Which behavior is characteristic of a marginally stable discrete-time system with a pole on the unit circle?

• A. Stable with decaying amplitude
• B. Unstable with exponential growth
• C. Unstable with oscillations that dampen
• D. Sustained oscillations at the pole frequency

Answer: (D)

In a discrete-time system, where the poles sit in the z-plane determines how the natural response behaves. A pole on the unit circle has magnitude 1, so it does not cause decay or growth. If that pole is simple, its contribution to the response is proportional to e^{jω0 n}, which forms a real sinusoid with constant amplitude. In other words, the system output keeps oscillating at the frequency corresponding to the pole, without dying out or blowing up. That is exactly sustained oscillation at the pole frequency. If the pole were inside the unit circle, you’d get decaying oscillations; if outside, exponential growth; and a repeated pole on the unit circle would lead to growth over time, not stable behavior.