Cascade realization of a class of two-variable transfer functions

Necessary and sufficient conditions under which a two-variable rational function may be realized as the transfer function of a doubly terminated cascade of single-variable lumped, lossless reciprocal two-ports, of which each two-port is with respect to p₁ or p₂, are derived. It is noted that a set of realizability conditions presented and the approach to the realization are entirely different from those in previous contributions. The derived conditions are based on the factorability of three polynomials determined from the denominator and numerator polynomials of the transfer function under consideration. The proposed approach is based on the synthesis of two single-variable driving point impedance functions, which are defined at the junction of single-variable two-ports, each of which depends only on p₁ or p₂. Moreover, a realization procedure is also derived from the viewpoint of reciprocal realization of the transfer function presented. In addition, a theorem on a cascade realization of multivariable positive real functions is extended