NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Introduction to Digital Filters with Audio Applications", by Julius O. Smith III, Copyright © 2017-11-26 by Julius O. Smith III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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Relation to Schur Functions
Definition. A Schur function
is defined as a complex function analytic and of modulus not exceeding unity in
.
Theorem. The function
is a Schur function if and only ifis positive real.
Proof.
Suppose
is positive real. Then for
,
is PR. Consequently,
is minimum phase which implies all roots of
lie in the unit circle. Thus
is analytic in
. Also,
By the maxmimum modulus theorem,takes on its maximum value in
on the boundary. Thus
is Schur.
Conversely, suppose
is Schur. Solving Eq. (1.2) for
and taking the real part on the unit circle yields
(33) (34) (35) (36) (37)
Ifis constant, then
is PR. If
is not constant, then by the maximum principle,
for
. By Rouche's theorem applied on a circle of radius
,
, on which
, the function
has the same number of zeros as the function
in
. Hence,
is minimum phase which implies
is analytic for
. Thus
is PR.