Prerequisites

Before You Begin

This chapter builds on linear algebra (Chapter 1), OFDM modulation (Chapter 4), random processes and noise (Chapter 8), and information-theoretic capacity (Chapter 11). The reader should be familiar with the concept of bandwidth, power spectral density, the Gaussian channel capacity formula, and basic probability including Poisson processes. Knowledge of multi-user MIMO (Chapter 17) is helpful but not strictly required.

Orthogonal bases and inner products(Review ch01)
Self-check: Can you verify that two vectors $\mathbf{c}_i, \mathbf{c}_j$ of length $N$ are orthogonal by checking $\mathbf{c}_i^H \mathbf{c}_j = 0$ , and explain what happens when they are not?
OFDM and multi-carrier modulation(Review ch04)
Self-check: Can you describe how OFDM divides a wideband channel into $N$ narrow orthogonal subcarriers and explain the role of the cyclic prefix in eliminating inter-carrier interference?
Spectral characterisation of random processes(Review ch08)
Self-check: Can you define the power spectral density of a wide-sense stationary process and relate it to the autocorrelation function via the Wiener-Khinchin theorem?
Channel capacity and the water-filling solution(Review ch11)
Self-check: Can you state the Gaussian channel capacity $C = W\log_2(1 + \text{SNR})$ and explain how water-filling allocates power across parallel sub-channels to maximise the total rate?
Multi-user information theory basics(Review ch11)
Self-check: Can you describe the multiple-access channel (MAC) capacity region for two users and explain why the sum-rate point requires successive decoding rather than treating interference as noise?

Chapter 19 Notation

Key symbols introduced or heavily used in this chapter.

Symbol	Meaning	Introduced
$K$	Number of users sharing the channel	s01
$W$	Total system bandwidth (Hz)	s01
$T$	Frame duration (seconds)	s01
$P$	Total transmit power budget	s01
$P_k$	Transmit power allocated to user $k$	s01
$R_k$	Achievable rate of user $k$ (bits/s/Hz)	s01
$N$	Processing gain (spreading factor) in CDMA	s03
$\mathbf{c}_k$	Spreading code vector for user $k$	s03
$G$	Offered load (Erlangs) in random access	s04
$S$	Throughput of random access protocol	s04
$T_c$	Coherence time (in symbols)	s05
$M$	Number of base-station antennas	s05
$\sigma^2$	Noise variance	s01
$\|h_k\|^2$	Channel power gain for user $k$	s01

← Ch 18 Orthogonal Multiple Access (FDMA/TDMA/OFDMA)