Prerequisites & Notation

Before You Begin

This chapter depends on the signal-space geometry and error-probability machinery of Chapter 1 plus two staples from classical coding theory (convolutional codes and the Viterbi algorithm). If any item feels unfamiliar, revisit the linked material before proceeding.

Chapter 1 — Signal-space view, coding gain, pairwise error probability(Review ch01)
Self-check: Can you write the pairwise error probability for an AWGN constellation in terms of the Euclidean distance between two points?
Convolutional codes: shift-register structure, generator polynomials, trellis representation
Self-check: Given a rate- $1/2$ convolutional code with $K=3$ and $(g_0, g_1) = (7, 5)_{\rm oct}$ , can you draw the encoder, list the states, and compute the free Hamming distance?
Viterbi algorithm: forward path metrics, survivor paths, traceback
Self-check: Can you run Viterbi by hand on a trellis with 4 states for 4 time steps, tracking all survivor paths?
Standard constellations: BPSK, QPSK, 8-PSK, 16-QAM — coordinates and minimum distance
Self-check: Can you compute $d_{\min}^2$ for 8-PSK (uniform on the unit circle) and for 16-QAM (square, unit average energy)?
Q-function asymptotics and the AWGN error exponent
Self-check: At high SNR, does a coding scheme with $d_{\rm free}^2 = 3 \cdot d_{\rm uncoded}^2$ gain $10\log_{10} 3 \approx 4.77$ dB over the uncoded baseline?

Notation for This Chapter

Symbols introduced in this chapter. See also the NGlobal Notation Table master table in the front matter.

Symbol	Meaning	Introduced
$\mathcal{X}$	Signal constellation (finite subset of $\mathbb{C}$ or $\mathbb{R}^2$ )	s01
$M$	Constellation size, $M = \|\mathcal{X}\|$	s01
$\Delta_i^2$	Minimum squared Euclidean distance within any subset at partition level $i$ (intra-subset MSED)	s01
$D_i$	A subset (coset) at partition level $i$	s01
$s$	Trellis state	s03
$N_s$	Number of trellis states, $N_s = 2^\nu$ for convolutional memory $\nu$	s03
$\nu$	Convolutional-encoder memory (number of delay cells)	s03
$K$	Constraint length of a convolutional code, $K = \nu + 1$	s03
$g_0, g_1$	Generator polynomials of a rate- $1/2$ convolutional code (octal notation)	s03
$d_H$	Hamming distance between two binary sequences	s03
$d_{\rm free}$	Free Euclidean distance of a TCM scheme (minimum Euclidean distance between any two distinct code paths through the trellis)	s03
$d_{\rm free}^2$	Squared free Euclidean distance	s03
$d_{\rm uncoded}$	Minimum distance of the uncoded constellation at the same spectral efficiency (baseline)	s05
$\gamma_c$	Asymptotic coding gain: $\gamma_c = d_{\rm free}^2 / d_{\rm uncoded}^2$	s05
$m$	Number of information bits per modulation symbol	s02
$\tilde{m}$	Number of coded bits per symbol used to select a subset, $\tilde{m} \le m$	s02
$P_e$	Error probability (symbol or bit, context-dependent)	s05
$E_s$	Average symbol energy	s01
$N_0$	One-sided noise power spectral density	s05

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