# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1706.05972