[Paper Review] Kurtosis-limited Sphere Shaping for Nonlinear Interference Noise Reduction in Optical Channels

Yunus Can Gültekin, Alex Alvarado+5 more|arXiv (Cornell University)|May 31, 2021

Optical Network Technologies58 references41 citations

TL;DR

This paper proposes Kurtosis-Limited Sphere Shaping (K-ESS), a modified enumerative sphere shaping algorithm that reduces the kurtosis of probabilistically shaped inputs to mitigate nonlinear interference (NLI) in optical fiber channels. By constraining the fourth-order moment (kurtosis) of the input distribution, K-ESS achieves a 0.4 dB effective SNR gain and a twofold reduction in frame error rate compared to conventional sphere shaping in single-span, 400 Gbit/s systems.

ABSTRACT

Nonlinear interference (NLI) generated during the propagation of an optical waveform through the fiber depends on the fourth order standardized moment of the channel input distribution, also known as kurtosis. Probabilistically-shaped inputs optimized for the linear Gaussian channel have a Gaussian-like distribution with high kurtosis. For optical channels, this leads to an increase in NLI power and consequently, a decrease in effective signal-to-noise ratio (SNR). In this work, we propose kurtosis-limited enumerative sphere shaping (K-ESS) as an algorithm to generate low-kurtosis shaped inputs. Numerical simulations at a shaping blocklength of 108 amplitudes demonstrate that with K-ESS, it is possible to increase the effective SNRs by 0.4 dB in a single-span single-channel scenario at 400 Gbit/s. K-ESS offers also a twofold decrease in frame error rate with respect to Gaussian-channel-optimal sphere shaping.

Motivation & Objective

• To address the performance penalty in optical communication systems caused by high-kurtosis probabilistically shaped inputs.
• To mitigate nonlinear interference (NLI) in fiber-optic channels, which increases with input kurtosis.
• To develop a constructive shaping algorithm that explicitly limits kurtosis while maintaining high spectral efficiency.
• To demonstrate that kurtosis-limited shaping outperforms Gaussian-channel-optimized shaping in nonlinear optical fiber systems.
• To enable practical implementation of kurtosis-constrained shaping via a modified enumerative sphere shaping algorithm.

Proposed method

• K-ESS modifies the standard enumerative sphere shaping (ESS) algorithm by introducing a kurtosis constraint (K•) on the input distribution.
• The algorithm excludes sequences with high kurtosis from the shaping set, prioritizing low-kurtosis sequences within a bounded energy shell.
• The shaping set is defined by a joint constraint on energy (E•) and kurtosis (K•), with the energy threshold E•_k increased to compensate for the reduced number of valid sequences.
• The method uses a lookup table-based enumeration to efficiently map input bits to shaped signal sequences satisfying both energy and kurtosis constraints.
• The algorithm is implemented at a blocklength of 108 amplitudes for numerical evaluation in a single-span, single-channel scenario.
• The approach is generalizable to include constraints on higher-order moments beyond kurtosis, enabling future extensions for improved nonlinear tolerance.

Experimental results

Research questions

• RQ1 Can kurtosis-limited shaping reduce nonlinear interference (NLI) in optical fiber channels more effectively than conventional Gaussian-channel-optimized shaping?
• RQ2 How does the performance of kurtosis-limited sphere shaping (K-ESS) compare to standard sphere shaping in terms of effective SNR and frame error rate?
• RQ3 What is the impact of blocklength on the performance gain of K-ESS, particularly in short- and single-span transmission?
• RQ4 To what extent does kurtosis limitation improve spectral efficiency and nonlinear tolerance in high-speed optical systems?
• RQ5 Can K-ESS be generalized to include constraints on higher-order moments (e.g., sixth-order) to further enhance nonlinear noise mitigation?

Key findings

• K-ESS achieves a 0.4 dB effective SNR gain over conventional sphere shaping in a single-span, 400 Gbit/s optical transmission system.
• K-ESS reduces the frame error rate by a factor of two compared to Gaussian-channel-optimized sphere shaping.
• The performance gain of K-ESS is most pronounced at longer blocklengths and in systems with significant channel memory.
• The kurtosis constraint in K-ESS effectively reduces the fourth-order moment of the input distribution, directly mitigating NLI power.
• The gains from K-ESS diminish at very short blocklengths (e.g., N < 18), where the kurtosis of standard ESS already approaches the optimal value.
• K-ESS can be extended to include constraints on higher-order moments, such as the sixth-order moment, to further improve nonlinear tolerance in future systems.