Projecting Filters for Recursive Prediction of Discrete-Time Processes
A. Gersho, D.J. Goodman · Bell System Technical Journal · 1970
We consider the design of time-invariant recursive filters of constrained order for one-step prediction of discrete-time stationary processes. For this purpose, we introduce the projecting-filter concept. An nth-order projecting filter for a given process has the characterizing property that with the process as input, the output at each instant is the optimal linear combination of the n previous output and n latest input samples. This definition implies that (i) the filter is stable, (ii) any n + 1 consecutive samples of the prediction error sequence are mutually uncorrelated, (iii) the mean-square prediction error is at least as low as that of the best nth order nonrecursive predictor, and (iv) if the spectral density of the process is rational of order 2n or less, then the nth-order projecting filter coincides with the optimal (unconstrained) linear predictor. A design algorithm for nth-order projecting filters iteratively generates successive sets of coefficients of a time-varying nth-order recursive filter which asymptotically approaches the desired time-invariant filter. The only input data needed for the algorithm are the autocovariance coefficients of the process to be predicted. When the order of the filter is matched to the order of the process, the time-varying filter is the same as the Kalman predictor. The algorithm has yielded effective projecting filters for several specific processes. Our results indicate that near optimal prediction may often be obtained with filters of order lower than that of the optimal unconstrained predictor.