Probabilistic Amplitude Shaping

Probabilistic amplitude shaping (PAS) proposed in Bocherer, Steiner, Schulte [24] is a practical architecture for combining non-uniform distributions on higher-order constellations with off-the-shelf forward error correction (FEC) codes. PAS consists of a distribution matcher (DM) that imposes a desired distribution on the signal point amplitudes, followed by systematic FEC encoding, preserving the amplitude distribution. FEC encoding generates additional parity bits, which select the signs of the signal points. At the receiver, FEC decoding is followed by an inverse DM. PAS quickly had a large industrial impact, in particular in fiber-optic communications. This monograph details the practical considerations that led to the invention of PAS and provides an information-theoretic assessment of the PAS architecture. Because of the separation into a shaping layer and an FEC layer, the theoretic analysis of PAS requires new tools. On the shaping layer, the cost penalty and rate loss of finite length DMs is analyzed. On the FEC layer, achievable FEC rates are derived. Using mismatched decoding, achievable rates are studied for decoding metrics of practical importance. Combining the findings, it is shown that PAS with linear codes is capacity-achieving on a class of discrete input channels. Open questions for future study are discussed.