Prerequisites & Notation

Before You Begin

This chapter takes the BICM framework of Chapters 5-6 and follows it into the modulation-and-coding (MCS) sections of the actual 3GPP NR, IEEE Wi-Fi, and ETSI DVB standards. The treatment is applied: we quote rates, code-block sizes, and SNR thresholds directly from the specifications and show how they are derived from the BICM capacity formulas and Caire-Taricco-Biglieri diversity analysis. The reader should already be comfortable with the BICM encoder-decoder block diagram, with the capacity formula CBICM(μ)=∑ℓI(Y;Bℓ)C_{\rm BICM}(\mu) = \sum_\ell I(Y; B_\ell), and with the PEP bound P(c→c^)≤exp⁡(−dH⋅davg2(μ)Es/(4N0))P(\mathbf{c} \to \hat{\mathbf{c}}) \le \exp(-d_H \cdot d^2_{\rm avg}(\mu) E_s/(4 N_0)) from Chapter 6. Familiarity with the Gray-labelling near-optimality result on AWGN is essential.

  • BICM paradigm: one binary code + interleaver + mapper(Review ch05)

    Self-check: Can you draw the BICM encoder-decoder block diagram and explain why the decoder sees a scalar binary channel regardless of the constellation?

  • BICM capacity formula and Gray near-optimality(Review ch05)

    Self-check: Can you write CBICM(μ)=∑ℓ=0L−1I(Y;Bℓ)C_{\rm BICM}(\mu) = \sum_{\ell=0}^{L-1} I(Y; B_\ell) and state the empirical gap to CCMC_{\rm CM} for Gray-labelled 16-QAM on AWGN?

  • BICM pairwise error probability and diversity(Review ch06)

    Self-check: Can you state the PEP bound P(c→c^)≤exp⁡(−dHdavg2(μ)Es/(4N0))P(\mathbf{c} \to \hat{\mathbf{c}}) \le \exp(-d_H d^2_{\rm avg}(\mu) E_s/(4N_0)) and identify the BICM diversity order on fully-interleaved Rayleigh fading?

  • LDPC codes, sum-product decoding, density evolution(Review ch11)

    Self-check: Can you sketch a Tanner graph, describe the sum-product message-passing rule, and state what density evolution predicts?

  • QAM constellations, Gray labelling, spectral efficiency(Review ch01)

    Self-check: Can you compute the average energy per symbol EsE_s for 16-QAM, 64-QAM, 256-QAM, 1024-QAM with unit-spacing Gray mapping, and state the corresponding spectral efficiency η=log⁡2M\eta = \log_2 M bits/2D?

  • AWGN capacity and the Shannon limit(Review ch09)

    Self-check: Can you write C(SNR)=log⁡2(1+SNR)C(\text{SNR}) = \log_2(1 + \text{SNR}) and locate the Shannon limit of rate-1/21/2 transmission at SNR≈0\text{SNR} \approx 0 dB?

Notation for This Chapter

Symbols specific to the modulation-and-coding (MCS) engineering of this chapter. The BICM notation of Chapters 5-6 continues to apply.

SymbolMeaningIntroduced
Qm=log⁡2MQ_m = \log_2 MModulation order (bits per constellation point). Qm∈{2,4,6,8,10,12}Q_m \in \{2, 4, 6, 8, 10, 12\} for QPSK, 16-/64-/256-/1024-/4096-QAM respectivelys01
RRCode rate of the LDPC code. Typically R∈[0.1,0.93]R \in [0.1, 0.93] after rate matchings01
η=QmR\eta = Q_m RSpectral efficiency in bits per 2D symbol. The BICM throughput before framing overheads01
MCSindexMCS indexInteger index into a standards table pairing (Qm,R)(Q_m, R). 3GPP NR uses indices 0-27 for data and 28-31 for retransmissions01
BG1,BG2\mathrm{BG}_1, \mathrm{BG}_25G NR LDPC base graphs. BG1 for large blocks and high rates; BG2 for short blocks and low ratess01
CQICQIChannel Quality Indicator. 4-bit feedback from UE to gNB indicating the highest MCS that gives BLER ≤10%\le 10\%s04
HARQ−IR,HARQ−CCHARQ-IR, HARQ-CCHybrid ARQ with Incremental Redundancy (different code bits retransmitted) vs Chase Combining (same bits, soft-combined at receiver)s01
TBS\mathrm{TBS}Transport Block Size: the number of information bits packed into one MAC-layer transmission unit. Derived from MCS, bandwidth, and layer counts01
BLER\mathrm{BLER}Block Error Rate after all HARQ retransmissions. Design target: BLER=10%\mathrm{BLER} = 10\% for eMBB, 10−510^{-5} for URLLCs04
APSKAPSKAmplitude-and-Phase-Shift Keying: constellation with points on concentric rings, optimised for nonlinear satellite amplifierss03
λ\lambdaMaxwell-Boltzmann shaping parameter. p(x)∝exp⁡(−λ∣x∣2)p(x) \propto \exp(-\lambda |x|^2), with λ>0\lambda > 0s05
CCDMCCDMConstant-Composition Distribution Matcher. The block-arithmetic-coding algorithm used in PAS to realise MB-shaped inputss05
PASPASProbabilistic Amplitude Shaping (Bocherer-Steiner-Schulte 2015). Architecture = distribution matcher + systematic BICM encoder + amplitude/sign mappers05
GsG_sShaping gain. Gs→πe/6≈1.53G_s \to \pi e / 6 \approx 1.53 dB (0.25 bit/2D) at high SNR for MB-shaped QAM over uniform QAMs05
← Ch 85G NR: LDPC + QAM + HARQ