Information Transmission using the Nonlinear Fourier Transform, Part I: Mathematical Tools

TL;DR

This paper introduces the mathematical tools of the nonlinear Fourier transform, a method for solving integrable PDEs like the nonlinear Schrödinger equation, enabling nonlinear signal processing in fiber-optic communication.

Contribution

It provides a detailed explanation of the mathematical framework of the nonlinear Fourier transform for use in nonlinear channel data transmission.

Findings

NFT decorrelates signal degrees-of-freedom in nonlinear media

Enables direct handling of dispersion and nonlinearity in fiber-optic channels

Lays foundation for nonlinear frequency-division multiplexing

Abstract

The nonlinear Fourier transform (NFT), a powerful tool in soliton theory and exactly solvable models, is a method for solving integrable partial differential equations governing wave propagation in certain nonlinear media. The NFT decorrelates signal degrees-of-freedom in such models, in much the same way that the Fourier transform does for linear systems. In this three-part series of papers, this observation is exploited for data transmission over integrable channels such as optical fibers, where pulse propagation is governed by the nonlinear Schr\"odinger equation. In this transmission scheme, which can be viewed as a nonlinear analogue of orthogonal frequency-division multiplexing commonly used in linear channels, information is encoded in the nonlinear frequencies and their spectral amplitudes. Unlike most other fiber-optic transmission schemes, this technique deals with both dispersion and nonlinearity directly and unconditionally without the need for dispersion or nonlinearity compensation methods. This first paper explains the mathematical tools that underlie the method.