Synthesis of 3-D lossless first-order one ports with lumped elements

A special class of 3-D lossless one ports is considered. The author restricts his investigation to all-pass functions where the partial degree, with respect to every complex variable p_i , i=1 to 3, of its partial denominator polynomial does not exceed 1. The synthesis is achieved by reducing the given problem to the known synthesis of 2-D lossless multiports by the extraction of p₁-type reactive elements. The author makes use of the fact that a real, nonnegative, definite, two-variable polynomial can be written as a sum of squares of real polynomials if its total degree does not exceed four. No restrictions are made concerning the coefficients of the rational reflectance of the desired network: they can be either real or complex, so as to include even complex networks which are of special interest for multidimensional wave digital filters