## 4 Simulation Results and Discussions  
In this part, the performance analysis of the proposed MCIM-SR-DCSK-CC system over the AWGN channel and the multipath Rayleigh fading channel is shown. We adopt the three-path Rayleigh fading channel, which is usually utilized in many chaos-based communication systems. The delays of the propagation paths in this three-path Rayleigh fading channel are \(\tau_{1}\) = 0, \(\tau_{2}\) = 2, and \(\tau_{3}\) = 3. And the power gains are equal with \(E\left[\alpha_{1}^{2}\right]=E\left[\alpha_{2}^{2}\right]=E\left[\alpha_{3}^{2 }\right]=\frac{1}{3}\). Additionally, the channel models used in the \(S\to R\) link and the \(S\to D\) and \(R\to D\) links are the same. When the length of the Walsh codes increases, the noise variance will decrease. Then, the BER performance improves. Therefore, the BER performance improves as the length of the Walsh codes increases. However, the system complexity increases when the length of the Walsh codes increases. But the BER performance and the bit rate improve. Therefore, the trade-off between performance and system complexity can be selected according to actual needs. In this paper, we consider both not too much complexity and better system performance, so we set the length of the Walsh codes to 16 or 64. In Figure 7, the BER performance comparison between the simulated and the theoretical results of the proposed MCIM-SR-DCSK-CC system is given with \(M=4,N=10\), \(J=4,P=16\), and different \(U\). Intuitively, the consistency between the simulation results and the theoretical results serves as additional evidence for the correctness and effectiveness of the theoretical analysis. As the length of the chaotic signal shortens, the BER performance of the proposed system gets better. This is because a longer chaotic signal would lead to a more pronounced noise component in the noise-noise correlation term, thereby degrading the BER performance. In Figure 8, the effect of different lengths of the chaotic signal on the BER of the proposed system over the AWGN channel and multipath Rayleigh fading channel is demonstrated when \(\frac{E_{b}}{N_{0}}\) is fixed. It is evident that the BER initially decreases with U, hits an optimal point, and subsequently begins to increase. It indicates that an optimal choice of U exists in the proposed system. When the length of the chaotic signal increases, the noise component involved in the noise-noise correlation term becomes more significant, resulting in a poor BER performance. However, the computational complexity of the system will improve when the length of the chaotic signal increases. Furthermore, it can be seen from Figure 8 that the optimal selection of U is approximately 8 in the AWGN channel and the multipath Rayleigh fading channel. Figure 7: The BER performance comparison between the simulated and the theoretical results of the proposed MCIM-SR-DCSK-CC system. In Figure 9, the effect of different numbers of parallel groups on the BER of the proposed system over the AWGN channel and the multipath Rayleigh fading channel is demonstrated when \(\frac{E_{h}}{N_{0}}\) is fixed. The BER performance improves as the number of groups increases; however, it does not improve as the number of groups increases indefinitely. A compromise number of the parallel groups is 10. In Figure 10, the comparison of the BER performance with different \(M\) over the AWGN channel and multipath Rayleigh fading channel with \(U=100,P=64,J=4,N=10\) is shown. Obviously, the BER performance is best when M = 4. In Figure 11, the comparison of the normalized throughputs between the proposed system and the CIM-SR-DCSK-CC system over the AWGN channel and multipath Rayleigh fading channel is shown. The normalized throughput of the proposed MCIM-SR-DCSK-CC system greatly exceeds those of the CIM-SR-DCSK-CC system. For example, at \(E_{T}/N_{0}=30\) dB, the normalized throughput of the proposed MCIM-SR-DCSK-CC system is 1 over the AWGN and multipath Rayleigh fading channels. Compared with this, those of the CIM-SR-DCSK-CC system is 0.3.