Characterizing finite-dimensional filters for the linear innovations of continuous-time random processes

We examine finite-dimensional realizability issues associated with certain linear filtering problems which depend only on second-order statistics of the underlying random process. The optimal filters are determined by linear integral (Wiener-Hopf) equations. We exploit linearity of the filter input/output map to obtain conditions which apply to a fairly general class of realizations, including linear and nonlinear ones. We show that a "semiseparable" covariance structure is necessary and sufficient for finite dimensionality.