Probabilistic Eigenvalue Shaping for Nonlinear Fourier Transform Transmission

Abstract

We consider a nonlinear Fourier transform (NFT)based transmission scheme, where data is embedded into the imaginary part of the nonlinear discrete spectrum. Inspired by probabilistic amplitude shaping, we propose a probabilistic eigenvalue shaping (PES) scheme as a means to increase the data rate of the system. We exploit the fact that for an NFTbased transmission scheme, the pulses in the time domain are of unequal duration by transmitting them with a dynamic symbol interval and find a capacity-achieving distribution. The PES scheme shapes the information symbols according to the capacity-achieving distribution and transmits them together with the parity symbols at the output of a low-density parity-check encoder, suitably modulated, via time-sharing. We furthermore derive an achievable rate for the proposed PES scheme. We verify our results with simulations of the discrete-time model as well as with split-step Fourier simulations.

On time-bandwidth product of multi-soliton pulses

“…The basic idea of eigenvalue modulation for 1−solitons dates back to [20] where 1-out-of-N imaginary eigenvalues were selected for transmission. A corresponding lower bound on the capacity per soliton with eigenvalue modulation was derived in [21] and a capacity achieving probabilistic eigenvalue shaping scheme was shown in [22]. A noise model for the discrete spectrum was first introduced in [23].…”

On the Capacity of Amplitude Modulated Soliton Communication over Long Haul Fibers

“…In [ 18 ], AIR is estimated for a more complicated system that modulates both the eigenvalue and the norming constant in the discrete spectrum. Assuming a receiver capable of detecting variable pulse duration, in [ 25 ], the time-scaled mutual information (MI) is numerically optimized considering the memoryless channel model for soliton communication.…”

“…This is essentially equivalent to the amplitude modulated soliton communication in [ 26 ]. As mentioned above, a number of capacity bounds for such channel has been derived previously [ 18 , 23 , 24 ], and AIR in bits per second were also discussed in [ 25 ]. However, some intrinsic limitations, such as dependence of bandwidth on soliton amplitude and the interaction between neighboring soliton pulses have been ignored.…”

“…However, some intrinsic limitations, such as dependence of bandwidth on soliton amplitude and the interaction between neighboring soliton pulses have been ignored. Compared to the state-of-art works in the literature (e.g., [ 23 , 25 ]), we investigate the effect of channel memory induced by solitonic interaction, which is mostly ignored in the literature. In order to incorporate the time-bandwith degrees of freedom into the capacity problem formulation, we study the maximization of time-scaled MI similar to [ 25 ] but by assuming a more practical communication system that uses a fixed symbol duration (i.e., soliton pulse width).…”

Protograph-Based LDPC Code Design for Probabilistic Shaping with On-Off Keying

“…Using a binary FEC code of block length n and code rate R C with systematic encoding, a number of R C · n OOK symbols is transmitted with a non-uniform distribution, while the remaining (1 − R C ) · n parity bits are sent with a uniform distribution. A similar approach for coherent higher order modulations with non-binary low-density parity-check (LDPC) codes was suggested in [12], while binary codes were investigated for a nonlinear Fourier transform based, optical transmission system in [13] without a tailored LDPC code design.…”