Prerequisites

Before You Begin

This chapter builds on linear algebra and optimisation (Chapters 1 and 3), information-theoretic capacity of fading channels (Chapter 11), OFDM fundamentals (Chapter 14), and multi-user MIMO or network foundations from Chapter 19. The reader should be comfortable with convex optimisation, water-filling, ergodic capacity, and the basics of OFDM subcarrier structure.

Convex optimisation, KKT conditions, and water-filling(Review ch03)
Self-check: Can you solve a power allocation problem of the form $\max \sum_{k} \log(1 + p_k h_k)$ subject to $\sum_k p_k \leq P$ using the KKT conditions and state the water-filling solution $p_k = (\mu - 1/h_k)^+$ ?
Ergodic and outage capacity of fading channels(Review ch11)
Self-check: Can you write the ergodic capacity $C = \mathbb{E}_h[\log_2(1 + \text{SNR}\,|h|^2)]$ for a Rayleigh fading channel and explain the difference between ergodic capacity and $\varepsilon$ -outage capacity?
OFDM modulation, subcarrier structure, and cyclic prefix(Review ch14)
Self-check: Can you explain how OFDM converts a frequency-selective channel into $N$ parallel flat-fading subchannels and state the per-subcarrier signal model $Y_n = H_n X_n + W_n$ ?
Multi-user systems and interference management basics(Review ch19)
Self-check: Can you describe the downlink broadcast channel model with $K$ users and explain why time-division among users is generally suboptimal compared to superposition coding?
Order statistics and extreme-value distributions(Review ch01)
Self-check: Given $K$ i.i.d. random variables $X_1, \ldots, X_K$ with CDF $F(x)$ , can you write the CDF of the maximum $X_{(K)} = \max_k X_k$ as $F_{(K)}(x) = [F(x)]^K$ and explain how the expected maximum grows with $K$ ?

Chapter 20 Notation

Key symbols introduced or heavily used in this chapter.

Symbol	Meaning	Introduced
$K$	Number of users in the cell	s01
$h_k[t]$	Channel gain of user $k$ in time slot $t$	s01
$R_k[t]$	Instantaneous achievable rate of user $k$ in slot $t$ : $R_k[t] = \log_2(1 + \text{SNR}\,\|h_k[t]\|^2)$	s01
$\bar{T}_k[t]$	Exponentially weighted moving-average throughput of user $k$	s02
$N$	Number of OFDMA subcarriers	s03
$\pi(n)$	User assigned to subcarrier $n$ in the OFDMA allocation	s03
$p_{k,n}$	Power allocated to user $k$ on subcarrier $n$	s03
$\gamma_k$	Post-equalisation SINR reported by user $k$ (CQI)	s04
$\alpha$	Fairness parameter in the $\alpha$ -fair utility $U_\alpha(x) = x^{1-\alpha}/(1-\alpha)$	s02
$\Delta$	Frequency reuse factor in ICIC	s05
$P$	Total transmit power budget at the base station	s01
$\text{BLER}$	Block error rate after decoding	s04

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