有限块长率失真理论：伯努利源与汉明失真：教程
Finite Block Length Rate-Distortion Theory for the Bernoulli Source with Hamming Distortion: A Tutorial
AI总结：
本文介绍了伯努利源在有限块长下的率失真理论，推导了经典率失真函数并讨论了有限块长对率失真性能的影响。
摘要：
失真数据压缩是现代通信和存储系统的核心。香农的率失真理论提供了在给定保真度下源可压缩的极限，但其假设无限长的块长度，这在实践中从未实现。本文提供了一个自包含的教程，介绍最简单的非平凡源——伯努利(p)序列与汉明失真下的率失真理论。我们从基本原理推导出经典的率失真函数RD = Hp - HD，通过Blahut-Arimoto算法说明其计算，然后发展有限块长的改进，描述最小可实现率如何随着块长度n的增长接近香农极限。这一改进中的核心量是率失真扩散V(D)，它决定了在有限块长度下操作的O(1/√n)惩罚。我们通过配套的Python脚本生成数值示例和图表来配合所有理论发展。
英文摘要：
Lossy data compression lies at the heart of modern communication and storage systems. Shannon's rate-distortion theory provides the fundamental limit on how much a source can be compressed at a given fidelity, but it assumes infinitely long block lengths that are never realized in practice. We present a self-contained tutorial on rate-distortion theory for the simplest non-trivial source: a Bernoulli$(p)$ sequence with Hamming distortion. We derive the classical rate-distortion function $RD = Hp - HD$ from first principles, illustrate its computation via the Blahut-Arimoto algorithm, and then develop the finite block length refinements that characterize how the minimum achievable rate approaches the Shannon limit as the block length $n$ grows. The central quantity in this refinement is the \emph{rate-distortion dispersion} $V(D)$, which governs the $O(1/\sqrt{n})$ penalty for operating at finite block lengths. We accompany all theoretical developments with numerical examples and figures generated by accompanying Python scripts.
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