## 4 Results and Discussion
### Simulation Results  
The first step in the simulation study was to validate the model that we implemented. For this matter, we performed \(Q^{2}\)-factor evaluation as a function of the input power per band after propagation through 10 \(\times\) 100 km of an SSMF, using 1.8 SpS for the equalization process. All NLEs were optimized, as we did for the experimental study, through their adjustable parameters [9; 16]. The simulation results are shown in Figure 6. The measured \(Q^{2}\)-factor improvements, compared to linear equalization, are \(\sim\)0.32 dB (i.e., BER = 5.9 \(\times\) 10\({}^{-3}\) w. NLE and BER = 4.6 \(\times\) 10\({}^{-3}\) w. NLE) for the 3 rd-order IVSTF-NLE, \(\sim\)0.3 dB (i.e., BER = 5.9 \(\times\) 10\({}^{-3}\) w. NLE and BER = 4.6 \(\times\) 10\({}^{-3}\) w. NLE) for the DBP-SSF\({}_{1}\) NLE, and \(\sim\)0.35 dB (i.e., BER =5.9 \(\times\) 10\({}^{-3}\) w. NLE and BER = 4.5 \(\times\) 10\({}^{-3}\) w. NLE) for the DBP-SSF\({}_{2,8}\) NLEs. There is a very good agreement with the experimental results represented in Figure 5c. Thus, our model is a fair representation of the experimental setup.  
Next, we compare the maximum reach versus input power per band, with and without the NLEs, after signal transmission through an SSMF and a LEAF. Figure 7a shows that the maximum reach is achieved for input power per band equal to \(-\)3 dBm in SSMF, in all cases. At the FEC limit (i.e.,  
Figure 6: Simulation results on \(Q^{2}\)-factor vs. input power per band without/with IVSTF-NLE and DBP-SSF\({}_{1,2,8}\) NLEs for the 3 rd-band of the central OFDM in a WDM system with three wavelengths, 4-band DP-16 QAM OFDM super-channels after 10 \(\times\) 100 km of SSMF.  
BER = 2 \(\times\) 10\({}^{-2}\) or \(Q^{2}\) = 6.25 dB), the maximum reach is 1400 km with linear equalization only. After nonlinear compensation, the maximum reach was extended \(\sim\)3% for the 3 rd-order IVSTF-NLE, \(\sim\)3% for the DBP-SSF\({}_{1}\) NLE, \(\sim\)4.6% for the DBP-SSF\({}_{2}\) NLE and \(\sim\)7% for the DBP-SSF\({}_{8}\) NLE, with respect to linear equalization. We observe that the DBP-SSF method slightly outperforms the 3 rd-order IVSTF-NLE, especially if the number of steps per span increases.  
Then, we replace the SSMF with LEAF to investigate how the maximum reach is affected when the nonlinear impairments become stronger. In Figure 7b, the maximum reach without NLE is 1100 km in LEAF, which is 300 km shorter compared to SSMF. The maximum reach is extended by \(\sim\)6.3% for the 3 rd-order IVSTF-NLE, \(\sim\)5.4% for the DBP-SSF\({}_{1}\) NLE, \(\sim\)6.3% for the DBP-SSF\({}_{2}\) NLE, and \(\sim\)7.3% for the DBP-SSF\({}_{8}\) NLE compared to linear case. The DBP-SSF method with eight steps per span is once more the NLE with the best performance, compared to all the aforementioned NLEs, but at the vast expense of computational complexity, as shown in the following sub-section.  
The modest reach extensions provided by the NLEs provide valuable information about their physical limits while the nonlinearities increase gradually. Especially from Figure 7b, it is observed that even though the nonlinear effects have been further increased, due to the replacement of the SSMF with a LEAF, the performance of the single-step-step-span 3 rd-order IVSTF-NLE has been improved (i.e., from \(\sim\)3% in SSMF increased to \(\sim\)6.3% in LEAF). On the contrary, the DBP-SSF method needs eight-steps-per-span in order to provide the maximum reach extension (i.e., \(\sim\)7% in both, SSMF and LEAF, cases). Thus, in LEAF case, the performance of the 3 rd-order IVSTF-NLE differs only slightly compared to the performance of DBP-SSF\({}_{8}\) NLE while the former is almost eight times less complex.