Analysis of Nonlinear Fiber Interactions for Finite-Length Constant-Composition Sequences

In order to realize probabilistically shaped signaling within the probabilistic amplitude shaping (PAS) framework, a shaping device outputs sequences that follow a certain nonuniform distribution. In case of constant-composition (CC) distribution matching (CCDM), the sequences differ only in the ordering of their constituent symbols, whereas the number of occurrences of each symbol is constant in every output block. Recent results by Amari \textit{et al.} have shown that the CCDM block length can have a considerable impact on the effective signal-to-noise ratio (SNR) after fiber transmission. So far, no explanation for this behavior has been presented. Furthermore, the block-length dependence of the SNR seems not to be fully aligned with previous results in the literature. This paper is devoted to a detailed analysis of the nonlinear fiber interactions for CC sequences. We confirm in fiber simulations the inverse proportionality of SNR with CCDM block length and present two explanations. The first one, which only holds in the short-length regime, is based on how two-dimensional symbols are generated from shaped amplitudes in the PAS framework. The second, more general explanation relates to an induced shuffling within a sequence, or equivalently a limited concentration of identical symbols, that is an inherent property for short CC blocks, yet not necessarily present for long blocks. This temporal property results in weaker nonlinear interactions, and thus higher SNR, for short CC sequences. For a typical multi-span fiber setup, the SNR difference is numerically demonstrated to be up to 0.7dB. Finally, we evaluate a heuristic figure of merit that captures the number of runs of identical symbols in a concatenation of several CC sequences. For moderate block lengths up to approximately 100 symbols, this metric suggests that limiting the number of identical-symbol runs can be beneficial.