Recursion formulas for growing memory digital filters

A growing memory digital filter is defined by considering the input -output relationship in the form where is the weighting sequence of a linear time varying digital filter. Contained herein are a derivation of an optimum growing memory smoothing and prediction filter in the least squares sense for polynomial input functions, (of degree = ) and a theorem on the class of time invariant sequence , which are solutions of a difference equation of tiite order, and an application of the theorem to the synthesis of sampled correlated noise by digital processes, using recursion formulas. The recursion formulation represents a practical solution to the generation of a correlated noise sequence on line during simulation studies on digital computers.