Minimum Phase (MP) polynomials in

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Physical Audio Signal Processing

Positive Real Functions

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Physical Audio Signal Processing

Positive Real Functions

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Minimum Phase (MP) polynomials in

All properties of MP polynomials apply without modification to marginally stable allpole transfer functions (cf. Property 2):

Every first-order MP polynomial is positive real.
Every first-order MP polynomial $b(z)=1+b_1\,z^{-1}$ is such that $\frac{1}{ b(z)} - \frac{1}{ 2}$ is positive real.
A PR second-order MP polynomial with complex-conjugate zeros,

$\begin{eqnarray*} H(z)&=& 1+b_1z^{-1}+b_2z^{-2}\\ &=& 1-(2R\cos\phi)z^{-1}+R^2z^{-2},\quad R\leq 1 \end{eqnarray*}$

satisfies

$\displaystyle R^2 + \frac{\cos^2\phi}{ 2} \leq 1.$

If $2R^2+\cos^2\phi=2$ , then re $\left\{H(\ejo)\right\}$ has a double zero at

$\begin{eqnarray*} \omega &=& \cos^{-1}\left(\pm \sqrt{ \frac{1-R^2}{ 2R^2}}\right) = \cos^{-1}\left(\pm \frac{\cos\phi}{ 2R}\right)\\ &=& \cos^{-1}\left(\pm \frac{1}{ \sqrt{2}} \frac{\cos\phi}{ \sqrt{1+\sin^2\phi}}\right). \end{eqnarray*}$
All polynomials of the form

$\displaystyle H(z)=1+R^nz^{-n},\quad R\leq 1$

are positive real. (These have zeros uniformly distributed on a circle of radius .)

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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4
Copyright © 2024-06-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

Minimum Phase (MP) polynomials in

``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4 Copyright © 2024-06-28 by Julius O. Smith III Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4
Copyright © 2024-06-28 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA), Stanford University