Order Bound for the Realization of a Combination of Positive Filters

Abstract

In a problem on the realization of digital filters, initiated by Gersho and Gopinath [8], we extend and complete a remarkable result of Benvenuti, Farina and Anderson [4] on decomposing the transfer function t(z) of an arbitrary linear, asymptotically stable, discrete, time-invariant SISO system as a difference t(z) = t 1 (z) − t 2 (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 [4]. One of the appearing positive, asymptotically stable systems is always 1-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.

Positive Decomposition of Transfer Functions with Multiple Poles

“…Later in [9], the open case of nonnegative multiple poles was settled. Recently, in [10], a universal algorithm was found, providing a solution to the positive decomposition problem for any asymptotically stable transfer function t(v) (see Theorem 4 in [10]). In fact, keeping t 2 (v) 1-dimensional, a unified, easy-to-compute method was given to find a positive system t 1 (v), while keeping good control of the dimension N 1 .…”

“…In most cases, however, minimality of the dimension N 1 could not be claimed (nor was it claimed in [4]). In this paper we consider a special class of transfer functions with complex multiple poles for which the dimension N 1 given in [10] can be improved. Our approach here, however, does not cover all transfer functions.…”

“…In fact, the class of transfer functions considered in this note is quite narrow, so that we regard the main contribution here not as much in the achieved results but rather in the essentially new ideas behind the construction of Theorem 2 below. With further research in the future these ideas may well lead to a general improvement of the algorithm of [10].…”

“…We also remark that this 'geometric' lemma is utilized in most constructions concerning positive realizations (see e.g. [1,4,10]). Lemma 1.…”

“…Recently, in [10], a universal approach was found, providing a decomposition for any asymptotically stable t(z). Our approach here gives lower dimensions than [10] in certain cases but, unfortunately, at present it can only be applied to a relatively small class of transfer functions, and it does not yield a general algorithm. …”