Coupling Matrix Diagnosis by the Levenberg–Marquardt Algorithm With Homotopy Continuation

This article introduces a novel coupling matrix diagnosis technique based on the Levenberg–Marquardt algorithm (LMA). Unlike many existing coupling matrix extraction approaches that involve fitting measurement or simulation data with rational functions, synthesizing a transversal coupling matrix, and transforming the matrix as intermediate steps, the proposed method can directly extract a lossy coupling matrix in the desired form together with frequency-dependent phase shift at ports. However, the direct LMA approach suffers from the issue of being trapped in a local minimum if the initial guess is far from the optimal solution. Therefore, the homotopy continuation method is introduced to alleviate the reliance on high-quality initial values. This article details the LMA formulation with the homotopy continuation for the coupling matrix diagnosis problem. Measurement data of a dual-mode rectangular waveguide filter and a filter with a second-order dangling network at one terminal are used to demonstrate the proposed technique. In particular, coupling matrix diagnosis results with and without the homotopy method are presented. The comparison validates the effectiveness of the LMA with homotopy continuation. The proposed coupling matrix diagnosis technique applies to coupled-resonator networks of different types of resonators and coupling topologies.