Location of Repository

Statistical Fourier analysis : Clarifications and interpretations

By D.S.G. Pollock

Abstract

This paper expounds some of the results of Fourier theory that are essential to the statistical analysis of time series. It employs the algebra of circulant matrices to expose the structure of the discrete Fourier transform and to elucidate the filtering operations that may be applied to finite data sequences.\ud An ideal filter with a gain of unity throughout the pass band and a gain of zero throughout the stop band is commonly regarded as incapable of being realised in finite samples. It is shown here that, to the contrary, such a filter can be realised both in the time domain and in the frequency domain.\ud The algebra of circulant matrices is also helpful in revealing the nature of statistical processes that are band limited in the frequency domain. In order to apply the conventional techniques of autoregressive moving-average modelling, the data generated by such processes must be subjected to anti-aliasing filtering and sub sampling. These techniques are also described.\ud It is argued that band-limited processes are more prevalent in statistical and econometric time series than is commonly recognised

Publisher: Dept. of Economics, University of Leicester
Year: 2008
OAI identifier: oai:lra.le.ac.uk:2381/7576

Suggested articles

Preview

Citations

  1. (1999). A Handbook of Time-Series Analysis, Signal Processing and Dynamics, doi
  2. (1925). A Historical Note on doi
  3. (1966). An Introduction to Linear Analysis, doi
  4. (1962). An Introduction to the Theory of Stationary Random Functions, Prentice-Hall, Englewood Cliffs, New Jersey. Appendix: The Spectral Representation of a Stationary Process
  5. (1979). Analysis of Economic Time Series: a Synthesis, doi
  6. (1979). Circulant Matrices, doi
  7. (1989). Digital Filters: Third Edition, Prentice-Hall, Englewood Cliffs, doi
  8. (1941). Extrapolation, Interpolation and Smoothing of Stationary Time Series. Report on the Services Research Project DIC-6037.
  9. (2001). Filters for Short Non-stationary Sequences, doi
  10. (2005). Fourier Analysis, doi
  11. (1930). Generalised Harmonic Analysis, doi
  12. (1941). Interpolation und Extrapolation von stationa¨ren zuffaligen Folgen, Bulletin de l’academie des sciences de U.S.S.R.,
  13. (1937). Introduction to the Theory of Fourier Integrals, doi
  14. (1930). Introduction to the Theory of Fourier’s Series and Integrals, Third Edition, doi
  15. (1981). Quantum Mechanics, doi
  16. (1846). Re´cherches sur les De´terminants,
  17. (1968). Spectral Analysis and its Applications, Holden-Day,
  18. (1981). Spectral Analysis and Time Series, doi
  19. (1964). Spectral Analysis of Economic Time Series, doi
  20. (1969). Spectral Methods in Econometrics, doi
  21. (1967). Stationary Random Processes, Holden-Day, doi
  22. (1985). Stationary Sequences and Random Fields, doi
  23. (1941). Stationary Sequences in a Hilbert Space, doi
  24. (1957). Statistical Analysis of Stationary Time Series, doi
  25. (1990). Statistical Fourier Analysis D.S.G. POLLOCK: Statistical Fourier Analysis D.S.G. POLLOCK: Statistical Fourier Analysis D.S.G. POLLOCK: Statistical Fourier Analysis D.S.G. POLLOCK: Statistical Fourier Analysis D.S.G. POLLOCK: Statistical Fourier Analysis doi
  26. (1976). Statistical Fourier Analysis Fuller,
  27. (1988). The Fast Fourier Transform and its Applications, Prentice-Hall, Englewood Cliffs,
  28. (1958). The Principles of Quantum Mechanics, Fourth Edition, doi
  29. (2006). The Realisation of Finite-Sample Frequency-Selective Filters. Discussion paper of the
  30. (1974). The Spectral Analysis of Time Series, doi
  31. (1911). The Theory of Circulants from 1861 to 1880, doi
  32. (1915). The Theory of Circulants from 1880 to 1900, doi
  33. (1923). The Theory of Circulants from 1900 to 1920, doi
  34. (1906). The Theory of Circulants in the Historical Order of Development up to 1860, doi
  35. (1966). The Typical Spectral Shape of an Economic Variable, doi
  36. (1994). Time Series Analysis, doi
  37. (1976). Time Series Analysis: Forecasting and Control: Revised Edition, Holden-Day, doi
  38. (1975). Time Series: Data Analysis and Theory, doi
  39. (1987). Time Series: Theory and Methods, doi
  40. (1960). Time-series Analysis, doi
  41. (2002). Toeplitz and Circulant Matrices: A Review, doi
  42. (1997). Towards an Understanding of Some Business Cycle Characteristics, doi
  43. (2000). Trend Estimation and De-Trending via Rational Square-Wave Filters, doi
  44. (2000). Wavelet Methods for Time Series Analysis, doi

To submit an update or takedown request for this paper, please submit an Update/Correction/Removal Request.