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Embedded SystemsFPGAElectronics

Physical Audio Signal Processing
    Digital Waveguide Theory
       Properties of Passive Impedances
          Positive Real Functions
             Relation to Schur Functions

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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$.

Property. The function

$\displaystyle S(z) \isdef \frac{1-R(z)}{ 1+R(z)}$ (C.84)

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

Proof.

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.$ \,$(C.84) for $ R(z)$ and taking the real part on the unit circle yields

\begin{eqnarray*} R(z) &=& \alpha\,\frac{1-S(z)}{ 1+S(z)} \\ \mbox{re}\left\{R(... ...ert^2 }{ \left\vert 1+S(e^{j\omega})\right\vert^2}\\ &\geq& 0. \end{eqnarray*}

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$

Previous: Relation to Stochastic Processes
Next: Relation to functions positive real in the right-half plane

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


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