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 anarbitrarylinear, asymptotically stable, discrete, time-invariant single-input–single-output system as a differencet(z)=t₁(z)-t₂(z)of twopositive, asymptotically stablelinear systems. We give an easy-to-computealgorithm to handle the general problem, in particular, also the case of transfer functions$t(z)$withmultiple 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 beminimalin some cases and it can always be calculatedbeforecarrying out the realization.