Projecting Filters for Recursive Prediction of Discrete-Time Processes

Although the optimal linear predictor of a random process must m a k e use of the entire past of the process, any practical predictor can store only a finite number of data. One way to design a finite storage predictor is to determine the best linear combination of t h e n latest sample values of t h e process. However, for m a n y processes, a large 2377 2396 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1!)70 value of n is required to achieve a performance quality approaching t h a t of the unconstrained optimal linear predictor. An alternate approach is to find the best recursive predictor constrained to operate only on the n latest d a t a samples and the n latest predictions. This approach has the advantage of using condensed information from the entire past of the process with the consequence t h a t optimal or near optimal prediction can often be achieved with a relatively small amount of storage. T h e purpose of this paper is to introduce the projecting-filter approach to recursive prediction and to present an algorithm for the design of projecting filters t h a t has yielded effective low-order predictors not otherwise attainable. So far, a complete theory of projecting filters has not been established. We do not yet know how broad is t h e class of processes which possess projecting filters of a given order; nor have we determined the class of processes for which our design algorithm is effective. However, we can report very favorable experience in the design of projecting filters for a variety of specific processes.