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Lectures on Wiener and Kalman Filtering

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Lectures on Wiener and Kalman Filtering

Part of the book series: International Centre for Mechanical Sciences ((CISM,volume 140))

Abstract

Suppose we have two random variables X, Y with a known joint density function fx,y(.,.). Assume that in a particular experiment, the random variable Y can be measured and takes the value y. What can be said about the corresponding value, say x, of the unobservable variable X?

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References

  1. V.S. Pugachev, “The Determination of an Optimal System by Some Arbitrary Criterion”, Automation and Remote Control, vol. 19, pp. 513–532, June 1958. See also R.B. Street, IEEE Trans. on Automatic Control, vol. AC-8, pp. 375–376, October 1963. Also M. Zalcai, IEEE Trans. on Information Theory, vol. IT-10, pp. 95–96, January 1964.

    Google Scholar 

  2. T. Kailath, “The Innovations Approach to Detection and Estimation Theory”, Proc. IEEE, vol. 58, pp. 680–695, May 1970.

    MathSciNet  Google Scholar 

  3. T. Kailath, “A View of Three Decades of Linear Filtering Theory”, IEEE Transactions on Information Theory, vol. IT-20, no. 2, pp. 145–181, March 1974.

    MathSciNet  Google Scholar 

  4. T. Kailath, ed., “Linear Least-Squares Estimation”, Benchmark Papers in Electrical Engineering, vol. 17. Academic Press, New York, 1977.

    Google Scholar 

  5. N. Wiener, Extrapolation, Interpolation and Smoothing of Stationary Time Series, with Engineering Applications. New York: Technology Press and Wiley, 1949, (Originally issued in February 1942, as a classified Nat. Defense Res. Council Rep).

    Google Scholar 

  6. M.C. Yovits and J.L. Jackson, “Linear filter optimization with game theory considerations”, in IRE Nat. Cony. Rec., pt. 4, pp. 193–199, 1955.

    Google Scholar 

  7. N. Levinson, “A heuristic exposition of Wiener’s mathematical theory of prediction and filtering”, J. Math. Phys., vol. 15, pp. 110–119, July 1947; reprinted as an Appendix in [i].

    Google Scholar 

  8. M.G. Krein, “Integral equations on a half -axis with kernel depending on the difference of the arguments”, Usp. Math. Nauk., vol. 13, pp. 3–120, 1958, Amer. Math. Socy. TransL

    Google Scholar 

  9. H.W. Bode and C.E. Shannon, “A simplified derivation of linear least squares smoothing and prediction theory”, Proc. IRE, vol. 38, pp. 417–425, April 1950.

    MathSciNet  Google Scholar 

  10. L.A. Zadeh and J.R. Ragazzini, “An extension of Wiener’s theory of prediction”, J. Appl. Phys., vol. 21, pp. 645–655, July 1950.

    MathSciNet  ADS  Google Scholar 

  11. T. Kailath, Linear Systems, New Jersey: Prentice-Hall, 1980.

    MATH  Google Scholar 

  12. T. Kailath, “A view of three decades of linear filtering theory”, IEEE Trans. on Information Theory, vol. IT-20, No. 2, pp. 145–181, March 1974.

    Google Scholar 

  13. F. Smithies, Integral Equations, New York: Cambridge University Press, 1962.

    Google Scholar 

  14. J.A. Cochran, Analysis of Linear Integral Equations, New York: M Jraw-Hill, 1972.

    MATH  Google Scholar 

  15. H. Laning and R. Battin, Random Processes in Automatic Control, New York; McGraw-Hill, 1958.

    Google Scholar 

  16. H.L. Van Trees, Detection, Estimation, and Modulation Theory, Part I, New York: Wiley, 1968.

    MATH  Google Scholar 

  17. P. Whittle, Prediction and Regulation, New York: Van Nostrand Reinhold, 1963.

    Google Scholar 

  18. M. Shinbrot, “A generalization of a method for the solution of the integral equation arising in optimization of time-varying linear systems with nonstationary inputs”, IRE Trans. Inform. Theory, vol. IT-3, pp. 220–225, Dec. 1957.

    Google Scholar 

  19. J. Bendat, Principles and Applications of Random Noise Theory, New York: Wiley, 1958.

    Google Scholar 

  20. E.L. Peterson, Statistical Analysis and Optimization of Systems, New York: Wiley, 1961.

    MATH  Google Scholar 

  21. P. Swerling, “First-order error propagation in a stagewise smoothing procedure for satellite observations”, J. Astronaut. Sci., vol. 6, pp. 46–52, Autumn 1959; see also “A proposed stagewise differential correction procedure for satellite tracking and prediction”, RAND Corp. Rep. p. 1292, Jan. 1958.

    Google Scholar 

  22. R.E. Kalman, “A new approach to linear filtering and prediction problems”, J. Basic Eng., vol. 82, pp. 34–45, March 1960.

    Google Scholar 

  23. RE. Kalman and R.S. Bucy, “New results in linear filtering and prediction theory”, Trans. ASME, Ser. D, J. Basic Eng., vol. 83, pp. 95–107, Dec. 1961.

    MathSciNet  Google Scholar 

  24. R.L. Stratonovich, “Application of the theory of Markoff processes in optimal signal detection”, Radio Eng. Electron. Phys., vol. 1, pp. 1–19, 1960 (Translation from Russian).

    Google Scholar 

  25. T. Kailath, “A View of Three Decades of Linear Filtering Theory’ IEEE Trans. Inform. Theory, vol. IT-20, pp. 146–181, 1974.

