A discrete-time system is stable if all poles lie:

• A. On the real axis
• B. Outside the unit circle
• C. Inside the unit circle
• D. On the imaginary axis

Answer: (C)

A discrete-time system is considered stable when all of its poles are located inside the unit circle in the z-plane. This is due to the fact that the poles of a system's transfer function directly influence its behavior over time. When poles are positioned inside the unit circle, it ensures that the system's response to inputs diminishes over time, leading to a stable output that converges to a finite value. Specifically, poles that lie inside the unit circle indicate that any oscillation or transient response resulting from an input will eventually decay, resulting in a controlled and stable system behavior.

In contrast, if all poles were outside the unit circle, the system would exhibit an unstable response, where outputs would grow unbounded over time. Poles exactly on the unit circle would lead to marginal stability, where responses may oscillate indefinitely without growing or decaying. Meanwhile, poles on the real or imaginary axis do not guarantee stability, as their placement relative to the unit circle is what ultimately dictates the system's stability.

Thus, the critical condition for stability in a discrete-time system is that all poles must be situated within the unit circle.