Properties of ROC of z-transform

The region of convergence (ROC) of X(z) = Σ x[n] z⁻ⁿ is the set of z for which the sum converges absolutely, Σ |x[n]| |z|⁻ⁿ < ∞.

Properties

1. The ROC is a ring (annulus) centred at the origin: r₁ < |z| < r₂, where 0 ≤ r₁ and r₂ ≤ ∞. Convergence depends only on |z|, not on the angle.
2. The ROC contains no poles, since X(z) is infinite at a pole. It is bounded by poles (or extends to 0 or ∞).
3. Finite-duration sequence: the ROC is the entire z-plane, except possibly z = 0 and/or z = ∞. Example: x[n] = {1, 2, 3} for n = 0, 1, 2 gives X(z) = 1 + 2z⁻¹ + 3z⁻², ROC all z except z = 0.
4. Right-sided sequence (x[n] = 0 for n < N₁): the ROC is the exterior of a circle, |z| > r₁ (the largest pole magnitude); it includes z = ∞ if N₁ ≥ 0 (causal). Example: aⁿu[n] ↔ 1/(1 − az⁻¹), |z| > |a|.
5. Left-sided sequence (x[n] = 0 for n > N₂): the ROC is the interior of a circle, |z| < r₂ (the smallest pole magnitude); it includes z = 0 if N₂ ≤ 0. Example: −aⁿu[−n−1] ↔ 1/(1 − az⁻¹), |z| < |a|.
6. Two-sided sequence: the ROC is a ring r₁ < |z| < r₂, or empty if the right-sided and left-sided parts do not overlap. Example: (½)ⁿu[n] − 3ⁿu[−n−1] has ROC ½ < |z| < 3.
7. The ROC is a connected region (one ring). For a rational X(z) with several poles, the ROC is one of the rings between consecutive pole magnitudes, so different ROCs give different sequences for the same X(z).
8. Stability: x[n] is absolutely summable (its DTFT exists) if and only if the ROC includes the unit circle |z| = 1.
9. Causality of a system: for a causal LTI system the ROC of H(z) lies outside the outermost pole; for a stable and causal rational system, all poles are inside the unit circle.

Example: X(z) = 1/((1 − ½z⁻¹)(1 − 2z⁻¹)) has poles ½ and 2, so three possible ROCs: |z| < ½ (left-sided), ½ < |z| < 2 (two-sided, includes the unit circle, stable), |z| > 2 (right-sided, causal but unstable).