Analytical Coupling Matrix Synthesis for Lossy Filters with Arbitrary Transmission Zeros

An analytical approach for the synthesis of narrow-band lossy filters with arbitrary transmission zeros is proposed in this paper. The scattering parameters are deduced from filtering polynomials once the transmission zeros are assigned. The formulas defining the transformation from scattering matrix to admittance matrix can be used by reconstructing the non-paraconjugate transmission zeros as paraconjugate. A canonical N+2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N + 2$$\end{document} transversal or N×N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N \times N$$\end{document} coupling matrix is then identified using partial fraction expansion of the normalized admittance functions. An increased order of the final network (with respect to the order of the characteristic polynomials) is required for obtaining this result, provided there are non-paraconjugate transmission zeros. Because of its versatility, the proposed approach can also be employed for lossless networks that have a transfer function with symmetrical transmission zeros. Two typical examples are presented to validate the proposed technique.