Slepian functions and their use in signal estimation and spectral analysis
Slepian functions and their use in signal estimation and spectral analysis
复制标题
Slepian 函数及其在信号估计和频谱分析中的应用
DOI：
--
复制
发表时间：
2009
期刊：
影响因子：
--
通讯作者：
F. Simons
中科院分区：
文献类型：
--
作者：
F. Simons
这是一个众所周知的事实，即数学函数的时间限制（或空间限制）不能同时带宽限制（在频率）。然而，测量和计算的有限精度无疑限制了我们的观察和建模科学数据，我们通常只能访问或只对时间或空间有限的研究区域感兴趣。在地球科学中，我们可能对仅在某个时间间隔上定义的时间序列的频谱建模感兴趣，或者我们可能希望使用有效的带宽限制测量设备来描述观察到的特定地理区域。显然，如果能够找到“空间光谱”集中的函数基础，即同时在两个领域“局部化”的函数基础，将有助于分析和表示这类科学数据。在这里，我们给出了一个理论概述的一个特定的方法，这个“浓度”的问题，最初提出的时间序列的Slepian和同事，在20世纪60年代。我们展示了这个框架如何导致实用的算法和统计性能的方法，用于分析信号及其功率谱在一个和两个维度，并在一个球体的表面上。
It is a well-known fact that mathematical functions that are timelimited (or spacelimited) cannot be simultaneously bandlimited (in frequency). Yet the finite precision of measurement and computation unavoidably bandlimits our observation and modeling scientific data, and we often only have access to, or are only interested in, a study area that is temporally or spatially bounded. In the geosciences we may be interested in spectrally modeling a time series defined only on a certain interval, or we may want to characterize a specific geographical area observed using an effectively bandlimited measurement device. It is clear that analyzing and representing scientific data of this kind will be facilitated if a basis of functions can be found that are "spatiospectrally" concentrated, i.e. "localized" in both domains at the same time. Here, we give a theoretical overview of one particular approach to this "concentration" problem, as originally proposed for time series by Slepian and coworkers, in the 1960s. We show how this framework leads to practical algorithms and statistically performant methods for the analysis of signals and their power spectra in one and two dimensions, and on the surface of a sphere.