On the design of second-order state-space digital filter sections

A simplified parametric representation is introduced for the exploration of all real-scaled state-space realizations of a second-order transfer function having complex-conjugate poles. It is found that all real-scaled realizations can be represented as a function of a limited range of two real angles. It is shown that this representation can be used to obtain computationally efficient realizations with low roundoff noise and realizations which are roundoff-noise-optimal subject to L_∞-norm scaling. This latter class of realizations, which has not previously been synthesized, is found to be very similar to the realizations obtained by simply rescaling an optimal L₂-norm-scaled realization for L_∞ scaling. An algebraic technique is introduced for the synthesis of second-order state-space realizations. This technique provides realizations that are as computationally efficient as possible, subject to preserving low roundoff noise, low coefficient sensitivity, and freedom from limit cycles