Recursion formulas for growing memory digital filters

A growing memory digital filter is defined by considering the input <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(yν-u)</tex> -output <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(Z_m)</tex> relationship in the form <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Z_m = ∑ᵤ₌₀^m Wᵤₘ yₘ₋ᵤ, m = 0, 1, 2, ⋯</tex> where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Wᵤₘ</tex> 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 = <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</tex> ) and a theorem on the class of time invariant sequence <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">W_u</tex> , 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.