## 4 Results and Discussion
### Impact of Double Rayleigh Backscattering  
In the theoretical analysis and the simulation above, the DRB in the system was neglected. In this section we take the DRB into consideration, and analyze the impact of the DRB on the system. In the proposed theoretical model, we calculate the NSNI caused by the transceiver noise and spontaneous Raman scattering. These two kinds of noise can be treated as \"dot\" noise, which means that the noise is generated within a very small distance. However, the DRB in the system cannot be treated as the same way. It is obvious that the DRB at a point is relevant to the Raman gain and the signal power in the whole following fiber. To include the DRB to the proposed model, the DRB noise should be  
Figure 5: SNR vs. incident power. converted into \"dot\" noise. For simplicity, we firstly calculate the average power distribution of the DRB along the fiber [19]:  
\[P_{DRB}(z)=\int\limits_{z}^{L}P_{s}(z^{\prime})\zeta^{2}\exp\left(\int\limits_{ z^{\prime}}^{z}[-\alpha_{s}+g_{R}P_{p}(z^{\prime\prime})]dz^{\prime\prime}\right)dz^{\prime} \tag{9}\]  
where \(P_{DRB}\) is the power of the DRB, \(\zeta\) is the Rayleigh backscattering coefficient, \(\alpha_{s}\) is the fiber attenuation at the wavelength of the signal, \(g_{R}\) is the Raman gain coefficient, and \(P_{p}\) is the Raman pump power. Note that in Equation (9) the consumption of the pump due to the DRB is ignored. Then the DRB noise can be generated at each point in the fiber according to its statistical distribution. In the theoretical model and simulation the DRB is treated as Gaussian additive noise. Although the exact statistical characteristic of the DRB is not very clear yet, according to the central limit theorem, the Gaussian approximation of the DRB in a small section of the fiber is acceptable. After obtaining the power profiles of the signal and the pumps, the power profile of the DRB can be calculated using Equation (9). With the same parameters as the simulation above, we calculated the power profile of the DRB, which is shown in Figure 6. For comparison, the power profiles of the signal and the spontaneous Raman scattering are plotted in Figure 6 as well. As we can see from the figure, compared to the spontaneous Raman scattering, the DRB has higher power in the near-transmitter section, but lower power in the near-receiver section of the fiber. Because the signal power in the near-transmitter section of the fiber is much higher than that in the near-receiver section, the noise in the near-receiver side section has more severe impact on the optical signal-to-noise ratio (OSNR) of the signal. That means that the spontaneous Raman scattering is a more important noise source than the DRB. Figure 7 plots the OSNR degradation of the signal along the fiber. When only the spontaneous Raman was considered, in the near-transmitter section the OSNR of the signal decreased slowly, but in the near-receiver section of the fiber the OSNR of the signal exhibited a sharp decrease. On the other hand, in the case with only DRB, the OSNR of the signal exhibited a small drop in the near-transmitter section and decreased slowly in the rest section of the fiber. And at the end of the fiber, the OSNR of the case in which only spontaneous Raman scattering was considered is lower than the OSNR of the case with only DRB. This indicates that spontaneous Raman scattering has higher impact on the system than the DRB. Figure 6: Power profiles of the signal, spontaneous Raman scattering (spRs), and DRB. Next, we take the NSNI into consideration. According the analysis above, when the receiver-side DBP is applied, the noise in the near-receiver section causes more severe NSNI, because the noise in the near-receiver section is amplified in the DBP process. Thus, the NSNI caused by spontaneous Raman scattering is also higher than the NSNI caused by the DRB.