## 4 Results and Discussion
### Evaluation of Computational Complexity  
The computational complexity is one of the criteria used to decide whether a specific NLE is suitable for real-time implementation. Therefore, in this subsection, we evaluate the computational complexity of the various nonlinear compensation methods, in terms of the number of real multiplications per sample per polarization. In addition, we compute the maximum \(Q^{2}\)-factor provided by the various NLEs as a function of complexity, after transmission through SSMF and LEAF. First, we estimate the computational complexity of the 3 rd-order IVSTF-NLE in terms of real multiplications per sample per polarization following the method of [16]. However, in our calculations, and modeling, we used the same FFT block size (\(N_{FFT}\)) for both the linear and the nonlinear branches, while in [16] each of the nonlinear branches operate with \(N_{FFT}\)/2. We decided to use the same \(N_{FFT}\) in both, the linear and the nonlinear, branches because it was observed, through trial and error, that only then the 3 rd-order IVSTF-NLE exhibits its best performance in the worst case scenario (see Figure 3c)  
Figure 7: Maximum transmission reach at the FEC limit vs. launch power using IVSTF-NLE and DBP-SSF\({}_{1,2,8}\) NLEs for the 3 rd band of the center OFDM super-channel in the simulation case, after signal transmission through **(a)** SSMF and **(b)** LEAF. and after 1000 km of SSMF using only 1.8 SpS. We consider a radix-2 implementation of FFT. In this case, the number of real multiplications for every FFT/IFFT is equal to 2 log\({}_{2}N_{FFT}\)[16]. Therefore, the FFT and the IFFT at the input and at the output of the IVSTF-NLE, respectively, are realized with 4 log\({}_{2}N_{FFT}\) real multiplications [16]. The linear branch of the IVSTF-NLE, as depicted in Figure 1a, operates with 4 real multiplications. At each nonlinear branch (Figure 1b), CD compensation is performed with 4 real multiplications per sample, in the frequency domain. Next, an IFFT requires 2 log\({}_{2}N_{FFT}\) real multiplications in the time domain. Then, 5 real multiplications per sample per polarization are required for multiplying the total power of \(x\) and \(y\) polarizations with \(jc\) (i.e., \(C_{IVSTF}\)). The latter multiplication is common between the two polarizations, thus, only 2.5 multiplications are required per sample per polarization. An FFT follows, with 2 log\({}_{2}N_{FFT}\) real multiplications, for the compensation of the residual CD in the frequency domain, adding 4 more real multiplications. Totally, 4 + 4 log\({}_{2}N_{FFT}\) + \(N_{spans}\)\(\times\) (4 log\({}_{2}N_{FFT}\) + 10.5) real multiplications per sample per polarization are needed for the operation of the 3 rd-order IVSTF-NLE. For the DBP-SSF method, the required number of real multiplications per polarization per sample is \(N_{steps}\)\(\times\)\(N_{spans}\)\(\times\) (4 log\({}_{2}N_{FFT}\) + 10.5) [16]. From the computational complexity analysis, it is obvious that the lower the \(N_{FFT}\), the lower the computational complexity per block. At this point, the key question is if there is an optimum \(N_{FFT}\), at which the performance limit of each NLE can be reached with a minimum computational complexity per block. To answer this question, we applied, right before each NLE, the method of overlap and save (OS) to divide the received, uncompensated signal into small FFT blocks. After the full compensation of each block (i.e., linear and nonlinear), we aggregated them back together, at the output of each NLE, to form the output signal. Then, we measured the \(Q^{2}\)-factor as a function of the \(N_{FFT}\), after 14 \(\times\) 100 km of SSMF, at the optimum input power per band (i.e., \(-\)3 dBm). We chose this transmission reach because it corresponds to the FEC limit without nonlinear compensation (i.e., \(Q^{2}\) = 6.25 dB). Following a similar rationale, we repeated the simulation runs after 11 \(\times\) 100 km of LEAF, at optimum input power per band equal to \(-\)6 dBm.