Chapter Summary

Chapter Summary

Key Points

• 1.

  Maxwell-Boltzmann is the capacity-achieving input. On an average-power-constrained constellation, the capacity-achieving input distribution is $P_X(x) \propto e^{-\lambda|x|^2}$ — the discrete MB. At high SNR it converges to a Gaussian (Shannon's classical result); the 1.53 dB gap closes asymptotically.

• 2.

  PAS is the architecture that combines shaping with FEC. Probabilistic Amplitude Shaping (Böcherer-Steiner-Schulte 2015) shapes ONLY the amplitude bits; the sign bits come from a systematic LDPC output (already uniform). This composes shaping with any existing LDPC code via $R_{\rm PAS} = R_c \log_2 M + R_c - 1$.

• 3.

  CCDM is the practical distribution matcher. Constant- composition DM selects output sequences of identical histogram, achieving the MB rate with $O(\log n/n)$ rate loss. Arithmetic encoding realises the mapping in $O(n|\mathcal{A}|)$ time.

• 4.

  Geometric shaping is dual to probabilistic. Moving constellation points achieves the same asymptotic shaping gain as changing input probabilities. PS has won in cellular/optical standards for hardware compatibility; GS appears in ATSC 3.0.

• 5.

  Continuous rate adaptation via PAS. Tuning the MB parameter $\lambda$ gives continuous rate adaptation WITHOUT switching constellation or code. 400ZR optical coherent is the flagship deployment.

• 6.

  $\pi e / 6 \approx 1.53$ dB is the ultimate shaping gap. Whether via PS, GS, or any other shaping, the asymptotic ceiling is $\pi e / 6$ — the "sphere gain" of infinite-dimensional constellations. Current production systems capture 0.8-1.2 dB of this ceiling.