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.

这项研究探讨了基于Ramanujan几十年前引入的某些有女性思想的信号表示的新方向。详细研究了Ramanujan-sums在数字信号表示中的适用性。在许多科学领域获取并存储的信号，例如语音，音乐，医学，基因组和财务，tencontain隐藏的周期性模式包含重要信息。由于数据的大量数据和信息的极端本地化，这些信号的概要表示基于傅立叶和小波变换，因此很难识别和全国化。这项研究涉及使用Ramanujan-sums开发有效且经济的新陈述。这些提供了对教学和数字理论在科学和技术中信号表示的作用的见解，重点是特定的隐藏结构，例如周期性。预测这种表示对涉及大数据库的科学应用产生了重大影响。更具体地说，该研究深入涉及Ramanujan子空间，嵌套的周期性子空间和Ramanujan滤波器库，这对于周期序列的表示至关重要。这是解决信号处理中一些基本研究问题的一种。为了识别识别周期性的设计，应从离散的时间信号中收集的最低信息，这些数字过滤库的设计用作投影操作员的本质上，以及原子词典的最佳设计，以代表周期性的数字和不知名的integer期间。这项研究还涉及无网方法的理论和实施，以实现连续时间信号的超级估计，并扩展到二维信号。