Two Theorems on Positive-Real Functions and Their Application to the Synthesis of Symmetric and Antimetric Filters

It is first shown that the power gain of a filter which has been partitioned into two component parts may be expressed in terms of a formula involving only the two impedances seen looking to the left and the right of the common junction. By imposing the constraints of symmetry and antimetry this formula leads quite naturally to two global equations for positive-real (pr) functions. Theorems 1 and 2 present necessary and sufficient conditions for the existence of solutions. Moreover, the construction of these pr functions is made to depend on two algorithms of an extremely simple character. The theory is fully illustrated by means of four worked, nontrivial examples. Finally, it is pointed out that synthesis by bisection is often wasteful of reactances (especially in the symmetric case), and a careful count of elements is presented for antimetric filters.