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Channel Coding for Gaussian Channels with Multifaceted Power Constraints

Adeel Mahmood, Aaron B. Wagner

TL;DR

This paper develops a refined finite-blocklength theory for Gaussian channel coding under a broad family of multifaceted power constraints, extending beyond traditional mean or almost-sure cost models. The authors introduce a flexible cost framework with functions $f_i$ and thresholds $\Gamma_i$, and show that at rate $R = C(\Gamma) + \frac{r}{\sqrt{n}}$ the optimal error probability is characterized by a Gaussian-approximation expression averaged over a scalar distribution $P_U$ with $|\operatorname{supp}(P_U)| \le k+2$. The result unifies and extends prior second-order analyses (maximal cost, mean-variance cost) and demonstrates that the finite-blocklength performance is governed by a compact optimization over low-complexity discrete distributions, via Prokhorov-compactness and Bauer's maximization principle. The converse relies on a two-channel bound and a reduction to a finite-support extremal problem, while the achievability uses a sphere-based mixture input and a Berry-Esseen refinement to match the bound. Overall, the work provides a principled, general framework for analyzing refined power-constrained coding performance and opens avenues for discrete-memoryless extensions and other cost-approximation regimes.

Abstract

Motivated by refined asymptotic results based on the normal approximation, we study how higher-order coding performance depends on the mean power $Γ$ as well as on finer statistics of the input power. We introduce a multifaceted power model in which the expectation of an arbitrary 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.

Channel Coding for Gaussian Channels with Multifaceted Power Constraints

TL;DR

This paper develops a refined finite-blocklength theory for Gaussian channel coding under a broad family of multifaceted power constraints, extending beyond traditional mean or almost-sure cost models. The authors introduce a flexible cost framework with functions and thresholds , and show that at rate the optimal error probability is characterized by a Gaussian-approximation expression averaged over a scalar distribution with . The result unifies and extends prior second-order analyses (maximal cost, mean-variance cost) and demonstrates that the finite-blocklength performance is governed by a compact optimization over low-complexity discrete distributions, via Prokhorov-compactness and Bauer's maximization principle. The converse relies on a two-channel bound and a reduction to a finite-support extremal problem, while the achievability uses a sphere-based mixture input and a Berry-Esseen refinement to match the bound. Overall, the work provides a principled, general framework for analyzing refined power-constrained coding performance and opens avenues for discrete-memoryless extensions and other cost-approximation regimes.

Abstract

Motivated by refined asymptotic results 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 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.

Paper Structure

This paper contains 17 sections, 16 theorems, 145 equations.

Table of Contents

  1. Introduction
  2. Formulation and Result Statement
  3. Discussion
  4. Preliminaries to the Proof
  5. Converse
  6. Achievability
  7. Proof of Proposition \ref{['main_prop_converse']}
  8. Proof of Proposition \ref{['main_prop_achiev']}
  9. Concluding Remarks
  10. Excess Cost Probability Constraint
  11. Proof of Lemma \ref{['asymplemma']}
  12. Proof of Lemma \ref{['slightlybetterlemma']}
  13. Proof of Lemma \ref{['contrlemma']}
  14. Proof of Lemma \ref{['log_dist_eq']}
  15. Proof of Lemma \ref{['lemma_gen_lower_bnd_wo_opt']}
  16. ...and 2 more sections

Key Result

Theorem 1

Let $W(\cdot|\mathbf{x}) = \mathcal{N}(\mathbf{x}, N I_n)$ and let $r$ be any real number. Given any $f = (f_1, \ldots, f_k)$ and $\mathbf{\Gamma} = (\Gamma, \Gamma_1, \ldots, \Gamma_k)$, let $\mathcal{C}_{n, R, f, \mathbf{\Gamma}}$ be the set of $(n, R)$ channel codes as defined in Definition defco where

Theorems & Definitions (21)

  • Definition 1
  • Definition 2: Admissible distributions
  • Definition 3
  • Theorem 1
  • Definition 4
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • ...and 11 more