Explicit Solution to Exact Synthesis of Cross-Coupled Quadruplet Filter with a Real-Frequency Zero Pair

Explicit solution is derived for the exact synthesis of cross-coupled quadruplet filters with a real-frequency zero pair. Through a comparison of two-port S-parameters of the low-pass prototype circuit with the canonical generalized Chebyshev insertion loss function, a univariate quadratic function is formulated, and all element values of the circuit are explicitly expressed in terms of the designated zero and in-band ripple levels. The analytical expression of the sidelobe attenuation level is also obtained in terms of the zero, and vice versa. Results are checked against those from the existing literature.