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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^{-1}+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 stability criterion:

$\textstyle \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 §8.4, and yet another stability test (most often used in practice) is given in §8.4.1.

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``Introduction to Digital Filters with Audio Applications'', by Julius O. Smith III, (September 2007 Edition)
Copyright © 2024-05-20 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University