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

##### Author(s)

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

##### Date issued

2018-05##### Department

Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science##### Journal

IEEE Transactions on Information Theory

##### Publisher

Institute of Electrical and Electronics Engineers (IEEE)

##### Citation

Yang, Wei et al. "Beta–Beta Bounds: Finite-Blocklength Analog of the Golden Formula." IEEE Transactions on Information Theory 64, 9 (September 2018): 6236 - 6256 © 1963-2012 IEEE.

Version: Author's final manuscript

##### ISSN

0018-9448

1557-9654