Minimum roundoff noise digital filters with some power-of-two coefficients

Abstract

A new technique and numerical algorithm are introduced for synthesizing <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> th-order minimum roundoff noise state-space structures for nth-order fixed-point recursive digital filters. The technique yields structures which employ <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(n^{2} - n -1)</tex> trivial power-of-two multiplies and so require only <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(3n + 2)</tex> nontrivial multiplies. This compares to the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(n + 1)^{2}</tex> nontrivial multiplies required by other minimum-noise structures. Although the power-of-two structures do not satisfy theoretical conditions for roundoff noise optimality, their roundoff noise is found to be but negligibly higher than minimum. A numerical example illustrating the synthesis technique is provided.