## 1 Introduction  
The popularization of mobile multimedia applications and cloud computing, in combination with the emergence of the Internet of Things, is forcing telecommunications operators to increase the data capacity they can offer continuously. In this scenario, optical fibre infrastructures are progressively being extended, making them arriving closer to the end-user [1, 2]. Among the different alternatives, passive optical networks (PONs) have been extensively employed due to their low cost [3; 4; 5; 6]. Given the lack of amplification, however, fibre loss usually limits the bandwidth length product of these systems, especially when dispersion is compensated by predistortion techniques [7]. In order to extend the range of PON networks and, consequently, enable the resource concentration and cost reduction, two main approaches are commonly considered. On the one hand is the use of active splitting nodes that compensate the combined loss of fibre transmission and splitting [3; 4]. This approach, nevertheless, makes the distribution network architecture more complicated and hinders its maintenance, particularly in long reach (LR) PONs where the splitting node may be far from urbanized areas. On the other hand, the advent of high speed digital signal processors (DSPs) has enabled the implementation of cost efficient digital coherent receivers [6], which dramatically changed the way about which high capacity links are thought. This inflexion point is not only due to the mixing gain they offer, but also because they permit simultaneous phase and amplitude modulation and open the possibility to compensate system impairments in a completely new and efficient way [8]. With linear impairments such as dispersion, phase noise, polarization fluctuation, and polarization-mode dispersion elegantly compensated in the baseband electrical domain, the interplay between nonlinear distortion and noise emerges as the main capacity limitation [9]. In single channel systems where the transmitted signal is broadcast to several users, the received power is reduced compared to wavelength division multiplexed (WDM) systems where demultiplexers are used and, consequently, the impact of the receiver noise is more critical. In regards to nonlinear distortion, the Kerr effect is widely claimed as the dominant nonlinear effect in digital coherent systems [10; 11]. It is well known that the Kerr effect further leads to self-phase modulation (SPM), cross-phase modulation (XPM), and four wave mixing (FWM) processes [12]. In the case of multi-wavelength PON systems, it is expected that XPM and FWM will be the main nonlinear degradation mechanisms, but for those systems operating with a single channel, the nonlinear distortion is solely governed by SPM. SPM is particularly harmful in modulation formats with non-uniform amplitude, for instance 16 quadrature amplitude modulation (16-QAM) [13]. Since SPM causes a phase rotation that is proportional to the power of the symbol (\(\phi_{NL}\approx-\gamma PL_{eff}\), where \(\gamma\) is the nonlinear coefficient, \(P\) is the symbol power, and \(L_{eff}\) is the effective fibre length), for moderate and elevated launch optical power levels, the constellation points with higher amplitude suffer a larger phase rotation than those with lower amplitude, leading to a characteristic spiral-like shape constellation [14]. This rotation, nevertheless, cannot be completely corrected by a simple nonlinear phase rotation because, even in a low dispersion regime, and the interplay of SPM and chromatic dispersion leads to more complex distortion. This complex distortion can be understood by noting that the pulse broadening caused by the chromatic distortion leads to word dependent behaviour through intersymbol interference (ISI). Even if ISI can be efficiently compensated in the baseband domain by DSP processing, the different superposed pulses are nonlinearly mixed through SPM. In addition, the interaction between the linear and nonlinear impairments varies as the signal propagates through the fibre.