An Alternate Approach to the Prediction of Polynomial Signals in Noise from Discrete Data

Abstract
When the problem of predicting the past, present, or future value of a polynomial signal or any of its derivatives is considered, where the signal is in white Gaussian noise, the standard approach has been to minimize mean-square-error with constraints by use of Lagrange multipliers. In this paper an alternate approach is described, using results of Rao and Bhattacharyya from the statistical literature, which reduce the specified prediction problem to a simple one requiring no formal minimizations and no use of Lagrange multipliers. It further has the advantage of yielding the covariances between estimates of the polynomial and its derivates. Useful engineering formulas for smoothing and prediction are developed in the main part of the paper. These include both filter and covariance expressions. A tutorial discussion of the theory is given in two appendixes.

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