Chapter 7: BICM Capacity and Error Exponents

Learning Objectives

• State the Wachsmann–Fischer–Huber parallel-binary-channel model of BICM and reconcile it with the Caire-Taricco-Biglieri capacity formula $C_{\rm BICM} = \sum_\ell I(Y; B_\ell)$ derived in Chapter 5
• Recognise BICM as a mismatched decoder that applies the product bit metric $\prod_\ell p_{W_\ell}(y\mid b_\ell)$ in place of the true symbol likelihood $p(y\mid x)$, and define the decoder scaling parameter $s$
• Define the generalised mutual information $I^{\mathrm{GMI}}(s)$ and prove, via the Gallager random-coding argument, that the largest rate achievable by any random code with the BICM bit metric is $\sup_{s>0} I^{\mathrm{GMI}}(s)$
• Derive the BICM random-coding error exponent $E_r^{\mathrm{BICM}}(R)$ from the Gallager function $E_0(\rho)$ specialised to the product metric, show that $E_r^{\mathrm{BICM}}(R) \le E_r^{\mathrm{CM}}(R)$ everywhere, and quantify the gap under Gray labelling
• Compute the BICM cutoff rate $R_0^{\mathrm{BICM}} = E_0^{\mathrm{BICM}}(1)$ and contrast it with the CM cutoff rate; explain the operational role of $R_0$ for sequential / list decoders
• Apply the Martinez–Fàbregas–Caire 2006 saddle-point PEP bound and compare it with the classical Caire-Taricco-Biglieri union bound of Chapter 6