Nonlinear Fourier Transform in application to long-haul optical communications

The optical fibres form a basis of the long-haul transmissions systems, and is a significant component of the connectivity infrastructure. Rapidly growing demand in data traffic requires instantaneous imperative actions with long-term effect to meet the future expansion of the digital economy. The current optical networks resources are overstretched, and the further extensive utilisation will ultimately constrain the development of other economic sectors. The intelligent and effective usage of the installed infrastructure can shift forward the existent limitations, keeping the cost low because of avoiding of the reinstallation. One of the principal constrain, which bounds the further optical fibre capacity grows, is the existence of undesirable nonlinear phenomena, the so-called Kerr nonlinearity, causing self-phase, cross-phase modulation and four-wave mixing. The combination of advanced achievements of mathematical physics, together with communication engineering and information theory allowed to implement the so-called nonlinear Fourier transform (NFT) approach to optical communication. In its paradigm, the fibre nonlinearity is considered as a valuable part of the model, and the NFT mapping effectively (de)composes the signal to naturally non-interacting modes. The NFT concept can be applied to the signal propagation model with either vanishing or periodic boundary condition, which involves the different structures of parameters for manipulation. In this thesis, I focused on the investigation of boundary condition cases, discovering analytical properties, available degrees of freedom, developing numerical methods, and coding approaches; then examining their performance via the simulation of optical transmission systems. The results allow us to conclude the existence of several technical limitations, which limit the achievable transmission quality and data-rate. These include: the deviation of the channel model from the purely integrable, nonlinear and not explicit coupling of the resulting signal parameters, numerical methods accuracy and amplifiers noise accumulation. In spite of those, the simulations demonstrate the considerable performance of NFT-based communication systems

Contour integrals for numerical computation of discrete eigenvalues in the Zakharov–Shabat problem

We propose a novel algorithm for the numerical computation of discrete eigenvalues in the Zakharov–Shabat problem. Our approach is based on contour integrals of the nonlinear Fourier spectrum function in the complex plane of the spectral parameter. The reliability and performance of the new approach are examined in application to a single eigenvalue, multiple eigenvalues, and the degenerate breather’s multiple eigenvalue. We also study the impact of additive white Gaussian noise on the stability of numerical eigenvalues computation

Noise-induced signal corruption in nonlinear Fourier-based optical transmission system in the presence of discrete eigenvalues

We present the numerical analysis of the correlation properties of the amplifier spontaneous emission (ASE) noise transformed into the nonlinear Fourier (NF) domain, addressing the noise-induced corruptions in the communication systems employing the nonlinear Fourier transform (NFT) based signal processing. In our current work we deal with the orthogonal frequency division multiplexing (OFDM) modulation of a continuous NF spectrum and account for the presence of discrete (soliton) eigenvalues. This approach is aimed at extending our previous studies that referred to the modulation of continuous NF spectrum only. The effective noise covariance functions are obtained from numerical simulations for a range of propagation distances, values of discrete eigenvalue, and different effective signal power levels. We report the existence of the correlations between the continuous and discrete parts of the NF spectrum

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

Maximization or leveling: characterization of the trade-offs for the transmission throughput in ultrawideband optical transmission

In ultrawideband transmission, the overall noise comes from the amplification, fiber properties at different wavelengths, and stimulated Raman scattering, and its impact on channels across transmission bands is different. This requires a range of methods to mitigate the noise impact. Performing channel-wise power pre-emphasis and constellation shaping, one can compensate for the noise tilt and attain maximum throughput. In this work, we study the trade-off between the goals of maximizing the total throughput and leveling the transmission quality for different channels. We use an analytical model for multi-variable optimization and identify the penalty from constraining the mutual information variation

On the rigorous justification of b-modulation method and inclusion of discrete eigenvalues

Addressing the optical communication systems employing the nonlinear Fourier transform (NFT) for the data modulation/demodulation, we provide the explicit proof for the properties of the signals emerging in the so-called b-modulation method, the nonlinear signal modulation technique that provides the explicit control over the signal extent. Our approach ensures that the time-domain profile corresponding to the b-modulated data has a limited duration, including the cases when the bound states (discrete solitonic eigenvalues) are present. In particular, in contrast to the previous approaches, we show that it is possible to include the discrete eigenvalues with the specially chosen parameters into the b-modulation concept while keeping the signal localization property exactly

Nonlinear Fourier Spectrum Characterization of Time-Limited Signals

Addressing the optical communication systems employing the nonlinear Fourier transform (NFT) for the data modulation/demodulation, we provide an explicit proof for the properties of the signals emerging in the so-called $b$ -modulation method, the nonlinear signal modulation technique that provides explicit control over the signal extent. We present details of the procedure and related rigorous mathematical proofs addressing the case where the time-domain profile corresponding to the $b$ -modulated data has a limited duration, and when the bound states corresponding to specifically chosen discrete solitonic eigenvalues and norming constants, are also present. We also prove that the number of solitary modes that we can embed without violating the exact localisation of the time-domain profile, is actually infinite. Our theoretical findings are illustrated with numerical examples, where simple example waveforms are used for the $b$ -coefficient, demonstrating the validity of the developed approach. We also demonstrate the influence of the bound states on the noise tolerance of the b-modulated system

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

Signal modulation and processing in nonlinear fibre channels by employing the Riemann-Hilbert problem

Most of the nonlinear Fourier transform (NFT) based optical communication systems studied so far deal with the burst mode operation that substantially reduce achievable spectral efficiency. The burst mode requirement emerges due to the very nature of the commonly used version of the NFT processing method: it can process only rapidly decaying signals, requires zero-padding guard intervals for processing of dispersion-induced channel memory, and does not allow one to control the time-domain occupation well. Some of the limitations and drawbacks imposed by this approach can be rectified by the recently-introduced more mathematicallydemanding periodic NFT processing tools. However, the studies incorporating the signals with cyclic prefix extension into the NFT transmission framework have so far lacked the efficient digital signal processing (DSP) method of synthesising an optical signal, the shortcoming that diminishes the approach flexibility. In this work we introduce the Riemann-Hilbert problem (RHP) based DSP method as a flexible and expandable tool that would allow one to utilise the periodic NFT spectrum for transmission purposes without former restrictions. First, we outline the theoretical framework and clarify the implementation underlying the proposed new DSP method. Then we present the results of numerical modelling quantifying the performance of longhaul RHP-based transmission with the account of optical noise, demonstrating the good performance quality and potential of RHP-based optical communication systems

Communication system based on periodic nonlinear Fourier transform with exact inverse transformation

By performing the exact inverse transformation, a periodic solution to channel model is constructed and used in an NFT-based communication system. The achievable mutual information is calculated using the non-uniform probability distribution for transmitted symbols for different link lengths