Optimal input signal distribution for nonlinear optical fiber channel with small Kerr nonlinearity

TL;DR

This paper analyzes a nonlinear optical fiber channel with small Kerr nonlinearity, deriving the optimal input signal distribution and channel capacity using perturbation theory and information-theoretic calculations.

Contribution

It introduces a perturbative approach to compute channel capacity and optimal input signals for a nonlinear optical fiber channel with small Kerr nonlinearity.

Findings

Derived the first three terms of the conditional probability density function expansion.

Calculated the channel capacity and optimal input distribution in the small nonlinearity regime.

Provided a method to construct input signals with optimal statistics for given signal shapes.

Abstract

We consider the information channel described by Schr\"{o}dinger equation with additive Gaussian noise. We introduce the model of the input signal and the model of the output signal receiver. For this channel, using perturbation theory for the small nonlinearity parameter, we calculate the first three terms of the expansion of the conditional probability density function in the nonlinearity parameter. At large signal-to-noise power ratio we calculate the conditional entropy, the output signal entropy, and the mutual information in the leading and next-to-leading order in the nonlinearity parameter and in the leading order in the parameter $1/\mathrm{SNR}$. Using the mutual information we find the optimal input signal distribution and channel capacity in the leading and next-to-leading order in the nonlinearity parameter. Finally, we present the method of the construction of the input signal with the optimal statistics for the given shape of the signal.