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

Author

Bruce William Bomar

Date of Award

8-1983

Degree Type

Dissertation

Degree Name

Doctor of Philosophy

Major

Electrical Engineering

Major Professor

James C. Hung

Committee Members

Joe M. Mooge, Don W. Bouldin, W. Michael Farmer

Abstract

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.

Recommended Citation

Bomar, Bruce William, "State-space structures for the realization of low roundoff noise digital filters. " PhD diss., University of Tennessee, 1983.
https://trace.tennessee.edu/utk_graddiss/13008