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

研究人员BalanDMS-1109498和他的同事们研究了从非线性处理方案中恢复信号的新方法。最近，在更大的嵌入空间中，有两个深远的发现将非线性信息(帧系数的大小)与某些标量积联系起来。因此，初始问题基本上是非线性的，被重塑为一个线性重建问题和一个一阶逼近问题。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，相关张量算子继承这种不变性。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矢量来表示，而是由更大维类希尔伯特-施密特空间中的算符来表示，类似于量子态理论。这些方法使用了调和分析、算子理论和多项式代数等广泛数学领域的结果。该项目的研究成果在信号处理、光通信、量子计算、X射线结晶学等领域具有实际应用价值。除了提高对应用谐波分析的科学认识外，该项目还扩大了数学与电气工程之间的双向交流，促进了教学、培训和学习。这位研究员正在通过他与行业和国际研究实验室的联系，为具有全球竞争力的STEM力量培训研究生。该项目得到数学科学司和计算和通信基金会司的支持。

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.