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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}$:

$\textstyle \parbox{0.8\textwidth}{A polynomial of the form
\par\begin{center}\b...
...$\ 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 $h(n)$, $n\in{\bf Z}$, 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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About the Author: 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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