Direct nonlinear Fourier transform algorithms for the computation of solitonic spectra in focusing nonlinear Schrödinger equation

Starting from a comparison of some established numerical algorithms for the computation of the eigenvalues (discrete or solitonic spectrum) of the non-Hermitian version of the Zakharov–Shabat spectral problem, this article delivers new algorithms that combine the best features of the existing ones and thereby allays their relative weaknesses. Our algorithm is modelled within the remit of the so-called direct nonlinear Fourier transform (NFT) associated with the focusing nonlinear Schrödinger equation. First, we present the data for the calibration of existing methods comparing the relative errors associated with the computation of the continuous NF spectrum. Then each method is paired with different numerical algorithms for finding zeros of a complex-valued function to obtain the eigenvalues. Next we describe a new class of methods based on the contour integrals evaluation for the efficient search of eigenvalues. After that we introduce a new hybrid method, one of our main results: the method combines the advances of contour integral approach and makes use of the iterative algorithms at its second stage for the refined eigenvalues search. The veracity of our new hybrid algorithm is established by estimating the convergence speed and accuracy across three independent test profiles. Along with the development of a new approach for the computation of the eigenvalues, our study also addresses the problem of computation of the so-called norming constants associated with the eigenvalues. We show that our formalism effectively amounts to accurate and fast enough computation of residues of the reflection coefficient in the upper complex half-plane of the spectral parameter

Neural networks for computing and denoising the continuous nonlinear Fourier spectrum in focusing nonlinear Schrödinger equation

We combine the nonlinear Fourier transform (NFT) signal processing with machine learning methods for solving the direct spectral problem associated with the nonlinear Schrödinger equation. The latter is one of the core nonlinear science models emerging in a range of applications. Our focus is on the unexplored problem of computing the continuous nonlinear Fourier spectrum associated with decaying profiles, using a specially-structured deep neural network which we coined NFT-Net. The Bayesian optimisation is utilised to find the optimal neural network architecture. The benefits of using the NFT-Net as compared to the conventional numerical NFT methods becomes evident when we deal with noise-corrupted signals, where the neural networks-based processing results in effective noise suppression. This advantage becomes more pronounced when the noise level is sufficiently high, and we train the neural network on the noise-corrupted field profiles. The maximum restoration quality corresponds to the case where the signal-to-noise ratio of the training data coincides with that of the validation signals. Finally, we also demonstrate that the NFT b-coefficient important for optical communication applications can be recovered with high accuracy and denoised by the neural network with the same architecture

Nonlinear Fourier transform for optical data processing and transmission:advances and perspectives

Fiber-optic communication systems are nowadays facing serious challenges due to the fast growing demand on capacity from various new applications and services. It is now well recognized that nonlinear effects limit the spectral efficiency and transmission reach of modern fiber-optic communications. Nonlinearity compensation is therefore widely believed to be of paramount importance for increasing the capacity of future optical networks. Recently, there has been steadily growing interest in the application of a powerful mathematical tool-the nonlinear Fourier transform (NFT)-in the development of fundamentally novel nonlinearity mitigation tools for fiber-optic channels. It has been recognized that, within this paradigm, the nonlinear crosstalk due to the Kerr effect is effectively absent, and fiber nonlinearity due to the Kerr effect can enter as a constructive element rather than a degrading factor. The novelty and the mathematical complexity of the NFT, the versatility of the proposed system designs, and the lack of a unified vision of an optimal NFT-type communication system, however, constitute significant difficulties for communication researchers. In this paper, we therefore survey the existing approaches in a common framework and review the progress in this area with a focus on practical implementation aspects. First, an overview of existing key algorithms for the efficacious computation of the direct and inverse NFT is given, and the issues of accuracy and numerical complexity are elucidated. We then describe different approaches for the utilization of the NFT in practical transmission schemes. After that we discuss the differences, advantages, and challenges of various recently emerged system designs employing the NFT, as well as the spectral efficiency estimates available up-to-date. With many practical implementation aspects still being open, our mini-review is aimed at helping researchers assess the perspectives, understand the bottlenecks, and envision the development paths in the upcoming NFT-based transmission technologies

