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。基于线性向量空间理论，将观测到的信号视为一个矢量，发展了高度复杂的检测、估计、分类和信号优化设计技术，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，信号特征成为直接处理的对象。功率谱密度（PSD）矩阵就是其中一个富含二阶统计信息的特征。然而，直接处理PSD矩阵在信号分类方面并没有取得预期的成功。这使得申请人意识到PSD具有结构约束，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流形的表面进行，即使用黎曼距离（RD）。经过深入的研究，申请人得出了几个封闭形式的RD表达式，用于PSD歧管的测量，并在脑电图信号的分类上进行了测试，结果非常准确。这里的提案描述了一个研究计划，旨在基于PSD歧管的几何形状和申请人派生的RD的使用来开发加工技术。通过在新概念和新工具的基础上开发这些技术，预计重新审视现有的应用领域将导致更深入的见解和更好的信号处理算法。选择以下重要领域进行探索：***1。PSD矩阵的统计特性在信号处理中一个主要的必要条件是信号的统计特性。歧管上的处理没有什么不同。本项目研究PSD矩阵并探索其基本统计性质。这将极大地促进PSD矩阵的后续分析和处理技术的发展。* * 2。探测-对于工程系统，如雷达，声纳，通信，这是一个基本要求。我们提出探索基于PSD矩阵直接处理的信号检测方法。新技术将利用：*a) 1中得到的概率分布。建立一个似然比检验（LRT）来检测歧管。*b)不同类别的PSD矩阵之间的RD，通过比较与类别的相对接近度来检测。**3。线性估计和滤波-使用毕达哥拉斯定理和投影定理，参数估计和滤波在线性向量空间中成为可能。由于PSD矩阵的结构约束，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，我们寻求在PSD流形上建立等价概念，并探索PSD矩阵的最佳估计和滤波的可能性。* * 4。信号设计-近年来研究了基于欧氏距离的PSD矩阵的信号组设计。在这里，我们提出采用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函数作为基础来解决PSD的最佳信号设计问题。**上述提议的项目代表了对传统信号处理方法的根本背离，从向量空间转移到流形。随着对RD的测量更加精确，所开发的算法也有望更加准确。将这些处理技术扩展到其他特征的研究也将被考虑。

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.