一个经典信息论公式的初等证明
An Elementary Proof of a Classical Information-Theoretic Formula
摘要：
Shannon的著名信息理论公式将具有平稳高斯输入的白高斯信道的互信息率表示为信道输入的功率谱密度的简单函数的积分。本文给出了这个经典公式的一个严格而又初等的证明。相对于所有传统的方法，这要么依赖于沉重的数学机器，或不得不诉诸一些“外部”的结果，我们的证明，这取决于最近证明的采样定理，是基本的和独立的，只使用一些众所周知的事实，从基本微积分和矩阵理论。
英文摘要：
A renowned information-theoretic formula by Shannon expresses the mutual information rate of a white Gaussian channel with a stationary Gaussian input as an integral of a simple function of the power spectral density of the channel input. We give in this paper a rigorous yet elementary proof of this classical formula. As opposed to all the conventional approaches, which either rely on heavy mathematical machineries or have to resort to some "external" results, our proof, which hinges on a recently proven sampling theorem, is elementary and self-contained, only using some well-known facts from basic calculus and matrix theory.
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