@article{Alvarado2014,
title = {High-SNR Asymptotics of Mutual Information for Discrete Constellations With Applications to BICM},
author = {Alex Alvarado and Fredrik Brannstrom and Erik Agrell and Tobias Koch},
url = {http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=6671479
http://www.tsc.uc3m.es/~koch/files/IEEE_TIT_60%282%29.pdf},
issn = {0018-9448},
year = {2014},
date = {2014-01-01},
journal = {IEEE Transactions on Information Theory},
volume = {60},
number = {2},
pages = {1061--1076},
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.},
keywords = {additive white Gaussian noise channel, Anti-Gray code, bit-interleaved coded modulation, discrete constellations, Entropy, Gray code, high-SNR asymptotics, IP networks, Labeling, minimum-mean square error, Modulation, Mutual information, Signal to noise ratio, Vectors},
pubstate = {published},
tppubtype = {article}
}

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.

@inproceedings{Alvarado2013b,
title = {High-SNR Asymptotics of Mutual Information for Discrete Constellations},
author = {Alex Alvarado and Fredrik Brannstrom and Erik Agrell and Tobias Koch},
url = {http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=6620631},
issn = {2157-8095},
year = {2013},
date = {2013-01-01},
booktitle = {2013 IEEE International Symposium on Information Theory},
pages = {2274--2278},
publisher = {IEEE},
address = {Istanbul},
abstract = {The asymptotic behavior of the mutual information (MI) at high signal-to-noise ratio (SNR) 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. Asymptotics of the minimum mean-square error (MMSE) are also developed. It is shown that for any input distribution, the MI and the MMSE have an asymptotic behavior proportional to a Gaussian Q-function, whose argument depends on the minimum Euclidean distance of the constellation and the SNR. Closed-form expressions for the coefficients of these Q-functions are calculated.},
keywords = {AWGN channels, discrete constellations, Entropy, Fading, Gaussian Q-function, high-SNR asymptotics, IP networks, least mean squares methods, minimum mean-square error, MMSE, Mutual information, scalar additive white Gaussian noise channel, Signal to noise ratio, signal-to-noise ratio, Upper bound},
pubstate = {published},
tppubtype = {inproceedings}
}

The asymptotic behavior of the mutual information (MI) at high signal-to-noise ratio (SNR) 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. Asymptotics of the minimum mean-square error (MMSE) are also developed. It is shown that for any input distribution, the MI and the MMSE have an asymptotic behavior proportional to a Gaussian Q-function, whose argument depends on the minimum Euclidean distance of the constellation and the SNR. Closed-form expressions for the coefficients of these Q-functions are calculated.