On the problem of degree reduction of a scattering matrix by factorization

Abstract Belevitch ( 1 ) has shown that, starting from a given passive, rational, n x n scattering matrix S(p) of degree δ, one can proceed to a realization by factoring it in the form S(p) = S 1 (p)S 2 (p), where S 2 (p) is an n x n, lossless scattering matrix of degree one, while the degree of S 1 (p) is reduced to δ- 1 . Some sufficient conditions allowing the stated factorization were developed by Youla ( 2 ) and Belevitch ( 3 ) but complete necessary and sufficient conditions were not obtained. Complete conditions are derived here by two different and complementary methods, one based on Hankel matrices, the other on the Smith-MacMillan form. Moreover, several errors of the above-mentioned papers are corrected. The resulting conditions are quite simple and only involve the structure of the resistivity matrix of the given network in the neighborhood of a singularity. Finally, the conditions clarify certain aspects of the cascade synthesis of passive n-ports and increase the similarity of this process with the Darlington one-port synthesis.