Applied Harmonic Analysis and Wireless Communications

无线通信系统对数据速率和服务质量的要求越来越高，要求采用新技术来提高无线电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。数学是这些技术的基础，为有效的数值实现提供理论基础和手段。研究者推导了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，他发展了与时变系统相关的算子的近似对角化的定性和定量理论。这些理论结果为构建基于Krylov子空间技术的快速、可靠的数值均衡方法奠定了基础。研究者还研究了框架理论在无线通信中的应用。利用球填充和群论的概念，分析了格拉斯曼框架等特殊框架的理论性质。此外，他还开发了与多载波通信系统（如OFDM）相关的理论和数值方案。这包括设计具有特定性质的传输信号，使用广义的长球面波函数的概念。通过将谐波分析的最新工具引入无线通信领域，该项目使无线通信的进一步发展和突破成为可能。同时，它激发了应用数学的新研究领域，为应用数学和通信工程师之间的进一步互动铺平了道路。该项目的目标是开发无线通信技术的数学概念和计算方法。研究者将现代数学工具与信息理论和信号处理方法相结合，为无线通信中的关键技术（如编码、传输和均衡）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。通过提供提高无线电链路可靠性和提高数据速率的工具，该项目有助于满足对未来无线通信系统日益增长的要求。该项目以分析和构建无线传输系统的新数学方法的形式产生概念性成果。该项目还以数值算法的形式产生具体的可交付成果，用于科学和工业部门。

The increasing requirements on data rate and quality ofservice for wireless communications systems call for newtechniques to improve radio link reliability and to increasespectral efficiency. The three key technologies to achieve thesegoals are equalization, diversity, and channel coding.Mathematics is of fundamental importance to these technologies,providing the theoretical basis as well as the means forefficient numerical implementations. The investigator derives atheoretical and numerical framework for designing equalizationtechniques for time-varying channels. Using methods frompseudo-differential operator theory and time-frequency analysis,he develops a qualitative and quantitative theory for theapproximate diagonalization of operators associated withtime-varying systems. These theoretical results form a keystonein the construction of fast and reliable numerical equalizationmethods that are based on Krylov subspace techniques. Theinvestigator also studies the use of frame theory in wirelesscommunications. Using concepts from sphere packings and grouptheory, he analyzes theoretical properties of special frames suchas Grassmannian frames. Furthermore, he develops theoretical andnumerical schemes in connection with multi-carrier communicationsystems such as OFDM. This includes the design of transmissionsignals with specific properties using a generalization of theconcept of prolate spheroidal wave functions. By taking recenttools from harmonic analysis into the wireless communicationscommunity, this project enables further advances andbreakthroughs in wireless communications. At the same time itstimulates new research areas in applied mathematics and pavesthe road for further interactions between applied mathematiciansand communication engineers. The goal of this project is to develop mathematical conceptsand computational methods for wireless communications technology.The investigator combines modern tools from mathematics withmethods from information theory and signal processing to developnew concepts and algorithms for key technologies in wirelesscommunications, such as coding, transmission, and equalization.Mathematics is of fundamental importance to these technologies,because it provides the theoretical basis as well as the meansfor efficient numerical implementations. By providing tools toimprove radio link reliability and increase data rates, thisproject is instrumental in meeting the increasing requirements onfuture wireless communications systems. The project producesconceptual deliverables in the form of new mathematical methodsto analyze and construct wireless transmission systems. Theproject also produces concrete deliverables in the form ofnumerical algorithms for use in the scientific and industrialsector.