Statistical Processing on Signal Feature Manifolds

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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.