Dispersion of Infinite Constellations in Fast Fading Channels

TL;DR

This paper extends Poltyrev's framework to fast fading channels with CSI at the receiver, analyzing the dispersion and finite block length effects on the optimal codeword density and capacity.

Contribution

It derives the channel dispersion for fast fading channels with CSI, linking it to the high SNR power constrained channel and analyzing finite block length performance.

Findings

Derived the dispersion for fast fading channels with CSI.

Established the asymptotic gap between achievable NLD and capacity.

Connected dispersion results to error exponents in fading channels.

Abstract

In this work we extend the setting of communication without power constraint, proposed by Poltyrev, to fast fading channels with channel state information (CSI) at the receiver. The optimal codewords density, or actually the optimal normalized log density (NLD), is considered. Poltyrev's capacity for this channel is the highest achievable NLD, at possibly large block length, that guarantees a vanishing error probability. For a given finite block length n and a fixed error probability, there is a gap between the highest achievable NLD and Poltyrev's capacity. As in other channels, this gap asymptotically vanishes as the square root of the channel dispersion V over n, multiplied by the inverse Q-function of the allowed error probability. This dispersion, derived in the paper, equals the dispersion of the power constrained fast fading channel at the high SNR regime. Connections to the error exponent of the peak power constrained fading channel are also discussed.