Chapter Summary

Chapter Summary

Key Points

• 1.

  BICM is $L$ parallel binary channels (Wachsmann-Fischer-Huber view). The BICM capacity $\sum_\ell I(Y; B_\ell)$ of Ch. 5 is also the sum capacity of the $L$ parallel per-position bit channels $W_\ell$ (Thm. TBICM Capacity from Parallel Binary Channels). The parallel-channel viewpoint is the natural stage for the error- exponent and cutoff-rate analyses of this chapter.

• 2.

  BICM is mismatched decoding. The receiver uses the product bit metric $q_{\rm BICM}(y, x) = \prod_\ell p_{W_\ell}(y\mid b_\ell)$ rather than the true symbol likelihood $p(y\mid x)$ — a mismatched decoder (Thm. $M \ge 16$" data-ref-type="theorem">TThe BICM Product Metric Is Strictly Mismatched for $M \ge 16$). The mismatch is strict for $M \ge 16$ on AWGN and zero for QPSK / Gray-BPSK.

• 3.

  The achievable rate is the GMI (Guillén-Martínez-Caire 2008). The rate achievable by a random code with the mismatched BICM decoder is the generalised mutual information $I^{\mathrm{GMI}}(s)$, a concave function of the decoder scaling $s > 0$. The largest achievable rate is $\sup_s I^{\mathrm{GMI}}(s)$ (Thm. TBICM Achievability via GMI (Guillén-Martínez-Caire)).

• 4.

  At $s = 1$, GMI = CTB capacity. The GMI of the BICM product metric at scaling $s = 1$ reduces exactly to the Caire-Taricco- Biglieri 1998 BICM capacity $\sum_\ell C_\ell$ (Thm. $s = 1$ Equals the CTB Capacity" data-ref-type="theorem">TBICM GMI at $s = 1$ Equals the CTB Capacity). For Gray-QAM at high SNR, $s^\star \to 1$ and the CTB formula is tight; at low SNR or non-Gray labellings $s^\star \ne 1$ gives a measurable rate gain.

• 5.

  BICM exponent $\le$ CM exponent. The random-coding exponent of BICM is pointwise below the CM exponent: $E_r^{\mathrm{BICM}}(R) \le E_r^{\mathrm{CM}}(R)$ for all $R \le I^{\mathrm{BICM}}$, with equality at $R = 0$ (Thm. $\le$ CM Exponent" data-ref-type="theorem">TBICM Random-Coding Exponent $\le$ CM Exponent). Under Gray on AWGN the exponent ratio is $\gtrsim 0.85$ — a $\lesssim 1.18\times$ blocklength penalty, practically acceptable. Under SP it drops to $\sim 0.5$, doubling the required blocklength and explaining why SP- BICM never became a standard.

• 6.

  Cutoff rate $R_0 = E_0(1)$ is below capacity but above practical thresholds. $R_0^{\mathrm{BICM}} \le R_0^{\mathrm{CM}}$, mirroring the capacity ordering. For sequential / list decoders, $R_0$ is the rate above which complexity grows exponentially. LDPC+BP is a notable exception — it can operate above $R_0$, which is why 5G NR achieves within $\sim 0.3$ dB of Shannon despite using BICM.

• 7.

  Martinez-Fàbregas-Caire saddle-point PEP bound. The pairwise error probability of a BICM codeword pair at Hamming distance $d_H$ satisfies the saddle-point bound $\min_s \prod_n \mathbb{E}[(p_{W_\ell}(y\mid \bar b) / p_{W_\ell}(y\mid b))^s]$ (Thm. TMartinez-Fàbregas-Caire Saddle-Point PEP Bound), $\sim 1$–$2$ dB tighter than the Caire-Taricco-Biglieri 1998 union bound (which corresponds to $s = 1/2$). The saddle-point optimisation is the same $s$ knob as the GMI scaling — the mismatched-decoding framework is internally coherent.

• 8.

  CommIT contribution. Guillén i Fàbregas, Martínez, and Caire's 2008 Foundations and Trends monograph (vol. 5, pp. 1–153) is the definitive information-theoretic treatment of BICM. It recasts the 1998 Caire-Taricco-Biglieri formula as the GMI at $s = 1$, derives the random-coding exponent via the product-metric Gallager function, extends to fading and MIMO, and provides the theoretical foundation for all subsequent BICM research. The companion 2006 paper (Martinez- Fàbregas-Caire, IEEE Trans. IT) gives the tight saddle-point PEP bound. Together these constitute the mature information-theoretic framework of BICM.