Synthesis of cascaded quadruplet filters involving complex transmission zeros

Cascade synthesis approach in transformed frequency domain (z-domain) is reformulated to cover the extraction of complex (s = sigma +j omega) and sigma -axis transmission zeros (TZs) in addition to the usual j omega -axis TZs. It is shown that complex conjugate quadruplets of TZs (s = +/- sigma +/- j omega) can be extracted as fourth order sections by applying zero shifting technique to the TZs at both s=0 and s=infinity. The resulting fourth order sections are then transformed into cross-coupled forms with four coupled resonators, termed as linear phase Cascade Quadruplet (CQ) block. Such CQ blocks can be cascaded with the j omega -axis finite Triplet) blocks to realize linear phase filters with symmetric or asymmetric amplitude response.