Information Transmission using the Nonlinear Fourier Transform, Part II: Numerical Methods

TL;DR

This paper introduces and compares numerical methods for computing the nonlinear Fourier spectrum of signals in optical fiber communication channels, highlighting effective schemes and spectral behaviors under modulation.

Contribution

It proposes and evaluates numerical techniques for spectrum estimation of signals in integrable channels, with insights into eigenvalue dynamics during modulation.

Findings

Layer-peeling and spectral methods provide accurate spectrum estimates.

Eigenvalues can collide and change trajectories under parameter variations.

Real axis eigenvalues often originate or are absorbed during spectral evolution.

Abstract

In this paper, numerical methods are suggested to compute the discrete and the continuous spectrum of a signal with respect to the Zakharov-Shabat system, a Lax operator underlying numerous integrable communication channels including the nonlinear Schr\"odinger channel, modeling pulse propagation in optical fibers. These methods are subsequently tested and their ability to estimate the spectrum are compared against each other. These methods are used to compute the spectrum of various signals commonly used in the optical fiber communications. It is found that the layer-peeling and the spectral methods are suitable schemes to estimate the nonlinear spectra with good accuracy. To illustrate the structure of the spectrum, the locus of the eigenvalues is determined under amplitude and phase modulation in a number of examples. It is observed that in some cases, as signal parameters vary, eigenvalues collide and change their course of motion. The real axis is typically the place from which new eigenvalues originate or are absorbed into after traveling a trajectory in the complex plane.