Hierarchical Distribution Matching for Probabilistically Shaped Coded Modulation

TL;DR

The paper tackles the challenge of integrating distribution matching with reverse concatenation probabilistic shaping by introducing Hierarchical DM (HiDM), a hardware-friendly LUT-tree approach that replaces complex CCDM while maintaining near Maxwell–Boltzmann distributions. HiDM enables parallel, pipelined DM/invDM processing, dramatically reducing memory and hardware complexity at the cost of modest rate loss and slightly higher SNR requirements for a given post-FEC BER. Through analysis and simulations for high-throughput optical links, it shows HiDM achieves competitive performance with far lower hardware resource demands and offers a practical means to estimate post-invDM BER from post-FEC BER. The work demonstrates that reverse-concatenation PS can be made viable in practice, with significantly reduced DM/storage requirements and a clear design methodology for LDPC-based optical systems.

Abstract

The implementation difficulties of combining distribution matching (DM) and dematching (invDM) for probabilistic shaping (PS) with soft-decision forward error correction (FEC) coding can be relaxed by reverse concatenation, for which the FEC coding and decoding lies inside the shaping algorithms. PS can seemingly achieve performance close to the Shannon limit, although there are practical implementation challenges that need to be carefully addressed. We propose a hierarchical DM (HiDM) scheme, having fully parallelized input/output interfaces and a pipelined architecture that can efficiently perform the DM/invDM without the complex operations of previously proposed methods such as constant composition DM (CCDM). Furthermore, HiDM can operate at a significantly larger post-FEC bit error rate (BER) for the same post-invDM BER performance, which facilitates simulations. These benefits come at the cost of a slightly larger rate loss and required signal-to-noise ratio at a given post-FEC BER.

Figures (7)

• Figure 1: System model including signal notation and performance metrics.
• Figure 2: Example of the proposed hierarchical DM (HiDM), where $\uline{N_\text{u}=15}$ uniform input bits $\boldsymbol{A}'$ are converted into $\uline{m N_\text{s}=20}$ output bits $\boldsymbol{D}$ (4 uniform and 16 shaped bits). The non-shaped (uniform) bits will be used as sign bits of the 32-PAM symbols with $N_\text{s}=4$. The bar charts illustratebitprobabilities of being '1' for the respective bits.
• Figure 3: Line-side fFrame structure with (a) CCDM or with (b) HiDM. One blue (filled) rectangle shows one DM word.
• Figure 4: Simulation results of (a) CCDM-based PS-256-QAM, (b) HiDM-based PS-256-QAM, and (c) BICM-based 128-QAM.
• Figure 5: Simulation setup for DM to invDM back-to-back error insertion test.