For a discrete-time system, a pole on the unit circle corresponds to which stability behavior?

• A. Stable with decaying amplitude
• B. Unstable with exponential growth
• C. Unstable with oscillations that dampen
• D. Marginal stability; sustained oscillations may occur at the pole frequency

Answer: (D)

In discrete-time systems, stability hinges on where the poles lie relative to the unit circle. If all poles are strictly inside the unit circle, the system is asymptotically stable and responses decay. If a pole sits exactly on the unit circle, the amplitude no longer decays; the system is on the edge—this is marginal stability. When the pole is on the unit circle at a frequency e^{jω0} (or comes as a complex conjugate pair on the circle), the time-domain response contains terms like e^{j n ω0} or cos(nω0), which produce oscillations whose amplitude stays constant over time. Thus the system does not blow up, but it does not decay either, yielding sustained oscillations at the pole frequency. That’s why the correct description is marginal stability with the possibility of sustained oscillations at the pole frequency. The other scenarios—poles inside the circle giving decaying responses, or poles outside causing growth—don’t match the behavior on the unit circle.