Chapter Summary

Chapter Summary

Key Points

• 1.

  BICM PEP on AWGN factorises into code and labelling contributions. The Chernoff/Bhattacharyya bound on the bit metric yields $P(\mathbf{c} \to \hat{\mathbf{c}}) \le Q(\sqrt{\frac{1}{2} d_H \cdot d^2_{\rm avg}(\mu) \cdot E_s/N_0})$, where $d_H$ is the Hamming distance between the two codewords and $d^2_{\rm avg}(\mu)$ is the average squared Euclidean distance between pairs of constellation points differing in a single label bit. The code and the labelling can be designed independently — the architectural promise of BICM.

• 2.

  Gray labelling maximises $d^2_{\rm avg}(\mu)$ on square QAM. By distributing the Euclidean distance across bit positions evenly, Gray avoids the "weak bit" failure mode of set-partition labelling. SP concentrates large distances on the MSB but leaves the LSB close, and the Bhattacharyya PRODUCT is dominated by the weakest bit. On AWGN, use Gray.

• 3.

  The BICM diversity formula on fully-interleaved Rayleigh fading is $d_{\rm BICM}(\mu) = d_H \cdot L_{\min}(\mu)$ (Caire–Taricco–Biglieri 1998, Thm. 3). For Gray on square QAM, $L_{\min}(\mu_G) = 1$, so the diversity equals the binary code's Hamming distance. This reduces fading BICM design to binary code design — the cleanest operational conclusion in coded modulation theory. CommIT contribution.

• 4.

  The union bound gives the BER as $P_b \le \frac{1}{k} \sum_d W_d c_d P(d)$, where $W_d$ is the weight enumerator, $c_d$ is the input-weight multiplicity, and $P(d)$ is the diversity-$d$ PEP. At high SNR the sum is dominated by the $d = d_H$ term and $P_b \propto \text{SNR}^{-d_H L_{\min}(\mu)}$. The union bound is tight at the error floor but loose in the waterfall — tighter bounds (Ch. 7) are needed for waterfall analysis.

• 5.

  A finite interleaver of length $N$ on a channel with coherence $T_c$ caps the diversity at $N_{\rm eff} = \lceil N/T_c \rceil$: the effective diversity is $L_{\min}(\mu) \cdot \min(d_H, N_{\rm eff})$. The design rule is $N \ge d_H T_c$, with modest margin. This single formula drives interleaver length choices across five generations of cellular standards — GSM, IS-95, HSPA, LTE, 5G NR — and two decades of DVB satellite systems.

• 6.

  Design criterion summary. On AWGN, maximise $d_H \cdot d^2_{\rm avg} (\mu)$ — pick a code with large $d_H$ and a Gray labelling. On fully- interleaved fading, maximise $d_H \cdot L_{\min}(\mu)$ — the labelling is essentially free on symmetric QAM, so pick the code. On finite- coherence fading, additionally require $N \ge d_H T_c$. These three rules — together with the capacity rule from Ch. 5 — constitute the complete BICM design framework.