Application of the Non-Hermitian Singular Spectrum Analysis to the Exponential Retrieval Problem
https://doi.org/10.32603/1993-8985-2020-23-3-6-24
Аннотация
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.
При поддержке: We would like to thank A. Rybkin and V. Romanovsky for all their advice, patience, critique and reassurances along the way. We also appreciate comments and suggestions by the anonymous reviewer, who greatly helped to improve the quality of this manuscript. This research was funded by ARCSS Program and by the Polar Earth Science Program, Office of Polar Programs, National Science Foundation and by the State of Alaska.
Об авторах
D. J. Nicolsky
Geophysical Institute, University of Alaska Fairbanks
Соединённые Штаты Америки
Соединённые Штаты Америки
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
G. S. Tipenko
Institute of Environmental Geoscience Russian Academy of Sciences
Россия
Россия
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
Список литературы
1. Burg J. Maximum Entropy Spectrum Analysis. Proc. 37th annual international meeting of the society of the exploration geophysicists, International Meeting of the Exploration Geophysicists. Modern Spectrum Analysis. Ed. by D. G. Childers. Oklahoma City, Okla. Piscataway, IEEE Press, 1978 (1967), pp. 42‒48.
2. Schmidt R. Multiple Emitter Location and Signal Parameter Estimation. IEEE Transactions on Antennas and Propagation. 1986, vol. 34, no. 3, pp. 276‒280. doi: 10.1109/TAP.1986.1143830
3. Tufts D., Kumaresan R. Singular Value Decomposition and Improved Frequency Estimation Using Linear Prediction. IEEE Transactions on Acoustics, Speech and Signal Processing. 1982, vol. 30, iss. 4, pp. 671‒675. doi: 10.1109/TASSP.1982.1163927
4. Kay S. Modern Spectral Estimation: Theory and Application. Englewood Cliffs, NJ, Prentice-Hall, 1988, 543 p.
5. Roy R., Kailath T. ESPRIT ‒ Estimation of Signal Parameters via Rotational Invariance Techniques. IEEE Transactions on Acoustics, Speech and Signal Processing. 1989, vol. 37, iss. 7, pp. 984‒995. doi: 10.1109/29.32276
6. Hua Y., Sarkar T. Matrix Pencil Method for Estimating Parameters of Exponentially Damped/Undamped Sinusoids in Noise. IEEE Transactions on Acoustics, Speech and Signal Processing. 1990, vol. 38, iss. 5, pp. 814‒824. doi: 10.1109/29.56027
7. Kumaresan R. On the Zeros of the Linear Prediction-Error Filter for Deterministic Signals. IEEE Transactions on Acoustics, Speech and Signal Processing. 1983, vol. 31, iss. 1, pp. 217‒220. doi: 10.1109/TASSP.1983.1164021
8. Van Der Veen A., Deprettere E., Swindlehurst A. Subspace Based Signal Analysis Using Singular Value Decomposition. Proc. of the IEEE. 1993, vol. 81, iss. 9, pp. 1277‒1308. doi: 10.1109/5.237536
9. Golub G. H., Loan C. F. V. Matrix Computations. 4th ed. Baltimore, Maryland, The John Hopkins University Press, 2013, 784 p.
10. Buhren M., Pesavento M., Bohme J. A New Approach to Array Interpolation by Generation of Arti Cial Shift Invariances: interpolated ESPRIT. Proc. IEEE Int. Conf. Acoust., Speech and Signal Processing (ICASSP). 2003, vol. 5. doi: 10.1109/ICASSP.2003.1199904
11. Marchi S. D. On Computing the Factors of Generalized Vandermonde Determinants. Recent Advances in Applied and Theoretical Mathematics. 2000, pp. 140‒144.
12. Heineman E. R. Generalized Vandermonde Determinants. Transactions of the American Mathematical Society. 1929, vol. 31, no. 3, pp. 464‒476.
