Nonlinear Fourier transform using multistage perturbation technique for fiber-optic systems

A neural network architecture is proposed that allows a continuous nonlinear spectrum of optical signals to be predicted and an inverse nonlinear Fourier transform (NFT) to be performed for signal modulation. The average value of the relative error in predicting the continuous spectrum by the neural network when calculating the direct NFT is found to be 2.68 × 10−3, and the average value of the relative error in predicting the signal for the inverse NFT is 1.62 × 10−4.

Nonlinear Fourier transform (NFT) (also known in the mathematical and nonlinear science community as the inverse scattering transform [1] ) has recently attracted a great deal of attention in the context of optical transmission in fiber channels [2] , that can be approximated by the nonlinear Schrodinger equation. Within the NFT-based¨ transmission approach, we modulate the parameters of the nonlinear spectrum (NS) and generate the respective information signal in time domain using inverse NFT. Both discrete and continuous parts of NS can be used, here we focus on the continuous spectrum only. Then, the signal is launched into the fiber, and at the receiver we apply direct NFT to the received signal’s to retrieve the information encoded in NS. In this work we demonstrate that the high-accuracy computation of the continuous NS can be performed by using artificial neural networks (NN). The NS of a given localized signal q ( t ) containing no solitonic (discrete) components is represented by the continuous complex-valued function r ( ξ ) (the reflection coefficient) of the real spectral parameter ξ , where the latter plays the role of nonlinear frequency. The signals in our work have been specifically pre-selected to ensure that they contain no discrete spectrum. In the time domain, considered (normalized) symbol is given as the sum of independent sub-carriers [3] : $q(t) = \frac{1}{Q}\sum\nolimits_{k = 1}^M {{C_k}} {e^{i{\omega _k}t}}f(t)$ where M is a number of frequency channels, ω k is a carrier frequency of the k -th channel, C k corresponds to the digital data in k -th channel, and T defines the symbol interval; f ( t ) is the return-to-zero carrier support waveform. To assess the quality of NN prediction, we use the following formula for the relative error: $\eta (\xi ) = \frac{{\left\langle {\left| {{r_{predicted{\text{ }}}}(\xi ) - {r_{actual{\text{ }}}}(\xi )} \right|} \right\rangle }}{{\left\langle {\left| {{r_{actual{\text{ }}}}(\xi )} \right|} \right\rangle }}$ , where 〈•〉 denotes the mean over the spectral interval, the "predicted" and "actual" indices refer to the NN-predicted and precomputed values of the reflection coefficient r ( ξ ), respectively.

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.

Nonlinear Fourier transform (NFT) has attracted in the past several years a renewed interest as a potential tool for mitigation of nonlinear signal distortions in optical communications [1]. The NFT exploits the integrability of the master equation for the signal evolution in the optical fibre — nonlinear Schroedinger equation (NLSE). The NFT maps the solution of the NLSE to the domain of the complex-valued spectral parameter ξ, the nonlinear analogue of frequency. The complete set of nonlinear spectral data contains continuous functions α(ξ), b (ξ), and discrete spectrum, consisting of two complex-valued parameters for each discrete degree of freedom: eigenvalue ξ eig , IM ξ eig > 0, and spectral amplitude C(ξ eig ) associated to a discrete soliton eigenvalue [1].

The nonlinear Fourier transform (NFT) has attracted great interest recently as a new technique for nonlinearity mitigation in fibre-optic transmission. Most of the attention has so far been paid to the NFT for time-decaying signals while periodic signals can potentially ameliorate some difficulties of conventional NFT, offering a better system performance. We consider two communication systems based on the usage of periodic NFT (PNFT): with signal processing based (i) on the reduced algebro-geometric and (ii) on Riemann-Hilbert problem (RHP) approaches. Simulation results show that both methods can be used for nonlinearity mitigation. However, the latter approach is expandable without considerable penalty in processing complexity, which makes it a more appealing option for implementation.

We apply both the unsupervised and supervised machine learning (ML) methods, in particular, the k-means clustering and support vector machine (SVM) to improve the performance of the optical communication system based on the nonlinear Fourier transform (NFT). The NFT system employs the continuous NFT spectrum part to carry data up to 1000 km using the 16-QAM OFDM modulation. We classify the performance of the system in terms of BER versus signal power dependence. We show that the NFT system performance can be improved considerably by means of the ML techniques and that the more advanced SVM method typically outperforms the k-means clustering.

We investigate the computational cost of the nonlinear Fourier transform (NFT) based on the Zakharov–Shabat scattering problem as a nonlinear compensation technique for quadrature-phase-shift keyed (QPSK) signals with raised cosine frequency characteristic in optical fiber transmission systems with normal dispersion fibers. We show that the primary sources of computational errors that arise from the use of the NFT is the finite eigenvalue resolution of the left and the right reflection spectra. We show that this effect and, consequently, the computational cost of the NFT as a nonlinear mitigation technique in the normal dispersion regime increases exponentially or faster with both the channel power and the number of symbols per data frame even using the most efficient NFT algorithms that are currently known. We find that the computational cost of this approach becomes unacceptably large at data frame lengths and powers that are too small for this approach to be competitive with standard transmission methods. We explain the physical reasons for these limits.

