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CIF:Small: Ramanujan-sums in signal representation

CIF:Small：信号表示中的拉马努金求和

基本信息

批准号：
1712633
负责人：
Palghat Vaidyanathan
金额：
$ 45万
依托单位：
California Institute of Technology
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-07-01 至 2021-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1712633&HistoricalAwards=false
关键词：
CIF Small Ramanujan sums signal

项目摘要

This research explores new directions in the representation of signals, based on certainmathematical ideas introduced by Ramanujan many decades ago. In particular the applicability ofRamanujan-sums in the representation of digital signals is studied in detail. Signals acquired andstored in many scientific areas such as speech, music, medical, genomic, and finance oftencontain hidden periodic patterns bearing important information. These are difficult to identify andlocalize because of the large volume of data, and extreme localization of information.Conventional representations of signals based on Fourier and wavelet transforms are not efficientfor this purpose. This research involves the use of Ramanujan-sums to develop newrepresentations that are efficient and economical. These offer insights into the role ofmathematics, especially number theory, in representation of signals in science and technology,with emphasis on particular hidden structures such as periodicities. Such representations areexpected to have significant impact in scientific applications involving large data bases.More specifically, the research goes deep into Ramanujan subspaces, nested periodic subspaces,and Ramanujan filter banks, which are crucial to the representation of periodic sequences. Thisenables one to address some fundamental research problems in signal processing. This includesthe minimum information that should be gathered from a discrete-time signal in order to identifymultiple periodicities, the design of digital filter banks that serve as projection operators ontoperiodicity subspaces, and optimal design of dictionaries of atoms to represent sums of periodicsignals with unknown integer periods. The research also involves the theory and implementationof gridless methods to achieve super resolution in period-estimation for continuous-time signals,and extension to two- and higher-dimensional signals.

这项研究基于Ramanujan几十年前引入的某些数学思想，探索了信号表示的新方向。特别研究了Ramanujan和在数字信号表示中的适用性。在许多科学领域中获取和存储的信号，如语音、音乐、医学、基因组和金融，往往包含隐藏的周期模式，承载着重要的信息。传统的基于傅里叶变换和小波变换的信号表示方法不能有效地识别和定位这些信号，因为数据量大，信息的局部化程度极高。这项研究涉及到使用Ramanujan和来开发高效和经济的新表示。这些课程提供了对数学，特别是数论在科学和技术中表示信号方面的作用的洞察，并强调了特定的隐藏结构，如周期性。更具体地说，该研究深入到Ramanujan子空间、嵌套周期子空间和Ramanujan滤波器组，它们对于周期序列的表示是至关重要的。这使得人们能够解决信号处理中的一些基础研究问题。这包括为了识别多个周期而应从离散时间信号中收集的最少信息，作为周期子空间的投影算子的数字滤波器组的设计，以及表示具有未知整数周期的周期信号和的原子字典的优化设计。该研究还涉及在连续时间信号的周期估计中实现超分辨率的无网格方法的理论和实现，并将其推广到二维和更高维信号。