Computing Continuous Nonlinear Fourier Spectrum of Optical Signal with Artificial Neural Networks

Abstract

Listen

Translate article

Nonlinear Fourier transform (NFT) (also known in the mathematical and nonlinear science community as the inverse scattering transform [1] ) has recently attracted a great deal of attention in the context of optical transmission in fiber channels [2] , that can be approximated by the nonlinear Schrodinger equation. Within the NFT-based¨ transmission approach, we modulate the parameters of the nonlinear spectrum (NS) and generate the respective information signal in time domain using inverse NFT. Both discrete and continuous parts of NS can be used, here we focus on the continuous spectrum only. Then, the signal is launched into the fiber, and at the receiver we apply direct NFT to the received signal’s to retrieve the information encoded in NS. In this work we demonstrate that the high-accuracy computation of the continuous NS can be performed by using artificial neural networks (NN). The NS of a given localized signal q ( t ) containing no solitonic (discrete) components is represented by the continuous complex-valued function r ( ξ ) (the reflection coefficient) of the real spectral parameter ξ , where the latter plays the role of nonlinear frequency. The signals in our work have been specifically pre-selected to ensure that they contain no discrete spectrum. In the time domain, considered (normalized) symbol is given as the sum of independent sub-carriers [3] : $q(t) = \frac{1}{Q}\sum\nolimits_{k = 1}^M {{C_k}} {e^{i{\omega _k}t}}f(t)$ where M is a number of frequency channels, ω k is a carrier frequency of the k -th channel, C k corresponds to the digital data in k -th channel, and T defines the symbol interval; f ( t ) is the return-to-zero carrier support waveform. To assess the quality of NN prediction, we use the following formula for the relative error: $\eta (\xi ) = \frac{{\left\langle {\left| {{r_{predicted{\text{ }}}}(\xi ) - {r_{actual{\text{ }}}}(\xi )} \right|} \right\rangle }}{{\left\langle {\left| {{r_{actual{\text{ }}}}(\xi )} \right|} \right\rangle }}$ , where 〈•〉 denotes the mean over the spectral interval, the "predicted" and "actual" indices refer to the NN-predicted and precomputed values of the reflection coefficient r ( ξ ), respectively.