Educative Explanation to the Alternating Pole Synthesis Technique in Filter Theory

In this paper, the alternating pole synthesis technique in filter theory is revisited and an educative explanation is presented. The parametric variation in the insertion loss response provides further insights of the pole movement in the complex plane p = σ +jω of the normalized frequency of the prototype filter. Moreover, the introduction of a weight in the description enables an analytic continuation of the parametric variation. This step enables the analysis of the behaviour at the transition to the jω-axis. In the case of an equiripple passband performance, it can simply be shown, that the required alternating slope of the equiripple function at its roots leads to alternating movement of the poles in the complex plane to ±σ in the vicinity of the roots. Further examples with multiple roots and complex conjugate roots are also discussed.