# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1901.11419