Define Region of Convergence

The Region of Convergence (ROC) of the z-transform X(z) = Σ(n = −∞ to ∞) x[n] z⁻ⁿ is the set of all complex values z for which the series converges (the sum is finite), i.e. Σ |x[n]| |z|⁻ⁿ < ∞.

It is generally a ring r₁ < |z| < r₂ in the z-plane, bounded by poles, and it contains no poles. The ROC is needed to specify X(z) completely: the same expression, for example 1/(1 − az⁻¹), corresponds to aⁿu[n] if |z| > |a| and to −aⁿu[−n−1] if |z| < |a|.

Example: x[n] = (½)ⁿ u[n] gives X(z) = 1/(1 − ½z⁻¹) with ROC |z| > ½.