## Minimum-Phase Polynomials

A filter is minimum phase if both the numerator and denominator of its
transfer function are
*minimum-phase polynomials*
in :

The case is excluded because the polynomial cannot be minimum phase in that case, because then it would have a zero at 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*:

Note that *every stable all-pole filter
is
minimum phase*, because stability implies that is minimum
phase, and there are ``no zeros'' (all are at ).
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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Maximum Phase Filters

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Definition of Minimum Phase Filters