Channel Coding for Gaussian Channels with Multifaceted Power Constraints

TL;DR

This paper analyzes how higher-order coding performance over Gaussian channels depends on detailed power statistics, introducing a generalized power constraint model and providing exact error probability characterizations.

Contribution

It introduces a multifaceted power model that generalizes existing constraints and characterizes the minimum error probability with higher-order power statistics.

Findings

Exact characterization of error probability as a function of first- and second-order rates.

Generalization of power constraints to include arbitrary functions of normalized power.

Provides benchmarks for practical modulation schemes with complex amplitude and shaping.

Abstract

Through refined asymptotic analysis based on the normal approximation, we study how higher-order coding performance depends on the mean power as well as on finer statistics of the input power. We introduce a multifaceted power model in which the expectation of an arbitrary (but finite) number of arbitrary functions of the normalized average power is constrained. The framework generalizes existing models, recovering the standard maximal and expected power constraints and the recent mean and variance constraint as special cases. Under certain growth and continuity assumptions on the functions, our main theorem gives an exact characterization of the minimum average error probability for Gaussian channels as a function of the first- and second-order coding rates. The converse proof reduces the code design problem to minimization over a compact (under the Prokhorov metric) set of probability distributions, characterizes the extreme points of this set and invokes the Bauer's maximization principle. Our results for the multifaceted power model serve as more precise benchmarks for practical modulation schemes with multiple amplitude levels, probabilistic shaping and nonuniform constellation geometries.