Hierarchical Distribution Matching: a Versatile Tool for Probabilistic Shaping

Abstract

The hierarchical distribution matching (Hi-DM) approach for probabilistic shaping is described. The potential of Hi-DM in terms of trade-off between performance, complexity, and memory is illustrated through three case studies. © 2020 The Author(s) OCIS codes: 060.1660, 060.2330, 060.4080 IntroductionRecently, probabilistic shaping (PS) techniques have been widely investigated to improve the performance and the flexibility of optical fiber networks. By assigning different probabilities to the constellation symbols-e.g., trying to approximate the capacity-achieving Gaussian distribution for the additive white Gaussian noise (AWGN) channel-PS allows both to finely adapt the information rate (IR) to the available signal to noise ratio (SNR) and to reduce the gap between the IR achievable with uniform QAM constellations and the channel capacity [1,2].An effective PS approach, named probabilistic amplitude shaping (PAS) and based on a proper concatenation of a fixed-to-fixed-length distribution matcher (DM) and a systematic forward error correction (FEC) encoder, has been proposed in [3]. The key element of PAS is the DM, which maps k uniformly distributed bits on N shaped (according to a desired distribution) amplitudes from the alphabet A = {1, 3, . . . , 2M − 1} with rate R = k/N. This map induces a specific structure on the output sequence, whose elements are, therefore, not independent. Consequently, the rate R of the DM is lower than the entropy rate H that would be obtained with a sequence of i.i.d. amplitudes with the same target distribution, yielding the rate loss R loss = H − R ≥ 0.Different methods to realize a DM with a low rate loss have been recently proposed [4][5][6][7]. While the rate loss usually tends to zero when the block length N increases (but with different convergence speed for different DMs), this happens at the expense of increased computational cost, memory, and/or latency. To address this issue, an effective solution based on a hierarchical DM (Hi-DM) structure based on look up table (LUT)s has been recently proposed [6,8]. Elaborating on the latter idea, our work investigates a generalized Hi-DM approach in which several short DMs are hierarchically combined to obtain a longer DM with reduced rate loss and limited complexity, memory, and latency. The proposed approach can be tailored to look for the best trade-off between the mentioned parameters, depending on the system requirements. To show the potential of Hi-DM, we illustrate three different structures. Hierarchical distribution matcher

Sequence-Selection-Based Constellation Shaping for Nonlinear Channels

“…In practice, a DM with a reasonably low rate loss requires a long block length (typically, hundreds of symbols), making its implementation based on a single look-up-table (LUT) unfeasible. Finding a good trade-off between rate loss and implementation complexity is a key aspect in the design of DMs.Different techniques for the implementation of DMs have been proposed in the past years, including the enumerative sphere shaping (ESS) and its variations [24]- [26]; the constant composition distribution matcher (CCDM) and its variations [10], [22]; and the hierarchical DM (HiDM), which, using multiple DM layers (each based, for instance, on ESS, CCDM, or even LUTs), achieves a lower rate loss compared to the equivalent-complexity single-layer DM [12], [13], [27]. We refer to [20] and references therein for more details about DM implementations.…”

Compressed Shaping: Concept and FPGA Demonstration

“…We do not need an operational mode change such as updating the source statistics based on prior knowledge, although this would be a kind of data compression. Our previously proposed look-up table (LUT)-based hierarchical DM [18] and following works [29], [30] are applicable for this purpose without significantly increased complexity, but rather a reordering of the LUT entries. The proposed technique can reduce the rate losses associated with source and channel coding compared with state-of-the-art DM schemes such as constant-composition DM (CCDM) [15], or reduce the power consumption in the FEC at a given information rate and signal-to-noise ratio (SNR) by relaxing the FEC performance.…”

Probabilistic Shaping for Finite Blocklengths: Distribution Matching and Sphere Shaping

“…In [ 73 ], a shaper based on ESS was introduced to shape a subset of the amplitude bit-levels, which is referred to as partial ESS. The authors of [ 74 ] introduced the “hierarchical” DM which realizes a nonuniform distribution with hierarchical lookup tables (LUTs) [ 75 ]. An approximate sphere shaping implementation based on Huffman codes was proposed in [ 76 ].…”