Abstract

In this article, for the first time, a full-spectrum periodic nonlinear Fourier transform (NFT)-based communication system with the inverse transformation at the transmitter performed by using the solution of Riemann-Hilbert problem (RHP), is proposed and studied. The entire control over the nonlinear spectrum rendered by our technique, where we operate with two qualitatively different components of this spectrum represented, correspondingly, in terms of the main spectrum and the phases, allows us to design a time-domain signal tailored to the characteristics of the transmission channel. In the heart of our system is the RHP-based signal processing utilised to generate the time-domain signal from the modulated nonlinear spectrum. This type of NFT processing leads to a computational complexity that scales linearly with the number of time-domain samples, and we can process signal samples in parallel. In this article, we suggest the way of getting an exactly periodic signal through the correctly formulated RHP, and present evidence of the analogy between band-limited (in ordinary Fourier sense) signals and finite-band (in RHP sense) signals. Also, for the first time, we explain how to modulate the phases of individual periodic nonlinear modes. The performance of our transmission system is evaluated through numerical simulations in terms of bit error rate and Q <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> -factor dependencies on the transmission distance and power, and the results demonstrate the good potential of the approach.

Highlights

  • S INCE the advent of inverse scattering transform [1] and nonlinear Fourier transform (NFT)-based signal processing and eigenvalue communication dating back to the pioneering work of Hasegawa and Manuscript received September 15, 2019; revised January 30, 2020 and February 28, 2020; accepted March 5, 2020

  • The main spectrum structure chosen in our work allows us to manipulate the signal parameters, such that we can the desirable values of power, while phases attributed to each nonlinear mode will bear the encoded information

  • We developed a systematic way of modulating the parameters for periodic finite-band solutions to the nonlinear Schrödinger equation (NLSE), addressed the issue of exact periodicity for the carrier wave-forms, and presented the approach for controlling the signal characteristics

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Summary

INTRODUCTION

S INCE the advent of inverse scattering transform [1] and NFT-based signal processing and eigenvalue communication dating back to the pioneering work of Hasegawa and Manuscript received September 15, 2019; revised January 30, 2020 and February 28, 2020; accepted March 5, 2020. The difficulty here is that the evolution of the auxiliary NS part is governed by a set of coupled nonlinear differential equations [14], such that when using them directly we effectively loose the very advantage of the NFT-based methods Even though these equations can be turned into linear relations using the algebro-geometric construction called the Abel map [23], the conventional algebro-geometric approach to the inverse PNFT, i.e. to the construction of a (quasi-) periodic signal from its main and transformed auxiliary NS, entails the evaluation of computationally-expensive functions – the multidimensional Riemann theta-functions [23]. Throughout this text we use l.h.s and r.h.s acronyms to point the left-hand-side and right-hand-side of equations

PERIODIC NONLINEAR FOURIER TRANSFORM
A PNFT-BASED COMMUNICATION SYSTEM
Inverse Transform
Periodicity of the Resulting Finite-Band Signal
NUMERICAL SIMULATIONS AND RESULTS
Signal Characteristics Determined by the Main Spectrum
Accuracy and Limitations of the Numerical Routines
Simulation Results for a PNFT-Based Communication System
DISCUSSION AND CONCLUSION
An Iterative Approach Starting From a Signal With Infinitesimal Power
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