13. Akaike H. A New Look at the Statistical Model Identification. IEEE Transactions on Automatic Control. 1974, vol. 19, iss. 6, pp. 716‒723. doi: 10.1109/TAC.1974.1100705
14. Schwartz G. Estimating the dimension of a model. Annals of Statistics. 1978, vol. 6, no. 2, pp. 461‒464.
15. Rissanen J. Modeling by Shortest Data Description. Automatica. 1978, vol. 14, pp. 465‒471.
16. Wax M., Kailath T. Detection of Signals by Information Theoretic Criteria. IEEE Transactions on Acoustics Speech and Signal Processing. 1985, vol. ASSP-33, no. 2, pp. 387‒392. doi: 10.1109/TASSP.1985.1164557
17. Zhao L. C., Krishnaiah P. R., Bai Z. D. On Detection of the Number of Signals in Presence of White Noise. J. of Multivariate Analysis. 1986, vol. 20, iss. 1, pp. 1‒25. doi: 10.1016/0047-259X(86)90017-5
18. Fuchs J. Estimating the Number of Sinusoids in Aditive White Noise. IEEE Transactions on Acoustics, Speech and Signal Processing. 1998, vol. 36, iss. 12, pp. 1846‒1853. doi: 10.1109/29.9029
19. Badeau R., David B., Richard G. Selecting the Modeling Order for the ESPRIT High Resolution Method: an Alternative Approach. Proc. IEEE Int. Conf. Acoust., Speech and Signal Processing (ICASSP). Montreal, Canada, 1721 May 2004. Piscataway, IEEE, 2004, vol. II, pp. 1025‒1028. doi: 10.1109/ICASSP.2004.1326435
20. Kumaresan R., Tufts D. Estimating the Parameters of Exponentially Damped Sinusoids and Pole-Zero Modeling in Noise. IEEE Transactions on Acoustics, Speech and Signal Processing. 1982, vol. 30, iss. 6, pp. 833‒840. doi: 10.1109/TASSP.1982.1163974
21. Tretter S. Estimating the Frequency of a Noisy Sinusoid by Linear Regression. IEEE Transactions on Information Theory. 1985, vol. 31, iss. 6, pp. 832‒835. doi: 10.1109/TIT.1985.1057115
22. Stoica P., Moses R. L., Friedlander B., Soderstrom T. Maximum likelihood estimation of the parameters of multiple sinusoids from noisy measurements. IEEE Transactions on Acoustics, Speech and Signal Processing. 1989, vol. 37, iss. 3, pp. 378‒392. doi: 10.1109/29.21705
23. Kay S. A Fast and Accurate Single Frequency Estimator. IEEE Transactions on Acoustics, Speech and Signal Processing. 1989, vol. 37, iss. 12, pp. 1987‒1990. doi: 10.1109/29.45547
24. Stoica P., Moses R. Spectral Analysis of Signals. Prentis-Hall, Upper Saddle River, NJ, 2005, 452 p.
25. Cedervall M., Stoica P., Moses R. Mode-Type Algorithm for Estimating Damped, Undamped, or Explosive Modes. Circuits, Systems and Signal Processing. 1997, vol. 16, iss. 3, pp. 349‒362. doi: 10.1109/ACSSC.1995.540870
26. So H., Chan K., Chan Y., Ho K. Linear Prediction Approach for Efficient Frequency Estimation of Multiple Real Sinusoids: Algorithms and Analyses. IEEE Transactions on Signal Processing. 2005, vol. 53, iss. 7, pp. 2290‒2305. doi: 10.1109/TSP.2005.849154
27. Qian F., Leung S., Zhu Y., Wong W., Pao D., Lau W. Damped Sinusoidal Signals Parameter Estimation in Frequency Domain. Signal Processing. 2012, vol. 92, iss. 2, pp. 381‒391. doi: 10.1016/j.sigpro.2011.08.003
28. Li T.-H. Time Series with Mixed Spectra. CRC Press, 2014, 680 pp.
29. Zhao S., Loparo K. Forward and Backward Extended Prony Method for Complex Exponential Signals with/without Additive Noise. Digital Signal Processing. 2019, vol. 86, pp. 42‒54. doi: 10.1016/j.dsp.2018.12.012
30. Stewart G. W. Perturbation Theory for the Singular Value Decomposition. SVD and Signal Processing, II: Algorithms, Analysis and Applications. 1991, pp. 99‒109.
