On the interpolation and prediction of signals plus noise for infinite and finite smoothing times

The first part of the paper is concerned with the choice of the optimum filter for the extrapolation or interpolation of a signal which is contaminated by noise. The problem has been completely solved by Wiener in his book on the subject. The method used here for solving the integral equations is different and leads to the solution for the lagging case without having to approximate ¿¿p¿ by the ratio of polynomials in p. Laplace transformations are used and lead directly to the solution. The second part of the paper deals with the solution of the optimum filter when the error is minimized at a particular time after switching on, the averaging being taken over an ensemble. Examples of each problem are given.