Nonlinear Signal Processing and Wireless Communications using Frames and Operators Theory

天平DMS-0807896 研究者为非线性信号处理和无线通信信道开发了一个新的数学框架。这里解决的非线性信号处理问题是从冗余线性表示的幅度进行信号重建。 当线性冗余表示与群表示（如Weyl-Heisenberg群）相关联时，相关的Hilbert-Schmidt算子继承了这一不变性。 因此，快速（非线性）重建算法似乎是可能的。 无线通信信道被建模为线性算子，描述了发送信号如何传播到接收器。 对于超宽带（UWB）信号，多普勒效应不再能被建模为后移。 相反，它被捕获为时间膨胀算子。 采用时间尺度位移的连续叠加来模拟UWB通信信道，并分析了微分算子理论的结果。 研究者提出了两个与信号处理有关的问题。 首先，他考虑了如何以一种允许从有限信息中更有效地重建信号的方式来表示信号。 第二章分析了无线通信信道的特性，旨在改进传输。 这些问题的解决方案对信息技术的战略领域产生了强烈的影响。信号处理、X射线晶体学和量子计算等重要应用都受到第一个问题的解决方案的影响。 与第二个问题相关的高影响力应用包括超宽带穿墙成像系统、更高吞吐量的802.15无线个人区域网络和无线传感器网络。

BalanDMS-0807896 The investigator develops a new mathematical framework fornonlinear signal processing and wireless communication channels. The nonlinear signal processing problem addressed here is signalreconstruction from the magnitude of a redundant linearrepresentation. When the linear redundant representation isassociated to a group representation (such as the Weyl-Heisenberggrgoup), the relevant Hilbert-Schmidt operators inherit thisinvariance property. Thus a fast (nonlinear) reconstructionalgorithm seems possible. A wireless communication channel ismodeled as a linear operator that describes how transmittedsignals propagate to a receiver. For ultrawide band (UWB)signals, the Doppler effect no longer can be modeled as afrequency shift. Instead it is captured as a time dilationoperator. A continuous superposition of time-scale shifts isused to model a UWB communication channel, and consequences topseudo-differential operator theory are analyzed. The investigator takes up two problems related to signalprocessing. In the first he considers how to represent signalsin ways that allow more effective reconstructon of them fromlimited information. In the second he analyzes properties ofwireless communication channels, aiming at improvements intransmission. The solutions to these problems have a strongimpact in the strategic area of information technology. Important applications such as signal processing, X-raycrystallography, and quantum computing are affected by solutionsto the first problem. High-impact applications related to thesecond problem include ultrawide-band through-wall imagingsystems, higher-throughput 802.15 Wireless Personal AreaNetworks, and wireless sensor networks.