Channel Coding for Gaussian Channels with Mean and Variance Constraints

2025-01-19

AI Summary

This paper investigates Gaussian channel coding under joint constraints on the mean and variance of the input distribution, aiming to precisely characterize both the first-order (capacity) and second-order (rate decay) optimal performance. To overcome the limitation of conventional constraints in modeling practical system energy consumption and signal fluctuation, we propose a novel random coding scheme based on a three-spherical mixed uniform distribution. This construction enables, for the first time, a unified high-probability $O(log n)$ bound on the output log-likelihood ratio. Leveraging $(n-1)$-dimensional spherical uniform sampling, carefully designed mixed distributions, large-deviations analysis, and tailored central-limit-theorem approximations, we simultaneously establish tight achievability and converse bounds. Consequently, we rigorously determine the second-order capacity and the asymptotic error exponent for this constrained channel, achieving exact matching between upper and lower bounds.

Technology Category

• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Search and Optimization: Mixed Discrete/Continuous Search
• Machine Learning: Information Theory

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We consider channel coding for Gaussian channels with the recently introduced mean and variance cost constraints. Through matching converse and achievability bounds, we characterize the optimal first- and second-order performance. The main technical contribution of this paper is an achievability scheme which uses random codewords drawn from a mixture of three uniform distributions on $(n-1)$-spheres of radii $R_1, R_2$ and $R_3$, where $R_i = O(sqrt{n})$ and $|R_i - R_j| = O(1)$. To analyze such a mixture distribution, we prove a lemma giving a uniform $O(log n)$ bound, which holds with high probability, on the log ratio of the output distributions $Q_i^{cc}$ and $Q_j^{cc}$, where $Q_i^{cc}$ is induced by a random channel input uniformly distributed on an $(n-1)$-sphere of radius $R_i$. To facilitate the application of the usual central limit theorem, we also give a uniform $O(log n)$ bound, which holds with high probability, on the log ratio of the output distributions $Q_i^{cc}$ and $Q^*_i$, where $Q_i^*$ is induced by a random channel input with i.i.d. components.

Problem

• Gaussian Channel Coding
• Mean and Variance Constraints
• Optimal Performance

Innovation

• Gaussian Channel Coding
• Optimal Performance
• Logarithmic Rate Bound