Journal of Statistical Physics

, Volume 13, Issue 6, pp 491–514 | Cite as

Dynamic observation models

  • Michael Schilder


It is shown that radar and quantum mechanics may be modeled using the Kalman-Bucy state-equation observation approach. A method is given for realizing the optimal position filters.

Key words

Radar quantum mechanics nonlinear filter Doppler shift optimal filter 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    E. Nelson,Dynamical Theories of Brownian Motion, Princeton University Press, Princeton, New Jersey (1967).Google Scholar
  2. 2.
    R. E. Kaiman, “A New Approach to Linear Filtering and Prediction Problems,”J. Basic Eng. (ASME Trans.) 82D:35–45 (1960).Google Scholar
  3. 3.
    D. E. Vakman,Sophisticated Signals and The Uncertainty Principle in Radar, Springer Verlag, New York (1968).Google Scholar
  4. 4.
    J. DiFranco and W. L. Rubin,Radar Detection, Prentice Hall, Englewood Cliffs, New Jersey (1968).Google Scholar
  5. 5.
    R. P. Wishner, R. E. Larson, and M. Athens, “Status of Radar Tracking Algorithms,” inProc. Symposium on Nonlinear Estimation Theory and its Applications, Univ. of California at San Diego, September 1970.Google Scholar
  6. 6.
    R. T. Feynman and R. Hibbs,Quantum Mechanics and Path Integrals, McGraw-Hill, New York (1965).Google Scholar
  7. 7.
    J. M. F. Moura, H. L. Van Trees, and A. B. Baggeroer, “Space/Time Tracking with a Passive Observer,” inProceedings of the Fourth Symposium on Nonlinear Estimation Theory and its Applications, Univ. of California at San Diego, September 1973.Google Scholar
  8. 8.
    R. H. Cameron and D. A. Storvick, “An Operator Valued Function Space Integral Applied to Integrals of Functions of ClassL 2,”J. Math. Anal. Appl. 42:330–372 (1973).Google Scholar
  9. 9.
    G. W. Johnson and D. L. Skoug, “Feynman Integrals of Non-factorable Finite-Dimensional Functional,”Pacific J. Math. 45:257–267 (1973).Google Scholar
  10. 10.
    G. W. Johnson and D. L. Skoug, “Operator Valued Feynman Integrals of Finite Dimensional Functions,”Pacific J. Math. 34:415–425 (1970).Google Scholar
  11. 11.
    J. L. Doob,Stochastic Processes, Wiley, New York (1953).Google Scholar
  12. 12.
    M. Schilder, “New Applications of Wiener Integrals to Engineering and Physics,”J. Stat. Phys. 1(3):475–516 (1969).Google Scholar
  13. 13.
    R. H. Cameron, “Nonlinear Volterra Functional Equations,”J. d'Analyse Math. 5:470–510 (1957).Google Scholar
  14. 14.
    M. Kac, “Wiener and Integration on Function Spaces,”Bull. Am. Math. Soc. 75(1, II) (January 1966).Google Scholar
  15. 15.
    R. E. Mortensen, “Optimal Control of Continuous Time Stochastic Systems,” Report ERL-66-1, College of Engineering, University of California, Berkeley, August 1966.Google Scholar
  16. 16.
    R. S. Bucy and P. D. Josephs,Filtering for Stochastic Processes with Applications to Guidance, Wiley, New York (1968).Google Scholar
  17. 17.
    M. Zakai, “On the Optimal Filtering of Diffusion Processes,”Z. Wahrschein-lichkeitstheorie II:230–243.Google Scholar
  18. 18.
    I. V. Girsanov, “On the Transformation of a Class of Stochastic Processes by Means of an Absolutely Continuous Change of Measure,” inTheory of Probability and its Applications (1960), pp. 314–330.Google Scholar
  19. 19.
    H. P. McKean, “Wiener's Theory of Nonlinear Noise,” inStochastic Differential Equations, Vol. VI, SIAM-AMS Proceedings, American Math. Soc, Providence, Rhode Island (1973).Google Scholar
  20. 20.
    Symposia on Nonlinear Estimation Theory, San Diego, Vols. I, II, III, and IV, Western Periodicals Co., North Hollywood, California (1970–1973).Google Scholar
  21. 21.
    M. Schilder, “A Power Series Nonlinear Filter,” inFourth Symposium on Nonlinear Estimation Theory, San Diego, September 1973.Google Scholar
  22. 22.
    M. Kac,Probability and Related Topics in the Physical Sciences, Interscience Publishers, London (1959).Google Scholar
  23. 23.
    E. B. Dynkin,Markov Processes, Academic Press, New York (1965).Google Scholar
  24. 24.
    Reference Data For Engineers, 4th ed., International Telephone and Telegraph Corporation.Google Scholar
  25. 25.
    N. Wiener,Nonlinear Problems in Random Theory, M.I.T. Press, Cambridge, Massachusetts (1958).Google Scholar
  26. 26.
    H. P. McKean, “Geometry of Differential Space,”Ann. Probability 1(2):197–206 (1973).Google Scholar
  27. 27.
    R. King, “Stochastic Integrals and Metadistributions: Applications to Stochastic Partial Differential Equations and Quantum Field Theory,” Ph.D Thesis, Cornell University (1975).Google Scholar
  28. 28.
    W. Feller,An Introduction to Probability Theory and Its Applications, Vol. I, 2nd ed. Wiley, New York (1957).Google Scholar
  29. 29.
    N. Wiener,Extrapolation, Interpolation, and Smoothing of Stationary Time Series, M.I.T. Paperback, Cambridge, Massachusetts (1964).Google Scholar
  30. 30.
    H. Kushner,Stochastic Stability and Control, Academic Press, New York (1967).Google Scholar
  31. 31.
    M. Schilder, “A Solution to the Generalized Heat Equation Along its Generalized Characteristics,” Presented to the American Math. Society, March 1972, New York.Google Scholar

Copyright information

© Plenum Publishing Corporation 1975

Authors and Affiliations

  • Michael Schilder
    • 1
  1. 1.Department of MathematicsCornell UniversityIthaca

Personalised recommendations