31. Li F., Vaccaro R. J. Performance Degradation of DOA Estimators Due to Unknown Noise Fields. IEEE Transactions on Signal Processing. 1992, vol. 40, iss. 3, pp. 686‒690. doi: 10.1109/78.120813
32. Broomhead D. S., King G. P. Extracting Qualitative Dynamics from Experimental Data. Physica D. 1986, vol. 20, iss. 2‒3, pp. 217‒236. doi: 10.1016/0167-2789(86)90031-X
33. Vautard R., Ghil M. Singular-Spectrum Analysis in Nonlinear Dynamics, with Applications to Paleoclimatic Time Series. Physica D. 1989, vol. 35, iss. 3, pp. 395‒424. doi: 10.1016/0167-2789(89)90077-8
34. Ghil M., Vautard R. Interdecadal Oscillations and the Warming Trend in Global Temperature Time Series. Nature. 1991, vol. 350, pp. 324‒327.
35. Allen M., Read P., Smith L. Temperature Time Series. Nature. 1992, vol. 355, p. 686.
36. Allen M., Read P., Smith L. Temperature Oscillation. Nature. 1992, vol. 359, p. 679.
37. Yiou P., Sornette D., Ghil M. Data-Adaptive Wavelets and Multi-Scale SSA. Physica D. 2000, vol. 142, pp. 254‒290. doi: 10.1016/S0167-2789(00)00045-2
38. Varadi F., Pap J. M., Ulrich R. K., Bertello L., Henney C. J. Searching for Signal in Noise by Random-Lag Singular Spectrum Analysis. The Astronomical J. 1999, vol. 526, pp. 1052‒1061.
39. Golyandina N., Shlemov A. Variations of Singular Spectrum Analysis for Separability Improvement: Nonorthogonal Decompositions of Time Series. Statistical Interface. 2014, vol. 8, iss. 3, pp. 277‒294. doi: 10.4310/SII.2015.v8.n3.a3
40. Goljandina N., Nekrutkin V., Zhigljavsky A. Analysis of Time Series Structure: SSA and related techniques. London, Chapman and Hall, 2001, 320 p.
41. Fowler M. Phase-Based Frequency Estimation: A Review. Digital Signal Processing. 2002, vol. 12, iss. 4, pp. 590‒615. doi: 10.1006/dspr.2001.0415
42. Golub G., Milanfar P., Varah J. A Stable Numerical Method for Inverting Shape from Moments. SIAM Journal on Scientific Computing. 1999, vol. 21, iss. 4, pp. 1222‒1243. doi: 10.1137/s1064827597328315
43. Kundu D., Mitra A. Estimating the Parameters of Exponentially Damped/Undamped Sinusoids in Noise: A Non-Iterative Approach. Signal Processing. 1995, vol. 46, iss. 3, pp. 363‒368. doi: 10.1016/0165-1684(95)00094-6
44. Avdonin S., Bulanova A., Nicolsky D. Boundary Control Approach to the Spectral Estimation Problem. The Case of Simple Poles. Sampling Theory in Signal and Image Processing. 2009, vol. 8, iss. 3, pp. 225‒248.
Рецензия
Для цитирования:
Nicolsky D.J., Tipenko G.S. Application of the Non-Hermitian Singular Spectrum Analysis to the Exponential Retrieval Problem. Известия высших учебных заведений России. Радиоэлектроника. 2020;23(3):6-24. https://doi.org/10.32603/1993-8985-2020-23-3-6-24
For citation:
Nicolsky D.J., Tipenko G.S. Application of the Non-Hermitian Singular Spectrum Analysis to the Exponential Retrieval Problem. Journal of the Russian Universities. Radioelectronics. 2020;23(3):6-24. https://doi.org/10.32603/1993-8985-2020-23-3-6-24
Просмотров: 436
Послать статью по эл. почте 