On the relevance of noncommutative power series in spatial filters realization

Several properties of noncommutative rational power series are relevant in realizing spatial filters. After a brief survey of previous results, this paper presents an extension of Ho´s algorithm, which provides a representation technique for noncommutative rational power series, and outlines the solution to the problem of the partial representation. Spatial filters, as input/output maps, and doubly indexed dynamical systems are then considered, and a characterization of realizable filters is derived. Finally an algorithm is constructed, which provides all the minimal realizations of a prescribed filter by exploiting the generalized Ho´s algorithm.