On the approximation of optimal realizable linear filters using a Karhunen-Loeve expansion (Corresp.)

The Karhunen-Loève expansion of a random process is used to derive the impulse response of the optimal realizable linear estimator for the process. The expansion is truncated to yield an approximate state-variable model of the process in terms of the first
N
eigenvalues and eigenfunctions. The Kalman-Bucy filter for this model provides an approximate realizable linear estimator which approaches the optimal one as
N \rightarrow \infty
. A bound on the truncation error is obtained.