Prerequisites & Notation

Before You Begin

This chapter builds the modern theory and practice of shaping — the last dB of the gap to AWGN capacity for a coded modulation system. The foundation from Chapter 4 (Forney-style lattice shaping) and Chapter 9 Section 5 (Probabilistic Amplitude Shaping in 400G optical) is essential: we assume the reader can already state the $\pi e / 6 \approx 1.53$ dB asymptotic shaping bound and has seen the PAS block diagram. Chapter 19 now puts the machinery under the microscope. We derive the Maxwell-Boltzmann distribution from first principles (KKT on a max-entropy Lagrangian), build a constant-composition distribution matcher (CCDM) by arithmetic coding, quantify its short-block rate loss, introduce geometric shaping as the dual option, and conclude with rate-adaptive transmission — the operational reason the industry cares. Familiarity with LDPC codes (Telecom Chapter 11), BICM (Chapter 5), and the AWGN-channel capacity formula is assumed.

Sphere-shaping gain bound $\gamma_s \le \pi e / 6$ (Review ch04)
Self-check: Can you state the shaping gain ceiling $\gamma_s \le \pi e / 6 \approx 1.53$ dB and explain why it is the asymptotic limit of spherical-vs-cubic second-moment ratio?
Probabilistic Amplitude Shaping (PAS) block diagram(Review ch09)
Self-check: Can you draw the PAS pipeline (info bits $\to$ CCDM $\to$ amplitude bits $+$ systematic LDPC $\to$ QAM mapper) and explain why the sign bits can remain uniform while the amplitude bits carry the MB distribution?
BICM capacity and Gray-labelling near-optimality(Review ch05)
Self-check: Can you write $C_{\rm BICM}(\mu) = \sum_\ell I(Y; B_\ell)$ for a Gray-labelled $M$ -QAM and state the $< 0.1$ dB empirical gap to $C_{\rm CM}$ at high SNR?
LDPC codes and systematic-form encoding(Review ch11)
Self-check: Can you explain what a systematic LDPC code is (information bits appear verbatim in the codeword) and why PAS requires this property?
Gaussian maximum-entropy theorem(Review ch09)
Self-check: Can you prove that among distributions on $\mathbb{R}$ with variance $\sigma^2$ , the entropy-maximising one is $\gauss(0, \sigma^2)$ ?
Lagrangian optimisation and KKT conditions
Self-check: Given a concave objective $f(p)$ on a probability simplex with a linear constraint $\mathbb{E}_p[g(X)] = E$ , can you write the Lagrangian and solve $\partial L / \partial p(x) = 0$ for the stationary distribution?
Arithmetic coding(Review ch05)
Self-check: Can you sketch how arithmetic coding maps a binary input stream into a real interval and state why the compression rate approaches the source entropy in the limit of long blocks?
Adaptive modulation and coding (AMC) MCS tables(Review ch09)
Self-check: Can you explain why 5G NR uses 28 discrete MCS indices (staircase) for rate adaptation and what the typical gap between adjacent staircase steps is in bits per symbol?

Notation for This Chapter

Symbols specific to Chapter 19. We follow the Bocherer-Steiner-Schulte convention for PAS and the Forney convention for shaping gain.

Symbol	Meaning	Introduced
$\lambda$	Maxwell-Boltzmann shaping parameter, $\lambda > 0$ . The MB distribution is $p_\lambda(x) \propto \exp(-\lambda \|x\|^2)$	s01
$Z(\lambda)$	Partition function, $Z(\lambda) = \sum_{x \in \mathcal{X}} \exp(-\lambda \|x\|^2)$	s01
$H(X)$	Entropy of the constellation input distribution, in bits	s01
$\gamma_s$	Shaping gain in dB. $\gamma_s \to \pi e / 6 \approx 1.53$ dB at high SNR	s01
$R_c$	LDPC code rate. $R_c = k / n$ where $k$ information bits produce $n$ codeword bits	s02
$R$	Effective PAS transmission rate in bits per 2D QAM symbol. $R = R_c \log_2 M + (R_c - 1)$	s02
$R_{\rm DM}$	Distribution-matcher output rate in bits per amplitude symbol; equals $H(A)$ minus rate loss	s03
$\mathcal{A}$	Amplitude alphabet. For square $M$ -QAM, $\mathcal{A} = \{1, 3, \ldots, \sqrt{M}-1\}$ per in-phase/quadrature	s02
$p_A(a)$	Target distribution on amplitudes. MB-derived: $p_A(a) \propto \exp(-\lambda a^2)$	s02
$n$	CCDM block length (amplitude symbols per shaping block)	s03
$\mathrm{CCDM}$	Constant-composition distribution matcher. Arithmetic-coded invertible bit-to-amplitude mapping	s03
$\eta$	Spectral efficiency in bits/s/Hz. Under PAS, $\eta$ is continuous in $\lambda$ and $R_c$	s05
$M$	Constellation size (QAM order). Standard values: $16$ , $64$ , $256$ , $1024$	s01

Key Takeaway

Shaping is the last dB of the Shannon gap that coding cannot close. Chapter 4 taught us that sphere shaping saves up to $\pi e / 6 \approx 1.53$ dB in the asymptotic, large-dimension limit. Chapter 9 showed the industry implementation via PAS. Chapter 19 completes the theory: the Maxwell-Boltzmann distribution is the optimal finite-alphabet input, the CCDM realises it with $O(\log n / n)$ rate loss, geometric shaping is the dual of probabilistic shaping and achieves the same asymptotic, and all of it enables rate-adaptive transmission with a single fixed modulation and code — the feature that made 400ZR a commercial reality.

← Ch 18 Maxwell-Boltzmann Shaping