Realization of general extended state-space fixed-point digital filters with minimum roundoff noise

In this study a technique for determining a minimum roundoff noise extended state-space (e-state) realization of fixed-point recursive digital filters is developed. Previous efforts developed such e-state structures for the second order e-state equa- tion only. This new algorithm determines a minimum roundoff e-state structure for the general order e-state equation. The technique determines a linear transforma- tion which changes the original structure into an e-state structure with minimum roundoff noise. A combination of a conjugate gradient algorithm and a variable metric algorithm is employed to determine the linear transformation. The roundoff noise characteristics of e-state structures are investigated by studying 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 N_th 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]. Compared to other realizations, the e- state structure uses fewer, but longer inner products which pipelined digital signal processors are designed to handle very efficiently.