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

## Author(s): Yang, Wei; Collins, Austin; Durisi, Giuseppe; Polyanskiy, Yury; Poor, H Vincent

DownloadTo refer to this page use:

`http://arks.princeton.edu/ark:/88435/pr1cf9j66b`

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 finiteblocklength channel-coding converse bound by Polyanskiy and Verdú, which involves the ratio of two Neyman-Pearson β functions (beta-beta converse bound); and 2) a novel beta-beta channel-coding 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 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. |

Publication Date: | 16-May-2018 |

Citation: | Yang, Wei, Collins, Austin, Durisi, Giuseppe, Polyanskiy, Yury, Poor, H Vincent. (2018). Beta–Beta Bounds: Finite-Blocklength Analog of the Golden Formula. IEEE Transactions on Information Theory, 64 (9), 6236 - 6256. doi:10.1109/tit.2018.2837104 |

DOI: | doi:10.1109/tit.2018.2837104 |

ISSN: | 0018-9448 |

EISSN: | 1557-9654 |

Pages: | 6236 - 6256 |

Type of Material: | Journal Article |

Journal/Proceeding Title: | IEEE Transactions on Information Theory |

Version: | Author's manuscript |

Items in OAR@Princeton are protected by copyright, with all rights reserved, unless otherwise indicated.