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首页> 外文期刊>Information Theory, IEEE Transactions on >Relations Between Redundancy Patterns of the Shannon Code and Wave Diffraction Patterns of Partially Disordered Media
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Relations Between Redundancy Patterns of the Shannon Code and Wave Diffraction Patterns of Partially Disordered Media

机译：香农码的冗余模式与部分无序介质的波衍射模式之间的关系

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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 nonvanishing, 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.
机译：香农码的平均冗余$ R_ {n} $作为块长度$ n $的函数，根据源的某些参数的合理性或不合理性，它表现出两种截然不同的行为类型：它要么随着$ n $的增长而无收敛收敛到$ 1/2 $，要么对于所有大的$ n $而言，在值$ 1/2 $附近可能都没有消失的，振荡的（（准））周期性模式。在本文中，我们试图通过与波传播的物理领域（尤其是散射和衍射的基本理论）进行类比，以深入了解$ R_ {n} $的这种不稳定行为。事实证明，由晶体形成的波衍射图案有两种类型的行为，它们分别类似于$ R_ {n} $的两种类型的图案。当晶体是完美的时，衍射强度谱在完全相长干涉的波长处表现出非常尖锐的峰，也称为布拉格峰。这些波长对应于振荡模式$ R_ {n} $的谐波频率。另一方面，当晶体不完美并且其结构存在相当程度的无序时，布拉格峰消失，并且该模式的行为类似于其中$ R_ {n} $收敛的模式。

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