Chapter 19: Probabilistic Shaping and Geometric Shaping

Learning Objectives

• State the Maxwell-Boltzmann optimality theorem: on a finite constellation $\mathcal{X}$ subject to an average-power constraint, the capacity-achieving input distribution is $p_\lambda(x) \propto \exp(-\lambda |x|^2)$, and derive $\lambda(E)$ from the KKT conditions for the max-entropy-at-fixed-second-moment problem
• Derive the Bocherer-Steiner-Schulte (2015) Probabilistic Amplitude Shaping (PAS) rate $R = R_c \log_2 M + (R_c - 1)$ bits/symbol and explain why the systematic-LDPC sign/parity decomposition preserves the MB distribution on the amplitude bits
• Construct a constant-composition distribution matcher (CCDM) via arithmetic coding, state its $O(\log n / n)$ rate-loss bound, and quantify the short-block penalty that motivates hierarchical and product distribution matchers
• Design a geometrically shaped non-uniform constellation for a target SNR by solving $\max_{\mathcal{X}} I(X;Y)$ over point locations, and prove that at high SNR geometric and probabilistic shaping achieve the same asymptotic gap to capacity ($\pi e / 6 \approx 1.53$ dB)
• Operate a rate-adaptive PAS transceiver that varies the information rate continuously over $R \in [0, \log_2 M]$ by tuning $\lambda$ and the LDPC rate $R_c$, without changing the modulation format, and compare it to the discrete MCS-staircase approach
• Identify the three deployment domains where probabilistic shaping is mandatory or proposed — OIF 400ZR coherent optical (mandatory), DVB-S2X (optional), ATSC 3.0 (geometric), 6G eMBB (proposed) — and argue from system constraints why each standard picked the shaping variant it did