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In digital computations, errors resulting from sampling and amplitude quantization (round off) are unavoidable. This work evaluates the mean-square error caused by sampling and quantization at the output of a linear network which contains a single quantizer. A detailed answer is given to the question, "Given quantized samples of a signal which is a sample function of a random process, what is the optimum linear filter for recovering the signal from its samples?" This filter is determined and its characteristics are summarized graphically for a specific example. A comparison with the conventional hold circuit shows that the optimum filter is much better if high accuracy is required and quantization is coarse. The difference in performance between the two filters is small when the accuracy requirement is low and the quantization is fine. Also included as Appendix V is a survey of the general quantization errors problem, as it appears in the areas of digital computation and numerical analysis, and a study of multiquantizer networks. It is found that extension of the method to networks which contain more than one quantizer is impractical, if not impossible.