Electronics Engineering (ELEX) Board Practice Exam

If all the poles of a transfer function H(z) are outside the unit circle, what can be concluded about the system?

1. The system is stable

2. The system is unstable

3. None of the choices

4. The system is marginally stable

The correct answer is: None of the choices

When analyzing the stability of a discrete-time system characterized by its transfer function \( H(z) \), the position of the poles in relation to the unit circle in the z-plane is crucial. If all the poles of the transfer function are outside the unit circle, it directly indicates that the system is unstable. In discrete-time systems, stability is defined by the location of the poles: for a system to be stable, all poles must lie inside the unit circle. When poles are located on or outside the unit circle, the response of the system can either grow unbounded (which corresponds to instability) or settle indefinitely without oscillation (which describes marginal stability). Thus, if all poles are found outside the unit circle, it confirms that the system's output will not converge to a finite value over time; instead, it will diverge. Therefore, the appropriate conclusion about the stability of the system is that it is unstable. Hence, the correct reflection of this situation is that the system is unstable, providing clarity on the dynamics involved. Options suggesting that the system is stable or marginally stable misinterpret the implications of pole locations on system behavior. Having all poles outside definitively leads to instability, which aligns with basic control theory principles.