  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.
