Advanced Signal Processing Techniques on a Riemannian Manifold

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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.