Elementary Digital Filter Theory



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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.

Proof.

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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Copyright © 2000-09-19 by Julius O. Smith III, Adpated from Techniques for Digital FilterDesign and System Identification, with Application to the Violin, Julius O. Smith III, Ph.D. Dissertation, CCRMA, Department of Electrical Engineering, Stanford University, June 1983.. This is an original HTML version reproduced by permission.
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