Beta-Beta Bounds: Finite-Blocklength Analog of the Golden Formula

TL;DR

This paper extends the golden formula to finite-blocklength regimes using beta-beta bounds, providing new converse and achievability bounds for channel coding, with applications to wideband-slope, exponential noise, and MIMO channels.

Contribution

It introduces a novel finite-blocklength extension of the golden formula via beta-beta bounds, unifying converse and achievability results in information theory.

Findings

Derived a finite-blocklength extension of Verdú's wideband-slope approximation.

Provided the tightest finite-blocklength achievability bound for MIMO Rayleigh-fading channels.

Characterized channel dispersion for additive exponential-noise channels.

Abstract

It is well known that the mutual information between two random variables can be expressed as the difference of two relative entropies that depend on an auxiliary distribution, a relation sometimes referred to as the golden formula. This paper is concerned with a finite-blocklength extension of this relation. This extension consists of two elements: 1) a finite-blocklength channel-coding converse bound by Polyanskiy and Verd\'{u} (2014), which involves the ratio of two Neyman-Pearson $\beta$ functions (beta-beta converse bound); and 2) a novel beta-beta channel-coding achievability bound, expressed again as the ratio of two Neyman-Pearson $\beta$ functions. To demonstrate the usefulness of this finite-blocklength extension of the golden formula, the beta-beta achievability and converse bounds are used to obtain a finite-blocklength extension of Verd\'{u}'s (2002) wideband-slope approximation. The proof parallels the derivation of the latter, with the beta-beta bounds used in place of the golden formula. The beta-beta (achievability) bound is also shown to be useful in cases where the capacity-achieving output distribution is not a product distribution due to, e.g., a cost constraint or structural constraints on the codebook, such as orthogonality or constant composition. As an example, the bound is used to characterize the channel dispersion of the additive exponential-noise channel and to obtain a finite-blocklength achievability bound (the tightest to date) for multiple-input multiple-output Rayleigh-fading channels with perfect channel state information at the receiver.