Table of Contents Author Guidelines Submit a Manuscript
Journal of Applied Mathematics
Volume 2017 (2017), Article ID 5108946, 27 pages
Review Article

A Guide on Spectral Methods Applied to Discrete Data in One Dimension

Helmholtz-Zentrum Dresden-Rossendorf, Bautzner Landstraße 400, 01328 Dresden, Germany

Correspondence should be addressed to Martin Seilmayer

Received 2 January 2017; Accepted 10 April 2017; Published 24 July 2017

Academic Editor: Zhichun Yang

Copyright © 2017 Martin Seilmayer and Matthias Ratajczak. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. R. Core Team, R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing, Vienna, Austria, 2016,
  2. M. Seilmayer, “Spectral: Common Methods of Spectral Data Analysis,” 2016, R package version 1.0.,
  3. S. W. Smith, The Scientist & Engineer’s Guide to Digital Signal Processing, California Technical Pub, San Diego, Calif, USA, 1st edition, 1997.
  4. L. Cohen, Time-Frequency Analysis, ser. Prentice Hall signal processing series, Prentice Hall PTR, Englewood Cliffs, N.J, 1995.
  5. I. N. Bronštejn and K. A. Semendjajew, Taschenbuch der Mathematik, Harri Deutsch, Thun, Switzerland, 5th edition, 2001.
  6. A. J. Jerri, “The shannon sampling theorem—its various extensions and applications: a tutorial review,” Proceedings of the IEEE, vol. 65, no. 11, pp. 1565–1596, 1977. View at Publisher · View at Google Scholar · View at Scopus
  7. J. D. Scargle, “Studies in astronomical time series analysis. II—statistical aspects of spectral analysis of unevenly spaced data,” The Astrophysical Journal, vol. 263, pp. 835–853, 1982. View at Publisher · View at Google Scholar
  8. N. E. Huang, Z. Shen, S. R. Long et al., “The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis,” Proceedings of the Royal Society A. Mathematical, Physical and Engineering Sciences, vol. 454, no. 1971, pp. 903–995, 1998. View at Publisher · View at Google Scholar · View at MathSciNet
  9. D. E. Vakman and L. A. Vaĭnshteĭn, “Amplitude, phase, frequency—fundamental concepts of oscillation theory,” Soviet Physics Uspekhi, vol. 20, no. 12, pp. 1002–1016, 1977. View at Google Scholar
  10. J. Duoandikoetxea Zuazo, Graduate Studies in Mathematics, vol. 29 of Fourier Analysis, American Mathematical Society, Providence, RI, USA, 2001.
  11. C. E. Shannon, “Communication in the Presence of Noise,” Proceedings of the IRE, vol. 37, no. 1, pp. 10–21, 1949. View at Publisher · View at Google Scholar · View at Scopus
  12. J. M. Whittaker, Interpolatory Function Theory, Cambridge University Press, Cambridge, UK, 1935.
  13. R. C. Gonzalez and R. E. Woods, Digital Image Processing, Prentice Hall, Upper Saddle River, NJ, USA, 2nd edition, 2002.
  14. T. H. Cormen, C. Stein, C. E. Leiserson, and R. L. Rivest, Introduction to Algorithms, The MIT Press, Cambridge, Mass, USA, 2nd edition, 2001.
  15. M. Frigo and S. G. Johnson, “The design and implementation of FFTW3,” Proceedings of the IEEE, vol. 93, no. 2, pp. 216–231, 2005. View at Publisher · View at Google Scholar
  16. L. Debnath and F. A. Shah, Wavelet Transforms And Their Applications, Birkhäuser Boston, Boston, Mass, USA, Second edition, 2015.
  17. R. J. Marks, Handbook of Fourier Analysis & Its Applications, vol. 800, Oxford University Press, Oxford, Conn, USA, 2009.
  18. N. Wiener, Extrapolation, Interpolation, and Smoothing of Stationary Time Series, vol. 2, MIT Press, Cambridge, Mass, USA, 1949.
  19. M. I. Skolnik, Introduction to Radar Systems, McGraw-Hill, New York, NY, USA, 2nd edition, 1980.
  20. A. Cumming, G. W. Marcy, and R. P. Butler, “The lick planet search: detectability and mass thresholds,” The Astrophysical Journal, vol. 526, no. 2, pp. 890–915, 1999. View at Publisher · View at Google Scholar · View at Scopus
  21. A. Mathias, F. Grond, R. Guardans et al., “Algorithms for spectral analysis of irregularly sampled time series,” Journal of Statistical Software, vol. 11, no. 2, pp. 1–30, 2004. View at Google Scholar · View at Scopus
  22. K. Hocke and N. Kämpfer, “Gap filling and noise reduction of unevenly sampled data by means of the Lomb-Scargle periodogram,” Atmospheric Chemistry and Physics, vol. 9, no. 12, pp. 4197–4206, 2009. View at Publisher · View at Google Scholar · View at Scopus
  23. N. R. Lomb, “Least-squares frequency analysis of unequally spaced data,” Astrophysics and Space Science, vol. 39, no. 2, pp. 447–462, 1976. View at Publisher · View at Google Scholar · View at Scopus
  24. R. Penrose and J. A. Todd, “A generalized inverse for matrices,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 51, no. 3, p. 406, 1955. View at Google Scholar
  25. R. H. D. Townsend, “Fast calculation of the lomb-scargle periodogram using graphics processing units,” The Astrophysical Journal, Supplement Series, vol. 191, no. 2, pp. 247–253, 2010. View at Publisher · View at Google Scholar · View at Scopus
  26. M. Zechmeister and M. Kürster, “The generalised lomb-scargle periodogram—a new formalism for the floating-mean and keplerian periodograms,” Astronomy & Astrophysics, vol. 496, no. 2, pp. 577–584, 2009. View at Publisher · View at Google Scholar · View at Scopus
  27. P. Stoica, J. Li, and H. He, “Spectral analysis of nonuniformly sampled data: a new approach versus the periodogram,” IEEE Transactions on Signal Processing, vol. 57, no. 3, pp. 843–858, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  28. R. Hoffmann, Signalanalyse Und -Erkennung: Eine Einführung Für Informationstechniker, Springer Berlin Heidelberg, Heidelberg, Germany, 1998. View at Publisher · View at Google Scholar
  29. E. M. Stein and G. L. Weiss, Introduction to Fourier analysis on Euclidean Spaces, vol. 32 of Princeton mathematical series, Princeton University Press, Princeton, NJ, USA, 1975. View at MathSciNet