Computational Cost Reduction for <italic>N+2</italic> Order Coupling Matrix Synthesis Based on Desnanot-Jacobi Identity

Abstract

Matrix inversion is routinely performed in computational engineering, with coupling matrix filter synthesis considered here as just one of many example applications. When calculating the elements of the inverse of a matrix, the determinants of the submatrices are evaluated. The recent mathematical proof of the Desnanot-Jacobi (also known as the “Lewis Carol”) identity shows how the determinant of an N+2 order square matrix can be directly computed from the determinants of the N+1 order principal submatrices and N order core submatrix. For the first time, this identity is applied directly to an electrical engineering problem, simplifying N+2 order coupled matrix filter synthesis (general case, which includes lossy and asymmetrical filters). With the general two-port network theory, we prove the simplification using the Desnanot-Jacobi identity and show that the N+2 coupling matrix can be directly extracted from the zeros of the admittance parameters (given by N+1 order determinants) and poles of the impedance parameters (given by the N order core matrix determinant). The results show that it is possible to decrease the computational complexity (by eliminating redundancy), reduce the associated cost function (by using less iterations), and under certain circumstances obtain different equivalent solutions. Nevertheless, the method also proves its practical usefulness under constrained optimizations when the user desires specific coupling matrix topologies and constrained coefficient values (e.g, purely real/imaginary/positive/negative). This can lead to a direct coupling matrix constrained configuration where other similar methods fail (using the same optimization algorithms).