimized degree profiles for optimal code performance. In addition, for efficient decoding, the belief propagation (BP) algorithm can be adopted for much better performance at the cost of higher complexity and larger latency. In our scheme, however, we use the polar code with simple and non-complex SC decoding at the cost of performance loss. Generally, polar codes are superior in performance to LDPC codes at short code lengths of less than 1000. Thus, they possess much better performance in the case of short codewords. In this experiment, however, due to the moderate code length, LDPC obtains a little better performance at the cost of more complexity and latency. The second aspect is the cooperation strategy used in our scheme.