Chapter Summary

Chapter Summary

Key Points

• 1.

  BICM-ID closes the loop left open by one-shot BICM. The demapper-with-a-priori computes refined bit LLRs using decoder- extrinsic information, and the SISO decoder computes refined a-posteriori LLRs using demapper-extrinsic information. The two boxes exchange EXTRINSIC LLRs only — subtracting the a-priori at each output prevents the positive-feedback failure mode in which the channel observation gets counted twice. This single architectural choice (Li–Ritcey 1997, ten Brink–Speidel 1998) is what makes iterative BICM decoding a convergent rather than a divergent process.

• 2.

  EXIT charts project density evolution onto mutual information. Under the consistent-Gaussian LLR assumption, each soft-in/soft- out box becomes a scalar map from a-priori MI $I_A$ to extrinsic MI $I_E$. The demapper curve $T_{\rm dem}(I_A, \mathrm{SNR})$ and the decoder curve $T_{\rm dec}(I_A, R)$ together define the iteration as a walk on the unit square. The J-function $J(\sigma) = I(B; \mathcal{N}(\pm \sigma^2/2, \sigma^2))$ and its inverse convert between MI and LLR variance and are the numerical workhorse of the whole machinery.

• 3.

  Convergence iff the tunnel is open. BICM-ID reaches BER = 0 if and only if the demapper curve strictly dominates the inverted decoder curve on $[0, 1)$. The convergence threshold $\mathrm{SNR}_{\rm conv}$ is the SNR at which the two curves just kiss; above threshold, a staircase trajectory climbs to $(1, 1)$ in 5–10 iterations, below threshold, the iteration stalls at the touching point and an error floor persists. The tunnel WIDTH is the SNR margin.

• 4.

  Set-partition labelling wins under iteration; Gray wins without. Ch. 5–6's Gray optimum is a one-shot conclusion. Under iterative decoding, what matters is the SHAPE of the entire demapper EXIT curve — especially its slope and its endpoint $T_{\rm dem}(1, \mathrm{SNR})$. SP's Ungerboeck chain rule doubles sub-constellation minimum distance at each bit level, which gives SP a steep slope and an endpoint $T_{\rm dem}(1, \mathrm{SNR}) \to 1$. Gray has flat levels and a flat curve. For 16-QAM at rate $1/2$, SP- BICM-ID beats Gray-BICM-ID by $\sim 1$ dB in convergence threshold. The optimal labelling depends on what the decoder can do.

• 5.

  EXIT matching turns code design into a linear programme. Given a demapper curve at a target SNR, the LDPC degree profile $(\lambda, \rho)$ is chosen to make the inverted decoder curve tuck just below the demapper everywhere. The matched rate $R^*(\gamma, \mu)$ is the largest code rate feasible subject to tunnel openness — a linear programme over the variable-node edge distribution (with alternating optimisation over $\rho$). This is how DVB-S2, DVB-S2X, 5G NR, and 802.11 LDPC codes are designed.

• 6.

  Engineering bottom line: BICM-ID buys 1–2 dB of the gap to CM capacity. At the cost of 5–10 iterations of receiver complexity, BICM-ID with SP (or designed BICM-ID labelling) and an EXIT- matched LDPC operates within 0.2–0.5 dB of the CM capacity limit. This is the SAME gap that separated one-shot Gray BICM from CM capacity in Ch. 5; iteration closes most of it. The remaining 0.2 dB is attributable to the Gaussian-LLR approximation plus finite- block-length effects. DVB-S2X's VL-SNR MODCODs deploy the full BICM-ID stack; 5G NR uses a reduced form (Gray, 2–3 iterations) optimised for throughput and latency.