Minimum uncorrelated unit noise in state-space digital filtering

Abstract

For the class of digital filters described by the state equations, new expressions for the roundoff noise and the dynamic-range constraint equation have been established. Then, minimization of the roundoff noise subject to the l <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> norm dynamic-range constraint is considered. Using the polar decomposition of a nonsingular matrix, the effects of an equivalent transformation on the noise are analyzed, then, a lower bound and the global minimum of the unit noise, i.e., the noise generated-under the assumption that each state equation contains exactly one noise source, is obtained. A method of realizing the minimum unit noise filter is given, and possible further optimization of the filter is discussed. A numerical example is given to illustrate the computational procedure.