Synthesis of minimum roundoff noise structures for extended state-space digital filter implementations

An algorithm for minimizing roundoff noise in extended state-space (e-state) realizations of recursive digital filters, where the order of the e-state equation is 2, is developed. It is shown that previous efforts to minimize roundoff noise in e-state structures have not provided a global minimum. The algorithm presented applies an unconstrained transformation matrix to an arbitrary starting state-space structure to produce an intermediate structure. A second matrix transforms the intermediate structure to e-state form. A conjugate-gradient optimization scheme is used to determine the coefficients of the first matrix that minimize the roundoff noise gain of the e-state structure produced by the second transformation. A numerical example illustrates that orders-of-magnitude improvement over previous results can be achieved with this approach