纤维丛在时变阵列信号跟踪与检测中的应用

The array processing has been widely applied to many fields, such as wireless communications, sensor networks, and radar systems. The classical signal processing technique is based on linear signal space, and it describes the signal structure by using the second-order moment. However, due to the fast time-varying signals and complicated interference, it is difficult for signals to maintain the linear structure. The fiber bundle in integrated differential geometry is a powerful tool in flexibly analyzing function space, which represents the globle invariants and has promoted the Gauge theory in physics. Therefore, this project will extend the related approaches from the traditional linear space to a curved space and utilize the fiber bundle to deal with several issues in time-varying array signal processing. Specifically, we firstly construct the local space and present three approaches to develop the local frame, which will provide a generalized analysis to local subspace. We then integrate fiber bundle with stochastic differential equation to spread the signal manifold in a linear space. In this way, we develop a generalized filtering approach to meet the requirements of tracking and detecting the time-varying signals. We finally apply the fiber bundle technique to the accumulative processing among data frames, and present a novel accumulative detection based on tracking the time-varying signals. The research in this project will provide a new theoretical tool for the applications of array signal processing, such as wireless communications, sensor networks, radar systems, and so on.

阵列处理已经在无线通信和雷达等领域得到了广泛应用，经典处理方法基于线性信号空间，采用二阶矩为测度描述信号的线性结构。然而信号的快速时变以及复杂干扰的存在使得信号的线性结构难以保持。整体微分几何中的纤维丛理论提供了空间分析的灵活性，反映了流形空间大范围的不变量，并推动了物理学中规范场论的发展。因此，课题将传统的线性信号空间拓展到流形信号空间，利用纤维丛理论解决时变阵列信号处理中的若干问题，研究内容包括：1）研究信号局部空间的建立问题，分别采用三种方法构造信号的局部标架，提炼广义的局部子空间分析方法；2）应用纤维丛技术结合随机微分方程，在线性空间中将信号流形展开，建立一种广义的滤波方法以满足时变信号跟踪和处理的要求；3）进一步将纤维丛技术应用于信号帧间的累积处理，基于时变信号的跟踪提出一种新的检测方法。课题研究将为无线通信、传感器网络以及雷达等阵列信号处理的应用提供新的理论工具和技术途径。

课题以动态信号分析与处理为出发点，实现对动态高维信号的跟踪、优化处理、检测等应用。采用流形建模动态高维信号，建立流形和线性切空间之间的映射关系，从现代几何学的角度，描述度量空间中信号的几何结构，根据信号的内蕴几何性质建立新的处理方式，提高对动态信号的滤波和检测能力。取得的主要结论有：1）观测信号流形等效为满足一定约束条件下的欧式空间的嵌入，采用平行传输理论，能够借助线性切空间通过优化手段描述流形上任意两点之间的测地线；2）优化过程给出了线性空间到流形上映射的实现手段，不仅描述了流形上的测地线同时也给出了相应的度量；3）设计了新的单步状态变化概率，导出了相应的DP-TBD检测统计量，提高了对运动目标的检测性能；4）研究了缓存影响下的多用户中继网络，依据动态认知网络性能分析给出了系统优化设计的准侧。