On the design of optimal state-space realizations of second-order digital filters

Simple design equations are derived for optimal (minimum roundoff noise) state-space realizations of second-order digital filter sections with complex poles. These relations permit easy computation of filter coefficients and unit noise gain directly from the filter transfer function. New parameters are identified that are easily computed from the transfer function, are invariant under certain frequency-scaling transformations, and which can be used to simply describe all of the design variables of optimal second-order filters, including state covariance matrix, unit noise matrix and second-order modes. The design relations are illustrated by computing the unit noise gains and eigenvalue sensitivities of optimal parallel realizations of Chebyshev lowpass filters.