    Google Scholar 

  26. J. Rissanen and T. Kailath, “Partial realization of stochastic processes”, Automatica, vol. 8, pp. 380–386, July 1972.

    Google Scholar 

  27. M. Gevers and T. Kailath, “An innovations approach to least-squares estimation — Part VI: Discrete-time innovations representations and recursive estimation”, IEEE Trans. Automat Contr., vol. AC-18, pp. 588–600, Dec. 1973.

    Google Scholar 

  28. T. Kailath and P. Frost, “Mathematical modeling for stochastic processes” in Stochastic Problems in Control”, (Proc. Symp. AACC), Amer. Soc. Mech. Engrs., 1968.

    Google Scholar 

  29. T. Kailath, “An Innovations Approach to Least-Squares Estimation, Part I: Linear Filtering in Additive White Noise”, IEEE Trans. Automatic Control, vol. AC-13, pp. 646–655, December 1968.

    Google Scholar 

  30. M. Gevers and T. Kailath, “An Innovations Approach to Least-Squares Estimation, Part VI: Discrete-Time Innovations Representations and Recursive Estimation”, IEEE Trans. Automatic Control, vol. AC-18, pp. 588–600, December 1973.

    Google Scholar 

  31. R.E. Kalman, “A New Approach to Linear Filtering and Prediction Problems”, J. Basic Eng., vol.. 82, pp. 34–45, March 1960.

    Google Scholar 

  32. L. Ljung, T. Kailath and B. Friedlander, “Scattering Theory and Linear Least-Squares Estimation, Part I: Continuous-Time Problems”, Proc. IEEE, vol. 64, No. 1, pp. 131–139, January 1976.

    MathSciNet  Google Scholar 

  33. B. Friedlander, T. Kailath and L. Ljung, “Scattering Theory and Linear Least Squares Estimation, Part II: Discrete-Time Problems”, J. of The Franklin Institute, vol. 301, Nos. 1 & 2, pp. 71–82, January-February 1976.

    Google Scholar 

  34. B. Friedlander, G. Verghese and T. Kailath, “Scattering Theory and Linear Least-Squares Estimation”,Part III: The Estimates, Proceedings 1977 IEEE Decision and Control Conference, New Orleans, La, Dec. 1977. Also IEEE Trans. Automat. Control, vol. AC-25, pp. 794–802, Aug. 1980.

    Google Scholar 

  35. M. Morf and T. Kailath, “Square-Root Algorithms for Least-Squares Estimation”, IEEE Trans. Automatic Cont ol, vol. AC-20, No. 4, pp. 487–497, August 1975.

    MathSciNet  Google Scholar 

  36. G. Bierman, Factorization Methods for Discrete Sequential Estimation, New York Academic Press, 1977.

    Google Scholar 

  37. N. Levinson, “The ’ Wiener rms (root-mean-square) Error Criterion in Filter Design and Prediction”, J. Math. Phys., vol. 25, pp. 261–278, January 1947.

    MathSciNet  Google Scholar 

  38. E.A. Robinson, Multichannel Time-Series Analysis with Digital Computer Programs, San Francisco, Ca.: Holden-Day 1967.

    MATH  Google Scholar 

  39. B. Friedlander, M. Morf, T. Kailath and L. Ljung, “New Inversion Formulas for Matrices Classified in Terms of Their Distance From Toeplitz Matrices”, J. Lin. Alg. and Appins., vol. 27, pp. 31–60, Oct. 1979.

    MathSciNet  MATH  Google Scholar 

  40. B. Friedlander, T. Kailath, M. Morf and L. Ljung, “Levinson — and Chandrasekhar — type Equations for a General Discrete-Time Linear Estimation Problem”, Proceedings 1976 IEEE Decision and Control Conference, Florida, December 1976. Also IEEE Trans. Automat. Control, pp. 653–659, Aug. 1978.

    Google Scholar 

  41. B. Dickinson, M. Morf and T. Kailath, “Canonical Matrix Fraction and State Space Description for Deterministic and Stochastic Linear Systems”, IEEE Trans. on Automatic Control, Special Issue on System Identification and Time-Series Analysis, vol. AC-19, No. 6, pp. 656–667, December 1974.

    MathSciNet  Google Scholar 

  42. P. Swerling, Uncertain Dynamic Systems, Englewood Cliffs: Prentice-Hall, 1973.

    Google Scholar 

  43. F. Schweppe, “Modern State Estimation Methods from the Viewpoint of the Method of Least Squares”, IEEE Trans. Automatic Control, Vol. AC-16, pp. 707–720, December 1971.

    Google Scholar 

  44. R.E. Kalman, “New Methods of Wiener Filtering Theory”, in Proc. 1st Symp. Engineering Applications of Random Function Theory and Probability, J.L. Bogdanoff and F. Kozin, Eds., New York: Wiley, 1963, pp. 270–388; also, RIAS, Baltimore, Md., Tech. Rep. 61–1, 1961.

    Google Scholar 

  45. R.E. Kalman and R.S. Bucy, “New Results in Linear Filtering and Prediction Theory”, Trans. ASME, Ser. D, J. Basic Eng., vol. 83, pp. 95–107, December 1961.

    MathSciNet  Google Scholar 

  46. T. Kailath, “An Innovations Approach to Least-Squares Estimation, Pt. I: Linear Filtering in Additive White Noise”, IEEE Trans. on Automatic Control, vol. AC-13, pp. 646–655, December 1968.

    Google Scholar 

  47. J. Lawson and G.E. Uhlenbeck, Threshold Signals, MIT Radiation Laboratory Series, vol. 24, McGraw-Hill Book Co., New York, 1948.

