Karhunen Loeve Expansion of Gaussian Processes

  • Bernard C. Levy


Covariance Function Discrete Cosine Transform Gaussian Process Wiener Process Eigenfunction Expansion 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    W. B. Davenport, Jr. and W. L. Root, An Introduction to the Theory of Random Signals and Noise. New York: McGraw-Hill, 1958. Reprinted by IEEE Press, New York, 1987.MATHGoogle Scholar
  2. 2.
    D. Middleton, An Introduction to Satistical Communication Theory. New York: McGraw-Hill, 1960. Reprinted by IEEE Press, New York, 1996.Google Scholar
  3. 3.
    H. L. Van Trees, Detection, Estimation and Modulation Theory, Part I: Detection, Estimation and Linear Modulation Theory. New York: J. Wiley & Sons, 1968. Paperback reprint edition in 2001.Google Scholar
  4. 4.
    J. M. Wozencraft and I. M. Jacobs, Principles of Communication Engineering. New York: J. Wiley & Sons, 1965. Reprinted by Waveland Press, Prospect Heights, IL, 1990.Google Scholar
  5. 5.
    N. Young, An Introduction to Hilbert Space. Cambridge, UK: Cambridge Univ. Press, 1988.Google Scholar
  6. 6.
    D. Bertsekas, Nonlinear Programming, Second Edition. Belmont, MA: Athena Scientific, 1999.MATHGoogle Scholar
  7. 7.
    L. N. Trefethen and D. Bau, III, Numerical Linear Algebra. Philadelphia, PA: Soc. for Industrial and Applied Math., 1997.CrossRefMATHGoogle Scholar
  8. 8.
    B. Jamison, “Reciprocal processes,” Z. Wahrscheinlichkeitstheorie verw. Gebiete, vol. M30, pp. 65–86, 1974.CrossRefMathSciNetGoogle Scholar
  9. 9.
    P. Lèvy, “A special problem of Brownian motion and a general theory of Gaussian random functions,” in Proc. 3rd Berkeley Symposium on Math. Statistics and Probability, vol. 2, (Berkeley, CA), pp. 133–175, Univ. California Press, 1956.Google Scholar
  10. 10.
    A. J. Krener, R. Frezza, and B. C. Levy, “Gaussian reciprocal processes and self-adjoint stochastic differential equations of second order,” Stochastics and Stochastics Reports, vol. 34, pp. 29–56, 1991.MATHMathSciNetGoogle Scholar
  11. 11.
    B. C. Levy, R. Frezza, and A. J. Krener, “Modeling and estimation of discrete-time Gaussian reciprocal processes,” IEEE Trans. Automatic Control, vol. 35, pp. 1013–1023, Sept. 1990.CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    J. M. Coleman, B. C. Levy, and J. Krener, “Gaussian reciprocal diffusions and positive definite Sturm-Liouville operators,” Stochastics and Stochastics Reports, vol. 55, pp. 279–313, 1995.MATHMathSciNetGoogle Scholar
  13. 13.
    S. Karlin and H. M. Taylor, A First Course in Stochastic Processes. New York: Academic Press, 1975.MATHGoogle Scholar
  14. 14.
    D. Slepian, “First passage times for a particular Gaussian processes,” Annals Math. Statistics, vol. 32, pp. 610–612, 1961.CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    C. Van Loan, Computational Frameworks for the Fast Fourier Transform. Philadelphia, PA: Soc. for Industrial and Applied Math., 1992.MATHGoogle Scholar
  16. 16.
    W. D. Ray and R. M. Driver, “Further decomposition of the Karhunen-Loève series representation of a stationary random process,” IEEE Trans. Informat. Theory, vol. 16, Nov. 1970.Google Scholar
  17. 17.
    N. Ahmed, T. Natarajan, and K. R. Rao, “Discrete cosine transform,” IEEE Trans. Computers, vol. 23, pp. 90–93, 1994.CrossRefMathSciNetGoogle Scholar
  18. 18.
    A. B. Baggeroer, “A state-variable approach to the solution of Fredholm integral equations,” IEEE Trans. Informat. Theory, vol. 15, pp. 557–570, Sept. 1969.CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    R. Frezza, Models of higher-order and mixed-order Gaussian reciprocal processes with application to the smoothing problem. PhD thesis, Univ. California, Davis, Grad. Program in Applied Math., 1990.Google Scholar
  20. 20.
    T. Kailath, A. H. Sayed, and B. Hassibi, Linear Estimation. Upper Saddle River, NJ: Prentice Hall, 2000.Google Scholar
  21. 21.
    K. Karhunen, “Uber lineare methoden in der wahrcheinlichkeitsrechnung,” Ann. Acad. Sci. Sci. Fenn., Ser. A, I, Math. Phys., vol. 37, pp. 3–79, 1947.Google Scholar
  22. 22.
    M. Loève, “Sur les fonctions alèatoires stationnaires du second ordre,” Revue Scientifique, vol. 83, pp. 297–310, 1945.MATHGoogle Scholar
  23. 23.
    M. Loève, Probability Theory, 3rd edition. Princeton, NJ: Van Nostrand, 1963.MATHGoogle Scholar
  24. 24.
    E. Wong and B. Hajek, Stochastic Processes in Engineering Systems. New York: Springer-Verlag, 1985.MATHGoogle Scholar
  25. 25.
    C. W. Helstrom, Elements of Signal Detection & Estimation. Upper Saddle River, NJ: Prentice-Hall, 1995.MATHGoogle Scholar
  26. 26.
    N. S. Jayant and P. Noll, Digital Coding of Waveforms – Principles and Applications to Speech and Video. Englewood Cliffs, NJ: Prentice-Hall, 1984.Google Scholar
  27. 27.
    N. Saito, “The generalized spike process, sparsity, and statistical independence,” in Modern Signal Processing (D. Rockmore and D. Healy, Jr., eds.), Cambridge, UK: Cambridge Univ. Press, 2004.Google Scholar
  28. 28.
    D. Slepian, “Some comments on Fourier analysis, uncertainty and modeling,” SIAM Review, vol. 25, pp. 379–393, July 1983.CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • Bernard C. Levy
    • 1
  1. 1.University of CaliforniaDavisUSA

Personalised recommendations