Computational Noncommutative Harmonic Analysis with Applications

无线通信系统对数据速率和服务质量的要求越来越高，需要新的技术来提高无线链路的可靠性和频谱效率。实现这些目标的三个关键技术是均衡、分集和信道编码。数学对这些技术至关重要，为有效的数值实现提供了理论基础和手段。研究人员将推导出设计时变信道均衡技术的理论和数值框架。利用伪微分算子理论和时频分析的方法，他将发展一种与时变系统相关的算子近似对角化的定性和定量理论。这些理论结果将为建立基于Krylov子空间技术的快速、可靠的数值均衡方法奠定基础。研究人员还将研究帧理论在无线通信中的应用。利用球面填充和群论中的概念，他将分析格拉斯曼框架等特殊框架的理论性质。此外，他还将开发与多载波通信系统(如ofdm)相关的理论和数值方案。这包括使用长椭球波函数的概念的推广来设计具有特定属性的传输信号。通过将谐波分析的最新工具引入无线通信界，这项研究活动将使无线通信领域取得进一步的进步和突破。同时，它将促进应用数学的新研究领域，并为应用数学家和通信工程师之间的进一步互动铺平道路。该项目的目标是为无线通信技术发展数学概念和计算方法。研究人员将把数学中的现代工具与信息论和信号处理的方法结合起来，为无线通信中的编码、传输和均衡等关键技术开发新的概念和算法。数学对这些技术至关重要，因为它为有效的数值实现提供了理论基础和手段。通过改善无线电链路可靠性和提高数据速率，这项研究活动将有助于满足对未来无线通信系统日益增长的需求。该项目将以新的数学方法的形式产生概念成果，以分析和构建无线传输系统。该项目还将以数值算法的形式产生具体的交付成果，供科学和工业部门使用。

The increasing requirements on data rate and quality of service for wireless communications systems call for new techniques to improve radio link reliability and to increase spectral efficiency. The three key technologies to achieve these goals are equalization, diversity, and channel coding. Mathematics is of fundamental importance to these technologies providing the theoretical basis as well as the means for efficient numerical implementations. The investigator will derive a theoretical and numerical framework for designing equalization techniques for time-varying channels. Using methods from pseudo-differential operator theory and time-frequency analysis he will develop a qualitative and quantitative theory for the approximate diagonalization of operators associated with time-varying systems. These theoretical results will form a key stone in the construction of fast and reliable numerical equalization methods which will be based Krylov subspace techniques. The investigator will also study the use of frame theory in wireless communications. Using concepts from sphere packings and group theory he will analyze theoretical properties of special frames such as Grassmannian frames. Furthermore he will develop theoretical and numerical schemes in connection with multi-carrier communication systems such as OFDM. This includes the design of transmission signals with specific properties using a generalization of the concept of prolate spheroidal wave functions. By taking recent tools from harmonic analysis into the wireless communications community this research activity will enable further advances and breakthroughs in wireless communications. At the same time it will stimulate new research areas in applied mathematics and pave the road for further interactions between applied mathematicians and communication engineers.The goal of this project is to develop mathematical concepts and computational methods for wireless communications technology. The investigator will combine modern tools from mathematics with methods from information theory and signal processing to develop new concepts and algorithms for key technologies in wireless communications such as coding, transmission, and equalization. Mathematics is of fundamental importance to these technologies, since it provides the theoretical basis as well as the means for efficient numerical implementations. By improving radio link reliability and increasing data rates this research activity will be instrumental in fulfilling the increasing requirements on future wireless communications systems. The project will produce conceptual deliverables in the form of new mathematical methods to analyze and construct wireless transmission systems. The project will also produce concrete deliverables in the form of numerical algorithms for use in the scientific and industrial sector.