In Chapter 10 we looked at linear-phase and zero-phase digital
filters. While such filters preserve waveshape to a maximum extent in
some sense, there are times when phase linearity 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*:

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, alternatively, the elements are minimum-phase
transfer functions, and the group operation is 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
.
^{12.1}

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University