On normal realizations of digital filters with minimum roundoff noise gain

It is well known that normal realizations are free of limit cycles and that a digital filter implemented with a state-space realization (A,B,C,d)(A,B,C,d) has no limit cycles if there exists some diagonal matrix D>0D>0 such that D-ATDA⩾0D-ATDA⩾0. In this brief, a method is proposed to check the existence of such a D for any given realization. It is also shown that the normal realizations have a minimal error propagation gain. More interestingly, the normal realizations are characterized, the minimum roundoff noise normal realization problem is formulated and solved analytically. An example is presented to test the efficiency of the proposed method and to demonstrate the performance of the proposed optimal normal realizations.