For a causal discrete-time LTI system, what condition on its poles ensures BIBO stability?

• A. All poles lie inside the unit circle (|p| < 1).
• B. All poles lie outside the unit circle (|p| > 1).
• C. Poles on the unit circle (|p| = 1).
• D. Pole locations do not affect stability.

Answer: (A)

For a causal discrete-time LTI system, BIBO stability means the impulse response must be absolutely summable. In the Z-domain, this corresponds to the ROC being outside the outermost pole, but also to the impulse response decaying fast enough so that sum of its magnitudes converges. This happens precisely when every pole lies strictly inside the unit circle (its magnitude is less than 1). In that case, the impulse response terms look like polynomials in n times p^n with |p|<1, which decay exponentially and make the series ∑|h[n]| converge. If any pole lies on or outside the unit circle, the impulse response either does not decay or grows, and the sum of magnitudes does not converge, breaking BIBO stability. Therefore, the correct condition is that all poles are inside the unit circle.