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

Abstract

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 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 in case of 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 0.7 dB. 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 identicalsymbol runs can be beneficial for reducing fiber nonlinearities and thus, for increasing SNR.

Huffman-Coded Sphere Shaping for Extended-Reach Single-Span Links

“…Hence, the temporal structure of the signal (e.g., ordering of transmitted symbols) may have a significant impact on nonlinear performance. By shortening the shaping sequence length, it it possible to introduce some advantageous changes in the temporal structure -e.g., reduced concentration and probability of high-power symbols [31]. Also, by increasing the dimensionality of symbol mapping, the effective shaped symbol sequence length can be reduced as well as the number of simultaneously interacting independent shaped sequences, which leads to reduced probability of high peak power values.…”

“…The dependence of nonlinear tolerance on shaping length was investigated using constant composition distribution matching (CCDM) [30], [31] and ESS [27], [32] for multi-span long-haul links. Also, the advantage of 2D symbol mapping for nonlinearity tolerance has been mentioned previously in [31], while the advantage of the short-length shaping regime for a nonlinear optical channel was theoretically investigated in [33], [34]. Significant shaping gain exceeding the theoretical gain for the AWGN channel was demonstrated in numerical simulations for single-span links by optimally combining linear and nonlinear shaping gains using SM with very short shaping lengths [34].…”

Nonlinearity Tolerant LUT-Based Probabilistic Shaping for Extended-Reach Single-Span Links

“…, where H(X) is the entropy of 4Dsymbols X, D accounts for mapping dimensionality (D = 4) and "1" accounts for the sign bit per dimension. This 4D mapping effectively shortens the length of the shaped 4Dsymbol sequence in the time-domain, compared with a 1D or 2D mapping [14], without increase in the rate loss.…”

“…Copyright (c) 2020 IEEE nonlinearities in long-haul transmission [11] and unrepeated transmission [12]. The advantage of short-length probabilistic shaping for the nonlinear channel was thus investigated for multi-span long-haul links using constant composition distribution matching (CCDM) [13], [14] and enumerative sphere shaping (ESS) [15], [16]. Furthermore, significant shaping gain exceeding the theoretical gain for the linear channel was demonstrated for single-span links by optimally combining linear and nonlinear shaping gain contributions using shell mapping with very short shaping length [17].…”

Mitigating Fiber Nonlinearities by Short-length Probabilistic Shaping

“…Such an effect had not been observed before, mainly due to random-draw based PAS emulation technique used previously. The block-length dependence of SNR has been studied in detail in [7] for variable-length outputs of a constantcomposition DM (CCDM) [1, Sec. V].…”