Coding in the Finite-Blocklength Regime: Bounds based on Laplace Integrals and their Asymptotic Approximations

TL;DR

This paper introduces new integral expressions and asymptotic approximations for finite blocklength bounds in communication channels, improving accuracy for short packet lengths where normal approximations fail.

Contribution

It develops a generalized Laplace integral approach to derive compact bounds and asymptotic formulas for finite blocklength regimes in various communication channels.

Findings

Provides new integral expressions for bounds

Offers simple asymptotic approximations

Improves accuracy for short packets

Abstract

In this paper we provide new compact integral expressions and associated simple asymptotic approximations for converse and achievability bounds in the finite blocklength regime. The chosen converse and random coding union bounds were taken from the recent work of Polyanskyi-Poor-Verdu, and are investigated under parallel AWGN channels, the AWGN channels, the BI-AWGN channel, and the BSC. The technique we use, which is a generalization of some recent results available from the literature, is to map the probabilities of interest into a Laplace integral, and then solve (or approximate) the integral by use of a steepest descent technique. The proposed results are particularly useful for short packet lengths, where the normal approximation may provide unreliable results.