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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 eigenvalues and eigenfunctions. The Kalman-Bucy filter for this model provides an approximate realizable linear estimator which approaches the optimal one as . A bound on the truncation error is obtained.