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* [1,9]. Very little, however, seems
to be available in the discrete time case. The purpose of this
note (an excerpt from [7]) is to collect together some
facts about positive real transfer functions for discrete-time linear
systems.

**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 . 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.^{1} 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).
From (2) we derive the facts below.

**Theorem. ** A real rational function is PR iff
.

**Proof. **Expressing in polar form gives

since the zeros of are isolated.

**Proof. **Assuming is PR, we have by Thm. (1),

**Theorem. ** A PR function is analytic and nonzero in
the strict outer disk.

**Proof. **(By contradiction)

Without loss of generality, we treat only order polynomials

The general (normalized) causal, finite-order, linear,
time-invariant transfer function may be written

where is the number of distinct poles, each of multiplicity ,and

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 ,

Therefore, approaching the pole at an angle gives

**Corollary. **In equation Eq.(1), .

**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* [3,4].

**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.(1), 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

In order to have near this pole, it is necessary that and .

**Corollary. **If is PR with a zero at
, then

**Proof. **We may repeat the above for .

**Theorem. **Every PR function has a causal inverse *z* transform .

**Proof. **This follows immediately from analyticity in the outer disk
[5, 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 Thm. (1). Note that this pole at infinity cannot be cancelled since otherwise

which contradicts the hypothesis that is non-causal.

**Theorem. **
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 Thm. (1) 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 re 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
Eq.(2) of near a pole on the unit circle at ,
we see that

since the residue is positive, and the net angle does not exceed . From Eq.(4) we can state that for points with modulus , we have For all , there exists such that . Thus is analytic in the strict outer disk, and continuous up to the unit circle which forms its boundary. By the maximum modulus theorem [2],

rere

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.

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