Optimum Power Control at Finite Blocklength

TL;DR

This paper characterizes the second-order term of the maximal channel coding rate at finite blocklengths under average power constraints for AWGN and fading channels, revealing it scales with n^{-1}ln n and proposing practical coding strategies.

Contribution

It provides a precise characterization of the second-order term in the finite blocklength regime for power-constrained channels and introduces a practical truncated channel inversion scheme.

Findings

Second-order term scales with n^{-1}ln n

Truncated channel inversion achieves the second-order rate in fading channels

Develops easy-to-evaluate approximations for maximal coding rate

Abstract

This paper investigates the maximal channel coding rate achievable at a given blocklength $n$ and error probability $\epsilon$, when the codewords are subject to a long-term (i.e., averaged-over-all-codeword) power constraint. The second-order term in the large-$n$ expansion of the maximal channel coding rate is characterized both for additive white Gaussian noise (AWGN) channels and for quasi-static fading channels with perfect channel state information available at both the transmitter and the receiver. It is shown that in both cases the second-order term is proportional to $\sqrt{n^{-1}\ln n}$. For the quasi-static fading case, this second-order term is achieved by truncated channel inversion, namely, by concatenating a dispersion-optimal code for an AWGN channel subject to a short-term power constraint, with a power controller that inverts the channel whenever the fading gain is above a certain threshold. Easy-to-evaluate approximations of the maximal channel coding rate are developed for both the AWGN and the quasi-static fading case.