## 2 Drawbacks and Deficiencies of Benchmark Fiber Non-Linearity Compensation Schemes  
Ms-OpcThis technique attempts to inverse the spectrum in the mid-distance of fiber transmission using the inversed spectrum to propagate the remaining half of the distance. By doing so, the non-linearity accumulated in the first half of the span will be automatically cancelled with those gained in the second half of the span [37]. The main drawback of this technique is that generating the \"inverted spectrum\" in the mid-span is complex and its application is limited to long range point-to-point transmission, because otherwise in an optical routed network the mid-span point is hard to identify [38]. MS-OPC also cannot compensate 2 nd order chromatic dispersion (CD) and requires symmetric dispersion map which can be partially achieved using expensive Raman amplification [39]. Multi-stage OPCs have been recently proposed to enhance flexibility and performance in next generation flex-grid optical networks [40]; however, this approach inevitably adds cost and complexity. PCSCThis technique has similar fundamental principle as MS-OPC and is a variation of the phase-conjugated twin-waves (PCTW) for single-carrier optical systems [41], where signals are polarization multiplexed with one polarization and a phase-conjugated signal against the other, and then after transmission, special DSP is designed to cancel nonlinearity by overlapping signals from the two polarizations. In PCSC, however, a portion of OFDM subcarriers (up to 50%) is transmitted with its phase conjugates, which is used at the receiver to estimate the nonlinear distortions in the respective subcarriers and other subcarriers, which are not accompanied by phase-conjugated subcarriers or pilots (PCPs) [42]. The nonlinearity cancellation is very effective with not much complexity added to the whole system. However, this method sacrifices spectral efficiency for both single- and dual-polarization CO-OFDM. Modified PCTW-based approaches have been proposed in [14; 15] by modulating one of the conjugated signals with additional bits or by diplexing the twin waves. However, in [14] spectral-efficiency is still sacrificed (from 20% to 50%), while in [15] the performance enhancement is not impressive (maximum of 1.2 dB in quality(Q)-factor). DBPAs the name suggests, this is a DSP method that attempts to re-wind the non-linear channel. In this method, the optical channel is rigorously numerically modelled and the received signals are digitally back-propagated through a modelled 'virtual' channel with the help of the split-step Fourier (SSF) method as shown in Figure 1[12]. In Figure 1, the \(\alpha\), \(\beta\), and \(\gamma\) terms refer to the loss, 2 nd order CD and fiber non-linearity, respectively. In this way, part of the non-linearity can be cancelled. However, the problem associated with this method is that the channel cannot be modelled very accurately due to random parameters during transmission such as random polarization mode dispersion (PMD) and the interaction between amplified spontaneous emission (ASE) noise from optical amplification with fiber non-linearity (also known as parametric noise amplification), which can only be statistically characterized. Additionally, it demonstrates impractically high complexity for real-time applications since a huge number of computation steps are needed to undo the non-linear interactions. For the latter, it has been shown in [11; 12] that a minimum 40 steps/span is required to eliminate non-linear distortions (also called as FS-DBP). It worth mentioning that DBP has been recently modified to account for PMD [43] and a stochastic-DBP was also designed to partly account for the ASE noise from optical amplification [44]. In [44], however, a maximum a posteriori principle was introduced with the help of Bayesian graphical models (a machine learning based approach) being combined with the deterministic DBP. Figure 1: Digital back-propagation (DBP) conceptual diagram [11].