几何均匀框架
Geometrically Uniform Frames
摘要：
我们引入了一类新的框架，称为几何一致框架（GU）的强对称性，这是定义在一个阿贝尔群的酉矩阵和生成的一个单一的生成向量。GU框架的概念，然后扩展到复合GU（CGU）框架，这是由一个阿贝尔群的酉矩阵使用多个生成向量。与GU框架相关联的对偶框架向量和规范紧框架向量被示出为GU，并且因此由单个生成向量生成，该生成向量可以使用在框架的生成群上定义的傅里叶变换来非常有效地计算。类似地，与CGU框架相关联的对偶框架向量和规范紧框架向量被示出为CGU。考虑了从GU帧中删除单个或多个元素的影响。一个系统的方法，从一组给定的框架向量，不是GU的最佳GU框架也开发。最后，讨论了GU框架的欧氏距离性质，并在酉矩阵的阿贝尔群上给出了GU框架具有严格正距离谱的条件，而与生成向量无关.
英文摘要：
We introduce a new class of frames with strong symmetry properties called geometrically uniform frames (GU), that are defined over an abelian group of unitary matrices and are generated by a single generating vector. The notion of GU frames is then extended to compound GU (CGU) frames which are generated by an abelian group of unitary matrices using multiple generating vectors. The dual frame vectors and canonical tight frame vectors associated with GU frames are shown to be GU and therefore generated by a single generating vector, which can be computed very efficiently using a Fourier transform defined over the generating group of the frame. Similarly, the dual frame vectors and canonical tight frame vectors associated with CGU frames are shown to be CGU. The impact of removing single or multiple elements from a GU frame is considered. A systematic method for constructing optimal GU frames from a given set of frame vectors that are not GU is also developed. Finally, the Euclidean distance properties of GU frames are discussed and conditions are derived on the abelian group of unitary matrices to yield GU frames with strictly positive distance spectrum irrespective of the generating vector.
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