    Google Scholar 

  48. K.J. Åstrom, Introduction to Stochastic Control Theory, Academic Press, New York, 1970.

    Google Scholar 

  49. T. Kailath, “A General Likelihood-Ratio Formula for Random Signals in Gaussian Noise,” IEEE Trans. on Information Theory, vol. IT-15, pp. 350–361, May 1969.

    Google Scholar 

  50. T. Kailath, “ A Note on Least-Squares Estimation by the Innovations Method,” J. SIAM Control, vol. 10 pp. 477–486, August 1972.

    MathSciNet  MATH  Google Scholar 

  51. T. Kailath, “Some Extensions of the Innovations Theorem,” Bell Syst. Tech. J., vol. 50, pp. 1487–1494, April 1971.

    MATH  Google Scholar 

  52. P.A. Meyer, “Sur un problème de filtration”, Séminaire de probabilités, Pt. VII, Lecture Notes in Mathematics, vol. 321, Springer, New York, pp. 223–247, 1973.

    Google Scholar 

  53. A. Segall, “Stochastic processes in estimation theory”, IEEE Trans. Inform. Theory, vol. IT-22, pp. 275–286, 1976.

    Google Scholar 

  54. T. Kailath and P. Frost, “An Innovations Approach to Least-Squares Estimation, Pt. II: Linear Smoothing in Additive White Noise,” IEEE Trans. on Automatic Control, vol. AC-13, pp. 655–660, December 1968.

    Google Scholar 

  55. S. Fujishige, “Minimum-Variance Estimation for a Linear Continuous-Discrete System with Noisy State-Integral Observation”, IEEE Trans. on Automatic Control, vol. AC-20, pp. 139–140, February 1975.

    Google Scholar 

  56. J. Mishra and V.S. Rajamani, “Least-Squares State Estimation in Time-Delay Systems with Colored Observation Noise: An Innovations Approach”, IEEE Trans. on Automatic Control, vol. AC-20, pp. 140–142, February 1975.

    Google Scholar 

  57. R. Redheffer, “On the Relation of Transmission-Line Theory to Scattering and Transfer”, J. Math. Phys., vol. 41, pp. 1–41, 1962.

    MathSciNet  MATH  Google Scholar 

  58. R. Redheffer, Difference Equations and Functional Equations in Transmission-Line Theory”, Ch. 12 in Modern Mathematics for the Engineer, 2nd series (E.F. Beckenbach, ed.), McGraw-Hill Book Co., New York, 1961.

    Google Scholar 

  59. G. Verghese, B. Friedlander and T. Kailath, “Scattering Theory and-Linear Least-Squares Estimation” Part III: The Estimates, Proceedings 1977 IEEE Decision and Control Conference, New Orleans, Dec. 1977. Also IEEE Trans. Automat. Contr., vol. AC-25, pp. 794–802, Aug. 1980.

    Google Scholar 

  60. L. Ljung, T. Kailath and B. Friedlander, “Scattering Theory and Linear Squares Estimation” Part I: Continuous-Time Problems, Proc. IEEE, vol. 64, pp. 131–139, January 1976.

    MathSciNet  Google Scholar 

  61. B. Friedlander, T. Kailath and L. Ljung, “Scattering Theory and Linear Least Squares Estimation” Part II: Discrete Time Problems, Journal of The Franklin Institute, vol. 301, Nos. 1&2, pp. 71–82, January-February 1976.

    Google Scholar 

  62. L. Ljung and T. Kailath, “A Unified Approach to Smoothing Formulas”, Automatica, vol. 12, pp. 147–157, March 1976.

    MathSciNet  MATH  Google Scholar 

  63. B.C. Lévy, Castanon, G.C. Verghese and A.S. Villsky, A Scattering Framework for Decentralized Estimation Problems, Proc. 1981 IEEE Decision and Control Conferetce, San Diego, Ca, Dec. 1981.

    Google Scholar 

  64. M.H.A. Davis, Linear Estimation and Stochastic Control, Halsted Press-Wiley, New York, 1977.

    MATH  Google Scholar 

  65. A.J. Viterbi, Principles of Coherent Communication, McGraw-Hill Book Co., New York.

    Google Scholar 

  66. H.L. Van Trees, Detection, Estimation, and Modulation Theory, Pt. I, J. Wiley & Sons, New York, 1968. Detection, Estimation and Modulation Theory, Pt. II: Nonlinear Modulation Theory, J. Wiley & Sons, New York 1971.

    Google Scholar 

  67. T. Kailath and L. Ljung, “Asymptotic Behaviour of Constant-Coefficient Riccati Differential Equations”, IEEE Trans. on Automatic Control, vol. AC-21, pp. 385–388, 1976.

    Google Scholar 

  68. T. Kailath, Linear Systems, Prentice-Hall, New Jersey, 1980.

    MATH  Google Scholar 

  69. V.M. Popov, “Hyperstability and Optimality of Automatic Systems with Several Control Functions”, Rev. Rolm. Sci. Tech., vol. 9, pp. 629–690, 1964.

    Google Scholar 

  70. R.E. Kalman, “When is a Linear Control System Optimal ?”, Trans. ASME, Ser. D., J. Basic Eng., vol. 86, pp. 51–60, June 1964.

    Google Scholar 

  71. T. Kailath, “A View of Three Decades of Linear Filtering Theory”, IEEE Transactions on Information Theory, vol. IT-20, pp. 145–181, March 1974.

