Exhaustive Synthesis Framework of Coupled Resonator Microwave Bandpass Filters

The synthesis of coupled resonator bandpass filters has been a classic theme of research for the past half century. Continuous efforts have been paid to explore new coupling topologies. Needless to say, the ultimate goal for filter synthesis is to find all the viable coupling topologies and their real-valued circuit models. In this article, a rigorous and straightforward framework for exhaustive search of not only all the viable coupling topologies but also all their real coupling matrices for a given filtering function is presented. The proposition that provides sufficient and necessary condition for a viable coupling topology with finite real coupling matrices is proved. The uniqueness of the folded coupling matrix is also proved and is used to establish the simultaneous equations for a well-behaved numerical solution search process. The special cases of the proposition are discussed with illustrative examples. Demonstration examples, including a prototyped filter in a novel “grid” coupling topology, have shown that the framework can systematically find all the viable coupling topologies and their real-valued solutions numerically, through which many new useful coupling configurations can be found, including the “trapezoid” configuration that is highly suitable for a dual-mode realization for a symmetric/asymmetric filter response.