Chapter Summary

Chapter Summary

Key Points

• 1.

  BICM = one binary code + bit interleaver + mapper. The fundamental BICM encoder (Def. DBICM Encoder) uses a single binary code $\mathcal{E}$, a bit interleaver $\pi$, and a mapper $\mu : \{0,1\}^L \to \mathcal{X}$ to drive an $M$-ary constellation. The receiver computes per-bit LLRs and feeds a single binary decoder. Modularity is the central design virtue.

• 2.

  The ideal interleaver makes the bit channels memoryless. Under a uniformly random interleaver (Thm. $\Rightarrow$ Memoryless Parallel Bit Channels" data-ref-type="theorem">TIdeal Interleaver $\Rightarrow$ Memoryless Parallel Bit Channels), the $M$-ary channel decomposes into a uniform mixture of $L$ binary sub-channels $W_\ell$. Each $W_\ell$ is characterised by the transition law $p_{W_\ell}(y \mid b) = (2/M) \sum_{x \in \mathcal{X}_\ell^{(b)}} p(y \mid x)$ — a scalar binary channel per label position.

• 3.

  BICM capacity is a sum of per-bit marginals. The main theorem of the chapter (Thm. TThe BICM Capacity Decomposition) states that $C_{\rm BICM}(\mu) = \sum_{\ell=0}^{L-1} I(Y; B_\ell)$. The formula is tight under exact marginal demapping (and upper-bounds any product-metric demapping). Achievability is a direct application of Shannon's theorem to the mixture bit channel; the converse is a mismatched-decoding GMI argument.

• 4.

  CM $\ge$ BICM, with the gap equal to a chain-rule residual. By the chain rule, $C_{\rm CM} - C_{\rm BICM}(\mu) = \sum_{\ell=1}^{L-1} I(B_\ell; B_{<\ell} \mid Y) \ge 0$, with equality iff the label bits are conditionally independent given $Y$ — a non-generic condition.

• 5.

  Gray labelling is near-optimal on AWGN. Theorem TGray Labelling Near-Optimality on AWGN guarantees that on square QAM with standard Gray labelling, the CM-BICM gap is bounded uniformly by $\lesssim 0.05$ bits over all SNRs and decays exponentially at high SNR. SP labelling is strictly suboptimal in BICM by roughly $0.4$ bits on 16-QAM and $\gtrsim 2$ dB of SNR equivalent on 64-QAM.

• 6.

  Gray is the right default but not a universal truth. Very low SNR admits slightly-better "anti-Gray" labellings (Thm. TGray Labelling Near-Optimality on AWGN(c)); iterative decoding (BICM-ID, Ch. 8) can make SP competitive or superior; fading channels (Ch. 6) introduce a diversity-order criterion that depends on labelling and interacts with code design.

• 7.

  BICM dominated the standards because of modularity and rate adaptation. A single LDPC base graph + rate matching drives every modulation in 5G NR (QPSK-1024-QAM), Wi-Fi 6/7 (BPSK-4096-QAM), and DVB-S2/S2X (QPSK-256-APSK). MLC would require $L$ codes per modulation — a prohibitive design burden. The $\sim 0.3$ dB capacity cost of Gray-BICM vs CM is swamped by the engineering gain.

• 8.

  CommIT contribution. Caire, Taricco, and Biglieri (1998, IEEE Trans. IT, vol. 44) is the foundational BICM paper and arguably the most influential coded-modulation result of the last three decades. It established BICM as a parallel-bit- channel model, proved Gray near-optimality, analysed PEP/ diversity on fading, and became the theoretical foundation for every modern wireless MCS design.