Heisenberg-Weyl Groups, Sequences, and Codes

该提案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组，信号设计领域定义了第二个组，新的数学框架是用一个组来表达另一个组的结果。经典的代数纠错码作为相位编码波形有了新的应用，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编码中，与Rudin-Shapiro序列相关的相位编码波形对实现了瞬时雷达偏振测量，它提供了对极化散射矩阵的所有四个维度的并发和相干访问，而不是像今天这样的串行和非相干访问。这个新原语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。作为一个培训项目，该提案为研究生提供了一个创造新数学的机会，为一个重要的应用领域带来根本性的变化。这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数量。研究生将学会在与不同学科相关的不同文化中蓬勃发展，培养连接不同世界的能力，这种经历将使他们在学术和非学术职业中都有优势。该提案可能会导致技术转让给工业合作伙伴，包括雷声公司，他们已经确认了瞬时雷达偏振法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。

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.