Dmitry J. Nicolsky, Interdisciplinary Ph.D. from the University of Alaska Fairbanks (2007), a Research Assistant Professor (2013). The author of more than 40 scientific publications. Area of expertise: understanding of coupled ground-atmosphere-ocean processes in the Arctic and numerical solution of partial differential equations.
Fairbanks, PO Box 757320, Fairbanks, AK 99775
Dmitry J. Nicolsky, Interdisciplinary Ph.D. from the University of Alaska Fairbanks (2007), a Research Assistant Professor (2013). The author of more than 40 scientific publications. Area of expertise: understanding of coupled ground-atmosphere-ocean processes in the Arctic and numerical solution of partial differential equations.
Fairbanks, PO Box 757320, Fairbanks, AK 99775
Gennadiy S. Tipenko, Cand. Sci. (Phys.-Math.) (1985), Assotiate Professor (1994), Leading Researcher in the Institute of Environmental Geoscience Russian Academy of Sciences. The author of more than 40 scientific publications. Area of expertise: spectral theory of differential operators; numerical modeling in geocryology problems.
13 Ulansky Pereulok, PO Box 145, Moscow 101000
Gennadiy S. Tipenko, Cand. Sci. (Phys.-Math.) (1985), Assotiate Professor (1994), Leading Researcher in the Institute of Environmental Geoscience Russian Academy of Sciences. The author of more than 40 scientific publications. Area of expertise: spectral theory of differential operators; numerical modeling in geocryology problems.
13 Ulansky Pereulok, PO Box 145, Moscow 101000
Introduction. In practical signal processing and its many applications, researchers and engineers try to find a number of harmonics and their frequencies in a time signal contaminated by noise. In this manuscript we propose a new approach to this problem. Aim. The main goal of this work is to embed the original time series into a set of multi-dimensional information vectors and then use shift-invariance properties of the exponentials. The information vectors are cast into a new basis where the exponentials could be separated from each other. Materials and methods. We derive a stable technique based on the singular value decomposition (SVD) of lagcovariance and cross-covariance matrices consisting of covariance coefficients computed for index translated copies of an original time series. For these matrices a generalized eigenvalue problem is solved. Results. The original time series is mapped into the basis of the generalized eigenvectors and then separated into components. The phase portrait of each component is analyzed by a pattern recognition technique to distinguish between the phase portraits related to exponentials constituting the signal and the noise. A component related to the exponential has a regular structure, its phase portrait resembles a unitary circle/arc. Any commonly used method could be then used to evaluate the frequency associated with the exponential. Conclusion. Efficiency of the proposed and existing methods is compared on the set of examples, including the white Gaussian and auto-regressive model noise. One of the significant benefits of the proposed approach is a way to distinguish false and true frequency estimates by the pattern recognition. Some automatization of the pattern recognition is completed by discarding noise-related components, associated with the eigenvectors that have a modulus less than a certain threshold.
Introduction. In practical signal processing and its many applications, researchers and engineers try to find a number of harmonics and their frequencies in a time signal contaminated by noise. In this manuscript we propose a new approach to this problem. Aim. The main goal of this work is to embed the original time series into a set of multi-dimensional information vectors and then use shift-invariance properties of the exponentials. The information vectors are cast into a new basis where the exponentials could be separated from each other. Materials and methods. We derive a stable technique based on the singular value decomposition (SVD) of lagcovariance and cross-covariance matrices consisting of covariance coefficients computed for index translated copies of an original time series. For these matrices a generalized eigenvalue problem is solved. Results. The original time series is mapped into the basis of the generalized eigenvectors and then separated into components. The phase portrait of each component is analyzed by a pattern recognition technique to distinguish between the phase portraits related to exponentials constituting the signal and the noise. A component related to the exponential has a regular structure, its phase portrait resembles a unitary circle/arc. Any commonly used method could be then used to evaluate the frequency associated with the exponential. Conclusion. Efficiency of the proposed and existing methods is compared on the set of examples, including the white Gaussian and auto-regressive model noise. One of the significant benefits of the proposed approach is a way to distinguish false and true frequency estimates by the pattern recognition. Some automatization of the pattern recognition is completed by discarding noise-related components, associated with the eigenvectors that have a modulus less than a certain threshold.
The authors declare that there are no conflicts of interest present.