Z-Transform ROC Combination: For two causal sequences x1[n] and x2[n] having Z-transforms X1(z) with ROC = R1 and X2(z) with ROC = R2, determine the region of convergence (ROC) for x1[n] + x2[n].

Introduction / Context:
In discrete-time signal processing, the Z-transform's region of convergence (ROC) determines where the transform exists and whether the associated time sequence is causal/stable. When adding two signals, understanding how their individual ROCs combine is essential for correct analysis and implementation.

Given Data / Assumptions:

• X1(z) has ROC = R1.
• X2(z) has ROC = R2.
• x1[n] and x2[n] are specified as causal sequences.
• We seek the ROC for Xsum(z) = X1(z) + X2(z) corresponding to x1[n] + x2[n].

Concept / Approach:

For a given rational Z-transform, the ROC is the annulus in the z-plane that excludes poles and is determined by sequence sidedness (right-sided for causal). The Z-transform of a sum is the sum of transforms: Xsum(z) = X1(z) + X2(z). The ROC for the sum must include any z where both X1 and X2 individually converge; hence it must at least be the intersection of R1 and R2 (and can be larger only if exact pole-zero cancellations occur).

Step-by-Step Solution:

Verification / Alternative check:

Consider examples: x1[n] = a^n u[n] ⇒ ROC |z| > |a|, x2[n] = b^n u[n] ⇒ ROC |z| > |b|. Their sum is (a^n + b^n)u[n], whose ROC is |z| > max(|a|, |b|) = R1 ∩ R2, matching the rule.

Why Other Options Are Wrong:

• R1 ∪ R2: Union includes points where only one transform converges, but the sum then diverges—incorrect.
• (R1 ∪ R2) ∩ R1 and (R1 ∩ R2) ∪ R1: Both simplify to R1, which ignores R2 and is not generally valid.
• “Either R1 or R2 depending on cancellations”: Cancellations can enlarge the ROC, but without proof they do not replace the guaranteed intersection rule.

Common Pitfalls:

• Confusing union with intersection when combining transforms.
• Forgetting sidedness: causal sequences imply a right-sided ROC.

Final Answer:

R1 ∩ R2