Skip to main content

Harmonizable Signal Extraction, Filtering and Sampling

  • Chapter
Topics in Non-Gaussian Signal Processing

Abstract

The purpose of this paper is to describe some aspects of analysis on a class of non-stationary and non Gaussian processes dealing with linear filtering, signal extraction from observed data, and sampling the process. The class to be considered consists of harmonizable processes which uses some suitably generalized spectral methods of the classical theory. Let us elaborate these statements.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Bochner, S. (1953), “Fourier transforms of time series,” Proc. Nat. Acad. Set., Vol. 39, 302–307.

    Article  MathSciNet  Google Scholar 

  2. Bochner, S. (1956), “Stationarity, boundedness, almost periodicity of random valued functions,” Proc. Third Berkeley Symp. Math. Statist, and Prob., Vol. 2, 7–27.

    MathSciNet  Google Scholar 

  3. Cartan, H. (1963), Elementary Theory of Analytic Functions of One and Several Complex Variables, Hermann, Paris.

    MATH  Google Scholar 

  4. Chang, D.K., and Rao, M.M. (1983), “Bimeasures and sampling theorems for weakly harmonizable processes,” Stoch. Anal. Appl. Vol. 1, 21–55.

    Article  MathSciNet  MATH  Google Scholar 

  5. Chang, D.K. (1983), Bimeasures, Harmonizable Processes and Filtering, Dissertation, University of California, Riverside, CA.

    Google Scholar 

  6. Chatterji, S. D. (1982), “Orthogonally scattered dilation of Hilbert space valued set functions,” ( Oberwolfach Proceedings ), Lect. Notes Math. Vol. 945, 269–281.

    Article  MathSciNet  Google Scholar 

  7. Cramér, H. (1951), “A contribution to the theory of stochastic processes,” Proc. Second Berkeley Symp. Math. Statist, and Prob., 329–339.

    Google Scholar 

  8. Cramér, H. (1971), Structural and Statistical Problems for a Class of Stochastic Processes, Princeton University Press.

    MATH  Google Scholar 

  9. Dolph, C.L., and Woodbury, M.A. (1952), “On the relation between Green’s func¬tions and covariances of certain stochastic processes and its application to unbiased linear prediction,” Trans. Amer. Math. Soc., Vol. 72, 519–550.

    MathSciNet  MATH  Google Scholar 

  10. Dunford, N., and Schwartz, J.T. (1958), Linear Operators: Part I, General Theory, Wiley-Interscience, New York.

    Google Scholar 

  11. Gikhman, I.I., and Skorokhod, A.V. (1969), Introduction to the Theory of Random Processes, W.B. Saunders Company, Philadelphia, PA.

    Google Scholar 

  12. Grenander, U. (1950), “Stochastic processes and statistical inference,” Ark. f. Mat., Vol. 1, 195–277.

    Google Scholar 

  13. Hille, E., and Phillips, R.S. (1957), Functional Analysis and Semigroups, Amer. Math. Soc. Collq. PubL, New York.

    Google Scholar 

  14. Kampe de Feriet, J., and Frenkiel, F.N. (1962), “Correlation and spectra for nonsta- tionary random functions,” Math. Comp., Vol. 16, 1–21.

    Google Scholar 

  15. Karhunen, K. (1947), “Lineare Transformationen stationarer stochastischer Prozesse,” Den. 10 Skandinaviske Matematiker Kongres i Kobenhaven, 320–324.

    Google Scholar 

  16. Kelsh, J.P. (1978), Linear Analysis of Harmonizable Time Series, Dissertation, University of California, Riverside, CA.

    Google Scholar 

  17. Lloyd, S.P. (1959), “A sampling theorem for stationary (wide sense) stochastic processes,” Trans. Amer. Math. Soc., Vol. 92, 1–12.

    Google Scholar 

  18. Loeve, M. (1948), “Fonctions aleatories du second order,” Note in P. Levy’s Processes Stochastique et Movement Brownien, Gauthier-Villars, Paris, 228–352.

    Google Scholar 

  19. Loeve, M. (1955), Probability Theory, D. Van Nostrand Company, Princeton, NJ.

    Google Scholar 

  20. Morse, M., and Transue, W. (1956), “ (rbimeasures and their integral extensions,” Ann. Math., Vol. 64, 480–504.

    Google Scholar 

  21. Nagabhushanam, K. (1951), “The primary process of a smoothing relation,” Ark. f. Math., Vol. 1, 421–488.

    Google Scholar 

  22. Niemi, H. (1975), “Stochastic processes as Fourier transforms of stochastic meas¬ures,” Ann. Acad. Scient. Fennicae, Ser. A I, No. Vol. 591, 1–47.

    Google Scholar 

  23. Niemi, H. (1977), “On orthogonally scatterred dilations of bounded vector meas¬ures,” Ann. Acad. Scient. Fennicae, Ser. A I. Math., Vol. 3, 43–52.

    Google Scholar 

  24. Parzen, E. (1962), “Spectral analysis of asymptotically stationary time series,” Bull. Inst. Internat. Statist., Vol. 39, 87–103.

    Google Scholar 

  25. Piranashvili, Z. (1967), “On the problem of interpolation of random processes,” Theor. Probl. Appl., Vol. 7, 647–659.

    Google Scholar 

  26. Pourahmadi, M. (1983), “A sampling theorem for multivariate stationary processes,” J. Multivar. Anal., Vol. 13, 177–186.

    Google Scholar 

  27. Rao, M.M. (1967), “Inference in stochastic processes-Ill,” Z. Wahrs. Vol. 8, 49–72.

    Google Scholar 

  28. Rao, M.M. (1982), “Domination problem for vector measures and applications to nonstationary processes,” (Oberwolfach Proceedings) Lect. Notes Math., Vol. 945, 296–313.

    Google Scholar 

  29. Rao, M.M. (1982), “Harmonizable processes: structure theory,” L’Enseign. Math., Vol. 28, 295–351.

    MATH  Google Scholar 

  30. Rao, M.M. (1984), “The spectral domain of multivariate harmonizable processes,” Proc. Nat. Acad. Sci., Vol. 82, 4611–4612.

    Article  Google Scholar 

  31. Rozanov, Yu. A. (1959), “Spectral analysis of abstract functions,” Theor. Prob. Appl., Vol. 4, 271–287.

    Article  MathSciNet  MATH  Google Scholar 

  32. Rozanov, Yu. A. (1967), Stationary Random Processes, Hold en-Day, Inc., San Fran¬cisco, CA.

    MATH  Google Scholar 

  33. Yaglom, A.M. (1962), Introduction to the Theory of Stationary Random Functions, Prentice-Hall, Englewood Cliffs, NJ.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 1989 Springer-Verlag New York Inc.

About this chapter

Cite this chapter

Rao, M.M. (1989). Harmonizable Signal Extraction, Filtering and Sampling. In: Wegman, E.J., Schwartz, S.C., Thomas, J.B. (eds) Topics in Non-Gaussian Signal Processing. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-8859-3_7

Download citation

  • DOI: https://doi.org/10.1007/978-1-4613-8859-3_7

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4613-8861-6

  • Online ISBN: 978-1-4613-8859-3

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics