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

  $$R^2 + \frac{\cos^2\phi}{ 2} \leq 1.$$
  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
  $$H(z)=1+R^nz^{-n},\quad R\leq 1$$
  are positive real. (These have zeros uniformly distributed on a circle of radius $R$ .)