Positive Real Functions

Any passive driving-point impedance, such as the impedance of a violin bridge, is positive real. Positive real functions have been studied extensively in the continuous-time case in the context of network synthesis [70,532]. Very little, however, seems to be available in the discrete time case. This section (reprinted from [437]) summarizes the main properties of positive real function in the plane (i.e., the discrete-time case).

Definition. A complex valued function of a complex variable is said to be positive real (PR) if

1. 
2.    re

We now specialize to the subset of functions representable as a ratio of finite-order polynomials in . This class of ``rational'' functions is the set of all transfer functions of finite-order time-invariant linear systems, and we write to denote a member of this class. We use the convention that stable, minimum phase systems are analytic and nonzero in the strict outer disk.^M.5 Condition (1) implies that for to be PR, the polynomial coefficients must be real, and therefore complex poles and zeros must exist in conjugate pairs. We assume from this point on that satisfies (1).