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Minimum-Phase Polynomials

A filter is minimum phase if both the numerator and denominator of its transfer function are minimum-phase polynomials in $ z^{-1}$ :

$\textstyle \parbox{0.8\textwidth}{A polynomial of the form
B(z) &=& b_0 + b_1 z^{-1}+ b_2 z^{-2} + \cdots + b_M z^{-M}\\
&=& b_0(1-\xi_1z^{-1})(1 - \xi_2 z^{-1})\cdots(1-\xi_Mz^{-1}),
\end{eqnarray*}\end{center}\par {}
where $b_0\ne 0$, is said to be \emph{minimum phase} if all of its
roots $\xi_i$\ are inside the unit circle, \textit{i.e.}, $\left\vert\xi_i\right\vert<1$.
The case $ b_0=0$ is excluded because the polynomial cannot be minimum phase in that case, because then it would have a zero at $ z=\infty$ unless all its coefficients were zero.

As usual, definitions for filters generalize to definitions for signals by simply treating the signal as an impulse response:

$\textstyle \parbox{0.8\textwidth}{A signal $h(n)$, $n\in\mathbb{Z}$, is said to be minimum phase
if its {\it z} transform\ $H(z)$\ is minimum phase.

Note that every stable all-pole filter $ H(z)=b_0/A(z)$ is minimum phase, because stability implies that $ A(z)$ is minimum phase, and there are ``no zeros'' (all are at $ z=0$ ). Thus, minimum phase is the only phase available to a stable all-pole filter.

The contribution of minimum-phase zeros to the complex cepstrum was described in §8.8.

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``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (September 2007 Edition)
Copyright © 2024-05-20 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University