Prerequisites & Notation

Before You Begin

This chapter steps from finite constellations (Ch. 1–3) to a more powerful geometric language — lattices — that will carry us through the rest of the book and into Part IV. The tools below should be at your fingertips. If any feels shaky, the indicated section will bring it back.

The set-partitioning principle and coding gain $\gamma_c$ (Review ch02)
Self-check: Can you state Ungerboeck's partition principle — each partition step doubles the squared intra-subset distance — and relate it to the coding-gain formula $\\gamma_c = d_{\\rm free}^2 / d_{\\rm uncoded}^2$ ?
Multilevel codes and the capacity rule $C = \sum_i C_i$ (Review ch03)
Self-check: Can you explain why a multilevel code with rates $R_i = C_i$ achieves the CM capacity of any constellation under partition-based labelling?
Gaussian AWGN capacity $C = \tfrac12 \log_2(1 + \text{SNR})$ and the Shannon limit(Review ch09)
Self-check: Can you explain why the AWGN capacity depends only on $\\text{SNR}$ , and derive the $\\pi e / 6$ shaping-gain ceiling from it?
Linear algebra: full-rank matrices, determinants, and volumes(Review ch01)
Self-check: Given a generator matrix $\\mathbf{G} \\in \\mathbb{R}^{n \\times n}$ , can you compute the fundamental volume $|\\det \\mathbf{G}|$ and explain its geometric meaning?
Differential entropy and the entropy power inequality (EPI)(Review ch08)
Self-check: Can you state the Gaussian maximum-entropy theorem and compute the differential entropy of a uniform distribution over a bounded region?
Elementary group theory: cosets and quotient groups
Self-check: Given a lattice $\\Lambda$ and a sublattice $\\Lambda' \\subset \\Lambda$ , can you explain what the coset $\\lambda + \\Lambda'$ means and why the number of cosets equals $|\\Lambda / \\Lambda'|$ ?

Notation for This Chapter

The lattice and shaping symbols used throughout the chapter. Book-wide symbols (SNR $\\text{SNR}$ , energy $\E_s$ , noise density $\N_0$ , noise variance $\\sigma^2$ , noise vector $\\mathbf{w}$ ) follow the global CM notation.

Symbol	Meaning	Introduced
$\Lambda$	A lattice in $\mathbb{R}^n$ : a discrete additive subgroup spanned by $n$ linearly independent vectors	s01
$\mathbf{G}$	Generator (basis) matrix of $\Lambda$ ; $\Lambda = \{\mathbf{G} \mathbf{z} : \mathbf{z} \in \mathbb{Z}^n\}$	s01
$V(\Lambda)$	Fundamental volume of $\Lambda$ : $V(\Lambda) = \|\det \mathbf{G}\|$	s01
$\mathcal{V}(\Lambda)$	Voronoi region of $\Lambda$ around the origin: $\{\mathbf{x} \in \mathbb{R}^n : \\|\mathbf{x}\\| \le \\|\mathbf{x} - \lambda\\| \; \forall \lambda \in \Lambda\}$	s01
$K(\Lambda)$	Kissing number of $\Lambda$ : the number of nearest neighbours of the origin	s01
$\Lambda'$	A sublattice of $\Lambda$ ( $\Lambda' \subset \Lambda$ ); generically used as the finer partition	s01
$\|\Lambda / \Lambda'\|$	Index of the partition: the number of cosets of $\Lambda'$ in $\Lambda$ , equal to $V(\Lambda')/V(\Lambda)$	s01
$\Lambda_c$	Coding lattice: the points the encoder selects	s02
$\Lambda_s$	Shaping lattice: a sublattice of $\Lambda_c$ whose Voronoi region defines the constellation boundary	s03
$G(\Lambda)$	Normalised second moment of $\Lambda$ : $G(\Lambda) = \sigma^2(\Lambda) / V(\Lambda)^{2/n}$ , where $\sigma^2(\Lambda)$ is the per-dimension second moment of a uniform distribution over $\mathcal{V}(\Lambda)$	s03
$\gamma_c$	Coding gain (in dB): distance-advantage of the coded constellation over an uncoded reference at the same rate	s02
$\gamma_s$	Shaping gain (in dB): energy-advantage from shaping the constellation boundary, $\gamma_s = 1/(12 G(\Lambda_s))$	s03
$C(\Lambda / \Lambda'; \mathcal{C})$	Forney coset code: the set of points $\lambda' + \mathbf{c}$ with $\lambda' \in \Lambda'$ and $\mathbf{c}$ a coset representative selected by the binary code $\mathcal{C}$	s02

← Ch 3 Lattices and Lattice Partitions