Cross-Coupling Matrix Reconfiguration Using the Levenberg–Marquardt Algorithm on Orthogonal Groups

This letter introduces a Levenberg-Marquardt (LM) algorithm on the orthogonal group to reconfigure the coupling matrix (CM) for cross-coupled resonator filters of general topology. By leveraging the framework of Lie groups and the exponential map, we perform LM steps locally in the linear space of skew-symmetric matrices, the Lie algebra associated with the orthogonal group. In each LM step, we derive and explicitly formulate a regularized least-squares (LSs) problem in matrix-vector form, and then map the solution back to the orthogonal matrix to transform the CM toward the one with the desired structure. In addition, we integrate the proposed LM algorithm with homotopy optimization to enhance global convergence, particularly for lossy CM reconfiguration. The proposed algorithm demonstrates fast convergence and high accuracy, as evidenced by the results of filter synthesis and fabrication.