Prerequisites

Before You Begin

This chapter builds on information-theoretic foundations, OFDM/OFDMA system design, diversity techniques, and multi-user access. If any item feels unfamiliar, revisit the linked material first.

Entropy, mutual information, and channel capacity (Shannon limit)(Review ch02)
Self-check: Can you state the capacity of an AWGN channel and explain what happens as blocklength $n \to \infty$ ?
OFDM system model, cyclic prefix, and subcarrier allocation(Review ch11)
Self-check: Can you describe how OFDM converts a frequency-selective channel into parallel flat-fading subchannels?
Diversity techniques: spatial, frequency, and time diversity; combining methods(Review ch12)
Self-check: Can you derive the outage probability scaling $P_{\mathrm{out}} \propto \text{SNR}^{-L}$ for $L$ -branch MRC in Rayleigh fading?
Multiple access schemes: TDMA, FDMA, CDMA, OFDMA, and random access (ALOHA)(Review ch19)
Self-check: Can you compute the throughput of slotted ALOHA and explain why it peaks at $1/e$ ?
Basic queueing theory: M/M/1 queue, arrival rate, service rate, stability condition
Self-check: Can you write the mean delay for an M/M/1 queue with arrival rate $\lambda$ and service rate $\mu$ ?
Compressed sensing basics: sparsity, RIP, and $\ell_1$ recovery
Self-check: Do you know that $O(s \log(n/s))$ measurements suffice to recover an $s$ -sparse vector in $\mathbb{R}^n$ ?

Notation for This Chapter

Symbols introduced in this chapter. See also the NGlobal Notation Table master table in the front matter.

Symbol	Meaning	Introduced
$n$	Blocklength (number of channel uses per codeword)	s01
$\varepsilon$	Block error probability (decoding failure probability)	s01
$C$	Shannon capacity (bits per channel use)	s01
$V$	Channel dispersion (bits $^2$ per channel use)	s01
$Q^{-1}(\cdot)$	Inverse of the Gaussian Q-function	s01
$R^*(n, \varepsilon)$	Maximum achievable rate at blocklength $n$ and error probability $\varepsilon$	s01
$L$	Diversity order (number of independent diversity branches)	s02
$K_{\mathrm{total}}$	Total number of mMTC devices in the cell	s03
$K_a$	Number of active devices in a given slot	s03
$\mathbf{A}$	Pilot (sensing) matrix for compressed sensing activity detection	s03
$\Delta(t)$	Age of Information at the monitor at time $t$	s04
$\bar{\Delta}$	Time-average Age of Information	s04
$\rho = \lambda / \mu$	Server utilisation (traffic intensity)	s04
$B$	Message size in bits (unsourced random access)	s03

← Ch 31 The Finite Blocklength Regime