State-space structures for the realization of low roundoff noise digital filters

This study deals with the synthesis of computationally efficient, low-roundoff-noise, L2-norm scaled, state-space realizations for fixed-point recursive digital filters. In addition to low roundoff noise, such realizations also possess low coefficient sensitivity and low likelihood of sustaining overflow oscillations.

The theory of state-space structures is reviewed, techniques for synthesizing roundoff-noise statespace structures are discussed, and a new technique for t h synthesizing n^th-order minimum-noise structures is introduced. The new technique yields structures which employ n²-n-1 trivial power-of-two multiplies and so require only 3n+2 nontrivial multiplies. This compares to (n+1)² nontrivial multiplies for 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. Extension of the technique to state-decimation realizations is considered.

Two new n^th-order state-space structures are proposed. One is based on transforming the filter system matrix to Hessenberg form and the other on requiring orthogonal state-variable unit-pulse responses. The former structure requires n(n+7)/2 multiplies and the 2 latter n²+n+2. Both have a roundoff-noise performance nearly as good as minimum-noise structures but require fewer multiplies.

Several new second-order filter structures are also introduced. These structures achieve, with two fewer multiplies, almost the same low level of roundoff noise as the minimum-noise and normal second-order structures. Realizations composed of subfilters employing the new structures require only about 3n multiplies. An approach to developing simple, algebraic, real-arithmetic design equations for scaled second-order structures is presented and then applied to deriving design equations for the new structures and for the minimum-noise and normal structures.

Many numerical examples are provided to demonstrate the synthesis and analysis of high-performance realizations. Computer programs in FORTRAN for synthesizing minimum-noise structures both with and without power-of-two coefficients are included.