    Google Scholar 

  72. B.D.O. Anderson and S. Vongpanitlerd, Network Analysis and Synthesis — A Modern Systems Theory Approach, Prentice-Hall, Inc., New Jersey, 1973.

    Google Scholar 

  73. V.M. Popov, Hyperstability of Control Systems, Springer-Verlag, New York, 1973. ( Original Rumanian edition, Bucharest, 1966 ).

    Google Scholar 

  74. P. Van Dooren, “A generalized eigenvalue approach for solving Riccati equations”, SIAM J. Scient. and Statist. Computing, vol. 2, pp. 121–135, June 1981.

    MATH  Google Scholar 

  75. R.E. Kalman, “Linear Stochastic Filtering-Reappraisal and Outlook Proc. Symp. System Theory, (J. Fox, ed.), Polytechnic Institute of Brooklyn Press, Brooklyn, New York, 1965.

    Google Scholar 

  76. P. Faurre, “Representation of Stochastic Processes”, Ph.D. Dissertation, Dept. of Electrical Engineering, Stanford University, Stanford, Ca., 1967.

    Google Scholar 

  77. P. Faurre, “Identification par minimisation d’une représentation Markovienne de processus aléatoire”, Lecture Notes in Mathematics, vol. 132, pp. 85–106, Springer-Verlag, New York, 1970.

    Google Scholar 

  78. T. Kailath, and R. Geesey, “An Innovations Approach to Least-Squares Estimation, Pt. IV: Recursive Estimation Given Lumped Covariance Functions”, IEEE Trans. on Automatic Control, vol. AC-16, pp. 720–727, December 1971.

    Google Scholar 

  79. T. Kailath, “A View of Three Decades of Linear Filtering Theory”, IEEE Trans. on Information Theory, vol. IT-20, pp. 145–181, March 1974.

    Google Scholar 

  80. T. Kailath, “Fredholm Resolvents, Wiener-Hopf Equations and Riccati Differential Equations”, IEEE Trans. on Information Theory, vol. IT-15, pp. 665–672, November 1969.

    Google Scholar 

  81. B.D.O. Anderson and T. Kailath, “The Choice of Signal Process Models in Kalman-Bucy Filtering”, J. Math. Anal. Appl., vol. 35, pp. 659–668, September 1971.

    MathSciNet  MATH  Google Scholar 

  82. T. Kailath, R. Geesey and H. Weinert, “Some Relations Between RKHS Norms, and Fredholm Equations, ands Innovations Representations”, IEEE Trans. on Information Theory, vol. IT-18, pp. 341–348, May 1972.

    Google Scholar 

  83. A. Schumitzky, “On the Equivalence Between Matrix Riccati Equations and Fredholm Resolvents”, J. Comp. Syst. Sei., vol. 2, pp. 76–87, June 1968.

    MathSciNet  MATH  Google Scholar 

  84. T. Kailath and R. Geesey, “An Innovations Approach to Least-Squares Estimation, Pt. V: Innovations Representations and Recursive Estimation in Colored Noise”, IEEE Trans. on Automatic Control, vol. AC-18, pp. 435–453, October 1973.

    Google Scholar 

  85. J.B. Moore and B.D.O. Anderson, “Spectral Factorization of Time-Varying Covariance Functions: The Singular Case”, Mathematical System Theory, vol. 4, pp. 10–23, 1970.

    MathSciNet  MATH  Google Scholar 

  86. B.D.O. Anderson and J.B. Moore, “The Kalman-Bucy Filter as a True Time-Varying Wiener Filter”, IEEE Trans. Syst., Man, Cybernetics, vol. SMC-1, pp. 119–128, 1971.

    Google Scholar 

  87. M.G. Krein, “On Integral Equations Leading to Differential Equations of Second Order”, Dokl. Akad. Nauk SSSR, 97, pp. 21–24, 1954.

    MathSciNet  Google Scholar 

  88. T. Kailath, “Application of a Resolvent Identity to a Linear Smoothing Problem”, SIAM J. Contr., vol. 7, pp. 68–74, 1969.

    MathSciNet  MATH  Google Scholar 

  89. V.V. Sobolev, A Treatise on Radiative Transfer, D. Van Nostrand, Princeton, N.J., 1963, (Russian edition 1956 ).

    Google Scholar 

  90. J.L. Casti, RE. Kalaba and V.K. Murthy, “ A new initial-value method for on-line filtering and estimation,” IEEE Trans. Inform. Theory, vol. IT-18, pp. 515–518, July 1972.

    Google Scholar 

  91. S. Chandrasekhar, “On the radiative equilibrium of a stellar atmosphere, Pt XXI,” Astrophys. J., vol. 106, pp. 152–216, 1947; Pt. XXII, ibid, vol. 107, pp. 48–72, 1948.

    Google Scholar 

  92. V.A. Ambartsumian, “Diffuse reflection of light by a foggy medium”, Dokl. Akad. Sci. SSSR, vol. 38, pp. 229–322, 1943.

    Google Scholar 

  93. V.V. Sobolev, A Treatise on Radiative Transfer, New York: Van Nostrand, 1963 (translated from Russian original, 1958 ).

    Google Scholar 

  94. T. Kailath, “Some Chandrasekhar-type algorithms for quadratic regulator problems”, in Proc. IEEE Conf. Decision and Control and 11th Symp. Adaptive Processes, IDec. 1972, pp. 219–223. Also Technical Report, Computing and Control Dept., Imperial College, London, Aug. 1972.

    Google Scholar 

  95. T. Kailath, “Some new algorithms for recursive estimation in constant linear systems”, IEEE Trans. Inform. Theory, vol. IT-19, pp. 750–760, Nov. 1973.

