Prerequisites & Notation

Before You Begin

This chapter introduces channels whose transition law depends on a random state sequence. The results build on the DMC channel coding theorem (Chapter 9) and the Gaussian channel capacity (Chapter 10). The Gel'fand-Pinsker theorem and Costa's dirty paper coding require comfort with random binning arguments and Gaussian mutual information calculations.

Channel coding theorem for DMCs: $C = \max_{P_X} I(X;Y)$ (Chapter 9)(Review ch09)
Self-check: Can you outline the achievability and converse proofs?
AWGN channel capacity and the role of Gaussian inputs (Chapter 10)(Review ch10)
Self-check: Can you derive $C = \frac{1}{2}\log(1 + P/N)$ ?
Joint typicality, packing lemma, and covering lemma (Chapter 3)(Review ch03)
Self-check: Can you state the joint typicality lemma?
Random binning: assigning codewords to bins uniformly at random(Review ch03)
Self-check: How does random binning enable distributed source coding (Slepian-Wolf)?
Gaussian mutual information: $I(X;Y) = \frac{1}{2}\log(1 + \text{SNR})$ for jointly Gaussian $(X, Y)$ (Review ch10)
Self-check: Can you compute $I(X; X + \alpha S + Z)$ for independent Gaussian $X, S, Z$ ?

Notation for This Chapter

Key symbols for channels with state. The state $S$ is a random variable that affects the channel transition law and may be known (causally or non-causally) at the encoder, the decoder, or both.

Symbol	Meaning	Introduced
$S$ , $S^n$	Channel state (scalar and sequence)	s01
$P_{Y\|X,S}$	Channel transition law given input $X$ and state $S$	s01
$U$	Auxiliary random variable (Gel'fand-Pinsker, Costa)	s02
$\alpha$	Costa's optimal coefficient: $\alpha^* = P/(P+N)$	s03
$Q$	State (interference) power in the Gaussian DPC model	s03

← Ch 11 Channels with Causal State Information at the Encoder