Prefix-Free Code Distribution Matching for Probabilistic Constellation Shaping

In this work, we construct optimal or near-optimal perfix-free codes that achieve a desired entropy rate with the minimum energy per code letter. In particular, we find many optimal binary-input variable-to-fixed length codes of size not larger than 4096 for 2-, 4-, 8-, and 16-ary amplitude shift keying (ASK) code alphabets, all optimal binary-input fixed-tovariable length codes of size not larger than 32 for 2- and 4-ASK code alphabets, and many near-optimal binary-input variable-tovariable length codes of size not larger than 32 for 2- and 4-ASK code alphabets. It is shown that the constructed codes under finite size constraints approach the unconstrained theoretical lower limit of energy for the same entropy rates to within a few tenths of a dB across a wide range of entropy rates with a very fine rate granularity. We then present a framing method for variable-length prefix-free codes, with which fixed-rate transmission can be achieved on a frame basis. A fixed-length penalty imposed by framing is studied by Monte-Carlo simulations and by Gaussian approximation of the entropy rate that is inherently probabilistic. It is shown that the fixed-length penalty is a monotonically decreasing function of the frame length, and reaches below 0.2 dB asymptotically in frame length for selected prefix-free codes. The performance of prefix-free codes is numerically evaluated in an additive white Gaussian noise channel, when used to implement distribution matching in probabilistic constellation shaping. The results show that prefix-free code distribution matching with framing offers fine-grained rate adaptation and achieves 1.1 dB and 0.65 dB of shaping gains over uniform 8- and 16-quadrature amplitude modulation, respectively, for a channel code rate of 5/6.