Stability and Feedback
Stability
A system is asymptotically stable if the region of convergence includes the unit circle.
As a result, a causal system is asymptotically stable if all of its poles are inside the unit circle (i.e., |z| < 1). An anti-causal system is asymptotically stable if all of its poles are outside the unit circle (i.e., |z| > 1).
If the system is asymptomatically stable, the impulse response $h[n]$ converses to $0$ as $t \rightarrow \infty$. The figures below show examples of stable and unstable systems.
Impulse Responses and Pole-Zero Plots
We can often infer the behavior of a system impulse response by the locations of pole. Poles on the unit circle observe no increase or decrease in amplitude over time. As the poles get further from the unit circle, the impulse response either increases or decreases with time (depending on the type of causality and location of the pole). Below are examples of impulse responses and pole-zero plots.
Example: Simple Feedback System and Stability
Consider the difference equation $$y[n] = \alpha y[n-1] + \alpha x[n]$$ The transfer function for the feedback system shown above is defined by $$\begin{eqnarray*} H(z) = \frac{\alpha}{1 - \alpha z^{-1}} \; . \end{eqnarray*}$$
The causal impulse response of this system is $$h[n] = \alpha^{n+1} u[n] \; .$$ As long as $|\alpha| < 1$, the causal impulse response is asymptotically stable.
The anti-causal impulse response of this system is $$h[n] = - \alpha^{n+1} u[-n-1] \; .$$ As long as $|\alpha| > 1$, the anti-causal impulse response is asymptotically stable.