The synthesis of two-dimensional passive n-ports containing lumped elements

Two-dimensional (2-D) passive networks are of interest e.g. for use as reference filters for two-dimensional wave digital filters. Necessary properties of the impedance matrix and scattering matrix, respectively, of such n-ports have been established, but not yet been shown to be also sufficient for a given two-variable rational matrix to be the impedance matrix or scattering matrix, respectively, of a passive network containing lumped elements. In the design of 2-D passive n-ports it will be however of great interest whether this mentioned feature can be used as a basis for ageneral synthesis procedure.In this paper it is shown that this is the case. The method presented for the synthesis of 2-D multiports is based mainly on a paraunitary bordering of the given scattering matrix of the desired network in order to obtain the scattering matrix of alossless 2-D multiport, which can be realized by using known procedures. The socalled spectral factorization of a two-variable para-Hermitian polynomial matrix which is nonnegative definite forp =j w plays a crucial role in the design approach presented. No restrictions are made concerning the coefficients of the given rational scattering matrix; they may be either real or complex, so as to include even complex networks which are of special interest for multidimensional systems.