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

TL;DR

This paper analytically determines the optimal input distribution for an additive uniform noise channel under peak and power constraints, revealing conditions under which the distribution is discrete or continuous.

Contribution

It provides a comprehensive analysis of the capacity-achieving input distribution considering different cost functions and constraints, filling a gap in channel capacity theory.

Findings

Discrete distribution is optimal under tight, concave cost constraints.

Continuous distribution spans entire interval under convex cost functions.

Analytical expressions depend on noise level and constraint parameters.

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.