Minimum Phase (MP) polynomials in $ z$

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(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 $ R$.)


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Special cases and examples