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 ($\vert z\vert=1$ ) in the complex $z$ plane. In particular, a pole $p$ outside the unit circle ($\vert p\vert>1$ ) gives rise to an impulse-response component proportional to $p^n$ which grows exponentially over time $n$ . We also saw in §6.2 that the z transform of a growing exponential does not converge on the unit circle in the $z$ plane. However, this was the case for a causal exponential $u(n)p^n$ , where $u(n)$ 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 $u(-n)p^n$ , 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.