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: = white noise, with discrete-time observations of and continuous-time observations of .