On the asymptotical separation of linear signals from harmonics by singular spectrum analysis
DOI:
https://doi.org/10.21638/spbu01.2022.206Abstract
The general theoretical approach to the asymptotic extraction of the signal series from the additively perturbed signal with the help of singular spectrum analysis (briefly, SSA) was already outlined in Nekrutkin (2010), SII, vol. 3, 297–319. In this paper we consider the example of such an analysis applied to the linear signal and the additive sinusoidal noise. It is proved that in this case the so-called reconstruction errors r_i(N) of SSA uniformly tend to zero as the series length N tends to infinity. More precisely, we demonstrate that max_i |r_i(N)| = O(N^(−1)) if N → ∞ and the “window length” L equals (N + 1)/2. It is important to mention, that the completely different result is valid for the increasing exponential signal and the same noise. As it is proved in Ivanova, Nekrutkin (2019), SII, vol. 12, 1, 49–59, in this case any finite number of last terms of the error series does not tend to any finite or infinite values.Keywords:
signal processing, singular spectral analysis, separability, linear signal, asymptotical analysis
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References
Литература
1. Golyandina N., Nekrutkin V., Zhigljavsky A. Analysis of Time Series Structure. SSA and Related Techniques. In Ser.: Monographs on Statistics and Applied Probability, vol. 90. Boca Raton, London, New York, Washington D. C., Chapman & Hall/CRC (2001).
2. Golyandina N., Zhigljavsky A. Singular Spectrum Analysis for Time Series, 2nd ed. In Ser.: Springer Briefs in Statistics. Springer (2020).
3. Nekrutkin V. Perturbation expansions of signal subspaces for long signals. Statistics and Its Interface 3, 297–319 (2010).
4. Golub G.H., Van Loan Ch.F. Matrix computations. 4th ed. Johns Hopkins University Press (2013).
5. Като Т. Теория возмущений линейных операторов, пер. с англ. Москва, Мир (1972).
6. Ivanova Е., Nekrutkin V. Two asymptotic approaches for the exponential signal and harmonic noise in Singular Spectrum Analysis. Statistics and Its Interface 12 (1), 49–59 (2019).
References
1. Golyandina N., Nekrutkin V., Zhigljavsky A. Analysis of Time Series Structure. SSA and Related Techniques. In Ser.: Monographs on Statistics and Applied Probability, vol. 90. Boca Raton, London, New York, Washington D. C., Chapman & Hall/CRC (2001).
2. Golyandina N., Zhigljavsky A. Singular Spectrum Analysis for Time Series, 2nd ed. In Ser.: Springer Briefs in Statistics. Springer (2020).
3. Nekrutkin V. Perturbation expansions of signal subspaces for long signals. Statistics and Its Interface 3, 297–319 (2010).
4. Golub G.H., Van Loan Ch.F. Matrix computations. 4th ed. Johns Hopkins University Press (2013).
5. Kato T. Perturbation theory for linear operators. Berlin, Heidelberg, New York, Springer-Verlag (1966). [Rus. ed.: Kato T. Teorija vozmushhenij linejnyh operatorov. Moscow, Mir Publ. (1972)].
6. Ivanova Е., Nekrutkin V. Two asymptotic approaches for the exponential signal and harmonic noise in Singular Spectrum Analysis. Statistics and Its Interface 12 (1), 49–59 (2019).
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Published
2022-07-06
How to Cite
Zenkova, N. V., & Nekrutkin, V. V. (2022). On the asymptotical separation of linear signals from harmonics by singular spectrum analysis. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy, 9(2), 245–254. https://doi.org/10.21638/spbu01.2022.206
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Mathematics
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Articles of "Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy" are open access distributed under the terms of the License Agreement with Saint Petersburg State University, which permits to the authors unrestricted distribution and self-archiving free of charge.
