# A Novel Optical-Based Methodology for Improving Nonlinear Fourier Transform
###### Abstract  
The increasing demand for bandwidth and long-haul transmission has led to new methods of signal processing and transmission in optical fiber communication systems. The nonlinear Fourier transform is one of the most recent methods proposed, and is able to represent an integrable nonlinear Schrodinger equation (NLSE) channel in terms of its continuous and discrete spectrum, to overcome the limitation of the bandwidth imposed by the Kerr effect on silica fibers. In this paper, we will propose and investigate the Boffetta-Osburne method for the direct nonlinear Fourier implementation, and the Gel'fand-Levitan-Marchenko equation for the inverse nonlinear Fourier, as only the continuous part of the nonlinear spectrum will be used to encode information. A novel methodology is proposed to improve their numerical implementation with respect to the NLSE, and we analyze in detail how the improved algorithm can be used in a real optical system, by investigating three different modulation schemes. We report increased performance transmission and consistency in the numerical results when the proposed methodology is applied. Our results show that b-modulation will increase the Q-factor by 2 dB with respect to the other two modulations. The improvement results with our proposed methodology suggest that b-modulation applied only to a continuous part of the nonlinear spectrum is a very effective method for maximizing both the transmission bandwidth and distance in optical fiber communication systems. nonlinear Fourier transform; numerical algorithms; nonlinear Schrodinger equation; modulation 20221045133425334  
systems. From a mathematical point of view, the NFT is able to represent an integrable nonlinear dispersive partial differential equation (PDE) in terms of its continuous and discrete spectrum [5]. As the wave propagates over the channel under different nonlinear effects, the impact of the channel on its spectral components will be simply given by independent linear equations, as will be shown later. This technique is very effective in multi-user channels where the information can be encoded on the nonlinear part of the spectrum. This results in reduced inter-channel interference, improved signal to noise ratio, and better spectral efficiency [6; 7; 8; 9]. On the other hand, the implementation of this technique requires physical changes in the current deployed infrastructure [10]. The NLSE is most widely solved using the numerical method, and hence, a larger number of numerical solutions have been proposed for it. The main reason behind the ability for the NLSE ability to be resolved numerically is that it does not have direct solutions known in analytical form, except for some specific cases. [PERSON] et al. [5; 11] made a comparative analysis of the numerical methods and the finite difference schemes used to solve the problem associated with NLSE. The analysis included the Ablowitz-Ladik scheme developed using notions related to the Inverse Scattering Transform (IST), nowadays called NFT in the telecommunications system community. The logic behind NFT is quite simple. If the information signal would have to propagate through a highly nonlinear optical fiber governed by the NLSE equation, then its waveform at a desired distance could be associated with a linear operator that we need to find. This linear operator is an infinite dimensional matrix with a spectrum that consists of continuous and discrete eigenvalues. It turns out that these eigenvalues are invariant of the NLSE during the propagation through the fiber. The goal is to find a modulation scheme where we can modulate these eigenvalues, thus overcoming the nonlinear effect limitation. The first idea based on this simplified observation was published around 30 years ago, where eigenvalue communication was proposed to substitute for the on-off key solution for transmission in optical fiber communication systems [12]. The same idea was re-proposed and experimentally verified later in [13; 14; 15].