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 [#!Brune!#,#!VV!#]. Very little, however, seems to be available in the discrete time case. This section (reprinted from [#!JOST!#]) 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
We now specialize to the subset of functions representable as a ratio of finite-order polynomials in , and . Such ``rational functions'' are meromorphic, meaning that they are analytic for all except at a set of isolated points given by the zeros of the denominator polynomial . This class of meromorphic functions represents all filter transfer functions for 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 . 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 forward that satisfies (1). From (2) we derive the facts below.
Property 1. A real rational function is PR iff .
Proof. Expressing in polar form gives
Since the zeros of are isolated (because is analytic for ), the phase is unrestricted only at isolated points (the zeros of ). We may define the phase arbitrarily when , and the most natural definition is its limit as approaches the zero; such a definition satisfies the constraint because every point in any valid limiting sequence satisfies the constraint, and the domain of the constraint is closed (the image of a continuous function of a compact set is compact--if were replaced by , we could have a problem.) Therefore, we may conclude that for all , not just where is nonzero.
Property 2. is PR iff is PR.
Proof. Assuming is PR, we have by Property 1,
Property 3. A PR function is analytic and nonzero in the strict outer disk.
Proof. (By contradiction)
Without loss of generality, we treat only order polynomials
which are nondegenerate in the sense that . Since facts about are readily deduced from facts about , we set at no great loss.
The general (normalized) causal, finite-order, linear,
time-invariant transfer function may be written
Suppose there is a pole of multiplicity outside the unit circle. Without loss of generality, we may set , and with . Then for near , we have
Consider the circular neighborhood of radius described by . Since we may choose so that all points in this neighborhood lie outside the unit circle. If we write the residue of the factor in polar form as , then we have, for sufficiently small ,
which cannot be confined to satisfy Property 1 regardless of the value of the residue angle , or the pole angle ( cannot be zero by hypothesis). We thus conclude that a PR function can have no poles in the outer disk. By Property 2, we conclude that positive real functions must be minimum phase.
Corollary. In equation Eq.(C.80), .
Proof. If , then there are poles at infinity. As , , we must have .
Corollary. The log-magnitude of a PR function has zero mean on the unit circle.
This is a general property of stable, minimum-phase transfer functions which follows immediately from the argument principle [#!MG!#,#!Nehari!#].
Corollary. A rational PR function has an equal number of poles and zeros all of which are in the unit disk.
This really a convention for numbering poles and zeros. In Eq.(C.80), we have , and all poles and zeros inside the unit disk. Now, if then we have extra poles at induced by the numerator. If , then zeros at the origin appear from the denominator.
Corollary. Every pole on the unit circle of a positive real function must be simple with a real and positive residue.
Proof. We repeat the previous argument using a semicircular neighborhood of radius about the point to obtain
Corollary. If is PR with a zero at , then
Proof. We may repeat the above for .
Property. Every PR function has a causal inverse z transform .
Proof. This follows immediately from analyticity in the outer disk [#!Papoulis!#, pp. 30-36] However, we may give a more concrete proof as follows. Suppose is non-causal. Then there exists such that . We have,
Hence, has at least one pole at infinity and cannot be PR by Property 3. Note that this pole at infinity cannot be cancelled since otherwise
which contradicts the hypothesis that is non-causal.
Property. is PR iff it is analytic for , poles on the unit circle are simple with real and positive residues, and re for .
Proof. If is positive real, the conditions stated hold by virtue of Property 3 and the definition of positive real.
To prove the converse, we first show nonnegativity on the upper semicircle implies nonnegativity over the entire circle.
Alternatively, we might simply state that real implies re is even in .
Next, since the function
is analytic everywhere except at
, it follows that
is analytic wherever
is finite. There are no poles of
outside the unit
circle due to the analyticity assumption, and poles on the unit circle
have real and positive residues. Referring again to the limiting form
near a pole on the unit circle at
we see that, as
, we have
occurs on the unit circle. Consequently,
For example, if a transfer function is known to be asymptotically stable, then a frequency response with nonnegative real part implies that the transfer function is positive real.
Note that consideration of leads to analogous necessary and sufficient conditions for to be positive real in terms of its zeros instead of poles.