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

A new technique and numerical algorithm are introduced for synthesizing n th-order minimum roundoff noise state-space structures for nth-order fixed-point recursive digital filters. The technique yields structures which employ (n^{2} - n -1) trivial power-of-two multiplies and so require only (3n + 2) nontrivial multiplies. This compares to the (n + 1)^{2} 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.