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.