Lossy coupling matrix filter synthesis based on hyperbolic reflections

The paper introduces the concept of hyperbolic reflections in order to complement the hyperbolic rotation matrices used in the lossy coupling matrix filter synthesis. A complex transversal coupling matrix representing a lossy filter is chosen and it is seen how by applying a succession of similarity transformations based also on hyperbolic reflections one obtains different results than by using hyperbolic rotations which are nowadays used in the coupling matrix reconfiguration. Finally a lossy symmetrical filter is designed and simulated based on a folded coupling matrix by shifting the losses into the first and last resonators of the filter.