Explicit solutions to the linear optimal estimation problem of continuous time-invariant linear processes

The optimal estimation problem of linear continuous time-invariant processes is considered. Explicit simple expressions, in closed form, are obtained, in the case where the measurement record is infinitely long, for the constant Kalman gain and the corresponding minimum-error covariance matrix of the filtered estimate in terms of the zero structure of the power-spectrum density matrix of the measurement vector. These expressions are very easy to apply and they provide a geometric insight into the structure of the optimal filter. They are also applied in filtering problems with a measurement record of finite time and in problems of optimal smoothing where similar explicit results are obtained.