Statistics of the Nonlinear Discrete Spectrum of a Noisy Pulse

TL;DR

This paper develops a new method to analyze the statistical behavior of the nonlinear Fourier transform of noisy pulses, providing both numerical and analytical tools to understand eigenvalues and spectral amplitudes under Gaussian noise.

Contribution

It extends the Fourier Collocation method for computing the discrete spectrum and derives analytic expressions for their joint statistics in noisy conditions.

Findings

Fourier Collocation accuracy is comparable to existing NFT algorithms.

Analytic expressions closely match empirical statistics.

Provides a comprehensive framework for understanding NFT statistics under noise.

Abstract

In the presence of additive Gaussian noise, the statistics of the nonlinear Fourier transform (NFT) of a pulse are not yet completely known in closed form. In this paper, we propose a novel approach to study this problem. Our contributions are twofold: first, we extend the existing Fourier Collocation (FC) method to compute the whole discrete spectrum (eigenvalues and spectral amplitudes). We show numerically that the accuracy of FC is comparable to the state-of-the-art NFT algorithms. Second, we apply perturbation theory of linear operators to derive analytic expressions for the joint statistics of the eigenvalues and the spectral amplitudes when a pulse is contaminated by additive Gaussian noise. Our analytic expressions closely match the empirical statistics obtained through simulations.