    Google Scholar 

  96. T. Kailath, M. Morf and G.S. Sidhu, “Some new algorithms for recursive estimation in constant discrete-time linear systems”, in Proc. 7th Princeton Symp. Information and System Science, 1973. See also IEEE Trans. Automat. Contro., vol. AC-19, pp. 315–323, Aug. 1974.

    Google Scholar 

  97. B. Dickinson, M. Morf and T. Kailath, “Canonical matrix fraction and state space description for deterministic and stochastic linear systems”, IEEE Trans. on Automatic Control, Special Issue on System Identification and Time-Series Analysis, vol. AC-19, No. 6, pp. 656–667, Dec. 1974.

    MathSciNet  Google Scholar 

  98. M. Morf and T. Kailath, “Square-root algorithms for linear least-squares estimation and control”, IEEE Trans. Auto. Cont., vol. AC-20, pp. 487–497, Aug. 1975.

    Google Scholar 

  99. L.Ljung and T. Kailath, “A scattering theory framework for fast least-squares algorithms”, in Multivariate Analysis IV, ed by P.R. Krishnaiah, North-Holland, Amsterdam, 1977. [Presented at Symposium, June 1975 ].

    Google Scholar 

  100. L. Ljung, T. Kailath and B. Friedlander, “Scattering Theory and Linear Least-Squares Estimation, Part I — Continuous-time Problems, Proceedings of the IEEE, vol. 64, pp. 131–139, Jan. 1976.

    MathSciNet  Google Scholar 

  101. B. Friedlander, T. Kailath and L. Ljung, “Scattering Theory and Linear Least-squares Estimation”, Part II — Discrete-time Problems, J. Framblin Inst., vol. 301, pp. 71–82, Feb. 1976.

    MathSciNet  MATH  Google Scholar 

  102. G. Verghese, B. Friedlander and T. Kailath, “Scattering Theory and Linear Least-squares Estimation”, Part III — The Estimates, IEEE Trans. Automat. Contr., vol. AC-25, pp. 794–802, Aug. 1980.

    Google Scholar 

  103. T. Kailath and L. Ljung, “Efficient Change of Initial Conditions, Dual Chandrasekhar Equations, and Some Applications”, IEEE Trans. Autom. Contr., vol. AC-22, pp. 443–447, June 1977.

    Google Scholar 

  104. T. Kailath, A. Vieira and M. Morf, “Orthogonal Transformation, Square-root Implementations of Fast Estimation Algorithms”, Int. Symp. on Systems Optim. and Analysis, pp. 81–91, Lecture Notes in Control and Inform. Sciences, vol. 14, New York: Springer-Verlag, 1979.

    Google Scholar 

  105. T. Kailath, B. Lévy, L. Ljung and M. Morf, “The factorization and Representation of Operators in the Algebra Generated by Toeplity Operators”, SIAM J. Appl. Math., vol. 37, pp. 467–484, Dec. 1979.

    MATH  Google Scholar 

  106. R.S. Bucy and P.D. Joseph, Filtering for Stochastic Processes with Applications to Guidance. New York: Wiley, 1968.

    MATH  Google Scholar 

  107. R. Bellman, R. Kalaba and J. Wing, “Invariant imbedding and mathematical physics-I: Particle processes”, J. Math. Phys., vol. 1, pp. 280–308, 1960.

    MathSciNet  MATH  ADS  Google Scholar 

  108. G.C. Stokes, Collected Papers, vol. 2, pp. 91–93, Cambridge Univ. Press, 1883.

    Google Scholar 

  109. M. Morf, B. Lévy and T. Kailath, “Square-root Algorithms for the Continuous-time Least-squares Estimation Problem”, IEEE Trans. Automat. Contr., vol. AC-23, pp. 907–911, Oct. 1978.

    Google Scholar 

  110. T. Kailath, “A View of Three Decades of Linear Filtering Theory”, IEEE Trans. on Inform. Thy., IT-20, no. 2, pp. 145–181, March 1974.

    MathSciNet  Google Scholar 

  111. D. Jacobson, “Totally Singular Quadratic Minimization”, IEEE Trans. on Autom., Contr., AC-16, pp. 651–658, 1971.

    Google Scholar 

  112. N. Krasner, “Separable Kernels in Linear Estimation and Control”, Ph.D. Dissertation, Dept. of Elec. Engg., Stanford Univ., Stanford, CA, May 1974.

    Google Scholar 

  113. D.J. Clements and B.D.O. Anderson, Singular Optimal Control: The Linear-Quadratic Problem”, Lecture Notes in Control, vol. 5, Springer-Verlag, New York, 1978.

    Google Scholar 

  114. L. Silverman, “Discrete Riccati Equations: Alternative Algorithms, Asymptotic Properties and System Theory Interpretations”, in Advances in Control, vol. 12, C.T. Leondes, ed., pp. 313–3386, Academic Press, New York, 1976.

    Google Scholar 

  115. B.P. Molinari, “A Strong Controllability and Observability in Linear Multivariable Control”, IEEE Trans. on Autom. Contr., AC-21, pp. 761–764, October 1976.

    Google Scholar 

  116. G. Verghese, “Infinite-Frequency Behaviour in Generalized Dynamic Systems”, Ph.D. Dissertation, Dept. of Elec. Engg., Stanford Univ., Stanford, CA, December 1978.

    Google Scholar 

  117. A. Emami Naeini, “Application of the Generalized Eigenstructure Problem to Multivariable Systems and the Robust Servomechanism for a Plant which Contains an Implicit Internal Model”, Ph.D. Dissertation, Dept. of Elec., En._., Stanford Univ., Stanford, CA, May 1981.

