The partial realization problem for complex Hamburger series and complex lossless multiport networks

Abstract

A formal matrix power series comprised of the Markov parameters of a complex lossless distributed system is considered. It is shown that certain matrix Pade approximants or continued fraction approximants to such a series are realizable by means of lumped lossless elementary building blocks including imaginary elements and can indeed be considered as approximations to immittance functions associated with complex lossless distributed systems on a rigorous basis. Physical examples of such systems include generalized versions of networks containing transmission line elements. The study of convergence of the diagonal sequence of matrix Pade approximants to a distributed immittance function of specific type, the properties of a related sequence of matrix orthogonal polynomials and the properties of the poles and zeros of the partial realizations are undertaken. Additionally, a matricial integral representation of the Cauer type for the immittance function is obtained in the present context of complex distributed lossless multiports. Our treatment yields a new circuit theoretic interpretation of the matricial Hamburger moment problem. Earlier work on formal power series associated with RC or LC multiport networks can be viewed as a special case of this more general theory. From a different standpoint, our results can also be viewed as complexification of the theory of Hamburger series and the associated power moment problem set in the context of circuits and systems theory.