部分无序介质中Shannon码的冗余模式与波衍射模式的关系
Relations between redundancy patterns of the Shannon code and wave diffraction patterns of partially disordered media
摘要：
作为块长度$n$的函数的香农码的平均冗余度$R_n$已知表现出两种非常不同类型的行为，这取决于源的某些参数的合理性或不合理性：当$n$无限增长时，它要么收敛到1/2，要么它可能具有非零的，振荡的，对于所有大的$n$，在值1/2附近的（准）周期性图案。在本文中，我们试图通过与波传播的物理学领域，特别是散射和衍射的基本理论进行类比，来深入了解$R_n$的这种不稳定行为。结果表明，晶体形成的波衍射图样有两种行为，它们与$R_n$的两种图样相对应。当晶体是完美的，衍射强度谱表现出非常尖锐的峰，也称为。布拉格峰，在全相长干涉的波长处。这些波长对应于$R_n$的振荡模式的谐波的频率。另一方面，当晶体不完美且其结构中存在相当程度的无序时，布拉格峰消失，并且该模式的行为类似于$R_n$收敛的模式。
英文摘要：
The average redundancy of the Shannon code, $R_n$, as a function of the block length $n$, is known to exhibit two very different types of behavior, depending on the rationality or irrationality of certain parameters of the source: It either converges to 1/2 as $n$ grows without bound, or it may have a non-vanishing, oscillatory, (quasi-) periodic pattern around the value 1/2 for all large $n$. In this paper, we make an attempt to shed some insight into this erratic behavior of $R_n$, by drawing an analogy with the realm of physics of wave propagation, in particular, the elementary theory of scattering and diffraction. It turns out that there are two types of behavior of wave diffraction patterns formed by crystals, which are correspondingly analogous to the two types of patterns of $R_n$. When the crystal is perfect, the diffraction intensity spectrum exhibits very sharp peaks, a.k.a. Bragg peaks, at wavelengths of full constructive interference. These wavelengths correspond to the frequencies of the harmonic waves of the oscillatory mode of $R_n$. On the other hand, when the crystal is imperfect and there is a considerable degree of disorder in its structure, the Bragg peaks disappear, and the behavior of this mode is analogous to the one where $R_n$ is convergent.
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