    Google Scholar 

  118. W.L. Chan, “Variational Dualities in the Linear Regulator and Estimation Problems”, J. Inst. Math. Appl., 18, pp. 237–248, October 1976.

    MathSciNet  MATH  Google Scholar 

  119. U.B. Desai and H.L. Weinert, “Generalized Control-Estimation Duality and Inverse Projections”, Proc. 1979 Conf. on Information Sciences and Systems, pp. 115–120, John Hopkins Univ., Maryland, March 1979.

    Google Scholar 

  120. A. Lindquist, “Optimal Control of Linear Stochastic Systems with Applications to Time Lag Systems”, Information Sci., 5, pp. 81–126, 1973.

    Google Scholar 

  121. M.H.A. Davis Linear Estimation and Stochastic Control, Halsted Press, New York, 1977.

    Google Scholar 

  122. T. Kailath, “A General Likelihood Ratio Formula for Random Signals in Noise”, IEEE Trans. on Inform. Thy., IT-15, no. 3, pp. 350–361, May 1969.

    Google Scholar 

  123. T. Kailath, Likelihood Ratios for Gaussian Processes, IEEE Trans. on Inform. Thy., IT-16, no. 3, pp. 276–288, May 1970.

    MathSciNet  Google Scholar 

  124. M.H.A. Davis and E. Andreadakis, “Exact and Approximate Filtering in Signal Detection: An Example”, IEEE Trans. on Inform. Thy., IT-23, pp. 768–772, Nov. 1977.

    Google Scholar 

  125. T. Kailath, B. Levy, L. Ljung and M. Morf, “Fast Time-Invariant Implementations of Gaussian Signal Detectors”, IEEE Trans. Inform. Thy., IT-24, no. 4, pp. 469–477, July 1978.

    MathSciNet  Google Scholar 

  126. B. Levy, T. Kailath, L. Ljung and M. Morf, “Fast Time-Invariant Implementations for Linear Least-Squares Smoothing Filters”, IEEE Trans. Autom. Contr., 24, no. 5, pp. 770–774, Oct. 1979.

    MathSciNet  MATH  Google Scholar 

  127. T. Kailath, A. Vieira and M. Morf, “Inverses of Toeplitz Operators, Innovations, and Orthogonal Polynomials”, SIAM Rev., 20, no. 1, pp. 106–119, Jan. 1978.

    MathSciNet  MATH  Google Scholar 

  128. B. Friedlander, M. Morf, T. Kailath and L. Ljung, “New Inversion Formulas for Matrices Classified in Terms of their Distance from Toeplitz Matrices”, Linear Algebra and Its Appls., pp. 31–60, Oct. 1979.

    Google Scholar 

  129. M. Morf, “Doubling Algorithms for Toeplitz and Related Equations”, Proc. Int.’I. Conf. on Acoustics, Speech and Signal Proc., pp. 954–959, Denver, Co., 1980.

    Google Scholar 

  130. J.M. Delosme, “Fast Algorithms for Finite Shift-Rank Processes”, Ph.D. Dissertation, Stanford Univ., Dept. of Elec. Engg., Stanford, CA, Dec. 1981.

    Google Scholar 

  131. H. Lev-Ari, “Modeling and Parametrization of Nonstationary Stochastic Processes”, Ph.D. Dissertation, Stanford Univ., Dept. of Elec., Engg., Stanford, CA, 1982.

    Google Scholar 

  132. B. Porat, “Contributions to the Theory and Applications of Lattice Filters”, Ph.D. Dissertation, Stanford Univ., Dept. of Elec. Eny:., Stanford, CA, Dec. 1982.

    Google Scholar 

  133. T. Kailath, “Equations of Wiener-Hopf Type in Filtering Theory and Related Applications”, in Norbert Wiener: Collected Works, vol. III, edited by P.R. Masani MIT Press, 1982.

    Google Scholar 

  134. C.W. Helstrom, “Solution of the Detection Integral Equation for Stationary Filtered White Noise”, IEEE Trans. on Inform. Thy., IT-11, pp. 335–339, 1965.

    Google Scholar 

  135. T. Kailath, “Fredholm Resolvents, Wiener-Hopf Equations and Riccati Differential’ Equations”, IEEE Trans. on Inform. Thy., IT-15, no. 6, pp. 655–672, November 1969.

    Google Scholar 

  136. B. Anderson and T. Kailath, “Some Integral Equations with Nonsymmetric Separable Kernels”, SIAM J. Appl. Math., 20, no. 4, pp. 659–669, June 1971.

    MathSciNet  Google Scholar 

  137. T. Kailath, R. Geesey and H. Weinert, “Some Relations Between RKHS Norms, Innovation Representations and Fredholm Equations”, IEEE Trans. on Inform. Thy., IT-18, no. 3, pp. 341–348, May 1972.

    MathSciNet  Google Scholar 

  138. T. Kailath, L. Ljung and M. Morf, “Generalized Krein-Levinson Equations for Efficient Calculation of Fredholm Resolvents of Nondisplacement Kernels”, in Topics in Functional Analysis, pp. 169–184, I.C. Gohberg and M. Kac, eds., Academic Press, New York, 1978.

    Google Scholar 

  139. A.J. Berkhout, “Stability and Least-Squares Estimation”, Automatica, 11, pp. 633–638, 1975.

    Google Scholar 

  140. J.C. Willems, “Dissipative Dynamical Systems I: General Theory; II: Linear Systems with Quadratic Supply Rates”, Archive for Rational Mechanics and Analysis, 45, pp. 321–343, 1972.

