Prerequisites & Notation

Before You Begin

This chapter builds on the degraded broadcast channel theory of Chapter 15 and the dirty-paper coding result of Chapter 12. We also use the MAC capacity region from Chapter 14 to establish the MAC-BC duality. The reader should be comfortable with superposition coding, random binning, and the Gel'fand-Pinsker theorem before proceeding.

Superposition coding and the degraded BC capacity region(Review ch15)
Self-check: Can you state the capacity region of the two-user degraded Gaussian BC with power splitting parameter $\alpha$ ?
Dirty-paper coding (Costa's theorem)(Review ch12)
Self-check: Can you state Costa's result and explain why the optimal auxiliary is $U = X + \alpha S$ with $\alpha = P/(P+N)$ ?
Random binning and Gel'fand-Pinsker coding(Review ch12)
Self-check: Can you describe how binning creates correlation between codewords without shared randomness?
MAC capacity region and successive cancellation(Review ch14)
Self-check: Can you write the $K$ -user MAC capacity region and identify the corner points achieved by SIC?
MIMO channel model and SVD decomposition(Review ch13)
Self-check: Can you state the MIMO capacity with CSIR and explain the water-filling solution over singular values?
Convex optimization, KKT conditions, Lagrangian duality
Self-check: Can you state the KKT conditions and explain why they are sufficient for convex problems?

Notation for This Chapter

Key symbols introduced or used heavily in this chapter. We work with the general two-user broadcast channel and its MIMO Gaussian specialization.

Symbol	Meaning	Introduced
$P_{Y_1, Y_2 \| X}$	Broadcast channel transition probability	s01
$U, V$	Auxiliary random variables in Marton's coding	s01
$R_{1}, R_{2}$	Rates to receiver 1 and receiver 2	s01
$\mathbf{H}_{k}$	Channel matrix to user $k$ in the MIMO BC	s03
$\mathbf{K}_k$	Input covariance matrix for user $k$ (DPC encoding order)	s03
$\sigma^2_{k}$	Noise variance at receiver $k$	s01
$C_{\text{BC}}$	Capacity region of the broadcast channel	s01
$C_{\text{MAC}}$	Capacity region of the dual MAC	s04
$P$	Total transmit power constraint	s01
$\mathbf{B}_k$	DPC precoding matrix for user $k$	s03

← Ch 15 Marton's Coding Scheme