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* [68,526]. Very little, however, seems
to be available in the discrete time case. This section (reprinted
from [432]) 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

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

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* [299,329].

**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

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

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

must satisfy

**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
[345, 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
Eq.
(C.81) of
near a pole on the unit circle at
,
we see that, as
, we have

since the residue is positive, and the net angle does not exceed . From Eq. (C.83) we can state that for points with modulus , we have that 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 [83],

occurs on the unit circle. Consequently,

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.

- Relation to Stochastic Processes
- Relation to Schur Functions
- Relation to -plane Positive-Real Functions
- Special cases and examples
- Minimum Phase (MP) polynomials in
- Miscellaneous Properties

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University