Achievable Rates for Probabilistic Shaping

For a layered probabilistic shaping (PS) scheme with a general decoding metric, an achievable rate is derived using Gallager’s error exponent approach. Several instances for specific decoding metrics are discussed, including bit-metric decoding, interleaved coded modulation, and hard-decision decoding. It is shown that important previously known achievable rates can also be achieved by layered PS. A practical instance of layered PS is the recently proposed probabilistic amplitude shaping (PAS).