High-SNR Asymptotics of Mutual Information for Discrete Constellations With Applications to BICM
Artikel i vetenskaplig tidskrift, 2014

Asymptotic expressions of the mutual information between any discrete input and the corresponding output of the scalar additive white Gaussian noise channel are presented in the limit as the signal-to-noise ratio (SNR) tends to infinity. Asymptotic expressions of the symbol-error probability (SEP) and the minimum mean-square error (MMSE) achieved by estimating the channel input given the channel output are also developed. It is shown that for any input distribution, the conditional entropy of the channel input given the output, MMSE, and SEP have an asymptotic behavior proportional to the Gaussian Q-function. The argument of the Q-function depends only on the minimum Euclidean distance (MED) of the constellation and the SNR, and the proportionality constants are functions of the MED and the probabilities of the pairs of constellation points at MED. The developed expressions are then generalized to study the high-SNR behavior of the generalized mutual information (GMI) for bit-interleaved coded modulation (BICM). By means of these asymptotic expressions, the long-standing conjecture that Gray codes are the binary labelings that maximize the BICM-GMI at high SNR is proven. It is further shown that for any equally spaced constellation whose size is a power of two, there always exists an anti-Gray code giving the lowest BICM-GMI at high SNR.

discrete constellations



Gray code

bit-interleaved coded modulation

minimum-mean square error


Anti-Gray code

mutual information

high-SNR asymptotics


additive white Gaussian noise channel




A. Alvarado

University of Cambridge

Fredrik Brännström

Chalmers, Signaler och system, Kommunikation, Antenner och Optiska Nätverk

Erik Agrell

Chalmers, Signaler och system, Kommunikation, Antenner och Optiska Nätverk

T. Koch

Universidad Carlos III de Madrid

IEEE Transactions on Information Theory

0018-9448 (ISSN)

Vol. 60 2 1061-1076 6671479


Elektroteknik och elektronik



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