### 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 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 language | English (US) |
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Article number | 8360156 |

Pages (from-to) | 6236-6256 |

Number of pages | 21 |

Journal | IEEE Transactions on Information Theory |

Volume | 64 |

Issue number | 9 |

DOIs | |

State | Published - Sep 2018 |

### All Science Journal Classification (ASJC) codes

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

### Keywords

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

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## Cite this

*IEEE Transactions on Information Theory*,

*64*(9), 6236-6256. [8360156]. https://doi.org/10.1109/TIT.2018.2837104