Alternate Stability Criterion

In §5.6 (page ), a filter was defined to be stable if its impulse response $h(n)$ decays to 0 in magnitude as time $n$ goes to infinity. In §6.8.5, we saw that the impulse response of every finite-order LTI filter can be expressed as a possible FIR part (which is always stable) plus a linear combination of terms of the form $a_i(n)p_i^n$, where $a_i(n)$ is some finite-order polynomial in $n$, and $p_i$ is the $i$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 $z$ plane, i.e., when $\vert p_i\vert<1$. 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, $H(z) = (1+z^{-1})/(1-z^{-1})$ is irreducible, while $H(z) = (1-z^{-2})/(1-2z^{-2}+z^{-2})$ is reducible, since there is the common factor of $(1-z^{-1})$ in the numerator and denominator. Using this terminology, we may state the following equivalent stability criterion:

$\parbox{0.8\textwidth}{\emph{An irreducible transfer function $H(z)$\ is stable if and only if its poles have magnitude less than one.}}$

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.