Unstable Poles--Unit Circle Viewpoint

We saw in §8.4 that an LTI filter is stable if and only if all of its poles are strictly inside the unit circle () in the complex plane. In particular, a pole outside the unit circle () gives rise to an impulse-response component proportional to which grows exponentially over time . We also saw in §6.2 that the z transform of a growing exponential does not not converge on the unit circle in the plane. However, this was the case for a causal exponential , where is the unit-step function (which switches from 0 to 1 at time 0). If the same exponential is instead anticausal, i.e., of the form , then, as we'll see in this section, its z transform does exist on the unit circle, and the pole is in exactly the same place as in the causal case. Therefore,to unambiguously invert a z transform, we must know its region of convergence. The critical question is whether the region of convergence includes the unit circle: If it does, then each pole outside the unit circle corresponds to an anticausal, finite energy, exponential, while each pole inside corresponds to the usual causal decaying exponential.

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Subsections

• Geometric Series
• One-Pole Transfer Functions

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.