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

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，研究人员创建数学概念和数值方法的信息技术，通信工程和数据分析。研究者使用伪微分算子理论、时频分析、随机矩阵理论和Banach代数理论的工具，在严格规定的条件下产生具有严格建立的性质的高效数值算法。这项研究工作的一些具体主题是：（i）发展经典和量子信息理论中关键问题的理论框架。具体而言，研究人员认为，在时变通信和量子通信的信道容量问题;（ii）稀疏表示和压缩感知在X射线晶体学，通信和雷达。建立非线性压缩传感框架的初步步骤;（iii）从计算效率和快速算法的发展的观点出发，非交换调和分析和伪微分算子，特别注意非交换设置中算子的谱因子分解，以及它们在信号处理和无线通信中的应用。在已有的坚实成果的基础上，对本项目的成功寄予厚望在这项工作中提出的研究是尖端数学与最先进的工程的几个领域的婚姻，寻求将抽象和应用谐波分析的先进技术带到通信工程，信号处理，通过将现代谐波分析方法引入工程界，这一研究活动将在重要应用中取得进一步的进展和突破。同时，它将刺激应用数学的新研究领域，并为应用数学家和工程师之间的进一步互动铺平道路。这一研究工作对整个社会的回报是多方面的，从新的信息技术能力和复杂的工具，以处理今天的大量数据。有财务效率，以获得这将伴随着为公众提供更好和更多的通信服务。其他潜在的好处包括改进医学成像和生物医学工程的方法。除了该项目广泛的技术影响外，它还可以作为数学/工程前沿研究和教育的关键跨学科活动的典范。因此，这项研究工作有助于培养研究生数学发展和提高技能，是至关重要的，迫切需要在一个高科技为导向的社会。