Computational Harmonic Analysis in Information Theory, Signal Processing, and Data Analysis

信息论、信号处理和数据分析中的计算谐波分析

• 批准号: 0811169
• 负责人: Thomas Strohmer
• 金额: $ 42万
• 依托单位: University of California-Davis
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-09-15 至 2012-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0811169&HistoricalAwards=false
• 关键词: Computational; Harmonic; Analysis; Information; Theory; Computational; Harmonic; Analysis; Information; Theory

In this research effort the investigator creates mathematical concepts and numerical methods for information technology, communications engineering, and data analysis. The investigator uses tools from pseudodifferential operator theory, time-frequency analysis, random matrix theory, and Banach algebra theory, yielding efficient numerical algorithms with rigorously-established properties under carefully stated conditions.Some concrete topics of this research effort are: (i) Development of a theoretical framework for key problems in classical and quantum information theory. Specifically, the investigator considers the channel capacity problem in time-varying communications and quantum communications;(ii) Sparse representations and compressed sensing in X-ray crystallography, communications, and radar. Initial steps toward building a framework for nonlinear compressed sensing; (iii) Noncommutative harmonic analysis and pseudodifferential operators from the point of view of computational efficiency and the development of fast algorithms.Particular attention is paid to spectral factorization for operators in a noncommutative setting, and their application in signal processing and wireless communications.Strong expectation for success of this project can be based on existing solid achievements by the investigator in each of the described areas.The research proposed in this effort is a marriage of several areas of cutting edge mathematics with state-of-the-art engineering, seeking to bring advanced techniques from abstract and applied harmonic analysis to communications engineering, signal processing, and data analysis in form of fast and efficient computational methods.By taking the modern harmonic analysis methodology into the engineering community this research activity will enable further advances and breakthroughs in important applications.At the same time it will stimulate new research areas in applied mathematics and pave the road for further interactions between applied mathematicians and engineers.The payoffs of this research effort for society at large are many, ranging from new information technology capabilities and sophisticated tools to deal with today's massive volumes of data.There are financial efficiencies to be gained by communications providers which will be accompanied by better and increased communications services for the public. Other potential benefits include improved methods for medical imaging and biomedical engineering.Beyond the project's broad technological impact, it serves as a model for the kind of cross-disciplinary activity critical for research and education at the mathematics/engineering frontier. Hence this research effort helps to train graduate students in mathematics to develop and enhance skills that are crucial and urgently needed in a high-tech oriented society.

在这项研究工作中，研究人员为信息技术，通信工程和数据分析创建了数学概念和数值方法。研究人员使用伪差异操作员理论，时频分析，随机矩阵理论和Banach代数理论的工具，在精心陈述的条件下具有严格确定的属性，得出有效的数值算法。这项研究工作的一种具体主题是：（i）（i）在类似和量子问题中的理论框架开发的构图。具体而言，研究人员考虑了随时间变化的通信和量子通信中的通道容量问题；（ii）X射线晶体学，通信和雷达中的稀疏表示和压缩感应。建立非线性压缩感应框架的初步步骤； （iii）从计算效率的角度来看和快速算法的发展，非共同的谐波分析和假数分化的运算符。在非交互环境中对运营商的频谱分解引起了特定的注意，以及他们在信号处理和无线沟通中的应用。基于现有的项目的努力。尖端数学几个领域与最先进的工程的婚姻，试图将高级技术从抽象和应用的谐波分析到通信工程，信号处理，信号处理和数据分析的形式，通过快速有效的计算方法形式。应用数学家和工程师之间的互动。这项研究工作的总体收益是许多人，从新的信息技术能力和复杂的工具到应对当今大量数据的相互作用，这些效率将由通信提供者所获得的财务效率，这些效率将伴随着更好的公众服务，并增加了公众的沟通服务。其他潜在的好处包括改进的医学成像和生物医学工程的方法。该项目的广泛技术影响，它是一种跨学科活动的模型，对于数学/工程领域的研究和教育至关重要。因此，这项研究工作有助于培训数学研究生，以发展和增强在面向高科技的社会中至关重要且急需的技能。