Balanced parametrizations of stable SISO all-pass systems in discrete time

A balanced canonical form for discrete-time stable SISO all-pass systems is obtained by requiring the realization to be balanced and such that the reachability matrix is upper triangular with positive diagonal entries, in analogy to the continuous-time balanced canonical form of Ober [O1]. It is shown that the resulting balanced canonical form can be parametrized by Schur parameters. The relation with the Schur parameters for stable AR systems is established. Using the structure of the canonical form it is shown that, for the space of stable all-pass systems of order less than or equal to a fixed number n, the topology of pointwise convergence and the topology induced by H(2) coincide. The topological space thus obtained has the structure of a hypersphere. Model reduction procedures based on truncation! which respect the canonical form, are discussed.