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A general theory of optimum linear estimation is considered in relation to the problem of reconstructing a nonstationary random signal sampled at arbitrary times and in the presence of a sampling noise. The resulting optimum filter predictor takes the form of a growing memory digital compensator. Presentation of the theory is tutorial, and a contrast to recursive estimation is discussed. Application is made to the use of external discrete position information in a long-term inertial navigator. A comparison between the optimized system and a reference (non-optimum) system is presented. Consideration is also given to truncation effects and the very important matter of the effect of poorly estimated problem statistics on performance of the optimized system. Digital computer simulation studies are presented.