Formulation of realizable scattering matrices for multiplexers with contiguous bands

Compact multiplexers are a class of multiport networks entirely formed by coupled resonators, with no additional elements, as junction structures or transmission line lengths. The synthesis of their coupling matrices require the formulation of scattering coefficients as rational functions that verify the realizability conditions. A solution is to approximate the reflection coefficients as the ones of isolated filters, and then to reconstruct the transmission coefficients. This approach is quite robust when the pass bands are clearly separated, but additional care must be taken when they are contiguous. In particular, phase offsets should be applied to the isolated filters responses. The effect of these phase shifts can be compensated after the coupling matrix synthesis.