    MathSciNet  MATH  ADS  Google Scholar 

  141. A. Vieira and T. Kailath, “On Another Approach to the Schur-Cohn Criterion”, IEEE Trans. Circuits and Systems, 24, pp. 218–220, April 1977.

    MathSciNet  MATH  Google Scholar 

  142. F.J. Dyson, “Old and New Approaches to the Inverse Scattering Problem”, in Studies in Mathematical Physics, Essays in Honor of Valentine Bargmann, pp. 151–167, E.H. Lieb, B. Simon and A.S. Wightman, eds., Princeton Univ. Press, 1976.

    Google Scholar 

  143. B.D. Anderson and T. Kailath, “Fast Algorithms for the Integral Equations of the Inverse Scattering Problems”, Integral Eqs. and Operator Thy., 1, no. 1, pp. 132–136, 1978.

    MathSciNet  MATH  Google Scholar 

  144. K.M. Case, “Inverse Scattering, Orthogonal Polynomials and Linear-Estimation”, in Topics in Functional Analysis, pp. 25–43, I. Gohberg and M. Kac, eds., Academic Press, New York, 1978.

    Google Scholar 

  145. H. Dym, “Applications of Factorization Theory to the Inverse Spectral Problem”, Proc. Intl. Symp. on Math. Thy. of Networks and Systems, pp. 188–193, Delft, Holland, July 1979. Published by Western Periodicals Co., North Hollywood, CA.

    Google Scholar 

  146. P. Dewilde, J. Fokkema and I. Widya, “Inverse Scattering and Linear Prediction: The Continuous-Time Case”, in Stochastic Systems, M. Hazewinkel and J.C. Willems, eds., Reidel, 1981.

    Google Scholar 

  147. H. Akaike, “Stochastic Theory of Minimal Realization”, IEEE Trans. Autom. Contr., AC-19, pp. 667–674, Dec. 1974.

    Google Scholar 

  148. H. Akaike, “Markovian Representation of Stochastic Processes by Canonical Variables”, SIAM J. Contr., 13, pp. 162–173, 1975.

    MathSciNet  MATH  Google Scholar 

  149. B.D.O. Anderson and T. Kailath, “The Choice of Signal Process Models in Kalman-Bucy Filtering”, J. Math. Anal. and Appls., 35, pp. 659–668, Sept. 1971.

    MathSciNet  MATH  Google Scholar 

  150. A. Lindquist and G. Picci, “On the Stochastic Realization Problem”, SIAM J. Contr. Optim., 17, pp. 365–389, 1979.

    MathSciNet  MATH  Google Scholar 

  151. G. Ruckebusch, “On the Theory of Markovian Representation”, in Lecture Notes in Mathematics, 695, pp. 77–88, Springer-Verlag, New York, 1978.

    Google Scholar 

  152. T. Kailath and L. Ljung, “Strict-Sense State-Space Realizations of Nonstationary Gaussian Processes”, to appear.

    Google Scholar 

  153. B.D.O. Anderson and S. Vongpanitlerd, Network Analysis and Synthesis, A Modern Systems Theory Approach, Prentice-Hall, New Jersey, 1973.

    Google Scholar 

  154. P. Dewilde, A. Vieira and T. Kailath, “On a Generalized Szego-Levinson Realization Algorithm for Optimal Linear Prediction Based on a Network Synthesis Approach”, IEEE Trans. Circuits and Systems, CAS-25, pp. 663–675, Sept. 1978.

    Google Scholar 

  155. B.D.O. Anderson and T. Kailath, “Forwards and Backwards Models for Finite-State Markov Processes”, Adv. Appl. Probl., 11, pp. 118–133, 1979.

    MathSciNet  MATH  Google Scholar 

  156. P. Dewilde and H. Dym, “Schur Recursions, Error Formulas and Convergence of Rational Estimators for Stationary Stochastic Sequences”, IEEE Trans. Inform. Thy., IT-pp. 446–461, 1981.

    Google Scholar 

  157. Y. Genin and P. Van Dooren, “On E -Lossless Transfer Functions and Related Questions”, Philips Res. Lab. Tech. Rept. R447, Brussels, Belgium, Nov. 1980.

    Google Scholar 

  158. D.Z. Arov, “Passive Linear Stationary Dynamic Systems”, Siberian Mat. J., 20, pp. 149–162, 1979.

    MATH  Google Scholar 

  159. G. Kimeldorf and G. Wahba, “Spline Functions and Stochastic Processes”, Sankhya, 132, pp. 173–180, 1970.

    MathSciNet  Google Scholar 

  160. H.L. Weinert and T. Kailath, “Stochastic Interpretations and Recursive Algorithms for Spline Functions”, Annals of Stat., 2, pp. 787–794, 1974.

    MathSciNet  MATH  Google Scholar 

  161. H.L. Weinert and G.S. Sidhu, “A Stochastic Framework for Recursive Computation of Spline Functions, Pt. I, Interpolating Splines”, IEEE Trans. Inform. Thy, IT-24, pp. 45–50, Jan. 1978.

    Google Scholar 

  162. R.J.P. de Figueireido and A. Caprihan, “An Algorithm for the Generalized Smoothing Spline with Application to System Identification”, Proc. 1977 Conf. Inform. Sciences and Systems, John Hopkins Univ., Baltimore, MD, April 1977.

    Google Scholar 

  163. E. Parzen, “Multiple Time-Series Modeling”, in Multivariate Analysis — II, P.R. Krishnaiah, ed., pp. 389–409, Academic Press, New York, 1969.

