Order Bound for the Realization of a Combination of Positive Filters

In a problem on the realization of digital filters, initiated by Gersho and Gopinath, we extend and complete a remarkable result of Benvenuti, Farina and Anderson on decomposing the transfer function t(z) of an arbitrary linear, asymptotically stable, discrete, time-invariant single-input-single-output system as a difference t(z)=t₁(z)-t₂(z) of two positive, asymptotically stable linear systems. We give an easy-to-compute algorithm to handle the general problem, in particular, also the case of transfer functions t(z) with multiple poles, which was left open in a previous paper. One of the appearing positive, asymptotically stable systems is always one-dimensional, while the other has dimension depending on the order and, in the case of nonreal poles, also on the location of the poles of t(z). The appearing dimension is seen to be minimal in some cases and it can always be calculated before carrying out the realization