Applied Harmonic Analysis and Wireless Communications

无线通信系统对数据传输速率和服务质量的要求不断提高，需要新的技术来提高无线链路的可靠性和频谱效率。 实现这些目标的三大关键技术是均衡、分集和信道编码，数学是这些技术的基础，它为这些技术的有效数值实现提供了理论基础和手段。 研究者推导出一个理论和数值框架，用于设计时变信道的均衡技术。 他运用微分算子理论和时频分析的方法，发展了一种定性和定量的理论，用于与时变系统相关的算子的近似对角化。 这些理论结果为构造基于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。 利用概念从球包装和群论，他分析了理论性质的特殊框架，如格拉斯曼框架。 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。 这包括设计传输信号的具体属性使用的概念的推广长椭球波函数。 通过将谐波分析的最新工具引入无线通信领域，该项目将使无线通信领域取得进一步的进步和突破。 同时，它激发了应用数学新的研究领域，并为应用数学家和通信工程师之间的进一步互动铺平了道路. 本项目的目标是发展无线通信技术的数学概念和计算方法。研究人员将现代数学工具与信息论和信号处理的方法相结合，为无线通信中的关键技术（如编码、传输和均衡）发展新的概念和算法。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。 通过提供工具来提高无线电链路的可靠性和增加数据速率，该项目是在满足对未来无线通信系统的日益增长的要求。 该项目以新的数学方法的形式产生概念交付成果，以分析和构建无线传输系统。 该项目还产生了具体的交付成果的形式ofnumerical算法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.