## 5 Numerical Simulations and Analysis of Results
### Simulations and Analyses of the Proposed PCSDF Cooperation  
In this section, experiment results are simulated to verify our analysis and evaluate the performance of the proposed PCSDF relay strategy. A multiple-relay cooperative communication model is adopted in simulations, and the parameters are set as follows: the information bit of a code word \(N=512\), and \(R=0.5\). Each channel is an independent Rayleigh channel, and the BPSK modulation is adopted. The transmission power of the source node equals that of each relay node. Figure 7 shows the OP curve of the cooperative system with different relay nodes with a collaboration threshold of 0.01. In Figure 7, the OP of the proposed PCSDF system decreases as the number of the relays increases. This phenomenon can be explained as follows: as the number of relay nodes increases, the probability of relays correctly decoding the messages increases; thus, the performance of the selected optimal relay from all candidate relays is improved. Obviously, the OP of the proposed scheme decreases as the number of relay nodes increases. On the other hand, the OP decreases as the SNR increases. It can be observed that the SNR of the signals has an effect on the OP of the system. The instantaneous SNR received at the destination node increases as the transmitted SNR increases. Therefore, the OP decreases as the SNR increases. The maximum channel capacity of the system is an important indicator for measuring the performance of a system. Equation (22) provides the theoretical channel capacity expression of the proposed cooperation system. The channel capacity curves with no relay nodes and with different numbers of relay nodes are shown in Figure 8.  
The results show that the channel capacity of the cooperative relay system is much larger than that of the simple direct transmission system. As the number of relay nodes increases, the channel capacity of the system gradually increases as well. In the case of a channel capacity of two bits/s, the four-relay system has an additional gain of 2 dB compared to the single-relay system. In our simulations with different numbers of relays, the channel capacity of our scheme with relays outperforms that without any relay. In addition, the differences in the proposed scheme with different numbers of relays is trivial. The introduction of relays greatly improves the channel capacity. Therefore, the channel capacity increases as the number of relays increases. Figure 7: The OP curves of different number relays with a cooperative threshold \(\theta=0.01\). Figure 9 shows the BER performance curves for the PCSDF scheme and other relay cooperation schemes. It can be seen from the figure that the performance of the scheme is much better than those of other existing schemes. In the PCSDF scheme, the received messages are forwarded only when the optimal relay channel quality is good. Through this phenomenon, error propagation is reduced and the performance of system is increased. A gain of about 0.9 dB and 0.8 dB is achieved compared with the non-cooperative and standard SDF schemes at a BER of \(10^{-3}\), with an approximate performance gain of 1.0 dB and 0.7 dB, respectively, at a BER of \(10^{-2}\). This phenomenon can be explained as follows. The proposed scheme uses an efficient channel capacity approaching the polar code to assist the SDF, making for better performance. Moreover, it adopts a compound relay link to exceed the cooperative threshold in order to reduce the chance of re-transmission, achieving better performance and less latency. Figure 8: Channel capacity with different numbers of relays, \(\theta=0.01\). Figure 9: BER performance of different schemes. Furthermore, the distance of link S-R\({}_{opt}\) has a deterministic impact on the performance of the relay system. In our simulations, the distance of link S-D is normalized as 1. The distance of link S-R\({}_{opt}\) is chosen as a coefficient in proportion to the normalized coefficient of link S-D, and is less than 1.