Coupling matrix synthesis of dissipative microwave filters

This study reviews the coupling matrix synthesis of dissipative microwave filters. It compares it to the synthesis of lossless transfer functions and shows that by correctly interpreting the mathematical operations in the lossless case, that is especially the matrix product of vectors and matrices, the same procedure can be adopted for the case including losses. Beyond this, necessary and sufficient conditions are provided guaranteeing that a proper coupling matrix, that is which is symmetric, is obtained at the end. Two examples show the application of the procedure and demonstrate that the correct filter transfer functions are indeed obtained.