Prerequisites & Notation

Before You Begin

This chapter opens Book CM by revisiting signal-space coding with the eyes of someone who already knows about codes, Gaussian channels, and capacity. The reader is expected to have been through the following material, or to have access to it while reading.

AWGN channel model, matched filtering, and signal-space representation(Review ch08)
Self-check: Can you state the discrete-time AWGN model $y = x + w$ with $w \sim \mathcal{N}(0, N_0/2)$ and explain the $E_s / N_0$ vs. $E_b / N_0$ normalization?
Shannon channel capacity for the AWGN channel and mutual information basics(Review ch09)
Self-check: Can you derive $C = W \log_2(1 + \text{SNR})$ and explain why this is an upper bound on any reliable communication rate?
Basic binary channel coding: Hamming distance, minimum distance, union bound(Review ch11)
Self-check: Can you state a union bound on the word-error probability of a linear binary code in terms of its weight enumerator and the pairwise error probability over AWGN?
M-PAM, M-PSK, and M-QAM constellations, Gray labeling, uncoded error probability(Review ch08)
Self-check: Can you sketch 16-QAM, write the uncoded BER under Gray labeling, and relate $d_{\min}^2$ to $E_s$ for a square QAM?
Probability, Gaussian tail bounds, and the $Q$ function(Review ch04)
Self-check: Can you state the Chernoff bound $Q(x) \leq \tfrac{1}{2} e^{-x^2/2}$ and differentiate it from the exact value?

Notation for This Chapter

Symbols used throughout Chapter 1. In later chapters of Book CM we will extend this with lattice-theoretic, MIMO, and coding-specific notation.

Symbol	Meaning	Introduced
$\mathcal{X} \subset \mathbb{R}^N$	Signal-space constellation (finite set of transmitted vectors)	s01
$M = \|\mathcal{X}\|$	Constellation size (number of code points)	s01
$N$	Signal-space dimension (real dimensions per code point)	s01
$R$	Information rate in bits per channel use (or bits/2D for 2D constellations)	s01
$\eta$	Spectral efficiency in bits/s/Hz (bits per 2D dimension pair)	s02
$E_s, E_b$	Average energy per symbol and per information bit; $E_s = R\, E_b$	s01
$N_0$	Noise one-sided power spectral density (noise variance per real dim. is $N_0/2$ )	s01
$\text{SNR}$	Signal-to-noise ratio, typically $E_s/N_0$ in this chapter	s01
$C$	Channel capacity (bits per channel use or bits/s/Hz)	s02
$d_{\min}, d_{\rm E}$	Minimum Euclidean distance of the constellation	s01
$\gamma_c$	Coding gain (dB) relative to a baseline uncoded constellation	s01
$\gamma_s$	Shaping gain (dB); ultimate limit $\pi e / 6 \approx 1.53$ dB	s03
$P_e$	Symbol (or codeword) error probability	s01
$Q(x)$	Gaussian tail: $Q(x) = \int_x^\infty \tfrac{1}{\sqrt{2\pi}} e^{-t^2/2}\, dt$	s01
$\mathbf{x} \to \hat{\mathbf{x}}$	Pairwise error event: transmitted $\mathbf{x}$ , decoded $\hat{\mathbf{x}}$	s04
$\boldsymbol{\Delta} = \mathbf{x} - \hat{\mathbf{x}}$	Error vector in signal space	s04
$\Lambda$	Lattice (infinite point set in $\mathbb{R}^N$ with group structure)	s03

Back to chapter overview Coding Gain vs. Bandwidth Efficiency