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Define z-transform and ROC

Short answer

Definition as above. ROC is set of z where sum converges. Right-sided (causal): |z|>R; left-sided: |z|<R; two-sided: ring. For a^k u(k), ROC is |z|>|a|; ROC never contains poles.

Define Z-transform and its region of convergence.

Z-transform

The z-transform of a sequence x(k) is

X(z) = Z[x(k)] = Σ (k = −∞ to ∞) x(k) z⁻ᵏ

For a causal sequence this is the one-sided transform Σ (k = 0 to ∞) x(k) z⁻ᵏ, where z is a complex variable.

Region of convergence (ROC)

The ROC is the set of values of z in the complex plane for which the series Σ x(k) z⁻ᵏ converges (to a finite value).

  • Right-sided (causal) sequence: ROC is |z| > R, outside a circle.
  • Left-sided sequence: ROC is |z| < R, inside a circle.
  • Two-sided sequence: ROC is a ring R₁ < |z| < R₂ (possibly empty).
  • Finite-length sequence: ROC is the whole z-plane except possibly z = 0 or ∞.

Example: x(k) = aᵏ for k ≥ 0 gives X(z) = Σ (a/z)ᵏ = z/(z − a), converging only when |a/z| < 1, i.e. ROC: |z| > |a|. The ROC never contains a pole of X(z).

✓ Answer checked by Claude Opus 5.5 (high effort)

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