AI 中文总结
研究平均输入功率约束下均匀噪声信道容量问题，通过刻画容量及对应输入输出分布解决长期问题，关键利用均匀噪声信道输出密度周期化恒等式及傅里叶分析技术。
AI 中文摘要
香农（1948 年）的基础工作将平均输入功率约束下加性噪声信道的容量确定为在二阶矩约束下输入密度上的互信息最大化问题。然而，即使对于高斯分布以外非常简单的噪声分布，对信道容量的定量理解也严重不足。特别是，确定噪声在中心区间均匀分布的信道容量是一个长期存在的问题。本文通过精确刻画此类信道的容量以及相应的达到容量的输入和输出分布来解决这个问题。在得出这些结果的过程中的一个关键观察是均匀噪声信道输出密度的某种周期化恒等式，这反过来又允许应用傅里叶分析技术。
英文摘要
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.
