Heisenberg-Weyl Groups, Sequences, and Codes

海森堡-韦尔群、序列和代码

• 批准号: 0701226
• 负责人: Arthur Calderbank
• 金额: $ 18万
• 依托单位: Princeton University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-09-01 至 2010-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0701226&HistoricalAwards=false
• 关键词: Heisenberg; Weyl; Groups; Sequences; Codes

This proposal develops a new mathematical framework for detection and estimation in applications as diverse as remote sensing and quantum computing. The application domain defines one Heisenberg-Weyl group, the signal design domain defines a second group, and the new mathematical framework results from expressing one group in terms of the other. Classical algebraic error correcting codes find new application as phase coded waveforms and the geometry of codewords provides fundamental limits on detection and estimation. In the second order Reed Muller code, pairs of phase coded waveforms associated with Rudin-Shapiro sequences enable instantaneous radar polarimetry which provides concurrent and coherent access to all four dimensions of the polarization scattering matrix rather than serial and non-coherent access as is the case today. The value of this new primitive is detection based on simple statistics in four dimensions rather than detection based on complicated statistics applied to lossy one-dimensional projections.As a training program, this proposal provides graduate students with an opportunity to create new mathematics that brings fundamental change to an important application domain. It will increase the pool of mathematicians who appreciate the power of mathematics and the important role that it plays in engineering disciplines. Graduate students will learn to flourish in the different cultures associated with the different academic disciplines, develop an ability to bridge different worlds and this experience will advantage them in both academic and non-academic careers. This proposal will likely result in technology transfer to industrial partners, including Raytheon, who have confirmed the value of instantaneous radar polarimetry in independent analysis. This engagement and others may lead to additional internship opportunities for US citizens and permanent residents.

这一提议为遥感和量子计算等各种应用中的检测和估计开发了一个新的数学框架。应用领域定义了一个Heisenberg-Weyl群，信号设计领域定义了第二个群，新的数学框架通过用一个群来表示另一个群而产生。经典的代数纠错码作为相位编码波形得到了新的应用，而码字的几何形状为检测和估计提供了基本的限制。在二阶Reed Muller码中，与Rudin-Shapiro序列相关联的成对的相位编码波形使得瞬时雷达极化能够提供对极化散射矩阵的所有四个维度的并行和相干访问，而不是像今天的情况那样的连续和非相干访问。这种新基元的价值是基于四维简单统计的检测，而不是基于应用于有损一维投影的复杂统计的检测。作为一种培训计划，这项提议为研究生提供了创造新数学的机会，使一个重要的应用领域发生根本性变化。它将增加认识到数学的力量及其在工程学科中所扮演的重要角色的数学家的队伍。研究生将学会在与不同学科相关的不同文化中茁壮成长，培养沟通不同世界的能力，这种经历将使他们在学术和非学术职业中都具有优势。这项提议可能会导致技术转让给工业合作伙伴，包括雷神公司，他们已经确认了瞬时雷达偏振测量在独立分析中的价值。这种接触和其他接触可能会为美国公民和永久居民带来更多的实习机会。