03. Relation to Schur Functions

GUIDE: Elementary Digital Filter Theory. Relation to Schur Functions

It appears that you are using AdBlocking software. The cost of running this website is covered by advertisements. If you like it please feel free to a small amount of money to secure the future of this website.

<< Previous page  TOC  INDEX  Next page >>

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.$\

<< Previous page  TOC  INDEX  Next page >>


© 1998-2017 – Nicola Asuni - Tecnick.com - All rights reserved.
about - disclaimer - privacy