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Linear prediction filters have recently been employed to obtain power spectral estimates which exhibit excellent resolution properties, particularly for the case of narrow band spectra. In this paper, we discuss an extension of linear prediction spectral analysis in which both previous and future values of the data sequence are used to estimate the sample of interest. Theoretical performance measures for this class of estimators are developed and used for comparison with linear prediction methods. It is shown that he new estimators, termed linear estimation filters, provide lower mean-square-error estimates in some problems of interest than can be achieved using linear prediction filters. The resulting power spectral estimates, however, are in general poorer than those provided by linear prediction . The conclusion drawn is that he mean-square-error criterion may not be the appropriate performance measure for this class of spectral estimator sand that additional criteria, such as aspectral flatness measure, should be investigated.