Prerequisites

Before You Begin

This chapter builds on mathematical foundations (Chapter 1), path-loss and propagation models (Chapter 5), information-theoretic capacity of fading channels (Chapter 11), and MIMO spatial multiplexing and diversity (Chapter 15). The reader should be comfortable with the capacity of point-to-point AWGN and fading channels, diversity order analysis, outage probability, and the basics of MIMO systems.

Probability theory and information-theoretic fundamentals(Review ch01)
Self-check: Can you state the capacity of the AWGN channel $C = \log_2(1 + \text{SNR})$ and explain the operational meaning of channel capacity in terms of reliable communication at rates below $C$ ?
Path-loss models and distance-dependent attenuation(Review ch05)
Self-check: Can you write the path-loss model $\ell(d) = d^{-\alpha}$ and compute the received SNR for a transmitter at distance $d$ with transmit power $P$ and noise power $\sigma^2$ ? Can you explain why $\alpha > 2$ in most terrestrial environments?
Capacity of fading channels, outage probability, and diversity(Review ch11)
Self-check: Can you derive the outage probability $p_{\text{out}} = \Pr[\log_2(1 + \text{SNR}|h|^2) < R]$ for a Rayleigh fading channel and show that it decays as $\text{SNR}^{-1}$ (diversity order 1)?
MIMO diversity and spatial multiplexing(Review ch15)
Self-check: Can you explain the diversity-multiplexing trade-off (DMT) and state that an $M \times N$ MIMO channel achieves maximum diversity order $MN$ ? Can you describe how Alamouti space-time coding achieves full transmit diversity with rate 1?

Chapter 22 Notation

Key symbols introduced or heavily used in this chapter.

Symbol	Meaning	Introduced
$S$	Source node	s01
$R$	Relay node	s01
$D$	Destination node	s01
$d_{SD}$	Source-to-destination distance	s01
$d_{SR}$	Source-to-relay distance	s01
$d_{RD}$	Relay-to-destination distance	s01
$C_{\mathrm{DF}}$	Achievable rate of decode-and-forward relaying	s01
$C_{\mathrm{AF}}$	Achievable rate of amplify-and-forward relaying	s01
$C_{\mathrm{CF}}$	Achievable rate of compress-and-forward relaying	s01
$C_{\mathrm{cut}}$	Cut-set upper bound on the relay channel capacity	s01
$d$	Diversity order: $d = -\lim_{\text{SNR}\to\infty}\frac{\log p_{\text{out}}}{\log \text{SNR}}$	s02
$f$	Relay position fraction: $d_{SR} = f \cdot d_{SD}$	s03
$T(n)$	Per-node throughput in a network of $n$ nodes	s05
$n$	Number of nodes in the wireless network	s05

← Ch 21 Relay Channel Fundamentals