Statistical Processing on Signal Feature Manifolds

信号特征流形的统计处理 信号处理是从物理测量中提取信息的过程。基于将观测信号视为向量的线性向量空间理论，信号检测、估计、分类、最优设计等技术已经发展起来，并有效地应用于雷达、声纳、通信、语音等工程系统。随着信号处理技术的进步，信号特征需要直接处理。功率谱密度(PSD)矩阵是一种富含二阶统计量信息的特征。然而，直接处理PSD矩阵并没有在信号分类中产生预期的成功。这使得申请人意识到PSD具有结构限制，因此，它们形成了一个流形(多维表面)。因此，这些特征的测量必须沿着PSD歧管的表面进行，即使用黎曼距离(RD)。经过深入的研究，申请人得出了几个用于在PSD流形上测量的RD的封闭形式，并在脑电信号的分类上进行了测试，从而获得了惊人的准确性。这里的提案描述了一项研究计划，旨在基于PSD歧管的几何形状和申请人推导的RD的使用来开发加工技术。通过基于新概念的此类技术的开发和新工具的使用，预计重新审视现有的应用领域将导致对信号处理的更深入的见解和更好的算法。选择了以下重要区域进行勘探： 1.PSD矩阵的统计特性-信号处理中的一个主要需求是信号的统计特性。在歧管上的处理也没有什么不同。这个项目研究了PSD矩阵，并探索了它们的基本统计性质。这将极大地便利后续对PSD矩阵的分析和处理技术的发展。 2.探测--对于雷达、声纳、通信等工程系统，这是一项基本要求。我们提出了基于PSD矩阵直接处理的信号检测方法。将使用以下技术开发新技术： A)上述1中获得的概率分布，以建立用于流形上检测的似然比检验(LRT)。 B)不同类PSD矩阵之间的Rd，并通过比较与类的相对贴近度来检测。 3.线性估计和滤波--利用勾股定理和投影定理，可以在线性向量空间中进行参数估计和滤波。由于PSD矩阵的结构限制，这里的“线性”加权运算必须特别定义。利用这种PSD矩阵的“线性”组合，我们试图在PSD流形上建立等价概念，并探索PSD矩阵的最优估计和滤波的可能性。 4.信号设计--基于欧氏距离PSD矩阵的信号组设计是近年来研究的热点。在这里，我们建议使用RD作为PSD矩阵之间差异的(更准确的)度量，并使用一组特殊的正交化函数作为基础来解决PSD的最优信号设计问题。 以上提出的项目从根本上背离了传统的信号处理方法，从向量空间转移到流形。随着RD的测量更加准确，因此开发的算法也有望更加准确。还将考虑将这些处理技术扩展到其他特征的调查。

Statistical Processing on Signal Feature Manifolds Signal processing is the extraction of information from physical measurements. Based on linear vector space theory which treats the observed signal as a vector, highly sophisticated techniques of detection, estimation, classification, optimum signal design have been developed and effectively applied to engineering systems such as radar, sonar, communications, speech, etc. As technology in signal processing advances, signal features are subject to direct processing. One such feature rich in second order statistics information is the power spectral density (PSD) matrix. However, direct processing of PSD matrices had not yielded expected successes in signal classification. This led the applicant to realize that PSD have structural constraints and thus, they form a manifold (multi-dimensional surface). Thus, measurements of these features must be carried out along the surface of the PSD manifold, i.e., using the Riemannian distance (RD). Intensive research led the applicant to arrive at several closed-form expressions of RD for measurement on the PSD manifold and these were tested on classification of EEG signals resulting in dramatic accuracy. The proposal here describes a research program aiming at developing processing techniques based on the geometry of the PSD manifold and on the use of the RD derived by the applicant. Through the development of such techniques founded on new concepts and through the use of new tools, it is expected that revisiting existing application areas will lead to deeper insights and to superior algorithms for signal processing. The following important areas are chosen for exploration: 1. Statistical Properties of the PSD Matrices – A major necessity in signal processing is the statistical properties of the signals. Processing on the manifold is no different. This project investigates the PSD matrices and explores their fundamental statistical properties. These will greatly facilitates ensuing analyses and development of processing techniques of the PSD matrices. 2. Detection – For engineering systems such as radar, sonar, communications, this is a fundamental requirement. We propose to explore the detection of signals based on the direct processing of the PSD matrices. New techniques will be developed using: a) The probability distributions obtained in 1. above to establish a likelihood ratio test (LRT) for detection on the manifold. b) The RD between the different classes of PSD matrices and detect by comparing the relative closeness to the classes. 3. Linear Estimation and filtering – Using the Pythagorean and the projection theorems, parameter estimation and filtering are made possible in a linear vector space. Due to the structural constraints of a PSD matrix, the “linear” weighting operation here has to be specially defined. Using such a “linear” combination of PSD matrices, we seek to establish equivalent concepts on the PSD manifold and explore the possibility of optimum estimation and filtering of the PSD matrix. 4. Signal Design – Signal bank design using the PSD matrix with Euclidean distance has been investigated in recent years. Here, we propose to employ RD as the (more accurate) measure for discrepancies between PSD matrices, and use a special set of orthonormal functions as basis to solve the problem of optimum signal design by PSD. 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 also expected to be more accurate. Investigations into extending these processing techniques to other features will also be considered.