Use of optimal estimation theory, in particular the Kalman filter, in data analysis and signal processing

The Kalman filter, a powerful and useful optimal estimation technique, does not seem to be widely known among physicists. Here we outline the derivation of the algorithm, and give three examples of its use: (a) in estimating the value of a constant, with both system and measurement noise, (b) in numerical differentiation of noisy data, and (c) in optimally estimating the amplitude of a signal with arbitrary but known time dependence superimposed on a noisy background.