# A Novel Optical-Based Methodology for Improving Nonlinear Fourier Transform
## 6 Conclusions  
In this work, we have improved the NFT and INFT algorithms by going into the numerical details of the integral equation, proposing a novel methodology for their implementation, validating the results, and applying them on three different modulated signals for an optical communication system. Only the continuous part of the nonlinear scattering data is investigated, as the spectral efficiency is higher and the complexity implementation is lower compared to the discrete nonlinear spectrum. We have considered the Boffetta-Osburne method for direct NFT, and the numerical solution of the GLM integral equation for INFT implementation. For both methods, we have simplified their complex mathematical derivation and have shown how to apply them step-by-step for an improved numerical implementation, as they are subject to various approximation errors that may drastically affect the final performance results. We have identified four parameters that affect the performances of our numerical implementations of NFT and INFT during the validation process. They consist of two temporal parameters: \(T\) function width and \(M\) discrete points, and two spectral parameters: resolution in frequency and the number of points \(N\). The temporal parameter, \(T\), requires the function to be considered zero outside of \(2T\), and this will be the source of additional overhead data that are used in a real modulated signal. The total discrete points, \(2M+1\), will define how many data points we can transmit at the same time, similar to the OFDM scheme. A lower number will affect the performance of the system. The number of points \(N\) was selected to be a high number with the assumption of having a large oversampling factor at the transmitter side. All these parameters should be optimized in order to gain the maximum performance from the nonlinear Fourier transform. We also observed that for large amplitudes, the performance drastically decreased because when truncating the integrals, due to the Gibbs effect, the errors will also increase. In addition, during each iteration, the errors are accumulated by affecting the performance of the system. In Section 3.1, we showed the steps of the algorithm to be used to properly represent the signal in the time and frequency domains, and how we should select the domain of the Marchenko system for INFT implementation. Figure 7: Q-factor versus transmission distances for all three modulations for a 50 Gbaud QPSK modulation. In blue line is shown the b-modulation, in black line, we show the modulation with a normalized reflection coefficient \(\eta_{c,n}\) given in Equation (64), and in red line, the normal modulation with reflection coefficient \(\eta_{c,n}\), given in Equation (63). We then conducted extensive numerical simulations on three different modulation schemes. Our results suggest that b-modulation is an effective method to be used in optical fiber communication systems. It shows better performance with a higher Q-factor at a fixed input average power and distance. The results presented in this paper show the feasibility of a continuous nonlinear optical-based spectrum with our proposed methodology for NFT/INFT implementation. Despite the fact that this is a significant step forward for numerical simulation, it needs further research targeting of least NFT/INFT complexity and higher accuracy algorithm. For future work, we plan to use only the Boffeta-Osburne method for both direct NFT and INFT using the duality principle, and we compare the results with a system where only the Marchenko system is used. Using only one algorithm without performance degradation will probably make the system less prone to numerical errors. Conceptualization, J.H. and E.D. ; methodology, J.H. ; software, J.H., E.D. and I.K. ; validation, J.H. and W.H.F.A. ; investigation, J.H. and M.A. ; visualization, I.K. and E.D. ; original draft preparation, J.H. and E.D. ; review and editing, W.H.F.A. and M.A. All authors have read and agreed to the published version of the manuscript.