Steady-State Behavior of Kalman Filter with Discrete- and Continuous-Time Observations

There is often a need for optimal mixing of continuous-time and Discrete-Time data. This can be readily accomplished by Kalman filtering, the theory of which is briefly reviewed. In the steady state, the filter gains for processing the continuous-time data are generally periodically varying functions of time and cannot be determined by simply solving either the Discrete-Time or the continuous-time filtering problem, but they can be determined with the aid of the solution of an equivalent Discrete-Time problem. An illustrative example is given for the system: x = white noise, with Discrete-Time observations of x and continuous-time observations of x.