New second-order state-space structures for realizing low-roundoff-noise digital filters with reduced number of multiplies

Efficient computational algorithms for realizing high-order digital filters can be constructed around parallel-of-subfilter or cascade-of-subfilter realizations, where each subfilter is either first or second-order. To this end, several new second-order state-space digital filter structures are developed and introduced. They are then analyzed with respect to their fixed-point-arithmetic roundoff noise performance. It is found by appropriately choosing from the new structures, nth-order realizations with low roundoff noise and only about 3n multiplies are possible. This compares favorably to the 4n multiplies required by minimum-noise or normal subfilters and to the (n+1)² multiplies required by an nth-order minimum-noise structure.

Design equations are developed for each of the new structures, These equations require only algebraic, real-arithmetic calculations and give designs with inherent ℓ₂ -norm probability-of-overflow scaling. Numerical examples are included to illustrate the design of filters using the new structures and a software realization is provided to demonstrate the advantages of the new structures.