Prefix-Free Code Distribution Matching for Probabilistic Constellation Shaping

In this paper, we construct variable-length prefix-free codes that are optimal (or near-optimal) in the sense that no (or few) other codes of the same cardinality can achieve a smaller expected energy per code symbol for the same resolution rate. Under stringent constraints of 4096 codewords or below per codebook, the constructed codes yield an energy per code symbol within a few tenths of a dB of the unconstrained theoretic lower bound, across a wide range of resolution rates with fine granularity. We also propose a framing method that allows variable-length codes to be transmitted using a fixed-length frame. The penalty caused by framing is studied using simulations and analysis, showing that the energy per code symbol is kept within 0.3 dB of the unconstrained theoretic limit for some tested codes with a large frame length. When the proposed method is used to implement probabilistic constellation shaping for communications in the additive white Gaussian noise channel, simulations show that between 0.21 dB and 0.98 dB of shaping gains are achieved relative to uniform 4-, 8-, 16-, and 32-ary quadrature amplitude modulation.