NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Introduction to Digital Filters with Audio Applications", by Julius O. Smith III, Copyright © 2017-11-26 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University
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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 ofnetwork synthesis [Brune 1931,Van Valkenburg 1960]. Very little, however, seems to be available in the discrete time case. The purpose of this home page is to collect some facts about positive real transfer functions for discrete-time linear systems.
Definition. A complex valued function of a complex variable
is said to bepositive 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*The strict outer disk is defined as the region
in the extended complex plane. 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
(1) (2)
since the zeros ofare isolated.
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Proof. Assuming
is PR, we have by Thm. (1),
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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
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
whereis 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
(7) (8)
Consider the circular neighborhood of radiusdescribed 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 poleat an angle
gives
which cannot be confined to satisfy Thm. (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 Thm. (1), we conclude that positive real functions must be minimum phase.
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Corollary. In equation Eq. (6),
.
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 [Markel and Gray 1976,Nehari 1952].
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. (6), 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 havenear this pole, it is necessary that
and
.
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Corollary. If
is PR with a zero at
, then
must satisfy
(9) (10)
Proof. We may repeat the above for
.
Theorem. Every PR function
has a causal inverse
-transform
.
Proof. This follows immediately from analyticity in the outer disk [Papoulis 1977, pp. 30-36] However, we may give a more concrete proof as follows. Suppose
is non-causal. Then there exists
such that
. We have,
(11) (12)
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
(13) (14) (15)
which contradicts the hypothesis thatis non-causal.
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Theorem.
is PR iff it is analytic for
, poles on the unit circle are simple with real and positive residues, and
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.
(16) (17) (18) (19) (20) (21)
Alternatively, we might simply state thatreal
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. (1) of
near a pole on the unit circle at
, we see that
since the residueis positive, and the net angle
does not exceed
. From Eq. (26) 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 [Churchill 1960],
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.