Alternate Stability Criterion
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-order LTI 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 stability criterion:
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.
Summary of the Partial Fraction Expansion