The challenge of probabilistic shaping is how to combine non-uniform signaling with forward error correction (FEC). Probabilistic amplitude shaping (PAS) solves this challenge by concatenating a distribution matcher with a systematic binary encoder:

• Distribution matcher (DM) generates amplitude sequence $A_1\dotsc A_{n_\text{c}}$ with distribution $P_A$.
• Labeling funcion $\boldsymbol{b}(\cdot)$ represents amplitude by $m-1$ bits.
• Parity matrix $\boldsymbol{P}$ of systematic rate $(m-1)n_\text{c}\times n_\text{c}m$ generator matrix $\boldsymbol{G}=[\boldsymbol{I}\,|\,\boldsymbol{P}]$ calculates $n_\text{c}$ check bits.
• Check bits are mapped to sign sequence $S_1\dotsc S_{n_\text{c}}$.
• $X_i=A_i\cdot S_i$ is scaled by $\Delta$ and transmitted.

G. Böcherer, F. Steiner, and P. Schulte, Bandwidth efficient and rate-matched low-density parity-check coded modulation, IEEE Trans. Commun., vol. 63, no. 12, pp. 4651–4665, Dec. 2015.