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

Wei Yang, Austin Collins, Giuseppe Durisi, Yury Polyanskiy, H. Vincent Poor

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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 Verdú, 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 Verdú's 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.

Original languageEnglish (US)
Article number8360156
Pages (from-to)6236-6256
Number of pages21
JournalIEEE Transactions on Information Theory
Issue number9
StatePublished - Sep 2018

All Science Journal Classification (ASJC) codes

  • Information Systems
  • Computer Science Applications
  • Library and Information Sciences


  • Channel coding
  • achievability bound
  • finite-blocklength regime
  • golden formula
  • hypothesis testing


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