## 4 Results and Discussion  
### Accuracy of the Theoretical Model  
In Section 2 we gave the expression of the final SNR of the signal in a bi-directional Raman amplified system with receiver-side DBP or split-DBP. In this section, we will verify the accuracy of the model through simulation and demonstrate the efficiency of split-DBP.  
First, to find the optimal pre-compensation length, different pre-compensation lengths were applied, and the SNRs of the signals were calculated using Equations (7) and (8) with different incident powers. Correspondingly, the transmission simulations were carried out with the same parameters. The SNRs of the signals in the transmission simulations were calculated using the error vector magnitude (EVM). As shown in Figure 4a, after calculating the average amplitude of each point of the signal constellation and the standard deviation of the noise, the EVM of the signal can be calculated, then the SNR of the signal can be calculated by SNR = 1/EVM\({}^{2}\). The simulation results were compared with the theoretical results, and both are shown in Figure 4b.  
Figure 4: **(a)** The calculation method of the error vector magnitude (EVM) of the signal, **(b)** signal-to-noise ratio (SNR) vs. The pre-compensation length.  
In Figure 4b, the black lines are the theoretical calculation results, the colorful marks are the results of simulations, and the different curves denote different incident powers. Comparing the results of theory and simulations, it can be found that the results of the theoretical calculation fit well with the simulations when the pre-compensation length was less than 100 km. When the pre-compensation length was more than 100 km, the SNR of the theoretical calculation was larger than the simulation, which we attributed to the influence of the up-sampling and the low-pass filter, which brings error to the transmitter-side DBP.  
As we can see, when the pre-compensation was 0, representing the receiver-side DBP, with the incident power increasing from 6 dBm to 8 dBm, the SNR of the signal decreased. This indicates that the NSNI caused by the DBP affects the performance of the signal. When split-DBP was applied, the SNR of the signal increased first and then remained at a high level, with the pre-compensation length increasing. For different incident powers, the optimal pre-compensation lengths were all round 50 km, but had small differences, and the trend was that the optimal pre-compensation length decreased with an increase in incident power.  
Then, to determine the efficiency of the split-DBP, the pre-compensation length was set to 0 and 50 km, and the SNRs of the signals were calculated with different incident powers, as well as simulations. The results are shown in Figure 5. Similarly, in this figure the black lines denote the results of the theoretical calculation and the colorful marks denote the results of the simulations. Because the pre-compensation length was less than 100 km, the results of the theoretical calculation matched well with the simulations. And we can see, by adopting split-DBP, the optimal incident power increased from 6 dBm to 8 dBm, and the maximum SNR increased by about 1 dB. This demonstrates the efficiency of NSNI mitigation of the split-DBP in the bi-directional Raman amplified system. However, we can see that when the incident power was over 8 dBm, the SNR of the signal still decreased quickly, which indicates there are other complicated nonlinear effects that impact the signal.