## 1 Introduction  
In the last decade, the digital signal processing (DSP) algorithm has been developed and modern DSP-based coherent receivers can fully compensate for the linear channel impairments. Consequently, nonlinear impairments, which include mainly Kerr effects such as self-phase modulation (SPM), cross-phase modulation (XPM), and four-wave mixing (FWM), limit the transmission performance [1; 2; 3]. The digital back-propagation (DBP) algorithm is now commonly acknowledged as one of the most suitable candidates for joint linear and nonlinear impairments compensation [4; 5; 6]. In DBP, the received electric signals propagate in a simulated virtual fiber that has opposite parameters to the practical transmission fiber, and the dispersion and nonlinearity of the practical transmission fiber are eliminated. However, for the traditional DBP algorithms, noise in the system, such as Amplified Spontaneous Emission (ASE), double Rayleigh scattering (DRB), and transceiver noise, is not taken into account.  
In practical optical fiber communication system, such noises in the fiber would interact with the signal through the Kerr effect, which is called nonlinear signal-to-noise interaction (NSNI). As the noises are not considered in the traditional DBP algorithm, the NSNI induced by the noises cannot be dealt with. Recently, an algorithm called stochastic DBP was proposed, in which the complex probability method was adapted to compensate for the NSNI caused by the Erbium-doped fiber amplifiers (EDFAs) in the link [7; 8]. Moreover, once the DBP algorithm was applied to the system, the noise would continue interacting with the signal during the DBP process, thus causing extra NSNI. There has been a study of the impact of the transceiver noise on the performance of DBP, which indicated that the transceiver noise would induce extra NSNI in the receiver-side DBP, thus limiting the performance of DBP [9]. To mitigate the extra NSNI, a new approach called split-DBP was proposed and studied, which involves moving a part of DBP from the receiver side to the transmitter side. By this means, part of the NSNI can be avoided and the performance is improved [10; 11]. In Reference [10], [PERSON] et al. demonstrated that the split nonlinear compensation scheme is better than both post- and pre-compensation schemes in EDFA amplified systems through simulation.  
Now, distributed Raman amplification is regarded as a mature and promising amplification scheme for next-generation fiber optical communication systems [12; 13]. Actually, the DBP algorithm has been applied in Raman amplified unrepeetered systems [14; 15], but the NSNI caused by the distributed Raman gain and the spontaneous Raman scattering is in these cases ignored as well. Unlike in the EDFA amplified system, the gain in the Raman amplified systems is distributed along the fiber; consequently, the noise, which includes ASE from EDFAs, spontaneous Raman scattering, and double Rayleigh scattering, is amplified along the fiber. As a result, the NSNI in a Raman amplified system is worse than that in EDFA amplified systems.  
In this paper, we theoretically deduced the NSNI in a bi-directional Raman amplified unrepeetered system with receiver-side DBP and split-DBP, considering the effects of the distributed Raman gain, transceiver noise, DRB, and the spontaneous Raman scattering along the fiber. The origin of the extra NSNI in the system was studied. Theoretical analysis indicated that the extra NSNI can be effectively mitigated by adopting split-DBP. Then the performance of the split-DBP was investigated through the theoretical model and simulations, using a bi-directional Raman amplified system with 32 Gbd 16 QAM modulation. The impact of the DRB on the system was also studied.