## 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