Kurtosis-constrained Optimum Enumeration Sphere Shaping for Probabilistic Shaping Optical Transmission System

Enumeration sphere shaping (ESS) is an amplitude shaping architecture designed to reduce rate loss at short block lengths. However, ESS experiences unexpected energy efficiency degradation caused by the sorting of sequences in lexicographical order. Additionally, the input distribution is affected by the fourth order normalized moment, namely kurtosis, of the input distribution. This increase in kurtosis leads to higher nonlinear interference (NLI) power, resulting in a reduced effective signal-to-noise ratio (SNR). In this paper, we propose a kurtosis-constrained optimum enumeration sphere shaping (KO-ESS) scheme that recovers the effective SNR-penalty caused by the input distribution with high kurtosis while achieve the ultimate energy efficiency. The performance of the KO-ESS is demonstrated in a 28 GBaud PS-64 QAM simulation system. Simulation results indicate that KO-ESS achieves lower rate loss at various shaping rates. Compared with the traditional ESS scheme, the optical signal-to-noise ratio (OSNR) of 0.41 dB is obtained when the normalized generalized mutual information (NGMI) threshold is reached, and the OSNR gain of 0.27 dB is obtained when the achievable information rate (AIR) is 4 bit/sym.