For a stable LTI system, where are its poles located in the complex plane?

• A. All poles lie in the left-half plane (Re(s) < 0).
• B. All poles lie on the imaginary axis.
• C. Poles can be anywhere as long as the ROC includes the imaginary axis.
• D. All poles lie in the right-half plane (Re(s) > 0).

Answer: (A)

The system’s stability in continuous time is tied to where its poles sit in the complex s-plane. For BIBO stability, every pole must contribute a decaying term to the impulse response. Each pole p yields a factor e^{p t} (times possibly t^k if the pole is repeated). If Re(p) < 0, these terms decay as time grows, so bounded inputs produce bounded outputs. If any pole lies on or to the right of the imaginary axis (Re(p) ≥ 0), you get non-decaying or growing responses for some inputs, which breaks stability. Therefore, to ensure a stable LTI system, all poles must be in the left-half plane, Re(s) < 0. The other possibilities would lead to persistent oscillations or growth, or rely on the transfer function’s ROC in a way that doesn’t guarantee stability.