In §5.6 (page ), a filter was defined to
be stable if its impulse response decays to 0 in
magnitude as time goes to infinity. In §6.8.5, we saw that
the impulse response of every finite-orderLTI filter can be expressed
as a possible FIR part (which is always stable) plus a linear
combination of terms of the form
, where is some
finite-order polynomial in , and is the th pole of the
filter. In this form, it is clear that the impulse response always
decays to zero when each pole is strictly inside the unit circle of
the plane, i.e., when . Thus, having all poles strictly
inside the unit circle is a sufficient criterion for filter
stability. If the filter is observable (meaning that there are
no pole-zero cancellations in the transfer function from input to
output), then this is also a necessary criterion.

A transfer function with no pole-zero cancellations is said to be
irreducible. For example,
is
irreducible, while
is reducible, since
there is the common factor of
in the numerator and
denominator. Using this terminology, we may state the following
equivalent stability criterion:

This characterization of stability is pursued further in §8.3,
and yet another stability test (most often used in practice)
is given in §8.3.1.