Gabor Frames in Finite Dimensions
Gabor Frames in Finite Dimensions
复制标题
DOI：
10.1007/978-0-8176-8373-3_6
复制
发表时间：
2013
期刊：
影响因子：
--
通讯作者：
G. Pfander
中科院分区：
文献类型：
--
作者：
G. Pfander
在过去的30年里，Gabor框架在时频分析中得到了广泛的研究。它们通常用于科学和工程中，以从时间和频率上局部化的构建块合成信号或将信号分解为构建块。本章包含了有限维复向量空间上的Gabor框架的基本和独立的介绍。在这种情况下，我们给出了最大可能的一般性的Gabor框架的中心结果的初等证明，也就是说，我们认为Gabor框架对应于任意有限阿贝尔群中的格。在本章的后半部分，我们回顾了有限维Gabor系统几何的最新结果：其成员子集的线性独立性，它们的相互相干性，以及此类系统的限制等距性。我们将这些结果应用于稀疏信号的恢复，并讨论了有限维Gabor系统的几何上的开放性问题。
Gabor frames have been extensively studied in time-frequency analysis over the last 30 years. They are commonly used in science and engineering to synthesize signals from, or to decompose signals into, building blocks which are localized in time and frequency. This chapter contains a basic and self-contained introduction to Gabor frames on finite-dimensional complex vector spaces. In this setting, we give elementary proofs of the central results on Gabor frames in the greatest possible generality; that is, we consider Gabor frames corresponding to lattices in arbitrary finite Abelian groups. In the second half of this chapter, we review recent results on the geometry of Gabor systems in finite dimensions: the linear independence of subsets of its members, their mutual coherence, and the restricted isometry property for such systems. We apply these results to the recovery of sparse signals, and discuss open questions on the geometry of finite-dimensional Gabor systems.