Circuit models for prediction, Wiener filtering, Levinson and Kalman filters for ARMA time series

ARMA models for random process estimation are generally much more convenient than all-pole forms, since they lead to structures having a minimum number of coefficients for fitting arbitrary autocorrelation or power spectral density forms. In the presence of observation noise, they are mandatory, to account for the zeros introduced into autoregressive spectra. In this paper, a novel derivation of optimum estimators, based on Z transforms, demonstrates that the use of ARMA models leads to extremely simple system block diagrams for signal prediction filters, Wiener filters, and other optimum filters related by linear transformations. It is shown that Kalman-ARMA filters are identical, in the steady state, to classical Wiener filters and that the Levinson-Durbin Recursion produces a time-varying filter identical to the Kalman Predictor. It is shown that a simple feedback system yields both the innovations process and the optimum predictor, the latter being determined uniquely by the covariance of the noisy observed signal. Because of space constraints, the derivations are necessarily brief. A more comprehensive paper is in process.