High-SNR asymptotics of mutual information for discrete constellations with applications to BICM

The high-signal-to-noise ratio (SNR) asymptotic behavior of the mutual information (MI) for discrete constellations over the scalar additive white Gaussian noise channel is studied. Exact asymptotic expressions for the MI for arbitrary one-dimensional constellations and input distributions are presented in the limit as the SNR tends to infinity. Using the relationship between the MI and the minimum mean-square error (MMSE), asymptotics of the MMSE are also developed. It is shown that for any input distribution, the MI, MMSE and symbol-error probability have an asymptotic behavior proportional to a Gaussian Q-function whose argument depends only on the minimum Euclidean distance of the constellation. The developed expressions are then used to study the high-SNR behavior of the generalized mutual information (GMI) for bit-interleaved coded modulation (BICM). The long-standing conjecture that, at high SNR, Gray codes are the binary labelings that maximize the BICM-GMI, is proven. It is also shown that for any equally spaced M-ary constellation, there always exists an anti-Gray code that gives the lowest BICM-GMI at high SNR