Advanced Signal Processing Techniques on a Riemannian Manifold

黎曼流形上的高级信号处理传统的信号处理技术基于线性向量空间。应用统计学和线性代数，检测，估计和设计中的复杂理论和算法已经发展并应用于语音，雷达，声纳，通信等。一个这样的特征是功率谱密度（PSD）矩阵。然而，PSD具有结构约束，使其形成流形，并且必须使用黎曼距离（RD）沿流形表面沿着进行测量。使用提升和投影的理论，申请人推导出RD的闭合形式表达式，其在信号处理应用上进行测试，产生显著改善的结果。本研究建议描述了调查，旨在开发加工技术和应用程序的基础上的几何形状和RD的歧管。通过这些建立在新概念上的技术，可以出现用于信号处理的上级算法。选择以下研究主题进行探索。1. PSD矩阵的概率分布及其在信号检测中的应用-信号处理中的一个主要必要条件是信号的统计特性。这个项目探讨了PSD矩阵在流形上的概率分布。然后，这些将被用来建立似然比测试（LRT），用于检测噪声中的信号。2.最佳信号设计-许多信号设计问题可以简化为PSD矩阵的设计。我们使用RD作为测量理想PSD矩阵和信号PSD矩阵之间的差异。因此，信号和滤波器组的设计可以用公式表示为流形上PSD矩阵之间的RD的优化。我们应用申请人开发的交替提升和投影技术来解决此类问题。MIMO雷达信号设计和通信中的脉冲整形是可能受益于该项目的两个应用。3.最小黎曼距离（MRD）估计和滤波-我们寻求PSD矩阵的最佳组合，并在流形上建立MRD估计的原理，在概念上与MMSE滤波平行。我们将应用这种新的MRD估计技术的自适应滤波，自适应阵列处理，自适应波束形成。4. PSD矩阵的kth均值和顺序统计量-我们将建立数学上严格的定义，这些统计量的PSD矩阵的RD和开发算法来定位它们。我们还将得出这些统计量的最佳组合，使信号PSD矩阵的鲁棒估计可以促进。上述提出的项目代表了从传统信号处理方法的根本出发，从向量空间转移到流形。随着RD的测量更加准确，因此开发的算法可能更加准确。

Advanced Signal Processing On a Riemannian Manifold Traditional signal processing techniques are based on linear vector space. Applying statistics and linear algebra, sophisticated theory and algorithms in detection, estimation, and design have been developed and applied to speech, radar, sonar, communications, etc. As signal processing advances, signal features are subject to processing. One such feature is the power spectral density (PSD) matrix. However, PSD have structural constraints making them form a manifold, and measurement must be carried out along the surface of manifold using Riemannian distance (RD). Using a theory of lifting and projecting, the applicant derived closed-form expressions of RD which were tested on signal processing applications yielding results of dramatic improvement. The present research proposal describes investigations aiming at developing processing techniques and applications based on the geometry and RD of the manifold. Through these techniques founded on new concepts, superior algorithms for signal processing may emerge. The following topics of research are chosen for exploration. 1. Probability Distribution of the PSD Matrices and Application to Signal Detection - A major necessity in signal processing is the statistical properties of the signals. This project explores the probability distributions of the PSD matrices on the manifold. These will then be used to establish likelihood ratio tests (LRT) for detection of signals in noise. 2. Optimum Signal Designs - Many signal design problems can be reduced to the design of the PSD matrix. We use RD as the measure for discrepancies between the ideal and the signal PSD matrices. Thus, the design of signals and filter banks can be formulated as an optimization of the RD between PSD matrices on the manifold. We apply the alternating lift and project technique developed by the applicant to solve such problems. MIMO radar signal design and pulse shaping in communications are two applications that may benefit from this project. 3. Minimum Riemannian Distance (MRD) Estimation and Filtering - We seek an optimum combination of PSD matrices and establish the principle of MRD estimation on the manifold, parallel in concepts to MMSE filtering. We will apply this new MRD estimation technique to adaptive filtering, adaptive array processing, and adaptive beamforming. 4. kth Mean and Order Statistics of PSD Matrices - We will established mathematically rigorous definitions in terms of RD for these statistics of PSD matrices and develop algorithms to locate them. We will also derive optimum combinations of these statistics so that robust estimation of signal PSD matrices can be facilitated. The above proposed projects represent a fundamental departure from the traditional signal processing approach, shifting from the vector space to the manifold. With the more accurate measurement of RD, the algorithms so developed are likely to be more accurate.