Physical interpretation and implications of similarity transformations in coupled resonator filter design

This paper discusses the physical interpretation and implications of similarity transformations for applications in coupled resonator filter design. It is shown that certain similarity transformations contain important "global" information that can be used to get insight into the workings of microwave bandpass filters. Instead of focusing on the effect a similarity transformation has on the original coupling matrix, the change of basis that the similarity transformation represents is considered the main operation. For a filter of order N that is designed as a set of N coupled resonators, it is shown that the transversal matrix is equivalent to using the global eigenmodes of the entire structure as a basis. The transversal matrix emerges as a universal and natural representation of coupled resonator bandpass filters of arbitrary orders, responses, and topologies. It can be used as an equivalent circuit in the optimization and diagnosis of bandpass coupled resonator filters. The results of this study have far reaching implications not only for the theory and design of microwave filters, but other circuits as well.