Analog Computer Applications in Predictor Design

The theory of least mean-squared-error filtering and prediction of statistical time series, developed by Kolmogorov,1 Wiener,2 Bode and Shannon,3 and others,4 has recently found applications in a variety of control systems. The design of optimum (least mean squared-error) predictors is based on measurement or calculation of power spectra. The present paper describes equipment for measuring power spectra at very low frequencies. It is shown that analog circuits having the transfer functions of exact stagger-tuned triples, with flat response in the pass band and good skirt selectivity, are easily designed and utilized in conjunction with magnetic tape recordings. Although the optimum predictor transfer function can be computed directly from the power spectrum of the signal to be predicted, the computation is complicated and a checking procedure is desirable. An analog computation which permits such a check is described. Since the power spectrum defines the optimum predictor, it determines the mean-squared prediction error as well. However, if a nonoptimum network is arbitrarily selected for use as a predictor, it becomes desirable to determine the incremental error that results from use of this network. A simple analog computation which determines the error of the optimum predictor and the incremental error of a nonoptimum predictor is described. This analog technique is a specific adaptation to prediction of the impulsive response techniques of Bennett5 and Laning and Battin.