On partial realization and interpolation of models from orthogonal basis function expansions

In this paper we address the problem of computing a minimal state-space realization from partial knowledge of an expansion in terms of generalized rational orthogonal basis functions. The basis functions considered are generated by stable all-pass filters. It is shown how a minimal state-space realization can be found on the basis of complete knowledge of the expansion coefficients. Subsequently an algorithm is given that solves the partial realization problem, meaning that it computes a minimal realization on the basis of a finite number of expansion coefficients. The analysis also results in compact expressions for computing the Hambo transform underlying this basis expansion as well as its inverse. Finally it is shown how these realization problems are related to the interpolation problem of finding a rational model of minimal degree that interpolates to the derivatives of a transfer function in a given set of points outside the unit disc