Prerequisites

Before You Begin

OFDM builds on linear algebra (Chapter 1), signals and systems including the DFT (Chapter 4), wireless channel modelling (Chapter 6), digital modulation (Chapter 8), detection and estimation (Chapter 9), and equalisation concepts (Chapter 11). The DFT-based transceiver architecture relies heavily on matrix representations and circular convolution, while channel estimation and adaptive modulation require estimation theory and fading channel models.

Discrete Fourier transform (DFT) and its properties(Review ch04)
Self-check: Can you compute the $N$ -point DFT of a sequence $x[n]$ and state the circular convolution theorem?
Linear time-invariant systems and convolution(Review ch04)
Self-check: Can you compute the output of an LTI system with impulse response $h[n]$ given an input $x[n]$ via linear and circular convolution?
Matrix algebra and eigendecomposition(Review ch01)
Self-check: Can you diagonalise a circulant matrix using the DFT matrix and explain why circulant matrices have DFT eigenvectors?
Multipath fading channel models(Review ch06)
Self-check: Can you describe the tapped delay-line model for a frequency-selective channel and define coherence bandwidth $B_c$ ?
Digital modulation and constellation mapping(Review ch08)
Self-check: Can you map a bit sequence to QAM symbols and compute the average symbol energy for 16-QAM?
Maximum-likelihood and MMSE estimation(Review ch09)
Self-check: Can you derive the LS and MMSE estimators for a linear observation model $\mathbf{y} = \mathbf{H}\mathbf{x} + \mathbf{n}$ ?
Equalisation and ISI mitigation(Review ch11)
Self-check: Can you explain why a frequency-selective channel causes ISI and describe the ZF and MMSE equalisers?

Chapter 14 Notation

Key symbols introduced or heavily used in this chapter.

Symbol	Meaning	Introduced
$N$	Number of OFDM subcarriers	s01
$\Delta f$	Subcarrier spacing (Hz)	s01
$T_{\text{sym}}$	OFDM symbol duration (useful part, excluding CP): $T_{\text{sym}} = 1/\Delta f$	s01
$T_{\text{cp}}$	Cyclic prefix duration	s02
$T_{\text{total}}$	Total OFDM symbol duration: $T_{\text{total}} = T_{\text{sym}} + T_{\text{cp}}$	s02
$X[k]$	Frequency-domain data symbol on the $k$ -th subcarrier	s01
$x[n]$	Time-domain OFDM sample (IDFT output)	s02
$H[k]$	Channel frequency response at the $k$ -th subcarrier	s01
$h[l]$	Channel impulse response tap at delay $l$	s02
$L$	Number of channel taps (channel delay spread in samples)	s02
$N_{\text{cp}}$	Cyclic prefix length in samples	s02
$\mathbf{F}$	Normalised $N \times N$ DFT matrix with $[\mathbf{F}]_{k,n} = \frac{1}{\sqrt{N}} e^{-j2\pi kn/N}$	s02
$\epsilon$	Normalised carrier frequency offset (CFO), $\epsilon = \Delta f_{\text{offset}} / \Delta f$	s05
$\text{PAPR}$	Peak-to-average power ratio of the OFDM signal	s04
$\text{CCDF}$	Complementary cumulative distribution function: $\Pr(\text{PAPR} > \gamma)$	s04

← Ch 13 Principle of OFDM