Prerequisites & Notation

Before You Begin

This chapter examines coding constructions that approach AWGN capacity. We assume familiarity with the AWGN capacity formula (Chapter 10), the channel coding theorem for DMCs (Chapter 9), and basic concepts of linear algebra over $\mathbb{R}^n$ and $\mathbb{F}_2$ .

AWGN channel capacity $C = \frac{1}{2}\log(1 + \text{SNR})$ (Chapter 10)(Review ch10)
Self-check: Can you state the AWGN capacity and explain why Gaussian input is optimal?
Channel coding theorem: achievability via random coding, converse via Fano's inequality (Chapter 9)(Review ch09)
Self-check: Can you outline the random coding argument for the DMC?
Binary symmetric channel (BSC) and binary erasure channel (BEC) capacities (Chapter 9)(Review ch09)
Self-check: What is the capacity of a BSC with crossover probability $p$ ?
Linear algebra: lattice, basis, Voronoi region, volume of fundamental region
Self-check: Can you define a lattice in $\mathbb{R}^n$ and sketch its Voronoi region in 2D?
Basic coding theory: linear codes, generator and parity-check matrices
Self-check: Can you define a linear code via its parity-check matrix over $\mathbb{F}_2$ ?

Notation for This Chapter

Key symbols for coding constructions over the Gaussian channel.

Symbol	Meaning	Introduced
$\Lambda$	Lattice in $\mathbb{R}^n$	s01
$\mathcal{V}(\Lambda)$	Voronoi region of lattice $\Lambda$	s01
$\text{Vol}(\Lambda)$	Volume of the fundamental region of $\Lambda$	s01
$G(\Lambda)$	Normalized second moment of $\Lambda$ (measures quantization efficiency)	s01
$W$	Binary-input DMC (for polarization)	s03
$W_N^{(i)}$	$i$ -th bit-channel after polar transform of size $N$	s03
$I(W)$	Symmetric capacity of channel $W$	s03
$C_{\text{Sh}}$	Shaping gain (dB)	s04
$C_{\text{Cd}}$	Coding gain (dB)	s04

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