Nonlinear Signal Processing and Distributed Optimal Control using Frames and Operators Algebras

使用框架和算子代数的非线性信号处理和分布式最优控制

• 批准号: 1109498
• 负责人: Radu Balan
• 金额: $ 25.05万
• 依托单位: University of Maryland, College Park
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-10-01 至 2015-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1109498&HistoricalAwards=false
• 关键词: Nonlinear; Signal; Processing; Distributed; Optimal

BalanDMS-1109498 The investigator and his colleagues study new methods to recover a signal from a nonlinear processing scheme. Recently two far-reaching discoveries have been made that connected the nonlinear information (magnitudes of frame coefficients) to certain scalar products in larger embedding spaces. Thus the initial problem, which is fundamentally nonlinear, is recast into a linear reconstruction problem coupled with a rank-one approximation problem. When the linear redundant representation is associated with a group representation (such as Weyl-Heisenberg, or windowed Fourier transform), then the relevant tensor operators inherit this invariance property. Thus a fast (nonlinear) reconstruction algorithm is possible. This approach suggests a new signal representation model, where signals are not represented simply by vectors in a Hilbert space, but rather by operators in a larger dimensional Hilbert-Schmidt like-space, similar to the quantum state theory. These methods use results from a wide range of mathematical areas such as harmonic analysis, operator theory, and polynomial algebras. Results of this project have a practical application to areas such as signal processing, optical communication, quantum computing, and X-ray crystallography. Besides advancing the scientific understanding in applied harmonic analysis, this project broadens the two-way communication between mathematics and electrical engineering while promoting teaching, training and learning. The investigator is training graduate students for a globally competitive STEM force through his contacts with industry and international research labs. The project is supported by the Division of Mathematical Sciences and the Division of Computing and Communication Foundations.

BalanDMS-1109498 研究人员和他的同事研究了从非线性处理方案中恢复信号的新方法。 最近取得了两个影响深远的发现，将非线性信息（帧系数的大小）与较大嵌入空间中的某些标量积联系起来。 因此，本质上是非线性的初始问题被重新转换为线性重构问题以及一阶近似问题。 当线性冗余表示与群表示（例如 Weyl-Heisenberg 或加窗傅立叶变换）相关联时，相关张量算子继承此不变性。 因此，快速（非线性）重建算法是可能的。 这种方法提出了一种新的信号表示模型，其中信号不是简单地由希尔伯特空间中的向量表示，而是由更大维的希尔伯特-施密特类空间中的算子表示，类似于量子态理论。 这些方法使用调和分析、算子理论和多项式代数等广泛数学领域的结果。 该项目的成果在信号处理、光通信、量子计算和X射线晶体学等领域具有实际应用。 除了增进对应用谐波分析的科学理解外，该项目还拓宽了数学和电气工程之间的双向交流，同时促进了教学、培训和学习。 该研究人员正在通过与行业和国际研究实验室的联系，为研究生培养具有全球竞争力的 STEM 力量。 该项目得到了数学科学部和计算与通信基础部的支持。