In Chapter 11 we looked at linear-phase and zero-phase digital
filters. While such filters preserve waveshape, there are times when
the linearity of the phase response is not important. In such cases,
it is valuable to allow the phase to be arbitrary, or else to set it
in such a way that the amplitude response is easier to match. In many
cases, this means specifying *minimum phase*.

Definition.An LTI filter is said to beminimum phaseif all its poles and zeros are inside the unit circle (excluding the unit circle itself).

Note that minimum-phase filters are stable by definition since the
poles must be inside the unit circle. In addition, because the zeros
must also be inside the unit circle, the inverse filter is
also stable when is minimum phase. One can say that
minimum-phase filters form an algebraic *group* in which the
group elements are impulse-responses and the group operation is
convolution (or, more specifically, the elements may be transfer
functions of a given order, and the group operation multiplication).

A minimum phase filter is also *causal* since noncausal terms in
the impulse response correspond to poles at infinity. The simplest
example of this would be the unit-sample *advance*, ,
which consists of a zero at and a pole at .^{13.1}

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

Definition.A polynomial of the form

is said to beminimum phaseif all of its roots are inside the unit circle,i.e., .

We may also define a *minimum-phase signal* (or sequence) as the inverse
*z* transform of a minimum-phase polynomial:

Definition.A signal , , is said to be minimum phase if itsztransform is minimum phase.

Note that *every stable, all-pole, (causal) filter
is minimum phase*, because stability implies that
is minimum phase, and there are ``no zeros'' (all are at
). This is an indication that minimum phase is in some sense the
most ``natural'' phase for a digital filter to have, since it is the
only phase available to a stable, causal, all-pole filter.

The effect of non-minimum-phase zeros on the *complex cepstrum*
was described in §8.4.

The opposite of minimum phase is *maximum phase*:

If zeros of occur both inside and outside the unit circle, the filter is said to be a

Definition.An LTI filter is said to bemaximum phaseif all zeros of the polynomial are outside the unit circle.

If is minimum phase, then is maximum phase, and vice versa.

By the *flip theorem* for *z* transforms,

FLIP

Each zero of inside the unit circle becomes a zero of
**Example**

An easy case to classify is the set of all first-order FIR filters

Among all signals having the identical magnitude spectra, the minimum-phase signal has thefastest decayin the sense that

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