High-SNR Asymptotics of Mutual Information for Discrete Constellations with Applications to BICM

TL;DR

This paper derives high-SNR asymptotic expressions for mutual information, SEP, and MMSE in discrete constellations, proving Gray codes maximize BICM-GMI at high SNR and revealing the impact of constellation structure.

Contribution

It provides the first rigorous high-SNR asymptotic analysis of mutual information and BICM-GMI, confirming Gray codes' optimality and characterizing the behavior for various constellations.

Findings

Mutual information and error metrics decay proportionally to the Gaussian Q-function at high SNR.

Gray codes maximize BICM-GMI in the high-SNR regime.

Anti-Gray codes minimize BICM-GMI for certain constellations.

Abstract

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.