Prerequisites & Notation

Before You Begin

This chapter treats equalization as an inference problem. Readers should be comfortable with the material below; if any item feels unsteady, revisit the linked chapter before proceeding.

Maximum-likelihood estimation and detection under Gaussian noise(Review ch02)
Self-check: Can you derive the log-likelihood of a Gaussian observation and reduce ML detection to a quadratic form?
Wiener filtering and the orthogonality principle(Review ch09)
Self-check: Can you state the Wiener–Hopf equations and solve them in the frequency domain?
Discrete-time linear systems: convolution, $z$ -transform, DTFT
Self-check: Can you compute $H(e^{j\omega})$ given a causal FIR with taps $\{h[0], h[1], h[2]\}$ ?
Jointly Gaussian random vectors and conditional expectation(Review ch03)
Self-check: Can you write $\mathbb{E}[\mathbf{x}|\mathbf{y}]$ in closed form when $(\mathbf{x}, \mathbf{y})$ are jointly Gaussian?
Dynamic programming on a graph / shortest-path thinking
Self-check: Do you recognize Bellman's principle of optimality as the engine behind Viterbi?

Notation for This Chapter

Symbols introduced or used heavily in this chapter. The channel impulse response is denoted $\{h[k]\}_{k=0}^{L}$ and the channel memory is $L$ (one fewer than the number of taps). We write $M = |\mathcal{A}|$ for the size of the modulation alphabet.

Symbol	Meaning	Introduced
$h[k]$ , $L$	Discrete-time channel impulse response taps and channel memory (so $L+1$ taps total)	s01
$\mathbf{h}$	Column vector $[h[0], h[1], \ldots, h[L]]^T$ of channel taps	s01
$H(f)$ or $H(e^{j\omega})$	DTFT of the channel impulse response	s01
$x[n]$ , $y[n]$ , $w[n]$	Transmitted symbols, received samples, and AWGN samples	s01
$\mathcal{A}$ , $M$	Modulation alphabet and its size $M = \|\mathcal{A}\|$	s01
$\hat{\mathbf{x}}_{\text{ML}}$	Maximum-likelihood sequence estimate	s01
$s_k \in \mathcal{S}$	Trellis state at time $k$ ; $\|\mathcal{S}\| = M^L$	s02
$\lambda_k(s)$	Viterbi path metric: minimum cumulative squared error reaching state $s$ at time $k$	s02
$W_{\text{ZF}}(f)$ , $W_{\text{MMSE}}(f)$	Frequency responses of the ZF and MMSE linear equalizers	s03
$N_0$ , $\text{SNR}$	Noise power spectral density and input signal-to-noise ratio	s03
$\mathbf{w}_{\text{ff}}$ , $\mathbf{w}_{\text{fb}}$	Feedforward and feedback filter taps of the DFE	s04
$\Delta$	Decision delay of the equalizer (in symbols)	s04

← Ch 10 The ISI Channel as an Inference Problem