2-D Inductor-Capacitor Lattice Synthesis

We consider a general class of 2-D passive propagation media, represented as a planar graph where nodes are capacitors connected to a common ground and edges are inductors. Capacitances and inductances are fixed in time but vary in space. Kirchhoff´s laws give the time dynamics of voltage and current in the system. By harmonically forcing input nodes and collecting the resulting steady-state signal at output nodes, we obtain a linear, analog device that transforms the inputs to outputs. We pose the lattice synthesis problem: given a linear transformation, find the inductances and capacitances for an inductor-capacitor circuit that can perform this transformation. Formulating this as an optimization problem, we numerically demonstrate its solvability using gradient-based methods. By solving the lattice synthesis problem for various desired transformations, we design several devices that can be used for signal processing and filtering.