An Innovations Approach to Least-Squares Estimation—Part III: Nonlinear Estimation in White Gaussian Noise

IEEE Transactions on Automatic Control

Volumn 16, Issue 3, 1971, Pages 217-226

  Frost, Paul A ^a   Kailath, Thomas ^b  

^a LUCENT TECHNOLOGIES   (United States)

^b STANFORD UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

INFORMATION THEORY; NOISE;

ESTIMATION; WHITE NOISE;

AUTOMATIC CONTROL--OPTIMIZATION;

EID: 0015078019     PISSN: 00189286     EISSN: 15582523     Source Type: Journal    
DOI: 10.1109/TAC.1971.1099704     Document Type: Article

References (37)

1. T. Kailath, “An innovations approach to least-squares estimation-Part I: Linear filtering in additive white noise,” IEEE Trans. Automat. Contr., vol. AC-13, Dec. 1968, pp. 646–654.
2. T. Kailath and P. Frost, “An innovations approach to least-squares estimation-Part II: Linear smoothing in additive white noise,” IEEE Trans. Automat. Contr., vol. AC-13, Dec. 1968, pp. 655–660.
3. P. Masani, “Wiener’s contributions to generalized harmonic analysis, prediction theory and filtering theory,” Bull. Amer. Math. Soc., vol. 72, pt. II, Jan. 1966, pp. 73–125.
4. P. Masani and N. Wiener, “Nonlinear prediction,” in Proc. 4th Berkeley Symp. Mathematical Statistics and Probability, vol. 2, 1961, pp. 403–419.
5. P. Frost, “Nonlinear estimation in continuous time systems,” Ph.D. dissertation, Stanford Univ., Stanford. Calif. 1968.
6. T. Kailath, “A general likelihood ratio formula for random signals in Gaussian noise,” IEEE Trans. Inform. Theory, vol. IT-15, May 1969, pp. 350–361.
7. T. Kailath and P. Frost, “Mathematical modeling of stochastic processes,” in Stochastic Problems in Control, Symp. Amer. Automat. Contr. Council, 1968. pp. 1–38.
8. R.L. Stratonovich, “Conditional Markov process theory,” Theory Prob. Appl. (USSR), vol. 5, 1960, pp. 156–178.
9. H.J. Kushner, “On differential equations satisfied by conditional probability densities of Markov processes,” SIAM J. Contr., vol. 2, 1964, pp. 106–119.
10. H.J. Kushner, “On the dynamical equations of conditional probability density functions, with applications to optimal stochastic control theory.” J. Math. Anal. Appl., vol. 8, 1964, pp. 332–344.
11. W.M. Wonham, “Some applications of stochastic differential equations to optimal nonlinear filtering,” SIAM J. Contr., vol. 2, 1965, pp. 347–369.
12. R.S. Buc y, “Nonlinear filtertying,” IEEE Trans. Automat. Contr. (Corresp.), vol. AC-10, Apr. 1965, p. 198.
13. R.S. Bucy and P.D. Joseph, Filtering for Stochastic Processes with Applications to Guidance. New York: Interscience, 1968.
14. R.E. Mortensen, “Optimal control of continuous time stochastic systems,” Ph.D. dissertation. Univ. of Calif. Berkeley. Calif., 1966.
15. A.N. Shiryaev, “On stochastic equations in the theory of conditional Markov processes:” Theory Prob. Appl. (USSR), vol. 11, 1966, pp. 179–184.
16. A.N. Shiryaev, “Stochastic equation of nonlinear filtering of Markovian jump process.” Problems of Information Transmission, vol. 2. 1966. pp. 1–18.
17. R. Lipster and A.N. Shiryaev, “nonlinear interpolation of components of Markov diffusion processes.” Theory Prob. Appl. (USSR). vol. 13, no. 4, 1968.
18. T.E. Duncan, “Probability densities for diffusion processes with applications to nonlinear filtering theory and detection theory,” Ph.D. dissertation. Stanford Univ., Stanford, Calif., 1967.
19. M. Zakai, “On the optimal filtering of diffusion processes.” Z. Wahrscheinlichkeits Theorie verw. Geb., vol. 11. 1969. pp. 230–243.
20. G. Kallianpur and C. Striebel, “Estimation of stochastic systems: Arbitrary system process with additive white noise observation errors.” Ann. Math. Statist., vol. 39, 1968. pp. 785–801.
21. G. Kallianpur, “Stochastic differential equations occurring in the estimation of continuous-parameter stochastic processes,” Theory Prob. Appl. (USSR). vol. 14, 1969, pp. 567–594.
22. J.T. Lo, “On optimal nonlinear estimation - Part 1: continuous observation,” in Proc. 8th Allerton Conf. Circuit and System Theory, Oct. 1970.
23. N.K. Loh, “On the evaluation of conditional entropy,” Proc. UMR-Mervin J. Kelly Communications Conf., Oct. 1970.
24. G.M. Lee, “Nonlinear interpolation,” IEEE Trans. Inform. Theory, vol. IT-17, Jan. 1971, pp. 45–49.
25. J.M.C. Clark, “Conditions for one-to-one correspondence between an observation process and its innovation,” Center for Computing and Automation, Imperial College, London. Tech. Rep. 1, 1969.
26. J. Neveu, Mathematical Foundations of the Calculus of Probability. San Franciscao: Holden-Day, 1965.
27. J.L. Doob, Stochastic Processes. New York: Wiley. 1953.
28. K. Itô, “Multiple Wiener integral,” J. Math. Soc. Japan, vol. 3. May 1951.
29. A.V. Skorokhod, Studies in the Theory of Random Processes. Reading, Mass.: Addison-Wesley, 1961.
30. M. Hitsuda, “Representation of Gaussian processes equivalent to Wiener processes,” Osaka J. Math., vol. 5, 1968, pp. 299–312.
31. T. Kailath, “Likelihood ratios for Gaussian processes.” IEEE Trans. Inform. Theory, vol. IT-16, May 1970. pp. 276–288.
32. J.M.C. Clark, “The representation of functionals of Brownian motion by stochastic integrals,” Ann. Math. Statist., vo l. 4, 1970. pp. 1282–1295.
33. P. Frost, “The innovations process and its application to nonlinear estimation and detection of signals in additive white noise.” Proc. UMR-Mervin J. Kelly J. Kelly Communications Conf., Oct. 1970.
34. T. Kailath, “The structure of Radon-Nikodym derivatives with respect to Wiener and related measures,” Ann. Math, Statist., vol. 6, June 1971, to be published.
35. A. Papoulis, Probability, Random Variables and Stochastic Processes. New York: McGraw-Hill. 1965.
36. T. Kailath and R. Geesey, “An innovations approach to least squares estimation-Part IV: Recursive estimation given the covariance function,” IEEE Trans. Automat. Contr., to be published.
37. T. Kailath, “The innovations approach to detection and estimation theory,” Proc. IEEE, vol. 58, May 1970. pp. 680–695.