Prerequisites & Notation

Before You Begin

This chapter develops the Bit-Interleaved Coded Modulation (BICM) framework β€” the dominant coded-modulation paradigm in every modern wireless standard. The reader should be comfortable with the algebraic tools from Chapter 1 (signal-space constellations, dmin⁑d_{\min}, coding gain), with Ungerboeck's set partitioning and trellis-coded modulation from Chapter 2, and with the capacity-rule / multilevel-coding arithmetic of Chapter 3. A working knowledge of binary linear codes (especially LDPC and convolutional codes) and of mutual information is also assumed.

  • Signal-space constellations, dmin⁑d_{\min}, and the coding-gain criterion(Review ch01)

    Self-check: Can you compute dmin⁑2d_{\min}^{2} for 16-QAM at unit EsE_s, and state why at high SNR the pairwise error probability behaves as Q(dmin⁑/2Οƒ22)Q(d_{\min}/\sqrt{2{\sigma^2}^{2}})?

  • Ungerboeck set partitioning and TCM(Review ch02)

    Self-check: Can you draw the set-partitioning tree for 16-QAM down to four levels and read off the intra-subset minimum distance at each level?

  • MLC, MSD, and the capacity rule(Review ch03)

    Self-check: Can you state the capacity rule CCM=βˆ‘β„“I(Y;Bβ„“βˆ£B0,…,Bβ„“βˆ’1)C_{\rm CM} = \sum_\ell I(Y; B_\ell \mid B_0, \ldots, B_{\ell-1}) and explain why its unconditional counterpart is generally smaller?

  • Mutual information, chain rule, KL divergence(Review ch02)

    Self-check: Can you state the chain rule for mutual information and explain why I(Y;Bβ„“βˆ£B<β„“)β‰₯I(Y;Bβ„“)I(Y; B_\ell \mid B_{<\ell}) \ge I(Y; B_\ell)?

  • AWGN channel capacity and the BI-AWGN channel(Review ch09)

    Self-check: Can you evaluate the BI-AWGN capacity C(SNR)=1βˆ’E[log⁑2(1+eβˆ’2SNR Zβˆ’2SNR)]C(\text{SNR}) = 1 - \mathbb{E}[\log_2(1 + e^{-2\sqrt{\text{SNR}}\, Z - 2\text{SNR}})] with Z∼N(0,1)Z \sim \mathcal{N}(0,1) numerically, and identify its high- and low-SNR asymptotics?

  • Binary LDPC and convolutional codes; bit interleavers(Review ch11)

    Self-check: Can you sketch the Shannon limit for rate-1/21/2 BI-AWGN, and name two capacity-approaching binary code families (LDPC, polar, turbo) and their design targets?

Notation for This Chapter

Symbols specific to the BICM framework. Chapter 3's MLC/MSD notation continues to apply; the BICM-specific symbols below are introduced as needed. See also the book-level notation table in the front matter.

SymbolMeaningIntroduced
X\mathcal{X}Constellation (QAM, PSK, APSK), size M=∣X∣M = |\mathcal{X}|s01
L=log⁑2ML = \log_2 MNumber of label bits per constellation points01
μ:{0,1}L→X\mu : \{0,1\}^L \to \mathcal{X}Labelling map. Gray labelling μG\mu_G; Set-Partition (Ungerboeck) labelling μSP\mu_{\rm SP}s01
cβ„“,bβ„“βˆˆ{0,1}c_\ell, b_\ell \in \{0,1\}Coded bit / label bit at position β„“βˆˆ{0,…,Lβˆ’1}\ell \in \{0, \ldots, L-1\}s01
Ο€\piBit interleaver (random permutation of the coded stream)s01
Ξ»β„“(y)\lambda_\ell(y)Soft log-likelihood ratio for bit position β„“\ell, Ξ»β„“=log⁑P(Bβ„“=0∣y)P(Bβ„“=1∣y)\lambda_\ell = \log \frac{P(B_\ell = 0 \mid y)}{P(B_\ell = 1 \mid y)}s02
Xβ„“(b)\mathcal{X}_\ell^{(b)}Subset of X\mathcal{X} whose labels have bit β„“\ell equal to bb, for b∈{0,1}b \in \{0,1\}s02
Cβ„“C_\ellCapacity of the β„“\ell-th BICM bit-channel, Cβ„“=I(Y;Bβ„“)C_\ell = I(Y; B_\ell)s03
CBICM(ΞΌ)C_{\rm BICM}(\mu)BICM capacity βˆ‘β„“=0Lβˆ’1Cβ„“\sum_{\ell=0}^{L-1} C_\ell under labelling ΞΌ\mus03
CCMC_{\rm CM}Coded-modulation (constellation-constrained) capacity I(Y;X)I(Y; X)s03
dHd_HHamming distance between two binary labelss04
Pe,PbP_e, P_bCodeword (or symbol) error probability and bit error probability respectivelys05
← Ch 4The BICM Paradigm