06. Minimum Phase (MP) polynomials in Z

GUIDE: Elementary Digital Filter Theory - Julius O. Smith III. Minimum Phase (MP) polynomials in Z

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NOTE: THIS DOCUMENT IS OBSOLETE, PLEASE CHECK THE NEW VERSION: "Introduction to Digital Filters with Audio Applications", by Julius O. Smith III, Copyright © 2017-11-26 by Julius O. Smith III - Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

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Minimum Phase (MP) polynomials in $z$

All properties of MP polynomials apply without modification to marginallystable allpole transfer functions (cf. Thm. (1)).

  • Every first-order MP polynomial is positive real.
  • Every first-order MP polynomial $b(z)=1+b_1 z^{-1}$ is such that $ \ is positive real.
  • A PR second-order MP polynomial with complex-conjugate zeros,
    $\ $\ $\ (38)
      $\ $\ (39)

    satisfies
    \

    If $2R^2+\, then $\ has a double zero at
    \

  • All polynomials of the form
    \

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

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