Periodic nonlinear Fourier transform for fiber-optic communications, Part I:theory and numerical methods

In this work, we introduce the periodic nonlinear Fourier transform (PNFT) method as an alternative and efficacious tool for compensation of the nonlinear transmission effects in optical fiber links. In the Part I, we introduce the algorithmic platform of the technique, describing in details the direct and inverse PNFT operations, also known as the inverse scattering transform for periodic (in time variable) nonlinear Schrödinger equation (NLSE). We pay a special attention to explaining the potential advantages of the PNFT-based processing over the previously studied nonlinear Fourier transform (NFT) based methods. Further, we elucidate the issue of the numerical PNFT computation: we compare the performance of four known numerical methods applicable for the calculation of nonlinear spectral data (the direct PNFT), in particular, taking the main spectrum (utilized further in Part II for the modulation and transmission) associated with some simple example waveforms as the quality indicator for each method. We show that the Ablowitz-Ladik discretization approach for the direct PNFT provides the best performance in terms of the accuracy and computational time consumption

On nonlinear Fourier transform-based fibre-optic communication systems for periodic signals

As the demand for information rate grows on a daily basis, new ways of improving the efficiency of fibre-optic communication systems, the backbone of the global data network,are highly anticipated. Nonlinear Fourier transform (NFT) is one of the newly emerged techniques showing promising results in recent studies both in simulation and experiment. Along this path, this method has shown its potential to overcome some difficulties of the fibre-optic communication regarding nonlinear distortions, especially the crosstalk between the user’s bands in wavelength division multiplexing (WDM) systems. NFT-based systems, however, in the conventional, widely considered case of vanishing boundary signals, have exhibited some drawbacks related to the computational complexity and spectral efficiency. Both problems are the direct consequences of large signal duration ensued from the vanishing boundary condition. Considering periodic solutions to the nonlinear Schrödinger equation is among attempts to solve this problem. It helps to decrease the processing window at the receiver and gives full control over the communication-related parameters of the signal. Periodic NFT (PNFT) can also be implemented through fast numerical methods which makes it yet more appealing. In this thesis, a general framework to implement PNFT in fibre-optic communication systems is proposed. As the most challenging part of such a system, the inverse transformation stage is particularly taken attention to, and a few ways to perform it are put forward. From the simplest signals with analytically known nonlinear spectrum to a complete periodic solution with arbitrary, finite number of degrees of freedom, several system configurations are conferred and evaluated in terms of their performance. Common measures such as bit error rate, quality factor and mutual information are considered in scrutinising the systems.Based on simulation results, we conclude that the PNFT can, in fact, improve the mutual information by overcoming some shortcomings of the vanishing boundary NFT

Machine Learning For Performance Improvement of Long-Haul End-to-End Optical Transmission Systems

The thesis focuses on addressing the challenges faced by optical fiber networks in keeping up with the growing demand for data transfer, especially with the advent of 5G/6G and the Internet of Things (IoT). The rapid expansion in data transfer requirements highlights the limitations of current optical fiber networks and the necessity for improvements in data encoding techniques, spectrumutilization, and signal clarity over long distances. The thesis contributes to this field by developing new methods for applying the Nonlinear Fourier Transform (NFT) to continuous signals, improving signal processing algorithms, and using Machine learning (ML) to understand complex patterns and make data-driven decisions to optimize optical communication systems. The work is divided into two primary sections. The first section delves into advanced NFT techniques, including their application in optical fiber channel modeling for single and dualpolarization systems, signal processing with a sliding window technique combined with NFT, exploring solitonic components in optical signals, and the use of neural networks for NFT to work with noisy signals. The second section is dedicated to the role of ML in optimizing optical communication systems, discussing the new High-Performance COMmunication library (Hp-Com) framework for simulating optical channels, the use of Gradient Boosting for nonlinear equalization, studying received symbol distributions using the GaussianMixtureModel, and summarizing findings with insights for future research. The thesis outlines the creation of innovative techniques to improve optical fiber systems, thus aiding the continued development of the digital world by handling the ever-increasing demands for data transmission