A Fresh Look at the Kalman Filter

In this paper, we discuss the Kalman filter for state estimation in noisy linear discrete-time dynamical systems. We give an overview of its history, its mathematical and statistical formulations, and its use in applications. We describe a novel derivation of the Kalman filter using Newton's method for root finding. This approach is quite general as it can also be used to derive a number of variations of the Kalman filter, including recursive estimators for both prediction and smoothing, estimators with fading memory, and the extended Kalman filter for nonlinear systems.