Finding Well-Separated Synthesis Solutions of Underdetermined Filter Topologies Within a Limited Range

In this article, the synthesis of underdetermined filter topologies, which feature infinite solutions, is discussed comprehensively for the first time. The slack variables are defined to limit the synthesis in a prescribed range dependent on physical and technological constraints. This aims to avoid wasting the computation resources on the synthesis solutions that cannot be implemented. To handle the singular Jacobian matrix encountered when solving the underdetermined synthesis equations, a modified two-step Levenberg-Marquardt (TSLM) algorithm is introduced to improve the synthesis robustness and efficiency significantly. Besides the classic trial step in the iterations, TSLM employs an extra approximate step in the trust region and adaptive parameters to accelerate the convergence. Compared with other algorithms, TSLM can save more than 2/3 of the computation time. Moreover, to prevent the synthesis solution vectors from clustering in a particular subspace, the least influential variables are initialized and fixed in a grid-like manner to search for multiple well-separated solutions in the global region. These well-separated multisolutions can provide distinct schemes for physical implementation. Multiple synthesis examples are presented, including cross-coupled topologies, topologies incorporating frequency-dependent couplings (FDCs), and networks with nonresonating nodes (NRNs). Finally, one fourth-order combline filter with all positive couplings is manufactured and measured to validate the proposed synthesis theories.