Adaptive Geometric Constellation Shaping in a Transmission System with a Real-time Optimisation Loop

We demonstrate the real-time performance of an adaptive intelligent transceiver, tailoring the constellation shape to the transmission system by iteratively maximising the information throughput, quantified by the GMI

Properties of the effective noise in the nonlinear Fourier transform-based transmission

We investigate the correlation properties of optical noise in nonlinear Fourier domain for communication systems using the nonlinear Fourier transform. Effective covariance functions are obtained numerically and compared with theoretical predictions

Fixed-point realization of fast nonlinear Fourier transform algorithm for FPGA implementation of optical data processing

The nonlinear Fourier transform (NFT) based signal processing has attracted considerable attention as a promising tool for fibre nonlinearity mitigation in optical transmission. However, the mathematical complexity of NFT algorithms and the noticeable distinction of the latter from the “conventional” (Fourier-based) methods make it difficult to adapt this approach for practical applications. In our work, we demonstrate a hardware implementation of the fast direct NFT operation: it is used to map the optical signal onto its nonlinear Fourier spectrum, i.e. to demodulate the data. The main component of the algorithm is the matrix-multiplier unit, implemented on field-programmable gate arrays (FPGA) and used in our study for the estimation of required hardware resources. To design the best performing implementation in limited resources, we carry out the processing accuracy analysis to estimate the optimal bit width. The fast NFT algorithm that we analyse, is based on the FFT, which leads to the O(N log^{2}_{2} N) method’s complexity for the signal consisting of N samples. Our analysis revealed the significant demand in DSP blocks on the used board, which is caused by the complex-valued matrix operations and FFTs. Nevertheless, it seems to be possible to utilise further the parallelisation of our NFT-processing implementation for the more efficient NFT hardware realisation

Study of Noise-Induced Signal Corruption for Nonlinear Fourier-Based Optical Transmission

We study the correlation properties of the amplifier spontaneous emission noise transformed into the nonlinear Fourier (NF) domain for communication systems employing the nonlinear Fourier transform (NFT) based signal processing with OFDM modulation of a continuous spectrum. The effective noise covariance functions are obtained from numerical simulations for propagation distances ∼ 1000 km and different effective NF “power” values. It is shown that the correlation between the continuous NF eigenmodes reveals a nontrivial dependence on both the power and propagation distance

A Closed-Form Expression for the Gaussian Noise Model in the Presence of Inter-Channel Stimulated Raman Scattering Extended for Arbitrary Loss and Fibre Length

A closed-form formula for the nonlinear interference (NLI) estimation using the Gaussian noise (GN) model in the presence of inter-channel stimulated Raman scattering (ISRS) is derived. The formula enables accurate estimation of the NLI evolution along any portion of the fibre span together with arbitrary values of optical fibre losses. The formula also accounts for wavelength-dependent fibre parameters, variable modulation formats and launch power profiles. The formula is suitable for ultra-wideband (UWB) optical transmission systems and its accuracy is assessed for a system with 20 THz optical bandwidth over the entire S-, C-, and L- band through comparison with numerical integration of the ISRS GN model and split-step Fourier method (SSFM) simulations in point-to-point transmission and inline NLI estimation scenarios

Challenges in Extending Optical Fibre Transmission Bandwidth beyond C+L Band and How to Get There

Recently, we demonstrated a record single-mode fibre net throughput of 178.08 Tbit/s. In this paper, we model this experiment, investigating the main limitations and challenges behind this total throughput, together with the details of some approaches to overcome them, and an outlook for the future ultra-wideband network design and optimisation

Phase computation for the finite-genus solutions to the focusing nonlinear Schrödinger equation using convolutional neural networks

We develop a method for retrieving a set of parameters of a quasi-periodic finite-genus (finite-gap) solution to the focusing nonlinear Schrödinger (NLS) equation, given the solution evaluated on a finite spatial interval for a fixed time. These parameters (named “phases”) enter the jump matrices in the Riemann-Hilbert (RH) problem representation of finite-genus solutions. First, we detail the existing theory for retrieving the phases for periodic finite-genus solutions. Then, we introduce our method applicable to the quasi-periodic solutions. The method is based on utilizing convolutional neural networks optimized by means of the Bayesian optimization technique to identify the best set of network hyperparameters. To train the neural network, we use the discrete datasets obtained in an inverse manner: for a fixed main spectrum (the endpoints of arcs constituting the contour for the associated RH problem) and a random set of modal phases, we generate the corresponding discretized profile in space via the solution of the RH problem, and these resulting pairs – the phase set and the corresponding discretized solution in a finite interval of space domain – are then employed in training. The method's functionality is then verified on an independent dataset, demonstrating our method's satisfactory performance and generalization ability

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

Analysis of laser radiation using the Nonlinear Fourier transform

Modern high-power lasers exhibit a rich diversity of nonlinear dynamics, often featuring nontrivial co-existence of linear dispersive waves and coherent structures. While the classical Fourier method adequately describes extended dispersive waves, the analysis of time-localised and/or non-stationary signals call for more nuanced approaches. Yet, mathematical methods that can be used for simultaneous characterisation of localized and extended fields are not yet well developed. Here, we demonstrate how the Nonlinear Fourier transform (NFT) based on the Zakharov-Shabat spectral problem can be applied as a signal processing tool for representation and analysis of coherent structures embedded into dispersive radiation. We use full-field, real-time experimental measurements of mode-locked pulses to compute the nonlinear pulse spectra. For the classification of lasing regimes, we present the concept of eigenvalue probability distributions. We present two field normalisation approaches, and show the NFT can yield an effective model of the laser radiation under appropriate signal normalisation conditions

Mutual Shaping and Pre-emphasis Gain Magnification in the Throughput Maximisation for Ultrawideband Transmission

We demonstrate that in ultrawideband transmisison over 20THz, the added value of joint probabilistic shaping and power pre-emphasis is a factor of two higher than their individual contributions and gives a 20% increase in total mutual information