Prerequisites & Notation

Before You Begin

This chapter builds the second canonical coded-modulation construction — multilevel coding (MLC) — and its natural companion receiver, multistage decoding (MSD). The reader is expected to be comfortable with the tools established in Chapters 1 and 2, and with the information-theoretic language of mutual information and the chain rule.

Signal-space constellations, $d_{\min}$ , and the coding-gain criterion(Review ch01)
Self-check: Can you compute $d_{\min}^{2}$ for 8-PSK at unit $E_s$ , and state the coding-gain formula $\gamma_c = d_{\min}^{2} / d_{\rm uncoded}^2$ ?
Ungerboeck set partitioning and trellis-coded modulation(Review ch02)
Self-check: Can you draw the set-partitioning tree for 8-PSK down to three levels and read off the intra-subset minimum distance at each level?
Mutual information, the chain rule, and conditional mutual information(Review ch02)
Self-check: Can you state the chain rule $I(Y; B_0, B_1, \ldots, B_{L-1}) = \sum_{i=0}^{L-1} I(Y; B_i \mid B_0, \ldots, B_{i-1})$ and explain why each term is non-negative?
AWGN channel capacity and the binary-input AWGN channel(Review ch09)
Self-check: Can you write the BI-AWGN capacity integral $C = 1 - \int p(y) \log_2(1 + e^{-2y/{\sigma^2}^{2}}) \, dy$ and explain why it is strictly below $\log_2(1 + \text{SNR})$ ?
Binary channel coding: convolutional codes, LDPC, and the Shannon gap(Review ch11)
Self-check: Can you sketch the Shannon limit for a BI-AWGN channel at rate $1/2$ , and explain what "capacity-approaching" means for a modern LDPC code?
The concept of coset decoding and parallel transitions in a trellis(Review ch02)
Self-check: Given an $M$ -PSK constellation partitioned into subsets of size $M/2$ , can you describe how the TCM trellis handles the two parallel transitions between each pair of states?

Notation for This Chapter

Symbols specific to the multilevel coding and multistage decoding framework. See the chapter-opener notation table of Book CM for the shared symbols (constellation $\mathcal{X}$ , energy $E_s$ , noise density $N_0$ , etc.).

Symbol	Meaning	Introduced
$L = \log_2 M$	Number of partition levels (and number of binary codes in MLC) for an $M$ -ary constellation	s01
$b_i \in \{0,1\}$	Label bit at level $i$ , $0 \le i \le L-1$ (level 0 is the most-significant, coarsest partition)	s01
$\mu : \{0,1\}^L \to \mathcal{X}$	Partition-based labelling map from the binary label $(b_0, \ldots, b_{L-1})$ to the constellation point	s01
$\mathcal{A}_i^{(b_0,\ldots,b_{i-1})}$	Coset at level $i$ indexed by the decoded history $(b_0, \ldots, b_{i-1})$	s01
$C_i$	Capacity of the $i$ -th binary sub-channel: $C_i = I(Y; B_i \mid B_0, \ldots, B_{i-1})$	s02
$R_i$	Rate of the binary code used at level $i$ ; the capacity rule sets $R_i = C_i$	s02
$C_{\rm CM}$	Coded-modulation capacity $I(Y; X)$ of the constellation $\mathcal{X}$ under uniform inputs	s04
$C_{\rm BICM}$	Bit-interleaved coded modulation capacity $\sum_i I(Y; B_i)$ (unconditional sum)	s04
$C_{\rm MLC/MSD}$	Achievable rate of MLC with multistage decoding; equals $C_{\rm CM}$	s04
$\eta$	Spectral efficiency, $\eta = \sum_i R_i$ bits per 2D symbol	s02
$I(Y; X)$	Mutual information between the channel output $Y$ and input $X$	s02

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