Minimization of fixed-point roundoff noise in extended state-space digital filters

A technique for determining a minimum roundoff noise extended state space (e-state) realization of fixed-point recursive digital filters is developed. Previous efforts have developed such minimum roundoff noise e-state structures for the second-order e-state equation only. This new algorithm determines a minimum roundoff e-state structure for the general order e-state equation. The technique obtains a linear transformation which, when applied to the original structure, produces a minimum noise e-state structure. A combination of a conjugate gradient algorithm and a variable metric algorithm is employed to determine the transformation coefficients. The e-state roundoff noise characteristics are illustrated by numerical examples. It is found that the minimum-noise e-state structure can have lower roundoff noise than the conventional minimum-noise structure, since fewer state variables are actually computed. In an e-state structure an Nth-order filter where N=LM can be implemented by M difference equations of order L resulting in a computational complexity of O[L(M+1)²]. One advantage of e-state structures, compared to other realizations, is the use of fewer but longer inner products which pipelined digital signal processors are designed to handle efficiently