03. Relation to Schur Functions

GUIDE: Elementary Digital Filter Theory - Julius O. Smith III. Relation to Schur Functions

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NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Introduction to Digital Filters with Audio Applications", by Julius O. Smith III, Copyright © 2017-11-26 by Julius O. Smith III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 $S(z)$is defined as a complex function analytic and of modulus not exceeding unity in $\.

Theorem. The function


is a Schur function if and only if $R(z)$ is positive real.


Suppose $R(z)$ is positive real. Then for $\, $\ is PR. Consequently, $1+R(z)$is minimum phase which implies all roots of $S(z)$ lie in the unit circle. Thus $S(z)$ is analytic in $\. Also,


By the maxmimum modulus theorem, $S(z)$ takes on its maximum value in $\on the boundary. Thus $S(z)$ is Schur.

Conversely, suppose $S(z)$ is Schur. Solving Eq. (1.2) for $R(z)$and taking the real part on the unit circle yields

$\ $\ $\ (33)
$\ $\ $\ (34)
  $\ $\ (35)
  $\ $\ (36)
  $\ $\ (37)

If $S(z)=\ is constant, then $R(z)=(1-\ is PR. If $S(z)$ is not constant, then by the maximum principle, $S(z)<1$ for $\. By Rouche's theorem applied on a circle of radius $1+\, $\, on which $\, the function $1+S(z)$ has the same number of zeros as the function $1$ in $\. Hence, $1+S(z)$ is minimum phase which implies $R(z)$ is analytic for $z\. Thus$R(z)$ is PR.$\

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