Synthesis of cascaded N-tuplet filters

In the theory of Darlington type filter synthesis extraction of transmission zero sections one by one, leading to a cascade of Darlington-A, B, C, D and Brune sections bears serious limitations like numerical accuracy problems, negative and/or extreme element values or impractical topologies for Darlington-C-D and Brune sections. In this paper all such problems are evaded/eased by extracting the transmission zeros as groups to form circuit sections which are readily convertible to multiple coupled resonator forms. In this approach, transmission zeros at s=0 and s= ∞ are extracted as groups to form doublets of resonators, the jω-axis and σ-axis transmission zeros are extracted as groups to form triplets of resonators, pairs of jω-axis and σ-axis transmission zeros and complex transmission zeros are extracted as groups to form quadruplets of resonators, etc. The classical element extraction procedure in transformed frequency domain is revised to cover extraction of complex transmission zeros as forth order section by zero shifting from both s=0 and s= ∞ leading to realisation as a quadruplet of resonators. Thus, the theory of cascade synthesis is generalised to include cascaded doublets, triplets, quadruplets and other N-tuplets of resonators as building blocks. It is found out that the limitations due to usual numerical accuracy problems are also reduced a lot and the resulting element value spreads can be controlled to avoid extreme values