Chapter 6: BICM Error Probability Analysis

Learning Objectives

• Derive the BICM pairwise error probability (PEP) upper bound on AWGN from the Chernoff/Bhattacharyya bound applied to the bit metric, and recognise it as the same proof pattern used for TCM PEP in Ch. 2
• Quantify the role of the labelling $\mu$ in the PEP exponent via the average squared intra-subset distance $d^2_{\rm avg}(\mu)$, and explain why Gray is near-optimal on AWGN while set partitioning is not
• State and prove the Caire–Taricco–Biglieri diversity theorem $d_{\rm BICM}(\mu) = d_H \cdot L_{\min}(\mu)$ on fully-interleaved Rayleigh fading, and interpret it as: the binary code's Hamming distance gets harvested as time diversity
• Assemble the union-bound BER formula $P_b \le \frac{1}{k} \sum_d W_d c_d P(d)$ from the weight enumerator of the binary code, and recognise where it is tight (error floor) and where it is loose (waterfall)
• Compute the diversity reduction caused by finite interleaver depth $N$ on a block-fading channel with coherence time $T_c$: the effective diversity is $\min(d_H, N/T_c)$
• Translate BICM diversity analysis into standards-level engineering choices: HARQ buffer sizing, interleaver depth in LTE and 5G NR, FEC frame size in DVB-S2