论文标题
平均环形晶格的连贯性
On average coherence of cyclotomic lattices
论文作者
论文摘要
我们通过类比在欧几里得空间中与这些框架上的这些概念引入了最大和平均连贯性。具有低相干性的晶格在信号处理中可能会引起人们的关注，而正交性缺陷的晶格对球体包装问题感兴趣。因此，相干性和正交性缺陷是晶格无法正交的程度的不同度量，并最大化其商（对尺寸相对于尺寸的最小向量的数量进行了归一化）提供了具有特别良好优化特性的晶格。虽然正交性缺陷是各种晶格家庭的相当古典且经过充分研究的概念，但连贯性并非如此。我们研究了来自环数字字段中整数环的一系列代数晶格家族的相干性能，证明了它们平均连贯性的简单公式。我们查看了此类晶格的一些示例，并将它们的相干性能与标准根晶格的相干性进行了比较。
We introduce maximal and average coherence on lattices by analogy with these notions on frames in Euclidean spaces. Lattices with low coherence can be of interest in signal processing, whereas lattices with high orthogonality defect are of interest in sphere packing problems. As such, coherence and orthogonality defect are different measures of the extent to which a lattice fails to be orthogonal, and maximizing their quotient (normalized for the number of minimal vectors with respect to dimension) gives lattices with particularly good optimization properties. While orthogonality defect is a fairly classical and well-studied notion on various families of lattices, coherence is not. We investigate coherence properties of a nice family of algebraic lattices coming from rings of integers in cyclotomic number fields, proving a simple formula for their average coherence. We look at some examples of such lattices and compare their coherence properties to those of the standard root lattices.
