Beta-beta bounds: Finite-blocklength analog of the golden formula
Journal article, 2018
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 (2014), which involves the ratio of two Neyman-Pearson β functions (beta-beta converse bound); and 2) a novel beta-beta channelcoding achievability bound, expressed again as the ratio of two Neyman-Pearson β 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 (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 exponentialnoise 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.