Second Order Asymptotics for Communication under Strong Asynchronism

TL;DR

This paper investigates how the second order rate terms in asynchronous communication are affected by sampling rate and delay constraints in the finite blocklength regime, revealing sensitivity to these parameters unlike capacity.

Contribution

It characterizes the second order asymptotics of asynchronous communication under finite blocklength, highlighting the impact of sampling rate and delay constraints on achievable rates.

Findings

Second order term is of order Θ(1/ρ) when d=n and ρ=O(1/√n).

Reliable communication requires ρ=ω(1/n) for second order terms to match full sampling case.

Second order term is O(√n) with dispersion depending on asynchronism level when ρ=ω(1/√n).

Abstract

The capacity under strong asynchronism was recently shown to be essentially unaffected by the imposed output sampling rate $\rho$ and decoding delay $d$---the elapsed time between when information is available at the transmitter and when it is decoded. This paper examines this result in the finite blocklength regime and shows that, by contrast with capacity, the second order term in the rate expansion is sensitive to both parameters. When the receiver must exactly locate the sent codeword, that is $d=n$ where $n$ denotes blocklength, the second order term in the rate expansion is of order $\Theta(1/\rho)$ for any $\rho=O(1/\sqrt{n})$---and $\rho =\omega(1/n)$ for otherwise reliable communication is impossible. However, if $\rho=\omega(1/\sqrt{n})$ then the second order term is the same as under full sampling and is given by a standard $O(\sqrt{n})$ term whose dispersion constant only depends on the level of asynchronism. This second order term also corresponds to the case of the slightly relaxed delay constraint $d\leq n(1+o(1))$ for any $\rho=\omega(1/n)$.