Statistical Processing on Signal Feature Manifolds

信号特征流形的统计处理

• 批准号: RGPIN-2014-04893
• 负责人: Wong, Kon
• 金额: $ 2.7万
• 依托单位: McMaster University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2014
• 资助国家: 加拿大
• 起止时间: 2014-01-01 至 2015-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=566814
• 关键词: Statistical; Processing; Signal; Feature; Manifolds

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.

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，将观测信号视为一个向量，高度复杂的检测，估计，分类，最佳信号设计技术已经开发出来，并有效地应用于工程系统，如雷达，声纳，通信，语音等，随着技术在信号处理的进步，信号特征进行直接处理。一个这样的富含二阶统计信息的特征是功率谱密度（PSD）矩阵。然而，PSD矩阵的直接处理在信号分类中没有取得预期的成功。这使得申请人认识到PSD具有结构约束，因此它们形成流形（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，黎曼距离（Riemannian Distance，RD）深入的研究使申请人获得了用于PSD流形上的测量的RD的几个封闭形式的表达式，并且这些表达式在EEG信号的分类上进行了测试，从而获得了惊人的准确性。这里的建议描述了一个研究计划，旨在开发基于PSD歧管的几何形状和申请人导出的RD的使用的处理技术。通过开发基于新概念的此类技术并通过使用新工具，可以预期，重新访问现有应用领域将导致更深入的见解和用于信号处理的上级算法。选择了以下几个重要领域进行探索：1。PSD矩阵的统计特性-信号处理中的一个主要必要条件是信号的统计特性。歧管上的处理也不例外。本项目研究PSD矩阵并探索其基本统计特性。这将极大地促进随后的PSD矩阵的分析和处理技术的发展。2.检测-对于雷达、声纳、通信等工程系统来说，这是一个基本要求。我们建议探索基于PSD矩阵的直接处理的信号检测。新的技术将开发使用：a）在1中获得的概率分布。以建立用于歧管上的检测的似然比检验（LRT）。B）PSD矩阵的不同类别之间的RD，并且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矩阵的结构约束，这里的“线性”加权操作必须被特别定义。使用这样一个“线性”组合的PSD矩阵，我们试图建立等价的概念上的PSD流形和探索的可能性，最佳估计和滤波的PSD矩阵。4.信号设计-近年来研究了使用PSD矩阵和欧几里德距离的信号库设计。在这里，我们建议采用RD作为（更准确的）测量PSD矩阵之间的差异，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的测量更加精确，因此开发的算法也被期望更加精确。将这些处理技术扩展到其他功能的调查也将被考虑。