    Google Scholar 

  164. J.P. Burg, “Maximum Entropy Spectral Analysis”, Ph.D. Dissertation, Dept. of Geophysics, Stanford Univ., Stanford, CA, 1975.

    Google Scholar 

  165. M. Morf, A. Vieira, D. Lee and T. Kailath, “Recursive Multichannel Maximum Entropy Spectral Estimation”, IEEE Trans. Geoscience Elect., GE-16, no. 2, pp. 85–94, April 1978.

    MathSciNet  ADS  Google Scholar 

  166. D.G. Childers, ed., Modern Spectrum Analysis, IEEE Press, New York, 1978.

    Google Scholar 

  167. S. Haykin, ed., Monlinear Spectral Analysis, Sprjnger-Verlag, New York, 1979.

    Google Scholar 

  168. K.J. Astrom and P. Eykhoff, “System Identification–A Survey”, Automatica, 7, pp. 123–162, 1971.

    MathSciNet  Google Scholar 

  169. M. Morf, T. Kailath and L. Ljung, “Fast Algorithms for Recursive Identifcation”, Proc. 1975 IEEE Conf. on Decision and Contr., pp. 916–921, Florida, Dec. 1976.

    Google Scholar 

  170. V. Solo, “Time Series Recursions and Stochastic Approximation”, Ph.D. Dissertation, Dept. of Statistics, Australian National Univ., Camberra, Australia, 1978.

    Google Scholar 

  171. R. Isermann, ed., Special Issue on Identification and System Parameter Estimation, Automatica, 17, pp. 1–259, Jan. 1981.

    Google Scholar 

  172. V. Strejc, “Trends in Identification”, Automatica, 17, pp. 1–22, Jan 1981.

    MathSciNet  Google Scholar 

  173. P. Young, “Parameter Estimation for Continuous-Time Models — A Survey”, Automatica, 17, pp. 23–40, Jan. 1981.

    MATH  Google Scholar 

  174. L. Ljung and T. Soderstrom, Theory and Practice of Recursive Identification, MIT Press, Cambridge, MA, 1982.

    Google Scholar 

  175. D. Messerschmitt, “A Geometric Theory of Intersymbol Interference, Pt. I: Zero-Forcing and Decision-Feedback Equalization”, Bell Syst. Tech. J., pp. 1483–1519, 1973.

    Google Scholar 

  176. D. Godard, “Channel Equalization Using a Kalman Filer for Fast Data Transmission”, IBM J. Res. Develop., pp. 267–273, May 1974.

    Google Scholar 

  177. D.D. Falconer and L. Ljung, “Application of Fast Kalman, Estimation to Adaptive Equalization”, IEEE Trans. Communications, COM-26, pp. 1439–1446, Oct. 1978.

    Google Scholar 

  178. L.E. Franks, ed., Data Communication, Benchmark Papers in Electrical Engineering and Computer Science, Halsted Press, New York, 1974.

    Google Scholar 

  179. E.H. Satorius and J.D. Pack, “Application of Least-Squares Lattice Algorithms to Adaptive Equalization”, IEEE Trans. on Communication, COM-29, pp. 136–142, Feb. 1981.

    Google Scholar 

  180. M.S. Mueller and J. Salz, “A Unified Theory of Data-Aided Equalization” Bell Syst. Tech. J., 1981.

    Google Scholar 

  181. V.U. Reddy, A.M. Peterson and T. Kailath, “Application of Modified Least-Squares Algorithms to Adaptive Echo Cancellation”, Proc. Int’L Symp. on Microwaves and Communication, Kharagpur, INDIA, Dec. 1981.

    Google Scholar 

  182. L.C. Wood and S. Treitel, “Seismic Signal Processing”, Proc. IEEE, 63, pp. 649–661, A April 1975.

    Google Scholar 

  183. J.D. Markel and A.H. Gray, Linear Prediction of Speech, Springer-Verlag, New York, 1976.

    MATH  Google Scholar 

  184. J. Makhoul, “Linear Prediction: A Tutorial Review”, Proc IEEE, 63, pp. 561–580, April 1975.

    ADS  Google Scholar 

  185. M. Morf and D.T. Lee, “Recursive Least-Squares Ladder Forms for Fast Parameter Tracking”, Proc. 1978 IEEE Conf. on Decision and Contr., pp. 1362–1367, San Diego, CA, Jan. 1979.

    Google Scholar 

  186. S. Wood and M. Morf, “A Fast Implementation of a Minimum Variance Estimator for Computerized Tomography Image Reconstruction”, Trans. Bio-Medical Engineering, BME-28, pp. 56–88, Feb. 1981.

    Google Scholar 

  187. S.R. Parker and L.J. Griffiths, eds., Joint Special Issue on Adaptive Signal Processing, IEEE Trans. on Acoustics, Speech and Signal Processing, ASSP-29, June 1981.

    Google Scholar 

  188. A. Benveniste and P. Dewilde, eds., Fast Algorithms for Linear Dynamical Systems, published by INRIA/CNRS, Paris, France, 1982. 48

    Google Scholar 

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  1. Department of Electrical Engineering, Stanford University, Stanford, California, 94305, USA

    Thomas Kailath

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Kailath, T. (1981). Lectures on Wiener and Kalman Filtering. In: Lectures on Wiener and Kalman Filtering. International Centre for Mechanical Sciences, vol 140. Springer, Vienna. https://doi.org/10.1007/978-3-7091-2804-6_1

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  • DOI: https://doi.org/10.1007/978-3-7091-2804-6_1

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