Prerequisites & Notation

Before You Begin

Chapters 5–7 gave the "one-shot" BICM story: a bit interleaver feeds a binary code's output into a constellation mapper, the receiver demaps each channel observation into per-bit LLRs, and the decoder takes those LLRs as final. Capacity, PEP, and error exponents were analysed under this sequential data flow. Chapter 8 closes the loop. The decoder's soft output is fed BACK to the demapper as a priori information, the demapper refines its LLRs, and the two boxes iterate to convergence. The reader should be comfortable with BICM's bit-metric LLR, with soft-in/soft-out (SISO) decoding of binary codes, and with the Gaussian-approximation machinery behind mutual-information analysis.

BICM encoder/decoder, bit metric, and LLR extraction(Review ch05)
Self-check: Can you write the BICM bit-metric LLR $\lambda_\ell(y) = \log \sum_{s \in \mathcal{X}_\ell^{(0)}} p(y \mid s) - \log \sum_{s \in \mathcal{X}_\ell^{(1)}} p(y \mid s)$ and explain why under a Gaussian channel it is a function of Euclidean distance?
Soft-in/soft-out (SISO) decoding of binary codes(Review ch05)
Self-check: Can you describe what a SISO decoder does: takes per-bit a priori LLRs and channel LLRs as input, returns per-bit a posteriori LLRs and their EXTRINSIC components (a posteriori minus input)?
Set-partition (Ungerboeck) labelling vs Gray labelling(Review ch02)
Self-check: Can you state the Ungerboeck set-partition chain rule and explain why SP labelling maximises intra-subset minimum distance while Gray maximises the average $d^2_{\\rm avg}$ ?
BICM union-bound BER and the role of the labelling(Review ch06)
Self-check: Can you explain why Gray labelling outperforms SP labelling under one-shot BICM decoding on AWGN, and why the conclusion flips on fully-interleaved fading?
BICM capacity and the gap to CM capacity(Review ch05)
Self-check: Can you sketch the $C_{\\rm BICM}(\\mu)$ versus $C_{\\rm CM}$ curves for 16-QAM and identify the SNR range where the gap is most visible?
Mutual information, entropy, and the data-processing inequality(Review ch02)
Self-check: Can you compute the mutual information between a binary input and a continuous output numerically, and state why post-processing a sufficient statistic does not reduce mutual information?
Binary-input AWGN channel and the consistent-Gaussian LLR model(Review ch03)
Self-check: Can you state that for BI-AWGN with SNR $\\gamma$ , the LLR is Gaussian with mean $\\pm 4\\gamma$ and variance $8\\gamma$ — the "consistent" Gaussian with mean = variance/2 that underlies the J-function?

Notation for This Chapter

Chapters 5–7's BICM notation (constellation $\mathcal{X}$ , labelling $\mu$ , bit subsets $\mathcal{X}_\ell^{(b)}$ , bit-metric LLR $\lambda_\ell$ ) carries over. The symbols below are the additional ones introduced for EXIT analysis and iterative decoding.

Symbol	Meaning	Introduced
$\lambda, \lambda_{\rm ch}, \lambda_A, \lambda_E$	LLR variables: total LLR $\lambda$ , channel LLR $\lambda_{\rm ch}$ , a-priori LLR $\lambda_A$ , extrinsic LLR $\lambda_E$ . $\lambda_E = \lambda - \lambda_A$	s01
$I_A, I_E$	A-priori mutual information $I_A = I(B; \lambda_A)$ and extrinsic mutual information $I_E = I(B; \lambda_E)$ , both valued in $[0, 1]$	s02
$J(\sigma)$	The J-function: mutual information between a binary $\pm 1$ input and a consistent-Gaussian LLR of standard deviation $\sigma$ (mean $\sigma^2/2$ )	s02
$T_{\mathrm{dem}}(I_A, \mathrm{SNR})$	Demapper EXIT curve: maps a-priori MI at the demapper input to extrinsic MI at its output, parametrised by $\mathrm{SNR}$	s02
$T_{\mathrm{dec}}(I_A, R)$	Decoder EXIT curve: maps a-priori MI at the decoder input (from the demapper) to extrinsic MI at its output, parametrised by code rate $R$	s02
$T_{\mathrm{dec}}^{-1}(\cdot, R)$	Inverse decoder EXIT curve, plotted on the same axes as $T_{\mathrm{dem}}$ for the convergence-tunnel visualisation	s03
$\Delta(I_A)$	Tunnel width at a-priori level $I_A$ : $\Delta(I_A) = T_{\mathrm{dem}}(I_A, \mathrm{SNR}) - T_{\mathrm{dec}}^{-1}(I_A, R)$	s03
$\mathrm{SNR}_{\mathrm{conv}}$ , $E_b/N_0\|_{\mathrm{conv}}$	Convergence threshold: the smallest SNR (or $E_b/N_0$ ) at which the tunnel is open, $\min_{I_A \in [0,1)} \Delta(I_A) > 0$	s03
$\mu_G, \mu_{\rm SP}, \mu_{\rm aG}$	Gray, set-partition, and anti-Gray labelings respectively; $\mu_{\rm aG}$ is the bitwise complement of $\mu_G$ on a rotated constellation	s04
$R^*(\mathrm{SNR}, \mu)$	Maximum achievable code rate at SNR under labelling $\mu$ such that the inverted decoder curve lies below the demapper curve — the EXIT-matched rate	s05
$\{\lambda_i\}_{i=0}^{d_v}$	Variable-node degree distribution of an LDPC code from the edge perspective: $\lambda_i$ is the fraction of edges attached to a variable node of degree $i$	s05
$\{\rho_j\}_{j=0}^{d_c}$	Check-node degree distribution of an LDPC code from the edge perspective	s05
$t$	Iteration index in BICM-ID, $t = 0, 1, 2, \ldots$ ; $I_E^{(t)}$ is the extrinsic MI after iteration $t$	s03

← Ch 7 Iterative Demapping: Closing the BICM Loop