# A Novel Optical-Based Methodology for Improving Nonlinear Fourier Transform
###### Abstract
However, many researchers observed that there is no need to have an unchanged shape in the optical pulse in the NFT domain, as the channel effect will be a linear one. Moreover, this effect can be easily equalized at the receiver side, or pre-compensated at the transmitter side. Several published papers exploit different degrees of freedom of the NFT and the Inverse NFT (INFT), using both discrete and continuous eigenvalues from the nonlinear spectrum. This results in a data transmission over the optical fiber similar to orthogonal frequency division multiplexing (OFDM) [16; 17]. Hence, multiple users can access the optical channel. Other researchers focused only on the discrete part of the spectrum and proved experimentally the reliability of the NFT and INFT [18; 19; 20; 21; 22; 23; 24]. The sole usage of the continuous part was studied in several papers [25; 26; 27; 28; 29]. Compared to using only the discrete part, this method resulted in increased performance in terms of spectral efficiency, transmission over long distances, and lower complexity at the receiver side. Dual polarization experimental transmission based on NFT was also reported in [30; 31]. Full spectrum modulation using both continuous and discrete nonlinear parts of the spectrum was reported in [32], but this was limited only to back-to-back transmission. Several numerical algorithms are proposed in the literature to solve the direct NFT and INFT [33]. To compute the direct NFT, we can generalize the algorithms into two main methods: (1) the Ablowitz-Ladik method [34; 35] and (2) the Boffetta-Osborne Method [36]. To compute the INFT, we can generalize the algorithms into two main methods: the Darboux transformation method [37] and the Gelfand-Levitan-Marchenko (GLM) method [38]. Fast non linear Fourier transform (FNFT) is also analyzed in several papers, and is published online as a tool to reduce the implementation complexity of the NFT [39; 40]. The purpose of FNFT is to develop an algorithm that computes a fast product between polynomials that are used to find the continuous spectrum. The NFT and INFT are relatively new methods, where the associated numerical implementations have not yet been fully addressed, especially its application in the field of optical fiber telecommunications. Analytical solutions of the NLS have also been recently proposed, in [41; 42; 43; 44; 45]. Unfortunately, they cannot be directly applied for optical communication, and numerical solutions with lower complexity are still the only alternative to solve NLS. In this paper, we propose a methodology to improve both the direct NFT and INFT numerical implementations in order to obtain only the continuous part of the nonlinear spectrum. The reason behind choosing this approach as a preliminary results shows a potential deployment in the future of the optical fiber transmission, reported in [9]. Compared to existing wavelength division multiplexing (WDM) systems, the continuous spectrum has a higher spectral efficiency and reduced complexity at the receiver side. For the direct NFT numerical implementation, we will investigate in detail the Boffetta-Osborne method, as it cannot be simplified using polynomials, in order to make use of the fast NFT (FNFT), as in [46; 47]. An explanation on how this method is improved for NFT, and how it will affect the INFT implementation without performance degradation, will be given below. For the INFT numerical implementation, we will solve and investigate numerically the GLM integral equations. These equations can represent a complexity that is comparable to FNFT. We have identified the parameters' values for maximizing the performance, and have used them to compare three different types of modulations that are proposed recently. In the main strength of our work, we present the complex mathematical tools that are involved in a simplified form, and minimize errors coming from multiple numerical approximations.