Mutual information in nonlinear communication channel. Preliminary analytical results in large SNR and small nonlinearity limit

TL;DR

This paper uses perturbation theory to derive analytical expressions for the mutual information in a nonlinear communication channel modeled by the nonlinear Schrödinger equation, focusing on high SNR and weak nonlinearity.

Contribution

It provides the first analytical analysis of mutual information corrections due to nonlinearity in the large SNR limit, demonstrating that singular SNR terms cancel out.

Findings

Corrections to Shannon's mutual information are quadratic in nonlinearity.

Singular terms in SNR vanish in the final mutual information expression.

Framework established for calculating nonlinear corrections in future work.

Abstract

Applying perturbation theory to the path-integral representation for the mutual information of the nonlinear communication channel described by the nonlinear Shr\"{o}dinger equation (NLSE) with the additive Gaussian noise we analyze the analytical expression for the mutual information at large signal-to-noise ratio ($\mathrm{SNR}$) and small nonlinearity. We classify all possible corrections to the mutual information in nonlinearity parameter and demonstrate that all singular in $\mathrm{SNR}$ terms vanish in the final result. Furthermore our analytical result demonstrates that the corrections to Shannon's contribution to the mutual information in the leading order in $\mathrm{SNR}$ are of order of squared nonlinearity parameter. We outline the way for the calculation of these corrections in the further investigations.