We evaluate improvement in the performance of the optical transmission systems operating with the continuous nonlinear Fourier spectrum by the artificial neural network equalisers installed at the receiver end. We propose here a novel equaliser designs based on bidirectional long short-term memory (BLSTM) gated recurrent neural network and compare their performance with the equaliser based on several fully connected layers. The proposed approach accounts for the correlations between different nonlinear spectral components. The application of BLSTM equaliser leads to a 16x improvement in terms of bit-error rate (BER) compared to the non-equalised case. The proposed equaliser makes it possible to reach the data rate of 170 Gbit/s for one polarisation conventional nonlinear Fourier transform (NFT) based system at 1000 km distance. We show that our new BLSTM equalisers significantly outperform the previously proposed scheme based on a feed-forward fully connected neural network. Moreover, we demonstrate that by adding a 1D convolutional layer for the data pre-processing before BLSTM recurrent layers, we can further enhance the performance of the BLSTM equaliser, reaching 23x BER improvement for the 170 Gbit/s system over 1000 km, staying below the 7% forward error correction hard decision threshold (HD-FEC).

We propose using the nonlinear Fourier transform (NFT) not as a method for integrating equations but as a tool for studying the properties of localized coherent structures in dissipative nonlinear systems. The effectiveness of using a discrete nonlinear spectrum for quantitatively describing the dynamics of optical pulses is demonstrated, even when the original model is not integrable. The application of this method is substantiated using the solution to the cubic Haus–Ginzburg–Landau equation (HGLE), which describes the generation of optical pulses from noise in passively mode-locked lasers. It is shown that stabilization of the discrete nonlinear spectrum serves as a reliable indicator of stable soliton generation. Additionally, algorithms for tracing individual discrete eigenvalues are proposed, including a machine-learning-based algorithm, which expands the toolbox for analyzing and automating the processing of data obtained using NFT.

A minimum mean squared error (MMSE) equalizer is a way to effectively increase transmission performance for nonlinear Fourier transform (NFT) based communication systems. Other equalization schemes, based on nonlinear equalizer approaches or neural networks, are interesting for NFT transmission due to their ability to deal with nonlinear correlations of the NFTs' eigenvalues and their coefficients. We experimentally investigated single- and dual-polarization long haul transmission with several modulation schemes and compared different equalization techniques including joint detection equalization and the use of neural networks. We observed that joint detection equalization provides range increases for shorter transmission distances while having low numeric complexity. We could further achieve bit error rates (BER) under HD-FEC for significant longer transmission distances in comparison to no equalization with different equalizers.

The impact of launch power, additive white Gaussian noise and fiber loss on the nonlinear Fourier transform (NFT) continuous spectrum is investigated. NFT is shown to undergo lower spectral distortion than the discrete Fourier transform.

Nonlinear frequency division multiplexing (NFDM) techniques encode information in the so called nonlinear spectrum which is obtained from the nonlinear Fourier transform (NFT) of a signal. NFDM techniques so far have been applied to the nonlinear Schrodinger equation (NLSE) that models signal propagation in a lossless fiber. Conventionally, the true lossy NLSE is approximated by a lossless NLSE using the path-average approach which makes the propagation model suitable for NFDM. The error of the path-average approximation depends strongly on signal power, bandwidth, and the span length. It can degrade the performance of NFDM systems and imposes challenges on designing high data rate NFDM systems. Previously, we proposed the idea of using dispersion decreasing fiber (DDF) for NFDM systems. These DDFs can be modeled by a NLSE with varying-parameters that can be solved with a specialized NFT without approximation errors. We have shown in simulations that complete nonlinearity mitigation can be achieved in lossy fibers by designing an NFDM system with DDF if a properly adapted NFT is used. We reported performance gains by avoiding the aforementioned path-average error in an NFDM system by modulating the discrete part of the nonlinear spectrum. In this article, we extend the proposed idea to the modulation of continuous spectrum. We compare the performance of NFDM systems designed with dispersion decreasing fiber to that of systems designed with a standard fiber with path-average model. Next to the conventional path-average model, we furthermore compare the proposed system with an optimized path-average model in which amplifier locations can be adapted. We quantify the improvement in the performance of NFDM systems that use DDF through numerical simulations.

Fast Fourier Transform(FFT) represents a common communication pattern shared by a large class of scientific and engineering problems and wavelength assignment is a key issue to increase efficiency and reduce cost in Wavelength Division Multiplexing (WDM) optical networks. In this paper, we propose a new scheme for the wavelength assignment of parallel FFT communication pattern on WDM linear arrays. By lattice embedding, the number of wavelengths required to realize parallel FFT communication pattern on WDM linear arrays significantly improves the known result. Our proposed embedding method also provides a new approach to the hypercube layout problem considering connections dimension by dimension rather than all connections as in the traditional approach.

Several non-linear fluid mechanical processes, such as wave propagation in shallow water, are known to generate solitons: localized waves of translation. Solitons are often hidden in a wave packet at the beginning and only reveal themselves in the far-field. With a special signal processing technique known as the non-linear Fourier transform (NFT), solitons can be detected and characterized before they emerge. In this paper, we present a new algorithm aimed at computing the phase shift of solitons in processes governed by the Korteweg–de Vries (KdV) equation. In numerical examples, the new algorithm is found to perform reliably even in cases where existing algorithms break down.

In this letter, we mathematically analyze the influence of the perturbation of the real and imaginary part of the eigenvalues on the amplitude and phase noise of the discrete spectrum based on the nonlinear Fourier transform (NFT). Considering the correlation between the multiple eigenvalues, the impact of the noise is more complex and difficult to be numerically processed. We propose a method to suppress the noise interference by applying neural network (NN) of the nonlinear spectrum domain as an equalizer at the receiver. As shown in simulation results, the proposed method could effectively mitigate the impact of noise on NFT-based multi-eigenvalue transmission. Compared with other method, the proposed method shows a significant performance gain. Our research provides a novel perspective for solving the coupling interference between the independently modulated spectrum corresponding to different eigenvalues.