The universality of the Kalman filter: a conditional characteristic function perspective

TL;DR

This paper offers a new theoretical perspective on the Kalman filter by analyzing its evolution through conditional characteristic functions and Ito calculus, revealing its universality and invariance under different stochastic interpretations.

Contribution

It develops three Kalman filtering theorems based on characteristic function evolution, providing a unified theoretical framework and formal proofs for the filter's universality.

Findings

Kalman filtering equations derive from conditional characteristic function evolution.

Kalman filtering results from stochastic evolution of characteristic functions in continuous measurements.

The structure of the Kalman filter is invariant under Ito and Stratonovich interpretations.

Abstract

The universality of the celebrated Kalman filtering can be found in control theory. The Kalman filter has found its striking applications in sophisticated autonomous systems and smart products, which are attributed to its realization in a single complex chip. In this paper, we revisit the Kalman filter from the perspective of conditional characteristic function evolution and Ito calculus and develop three Kalman filtering Theorems and their formal proof. Most notably, this paper reveals the following: (i) Kalman filtering equations are a consequence of the evolution of conditional characteristic function for the linear stochastic differential system coupled with the linear discrete measurement system. (ii) The Kalman filtering is a consequence of the stochastic evolution of conditional characteristic function for the linear stochastic differential system coupled with the linear continuous measurement system. (iii) The structure of the Kalman filter remains invariant under two popular stochastic interpretations, the Ito vs Stratonovich.