Observations on linear estimation

Heisey and Griffiths have proposed a generalization of linear prediction, called "linear estimation", in which both past and future data samples are used to predict (estimate) the present sample. They report that although the mean-square error from this formulation is usually smaller than from standard linear prediction, the corresponding spectral estimate is a poorer fit to the true spectrum. We give a general explanation for this apparent paradox in terms of the zeros of the estimated inverse filter and examine specifically the case of frequency estimation for a single complex sinusoid in noise. The intuitively appealing idea that future as well as past data should be included in the estimates is best implemented by a combined forward-backward prediction method.