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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 $ \vert z\vert\leq 1$.

Theorem. The function

$\displaystyle S(z) \isdef \frac{1-R(z)}{ 1+R(z)} \protect$ (5)

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


Suppose $ R(z)$ is positive real. Then for $ \vert z\vert\geq 1$, re$ \left\{R(z)\right\}\geq 0
\,\,\Rightarrow\,\,1+$re$ \left\{R(z)\right\}\geq 0 \,\,\Rightarrow\,\,1+R(z)$ 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 $ \vert z\vert\leq 1$. Also,

$\displaystyle \left\vert S(e^{j\omega})\right\vert = \frac{1-2\mbox{re}\left\{R...
...left\{R(e^{j\omega})\right\} + \left\vert R(e^{j\omega})\right\vert^2} \leq 1.

By the maximum modulus theorem, $ S(z)$ takes on its maximum value in $ \vert z\vert\geq 1$ on the boundary. Thus $ S(z)$ is Schur.

Conversely, suppose $ S(z)$ is Schur. Solving Eq.$ \,$(5) for $ R(z)$ gives

$\displaystyle R(z) = \frac{1-S(z)}{ 1+S(z)}

and taking the real part on the unit circle yields

\mbox{re}\left\{R(e^{j\omega})\right\} &=& \mbox{re}\left\{\fr...
...ert^2 }{ \left\vert 1+S(e^{j\omega})\right\vert^2}\\
&\geq& 0.

If $ S(z)=\alpha$ is constant, then $ R(z)=(1-\vert\alpha\vert^2)/\vert 1+\alpha\vert^2$ is PR. If $ S(z)$ is not constant, then by the maximum principle, $ S(z)<1$ for $ \vert z\vert>1$. By Rouche's theorem applied on a circle of radius $ 1+\epsilon $, $ \epsilon>0$, on which $ \vert S(z)\vert<1$, the function $ 1+S(z)$ has the same number of zeros as the function $ 1$ in $ \vert z\vert\geq 1+\epsilon $. Hence, $ 1+S(z)$ is minimum phase which implies $ R(z)$ is analytic for $ z\geq 1$. Thus $ R(z)$ is PR.$ \Box$

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``Properties of Positive Real Functions'', by Julius O. Smith III, from ``Techniques for Digital Filter Design and System Identification, with Application to the Violin,'' Julius O. Smith III, Ph.D. Dissertation, CCRMA, Department of Electrical Engineering, Stanford University, June 1983.
Copyright © 2009-12-01 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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