An innovations approach to least-squares estimation--Part III: Nonlinear estimation in white Gaussian noise

In Parts I and II of this paper, we presented the innovations approach to linear least-squares estimation in additive white noise. In the present paper, we show how to extend this technique to the nonlinear estimation (filtering and smoothing) of non-Gaussian signals in additive white Gaussian noise. The use of the innovations allows us to obtain formulas and simple derivations that are remarkably similar to those used for the linear case thereby distinguishing clearly the essential points at which the nonlinear problem differs from the linear one.

[1]  Kiyosi Itô.  Multiple Wiener Integral , 1951 .

[2]  R. L. Stratonovich.  CONDITIONAL MARKOV PROCESSES , 1960 .

[3]  W. Wonham.  Some applications of stochastic difierential equations to optimal nonlinear ltering , 1964 .

[4]  H. Kushner.  On the dynamical equations of conditional probability density functions, with applications to optimal stochastic control theory , 1964 .

[5]  R. Bucy.  Nonlinear filtering theory , 1965 .

[6]  P. Masani.  Wiener's contributions to generalized harmonic analysis, prediction theory and filter theory , 1966 .

[7]  A. Shiryaev.  Addendum: On Stochastic Equations in the Theory of Conditional Markov Processes , 1967 .

[8]  G. Kallianpur,et al.  Arbitrary system process with additive white noise observation errors , 1968 .

[9]  M. Hitsuda.  Representation of Gaussian processes equivalent to Wiener process , 1968 .

[10]  M. Zakai.  On the optimal filtering of diffusion processes , 1969 .

[11]  G. Kallianpur,et al.  Stochastic Differential Equations Occurring in the Estimation of Continuous Parameter Stochastic Processes , 1969 .

[12]  Thomas Kailath,et al.  A general likelihood-ratio formula for random signals in Gaussian noise , 1969, IEEE Trans. Inf. Theory.

[13]  T. Kailath.  The innovations approach to detection and estimation theory , 1970 .

[14]  J. M. Clark,et al.  The Representation of Functionals of Brownian Motion by Stochastic Integrals , 1970 .

[15]  Thomas Kailath,et al.  Likelihood ratios for Gaussian processes , 1970, IEEE Trans. Inf. Theory.

[16]  Gary M. Lee,et al.  Nonlinear interpolation , 1971, IEEE Trans. Inf. Theory.

[17]  T. Kailath,et al.  An innovations approach to least squares estimation--Part IV: Recursive estimation given lumped covariance functions , 1971 .

[18]  T. Kailath.  The Structure of Radon-Nikodym Derivatives with Respect to Wiener and Related Measures , 1971 .