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The optimal minimum-mean-square-error discrete filters are obtained for finite bandwidth signals in noise represented by a rational pole-zero model in the spectral domain. An analytical technique called the method of undetermined coefficients is used to solve the Wiener equation for the optimal prediction coefficients. These coefficients are then used to examine (a) the frequency response of the Wiener prediction filter, and (b) the maximum entropy spectral estimator. Special cases considered include the limiting case of narrow bandwidth signals and large prediction filter lengths.