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Bit-Metric Decoding

BICM channel

At the receiver, we use a bitwise demapper to calculate soft information for each bit-level and we pass these soft informations to a binary decoder. This combination of a bitwise demapper with a binary decoder is called bit-metric decoding.

  • Binary label of $X=S\cdot A$ is $b(S)\mathbf{b}(A)=:B_1B_2\dotsb B_m=:\boldsymbol{B}$.
  • We use Gray labeling.
  • Denote signal point with label $\boldsymbol{B}$ by $x_{\boldsymbol{B}}$.
  • The demapper calculates bitwise soft-information $$ L_j=\log\frac{P_{B_j}(0)}{P_{B_j}(1)}+\log\frac{p_{Y|B_j}(Y|0)}{p_{Y|B_j}(Y|1)},\quad j=1,\dotsc,m. $$
  • Achievable rate for bit-metric decoding: $$ \mathsf{R}_\text{BMD}(P_X,\text{SNR})=\Bigl[H(\boldsymbol{B})-\sum_{i=1}^m H(B_i|Y)\Bigr]^+. $$
  • No iterative demapping!

G. Böcherer, Achievable Rates for Shaped Bit-Metric Decoding, submitted to IEEE Trans. Inf. Theory, revised.