Recent developments in the synthesis of a class of lumped-distributed filters

In the light of several recent theoretical breakthroughs^1,2 it appears reasonably certain that the insertion-loss synthesis of a broad class of practical lumped-distributed filters will soon become a reality. In particular, the structure of filters composed of a finite cascade of lumped reactive 2-ports and equidelay ideal TEM lines is now almost completely understood and the related approximation problem not only seems soluble but also amenable to digital computer implementation. The advances in References 1 and 2 were sparked by a remarkable counter-example due to Rhodes and Marston³ and it is the purpose of the present summary to give a complete discussion and account of the concrete gains made up to this point without, of course, entering into proofs of material already published. Briefly, the order of ideas is as follows. In Reference 1 a fundamental synthesis theorem is established with the aid of the 2-variable positive-real concept and in Reference 2 explicit formulae are derived for the various junction reflection coefficients. In addition, Reference 2 reformulates the synthesis theorem in Reference 1 and succeeds in removing the 2-variable positive-reality requirement but only at the expense of introducing two extra conditions of the 1-variable type. Theorem 2 of the present note succeeds in eliminating the more complicated of these two conditions and has as consequence the final and definitive ‘Strong Theorem 1’. A full proof of theorem 3 is given in the Appendix and one illustrative example is worked out in detail.