Prerequisites

Before You Begin

This chapter builds on linear algebra (Chapter 1), signals and systems (Chapter 4), fading channels (Chapter 6), digital modulation (Chapter 8), and detection theory (Chapter 9). Equalization combines discrete-time filtering with statistical estimation to undo the distortion imposed by frequency-selective channels.

Matrix inversion, eigenvalues, and Toeplitz structure(Review ch01)
Self-check: Can you invert a $3 \times 3$ Toeplitz matrix and compute its eigenvalues? Do you know what positive definiteness means?
Discrete-time convolution and z-transforms(Review ch04)
Self-check: Can you compute the output of a discrete-time LTI system $y[n] = \sum_k h[k]\, x[n-k]$ and find $H(z)$ ?
Frequency-selective fading and multipath channels(Review ch06)
Self-check: Can you describe how multipath propagation leads to a frequency-selective channel with impulse response $h[n]$ ?
Digital modulation and pulse shaping(Review ch08)
Self-check: Can you describe how a sequence of symbols $\{a_k\}$ is transmitted through a pulse-shaping filter and sampled at the receiver?
Matched filter, ML detection, and sufficient statistics(Review ch09)
Self-check: Can you explain why the matched filter maximises output SNR and state the ML decision rule for AWGN channels?

Chapter 13 Notation

Key symbols introduced or heavily used in this chapter.

Symbol	Meaning	Introduced
$h[n]$	Discrete-time channel impulse response (channel taps)	s01
$L$	Channel memory (number of ISI-causing taps)	s01
$a_k$	Transmitted symbol at time $k$	s01
$y_k$	Received sample at time $k$ (after matched filtering)	s01
$w_k$	Equalizer tap coefficients	s02
$N_f$	Number of feedforward equalizer taps	s02
$N_b$	Number of feedback taps in a DFE	s03
$d_{\min}$	Minimum eye opening (eye diagram)	s01
$\sigma_w^2$	Noise variance at the equalizer input	s02
$J_{\mathrm{MMSE}}$	Minimum mean-square error of the equalizer	s02
$\mu$	LMS step size	s05
$\mathbf{R}_{yy}$	Autocorrelation matrix of the received signal	s02

← Ch 12 ISI and the Need for Equalization