Roundoff Noise Invariants in Normal Digital Filters

The unit noise gains for optimal and parallel normal realizations of digital filters can be expressed in terms of a set of noise gain parameters that are simply related to the pole locations and pole residues. These noise gain parameters are shown to be invariant under a class of frequency transformations, and for digital filter transfer functions derived by bilinear transformation of an analog transfer function, are independent of the frequency scaling parameter. As a result, the unit noise gains of normal realizations can be simply related to the performance characteristics of the filter, i.e., to filter order, passband ripple, and stopband gain. These simple relations make it easy for the filter designer to select a structure with acceptable roundbff error. Unit noise gain for normal realizations of Butterworth, Chebyshev, and elliptic filters are plotted for a range of performance characteristics, and compared with optimal state-space structures. These results show that there is no significant difference between the unit noise gains of optimal normal realizations and parallel normal realizations, and that the unit noise gains of optimal state-space structures are significantly lower than the normal forms only for high-order Butterworth filters.