BICM Mutual Information (Capacity)

By Theorem $\Rightarrow$ Memoryless Parallel Bit Channels" data-ref-type="theorem">TIdeal Interleaver $\Rightarrow$ Memoryless Parallel Bit Channels, the binary decoder sees a memoryless channel with transition $p_{W_{\rm BICM}}(y \mid b) = \frac{1}{L} \sum_\ell p_{W_\ell}(y \mid b)$. Shannon's noisy-channel coding theorem states that this channel's capacity is $$C_{W_{\rm BICM}} \;=\; I(B; Y) \;=\; \frac{1}{L} \sum_{\ell} I(Y; B_\ell),$$ where the second equality follows from the fact that $I$ is linear in the input distribution when the conditional is a mixture (by direct expansion of the mutual information or by noting that choosing $\ell$ uniformly at random acts as auxiliary randomness).

The BICM system sends $N$ coded bits per codeword and maps them to $N/L$ symbols. A code of rate $R_c$ delivers $K = R_c N$ information bits. The spectral efficiency is therefore $\eta = R_c L$ bits per symbol. Applying the binary-channel coding theorem gives $$R_c \;<\; C_{W_{\rm BICM}} \;=\; \frac{1}{L} \sum_\ell I(Y; B_\ell),$$ i.e.\ $\eta = R_c L < \sum_\ell I(Y; B_\ell) = C_{\rm BICM}(\mu)$.

The BICM decoder's metric factorises over bits: $M(\mathbf{y}, \mathbf{c}) = \prod_n p_{W_{\rm BICM}}(y_n \mid c_n)$. This is a mismatched likelihood with respect to the true joint symbol likelihood. The maximum reliable rate under any such mismatched decoder is upper-bounded by the generalised mutual information (GMI) — a standard result in the mismatched-decoding literature. For the product-form bit metric used in BICM, the GMI coincides with $\sum_\ell I(Y; B_\ell)$ (Caire-Taricco-Biglieri Thm. 1; a clean derivation is in Martínez–Guillén i Fàbregas–Caire 2008 and our Ch. 7). Hence no BICM scheme exceeds $C_{\rm BICM}(\mu)$ with vanishing error.

By the chain rule for mutual information, $$C_{\rm CM} = I(Y; B_0, B_1, \ldots, B_{L-1}) = \sum_{\ell = 0}^{L-1} I(Y; B_\ell \mid B_{<\ell}),$$ while by definition $C_{\rm BICM}(\mu) = \sum_\ell I(Y; B_\ell)$. Subtracting, $$C_{\rm CM} - C_{\rm BICM}(\mu) = \sum_{\ell = 1}^{L-1} \bigl[I(Y; B_\ell \mid B_{<\ell}) - I(Y; B_\ell)\bigr].$$ (The $\ell = 0$ term cancels.)

Using the identity $I(Y; B_\ell \mid B_{<\ell}) - I(Y; B_\ell) = I(B_\ell; B_{<\ell} \mid Y) - I(B_\ell; B_{<\ell})$ and the fact that the label bits are a priori independent (so $I(B_\ell; B_{<\ell}) = 0$ for i.i.d.\ uniform priors), we obtain $$I(Y; B_\ell \mid B_{<\ell}) - I(Y; B_\ell) = I(B_\ell; B_{<\ell} \mid Y) \ge 0.$$ Conditional mutual information is non-negative, and is zero iff $B_\ell$ and $B_{<\ell}$ are conditionally independent given $Y$. Hence $C_{\rm CM} \ge C_{\rm BICM}(\mu)$, with the equality condition stated in the theorem. $\blacksquare$

At large $\text{SNR}$ the posterior $p(X = x \mid y)$ concentrates on the ML-decoded $\hat{x}(y)$ and on its nearest neighbours. The probability of a symbol error is dominated by $\tfrac{1}{2} N_{\min} Q(\sqrt{\text{SNR}/2} \, d_{\min})$, where $N_{\min}$ is the average number of nearest neighbours and $d_{\min}$ is the minimum Euclidean distance.

Under Gray labelling, any nearest-neighbour pair differs in exactly one label bit. Therefore a symbol error at the ML decoder's margin corresponds to exactly one bit flip — and only one bit position has non-trivial residual uncertainty given $y$. That is, $$H(B_\ell \mid Y) \;\approx\; h_2\bigl(Q(\sqrt{\text{SNR}/2} \, d_{\min})\bigr) \big/ (2 \text{ or } 4),$$ where $h_2$ is the binary entropy. Summing over $\ell$ gives the same high-SNR decay as the joint entropy $H(X \mid Y)$, so the gap $C_{\rm CM} - C_{\rm BICM, Gray}$ vanishes exponentially.

Under SP labelling, a nearest-neighbour flip at the ML decoder changes $b_{L-1}$ with probability $\approx 1$, but also (with non-negligible probability) flips higher-level bits through neighbouring-coset confusion at higher intra-coset distances. The marginal entropies $H(B_\ell \mid Y)$ for small $\ell$ do not concentrate as tightly as their joint, and the gap $\sum_\ell I(Y; B_\ell) - I(Y; X)$ stays bounded away from zero. Detailed tables are in Caire-Taricco-Biglieri 1998, §V.B. $\blacksquare$