On the generalized DFIIt structure and its state-space realization in digital filter implementation

It is well known that for a digital filter of order p, the number of nontrivial parameters in the classical optimal state-space realizations is proportional to p², while the traditional shift operator z-based direct-form II transposed (zDFIIt) structure, though having poor numerical properties, is one of the most efficient structures, just possessing 3p+1 nontrivial parameters. In this paper, based on the concept of polynomial operators, a new structure is proposed for digital filter implementation, which is a generalization of the traditional zDFIIt and the prevailing δDFIIt structures. This structure, denoted as ρDFIIt, possesses 3p+1 nontrivial parameters plus p parameters at choice. Expressions for evaluating the sensitivity measure and the roundoff noise gain are derived for the ρDFIIt structure and its equivalent state-space realization that has the same structure complexity. It is shown that the state-space realization always yields a smaller roundoff noise gain than the ρDFIIt structure. One of the nice properties of these two structures is that for a given digital filter, they can be optimized with the p free parameters. The optimal structure problems can be solved with exhaustive researching under practical considerations. Numerical examples are presented to illustrate the design procedure.