Probabilistic Shaping for Finite Blocklengths: Distribution Matching and Sphere Shaping

Abstract

In this paper, we provide for the first time a systematic comparison of distribution matching (DM) and sphere shaping (SpSh) algorithms for short blocklength probabilistic amplitude shaping. For asymptotically large blocklengths, constant composition distribution matching (CCDM) is known to generate the target capacity-achieving distribution. As the blocklength decreases, however, the resulting rate loss diminishes the efficiency of CCDM. We claim that for such short blocklengths and over the additive white Gaussian channel (AWGN), the objective of shaping should be reformulated as obtaining the most energy-efficient signal space for a given rate (rather than matching distributions). In light of this interpretation, multisetpartition DM (MPDM), enumerative sphere shaping (ESS) and shell mapping (SM), are reviewed as energy-efficient shaping techniques. Numerical results show that MPDM and SpSh have smaller rate losses than CCDM. SpSh-whose sole objective is to maximize the energy efficiency-is shown to have the minimum rate loss amongst all. We provide simulation results of the end-toend decoding performance showing that up to 1 dB improvement in power efficiency over uniform signaling can be obtained with MPDM and SpSh at blocklengths around 200. Finally, we present a discussion on the complexity of these algorithms from the perspective of latency, storage and computations.

Comparison of Short Blocklength Sphere Shaping and Nonlinearity Compensation in WDM Systems

“…On the other hand, when these techniques (i.e., uniform signaling and ESS) are combined with DBP or VNLE, the CD compensation part is already included in the DBP and VNLE implementation. Bounded-precision ESS [15] is used in this work to reduce the storage requirements [20]. The CD compensation is implemented in frequency-domain using a fast Fourier transform (FFT)/Inverse-FFT method, as in [16].…”

Adaptive Coded Modulation for IM/DD Free-Space Optical Backhauling: A Probabilistic Shaping Approach

“…For the DM, the CCDM, implemented using arithmetic coding [36], requires k p iterations for the matching, while it needs n p for dematching. Each iteration involves M additions, multiplications, and comparisons [55]. Unfortunately, the algorithms for arithmetic coding are sequential in nature; hence, it is a challenging task to parallelize the implementation [39].…”

“…The potential codewords are on the surface or inside an n p -dimensional sphere [10]. The computational complexity of enumerative sphere shaping [56] and shell mapping [57, Algorithm 2] is O(n p ) and O(n p 3 ), respectively [55]. The rate loss for indirect signal shaping can be smaller than CCDM; however, the rate loss as a function of the sequence length is not the only parameter to judge the performance of the PS algorithms.…”