Minimum Phase (MP) polynomials in

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All properties of MP polynomials apply without modification to marginally stable allpole transfer functions (cf. Thm. (M.4)):

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 $\left\{H(e^{j\omega})\right\}$ has a double zero at

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

written by 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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