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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(\ejo)\right\vert = \frac{1-2\mbox{re}\left\{R(\ejo)\right\} + \left\vert R(\ejo\right\vert^2 }{ 1+2\mbox{re}\left\{R(\ejo)\right\} + \left\vert R(\ejo)\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(\ejo)\right\} &=&\alpha\,\mbox{re}\left\{\frac{1-S(\ejo)}{ 1+S(\ejo)}{1+S(e^{-j\omega})}{ 1+S(e^{-j\omega})}\right\} \\ &=&\alpha\,\mbox{re}\left\{\frac{1-S(\ejo)+S(e^{-j\omega})-\left\vert S(\ejo)\right\vert^2 }{ \left\vert 1+S(\ejo)\right\vert^2}\right\}\\ &=&\alpha\,\frac{1-\left\vert S(\ejo)\right\vert^2 }{ \left\vert 1+S(\ejo)\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$

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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4
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Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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