A Beta-Beta Achievability Bound with Applications

TL;DR

This paper introduces a new achievability bound based on Neyman-Pearson β functions, simplifying analysis for complex channels and deriving tight bounds for various communication scenarios.

Contribution

It presents a novel achievability bound dual to a known converse, applicable to channels with non-product output distributions, and derives several key bounds in information theory.

Findings

Derived bounds for channel dispersion in non-Gaussian noise channels

Established the channel dispersion for exponential-noise channels

Provided a tight second-order expansion for the minimum energy per bit in AWGN channels

Abstract

A channel coding achievability bound expressed in terms of the ratio between two Neyman-Pearson $\beta$ functions is proposed. This bound is the dual of a converse bound established earlier by Polyanskiy and Verd\'{u} (2014). The new bound turns out to simplify considerably the analysis in situations where the channel output distribution is not a product distribution, for example due to a cost constraint or a structural constraint (such as orthogonality or constant composition) on the channel inputs. Connections to existing bounds in the literature are discussed. The bound is then used to derive 1) an achievability bound on the channel dispersion of additive non-Gaussian noise channels with random Gaussian codebooks, 2) the channel dispersion of the exponential-noise channel, 3) a second-order expansion for the minimum energy per bit of an AWGN channel, and 4) a lower bound on the maximum coding rate of a multiple-input multiple-output Rayleigh-fading channel with perfect channel state information at the receiver, which is the tightest known achievability result.