## 4 Numerical Results and Discussion  
In this section, Monte Carlo computer simulation experiments are performed using MATLAB software to verify the reliability and accuracy of the theoretical formulas and illustrate the effects of various system parameters on the BER and outage probability of the system under binary phase shift keying modulation. In all simulations, the simulation process uses the following default settings unless mentioned otherwise: (1) The total system power is 2, \(P_{\text{S}}=P_{\text{R}}=1\). (2) To highlight the impact of channel fading and noise on the performance, we assume that the average SNR of the system channel is \(\Delta\), \(N_{\text{W}}=N_{\text{P}}=1/\Delta\), that is, \(\Delta_{\text{GR}}=\Delta_{\text{LD}}=\Delta\). (3) The system threshold \(R_{\text{th}}=0.2\). (4) The Bernoulli-Gaussian noise parameter \(p=0.1\), \(T=10\). Figure 6 shows the BER performance of the simulated and theoretical calculations when the fading parameters are set to \(\{m_{R},\sigma_{D}\}=\{1,\ 2.2\},\{1.6,\ 2\},\{2.1,\ 2.4\}\), the values of the Bernoulli-Gaussian noise parameters \(p\) and \(T\) are shown in the figure. The BER decreases as the system SNR increases, and the resulting curve matches well with the Monte Carlo simulations, thereby implying the correctness of the derived analytical expressions and the validity of the theoretical derivation of the two approximation algorithms. The coincidence degree between the theory and simulation of the L2G algorithm is likewise higher than that of the G2L algorithm in terms of algorithm comparison, especially at high SNR, which is caused by multiple approximations in the theoretical calculation of the latter algorithm. The L2G algorithm is highly accurate and suitable for arbitrary \(m\) values rather than special parameters, which are reflected in the BER curves of the three sets of parameters in the figure. However, the simulation process of the L2G algorithm is sensitive to the selection of parameters, and such sensitivity is related to the high complexity of the algorithm and the large proportion of integrals involved. Although the G2L algorithm is not accurate enough, the theoretical calculation is simply for end-to-end SNR and multiple approximations, which can be used to analyze many performance indicators and solve a wide range of problems. Figure 7 shows the simulation and theoretical comparison of the system outage probability and the average SNR of the two approximated algorithms under different thresholds \(R_{\text{th}}\). The setting of the three sets of fading parameters is the same as that in Figure 6, and the Bernoulli-Gaussian noise parameter is the default setting. The three sets of curves in the figure show that the outage probability of the different parameters increases continuously as the value increases from 0.2 to 0.4, and the theoretical curves of the two algorithms are extremely close to the simulation results, thus verifying the correctness of our analysis. Furthermore, the curve of the L2G algorithm almost coincides with the simulation curve and the accuracy is higher than that of the G2L algorithm, however, the coincidence degree is  
slightly reduced at high SNR. Figure 7 shows that the algorithm can obtain sufficient computational accuracy in a certain range of SNR based on the consumption complexity. Figure 8 illustrates the performance of BER in the two approximation algorithms with the different approximation accuracies caused by the value of the \(s\) vectors. The fading parameters are \(\{m_{R},\sigma_{D}\}=\{2.8,\ 2.6\}\), \(\{1,\ 2.2\}\), and the other parameters are set by default. Figure 8a,b shows that the coincidence between the theoretical algorithms and the simulation of the two approximation theoretical algorithms with the best \(s\) values is much higher than that with the random selection of \(s\) values.