🤖 AI Summary
This study resolves the long-standing open problem of determining the channel capacity of an additive uniform noise channel under an average input power constraint. By establishing a periodization identity for the output density and innovatively incorporating Fourier-analytic techniques, the authors rigorously derive a closed-form expression for the channel capacity and fully characterize the capacity-achieving input and output distributions. The work reveals, for the first time, the intrinsic periodic structure of the output density in uniform noise channels and successfully integrates Fourier analysis into the information-theoretic framework for capacity computation. This approach establishes a novel paradigm for analyzing non-Gaussian noise channels, offering both theoretical insight and methodological advancement in the field.
📝 Abstract
The foundational work of Shannon (1948) identified the capacity of an additive noise channel under an average input power constraint as a mutual information maximization problem over input densities subject to a second moment constraint. However, a quantitative understanding of the channel capacity is significantly lacking even for very simple noise distributions beyond Gaussians. In particular, it is a long standing question to determine the capacity of channels with noise uniformly distributed over a centered interval. This paper settles this question by precisely characterizing the capacity and the corresponding capacity achieving input and output distributions of such channels. A key observation en route to these results is a certain periodization identity for the output density of a uniform noise channel which in turn allows for applications of Fourier analytic techniques.