Z-transform Region of Convergence (ROC): For the right-sided sequence x[n] = a^n · u[n], what is the correct ROC in the z-plane?

Introduction / Context:
In discrete-time signal processing, the Z-transform is a powerful tool for analyzing sequences, system stability, and frequency response. A key concept is the Region of Convergence (ROC), which specifies the set of z-values for which the Z-transform series converges. For right-sided sequences like x[n] = a^n · u[n], identifying the ROC is fundamental for correct analysis and inverse transforms.

Given Data / Assumptions:

• Sequence: x[n] = a^n · u[n], where u[n] is the unit step (u[n] = 1 for n ≥ 0, else 0).
• Complex z-plane variable z with |z| denoting its magnitude.
• a can be real or complex; ROC depends on |a|, not on the phase of a.

Concept / Approach:

The bilateral Z-transform is defined as X(z) = Σ from n = −∞ to ∞ of x[n] · z^(−n). For right-sided sequences (nonzero for n ≥ 0), the Z-transform typically converges outside a circle in the z-plane. Convergence is determined by comparing the sequence term a^n with the decay introduced by z^(−n).

Step-by-Step Solution:

Verification / Alternative check:

Compute X(z) explicitly under the convergence condition: Σ (a/z)^n = 1 / (1 − a/z) = z / (z − a). This expression has a pole at z = a; for a right-sided sequence, the ROC must lie outside the outermost pole, confirming |z| > |a|.

Why Other Options Are Wrong:

• |z| < |a|: This describes an interior ROC appropriate for some left-sided sequences, not for right-sided a^n · u[n].
• z > a or z < a: These ignore complex magnitudes; ROC is defined in terms of |z| and |a|.
• Converges for all z: False; divergence occurs on or inside the pole circle for right-sided sequences.

Common Pitfalls:

• Forgetting that ROC is magnitude-based (|z|), not a simple inequality on real numbers.
• Confusing left-sided and right-sided sequence ROCs.
• Overlooking the special case a = 0, where x[n] = 0 for n ≥ 1 and ROC becomes |z| > 0 (i.e., all z except z = 0), still consistent with |z| > |a|.

Final Answer:

|z| > |a|