A general approach to discrete-time filtering for dynamical systems evolving in a Lie group is presented. The most general filtering solution is given by the signal probability density function conditioned on the measurements, which for nonlinear systems is usually nonparametric.

A study of the use of entropy as a criterion function for analyzing the performance of sampled-data estimating systems is presented, and performance bounds are obtained for a broad class of such systems. The general result is that the difference between the entropy of the signal to be estimated and the entropy of its estimate based on the output of a noisy sensor can never be larger than the mutual information between the sensor input and output. An interesting, but not totally satisfactory, sufficient condition for attainment of the bound is developed.

A new formulation of the multidimensional optimal nonlinear filtering problem is presented in this two-part paper. This formulation permits generalization and unification of some well-known recent results on optimal nonlinear filtering theory. [1]-[7] Specifically, the problem investigated is that of determining the conditional probability density function ofx(t)given\{y(\tau); t_{0} \leq \tau \leq t\}, wherex(t)is then-dimensional state vector of a non-linear system perturbed by an independent increment noise process, andy(t)is anm-dimensional measurement vector which is a nonlinear function ofxand contains an additive independent increment noise process. The results are obtained through use of characteristic functions and the theory of independent increment processes. The foundation for the treatment of general independent increment noise processes is given in Part I, but the final results in Part I are restricted to Gaussian independent increment noise processes. The extension to general independent increment noise processes is considered in Part II. It is shown in Part I that the results for the linear-Gaussian case can be obtained in two different ways, one of which cannot be used for the general case. Some important properties of general independent increment processes and a special property of Gaussian independent increment processes are discussed.

The main result obtained in Part I [1]--which is contained in Theorem1of Part I--is generalized and extended in several ways. First, a new basic stochastic integro-partial differential equation for the conditional probability density function for the state of a nonlinear dynamic system with disturbance noise given noisy nonlinear measurements of the state is derived under the less restrictive assumption that the disturbance noise is an arbitrary independent increment process with an infinitely divisible distribution, and the measurement noise is a Gaussian independent increment process with an infinitely divisible distribution. It is then shown that, under proper restrictions, this basic equation reduces to either the Fokker-Planck equation for diffusion processes or the Kolmogorov-Feller equation for jump processes. Also, it is shown that this basic equation contains the earlier results of Kushner [2] and Wonhamt [3] as special cases. Next, it is shown how the result represented by this basic equation can be easily extended to include the case where the disturbance noise is a Markov process of the type initially assumed for the state of the dynamic system. Finally, it is shown that, in contrast to results obtained by Bryson and Johansen [4] and Cox [5] for the linear, Gaussian case, it is not generally possible to extend the result represented by this basic equation to include either the case where the measurement noise is a Markov process of the type initially assumed for the state of the dynamic system, or the case where the covariance matrix of the Gaussian measurement noise is singular. However, some incomplete results indicating when such extensions might be possible are given.