Multi-Symbol Digital Signal Processing Techniques for Discrete Eigenvalue Transmissions Based on Nonlinear Fourier Transform

Abstract

Optical communications based on Nonlinear Fourier Transform (NFT) and digital coherent transceivers are proposed as a new theoretical framework for communications over the nonlinear optical fiber channel. For discrete eigenvalue transmissions (or soliton transmissions), one seeks to encode as much information as possible in each degree of freedom and shorten the distance between neighboring pulses to increase the overall bit rate. However, such attempts would result in nonlinear inter-symbol interference (ISI) across multiple symbols and significantly degrade transmission performance. In this paper, we investigated joint modulation of discrete eigenvalue λ and <b xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</b> -coefficents <b xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</b> (λ) and developed a suite of multi-symbol digital signal processing (DSP) techniques to exploit the statistical correlations between the continuous and discrete eigenvalues and <b xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</b> -coefficents to mitigate nonlinear distortions and improve detection performance. This include 1) jointly modulating both λ and <b xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</b> (λ) of pairs of 1-solitons so that the mean value of λ for solitons with odd index is α + 1 <b xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</b> while it is - α + 1 <b xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</b> for solitons with even index. This is followed by decoding superimposed received waveforms as 2-solitons with twice the INFT processing time window; 2) linear minimum mean squared error (LMMSE) estimation filters to mitigate noise in discrete eigenvalue λ using continuous eigenvalue; 3) multi-symbol (MS) LMMSE filters to mitigate noise in <b xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</b> (λ) using discrete eigenvalue noise and 4) approximate the received signal distributions of λ and <b xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</b> (λ) as Gaussians with mean and covariance matrices obtained empirically from experiments followed by Maximum Likelihood (ML) detection for each symbol or multi-symbol (MS)-joint ML detection of 2-soliton signals. We jointly modulate λ with 16-QAM and <b xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</b> (λ) with 16-APSK and a record single-polarization discrete eigenvalue transmission of 64 Gb/s (net 54 Gb/s) over 1200 km is experimentally demonstrated with the proposed multi-symbol DSP algorithms.