Prerequisites & Notation

Before You Begin

This chapter applies the capacity theory developed for AWGN channels (Chapter 10) and parallel Gaussian channels (Chapter 12) to wireless fading environments. We assume fluency with differential entropy, mutual information for continuous random variables, and the water-filling power allocation from Chapter 12. Some familiarity with basic wireless propagation is helpful but not strictly required.

Differential entropy and mutual information for continuous RVs(Review ch05)
Self-check: Can you compute $h(X)$ for $X \sim \mathcal{N}(0, \sigma^2)$ ?
AWGN channel capacity: $C = \frac{1}{2}\log(1 + \text{SNR})$ (Review ch10)
Self-check: Can you derive the capacity of the real AWGN channel from the entropy-maximizing input distribution?
Parallel Gaussian channels and water-filling(Review ch12)
Self-check: Can you state the water-filling solution and explain the KKT conditions that produce it?
Jensen's inequality for convex and concave functions(Review ch01)
Self-check: Can you determine whether $\mathbb{E}[f(X)] \leq f(\mathbb{E}[X])$ or vice versa for a given $f$ ?
Basic probability: expectation, conditional expectation, CDF
Self-check: Can you compute $\mathbb{E}[g(X)]$ for a function $g$ when the distribution of $X$ is known?
Linear algebra: eigenvalues, SVD, positive semidefinite matrices
Self-check: Can you state the SVD of an $m \times n$ matrix and explain what the singular values represent?

Notation for This Chapter

Symbols introduced in this chapter. In this chapter, $H$ (non-bold, scalar) denotes the fading gain random variable, while $\mathbf{H}$ (bold) denotes the MIMO channel matrix. The entropy function is always written with its argument: $H(X)$ .

Symbol	Meaning	Introduced
$H$	Scalar fading gain (random variable)	s01
$Y = HX + Z$	Scalar fading channel model	s01
$Z$	Additive noise, $Z \sim \mathcal{N}(0, N)$	s01
$\text{SNR}$	Signal-to-noise ratio $P/N$	s01
$CSIR$	Channel state information at the receiver	s01
$CSIT$	Channel state information at the transmitter	s01
$C_{\text{erg}}$	Ergodic capacity	s02
$P(H)$	Power allocation as a function of fading state $H$	s03
$\nu$	Water-filling level over fading states	s03
$P_{\text{out}}(R)$	Outage probability at rate $R$	s04
$C_\epsilon$	$\epsilon$ -outage capacity	s04
$\mathbf{H} \in \mathbb{C}^{n_r \times n_t}$	MIMO channel matrix	s05
$\mathbf{K}_x$	Input covariance matrix, $\mathbf{K}_x = \mathbb{E}[\mathbf{x}\mathbf{x}^H]$	s05
$n_t, n_r$	Number of transmit and receive antennas	s05

← Ch 12 The Fading Channel Model