Analysis and synthesis of strictly positive real transfer functions

The concept of strictly positive real (SPR) transfer functions is examined. It is shown that commonly used frequency domain conditions for SPR do not satisfy some of the most basic elements of the definition and properties of this class of functions. For a given Hurwitz polynomial a, a degree n, we find the set of all possible polynomials b that make the ratio ba SPR, and (i) proper, and (ii) improper. Further, we show that the set of all possible bs can be parametrized in terms of, respectively, n + 1 and n + 2 numbers that satisfy a simple constraint.