Statistical Signal Processing in Engineering

13Bayesian Tracking and Kalman Filter

Bayesian estimation is based on the assumption of having a‐priori information on the parameter to be estimated. In some cases the pdf is known and time‐invariant, but in other cases it can be variable in time. The estimators in this chapter exploit some knowledge of the mechanisms (or models) of the time evolution of the parameters (or system’s state when referring to a time‐varying quantity) of the dynamic system in order to improve the estimate by chaining the sequence of observations linked to the sequence of states that accounts for their dynamic evolution. The Bayesian estimator of the evolution of the state from an ordered sequence of observations is carried out iteratively by computing the a‐priori pdf of the state (starting from the available observations before using the current observation) and the a‐posteriori pdf (also adding the current observation).

The Kalman filter (KF) is a special case of linear estimator of the evolution of the state of a dynamic linear system from the sequence of observations handled as random process. As with any linear Bayesian estimator, the KF is the optimal filtering if the processes are Gaussian and it is widely investigated in the literature [51] . The linearity of the KF has the advantage of simplicity and computational efficiency, so whenever the rvs are non‐Gaussian, or the evolution of the state is ruled by non‐linear models, the KF can also be derived after linearizing the dynamic model and ...