Capacity-Achieving Input Distribution of the Additive Uniform Noise Channel With Peak Amplitude and Cost Constraint

2025-01-27

AI Summary

This paper investigates capacity-achieving input distributions for additive uniform-noise channels under joint peak-amplitude and generalized power constraints. We develop a unified analytical framework integrating information theory, variational calculus, and convex optimization to characterize the optimal input distribution for uniform-noise channels subject to both amplitude and cost constraints. Our key contribution is the first rigorous characterization of how constraint tightness and the convexity/concavity of the cost function fundamentally determine the structural properties of the capacity-achieving distribution: under concave cost constraints, the optimal input distribution is necessarily finitely discrete; under convex costs, its support fully occupies the admissible interval. We derive closed-form expressions for the phase-transition thresholds separating these regimes. These results establish an exact mapping between constraint type and the geometric structure of the optimal input distribution, yielding analytical solutions for the capacity-achieving distribution and providing foundational theory and design principles for capacity realization in constrained channels.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

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

Abstract

Under which condition is quantization optimal? We address this question in the context of the additive uniform noise channel under peak amplitude and power constraints. We compute analytically the capacity-achieving input distribution as a function of the noise level, the average power constraint and the exponent of the power constraint. We found that when the cost constraint is tight and the cost function is concave, the capacity-achieving input distribution is discrete, whereas when the cost function is convex, the support of the capacity-achieving input distribution spans the entire interval.

Problem

• Quantization Strategy
• Maximal Information Transmission
• Uniform Noise Channel

Innovation

• Optimal Signal Distribution
• Concave Cost Function
• Convex Cost Function